Untitled - Aerobib - Universidad Politécnica de Madrid

Aerothermochemistry
Gregorio Millán Barbany
Catedrático de Mecánica de Fluidos y Aerodinámica
de la Escuela de Ingenieros Aeronáuticos
Miembro de la Real Academia de Ciencias
Reedición conmemorativa del 50 ◦ aniversario de la edición original
Escuela Técnica Superior de Ingenieros Aeronáuticos
Universidad Politécnica de Madrid
Asociación de Ingenieros Aeronáuticos de España
Madrid, 2009

Aerothermochemistry, Gregorio Millán Barbany
Edición conmemorativa del 50 o aniversario de la edición original de 1958 publicada
por el INTA, abril 2009
Editores científicos: Manuel Rodríguez Fernández, Carlos Vázquez Espí
Diseño de la cubierta: Javier Leonés Ranz
c○ 2009 Escuela Técnica Superior de Ingenieros Aeronáuticos
Plaza del Cardenal Cisneros 3, 28040 Madrid, España
ISBN 13:978-84-86402-08-2
D.L. GU-114-2009

PRESENTACIÓN
Amable Liñán
Es un gran placer para mí presentar esta re-edición literal de la monografía
Aerothermochemistry de Gregorio Millán (1919-2004). La re-edición por la Escuela
de Ingenieros Aeronáuticos se produce cuando acaban de cumplirse 50 años de la
publicación por el INTA, en ciclostilo, de los 800 ejemplares de la edición original.
Cuando Gregorio Millán cursaba sus estudios de Ingeniería Aeronáutica, que
inició en 1941, se estaba produciendo un cambio revolucionario en esta ingeniería. Se
acaba de iniciar el desarrrollo de los aerorreactores, sin los cuales era impensable que
los aviones pudiesen alcanzar velocidades transónicas o supersónicas; simultáneamente,
para la viabilidad de los cohetes de sondeo o de los misiles balísticos, se impulsó el
desarrollo de los motores cohete. La Mecánica de Fluidos, disciplina central de las
Ciencias Aeronáuticas y determinante del diseño de estos motores, fue elegida por
Gregorio Millán como objeto de su actividad docente e investigadora.
En la formación de Gregorio Millán había tenido un papel crucial el inusual
ambiente docente de la Academia Militar de Ingenieros Aeronáuticos, que se reflejaba
en la preocupación del profesorado por el papel de las ciencias básicas y aplicadas en
el desarrollo de la Ingeniería Aeronáutica. Este ambiente docente era heredero del que
ya se daba en la antecesora de la Academia, la Escuela Superior Aerotécnica, desde
su creación en 1928, bajo la dirección de Emilio Herrera. Para las enseñanzas básicas,
Herrera había conseguido la colaboración, que se mantuvo en la Academia Militar, de
los profesores universitarios españoles más eminentes (muchos de ellos, igual que el
propio Emilio Herrera, miembros de la Academia de Ciencias).
Uno de estos profesores era Esteban Terradas, que fue Presidente del Patronato
del INTA desde su creación. Terradas se propuso impulsar el desarrollo en España
de las Ciencias Aeronáuticas, invitando a los científicos extranjeros mas prestigiosos
en estas ciencias a impartir ciclos de conferencias. Entre ellos Teodoro von Kármán
que en 1948 vino a España, por primera vez, para hablar de Aerodinámica Transónica
y Supersónica y sobre Turbulencia. De entonces nació la colaboración fructífera de
Gregorio Millán con von Kármán, quien orientó la actividad docente e investigadora
posterior de Millán y también las investigaciones del Grupo Español de Combustión.
Von Kármán, que se considera con justicia el padre de las Ciencias Aeronáuticas
americanas, había sido el Director de los Guggenheim Aeronautical Laboratories
del Instituto Tecnológico de California (Caltech). Poco antes de la última Guerra
Mundial inició su preocupación por el desarrollo de los cohetes, y después por los

aerorreactores, siendo el creador del Jet Propulsion Laboratory. Comprendiendo que
el análisis de los procesos de combustión era esencial para el diseño de estos motores y
que debía hacerse con el apoyo de la Dinámica de Fluidos, se embarcó en el proyecto
de establecer el marco multidiciplinar apropiado. Para ello, consiguió la colaboración
del Profesor Saul Pennner del Caltech y de Gregorio Millán, Ingeniero del INTA y
Profesor de la Academia (luego Escuela) de Ingenieros Aeronáuticos.
La Aerothermochemistry se basa en el ciclo de conferencias que Teodoro von
Kármán impartió en la Sorbona durante el curso 1951-1952, en cuya preparación y
desarrollo contó con la ayuda de Gregorio Millán. Por el interés suscitado por estas
conferencias de von Kármán, el Air Research and Development Command (ARDC)
de las Fuerzas Aéreas de Estados Unidos ofreció a Gregorio Millán, en 1954, un contrato
para apoyar la redacción y actualización, mediante un programa de investigación,
de las conferencias de la Sorbona. Para este proyecto, Gregorio Millán contó con la
colaboración de un grupo de ingenieros y profesores de la Escuela de Ingenieros Aeronáuticos,
que formaron el Grupo Español de Combustión. Este grupo incluía a Segismundo
Sanz Aránguez, Jesús Salas Larrazábal, Carlos Sánchez Tarifa, José Manuel
Sendagorta e Ignacio Da Riva. Al grupo se sumaron pronto los profesores Francisco
García Moreno y Pedro Pérez del Notario, y yo mismo que empecé como becario en
1958. Más tarde, creció notablemente el número de participantes (entre ellos, Enrique
Fraga, Antonio Crespo, José Luis Urrutia y Juan Ramón Sanmartín) en los proyectos
de investigación del Grupo. También se ampliaron las fuentes de subvención, que incluyeron
el US Forest Service del Departamento de Agricultura americano, así como
la European Space Research Organization (ESRO).
El ARDC facilitó la difusión internacional, a través de laboratorios universitarios
y centros de investigación, de los 800 ejemplares de la edición original de la
Aerothermochemistry. A pesar de ello, hace ya bastante tiempo que no se encuentran
disponibles ni accesibles copias del original. El valor histórico y la actualidad de la Aerotermoquímica
de Millán nos ha animado a hacer el esfuerzo de transcribir el original
en TEX y rehacer las figuras para una re-edición. Esta tarea no hubiese sido posible sin
la decisión espontánea de Manuel Rodríguez Fernández de iniciar esa transcripción,
llegando a completar casi la tercera parte. Este impulso inicial animó a las autoridades
académicas de la Escuela a finalizar la tarea, encargando a Carlos Vázquez Espí la
coordinación y supervisión de la transcripción y de las correcciones, labor en la que
colaboraron Eva Villacieros y los estudiantes de la ETSI Aeronáuticos Alfredo Giralda,
Ramón Lacruz y David Marchante. En la re-edición se han eliminado erratas del
original, se han rehecho la mayoría de las figuras con los métodos actuales de cálculo,

que no cambiaron los resultados, y se han incluido algunas anotaciones significativas.
Por todo el apoyo recibido, tenemos que agradecer a la Universidad Politécnica de
Madrid y a la Asociación de Ingenieros Aeronáuticos la publicación en forma de libro
de la monografía.
En la Aerothermochemistry aparece por primera vez el marco multidisciplinar
coherente, que es necesario para el análisis de los procesos de combustión. Este
análisis, como había anticipado von Kármán, no puede hacerse sin ampliar las leyes
de la Mecánica de Fluidos con las de la Teoría de los Fenómenos de Transporte, la
Termodinámica de Mezclas y la Cinética de las Reacciones Químicas. 1 Aunque la
monografía de Millán contribuyó decisivamente a establecer la necesidad del tratamiento
multidisciplinar en la investigación de los Procesos de Combustión y de la Aerodinámica
Hipersónica, el nombre Aerothermochemistry fue desplazado finalmente
por el de Ciencias de la Combustión.
La importancia de la Aerothermochemistry para la Escuela Española de Mecánica
de Fluidos ha sido trascendental, porque en ella se abordan problemas de frontera de
la Mecánica de Fluidos con un lenguaje moderno y unas técnicas avanzadas. Ha sido
ejemplar, tanto para la enseñaza de la Mecánica de Fluidos como para la investigación
y el uso de la dinámica de los fluidos en ámbitos multidisciplinares.
Por el cariño con el que von Kármán acogió y valoró las aportaciones del Grupo
Español de Combustión, von Kármán eligió Madrid como sede del Primer Congreso
Internacional de Ciencias Aeronáuticas, que se celebró también en 1958. A este
Congreso acudieron las personalidades más relevantes de las Ciencias Aeronáuticas;
entre ellos, Geoffrey Taylor, Leónidas Sedov y James Lighthill. Lighthill, por ejemplo,
habló de los efectos en flujos hipersónicos de las reacciones químicas de disociación
de las moléculas del aire. Von Kármán fue también esencial en la organización en
1960 de un Curso sobre Ciencia y Tecnología del Espacio, en el INTA, que precedió a
la creación de la CONIE y a nuestra participación en la ESRO. Por las aportaciones
de von Kármán a las Ciencias Aeronáuticas, poco antes de su muerte en Aachen en
1963, recibió de las manos del Presidente Kennedy la primera Medalla Nacional de
Ciencias americana y su imagen fue recogida en uno de sus sellos.
Gregorio Millán agradece en el prólogo la valiosa colaboración de Carlos Sánchez
Tarifa, José Manuel Sendagorta e Ignacio Da Riva a la Aerothermochemistry.
1 Conviene advertir al lector de la Aerothermochemistry que, por ejemplo, fue en las conferencias de la
Sorbona donde apareció por primera vez la forma general de la ecuación de la energía para la dinámica
de fluidos reactantes. En ella se incorporan explícitamente los intercambios de energía química, térmica y
cinética, el flujo de calor y el trabajo de las fuerzas de presión y de viscosidad; todos ellos involucrados en
el funcionamiento de los motores térmicos.

Los tres fueron determinantes en el desarrollo de la Escuela Española de Mecánica de
Fluidos. José Manuel Sendagorta (responsable de despertar en mí la vocación por la
investigación) sustituyó a Gregorio Millán en la enseñanza de la Mecánica de Fluidos,
que tuvo que dejar cuando la fundación y el desarrollo de Sener le absorbieron todo
su tiempo; pero consiguió convertir pronto a Sener en una de las grandes empresas
españolas de Ingeniería. Primero Sánchez Tarifa y más tarde Millán, cuando dejó la
dirección de Babcock & Wilcox, fueron llamados por Sendagorta para contribuir al
desarrollo de Sener. Sánchez Tarifa fue el responsable de la enseñanza de Propulsión
en nuestra Escuela, y director, en el INI, del proyecto y ensayo del primer motor de
reacción español; desde Sener dirigió nuestra participación en el diseño del motor del
Eurofighter. La labor investigadora experimental en Combustión de Sánchez Tarifa fue
seminal, inspirando la labor de algunos de los investigadores más distinguidos procedentes
de nuestra Escuela. Ignacio Da Riva, pronto catedrático de Aerodinámica, sólo
dejó su fructífera dedicación, incansable y ejemplar, a la enseñanza e investigación
cuando la muerte le sobrevino en clase. Fue mi maestro y compañero en la investigación,
y puntal esencial en la Escuela Española de Mecánica de Fluidos.
Las contribuciones científicas de Millán, que están recogidas en parte en este
libro, fueron muy importantes y contribuyeron a su elección en 1961 como Miembro
de la Real Academia de Ciencias. Su influencia en la enseñanza es difícil de imaginar
para las nuevas generaciones, pero es inolvidable para mí; que aprendí Mecánica de
Fluidos de él y de José Manuel Sendagorta, y adopté las notas para clase que nos
dejaron. Yo aprendí en la Aerothermochemistry la base teórica de la Combustión; y fui
iniciado en la investigación sobre las llamas de difusión por Gregorio Millán en las
reuniones semanales que los componentes del Grupo de Combustión teníamos con él
en su etapa (1957-1961) de Director General de Enseñanzas Técnicas. El lector puede
encontrar en la reseña necrológica 2 que escribí para la Revista de la Real Academia
de Ciencias, un breve resumen de las aportaciones de Millán a la política educativa y
también de su labor como gestor a nuestro desarrollo tecnológico.
Espero que esta re-edición de la Aerothermochemistry sirva de modesto homenaje
a los méritos de las aportaciones de Gregorio Millán y sus colaboradores a las
Ciencias Aeronáuticas Españolas.
2 In Memoriam: D. Gregorio Millán Barbany, Rev. R. Acad. Cienc. Exact. Fis. Nat., Vol 99, No. 1, pp.
183-185, 2005.

INSTITUTO NACIONAL DE TÉCNICA AERONÁUTICA
ESTEBAN TERRADAS
AEROTHERMOCHEMISTRY
(A report based on the course conducted by Prof. Theodore von Kármán
at the University of Paris)
BY
GREGORIO MILLÁN
PROFESSOR OF AERODYNAMICS
AT
ESCUELA TÉCNICA SUPERIOR DE INGENIEROS AERONÁUTICOS
MADRID (SPAIN)
January, 1958
The research reported in this document has been sponsored by the AIR
RESEARCH AND DEVELOPMENT COMMAND, UNITED STATES AIR
FORCE, under contract No. 61 (514)-441, through the European Office, ARDC.

PREFACE
by
Theodore von Kármán
I have great pleasure in introducing the report of Prof. Gregorio Millán on
“Aerothermochemistry”. This word refers to problems whose solution necessitates
the application of fundamental principles of Thermodynamics and Chemistry especially
chemical kinetics. In other words they are flow problems with exchange of heat
and production of heat by means of chemical reaction. The phenomena of high speed
flight made it necessary for the aeronautical engineer to be fully familiar in addition to
Fluid Mechanics, i.e. Aerodynamics proper, with the fundamentals and applications
of Thermodynamics. This need created the science of Aerothermodynamics. The development
of jet propulsion introduced problems which have direct bearing in modern
aircraft and missile design and necessitate the understanding of changes in the composition
of the flowing medium, as dissociation and chemical reactions. Thus Aerothermochemistry
deals with the interaction between chemical changes and the merely
aerodynamic phenomena as pressure and velocity distribution, momentum transfer
and alike and between the phenomena of Aerothermodynamics, as shockwaves, heat
transfer, diffusion etc. This novel extension in the scope of the Aeronautical Sciences
necessitates a thorough study of branches of Science, which the aeronautical engineer
in general gave little attention during his professional education. This report is written
for aeronautical engineers, in that the author does not assume that the reader is
acquainted with the kinetic theory of gases and with chemistry beyond quite general,
ideas.
A few chapters of the report are based on lectures which I gave as Fullbright
guest professor at the Sorbonne in Paris in the scholastic year 1951/52. This course
had the same general aim to acquaint the aeronautical engineer including myself with
the problems introduced by the phenomena of combustion occurring in flowing media.
The author however completed, and I believe very successfully, the presentation of the
subject by including a systematic treatment of the Thermodynamics of gas mixtures,
of the theory of chemical equilibrium, the elements of chemical kinetics, theory of
transfer phenomena in gases and gas mixtures. In addition the problems of flame stabilization,
combustion of liquid droplets and diffusion flames, similarity which were
only touched on in my lectures are treated in this report systematically.
I am sure that the report will be helpful to many aeronautics engineers in their

aim to become acquainted with Aerothermochemistry and it may also contribute some
suggestions to those engaged in pursing the subject in their further researches.
For this reason the report includes rather extended bibliographic material after
each chapter.
I would like to mention without commitment that if I can secure the valuable
collaboration of Prof. Millán, I am playing with the idea to publish a book on the
fundamentals of Aerothermochemistry.

FOREWORD
During the academic term 1951–52, Professor Theodore von Kármán conducted
a course on Aerodynamics of Combustion at the University of Paris. This
course was mainly intended for aeronautical engineers with the objective of encouraging
the study of the problems risen by the new propulsion systems developed by
Aeronautics, which necessitates the assistance of the theories and methods of Aerodynamics,
Thermodynamics and Chemistry of Combustion. The new Science born
to deal with these problems needed a name and Prof. von Kármán suggested it be
called AEROTHERMOCHEMISTRY for its analogy with other classical science of
Aeronautics, such as Aeroelasticity or Aerothermodynamics. In a few years this designation
has been widely accepted.
The author was invited by Prof. von Kármán to participate as his assistant in
developing this course. Due to the amount of bibliography that has to be reviewed in
addition to the numerous studies and exercises required in its preparation it was not
possible at the time to write the lectures.
Later, when the European Office of the ARDC was activated it Brussels and
in view of the general demand for the Lectures by Prof. von Kármán, the said Office
offered the author the opportunity of preparing a Report on Aerothermochemistry
based upon the course conducted by Prof. von Kármán, through a research contract
sponsored by ARDC.
Work was started in 1953 but for different reasons it had to be interrumped
in several occasions. At the same time investigation was progressing very rapidly in
the different fields of Aerothermochemistry and it was necessary to incorporate them
into this report, in order to keep it current. For this reason, initial planning had to be
continuously extended and finally carried far beyond its original intention.
The author wishes to express his deep appreciation to Prof. von Kármán who
was a constant source of inspiration and encouragement and whose advise in preparing
this report proved invaluable. At the same time he trusts that Prof. von Kármán will
excuse this report if it fails to reflect the excellencies of the course on which it is based.
Special acknowledgment is given to the contribution of the European Office
ARDC, and the Instituto Nacional de Técnica Aeronáutica Esteban Terradas for the
help received in preparing this report for publication.
Finally, I wish to acknowledge the valuable collaboration of Carlos Sánchez

Tarifa, Manuel de Sendagorta and Ignacio Da Riva, aeronautical engineers of the
INTA, in the research work that made possible the publication of this report, and to
Mrs. de los Casares for her technical translation.
Madrid, January 1958

Contents
1 Thermochemistry 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thermodynamic functions of an ideal gas. . . . . . . . . . . . . . . 6
1.3 Mixture of gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Calculation of the thermodynamic functions . . . . . . . . . . . . . 16
1.5 Chemical reactions in a mixture of gases . . . . . . . . . . . . . . . 21
1.6 Chemical equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Case of a mixture of ideal gases . . . . . . . . . . . . . . . . . . . . 24
1.8 Chemical kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 Transport phenomena in gas mixtures 37
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 Viscosities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4 Thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 54
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 General equations 59
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Equation of continuity . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 64
v

3.4 Energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5 General equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.6 Entropy variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.7 One-dimensional motions . . . . . . . . . . . . . . . . . . . . . . . 72
3.8 Stationary, one-dimensional motions . . . . . . . . . . . . . . . . . 75
3.9 The case of only two chemical species . . . . . . . . . . . . . . . . 77
3.10 Stationary, one-dimensional motion of ideal gases with heat addition 79
3.11 Appendix: Notation and repertoire of vectorial and tensorial formulae 84
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4 Combustion Waves 89
4.1 Detonation and deflagration . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Kinds of detonations and deflagrations . . . . . . . . . . . . . . . . 91
4.3 Velocity of the burnt gases . . . . . . . . . . . . . . . . . . . . . . . 94
4.4 Propagation velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.7 Indeterminacy of the solution . . . . . . . . . . . . . . . . . . . . . 103
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5 Structure of the combustion waves 105
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2 Wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3 Characteristic times . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.4 Limiting form of the wave equations . . . . . . . . . . . . . . . . . 111
5.5 Detonations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.6 Deflagrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.7 Transition from deflagration to detonation . . . . . . . . . . . . . . 126
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6 Laminar flames 131
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 Equation for the combustion wave (two chemical species) . . . . . . 134
6.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.4 Modification of the conditions at the “cold boundary” . . . . . . . . 137
6.5 Propagation velocity of the flame . . . . . . . . . . . . . . . . . . . 140
6.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.7 Reaction velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.8 Flame equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.9 Solution of the flame equations . . . . . . . . . . . . . . . . . . . . 148
6.10 Structure of the combustion wave . . . . . . . . . . . . . . . . . . . 159
6.11 Ignition temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.12 General equations for the combustion wave . . . . . . . . . . . . . . 162
6.13 Ozone decomposition flame . . . . . . . . . . . . . . . . . . . . . . 169
6.14 Hydrazine decomposition flame . . . . . . . . . . . . . . . . . . . . 176
6.15 Flame propagation in Hydrogen-Bromine mixtures . . . . . . . . . . 187
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7 Turbulent flames 207
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.2 Turbulent combustion theories . . . . . . . . . . . . . . . . . . . . . 210
7.3 Turbulence generated by the flame . . . . . . . . . . . . . . . . . . 218
7.4 Comparison with experimental results . . . . . . . . . . . . . . . . 219
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8 Ignition, flammability and quenching 221
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.2 Ignition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8.3 Flammability limits . . . . . . . . . . . . . . . . . . . . . . . . . . 225

8.4 Quenching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
9 Flows with combustion waves 231
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
9.2 Conditions that must be satisfied by the jump across a flame front. . . 231
9.3 Normal flame front . . . . . . . . . . . . . . . . . . . . . . . . . . 235
9.4 Inclined flame front . . . . . . . . . . . . . . . . . . . . . . . . . . 236
9.5 Entropy jump across the flame front . . . . . . . . . . . . . . . . . . 239
9.6 Vorticity across the flame . . . . . . . . . . . . . . . . . . . . . . . 242
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
10 Aerothermodynamic field of a stabilized flame 245
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
10.2 Tsien method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
10.3 Method of Fabri-Siestrunck-Fouré . . . . . . . . . . . . . . . . . . 255
10.4 Cylindrical chambers . . . . . . . . . . . . . . . . . . . . . . . . . 258
10.5 Chamber with slowly varying cross-section . . . . . . . . . . . . . . 259
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
11 Similarity in combustion. Applications 261
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
11.2 Dimensionless parameters of Aerothermochemistry . . . . . . . . . 262
11.3 Scaling of rockets . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
11.4 Scaling of rockets for non-steady conditions . . . . . . . . . . . . . 270
11.5 Flame stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
12 Diffusion flame 281
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
12.2 General equations for laminar diffusion flames . . . . . . . . . . . . 289

12.3 Boundary conditions on the flame . . . . . . . . . . . . . . . . . . . 291
12.4 Simplified equations . . . . . . . . . . . . . . . . . . . . . . . . . . 292
12.5 Solutions of the simplified system . . . . . . . . . . . . . . . . . . . 296
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
13 Combustion of liquid fuels 301
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
13.2 Atomization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
13.3 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
13.4 Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
13.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
13.6 Continuity equations . . . . . . . . . . . . . . . . . . . . . . . . . . 308
13.7 Energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
13.8 Diffusion equations . . . . . . . . . . . . . . . . . . . . . . . . . . 313
13.9 Combustion velocity of the droplet . . . . . . . . . . . . . . . . . . 314
13.10 Integration of the equations . . . . . . . . . . . . . . . . . . . . . . 316
13.11 Numerical application . . . . . . . . . . . . . . . . . . . . . . . . . 319
13.12 Comparison with experimental results and limitations of the theory . 322
13.13 Influence of convection . . . . . . . . . . . . . . . . . . . . . . . . 324
13.14 Combustion of fuel sprays . . . . . . . . . . . . . . . . . . . . . . . 325
13.15 Droplet evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . 325
13.16 Appendix: Application of Probert’s method . . . . . . . . . . . . . . 331
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

Chapter 1
Thermochemistry
1.1 Introduction
Aerothermochemistry deals with mixtures of reactant gases of variable composition.
Such variation in composition is due to the reactions between the various chemical
species forming the mixture and to their mutual diffusion. In general, the pressure
of these mixtures is not larger than several atmospheres 1 and their temperatures
range from ambient temperature to 3 000 or 4 000 K. Under such conditions the mean
distance between molecules is large respect to their size, and their potential energy
of interaction is negligible compared to the kinetic energy of the molecular motion.
Thereby, the mixture can generally be treated as an ideal gas. In those cases where it
is necessary to take into consideration the deviations in the behavior of the gas with
respect to that of an ideal gas, this can be attained by including correcting terms in the
thermodynamic equations of mixture. For example, its state equation can be approximated
by the virial equation proposed by Kamerlingh Onnes
p
ρR g T = 1 + ρB (T ) + ρ2 C (T ) + . . . (1.1)
where p, ρ and T are, respectively, the pressure, density and absolute temperature of
the gas. R g is its particular constant and B (T ), C (T ) , . . . are the second, third, etc.
coefficients of the virial, respectively. These can be calculated from the potential ϕ
of interaction of the molecules. For example, if both molecules belong to the same
species and if the potential depends only on the distance r between their centers of
gravity but not on their relative orientation, the second coefficient, which generally is
1 Some ballistic problems are excluded (See Ref. [1]) as well as detonations of condensed explosives
(See Ref. [2]) where pressures can be very large.
1

2 CHAPTER 1. THERMOCHEMISTRY
3.0
2.5
2.0
1.5
ϕ/ε
1.0
0.5
0.0
−0.5
−1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
r/σ
Figure 1.1: Dimensionless Interaction Potential of Lennard-Jones, ϕ/ε, versus dimensionless
distance between molecules, r/σ.
the only needed except for very high pressures, is expressed in the form 2
B (T ) = 2π N M
∫ ∞
0
(
1 − exp(− ϕ(r)
kT ) )
r 2 dr, (1.2)
where N = 6.0288 × 10 23 mol −1 is the Avogadro number, M is the molar mass
of the gas and k = 1.38047 × 10 −16
erg/grad is the Boltzmann constant. It has
experimentally been checked that the interaction potential of Lennard-Jones
( (σ ) 12 ( σ
) ) 6
ϕ (r) = 4ε − , (1.3)
r r
represents, suitably, the actual behavior of many gases. In Eq. (1.3), σ is the radius
of the molecules and ε is a constant which has energy dimensions. Fig. 1.1 represents
Eq. (1.3). There, it is seen that if the distance r between the molecules is larger than
several times the radius of the molecule, the molecules attract each other whilst if
r < 1.12σ they reject with a force that increases very rapidly as the molecules get
closer. By substituting (1.3) into (1.2) one verifies that B (T ) can be expressed in the
form
where
2 See Ref. [3].
B (T ) = b 0
M B∗ (T ∗ ) , (1.4)
b 0 = 2 3 πNσ3 (1.5)

1.1. INTRODUCTION 3
GAS ε/k (K) σ (Ȧ) b 0 (cm 3 /mol)
Air 99.2 3.522 55.11
N 2 95.05 3.698 63.78
O 2 117.5 3.580 57.75
CO 100.2 3.763 67.22
CO 2 189 4.468 113.90
NO 131 3.170 40.00
N 2 O 189 4.590 122.00
CH 4 148.2 3.817 70.16
CH-CH 185 4.221 94.86
CH 2=CH 2 199.2 4.523 116.70
C 2H 6 243 3.954 78.00
C 3H 8 242 5.367 226.00
n-C 4 H 10 297 4.971 155.00
i-C 4 H 10 313 5.341 192.18
n-C 5 H 12 345 5.769 242.19
n-C 6 H 14 413 5.909 260.25
n-C 7 H 16 282 8.880 884.00
n-C 8 H 18 320 7.451 521.79
n-C 9H 20 240 8.448 760.53
Cyclohexane 324 6.093 285.33
C 6H 6 440 5.270 84.62
CH 3 OH 507 3.585 58.12
C 2 H 5 OH 391 4.455 111.53
CH 3 Cl 855 4.455 48.49
CH 2 Cl 2 406 4.759 135.96
CHCl 3 327 5.430 201.95
C 2 N 2 339 4.380 105.99
COS 335 4.130 88.86
CS 2 488 4.438 110.26
Table 1.1: Force constants for the Lennard-Jones potential.
and
T ∗ = kT ε . (1.6)
Table 1.1, taken from Ref. [3], gives the values of b 0 and ε/k for several gases.
These values have been determined from experimentation. Fig. 1.2 gives the values
for B ∗ (T ∗ ) taken from the same reference. 3 To get an idea on the order of magnitude
of coefficients B (T ) and C (T ), and, consequently, on the deviations that are to be
expected in the actual behavior of gases respect to their ideal model, Figs. 1.3 and
1.4 give the values of these coefficients for some gases. As can be observed B (T ) is
first negative and then becomes positive, tending slowly to zero for T → ∞. Such
behavior of B (T ) is common for all gases. It is seen, for example, that in nitrogen
for p = 25 kg/cm 2 and T = 800 K, B (T ) = 1 cm 3 /gr. Therefore, the error made
3 Ref. [3] also includes a table for the calculation of C (T ).

4 CHAPTER 1. THERMOCHEMISTRY
with the assumption that in such a state nitrogen behaves as an ideal gas is one per
cent. Therefore under the conditions of the present study it is justified to consider the
gases as ideal. Hereinafter, unless the contrary is made explicit, such an assumption
is adopted.
0.5
0.0
T * = 100ξ
T * = 10ξ
−0.5
T * = ξ
B * (T * )
−1.0
−1.5
−2.0
−2.5
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
ξ
Figure 1.2: B ∗ as a function of T ∗ . Note the two changes of scale.
8
H 2
C 7
H 16
4
O 2
N 2
B(T) (cm 3 /gr)
0
−4
CH 4
CO 2
−8
−12
0 400 800 1200 1600 2000 2400 2800 3200 3600
T(K)
Figure 1.3: Virial coefficient B (cm 3 /gr) , as function of temperature for several gases.

1.1. INTRODUCTION 5
14
12
10
N 2
CH 4
O 2
CO 2
CO
C(T) (cm 6 /gr 2 )
8
6
4
2
H 2
(0.1 C(T))
C 7
H 16
(0.1 C(T))
0
0 400 800 1200 1600
T(K)
Figure 1.4: Virial coefficient C (cm 3 /gr) 2 as a function of temperature for several gases.
In the following paragraphs we will first briefly summarize the thermodynamic
functions of ideal gases and of their mixtures. A more extensive study of the subject
can be found, for example in the works mentioned in Refs. [4] and [5], where, the thermodynamic
functions for the case of air gas are also included. A complete repertoire
of thermodynamic functions of the gases and their mixtures including the influence of
some virial coefficients, can be found in Ref. [6]. Further on, the form in which these
thermodynamic functions can be obtained is briefly considered. Finally, reference is
made to the thermodynamic equilibrium and to the chemical kinetics of gas reactions.
Throughout the following paragraphs the mixture will be assumed to be in thermodynamic
equilibrium, 4 at rest and that its state and composition are homogeneous. The
effects of motion and of lack of homogeneity will be studied in chapter 2.
It is generally advantageous in Aerothermochemistry to refer the thermodynamic
functions to the unit mass, which is done in the following instead of referring
then to a mole, as usually done in Thermodynamics. The thermodynamic functions
(internal energy, enthalpy, entropy and free energy) when referred to the unit mass
will be denoted with small letters. If referred to a mole the same letters in capital will
be used.
4 Obviously except in the paragraph devoted to Chemical Kinetics.

6 CHAPTER 1. THERMOCHEMISTRY
1.2 Thermodynamic functions of an ideal gas.
The equation of state
The state equation of an ideal gas is given by the expression
p
ρ = R gT, (1.7)
where
R g = R M g
(1.8)
is the particular constant of the gas, where R = 8.3144 joule/mol/grad = 1.9872 cal/mol/grad
is the universal constant of the gases.
Internal Energy
The internal energy of the gas is
∫ T
u = u 0 + c v (T ) dT. (1.9)
T 0
Here u 0 is the internal energy of the gas at temperature T 0 , c v is the specific heat at
constant volume, which depends on temperature as will be analyzed in §4, and T 0 is a
reference temperature. In particular, if c v is constant from T 0 to T , we have
u = u 0 + c v (T − T 0 ) , (1.10)
otherwise, in many applications c v is substituted by a mean value ¯c v . In such case
(1.9) takes the approximate form
u = u 0 + ¯c v (T − T 0 ) . (1.11)
Enthalpy
The gas enthalpy h is defined by
h = u + p ρ . (1.12)
Making use of the state equation (1.8) and of equation (1.9), one obtains for (1.12)
h = h 0 +
∫ T
T 0
c p (T ) dT, (1.13)

1.2. THERMODYNAMIC FUNCTIONS OF AN IDEAL GAS. 7
where h 0 is the enthalpy of the gas at the temperature T 0 , and c p = c v +R g its specific
heat at constant pressure. As before, if c p is constant or is substituted by a mean value
¯c p , we have
h = h 0 + ¯c p (T − T 0 ) . (1.14)
Since only differences in energy can be measured a convention becomes necessary
to determine u 0 and h 0 . This convention is as follows:
1) A standard state is adopted for each chemical substance, defined by a value p 0
of the pressure and a value T 0 of the temperature.
2) The formation enthalpies of the chemical elements in their stable phase in the
standard state are zero.
3) If any of these substances is a gas its formation enthalpy for T = T 0 and p → 0
were equal to that of the actual gas.
For moderate pressures, this enthalpy differs slightly from the enthalpy of the actual
gas at temperature T 0 and at pressure p 0 . At present the values generally adopted for
p 0 and T 0 are those proposed by G. M. Lewis [7], namely T 0 = 298.16 K, that is
25 ◦ C, and p 0 = 1 atm. Formerly values of temperature slightly smaller were used.
The reduction from one state to another can be easily performed. The previous convention
fixes the values of the internal energy and of the enthalpy of any substance as
per the first principle of Thermodynamics. The value of h 0 can be obtained, for example,
by measuring the heat of reaction at pressure p 0 and at temperature T 0 , when the
substance forms from its elements or from other substances for which the formation
enthalpies are known. Therefore, in the preceding formulae u 0 and h 0 are the internal
energy and the formation enthalpy of gas respectively. Between both the following
relation exists
h 0 = u 0 + p 0
ρ 0
. (1.15)
Tables 1.2, 1.3 and 1.4 give the formation enthalpies of some substances and
their stable phase in the state of reference. Such values have been taken from Ref. [4],
pp. 98 and following, were additional information can be found. Tables NBS-NACA
of thermal properties of the gases [8] are particularly interesting.
Entropy
The entropy s of the gas is
( ) ∫ p T
c p (T )
s = s 0 − R g ln + dT, (1.16)
p 0 T 0
T

8 CHAPTER 1. THERMOCHEMISTRY
Inorganic Substances
Substance State h 0 s 0 ∆G 0
H 2 gas 0.00 5.482 0
H gas 51 675.60 27.176 48 575
Br 2 liquid 0.00 0.228 0
Br gas 3 342.25 0.543 19 690
HBr gas -107.01 0.586 -12 720
O 2 gas 0.00 1.531 0
O gas 3 697.44 2.404 54 994
O 3 gas 708.33 1.183 39 060
H 2O gas -3 208.15 2.504 -54 635
H 2 O liquid -3 792.02 0.928 -56 690
N 2 gas 0.00 1.634 0
N gas 6108.37 2.614 81 476
NH 3 gas -648.19 2.701 -3 976
NO gas 719.81 1.678 20 719
NO 2 gas 175.86 1.249 12 390
N 2 O 4 gas 25.09 0.790 23 491
N 2O gas 442.79 1.195 24 760
HNO 3 liquid -657.04 0.591 -19 100
C(graphite) solid 0.113 0
C(diamond) solid 37.72 0.049 685
CO gas -943.09 1.689 -32 808
CO 2 gas -2 137.06 1.16 -94 259
Table 1.2: Formation enthalpies (cal/gr),
(cal/mol). Inorganic substances.
entropies (cal gr −1 K −1 ) and free energies
where s 0 is the entropy of the gas at pressure p 0 and at temperature T 0 . The third law of
Thermodynamics assigns zero values to the entropies of all substances at the absolute
zero of temperature. This determines the values of s 0 without the need of a convention
as for the enthalpy case. 5
various substances.
Tables 1.2-1.4 also include the formation entropies of the
Let s 0 be the entropy of the gas at temperature T and standard pressure p 0 that
is
∫ T
s 0 c p (T )
= s 0 + dT. (1.17)
T 0
T
From this equation and Eq. (1.16) one obtains for s
( ) p
s = s 0 − R g ln , (1.18)
p 0
where s 0 depends only on T , once p 0 is selected.
5 See, i.e., Ref. [9] for complementary information and for discussion of the anomalies that appear in the
application of the Third Law.

1.2. THERMODYNAMIC FUNCTIONS OF AN IDEAL GAS. 9
Alcohols
Substance Formula State h 0 s 0 ∆G 0
Methanol CH 3OH g -1501.15 1.773 38 700
Methanol CH 3 OH l -1780.04 0.946 -39 750
Ethanol C 2H 5OH g -1220.80 1.463 -40 300
Ethanol C 2 H 5 OH l -1440.39 0.834 -41 770
1-Propanol C 3H 7OH l -1212.43 0.767 -40 900
2-Propanol C 3 H 7 OH l -1279.00 0.716 -44 000
1-Butanol C 4H 9OH l -1074.07 0.735 -40 400
2-methyl-2-Propanol (CH 3 ) 3 COH l -1206.29 0.611 -47 500
1-Pentanol C 5H 11OH l -976.33 0.691 -39 100
2-methyl-2-Butanol C 2 H 5 COH(CH 3 ) 2 l -1094.32 0.622 -47 700
Diphenyl Carbinol (C 6H 5) 2CHOH s -110.62 0.311 30 700
Triphenyl Carbinol (C 6 H 5 ) 3 COH s 15.21 0.302 69 700
Cydohexanol C 6H 11OH l -85.47 0.476 -34 300
Ethylene Glycol CH 2 OHCH 2 OH l -1749.37 0.643 -77 120
Glycerol CH 2OHCHOHCH 2OH l -1728.23 0.540 -113 600
Phenol C 6 H 5 OH s -407.72 0.362 -11 000
Thiophenol C 6 H 5 SH s 0.477
Pyrocatechol C 6 H 4 (OH) 2 s -795.31 0.326 -51 400
Resorcinol C 6 H 4 (OH) 2 s -795.31 0.321 -53 200
Hydroquinone C 6 H 4 (OH) 2 s -795.31 0.304 -52 700
Aniline C 6 H 5 NH 2 l 78.82 0.492 35 400
Table 1.3: Formation enthalpies (cal/gr),
(cal/mol). Alcohols.
entropies (cal gr −1 K −1 ) and free energies
Free Energy
Gibbs free energy
which can be written in the form
g = h − T s, (1.19)
g = g 0 + R g T ln
( p
p 0
)
, (1.20)
where
g 0 = h − T s 0 (1.21)
is the free energy at standard pressure.
gas
The free energy G = Mg per mole is also called chemical potential µ of the
( ) p
µ = G = G 0 + RT ln , (1.22)
p 0

10 CHAPTER 1. THERMOCHEMISTRY
where
G 0 = H − T S 0 = µ 0 , (1.23)
and H, S 0 and µ 0 are, respectively, the molar enthalpy and entropy of the gas and its
chemical potential at standard pressure.
Hydrocarbons
Substance Formula State h 0 s 0 ∆G 0
Methane CH 4 g -1 115.14 2.774 -12 140
Ethane C 2 H 6 g -673.01 1.824 -7 860
Propane C 3 H 8 g -562.89 1.463 -56 14
n-Butane C 4 H 10 g -512.94 1.275 -3 754
n-Pentane C 5H 12 g -485.13 1.154 -1 960
Ethylene C 2 H 4 g 445.49 1.87 1 6282
Propilene C 3 H 6 g 115.95 1.516 14 990
1-Butene C 4H 8 g 4.99 1.31 17 217
cis-2-Butene - g -24.28 1.282 16 046
trans-2-Butene - g -42.87 1.263 15 315
Isobutene - g -59.59 1.251 14 582
1-Pentene C 5H 10 g -71.30 1.185 18 787
Acetylene C 2 H 2 g 2081.5 1.843 50 000
Methylacetylene C 3 H 4 g 1106.26 1.48 46 313
Cyclopentane C 5H 10 l -360.76 0.696 8 700
Methylcyclopentane C 6 H 12 l -392.96 0.704 7 530
Cyclohexane C 6 H 12 l -443.7 0.58 6 370
Methylcyclohexane C 7H 14 l -462.92 0.604 4 860
Benzene C 6 H 6 g 253.75 0.824 30 989
Benzene C 6 H 6 l 150.02 0.529 29 756
Toluene C 7H 8 g 129.7 0.829 29 228
Toluene C 7H 8 l 31.12 0.57 27 282
o-Xylene C 8 H 10 g 42.77 0.794 29 177
o-Xylene C 8 H 10 l -55.02 0.555 26 370
m-Xylene - g 38.81 0.805 28 405
m-Xylene - g -57.22 0.568 25 730
p-Xylene - g 40.41 0.793 28 952
p-Xylene - l -54.99 0.557 26 310
Durene C 6 H 2 (CH 3 ) 4 s -242.68 0.437 19 000
Cumene C 6 H 5 CH(CH 3 ) 2 l -81.94 0.556 20 700
Mesitylene C 6 H 3 (CH 3 ) 3 l -126.34 0.544 24 830
Diphenyl C 6H 5-C 6H 5 s 135.34 0.319 57 400
Diphenylmethane C 6 H 5 -CH 2 -C 6 H 5 s 117.22 0.340 63 600
Triphenylmethane (C 6 H 5 ) 3 s 170.92 0.305 101 400
Naphthalene C 10H 8 s 124.53 0.311 45 200
Anthracene C 14 H 10 s 154.86 0.278 64 800
Phenanthrene C 14 H 10 s 129.62 0.284 60 000
Table 1.4: Formation enthalpies (cal/gr),
(cal/mol). Hydrocarbons.
entropies (cal gr −1 K −1 ) and free energies

1.3. MIXTURE OF GASES 11
Helmholtz free energy
Like in (1.20), here f can be expressed in the form
f = u − T s. (1.24)
( ) p
f = f 0 + R g T ln , (1.25)
p 0
where
f 0 = u − R g T s 0 (1.26)
is the free energy at standard pressure, which depends only on the gas temperature.
Remark
When desired to refer the thermodynamic functions to a mol, the previous formulae
should be changed as follows:
1) c v and c p are the molar heats.
2) R g must be substituted by the universal constant R.
3) In the state equation (1.7), ρ must be substituted by the number c of moles of the
gas per unit volume.
1.3 Mixture of gases
Now let us see the form taken by the thermodynamic functions in a mixture of gases.
For the purpose we shall consider a mixture of gases formed by l different chemical
species A i , (i = 1, 2, . . . , l). Let us assume that its state is such that the mixture
behaves as an ideal gas. The problem deals with the expression of the thermodynamic
functions by means of the corresponding functions of the chemical species of the
mixture.
Concentrations, mole fractions, densities and mass fractions
Let c and c i , (i = 1, 2, . . . , l), be the number of moles of the mixture and of each
species per unit volume respectively, and let X i be the mole fraction c i /c of species A i .

12 CHAPTER 1. THERMOCHEMISTRY
Between these magnitudes the following relations exist
∑
c j =c , (1.27)
j
∑
X j =1 . (1.28)
j
Let ρ be the density of the mixture, ρ i the partial density of species A i , Y i its
mass fraction and M i its molar mass. The following relations exist
ρ i = ρY i , (1.29)
∑ ∑
ρ j = ρ, Y j = 1 , (1.30)
j
j
c i = ρ M i
Y i , (1.31)
c = ρ ∑ j
Y j
M j
, (1.32)
The mean molar mass M m of the mixture is defined by
X i = ∑
Y i/M i
, (1.33)
Y j /M j
j
Y i = ∑
M iX i
. (1.34)
M j X j
j
M m = ∑ j
M j X j . (1.35)
From Eqs.
following expression
(1.30) and (1.35) one deduces for M m as a function of Y i the
1
M m
= ∑ j
Y j
M j
. (1.36)
Equation of state of the mixture
Since the number of moles of the mixture per unit volume is c, one has
p = cRT = RT ∑ j
c j = ∑ j
p j , (1.37)
where
p j = c j RT = R gj T = X j p (1.38)

1.3. MIXTURE OF GASES 13
is the partial pressure exerted by species A j when only this species occupies the volume
of the mixture at temperature T and R gj is the gas constant of the said species.
From (1.32), Eq. (1.37) can be written in the form
p
ρ = RT ∑ j
Y j
M j
, (1.39)
or else, in the form
where
R m = R ∑ j
p
ρ = R mT, (1.40)
Y j
M j
=
R M m
= ∑ j
R gj Y j (1.41)
is the gas constant of the mixture. Opposite to R g , R m changes with the composition
of the mixture. If the molar masses of the species are very different then the variations
of R m with the composition of the mixture can be very ”large”. Otherwise, these
variations are ”small”. For simplicity, in many aerothermochemical problems R m is
assumed to be constant, taken a mean value.
When needed to take into account the deviations of the state equation respect
to Eq. (1.41) this can be attained by using an equation similar to (1.1) for the mixture.
Thus, the second virial coefficient B m (T ) for the mixture is given by the expression 6
∑∑
B rs Y r Y s
B m = M m √ . (1.42)
Mr M s
r
s
Here, if r = s, B rr is the second virial coefficient of species A r previously
defined. If r ≠ s, B rs is the second virial coefficient for the interaction potential
between the molecules of species A r and A s . Since, in general, the constants relative
to those mixed potentials are not available, several formulae have been proposed for
B rs as a function of B rr and B ss . For example 7
B rs = 1 2
(√ √ )
Mr Ms
B rr + B ss . (1.43)
M s M r
The potential energy of an ideal gas due to interaction between molecules is
zero. Therefore, each species adds to the value of any thermodynamic function ψ per
unit mass of the mixture (internal energy, enthalpy, entropy, free energy, etc.) with
a value which is independent from the presence of other species. For A i , the said
contribution to the value ψ i of the corresponding thermodynamic function per unit
6 See Ref. [1], p. 153.
7 See Ref. [6], p. 211.

14 CHAPTER 1. THERMOCHEMISTRY
mass of the species at temperature T of the mixture and at partial pressure p i of A i
multiplied by its mass fraction Y i . That is
ψ = ∑ i
ψ i Y i . (1.44)
In the following, Eq. (1.44) is applied to the calculation of the thermodynamic
functions of the mixture.
Internal Energy
The internal energy u per unit mass of the mixture is
u = ∑ i
u i Y i . (1.45)
When taking into this expression the value
u i = u 0i +
∫ T
obtained when (1.9) is particularized for species A i , it results
u = u 0 +
where
∫ T
T 0
c vi (T ) dT (1.46)
T 0
c v (T ) dT, (1.47)
u 0 = ∑ u 0i Y i (1.48)
i
is the energy of the mixture at temperature T 0 and
c v = ∑ i
c vi Y i (1.49)
is the heat capacity at constant volume per unit mass of the mixture.
Enthalpy
Similarly
with
and
∫ T
h = h 0 + c p (T ) dT, (1.50)
T 0
h 0 = ∑ i
c p = ∑ i
h 0i Y i (1.51)
c pi Y i . (1.52)

1.3. MIXTURE OF GASES 15
Entropy
Similarly
∫ T
c p (T )
s = s 0 + dT − ∑
T 0
T
j
( )
pj
R gj Y j ln
p 0
(1.53)
with
s 0 = ∑ j
s 0j Y j . (1.54)
Equation (1.53) can also be written in the following form
( ) p
s = s 0 − R m ln − ∑ R gj Y j ln X j , (1.55)
p 0
j
where
s 0 = ∑ j
s 0 jY j . (1.56)
Once p 0 is fixed, this value depends only on T and on the composition of the mixture.
In (1.55), the two first terms of the right hand side give the entropy that the gas would
have at pressure p and at temperature T if instead of being a mixture it were a single
chemical species. The remaining term is the entropy of mixing whose contribution is
always positive.
Free Energy
Gibbs free energy
g = h − T s, (1.57)
where h and s are given by (1.50) and (1.55) respectively. g can also be expressed in
the form
( ) p
g = g 0 + R m T ln + T ∑ p 0
j
R gj Y j ln X j , (1.58)
where
g 0 = h − T s 0 . (1.59)
Helmholtz free energy
where u and s are given by (1.47) and (1.55) respectively.
f = u − T s, (1.60)

16 CHAPTER 1. THERMOCHEMISTRY
As in the previous cases, f can be expressed in the form
( ) p
f = f 0 + R m T ln + T ∑ p 0
j
R gj Y j ln X j , (1.61)
where
f 0 = u − T s 0 . (1.62)
Chemical Potentials
The chemical potential µ i of species A i in the mixture is G i
( )
( )
µ i = G i = G 0 pi
p
i + RT ln = G 0 i + RT ln + RT ln X i . (1.63)
p 0 p 0
1.4 Calculation of the thermodynamic functions
The preceding formulas show that all thermodynamic functions of a gas can be determined
provided that one of them and the standard values for the others are known. For
example, the heat capacity, either at constant pressure or at constant volume, and the
formation enthalpy and entropy of the gas determine all other thermodynamic functions.
The law of variation of heat capacity as function of T is generally complicated.
Fig. 1.5, for example, obtained from the tables in Ref. [10], shows the curves of c p
versus T for some gases. These curves are usually approximated with empirical formulae.
For example, the following Table 1.3, taken from Ref. [4] p. 28, gives some
of these parabolic formulae as well as their maximum deviations within the range
300 < T < 2 000K. The use of linear formulas is advisable for smaller ranges of T .
Gas c p (cal gr −1 K −1 ) Max. error (%)
H 2 3.440 + 0.033 × 10 −3 T + 0.1395 × 10 −6 T 2 1
N 2 0.225 + 0.065 × 10 −3 T - 0.0123 × 10 −6 T 2 2
O 2 0.196 + 0.086 × 10 −3 T - 0.0241 × 10 −6 T 2 1
CO 0.223 + 0.075 × 10 −3 T - 0.0164 × 10 −6 T 2 2
NO 0.207 + 0.081 × 10 −3 T - 0.0204 × 10 −6 T 2 2
H 2 O 0.383 + 0.182 × 10 −3 T - 0.0191 × 10 −6 T 2 2
CO 2 0.156 + 0.194 × 10 −3 T - 0.0562 × 10 −6 T 2 3
NH 3 0.348 + 0.527 × 10 −3 T - 0.1040 × 10 −6 T 2 1
C 2H 2 0.318 + 0.404 × 10 −3 T - 0.1020 × 10 −6 T 2 2 (for T > 400 K)
CH 4 0.211 + 1.119 × 10 −3 T - 0.2620 × 10 −6 T 2 2
Table 1.5: c p(T ) correlations for several gases.

1.4. CALCULATION OF THE THERMODYNAMIC FUNCTIONS 17
1.4
1.2
1.0
c p
(cal/gr K)
0.8
0.6
0.4
C 2
H 2
CH 4
CO 2
H 2
(0.1× c p
)
H 2
O
0.2
CO, N 2
O 2
0.0
0 400 800 1200 1600 2000 2400 2800
T(K)
Figure 1.5: Specific heat at constant pressure of several gases as function of temperature.
When working at normal or not too high temperatures the determination of
the thermodynamic functions can be done through calorimetric measurements. On
the other hand, when working at very high temperatures, as are those encountered
in combustion processes, such measurements are very difficult when not impossible.
In such case one has to resort to Statistical Mechanics and to the use of the results
provided by the spectroscopic analysis of the gas. This method gives excellent results
specially when dealing with simple molecules.
The method is based on the formation of the so-called partition function of the
gas. Once this function is known all thermodynamic functions are really computed. 8
Quantum Mechanics shows that the energy of a particle confined within a domain
can only take discrete set of values. These values are called “energy levels”. Let
ε j (j = 1, 2, . . .) be the energy levels of the molecules, when the gas occupies volume
V at temperature T . The following expression
Q = ∑ j
exp (−ε j /kT ) , (1.64)
is called “partition function” of the gas molecules. The summation is extended to all
energy levels. If one of them is degenerate, it must be counted as many times as corresponds
to its order of degeneracy. That is, the number of discernible states in which
this level can be attained. Fixing volume V and temperature T of the gas its partition
8 For a full study on this matter see Refs. [11] and [12].

18 CHAPTER 1. THERMOCHEMISTRY
function is determined. Then, the problem lies in expressing the thermodynamic functions
of the gas by means of its partition function. According to Statistical Mechanics
the solution is as follows.
a) Internal energy.
b) Enthalpy.
c) Entropy.
h = R g T
( ) ∂ ln Q
u = R g T 2 . (1.65)
∂T
V
[( ) ∂ ln Q
+
∂ ln T
V
s = R g
[ln Q + T
( ) ] ∂ ln Q
. (1.66)
∂ ln V
T
( ) ] ∂ ln Q
. (1.67)
∂ ln T
V
d) Free energy.
d.1) Gibbs:
g = R g T
[
ln Q −
( ) ] ∂ ln Q
. (1.68)
∂ ln V
T
d.2) Helmholtz: f = −R g T ln Q. (1.69)
e) Chemical Potential.
G = M g g = RT
[
ln Q −
( ) ] ∂ ln Q
. (1.70)
∂ ln V
T
Heat capacities at constant volume, c v , and at constant pressure, c p , are expressed
by u and h respectively in the form
c v = ∂u
∂T ,
c p = ∂h
∂T . (1.71)
This enables the calculation of their values as a function of Q by means of (1.65) and
(1.66).
Thus, the problem reduces to the determination of the energy levels for the formation
of Q. For the purpose, one must analyze the ways in which a gas molecule can
store energy. Except when working at very high temperatures, where it is necessary to
consider the states of electronic excitation in the molecule, this can be considered as
a system of material points which are its atoms. Then, the energy of each molecule is
the summation of the kinetic energies of its atoms plus the potential energy of the field
of forces that hold them together. Be n the number of atoms in each molecule. Then,
its number of degrees of freedom is 3n. Of which three are external degree of freedom
corresponding to the translational motion of the center of gravity of the molecule. The

1.4. CALCULATION OF THE THERMODYNAMIC FUNCTIONS 19
remaining 3n − 3 are internal degrees of freedom and they correspond to its rotational
and vibrational motions. Energy ε j of the molecule can always be expressed as
ε j = ε t j + ε i j. (1.72)
Here ε t j is the kinetic energy of the molecule due to the motion of its center of gravity,
and ε i j is the internal energy. Moreover, the translational levels εt j are independent
from the internal levels ε i j . Therefore, the combination of either one of the εt j with
either one of the ε i j gives the energy level ε j. Then, it is easily verified that the separation
(1.72) for ε j corresponds to a factorization
Q = Q t · Q i (1.73)
for Q, where
Q t = ∑ j
exp ( −ε t j/kT ) (1.74)
and
Q i = ∑ j
exp ( −ε i j/kT ) (1.75)
are the translational and internal partition functions of the molecule respectively. All
thermodynamic functions are linear and homogeneous in ln Q and in its derivatives.
Hence, when factorization (1.73) is substituted into them, these functions are separated
into contributions from Q t and Q i respectively. Let Φ be one of these functions. Then,
Φ = Φ t + Φ i , (1.76)
where Φ t is the contribution to Φ from Q t , and Φ i is the contribution from Q i . This
property simplifies the calculation of the thermodynamic functions.
Translational partition function Q t is the same for all gases
Q t = V
( ) 3
2πmkT
2
. (1.77)
h 2
Here, m is the mass of the molecule and h = 6.6238 × 10 −27 erg×s is the Planck’s
constant. On the other hand Q i depends on the structure of the molecule. At least in
first approximation, ε i j can also be separated into contributions from internal energy
of rotation ε r j and internal energy of vibration εv j
ε i j = ε r j + ε v j . (1.78)
Here, as before, ε r j and εv j are approximately independent from each other. Therefore,
separation (1.78) gives for Q i
Q i = Q r · Q v . (1.79)

20 CHAPTER 1. THERMOCHEMISTRY
Two cases can arise in the rotational motion of a molecule:
1) If the molecule is linear, that is, if all its atoms lie on a straight line, the moment
of inertia with respect to the axis of the molecule is zero and it cannot store
energy in such a degree of freedom. Hence, the rotational degrees of freedom of
the molecule reduce to two.
2) Whereas if the molecule is non-linear the number of its rotational degrees of
freedom is 3.
If the rotational coordinates are selected accordingly, 9 the energies corresponding
to the rotational degrees of freedom are also separable. Then, except at temperatures
under ambient, the distribution Q rj from the rotational degree of freedom j to
the rotational partition function Q r is
√
2kT Ij
Q rj = 2π , (1.80)
h
where I j is the moment of inertia of the molecule for such degree. Hence, we have:
a) For linear molecules
Q r = 8π2 kT I
h 2 , (1.81)
δ
where I is the moment of inertia of the molecule respect to an axis passing
through its center of gravity and normal to the axis of the molecule.
b) For non-linear molecules
Q r = 1 δ
( 8π 2 ) 3
2
kT √Ix
h 2 I y I z . (1.82)
Here I x , I y and I z are the three moments of inertia of the molecule with respect
to three orthogonal axis at its center of gravity.
In (1.81) and (1.82) δ is a symmetry factor, that accounts for undisguised stable
states due to symmetries of the molecule. For example, for linear molecules δ = 2.
Thus, Q r is determined when the moments of inertia of the molecule are known. Such
moments are obtained from spectroscopic analysis.
As for vibration energy a direct determination of its levels through spectroscopic
analysis is best. Independent contributions of the vibrational degrees of freedom
can, otherwise, be obtained by assuming that their corresponding energies are
separable. Which happens, for example, when vibration amplitudes are small enough
so that the potential energy of the molecule can be approximated by a quadratic function
of the deviations of the atoms respect to their equilibrium positions. For this
9 For which the principal axis of inertia of the molecule must be taken.

1.5. CHEMICAL REACTIONS IN A MIXTURE OF GASES 21
case, a normal system of coordinates can be adopted where the vibrational energy is
separated into contributions from the natural modes of the molecule. Be ν s the frequency
of one of these modes. Under the assumption that the molecule behaves as an
harmonic oscillator the contribution Q vs of this mode to Q v is
(
Q vs = exp − hν ) ( (
s
1 − exp − hν )) −1
s
. (1.83)
2kT
kT
Frequency ν s is obtained from the vibration spectra of the molecule.
Finally, for very high temperatures the contribution of electronic excitation
must be included in the value of Q. The corresponding energy levels are obtained
from the spectra.
The preceding analysis assumes that the rotational and vibrational energies are
separable. Yet, this does not always happen, mainly for high temperatures. In fact, rotation
stretches the molecule and this introduces coupling terms between the rotational
and vibrational degrees of freedom, preventing a separate study of both contributions.
Besides, the assumption that vibration is harmonic not always is justified, which forces
the introduction of correcting terms to account for anharmonicity. 10
1.5 Chemical reactions in a mixture of gases
Now, let us assume that the l species A i of the mixture can react according to a system
of r reaction equations of the form
∑
ν ijA ′ i ⇄ ∑
i
i
ν ′′
ijA i , (j = 1, . . . , r), (1.84)
where ν ij ′ and ν′′ ij are the stoichiometric coefficients of species A i in the reaction j for
the forward and backward reactions respectively. Let
These coefficients must satisfy the following conditions
ν ij = ν ′′
ij − ν ′ ij. (1.85)
∑
ν ij M i = 0, (j = 1, . . . , r), (1.86)
i
which express that the mass of the mixture is not changed by chemical reactions.
Let us consider the unit mass of the mixture and be ∆C ij the number of moles
of species A i produced by the reaction j. Due to (1.84) the following relations exist
10 For further information on this problem see Refs. [11] and [12].

22 CHAPTER 1. THERMOCHEMISTRY
between ∆C ij
∆C 1j
ν 1j
= ∆C 2j
ν 2j
= . . . = ∆C lj
ν lj
. (1.87)
Let ξ j be the common value of these relations. ξ j is called the degree of
advancement of reaction j (De Donder [13]). Thus defined, ξ j has the dimensions
mole/gr. The number of moles of species A i produced by reaction j is, therefore,
∆C ij = ν ij ξ j , (1.88)
and the number ∆C i of moles of A i produced by the r reactions is
∆C i = ∑ j
∆C ij = ∑ j
ν ij ξ j . (1.89)
If C i0 is the number of moles of species A i existing when the reactions start,
the number C i that will exists when the degrees of advancement of the reactions are
ξ j , (j = 1, 2, . . . , r), is
C i = C i0 + ∑ ν ij ξ j . (1.90)
j
Similarly the mass fraction Y i of species A i produced by the r reaction is
∑
Y i = M i ν ij ξ j , (1.91)
and if Y i0 is the initial mass fraction of this species, one has
∑
Y i = Y i0 + M i ν ij ξ j . (1.92)
j
j
1.6 Chemical equilibrium
Conditions of equilibrium
Once the reactions are initiated they continue up to the point where the mixture reaches
its state of equilibrium. Such an equilibrium is determined by additional conditions
which fix the state of the system. Let us assume, for example, that the process is
adiabatic and takes place at constant pressure. Such a case has a great practical interest
in the study of combustion processes. Then the First Law of Thermodynamics shows
that the enthalpy of the mixture must be constant. Therefore, the conditions that must
be satisfied by the mixture are as follows
h = h 0 = const., p = p 0 = const. (1.93)

1.6. CHEMICAL EQUILIBRIUM 23
Since the process is adiabatic the state of equilibrium is determined by the
condition that entropy s be maximum. The condition for this is that its first variation
be zero. Moreover, from (1.93) the variations of h and p must also be zero. Therefore,
the state of equilibrium is determined by conditions
δh = δp = δs = 0. (1.94)
Thermodynamics shows 11 that the entropy variations of the mixture is given by
the expression
T δs = δh − V δp − ∑ i
µ i
M i
δY i . (1.95)
Here V is the volume of the unit mass of the mixture and µ i is the chemical potential
of species A i in the mixture. This relation, by virtue of (1.94), reduces to
∑ µ i
δY i = 0. (1.96)
M i
i
Variations of δY i cannot be arbitrary since Y i must satisfy conditions (1.92). From
them one obtains
δY i = ∑ j
M i ν ij δξ j . (1.97)
This expression when taken into (1.96) gives
∑∑
µ i ν ij δξ j = ∑
i j
j
( ∑
i
µ i ν ij
)
δξ j = 0, (1.98)
and since the variations δξ j are arbitrary this condition breaks into the following system
of equilibrium equations
∑
µ i ν ij = 0, (j = 1, 2, . . . , r). (1.99)
i
The expression ∑ µ i ν ij is named affinity α j of reaction j. Therefore, the equilibrium
is determined by the condition that the affinities of the reactions become zero
i
α j = 0, (j = 1, 2, . . . , r). (1.100)
System (1.100), together with conditions (1.92) and (1.93), determine the values
for the degrees of advancement of the various reactions and for the corresponding
mass fractions of the species as well as for the temperature T of the mixture.
11 See Ref. [4], p. 68.

24 CHAPTER 1. THERMOCHEMISTRY
Remarks
If the reactions take place at pressure and temperature (instead of enthalpy) both constant
the conditions of equilibrium are
δp = δT = δg = 0. (1.101)
But
δg = δh − sδT − T δs. (1.102)
From Eqs. (1.95), (1.97), (1.101) and (1.102) conditions (1.99) are obtained again.
Thus conditions of equilibrium are the same for both cases, with the only difference
that here, temperature is given whilst in the previous case it must be calculated. The
computation is done by determining the conditions of equilibrium as a function of temperature,
and then looking for the value of T which satisfies the additional condition
h = h 0 .
are
Likewise, for constant volume adiabatic reactions the conditions of equilibrium
δu = δV = δs = 0, (1.103)
from which result again conditions (1.99).
1.7 Case of a mixture of ideal gases
For the case of a mixture of ideal gases the form taken by the conditions of equilibrium
is specially simple. In fact, as for (1.63) system (1.99) reduces to
∑
G i ν ij = 0, (j = 1, 2, . . . , r), (1.104)
i
or else
∑
G 0 i ν ij + RT ∑
i
i
( )
pi
ν ij ln = 0, (j = 1, 2, . . . , r), (1.105)
p 0
which together with (1.38) gives
∏
i
X ν ij
i =
( ) (
νj
p0
exp − ∆G0 j
p
RT
)
, (j = 1, 2, . . . , r), (1.106)
where
ν j = ∑ i
ν ij (1.107)

∆G 0 j = ∑ i
1.7. CASE OF A MIXTURE OF IDEAL GASES 25
and
(
Hi − T Si
0 )
νij (1.108)
is the increase in free energy due to reaction j at temperature T of the mixture and
at standard pressure p 0 . Thus ∆G 0 j depends only on T . Table 1.2 gives the values of
for several species corresponding to their formation from the chemical elements
∆G 0 j
at the standard state. Therefore Eq. (1.106) can be written
∏
( ) νj
X νij p0
i = Kj 0 (T ) , (j = 1, 2, . . . , r), (1.109)
p
i
where Kj 0 is called the equilibrium constant of reaction j depending only on temperature
T . The values of Kj 0 versus T are given in Fig. 1.6, for several typical reactions
of interest in combustion problems.
30
25
1
2H2 +OH ↔ H2O
20
log 10
(K p
)
15
10
5
0
CO
H 2 + 1 2O2 ↔ H2O
CO 2 +H 2 ↔ CO +H 2O
+ 1 2O2 ↔ CO2
1
2N2 +H2O ↔ NO +H2O
H 2O ↔ H 2 +O
1
2 N2 + 1 2O2 ↔ NO
1
2 H2 ↔ H
−5
1
2 N2 ↔ N
−10
0 400 800 1200 1600 2000 2400 2800 3200 3600
T(K)
Figure 1.6: Equilibrium constant of reaction vs temperature, for several typical reactions of
interest in combustion problems.
The chemical equations (1.84) used for this calculation can be any ones within
the following limitations: a) they must be linearly independent; b) their number must
be the maximum that can exist between species. This number can be determined when
the phase rule is applied. Be l ′ < l the number of components of the mixture, then
the number of independent reactions will be l − l ′ . In turn l ′ can be determined as
follows: let E j , (j = 1, 2, . . . , l), be the chemical elements forming the species and
a ij be the number of atoms of element E j in species A i . The number of components

26 CHAPTER 1. THERMOCHEMISTRY
of the mixture is the rank of the matrix of coefficients a ij .
The coefficients of a
principal minor of order l ′ of this matrix determine the species that can be adopted
as components of the mixture. A complete system of independent chemical reactions
can now be obtained by expressing each one of the remaining species as a linear
combination of the said components in the form
∑
A j = b ji A i , (j = l ′ + 1, l ′ + 2, . . . , l). (1.110)
l ′
i=1
Here, the first l ′ species are assumed to be the components of the mixture.
For example in the combustion of mixtures of hydrocarbons and some of their
compounds with oxygen and nitrogen, the combustion products are formed by the
species: CO 2 , CO, H 2 O, HO, H 2 , H, O 2 , O, NO, N 2 and N. Therefore, one has:
l = 11; l ′ = 4; and the number of independent chemical reactions is 7. The following
systems of reactions can be adopted (see Ref. [13]) since they are linearly
independent:
C O 2 + H 2 ⇄ C O + H 2 O
1
2 N 2 + H 2 O ⇄ N O + H 2
2 H 2 O ⇄ 2 H 2 + O 2
H 2 O ⇄ O + H 2
N 2 ⇄ 2 N
H 2 ⇄ 2 H
H 2 O ⇄ 1 2 H 2 + O H
(1.111)
The r equations (1.106), together with the l ′ equations which express the conservation
of components in the reactions, determine the equilibrium composition of
the mixture, once its temperature is known. Thus in the preceding example system
(1.111) must be completed with the four equations for the conservation of the number
of atoms of carbon, oxygen, hydrogen and nitrogen. It is not always necessary to consider
all possible reactions since depending on the state of the mixture the fractions of
some of the species may be neglected and the computation simplified.
For the case of adiabatic equilibrium the temperature of the products is unknown.
Hence, trial and error methods become necessary. In applying then, first a
temperature is assumed and the equilibrium composition is determined for it. Once
this is known the corresponding temperature is obtainable through equation (1.10),
h = h 0 , and if this temperature differs from the one previously assumed, the computation
should be reviewed. Gayden and Wolfhard [14] recommend the application of
Damköhler and Edse’s method mentioned in Ref. [15]. The fundamentals and some
of the applications of this method are described in the book by Gayden and Wolfhard.

1.7. CASE OF A MIXTURE OF IDEAL GASES 27
0.30
3200
T b
X i
0.25
0.20
0.15
H O 2
CO (X /2)
H CO CO 2 2
O (X /2) 2 O2
3000
2800
2600
T(K)
0.10
0.05
H
O
OH
2400
2200
0.00
2000
0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20
Y
O2
/ (Y
O2
) s
Figure 1.7: Compositions and temperature of the combustion products obtained when burning
a mixture of ethylene and oxygen as a function of the ratio of the mass
fraction of oxygen to the corresponding stoichiometric mixture.
Other methods such as those in Refs. [16] and [17] include diagrams in order to reduce
calculations as they are very cumbersome when the number of species is large.
Then, specially when many cases must be solved, it becomes necessary to systematize
the procedure. This is accomplished by giving computational schemes which enable
the use of electronic computers. Full description of these methods can be found, for
instance, in Refs. [18] and [19] where additional bibliography is included.
Fig. 1.7, taken from Ref. [14], shows the compositions and temperature of the
combustion products obtained when burning a mixture of ethylene and oxygen as a
function of the ratio of the mass fraction of oxygen to the corresponding stoichiometric
mixture.
Fig. 1.8, taken from Ref. [17], gives the theoretical combustion temperatures
for some hydrocarbons.
Table 1.6, taken from Ref. [14], gives the adiabatic combustion temperatures
for some typical mixtures, as well as these that would be obtained by neglecting dissociations.
The book by Lewis and von Elbe mentioned in Ref. [20] gives the experimental
adiabatic combustion temperatures for great number of mixtures. See Ref. [21]
for the experimental technique used for the determination of combustion temperatures.

28 CHAPTER 1. THERMOCHEMISTRY
2500
2400
acetylene
methylacetylene
2300
T b
(K)
2200
ethene
benzene
propylene
a:n−hexane
1−butene
b: n−pentane
2100
1−pentene
c: n−butane
d: propane
cyclopentene
metane a,b,c,d
2000
etane
0.8 0.9 1.0 1.1 1.2 1.3 1.4
Y / (Y ) F F s
Figure 1.8: Theoretical combustion temperatures of several hydrocarbons in the air at atmospheric
pressure.
Fuel CH 4 CH 4 C 3H 8 C 3H 8 C 8H 18 C 8H 18 C 2H 2 H 2
Mixture 8
>< Fuel 0.2
Y i O 2 0.8
> :
N 2 0
Products
H 2O
O 2
O
H 2
8> < OH
X i
CO
> :
CO 2
NO
N 2
H
0.3719
0.0691
0.0326
0.0690
0.1438
0.1512
0.1166
0
0
0.0458
0.1062
0.3865
0.5073
0.2735
0.0128
0.0041
0.0343
0.0402
0.0747
0.0818
0.0054
0.4527
0.0105
0.2245
0.7755
0
0.2988
0.0665
0.0375
0.0648
0.1378
0.2064
0.1372
0
0
0.0510
0.1738
0.6005
0.2257
0.2627
0.0483
0.0230
0.0495
0.1010
0.1612
0.1232
0.0100
0.1880
0.0331
0.2218
0.7782
0
0.2562
0.0880
0.0462
0.0538
0.1447
0.2167
0.1450
0
0
0.0493
0.1076
0.3777
0.5147
0.1853
0.0307
0.0073
0.0175
0.0453
0.0782
0.1259
0.0091
0.2923
0.0084
0.0918
0.2113
0.6969
0.0745
0.0015
0.0008
0.0170
0.0092
0.1159
0.0810
0.0019
0.6934
0.0048
0.0305
0.1360
0.8335
0.1886
0.00
0.00
0.1505
0.00
0
0
0.00
0.5609
Temperatures ( ◦ C)
Dissociation 2 737 2 410 2 776 2 689 2 809 2 447 2 300 1 358
No Dissociation 5 047 2 980 5 202 4 405 5 740 3 417 2 439 1 362
0.00
Table 1.6: Adiabatic combustion temperature for some typical mixtures.

1.8. CHEMICAL KINETICS 29
1.8 Chemical kinetics
In some of the Aerothermochemistry problems it is essential to know the reaction
rates of the species forming a mixture of gases. 12
determined:
1) The reactions that can take place between the species.
2) The governing laws of their reaction rates.
Such questions are the subject of Chemical Kinetics.
For which the following must be
In Ref. [22] an introduction to Chemical Kinetics can be found mainly from
the viewpoint of jet propulsion. For a more complete study on this subject see Refs.
[23], [24] and [25]. The present paragraph gives a brief introduction to this matter.
First let us consider the case where the species react according to a single
irreversible reaction
∑
ν iA ′ i → ∑
i
i
ν ′′
i A i , (1.112)
and let ξ be the degree of advancement of this reaction as was defined in paragraph 5.
The reaction rate of Eq. (1.112) is, by definition,
r = dξ
dt = ˙ξ. (1.113)
From here, keeping in mind (1.88), the number dC i / dt of moles of species A i
produced per unit mass of the mixture and per unit time is
dC i
dt
= (ν ′′
i − ν ′ i) dξ
dt . (1.114)
Likewise, the mass dm i / dt of these species produced per unit mass and time is
dm i
dt
= M i
dC i
dt
= M i (ν ′′
i − ν ′ i) dξ
dt . (1.115)
Aerothermochemistry deals, normally, with mass w i of A i produced per unit
volume and per unit time, which is
w i = ρ dm i
dt
= ρM i (ν ′′
i − ν ′ i) dξ
dt . (1.116)
The problem lies in determining the way in which w i depends on the state and composition
of the mixture. Such determination is done through the so-called law of mass
action of Guldberg and Waage [26].
12 See, i.e., Chap. 6.

30 CHAPTER 1. THERMOCHEMISTRY
This law states that the number dc i / dt of moles of A i produced within unit
volume during unit time is
dc i
dt = (ν′′ i − ν ′ i) k ∏ j
c ν′ j
j . (1.117)
Here, k is independent from the composition and pressure of the mixture and depends
only on its temperature. It is called specific rate or rate coefficient of the reaction and
its dimensions are ( cm 3 /mol ) ν−1
s −1 , where
ν = ∑ j
ν ′ j (1.118)
is called order or molecularity of the reaction.
Since the molar mass of A i is M i , we have for w i , from (1.117)
w i = M i
dc i
dt = M i (ν ′′
i − ν ′ i) k ∏ j
c ν′ j
j . (1.119)
This formula can also be expressed as a function of the mass fractions of the reacting
species, by virtue of (1.31), in the form
w i = ∏
ρ ν
M ν′ j
j
j
kM i (ν ′′
i − ν ′ i) ∏ j
Y ν′ j
j . (1.120)
For example, in the case of a bimolecular reaction between species A 1 and A 2 which
produces A 3 ,
one has, due to (1.120),
A 1 + A 2 → A 3 , (1.121)
w 1
= w 2
= − w 3
= dc 1
M 1 M 2 M 3 dt = dc 2
dt = − dc 3
dt = −
ρ2
kY 1 Y 2 . (1.122)
M 1 M 2
The way in which the specific rate k depends on temperature is given by Arrhenius’s
law
k = Ae −E/RT (1.123)
where E is called activation energy of the reaction and A is a constant or the product
of a constant by a power of the absolute temperature T
A = BT α . (1.124)
Here B is a constant called frequency factor and α is an exponent within zero and one.
Arrhenius [27] deduced his law empirically as a generalization of van’t Hoff’s
law for the variation of equilibrium constants as functions of temperature. Statistical

1.8. CHEMICAL KINETICS 31
Mechanics 13 enables a theoretical derivation of this law throughout the so-called Collisions
Theory. For this derivation it is enough to assume that for a reaction to occur
the following is needed:
1) Coincidence of reacting molecules in a collision.
2) The collision energy in certain degrees of freedom must be larger than a given
value which depends on the activation energy of reaction.
When calculating the number of collisions per unit volume and per unit time that satisfy
the preceding conditions, one obtains expressions similar to the Arrhenius law for
the influence of temperature and to the law of mass action for the influence of concentrations.
For example, in the simple case of a bimolecular reaction of the type (1.121)
the number n of collisions per unit volume and per unit time between molecules of
the species A 1 and A 2 is
n = N 2 σ 2 12c 1 c 2
√8πRT
( )
M1 + M 2
. (1.125)
M 1 M 2
Here σ 12 is the mean molecular diameter of both species. Let us assume that from the
n collisions (1.125) only are active, that is, only produce reaction, those that have a
relative kinetic energy of approximation larger than ε = E/N. 14 The number n a of
these active collisions is
n a = n e −E/RT . (1.126)
Hence, the velocity dc 1 / dt of the reaction, in moles per unit volume and per unit
time, is
That is, due to (1.125),
dc 1
dt = dc 2
dt = − dc 3
dt = −n a
N . (1.127)
√
dc 1
dt = dc 2
dt = − dc ( )
3
dt = M1 + M 2
−Nσ2 12c 1 c 2 8πRT
e −E/RT . (1.128)
M 1 M 2
Due to (1.31), this formula can be written in the form
dc 1
dt = dc 2
dt = − dc 3
dt =
−
√
ρ2
Nσ 2
M 1 M
12Y 1 Y 2 8πRT
2
( )
M1 + M 2
e −E/RT ,
M 1 M 2
(1.129)
13 See Ref. [23].
14 It is not essential that the kinetic energy of approximation of the collision be precisely the one larger
than ε. The formulae holds as long as the collision energy in two separated degrees of freedom is larger
than ε.

32 CHAPTER 1. THERMOCHEMISTRY
which can be identified with Eq. (1.122) by making
√ ( )
k = Nσ12
2 M1 + M 2
8πRT
e −E/RT , (1.130)
M 1 M 2
that is, from Eq. (1.124),
α = 1 2
(1.131)
and
B = Nσ 2 12
√
8πR
( )
M1 + M 2
. (1.132)
M 1 M 2
When comparing Collisions Theory with experimental results, an exact numerical
agreement between the reaction rates predicted and observed is not to be expected,
but an agreement of the orders of magnitude. In fact, Collisions Theory does not include,
among others, the influence of the relative orientation of the molecules as they
collide which can decide the success of the collision. Such an effect can be of importance
specially in molecules of complicated structure. It is taken into consideration
by including in Eq. (1.129) a numerical factor P of orientation equal or smaller than
unity
dc 1
dt = dc 2
dt = − dc 3
dt =
− P
√
ρ2
Nσ 2
M 1 M
12Y 1 Y 2 8πRT
2
( )
M1 + M 2
e −E/RT .
M 1 M 2
(1.133)
Experimental reaction rates agree, fairly, with those predicted by Collision Theory
only for a short number of reactions, yet in other cases experimental rates are as much
as 10 8 times larger or smaller than those predicted by theory. Such a discrepancy can
not be justified within the theory. These results that in the best cases Collision Theory
represents approximately the actual facts only for a short number of reactions. A more
correct formulation is provided by the so-called Theory of Absolute Reaction Rates.
In this theory it is assumed that the passing of the reacting species to the products
takes place through the formation of an activated complex resulting from the
assembly of the reacting species. This complex is considered as located at the top of
an energy barrier which separates the species from the products. The reaction rate
is determined by the velocity at which the activated complex crosses the barrier. It
is, furthermore, assumed that such complex stands in equilibrium with the reacting
species and that its dissociation in products is due to the action of one of the vibrational
degree of freedom. Then, by applying the laws of Statistical Mechanics the
reaction rate can be computed when the structure of the activated complex is known.

1.8. CHEMICAL KINETICS 33
This structure can be obtained from Quantum Mechanics or be postulated from the
structure of the molecules. The Absolute Reaction Rate Theory has been successfully
applied to the study of a great number of reactions well as to many other physical
and physicochemical phenomena. A full development of the theory can be found, for
instance, in the works mentioned in Refs. [24] and [25].
In the previous study it has been assumed that a single one directional reaction
takes place within the mixture. Generally, all reactions proceed simultaneously in both
senses at different rates. Thus, reaction (1.112) must be substituted by the following
∑
ν iA ′ i ⇄ ∑
i
i
ν ′′
i A i , (1.134)
and the rate w i of production of species A i is given by the expressions
(
)
∏ ∏
w i = M i (ν i ′′ − ν i)
′ k f c ν′ i
− k b c ν′′ i
. (1.135)
Here k f and k b are the specific rates corresponding to the forward and backward reactions
respectively.
i
On the other hand to be able to form the products, the reacting molecules must
contact one another. This reduces the molecularity of a single elementary reaction to
1, 2 or 3, since the probability of a collision of more than three molecules is practically
nil. Consequently, when the reaction rate depends in a complicated way on the
concentrations, it can be assured that it proceeds from the combination of several elementary
reactions. The basic problem of Chemical Kinetics is then the establishment
of the set of elementary reactions that produce within the mixture. For example, the
stoichiometric equation for the decomposition of ozone is
While the following is the actual system of reactions [28]:
i
i
i
2O 3 ⇄ 3O 2 . (1.136)
O 3 +G k fi
⇄
k b1
O+O 2 +G, (1.137)
O+O 3
k f2
⇄
k b2
2O 2 , (1.138)
O 2 +G k f3
⇄
k b3
2O+G. (1.139)
Here G is one of the atoms or molecules of the mixture. Each one of these reactants
acts at its own reaction rate and the rate of consumption of the ozone results from

34 CHAPTER 1. THERMOCHEMISTRY
the combination of the rates corresponding to elementary reactions. Denoting species
O, O 2 and O 3 with subscripts 1, 2 and 3 respectively, one obtains for the rate of
consumption of O 3
− w 3
M 3
= k f1 cc 3 − k b1 cc 1 c 3 + k f2 c 1 c 3 − k b2 c 2 2. (1.140)
This expression differs essentially from that obtainable from Eq. (1.136).
The complexity of the system of the elementary reactions that produce within
a mixture of reacting species originates an essential difficulty for the study of these
problems which depend on the reaction rates of the species. One of such problems
is, for example, the calculation of the propagation velocity of a flame throughout a
combustible mixture. 15 A great effort is being made by Chemical Kinetics towards the
enlightenment of the elementary reactions that produce in each case and of their corresponding
rates. Specially in the field of Combustion, the Fifth International Congress
held in 1954 devoted the major part of its work to the study of Combustion Kinetics. 16
In spite of the efforts the actual state of knowledge is still scant making impossible a
full study of the process except for a short number of particular cases. 17
Even when assuming that the system of reactions is known, it is usually too
complicated to be fully included in the study of an Aerothermochemical problem. For
this case the system must be simplified, for example with the steady state assumption,
18 or else by adopting global reaction rates for the mixture such as
w = Aρ n (1 − Y ) n e −E/RT , (1.141)
where Y is the mass fraction of products and n is the molecularity of the overall
reaction. In the following chapters frequent use will be made of similar expressions
for the approximate study of several problems.
References
[1] Corner, J.: Theory of Interior Ballistics of Guns. John Wiley and Sons, Inc.
New York, 1950.
[2] Taylor, J.: Detonation in Condensed Explosives. Oxford, at the Clarendon
Press, 1952.
15 See chapter 6.
16 See the Proceedings of the Congress, soon to be published.
17 See chapter 6.
18 See Ref. [28] where von Kármán and Penner have shown that the introduction of such an assumption
for the oxygen atoms in the ozone flame is justified. Yet, for other cases it appears less justified. For
example, in the Hydrogen–Bromine flame.

1.8. CHEMICAL KINETICS 35
[3] Hirschfelder, J. O., Curtiss, C. F. and Bird, R. B.: Molecular Theory of Gases
and Liquids. John Wiley and Sons, Inc. New York, 1954.
[4] Prigogine, I. and Defray, R.: Chemical Thermodynamics. Longmans, Green
and Co., New York, 1954.
[5] Guggenheim, E. A.: Thermodynamics. North Holland Publishing Comp. Amsterdam,
1950.
[6] Taylor, H. S. and Glastone: A Treatise on Physical Chemistry, Vol. II, States of
Matter. D. Van Nostrand Comp. Inc., New York, 1951.
[7] Lewis, G. N. and Randall, M.: Thermodynamics and the Free Energy of Chemical
Substances. McGraw-Hill Book Co., New York.
[8] National Bureau of Standards: Selected Values of Chemical Thermodynamic
Properties. 1950.
[9] Rossini, F. D.: Chemical Thermodynamics. John Wiley and Sons, Inc., New
York, 1950.
[10] The NBS-NACA Tables of Thermal Properties of Gases.
[11] Mayer, J. E. and Mayer, M. G.: Statistical Mechanics. John Wiley and Sons,
Inc., 1940.
[12] Pitzer, K. S.: Quantum Chemistry. Prentice-Hall, Inc., New York, 1953.
[13] Hirschfelder, J. O., McClure, F. T., Curtiss, C. F. and Osborne, D. W.: Thermodynamic
Properties of Propellant Gases. NDRC Rep. No. A-116, Nov. 23,
1942.
[14] Gayden, A. G. and Welfhard H. G.: Flames. Their Structure Radiation and
Temperature. Chapman and Hall Ltd., 1953.
[15] Damköhler, G. and Edse, R.: Zeitschrift für Elektrochemi, Vol. 49, 1943, p. 178.
[16] Huff, V. W. and Caluart, C. S.: Charts for the Computation of Equilibrium
Composition of Chemical Reactions in the Carbon-Hydrogen-Oxygen-Nitrogen
System at Temperatures from 2 000 to 5 000 K. NACA Technical Note No. 1653,
July 1948.
[17] Vichnievsky, R., Sale, B. and Marcadet, J.: Combustion Temperatures and Gas
Composition. Jet Propulsion, Vol. 25, No. 3, March 1955, p. 105.
[18] Huff, V. N., Gordon, S. and Morrell, V.: General Method and Thermodynamic
Tables for Computation of Equilibrium Composition and Temperature of Chemical
Reactions. NACA Technical Report No. 1037, 1951.
[19] Brinialey, S. R.: Computational Methods in Combustion Calculations. Combustion
Processes, Sec. C, Vol. II of High Speed Aerodynamics and Jet Propulsion,
Princeton University Press, 1956, p. 64.

36 CHAPTER 1. THERMOCHEMISTRY
[20] Lewis, B. and von Elbe, G.: Combustion, Flames and Explosions of Gases.
Academic Press., Inc. Publishers, New York, 1951.
[21] Bundy, F. P. and Streng, H. M.: Measurement of Flame Temperature, Pressure
and Velocity. Physical Measurement in Gas Dynamics and Combustion, Sec. I,
Vol. IX of High Speed Aerodynamics and Jet Propulsion, Princeton University
Press, 1954.
[22] Penner, S. S.: Introduction to Chemical Kinetics. AGARD AG 5/p2, December
1952.
[23] Hinshelwood, C. N.: The Kinetics of Chemical Change. Oxford University
Press, 1940.
[24] Glastone, S., Laidler, K. J. and Eyring, H.: The Theory of Rate Processes.
McGraw-Hill Book Company, Inc., New York, 1941.
[25] Laidler, K. J.: Chemical Kinetics. McGraw-Hill Book Company, Inc., New
York, 1950.
[26] Taylor, H. S.: Fundamentals of Chemical Kinetics. Combustion Processes, Sec.
D, Vol. II of High Speed Aerodynamics and Jet Propulsion, Princeton University
Press, 1956, p. 101
[27] Arrhenius, S.: Zeitschrift für Physikalische Chemic, 1889.
[28] von Kármán, Th. and Penner, S. S.: Fundamental Approach to Laminar Flame
Propagation. Selected Combustion Problems, Vol. I, AGARD, Paris, 1954,
pp. 5-41.

Chapter 2
Transport phenomena in gas
mixtures
2.1 Introduction
When the state or composition of a mixture of gases are not uniform, a transport
of mass, momentum and energy takes place between different points of the mixture
tending to level the initial differences.
At the neighborhood of each point transport depends on the state and composition
of the mixture and on the lack of uniformity from which it arises, according
to laws dealt with in the present chapter. Even though these phenomena arise from
molecular motion their description can be made through phenomenological variables
except when the gas is very rarefied or when the variations in state and composition
within distances comparable with the molecular mean free path are large. 1 If such
cases are excluded, gas can be assimilated to a continuous medium where at each of
its points density, pressure, temperature, mass fractions and velocities of the species
and mixture are definable. Under such conditions, let P , Fig. 2.1, be a point, fixed or
in motion at a given velocity ¯v, and dσ a surface element linked to it. Transport dF
of mass of one of the species, or of momentum or energy of the mixture, through dσ
during time dt is of the form
dF = f dσ dt, (2.1)
where f is the transport per unit surface and per unit time. f is a function of the orientation
of dσ, defined by unit vector ¯n normal to dσ, of the variables that determine
the state and composition of the mixture at the point and of their derivatives. If P
1 See chapter 3 §1.
37

38 CHAPTER 2. TRANSPORT PHENOMENA IN GAS MIXTURES
dσ
v
P
n
Figure 2.1: Schematic diagram to define the transport phenomena.
moves at velocity ¯v of the mixture at the point 2 and mass transport of the species is
considered, the phenomenon is called diffusion. By considering the transport of momentum
one obtains pressure and viscous stresses. Finally when energy transport is
considered heat transfer is obtained.
In the following paragraphs these three phenomena will be studied in sequence
adopting as a starting point the expressions given for them by the Kinetic Theory of
Gases. This theory states the problem as follows: let f i (¯v i , ¯x, t) be the velocity distribution
function of species A i of the mixture. By definition, the probable number
of molecules of species A i whose velocities lie between ¯v i and ¯v i + d¯v i , and which
are contained in volume element dV around point P of spatial coordinate ¯x, at instant
t, is f i (¯v i , ¯x, t) dV dv i1 dv i2 dv i3 , where v i1 , v i2 and v i3 are the three components
of ¯v i in a rectangular cartesian system of coordinates. Due to molecular collisions
a statistical equilibrium establishes which determines the values for f i . Such values
are determined by a system of Boltzmann integral equations. When the state of the
mixture is uniform, the solution to this system is Maxwell’s velocity distribution for
each species. Otherwise, the solution depends not only on the state at the point but
also on its rate of variation when passing from one point to another. This variation is
defined by the successive derivatives of the variables of state and composition at the
point. If such derivatives are small, that is, when relative variations of density, pressure,
temperature, velocities and mass fractions at the point are small within distances
of the order of the molecular mean free path, 3 an approximate solution to Boltzmann’s
system can be obtained by means of the method of the perturbations developed by Enskog
and Chapman. This method looks for solutions to Boltzmann’s system which
differ slightly from Maxwell’s fundamental solution. Solutions show that transport of
mass, momentum and energy depend linearly on the first order derivatives of the variables
of state and composition at the point. Coefficients of such linear functions are
those of diffusion, viscosity and thermal conductivity of the mixture depending only
on the state and composition at the point and on dynamics of molecular collisions. For
a detailed study on the subject see Refs. [1] and [2].
2 See §2 of this chapter for definition of ¯v.
3 Such is normally the case with exceptions as, for example, inside shock waves.

2.2. DIFFUSION 39
Hereinafter, notation in chapter 1 will be kept, adopting the vectorial and tensorial
notation defined in the Appendix to chapter 3.
2.2 Diffusion
Let ¯v i be the velocity of species A i at P . 4 ¯v i is generally different from velocity ¯v of
the mixture. The difference ¯v di is called diffusion velocity of A i
¯v di = ¯v i − ¯v. (2.2)
The following relations exist between ¯v, ¯v i and ¯v di
¯v = ∑ i
Y i¯v i , (2.3)
∑
Y i¯v di = ¯0. (2.4)
i
Flux dm i of A i through dσ during time dt, when P moves at velocity ¯v of the
mixture, is obviously
dm i = ρ i¯v di · ¯n dσ dt. (2.5)
The problem lies in calculating flux vector ¯f i ,
¯f i = ρ i¯v di . (2.6)
Let us first consider the case of a binary mixture formed by species A 1 and A 2 .
The problem has been solved by Enskog [3] and Chapman [4]. The flux of A 1 is
¯f 1 = ρY 1¯v d1 = −ρ M (
1M 2
Mm
2 D 12 ∇X 1 − (Y 1 − X 1 ) ∇p )
∇T
− D T 1
p T . (2.7)
Here D 12 and D T 1 are, respectively, the coefficients of diffusion and thermal diffusion
5 of the mixture. Flux ¯f 2 of A 2 is, from (2.4) and (2.6),
¯f 2 = − ¯f 1 . (2.8)
4¯v i is defined as follows: taking a volume element dV around P and forming the mean value of velocities
¯v i , of the molecules A i on it. Such value is ¯v i
¯v i =
1 X
¯v j i
N ,
i
where N i is the number of molecules of A i on dV . If dV is large respect to the molecular scale and
small respect to the macroscopic scale, the value ¯v i thus defined is independent from size and shape of dV .
Velocity ¯v of the mixture is, by definition
¯v = X i
j
Y i¯v i .
5 The coefficient of thermal diffusion must not be confused with the thermal diffusivity, Ed.

40 CHAPTER 2. TRANSPORT PHENOMENA IN GAS MIXTURES
Formula (2.7) shows that there are three causes for diffusion corresponding to the three
terms in the right hand side of the equation. Namely, the differences in composition,
pressure and temperature. 6
first of these three causes. 7
Elementary Theory of Diffusion only acknowledge the
In Aerothermochemical problems it is advantageous to eliminate mole fraction
X 1 from Eq. (2.7). Furthermore, by doing so a simplified expression is obtained.
Elimination is immediately attained by keeping in mind the following relations 8
Thus obtaining
X 1 = M m
Y 1 ,
M 1
(2.9)
1
= Y 1
+ Y 2
= 1 ( 1
+ − 1 )
Y 1 ,
M m M 1 M 2 M 2 M 1 M 2
(2.10)
( )
M1 − M 2
¯f 1 = −ρD 12 ∇Y 1 + ρ D 12 Y 1 (1 − Y 1 ) ∇p
M m p − D ∇T
T 1
T . (2.11)
The first term in the right hand side of this expression gives the diffusion flux corresponding
to differences in composition. This term express Fick’s law. The second
term gives the diffusion flux arising from differences in pressure. Since D 12 is always
positive, formula (2.11) shows that pressure diffusion tends to carry the heavier
molecules towards the higher pressures. The third term gives the diffusion flux arising
from differences in temperature. Thermal diffusion coefficient D T 1 can be positive,
negative or zero. Therefore, no general rules can be giving regarding the sens of this
flux.
Formula (2.11) shows that diffusion’s behaviour in a binary mixture of gases
is determined by the values of two coefficients: D 12 and D T 1 . Frequently D T 1 is
substituted by
k T =
M 2 m
M 1 M 2
D T 1
ρD 12
. (2.12)
Systematic measurements on diffusion coefficients of gas mixtures are not
available. Therefore theoretical computations become necessary. They can be performed
when knowing the interaction potential between molecules. If forces between
molecules are independent from their relative orientation, the interaction potential
takes the form
ϕ = ε 12 f
( r
σ 12
)
, (2.13)
6 Where selective fields of forces exists, that is, fields acting with different strength over the two species
of the mixture, they originate a new cause of diffusion. See Ref. [1], p. 143. Such occurs for example,
when there exist ionized molecules and the mixture is submitted to the action of an electromagnetic field.
7 See Ref. [5], p. 198.
8 See chapter 1, pp. 12-13.

2.2. DIFFUSION 41
where r is the distance between centers of the molecules, ε 12 is an intensity constant
and σ 12 is a collision diameter. Consequently, once the form of f is known, the values
of ε 12 and σ 12 determine the potential. When forces between molecules depend on
their relative orientation, as occurs whit polar molecules, more complicated expressions
than (2.13) are needed for ϕ. Such expressions must include the influence of
the relative orientation on the collision. An example of such potentials, which has
been used by Hirschfelder et al. for the computation of transport coefficients of polar
molecules [6], is Stockmayer potential
( (σ12 ) 12 ( σ12
) ) 6
ϕ = 4ε 12 − − µ2 12
r r r 3 f (θ 1, θ 2 , φ 2 − φ 1 ) . (2.14)
The first term in the right hand side is Lennard-Jones’s potential and it represents
the part of interaction independent from the relative orientation of molecules.
The second term gives the attraction between two dipoles whose relative orientation
is defined in Fig. 2.2. So far very few results are available on transport coefficients in
mixtures of polar molecules. 9
φ − φ
1 2
θ 1
θ 2
+
Figure 2.2: Schematic diagram of the angles defining the orientation of two polar
molecules.
The values for the coefficients of diffusion and thermal diffusion can be computed
through a method of successive approximations developed by Chapman and
Enskog.
For this, the form of the interaction potential between molecules as well as its
corresponding parameters must be known. Generally, a first approximation is enough
for practical applications. For D 12 first approximation [D 12 ] 1
, gives
√
[D 12 ] 1
= 3
( )
8 √ kT M1 + M 2 1
2π pσ12
2 RT
(2.15)
M 1 M 2 Ω (1,1)∗
12 (T12 ∗ ).
Here T ∗ 12 is a dimensionless temperature defined by the expression
9 See Ref. [2], p. 597.
+
T ∗ 12 = kT
ε 12
, (2.16)

42 CHAPTER 2. TRANSPORT PHENOMENA IN GAS MIXTURES
and Ω (1,1)∗
12 is a function of T ∗ 12 whose form depends on the potential of molecular
interaction. For instance, if the molecules behave as rigid spheres Ω (1,1)∗
12 = 1 for all
values of T ∗ 12.
For numerical computations it is more convenient to express [D 12 ] 1
in the form
[D 12 ] 1
= 0.002628
√
M1 + M 2
2M 1 M 2
T 3
(2.17)
pσ12 2 Ω(1,1)∗ 12 (T12 ∗ ),
where p is measured in atm, T in K, σ 12 in o A and [D 12 ] 1
in cm 2 /s.
The above mentioned theory does not determine the interaction potential that
should be applied in each case. To the contrary a comparison between the theoretical
results corresponding to several potentials and experimental measurements, can
furnish information on such potentials. In Ref. [2], pp. 589 and following, some examples
of such comparisons can be found showing the agreement that can be expected
between theoretical and experimental results. It is verified that for many cases the best
agreement is attained with a Sutherland interaction potential or with that of Lennard-
Jones. Sutherland’s potential is the one that produces between rigid spheres attracting
one another with a force inversely proportional to a given power of its distance r.
Therefore, it has the form
This potential gives for Ω (1,1)∗
12
r < σ 12 : ϕ(r) = ∞,
r > σ 12 : ϕ(r) = −ε 12
( σ12
r
) δ
.
(2.18)
Ω (1,1)∗
12 = 1 + S 12
T , (2.19)
where S 12 = g(δ) ε 12
is a constant whose value depends on the parameters of Eq. (2.18)
k
and generally is determined experimentally. Substituting Eq. (2.19) into (2.15) one
obtains
[D 12 ] 1
= 3
8 √ 2π
k
pσ 2 12
√ ( ) 5
M1 + M 2 T 2
R
M 1 M 2 S 12 + T . (2.20)
If the interaction potential is of Lennard-Jones type, no simple expression can
be given for Ω (1,1)∗
12 , whose values must be calculated by numerical integration. Table
2.1 gives the values calculated by Hirschfelder et al. [7].

2.2. DIFFUSION 43
T ∗ Ω (1,1)∗ Ω (2,2)∗ T ∗ Ω (1,1)∗ Ω (2,2)∗ T ∗ Ω (1,1)∗ Ω (2,2)∗
0.30 2.662 2.785 1.70 1.140 1.248 4.20 0.874 0.9600
0.35 2.476 2.628 1.75 1.128 1.234 4.30 0.8694 0.9553
0.40 2.318 2.492 1.80 1.116 1.221 4.40 0.8652 0.9507
0.45 2.184 2.368 1.85 1.105 1.209 4.50 0.8610 0.9464
0.50 2.066 2.257 1.90 1.094 1.197 4.60 0.8568 0.9422
0.55 1.966 2.156 1.95 1.084 1.186 4.70 0.8530 0.9382
0.60 1.877 2.065 2.00 1.075 1.175 4.80 0.8492 0.9343
0.65 1.798 1.982 2.10 1.057 1.156 4.90 0.8456 0.9305
0.70 1.729 1.908 2.20 1.041 1.138 5.00 0.8422 0.9269
0.75 1.667 1.841 2.30 1.026 1.122 6.00 0.8124 0.8963
0.80 1.612 1.780 2.40 1.012 1.107 7.00 0.7896 0.8727
0.85 1.562 1.725 2.50 0.9996 1.093 8.00 0.7712 0.8538
0.90 1.517 1.675 2.60 0.9878 1.081 9.00 0.7556 0.8379
0.95 1.476 1.629 2.70 0.9770 1.069 10.0 0.7424 0.8242
1.00 1.439 1.587 2.80 0.9672 1.058 20.0 0.664 0.7432
1.05 1.406 1.549 2.90 0.9576 1.048 30.0 0.6232 0.7005
1.10 1.375 1.514 3.00 0.9490 1.039 40.0 0.596 0.6718
1.15 1.346 1.482 3.10 0.9406 1.030 50.0 0.5756 0.6504
1.20 1.320 1.452 3.20 0.9328 1.022 60.0 0.5596 0.6335
1.25 1.296 1.424 3.30 0.9256 1.014 70.0 0.5464 0.6194
1.30 1.273 1.399 3.40 0.9186 1.007 80.0 0.5352 0.6076
1.35 1.253 1.375 3.50 0.9120 0.9999 90.0 0.5256 0.5973
1.40 1.233 1.353 3.60 0.9058 0.9932 100. 0.5170 0.5882
1.45 1.215 1.333 3.70 0.8998 0.9870 200. 0.4644 0.5320
1.50 1.198 1.314 3.80 0.8942 0.9811 300. 0.436 0.5016
1.55 1.182 1.296 3.90 0.8888 0.9755 400. 0.417 0.4811
1.60 1.167 1.279 4.00 0.8836 0.9700
1.65 1.153 1.264 4.10 0.8788 0.9649
Table 2.1: Values of Ω (1,1)∗ and Ω (2,2)∗ as a function of T ∗ for the Lennard-Jones potential.
When comparing theoretical values of the transport coefficients with those experimentally
obtained, it is seen that in many cases Lennard-Jones potential gives the
best agreement for a more extensive range of temperatures. 10
The preceding formulae require knowing the values for ε 12 and σ 12 , which
must be experimentally obtained. Generally, such values are known for potentials
between equal molecules but not for those of different kind. The approximate values of
the constants for the potentials between molecules of species A 1 and A 2 are obtainable
as follows.
10 See Ref. [2], p. 597.

44 CHAPTER 2. TRANSPORT PHENOMENA IN GAS MIXTURES
Let (ε 1 , σ 1 ) and (ε 2 , σ 2 ) be the values of the constants for species A 1 and A 2 .
The values of the constants for mixed potentials between both species are 11
ε 12 = √ ε 1 ε 2 ,
σ 12 = 1 2 (σ 1 + σ 2 ) .
(2.21)
Table 1.1 in chapter 1 gives the values of ε and σ for certain number of gases. If such
values where not available any of the following formulas can be used for approximate
values [7]:
ε/k = 0.77T c , σ = 0.710(V c /N) 1 3 ,
ε/k = 1.15T b , σ = 0.984(V b /N) 1 3 ,
(2.22)
ε/k = 1.92T m , σ = 1.031(V m /N) 1 3 ,
ε/k = 0.292T B , σ = 1.030(V z /N) 1 3 .
Here V is the molar volume. Subscripts c, m, b, B and z denote the critical, melting,
boiling, Boyle’s 12 and absolute zero points.
Table 2.2, taken from Ref. [2], gives the diffusion coefficients for some binary
mixtures. Theoretical values correspond to a Lennard-Jones interaction potential.
Constants for the potential have been obtained through Eq. (2.21). It is seen that
agreement between theoretical and experimental values is, generally, excellent.
Except for rigid spheres where Ω (1,1)∗
12 is constant, this function decreases when
T is increased. Hence, according to formula (2.15) [D 12 ] 1
increases with T slightly
faster than T 3/2 . Fig. 2.3 represents as an example the coefficient [D 12 ] 1
versus T in
a mixture of N 2 and CO 2 for rigid spheres and for an interaction potential of Lennard-
Jones. In combustion problems, where temperature can change in a ratio of ten to one
from one point to another, the influence of temperature on the values of the transport
coefficients is very important. Formula (2.15) shows, as well, that diffusion coefficient
is inversely proportional to the mixture’s pressure and is independent from the mass
fractions of the species in first approximation. In order to obtain the influence of mass
fractions one must resort to higher approximations.
11 See Ref. [2], pp. 589 and following
12 Boyle’s temperature T B is the temperature at which curve pV vs. p has a horizontal tangent for p = 0.

2.2. DIFFUSION 45
Gas Pair σ 12 ( A) o ε 12 /k (K) T (K) [D 12 ] 1
(cm 2 s −1 )
Calculated Experimental
N 2 -H 2 3.325 55.2
273.2
288.2
293.2
0.656
0.718
0.739
0.674
0.743
0.760
N 2-O 2 3.557 102
273.2 0.175 0.181
293.2 0.199 0.220
N 2 -CO 3.636 100 273.2 0.174 0.192
N 2-CO 2 3.839 132
273.2
288.2
293.2
298.2
0.130
0.143
0.147
0.152
0.144
0.158
0.160
0.165
N 2 -C 2 H 4 3.957 137 298.2 0.156 0.163
N 2 -C 2 H 6 4.050 145 298.2 0.144 0.148
N 2-nC 4H 10 4.339 194 298.2 0.0986 0.0960
N 2-iso-C 4H 10 4.511 169 298.2 0.0970 0.0908
H 2 -O 2 3.201 61.4 273.2 0.689 0.697
H 2 -CO 3.279 60.6 273.2 0.661 0.651
273.2 0.544 0.550
H 2-CO 2 3.482 79.5
288.2 0.597 0.619
293.2 0.616 0.600
298.2 0.634 0.646
H 2 -N 2 O 3.424 85.6 273.2 0.552 0.535
H 2 -SF 6 3.922 89.1 298.2 0.473 0.420
H 2-CH 4 3.425 67.4
273.2 0.607 0.625
298.2 0.705 0.726
H 2 -C 2 H 4 3.600 82.6 298.2 0.595 0.602
H 2 -C 2 H 6 3.693 87.5 298.2 0.556 0.537
CO-O 2 3.512 112 273.2 0.175 0.185
CO 2 -O 2 3.715 147
273.2 0.128 0.139
293.2 0.146 0.160
CO 2 -CO 3.793 145 273.2 0.128 0.137
CO 2 -N 2 O 3.938 204 273.2 0.092 0.096
CO 2-CH 4 3.939 161 273.2 0.138 0.153
Table 2.2: Comparison of diffusion coefficients for some binary mixtures. Theoretical values
correspond to Lennard-Jones interaction potential.
Approximation of order j can be expressed in the form 13
[D 12 ] j
= [D 12 ] 1
f j D 12
. (2.23)
For example, for the second approximation, fD 2 12
depends on the molar masses
of the species, on their mass fractions, viscosity coefficients and temperature of the
mixture. Its value is always close to one. For the Lennard-Jones potential, for instance,
it ranges from 1 to 1.03.
13 See Ref. [2], p. 539.

46 CHAPTER 2. TRANSPORT PHENOMENA IN GAS MIXTURES
4.0
3.5
3.0
2.5
[D 12
] 1
2.0
1.5
1.0
Rigid spheres
Lennard−Jones potential
0.5
0.0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
T(K)
Figure 2.3: Coefficient [D 12] 1
as a function of temperature in a mixture of N 2 and CO 2, for
rigid spheres and for an interaction potential of Lennard-Jones.
Hence, the influence of mass fractions on the value of D 12 is always small and
can, generally, be neglected by adopting for it the value given by the first approximation.
Frequently, the value given by elementary diffusion theory is approximate
enough. According to it,
D 12 ∼ T 3/2
p . (2.24)
Or else, an empirical formula of the form
D 12 ∼ T n
p , (2.25)
where n is an exponent ranging from 1.5 to 2 whose value is determined empirically.
With respect to thermal diffusion ratio k T , even in the first approximation it is
a complicated function of the molar masses, mass fractions of the species and of the
temperature of the mixture. It is also very much influenced by the laws of molecular
interaction. The expression of the first approximation of k T can be obtained from
Ref. [2], p. 541, where tables for the calculation of their values are also included as
well as a comparison between theoretical and experimental values for many binary
mixtures. It appears that the agreement is not so good for k T as for the other transport
coefficients. When comparing the successive approximations to the value of k T , it is
also verified that in first approximation the error is larger here than for other transport

2.2. DIFFUSION 47
coefficients. 14 The values of k T are generally small. Normally they do not exceed 0.1.
Thereby thermal diffusion is of secondary importance in Aerothermochemistry and its
influence is, usually, disregarded.
In Refs. [8], [9], [10] and [11] recent information on theoretical and experimental
values of transport coefficients can be found.
For mixtures containing more than two components, the problem has been
solved by Curtiss and Hirschfelder [2], [12]. In such case, the aforementioned three
causes of diffusion, namely, differences in composition, pressure and temperature, still
hold. Yet in this case diffusion flux of each species depends not only on the gradient
of its molar fraction but is a linear function of the gradients of all molar fractions.
Diffusion flux ¯f i of species A i (i = 1, 2, . . . , l) is
¯f i = ρY i¯v di = ρ M i
M 2 m
∑
M j D ij
′
j≠i
(
∇X j + (X j − Y j ) ∇p
p
)
− D T i
∇T
T . (2.26)
Here D ′ ij is the diffusion coefficient of species A i and A j in the mixture. For a binary
mixture, D ′ ij is diffusion coefficient D ij previously defined. For the case of more than
two components it differs. D T i is the thermal diffusion coefficient of species A i in
the mixture.
The l equations (2.26) are not independent from each other. In fact, diffusion
fluxes of l species must satisfy the condition
∑
¯f i = ¯0. (2.27)
i
Diffusion coefficients D ij ′ in system (2.26) can be expressed as a function of
binary diffusion coefficients D ij of the species taken in couples. Ref. [2], pp. 541
and 543, and Ref. [12] contain expressions for these coefficients as well as for those
of thermal diffusion. Expressions for D ′ ij and D T i are too complicated for practical
computations. Then it is advantageous to eliminate D ij ′ by substituting system (2.26)
with the following that makes explicit the influence of binary diffusion coefficients 15
∑ X i X j
(¯v dj − ¯v di ) =∇X i + (X i − Y i ) ∇p
D ij
p
j≠i
− ∑ (
X i X j DT j
− D ) (2.28)
T i ∇T
ρD ij Y j Y i T . j≠i
As was done for the case of binary diffusion, it is also convenient to eliminate
molar fractions of the species by introducing their corresponding mass fractions.
14 A critical study on the subject will be found in Ref. [6], p. 492 and following, as well as bibliography.
15 See Ref. [2]. p. 517.

48 CHAPTER 2. TRANSPORT PHENOMENA IN GAS MIXTURES
Making use of relations (1.33) and (1.36) in chapter 1, it is obtained
∑
Y j
D ij
j≠i
M j
[
¯vdi − ¯v dj
+ ∇Y i
− 1 ρ
− ∇Y j
Y j
Y i
(
DT j
Y j
+ M j − M i ∇p
M m p
]
= ¯0, (i = 1, 2, . . . , l).
− D T i
Y i
) ∇T
T
(2.29)
Diffusion of a species whose mass fraction is very small is particularly important
in certain problems. For example, diffusion of active particles which propagate
chemical reactions through a combustion wave. Then it becomes possible to give an
explicit approximate expression for the diffusion flux of such species. For diffusion
arising from differences in composition this expression is
(
∑ Y j ∇Yj
− ∇Y )
i
j≠iM j Y j Y i
¯f i = ρY i¯v di = ρY i
∑ Y j
. (2.30)
M j D ij
j≠i
System (2.29) is, generally, too complicated to be used in the solution of many
of the Aerothermochemistry problems. Therefore, in some of its applications, diffusion
fluxes of the species are substituted by approximate expressions under the assumption
that diffusion flux due to differences in composition ¯f iY of each species is
proportional to its gradient of molar fraction. That is, by adopting for such flux an
expression of the form
¯f iY = −ρ M i
M m
D im ∇X i . (2.31)
Here D im is a binary diffusion coefficient of the species through the mixture formed
by the rest. This expression is exact only when binary diffusion coefficients and molar
masses of the species are all equal. If mean molar mass of the mixture is also constant,
one has
¯f iY = −ρD im ∇Y i , (2.32)
by virtue of Eqs. (1.33) and (1.36) in chapter 1. This expressions will be used in
chapter 13. In aerothermochemical problems diffusion fluxes arising from differences
in pressure and temperature are, in general, negligible. C. R. Wilke [14] has given an
explicit approximate expression, empirically deduced, similar to (2.31) for fluxes due
to differences in composition.

2.3. VISCOSITIES 49
2.3 Viscosities
Mechanics of continuous media teaches that the state of stresses at each point of a
medium is determined by a symmetric tensor of second order
⎛
⎞
p 11 p 12 p 13
⎜
⎟
τ e ≡ ⎝ p 21 p 22 p 23 ⎠ , (2.33)
p 31 p 32 p 33
of components p ij = p ji in a rectangular cartesian system of reference. 16 The physical
meaning of this tensor is the following: p ij is the component parallel to axis x j of the
stress acting upon a surface element normal to axis x i , see Fig. 2.4.
x
p
12
p
13
p
11
3
p
23
p
p 22
21
p 1
33
p31
x 2 p 32
Figure 2.4: Schematic diagram showing the components of the stress tensor τ e.
x
Stress ¯f acting upon a surface element dσ, see Fig. 2.5, whose orientation is
defined by the unit vector ¯n, is given by the expression 17
¯f = ¯n · τ e , (2.34)
whose components are
f i = ∑ j
n j p ji , (i = 1, 2, 3), (2.35)
where n j are the components of ¯n.
Kinetic Theory of gases shows 18 that in a dilute gas the stress originates from
transfer of momentum of the gas through a surface element dσ moving at velocity ¯v of
16 See, i.e., Ref. [15].
17 See appendix to chapter 3 for notation.
18 See Ref. [1], p. 31.

50 CHAPTER 2. TRANSPORT PHENOMENA IN GAS MIXTURES
d σ
x
3
P
f
n
f
3
f 1
x
x 2
1
f 2
Figure 2.5: Schematic diagram showing the components of the stress ¯f = ¯n · τ e.
the gas. 19 Such transfer is produced by the thermal agitation of the molecules. 20 If the
state of gas is uniform, the velocity distribution of thermal agitation is a Maxwell’s
distribution. In such case transfer of momentum is normal to dσ and independent
from the orientation ¯n. The corresponding state of stresses is a pure compression,
defined by a scalar: the pressure p of the gas. Tensor of Eq. (2.33) reduces as follows
τ e = −pU, (2.36)
where U is the unit tensor whose components are Kronecker’s δ ij
U ≡
⎛
⎜
⎝
1 0 0
0 1 0
0 0 1
Stress ¯f acting upon an element dσ in Fig. 2.5, is
⎞
⎟
⎠ . (2.37)
¯f = −p¯n. (2.38)
When the state of motion of the gas is not uniform other stresses produce, which add
to those of pressure, Eq. (2.38). Such stresses are given by viscous stress tensor
⎛
⎞
τ 11 τ 12 τ 13
⎜
⎟
τ ev ≡ ⎝ τ 21 τ 22 τ 23 ⎠ . (2.39)
τ 31 τ 32 τ 33
The Chapman-Enskog method described in the introduction to the present chapter
shows that components τ ij of the viscous stress tensor have the form 21
( ) 2
τ ij = 2µγ ij −
3 µ − µ′ (∇ · ¯v) δ ij . (2.40)
19 Within dense gases stresses are partially due to transfer of momentum and partially to the forces of
molecular interaction whose effect is not negligible.
20 If ¯v is the gas velocity at a point and ¯v j the velocity of a molecule at the neighborhood of such point,
velocity ¯v a of thermal agitation is, by definition, ¯v a = ¯v j − ¯v.
21 Actually Enskog and Chapman’s theory was developed for monotonic dilute gases, where µ ′ = 0.

2.3. VISCOSITIES 51
Here µ and µ ′ are called viscosity coefficient and bulk viscosity coefficient respectively,
and
3
8π √ 2 γ ij = 1 ( ∂vi
+ ∂v )
j
2 ∂x j ∂x i
are the components of deformation velocity tensor
τ vd ≡
⎛
⎜
⎝
(2.41)
⎞
γ 11 γ 12 γ 13
⎟
γ 21 γ 22 γ 23 ⎠ , (2.42)
γ 31 γ 32 γ 33
which determines the deformation suffered by a gas element as it moves. 22
The method of Chapman and Enskog allows calculation of the viscosity coefficient
of a gas by means of successive approximations. Expressions of the form
µ j = [µ] 1
f j µ (2.43)
are obtained, where [µ] 1
is the first approximation, j is the computed order of the approximation
and f j µ is a function which differs slightly from unity. When comparing,
for example, the first and third approximations for the Lennard-Jones potential it is
verified that error in the first is smaller than 0.8%. Thereby, as for diffusion coefficients,
the first approximation is usually enough.
In the case of a pure gas this approximation is given by 23
[µ] 1
= 5
16 √ N −1√ MRT
π σ 2 Ω (2,2)∗ (T ∗ ) . (2.44)
Here, as for diffusion, Ω (2,2)∗ is a function of reduced temperature T ∗ = kT/ε, whose
law of variation depends on the form of the molecular interaction potential. For example,
for rigid spheres Ω (2,2)∗ = 1.
For a Sutherland potential, Eq. (2.18), is
Ω (2,2)∗ = 1 + S µ
T , (2.45)
where S µ = g µ (δ) ε . In practice, this value is determined experimentally. When
k
Eq. (2.45) is substituted into Eq. (2.44) one obtains
22 See Ref. [15].
23 See Ref. [2], p. 527.
[µ] 1
= 5
16 √ π
N −1√ MR
σ 2 T 3 2
S µ + T . (2.46)

52 CHAPTER 2. TRANSPORT PHENOMENA IN GAS MIXTURES
Values Ω (2,2)∗ as function of T ∗ for the Lennard-Jones potential have been
taken into Table 2.1. It is seen that for many gases this potential represents experimental
results better than others. 24
For numerical computations it is more convenient to express Eq. (2.44) as
√
MT
10 7 [µ] 1
= 266.93
σ 2 Ω (2,2)∗ (T ∗ ) , (2.47)
where all quantities are measured in the same units than in Eq. (2.17) and [µ] 1
is
expressed in gr cm −1 s −1 .
The preceding formulae show that the viscosity coefficient of a gas is independent
from pressure and increases with temperature slightly over √ T . Fig. 2.6 gives
the law of variation with temperature of viscosity coefficients for several gases. For
additional information, see Refs. [2], [8], [9], [11] and [16], where complementary
bibliography is listed.
7000
6000
O 2
N 2
5000
10 7 µ (g cm −1 s −1 )
4000
3000
2000
CO 2
CH 4
H 2
1000
0
0 200 400 600 800 1000 1200 1400 1600
T(K)
Figure 2.6: Viscosity coefficient as function of temperature, for several gases.
Often, for practical applications an empirical formula of the form
µ ∼ T n (2.48)
is used, where n is an exponent ranging from 0.5 to 1.
With respect to bulk viscosity coefficient µ ′ it is zero for dilute monatomic
gases. For polyatomic and dense gases µ ′ is generally different from zero, yet small.
24 See Ref. [2], pp. 589 and following.

2.3. VISCOSITIES 53
Its value depends on the easiness of energy transfer between external and internal
degrees of freedom of the molecules during collisions. Such facility is characterized
by a relaxation time for each internal degree of freedom. The value for µ ′ is expressed
as a function of these relaxation times. Generally, it is sufficiently approximate to
assume µ ′ = 0. Additional information on this problem can be found in Ref. [2],
pp. 501 and 710.
Viscosity coefficient of a mixture of gases is a complicated function of molar
masses, mass fractions and viscosity coefficients of the species as well as of the interaction
potentials between different species. 25 For binary mixture, for example, the
first approximation [µ] 1
is given by
where
1
[µ] 1
= X µ + Y µ
1 + Z µ
, (2.49)
X µ = X2 1
+ 2X 1X 2
+ X2 2
,
µ 1 µ 12 µ
[
2
]
Y µ = 3 X 2
5 A∗ 1 M 1
12 + 2X 1X 2 (M 1 + M 2 ) 2 µ 2 12
+ X2 2 M 2
,
µ 1 M 2 µ 12 4M 1 M 2 µ 1 µ 2 µ 2 M 1
[ X
2
1 M 1
(2.50)
Z µ = 3 5 A∗ 12
M 2
(
(M 1 + M 2 ) 2 (
µ12
+ 2X 1 X 2 + µ ) )
12
− 1
4M 1 M 2 µ 1 µ 2
]
+ X2 2 M 2
.
M 1
In these expressions, µ 1 and µ 2 are the first approximations of the viscosity coefficients
for both species. µ 12 is given by
( )
µ 12 = 5
N −1 M1 M 2
√2RT
16 √ M 1 + M 2
π σ12 2 Ω(2,2)∗ 12 (T12 ∗ ) , (2.51)
ε 12 and σ 12 are the constants of the interaction potential between molecules of both
species. A ∗ is a function of reduced temperature T ∗ 12 = kT
ε 12
, whose value is given in
Table 2.3 for the Lennard-Jones potential.
25 See Ref. [2], p. 529.
T12 ∗ 0.3 0.5 0.75 1 1.25 1.5 2
A ∗ 1.046 1.093 1.105 1.103 1.099 1.097 1.094
T12 ∗ 3 4 5 10 50 100 400
A ∗ 1.095 1.098 1.101 1.11 1.13 1.138 1.154
Table 2.3: The coefficient A ∗ as a function of T ∗ 12.

54 CHAPTER 2. TRANSPORT PHENOMENA IN GAS MIXTURES
The following Table 2.4 gives, as an example, the values of the theoretical and
experimental viscosity coefficients for a mixture of oxygen and carbon monoxide.
%O
T (K) 0 42.01 77.33
300.06
Experimental 1 776 1 900 1 998
Calculated 1 771 1 896 2 003
500.06
Experimental 2 548 2 741 2 908
Calculated 2 539 2 717 2 871
Table 2.4: Viscosity coefficient (10 7 µ gr cm −1 s −1 ) for a mixture of O 2 -CO.
For mixtures of more than two components Buddemberg and Wilke give the
following semi-empirical expression which represents experimental results with good
approximation
[µ] 1
= ∑ i
X 2 1
Xi
2 + 1.385 ∑ X i X j
[µ i ] 1 j≠i
. (2.52)
RT
pM i [D ij ] 1
This expression allows calculation of the viscosity coefficient of a mixture when those
corresponding to the species are known as well as the diffusion coefficients of the
same when taken by couples.
In the preceding formulae substitution of molar fractions by mass fractions is
straightforward by making use of relation
X i = M m
M i
Y i . (2.53)
Fig. 2.7 gives the law of variation as a function of the composition of the viscosity
coefficients for some gas mixtures.
2.4 Thermal conductivity
Theory of heat conduction shows 26 that heat transfer at a point of a material medium
is defined by a vector ¯q of heat flux
¯q = q 1 ī 1 + q 2 ī 2 + q 3 ī 3 . (2.54)
Physical meaning of ¯q is the following: q i is heat flux per unit surface and per unit
time through a surface element normal to axis x i .
26 See, i.e., Ref. [16], pp. 3 and following.

2.4. THERMAL CONDUCTIVITY 55
2000
1800
10 7 µ (gr cm −1 s −1 )
1600
1400
1200
H 2
− CO
H 2
− N 2
H 2
− CO 2
H 2
− N 2
O
H 2
− O 2
1000
800
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
X
H2
Figure 2.7: Viscosity for some gas mixture as function of composition.
Considering a surface element dσ, see Fig. 2.5, not parallel to any of the coordinate
planes and whose orientation is defined by unit vector ¯n, it is easily demonstrated
that heat flux Q through dσ per unit surface and per unit time is
Q = ¯q · ¯n = ∑ i
q i n i . (2.55)
Therefore, once the flux vector at a point is known, this determines heat transfer
through any surface element passing through it.
Kinetic Theory of Gases shows that within a dilute gas heat flux originates only
from molecular energy transfer when dσ moves at a velocity ¯v of the gas at the point.
Such transfer originates from the motion of molecular agitation and from diffusion
between species forming the gas if it is a mixture. When such transfer is calculated
one obtain the expression 27
¯q = −λ∇T + ρ ∑ i
Y i h i¯v di + M m RT ∑ i
∑
j
Y j D T i
M i M j D ij
(¯v di − ¯v dj ) , (2.56)
where λ is the thermal conductivity of the gas and h i is the total specific enthalpy
of species A i . Since diffusion fluxes of the species are produced by differences in
composition, pressure and temperature, it results according to (2.56) that this three
causes also produce heat transfer. In general, contributions from the third term in the
27 See Ref. [2], p. 498.

56 CHAPTER 2. TRANSPORT PHENOMENA IN GAS MIXTURES
right hand side of (2.56) is small compared to the other two and can be neglected.
Then ¯q reduces to
¯q = −λ∇T + ρ ∑ Y i h i¯v di . (2.57)
i
In a pure gas ¯q reduces to
¯q = −λ∇T. (2.58)
If, moreover, the gas is monatomic, thermal conductivity coefficient can be
computed through successive approximations as done for diffusion and viscosity. The
first approximation [λ] 1
is
√
T/M
10 7 [λ] = 1989.1
σ 2 Ω (2,2)∗ (T ∗ ) . (2.59)
Here [λ] is measured in cal cm −1 s −1 K −1 and the other magnitudes in the same units
as in Eq. (2.17).
When comparing (2.59) and (2.47) it is seen that for monatomic gases thermal
conductivity and viscosity coefficients are proportional. Similarly to what happens for
other transport coefficients the first approximation is, generally, sufficient for practical
applications.
Expression (2.59) does not take into account for polyatomic molecules energy
transfer between internal and external degrees of freedom during collisions. Such
transfer is not important in the study of viscosity and diffusion but is fundamental in
thermal conductivity. 28 In polyatomic molecules [λ] must be substituted by the following
approximate expressions, due to Eucken, which takes into account the influence
of such energy transfer
[λ] = 15
4
(
R 4
M [µ] 1
15
C v
R + 3 )
, (2.60)
5
where C v is molar heat of the gas at constant volume. Table 2.5 compares experimental
values with those given by Eucken’s formula (2.60) for several gases. 29
H 2 O 2 CO 2 CH 4 N 2
Calculated 4 140 615 386 741 619
Experimental 4 227 635 398 819 619
Table 2.5: Values of the thermal conductivity coefficient (10 7 [λ] cal cm −1 s −1 K −1 ) for
several gases at T=300 K .
It is seen that agreement is fair even if not as good as for other transport coefficients.
28 See Ref. [2], p. 489.
29 See Ref. [2], p. 574.

2.4. THERMAL CONDUCTIVITY 57
4000
3500
10 7 λ (cal cm −1 s −1 K −1 )
3000
2500
2000
1500
1000
500
H 2
− O 2
H 2
− N 2
H 2
− CO
H 2
− N 2
O
H 2
− CO 2
0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
X
H2
Figure 2.8: Thermal conductivity for some gas mixture as function of composition.
In Ref. [2], pp. 526 and following, theoretical formulae are given for the calculation
of thermal conductivity coefficients in mixtures of monatomic gases as well
as empirical formulae for polyatomic gases. Here agreement between theoretical and
experimental results is also not so good as for other transport coefficients. Fig. 2.8
gives, as an example, the influence of the composition on the value of λ for certain
binary mixtures at temperature 273.16 K.
Finally, it should be noted that Aerothermochemistry development is not yet
sufficiently advanced so that transport coefficients are needed to be exact in most of its
applications. Frequently a rudimentary approximation is enough, usually empirical,
where the influence of composition is neglected and that of temperature is approximately
taken into account. These approximations appear justified considering that
other features of the problems must be accounted for in a very rudimentary way, such
as frequently are chemical kinetics, and, in some cases, velocity field and shape of the
flame, etc.
References
[1] Chapman, S. and Cowling, T. G.: The Mathematical Theory on Non-Uniform
Gases. Cambridge University Press, 1939.
[2] Hirschfelder, J. O., Curtiss, C. F., and Bird, R.B.: Molecular Theory of Gases
and Liquids. John Wiley and Sons, Inc., New York, 1954.

58 CHAPTER 2. TRANSPORT PHENOMENA IN GAS MIXTURES
[3] Enskog, D.: Kinetische Theorie der Vorgänge in massig verdünnten Gasen. Dissertation,
Upsala, 1917.
[4] Chapman, S.: On the Kinetic Theory of a Gas, Part II, A Composite Monatomic
Gas: Diffusion, Viscosity and Thermal Conduction. Philosophical Transactions
of the Royal Society of London, A-217, 1917, p. 115.
[5] Jeans, J.: An Introduction to the Kinetic Theory of Gases. Cambridge University
Press, 1948.
[6] Jost, W.: Diffusion in Solids, Liquids and Gases. Academic Press, Inc., Publishers,
New York, 1952.
[7] Bird, R. B., Hirschfelder, J. O. and Curtiss, C. F.: The Theoretical Calculation
of the Equation of State and Transport Properties of Gases and Liquids. The
American Society of Mechanical Engineers, Heat Transfer Division, Annual
Meeting, New York. 29 Nov. to 4 Dec., 1953.
[8] Johnson, E. F.: Molecular Transport Properties of Fluids. Industrial and Engineering
Chemistry, Vol. 45, 1953, pp. 902-6.
[9] Baron, T. and Oppenheim, A. K.: Fluid Dynamics. Industrial and Engineering
Chemistry, Vol. 45, 1953, pp. 941-51.
[10] Pigford, R. L.: Mass Transfer. Industrial and Engineering Chemistry, Vol. 45,
1953, pp. 951-56.
[11] Hirschfelder, J. O., Bird, R. B. and Sportz, E. L.: Viscosity and Other Physical
Properties of Gases and Gas Mixtures. ASME, 1949, pp. 921-37.
[12] Hirschfelder, J. O., Bird, R. B. and Sportz, E. L.: The Transport Properties
for non Polar Gases. 1.- Journal of Chemical Physics, 1948, pp. 968-81.
2.- Chemical Review, 1949, pp. 205-31.
[13] Curtiss, C. F. and Hirschfelder, J. O.: Transport Properties of Multicomponent
Gas Mixtures. The Journal of Chemical Physics, Vol. 17, June 1949, pp. 550-
55.
[14] Fairbanks, D. F. and Wilke, C. R.: Diffusion Coefficients in Multicomponent
Gas Mixtures. Industrial and Engineering Chemistry, March 1950, pp. 471-75.
[15] Sommerfeld, A.: Mechanics of Deformable Bodies. Academic Press Inc., Publishers,
1950.
[16] Carslaw, H. S. and Jaeger, J. C.: Conduction of Heat in Solids. Oxford at the
Clarendon Press, 1950.

Chapter 3
General equations
3.1 Introduction
In the present chapter we shall establish the general equations of Aerothermochemistry,
that is, the general equations of transformation of a mixture of gases that react
one to another and are in motion. It is assumed herein that the gas can assimilate to
a continuous medium as previously done in the preceding chapter when analyzing the
transport coefficients. This assumption is well justified except in the following cases:
a) When a characteristic length of the macroscopic scale of the process, for example
the size of an obstacle submerged in the gas, is of the order of magnitude of the
molecular mean free path.
b) In the case where the phenomenological variables suffer considerable variations
within a distance of the order of magnitude of the said mean free path.
The first case occurs when dealing with a very rarefied gas (Knudsen gas). The
second case occurs, for example, in the analysis of the structure of a shock wave. In
fact, the thickness of a sock wave is only a few times larger than the molecular free
path, except when the wave is very weak. In such cases, to speak of the macroscopic
functions of a point has no sense, and it is necessary to take into consideration the
molecular structure of the gas. Such cases are excluded from the present study.
The equations that govern the transformations of a mixture of reacting gases are
similar to the general Gas Dynamics equations. Their difference is due to the fact that
while Gas Dynamics deal with gases of homogeneous chemical composition invariable
with time, in Aerothermochemistry the composition of the mixture varies. Such
variation is due to chemical reactions taking place between the various species that
form the mixture and to their mutual diffusion, and has the following consequences:
59

60 CHAPTER 3. GENERAL EQUATIONS
1) The need to establish a separate balance for each one of the species by means of
partial continuity equations for each different species. In each of these equations
the effects of the chemical reactions and of diffusion must be included.
2) The modifications due to diffusion of some of the transport coefficients of the
mixture. As previously seen in chapter 2, diffusion does not change the values
of the viscosity coefficients but modifies the heat flux, mainly due to the transport
of the enthalpy of the various species through diffusion.
3) The presence of a new form of energy: the chemical energy released or absorbed
in the chemical reactions.
n
q
f
Σ
V
F
Figure 3.1: Schematic diagram of the actions upon a isolated gas element.
In Gas Dynamics the equations are established by applying the laws of the conservation
of mass, momentum and energy to an ideally isolated gas element following
its motion. 1
The material element thus isolated (Fig. 3.1) is bounded by a fluid surface,
that is to say by a surface Σ, in which the velocity at each point is the velocity
¯v of the gas particle at the same point. This element evolves in accordance with the
aforementioned laws of conservation and under the action of the surrounding mass,
which acts upon its boundary Σ. This action result in the surface forces and in the
heat transport ¯q throughout each of its surface elements. Furthermore, the action of
the possible force fields acting upon the element must be included; for example, the
action ¯F of the gravity field. In this manner 2 we obtain:
a) The continuity equation, which expresses the laws of the conservation of mass.
b) The equation of motion which expresses that Newton’s Second Law of Mechanics
is satisfied.
c) The energy equation which expresses that the First Law of Thermodynamics is
satisfied.
1 See, i.e., Ref. [1], pp. 337 and following.
2 The same equations could be established by starting from the molecular structure of the gas and applying
the laws of the Kinetic Theory of gases. See, i.e., Ref. [2], p. 25.

3.2. EQUATION OF CONTINUITY 61
The system of equations thus obtained must be completed with the state equation
of the gas and with the law of variation of the viscosity coefficient and of the
remaining thermodynamic functions that deal with the problem (internal energy or
enthalpy, etc.) as a function of the gas state variables.
In order to establish the general equations of Aerothermochemistry, a similar
procedure can be followed, taking into account the three observations previously
stated. 3 In such a case it is to be understood that when speaking of the material element
we refer to an element bounded by a surface which moves with a velocity that at
each point concurs with the mixture velocity. This specification is due to the need of
distinguishing between the velocity of the mixture and the velocities of the different
species in cases where diffusion exists. These velocities are not equal (see chapter 2).
In the following, the equations thus obtained are given under the same notation used
in the preceding chapters.
Vectorial and tensorial notation is preferably used as it enables the concise
writing of the equations and a simple interpretation of their terms. A summary of the
symbols and formulae can be found in the Appendix.
3.2 Equation of continuity
Consider species A i , for which the partial density at a point is ρ i . The variations of
ρ i , following the motion of an element of the mixture, is Dρ i
, where the operator
Dt
D (·)
Dt
= ∂ (·)
∂t
is the substantial derivative of the motion of the mixture.
The variation of ρ i is due:
+ ¯v · ∇ (·) (3.1)
a) To the expansion ∇ · ¯v of the volume element, which decreases ρ i in ρ i ∇ · ¯v.
b) To the flux by diffusion of species A i , through the boundary of the element. If
¯v di is the diffusion velocity of species A i diffusion reduces ρ i in ∇ · (ρ i¯v di ).
c) To the mass of species A i produced by chemical reactions. If w i is the reaction
rate of species A i 4 under the conditions of state and composition that prevail at
the point, then the mass produced per unit volume and per unit time is w i .
3 The general equations for Aerothermochemistry may also be deduced from the molecular structure of
the mixture by applying the laws of Kinetic Theory.
4 See chapter 1.

62 CHAPTER 3. GENERAL EQUATIONS
Therefore, the continuity equation for species A i is
Dρ i
Dt = −ρ i∇ · ¯v − ∇ · (ρ i¯v di ) + w i . (3.2)
If there are l different species A i , there is an equation similar to Eq. (3.2) for
each of them. 5
Adding the l equations corresponding to the l different species and making use
of the obvious relation
and of
∑
ρ i = ρ (3.3)
i
∑
w i = 0, (3.4)
i
which expresses the condition that the chemical reactions do not change the mass, and
of
∑
ρ i¯v di = ¯0, (3.5)
which was deduced in chapter 2, the following expression is obtained
i
Dρ
+ ρ∇ · ¯v = 0, (3.6)
Dt
which is the continuity equation for the mixture and is identical to the continuity
equation of Gas Dynamics.
Equation (3.6) allows simplification in the form of (3.2). For this purpose it is
convenient to express Eq. (3.2) as a function of the mass fraction Y i = ρ i /ρ of species
5 The detailed deduction of Eq. (3.2) is as follows.
The mass m i of species A i contained in volume V is
ZZZ
m i = ρ i dV.
V
Its time variation is (see Appendix)
dm i
= d ZZZ ZZZ
ρ i dV =
dt dt V
V
ZZZ » –
∂ρ i
∂t
ZZΣ
dV + ∂ρi
ρ i (¯v · ¯n) dσ =
V ∂t + ∇ · (ρ i¯v) dV.
The mass diffusing from V through Σ per unit time, is
Z
ZZZ
ρ i¯v di · ¯n dσ = ∇ · (ρ i¯v di ) dV.
Σ
V
The mass production in V by the chemical reaction, per unit time, is
ZZZ
w i dV.
V
Then when establishing the balance and expressing that this condition must be satisfied for all V , we obtain
Eq. (3.2).
In all the other formulae that follow the argumentation is identical, and for this reason the expansion of
the complete calculation is not considered necessary.

3.2. EQUATION OF CONTINUITY 63
A i . By doing so and taking into account Eq. (3.6) the following simplified expression
is obtained, which replaces Eq. (3.2)
ρ DY i
Dt + ∇ · (ρY i¯v di ) = w i , (i = 1, 2, . . . , l). (3.7)
Aside from the density and velocity ¯v of the mixture, the latter implicitly contained
in the substantial derivative, this system shows the mass fractions Y i of the
different species as well as their corresponding diffusion velocities ¯v di and reaction
rates w i .
As seen in chapter 2, diffusion velocities are linear functions of the gradients
of Y i , p and T . Therein it was shown that the diffusion velocities ¯v ′ di (i = 1, 2, . . . , l)
corresponding to the gradient of Y i and p are given by the system
∑
(
Y i ¯v ′ di − ¯v ′ dj
+ ∇Y i
− ∇Y j
+ M )
j − M i ∇p
= ¯0,
M j D ij Y i Y j M m p
j≠i
∑
Y j ¯v ′ dj = ¯0,
j
(3.8)
where
1
= ∑ Y j
(3.9)
M m M
j j
is the mean mass of one mole of the mixture (see chapter 1).
Similarly it was shown that the diffusion velocities ¯ v ′′ di corresponding to the
temperature gradient are given by expressions of the form
¯ v ′′ di = − D T i
ρY i
∇T
T , (3.10)
where D T i is the thermal diffusion coefficient of species A i .
With respect to the reaction rate w i , it should be noted that the conclusions
in chapter 1 were established under the assumption that both the composition and
the state of the mixture are homogeneous throughout the mass. Therefore, when the
composition and state of the mixture are not homogeneous, the applicability of said
conclusions depends on the assumption that neither one varies appreciably within a
distance of the order of magnitude of a characteristic length of the chemical transformations.
This distance could be, for example, the average length of a chain when
chain reactions are produced. Within such a limitation the conclusions obtained in
chapter 1 can be applied here. In particular, if there are r different chemical reactions,
we have seen in the aforementioned chapter that w i is a linear combination of the r
reaction rates w ij corresponding to said chemical reactions
∑
w i = M i ν ij r j . (3.11)
j

64 CHAPTER 3. GENERAL EQUATIONS
3.3 Equations of motion
The equations of motion are obtained by applying Newton’s Second Law of Mechanics,
to the material element in Fig. 3.1.
Let V be the volume of said element. Then ρV will be the mass and ρV ¯v the
momentum, for which the time variation when following the element is
D (ρV ¯v)
Dt
= ρV D¯v
Dt , (3.12)
since ρV does not vary. Therefore, the variation of momentum of the mass contained
in the unit volume is ρ (D¯v/Dt). Such variation originates from:
1) The surface forces acting upon the unit volume.
2) The mass forces acting upon the unit volume.
Let us calculate both, separately.
Let ¯f be the force that acts upon the surface element dσ at a point of Σ, Fig. 3.1.
It has been shown in chapter 1 that ¯f is determined by the stress tensor τ e of components
τ ij and that as a function of the tensor it can be expressed as follows
¯f = ¯n · τ e dσ, (3.13)
where ¯n is the outward normal to the surface element dσ at the point. As it can easily
be proved, the resultant of all forces ¯f per unit volume is ∇ · τ e . 6
Let ¯F be the field intensity of the mass forces, that is to say, the force per unit
mass. The force per unit volume is, evidently, ρ ¯F . 7
Now, by expressing Newton’s Second Law of Mechanics, we obtain
ρ D¯v
Dt = ∇ · τ e + ρ ¯F . (3.14)
This vectorial equation is equivalent to three scalar equations corresponding to
the three components of the reference system. For example, in rectangular cartesian
coordinates, the component of this equation parallel to the coordinate axis x i (i =
1, 2, 3) is
ρ ∂v i
∂t + ρv ∂v i
j = ∂τ ij
+ ρF i , (3.15)
∂x j ∂x j
6 In fact, the resultant of the forces acting upon Σ is RR Σ ¯n · τ e dσ. Now, by applying Ostrogradsky’s
theorem we obtain RR Σ ¯n · τe dσ = RRR V ∇ · τe dV. Therefore, the force per unit volume is ∇ · τe.
7 If the force per unit mass depends on the species and ¯F i is the intensity for species A i , ρ ¯F must be
substituted by P ρ i ¯Fi = ρ P Y i ¯Fi .
i
i

3.4. ENERGY EQUATION 65
where the subscripts indicate components, and the repeated subscripts indicate, according
to Einstein’s rule, summation respect to them, from one to three.
As shown in chapter 2, the components τ ij of the stress tensor are expressed as
a function of the gas pressure p at the point, and of the velocity ¯v of the motion, in the
following form
( ∂vi
τ ij = −pδ ij + µ + ∂v )
j
+
(µ ′ − 2 )
∂x j ∂x i 3 µ (∇ · ¯v) δ ij , (3.16)
where δ ij is the Kronecker symbol and µ and µ ′ are the coefficients of viscosity and
volumetric viscosity of the mixture, respectively. These coefficients depend on the
state and composition of the mixture as indicated in the above mentioned chapter. In
particular, for the diluted monatomic gases µ ′ = 0 and in general, a good approximation
is obtained by assuming that it is also zero for all other cases.
In Eq. (3.14) the pressure and viscous stresses can be made explicit, by using
the expression
τ e = −pU + τ ev (3.17)
previously given in chapter 2. By taking this expression of τ e into Eq. (3.14), we
obtain
ρ D¯v
Dt = −∇p + ∇ · τ ev + ρ ¯F . (3.18)
This equation is identical to those obtained in Gas Dynamics for mixtures of uniform
composition with no chemical reaction. The variations in the composition of the
mixture act only through their influence on the values of the viscosity coefficients µ
and µ ′ .
3.4 Energy equation
This equation is obtained when expressing that the variations of the energy contained
in V , Fig.3.1, are due to the work done by the forces acting upon the element and the
heat received by the said element. Let us calculate, separately, the different terms in
this equation.
a) Energy.
Let u i be the internal energy per unit mass of species A i . The internal energy u
per unit mass of the mixture is
u = ∑ i
Y i u i , (3.19)

66 CHAPTER 3. GENERAL EQUATIONS
and the internal energy of the mass contained in V is ρV u. Similarly, the kinetic
energy of the mass contained in V is ρV 1 2 v2 . Therefore, the total energy 8 of the
mass contained in V is ρV ( u + 1 2 v2) and its time variation when following the
motion of V is
(
D
ρV (u + 1 )
Dt 2 v2 ) = ρV D Dt (u + 1 2 v2 ). (3.20)
Consequently, the time variation of the energy of the mass contained in the unit
volume is
b) Work.
The work done by ¯f per unit time is
ρ D Dt (u + 1 2 v2 ). (3.21)
¯v · ¯f = ¯v · (¯n · τ e ) = ¯n · (¯v · τ e ) , (3.22)
and the work per unit time done by the surface forces that act upon the unit
volume is ∇ · (¯v · τ e ). 9 The work per unit time done by the mass forces that act
upon the unit volume is ρ ¯F · ¯v. Consequently, the work per unit time done by all
the forces that act upon the unit volume is
∇ · (¯v · τ e ) + ρ ¯F · ¯v. (3.23)
c) Heat.
The heat received by the element V through dσ per unit time is −¯q · ¯n dσ, where
¯q is the vector of the heat flux defined in chapter 2. Therefore, the heat per unit
time received by the unit volume is
−∇ · ¯q. (3.24)
In the phenomena normally studied by the Aerothermochemistry the radiation
effects are of secondary importance. Furthermore, great difficulty is generally
encountered for the study of the influence of these effects. Nevertheless, if they
are to be taken into consideration, the vector ¯q r of the radiation flux must be
added to the vector ¯q of the heat flux. Information regarding the radiation flux
can be found in the work of Hirschfelder, Curtiss and Bird, mentioned in Ref.
[4], pp. 720 and following.
8 Other forms of energy different from the internal and the kinetic energies, as for example the radiation
energy, are not considered in the present study.
9 It can be proved in the same manner as for the calculation of the resultant of the surface forces.

3.4. ENERGY EQUATION 67
The energy equation is now obtained by expressing that the energy variation,
given by Eq. (3.21), must be equal to the work done by the exterior forces, given by
Eq. (3.23), plus the heat received, given by Eq. (3.24), thus obtaining
ρ D Dt (u + 1 2 v2 ) = ∇ · (¯v · τ e ) + ρ ¯F · ¯v − ∇ · ¯q. (3.25)
Expanding the first term of the right hand side of this expression, there results
∇ · (¯v · τ e ) = ¯v · (∇ · τ e ) + τ e : τ v , (3.26)
where τ v is the deformation velocity tensor, of components γ ij 10 defined by
γ ij = 1 2
( ∂vi
+ ∂v )
j
. (3.27)
∂x j ∂x i
In expression (3.26), the first term of the right hand side measures the work
done by the surface forces that act upon the unit volume when the surface moves with
no deformation with velocity ¯v. The second term measures the work done by the
surface forces in order to produce the deformation τ v .
Bringing forth the pressure and the viscous stresses in τ e by using expression
(3.17) for τ e , it can be verified that the deformation work (τ e : τ v ) consists of the
pressure work (−p∇ · ¯v) done during the expansion of the gas, and of the work
Φ = τ ev : τ v (3.28)
dissipated by the viscosity to produce the deformation τ v . Function Φ, which as it can
be easily proved is always positive, is the dissipation function of Lord Rayleigh. That
is
τ e : τ v = −p∇ · ¯v + Φ. (3.29)
In order to bring forth the variation of the internal energy u, the energy equation
(3.25) can be simplified by making use of the equation of motion (3.14). In fact, by
subtracting from Eq. (3.25) the scalar product of Eq. (3.14) by ¯v, taking into account
Eq. (3.26), there results
or else, making use of Eq. (3.29),
10 See chapter 2.
ρ Du
Dt
ρ Du
Dt = τ e : τ v − ∇ · ¯q, (3.30)
= −p∇ · ¯v + Φ − ∇ · ¯q. (3.31)

68 CHAPTER 3. GENERAL EQUATIONS
It has been shown in chapter 2 that ¯q is given by the expression
¯q = −λ∇T + ρ ∑ i
∑∑
Y j D T i
(¯v di − ¯v dj )
i j M i M j D ij
Y i h i¯v di + RT
∑ Y i
. (3.32)
M j
In this expression the first term of the right hand side gives the heat flux through conduction
due to temperature gradient. This is the only existing term in Gas Dynamics.
The second term gives the enthalpy flux of the various species through diffusion. The
third term gives the heat flux due to the Dufour effect of reciprocal effect of the thermal
diffusion in Onsager’s sens. 11 Such an effect is zero when all the molecules have
the same mass since, then, D T i is zero. In general, the Dufour effect can be neglected
and thereby ¯q reduces to the expression
j
¯q = −λ∇T + ρ ∑ i
Y i h i¯v di . (3.33)
Taken this expression of ¯q into Eq. (3.30), the following final expression for
the variation of internal energy of the mixture is obtained
(
ρ Du
Dt = −p∇ · ¯v + Φ + ∇ · (λ∇T ) − ∇ · ρ ∑ )
Y i h i¯v di . (3.34)
i
from:
This equation shows that the time variation of the internal energy originates
a) The work −p∇ · ¯v done by pressure to compress the gas.
b) The work Φ dissipated by the viscous forces to produce the deformation.
c) The heat ∇ · (λ∇T ) received through conduction.
d) The enthalpy flux −∇ · (ρ ∑ i Y ih i¯v di ) of the species through diffusion.
Forms a), b) and c) are the same as those that exist in Gas Dynamics when there is no
change in composition.
In many cases it is convenient to work with Eq. (3.25) which includes kinetic
energy, 12 bringing forth in this equation the enthalpy h of the mixture
h = u + p ρ . (3.35)
For this purpose we shall start by making the pressure explicit in the first term
of the right hand side of Eq. (3.25) by means of Eq. (3.17). Thus, the following is
11 See Ref. [5], p. 118.
12 See chapters 4 and 5.

3.5. GENERAL EQUATIONS 69
obtained for the said term
∇ · (¯v · τ e ) = −∇ · (p¯v) + ∇ · (¯v · τ ev ) . (3.36)
Now, if the first term of the right hand side of this equation is expanded and
combined with the continuity equation (3.6), the following is obtained
∇ · (¯v · τ e ) = ∂p
∂t − ρ D ( ) p
+ ∇ · (¯v · τ ev ) . (3.37)
Dt ρ
Substituting this equation into Eq. (3.25), taking into account Eq. (3.35), it finally
gives
ρ D Dt
(h + 1 2 v2 )
= ∂p
∂t + ∇ · (¯v · τ ev)
+ ρ ¯F · ¯v + ∇ · (λ∇T ) − ∇ ·
(
ρ ∑ i
Y i h i¯v di
)
,
(3.38)
where Eq. (3.33) has been used to eliminate ¯q.
3.5 General equations
As a summary of preceding paragraphs we shall once more write the general system
of equations that govern the transformation of a mixture of reactant gases in motion,
that is to say the general equations of Aerothermochemistry.
Continuity Equation
For the mixture:
For the species:
Dρ
+ ρ∇ · ¯v = 0. (3.6)
Dt
ρ DY i
Dt + ∇ · (ρY i¯v di ) = w i , (i = 1, 2, . . . , l). (3.7)
Equation of Motion
ρ D¯v
Dt = −∇p + ∇ · τ ev + ρ ¯F . (3.18)

70 CHAPTER 3. GENERAL EQUATIONS
Energy Equation
ρ Du
Dt = −p∇ · ¯v + Φ + ∇ · (λ∇T ) − ∇ · (
ρ ∑ i
Y i h i¯v di
)
. (3.34)
State Equation
p
ρ = R mT. (3.39)
Moreover the laws of variation of the thermodynamic functions, of the transport coefficients
and of the reaction rates as functions of the state and composition of the
mixture must be known.
3.6 Entropy variation
It is interesting to bring forth the entropy variation and therewith the causes of the
irreversibility of the process. Moreover, the entropy variations have a great influence
upon motion, for example, producing vortexes. 13
Thermodynamics teaches 14 that the variation dS of the entropy of a reactant
system corresponding to the variations dU, dV and dm i of its internal energy. volume
and masses of the chemical species respectively, is given by the expression
T dS = dU + p dV − ∑ i
µ i dm i , (3.40)
where µ i is the chemical potential of species A i .
If one applies this expression to the mass contained in the volume element V
of Fig. 3.1 following the motion, and bearing in mind that for this volume
S = ρV s, U = ρV u, m i = ρV Y i ,
1
V
DV
Dt
= ∇ · ¯v, (3.41)
where s is the entropy of the mixture per unit mass, the following expression is obtained
for the time variation of the entropy of the mass contained in the unit volume
(
ρ Ds
Dt = 1 ρ Du
T Dt + p∇ · ¯v − ρ ∑ i
µ i
DY i
Dt
)
. (3.42)
13 See chapter 9.
14 See chapter 1.

3.6. ENTROPY VARIATION 71
Taking into this expression the variations of u and Y i given by Eqs. (3.34) and
(3.7) respectively, one obtains
ρT Ds
Dt =Φ + ∇ · (λ∇T ) − ∇ · (ρ ∑ i
− ∑ i
µ i w i + ∑ i
Y i h i¯v di )
µ i ∇ · (ρY i¯v di ).
(3.43)
As can easily be verified, this equation can be written in the form
[
]
ρ Ds
Dt =∇ · λ ∇T
T
− ρ ∑
Y i (h i − µ i )¯v di + 1 (∇T )2
[Φ + λ
T
T T
− ρ ∇T
T
i
∑
Y i (h i − µ i )¯v di − ρ ∑
i
i
Y i¯v di · ∇µ i − ∑ i
µ i w i
] (3.44)
Then we have 15 µ i = h i − T s i , (3.45)
where h i and s i are, respectively, the enthalpy and the entropy per unit mass of species
A i at the partial pressure p i of the species and at the temperature T of the mixture.
When taking this expression into Eq. (3.44) one obtains
[
ρ Ds
Dt =∇ · λ ∇T
T
− ρ ∑ ]
Y i s i¯v di + 1 (∇T )2
[Φ + λ
T T
i
− ρ∇T ∑ Y i s i¯v di − ρ ∑ Y i¯v di · ∇µ i − ∑ ]
µ i w i .
i
i
i
(3.46)
This equation admits a simple interpretation. In fact the first term of the right hand
side of this equation gives the reversible entropy flux across the boundary of the unit
volume. This flux originates from: a) heat transport through conduction, and b) diffusion
of the species. The second and third terms originate from the entropy generated
in the gas per unit volume and unit time, due to irreversibilities of the process. Such
irreversibilities, shown in Eq. (3.46), arise from: a) viscosity, b) heat conductivity,
c) diffusion, and d) chemical reactions. Making use of Eq. (3.11) the term ∑ µ i w i
i
corresponding to d) may be written in the form
∑
µ i w i = − ∑ r j α j , (3.47)
i
j
where α j is the chemical affinity corresponding to reaction j. 16 All contributions to
the variation of the entropy from the irreversibilities of the process must be positive. 17
15 See chapter 1.
16 See chapter 1.
17 For further information, see Refs. [4] or [5].

72 CHAPTER 3. GENERAL EQUATIONS
3.7 One-dimensional motions
As an application we shall see the form taken by the general equations in the case of
one dimensional motions. These equations, in particular those relative to stationary
motions, will be widely used in the study of the detonations and flames. 18
It will be assumed that:
a) No mass forces exist.
b) The coefficient of volumetric viscosity µ ′ is zero.
c) The effects of thermal diffusion and radiation can be neglected.
We shall adopt a cartesian rectangular system with the x axis parallel to the
direction of motion. Then, the only independent variables of the motion are coordinate
x and time t, and the only velocity component different from zero is that parallel to
the x axis, which will be designated as v.
The substantial derivative (3.1) reduces in this case to
D (·)
Dt
= ∂ (·)
∂t
+ v ∂ (·)
∂x . (3.48)
Continuity equations
The continuity equations for the mixture reduces to
∂ρ
∂t + v ∂ρ
∂x + ρ ∂v = 0. (3.49)
∂x
Similarly, the continuity equations (3.7) for the various species, reduces to
ρ ∂Y i
∂t + ρv ∂Y i
∂x + ∂
∂x (ρY iv di ) = w i , (i = 1, 2, . . . , l). (3.50)
Equations of motion
In the vectorial Eq. (3.18) the only component not identically zero is that parallel to
x axis. Furthermore, the only component different from zero in the viscous stress
tensor τ ev is
τ xx = 4 3 µ ∂v
∂x , (3.51)
corresponding to viscous stress that acts upon the plane normal to the motion and
parallel to the x axis. This stress is subtracted from the pressure. Therefore, Eq. (3.18)
18 See chapters 5 and 6.

3.7. ONE-DIMENSIONAL MOTIONS 73
reduces to
ρ ∂v ∂v
+ ρv
∂t ∂x = − ∂p
∂x + 4 3
∂
∂x
(
µ ∂v )
. (3.52)
∂x
Energy equation
Equation (3.34) reduces to
ρ ∂u ∂u
+ ρv
∂t
∂x = − p ∂v
∂x + 4 ( ∂v
3 µ ∂x
− ∂ (
λ ∂T
∂x ∂x
) 2
)
− ∂
∂x
(
ρ ∑ i
Y i h i v di
)
Or if it is decided to use the total energy equation (3.38) it takes the form
ρ ∂ (h + 1 )
∂t 2 v2 + ρv ∂ (h + 1 )
∂x 2 v2 = ∂p
∂t + 4 (
∂
µv ∂v )
3 ∂x ∂x
+ ∂ (
λ ∂T ) (
− ∂ ρ ∑ )
Y i h i v di .
∂x ∂x ∂x
i
.
(3.53)
(3.54)
The system of equations (3.49), (3.50), (3.52) and (3.53), or (3.54), must be
completed with the following equations.
Total mass fraction equation
∑
Y i = 1. (3.55)
i
Diffusion equations
The system of equations (3.8) which gives the diffusion velocities reduces to
∑
[ (
Y j vdi − v dj 1 ∂Y i
+
M j D ij Y i ∂x − 1 )
∂Y j
Y j ∂x
j
+ M j − M i
M m
( )] 1 ∂p
= 0, (i = 1, 2, . . . , l),
p ∂x
∑
Y i v di = 0.
i
(3.56)
State equation
The state equation is, by assumption, that of ideal gases
p
ρ = R mT. (3.39)

74 CHAPTER 3. GENERAL EQUATIONS
State functions
The internal energy u i of species A i is 19
The internal energy of the mixture is
where
u = ∑ i
∫ T
u i = u i0 + c vi dT. (3.57)
T 0
Y i u i = ∑ i
∫ T
Y i u i0 + c v dT, (3.58)
T 0
c v = ∑ Y i c vi (3.59)
i
is the mixture heat capacity at constant volume.
Similarly, for the enthalpy we have
and for the enthalpy of the mixture
with
h = ∑ i
h i = h i0 +
Y i h i = ∑ i
c p = ∑ i
∫ T
T 0
c pi dT, (3.60)
∫ T
Y i h i0 + c p dT, (3.61)
T 0
Y i c pi . (3.62)
Transport coefficients
In system (3.56), D ij are the binary diffusion coefficients between species A i and
A j . For each pair of these species the coefficients depend on the state variables, as
indicated in chapter 2.
λ and µ are the coefficients of thermal conductivity and viscosity of the mixture,
respectively. These coefficients depend on the state variables and on the composition
of the mixture as indicated in chapter 2.
Reaction rates
The reaction rate of species A i is given in Eq. (3.11). In this equation the coefficients
ν ij and the velocities r j corresponding to the different elementary reactions of the
mixture are given by the formulae of chapter 1.
19 See chapter 1.

3.8. STATIONARY, ONE-DIMENSIONAL MOTIONS 75
3.8 Stationary, one-dimensional motions
If the motion is not only one-dimensional but stationary, the preceding equations are
considerably simplified. This type of motion, as previously stated, is of special interest
for the study of the structure of combustion waves.
The stationary motions are characterized by the condition
∂ (·)
= 0. (3.63)
∂t
Therefore, the only independent variable is x.
Let us see how the equations in the preceding paragraph can be simplified when
condition (3.63) is introduced therein.
Continuity Equations
The continuity equation (3.49) reduces to
v dρ
dx + ρ dv
dx ≡
This equation can be integrated, thus obtaining
d (ρv) = 0. (3.64)
dx
ρv = m, (3.65)
where m is an integration constant which gives the mass flux per unit surface normal
to the direction of motion.
Similarly, equations (3.50) reduces to
d
dx (ρY iv + ρY i v di ) = w i , (i = 1, 2, . . . , l) . (3.66)
By introducing the flux of the different species, these equations reduce to a
very simple form. In fact, let m i = mε i be the flux of species A i through the unit
surface normal to the direction of motion, that is to say that ε i is the fraction of the
total flux corresponding to species A i . Since the velocity v i of species A i is
v i = v + v di , (3.67)
evidently the said flux is
mε i = ρY i (v + v di ) . (3.68)
Now, when this expression is compared with the left hand side of Eq. (3.66) it
is seen that this system can be written in the form
m dε i
dx = w i, (i = 1, 2, . . . , l). (3.69)

76 CHAPTER 3. GENERAL EQUATIONS
The physical interpretation of this equations is obvious, m dε i is the difference
between the fluxes of species A i through two surfaces normal to the direction of motion,
separated one from another by the distance dx. But this difference has its origin
in the amount of the said species produced per unit time by the chemical reactions that
take place within the space that separates both surfaces, which is evidently w i dx.
The l equations of system (3.69) are not independent. In fact:
1) Formulae (3.11) shows that w i are linear combinations of the r reaction rates r j
corresponding to the different reactions that take place between the species of the
mixture. Consequently, the maximum number of independent w i is at most r.
2) The chemical reactions do not change the total number of atoms of the elements
that form the species. Let g be the number of different chemical elements of the
species and let a ij (j = 1, 2, ...., g) be the number of atoms of the element j in
species i. The conservation of the element in the chemical reactions imposes the
following system of conditions between w i
∑ w i
a ij = 0, (j = 1, 2, . . . , g). (3.70)
M i
i
It might occur however that not all these conditions are independent. In fact
the number of independent equations in (3.70) is the rank of the matrix of coefficients
a ij . This rank is the minimum number of components needed to form the l species
A i in the sense of the phase rule. Therefore, if g ′ ≤ g is the number of components of
the mixture, there exist g ′ independent linear relations between w i , due to Eq. (3.70),
and the number of independent w i is, at most, l − g ′ .
Thereby, the number l ′ < l of the independent w i is the smallest one of the
numbers r and l ′ − g ′ . 20
System (3.69) shows that there are as many ε i independent from each other as
there are w i , that is to say, l ′ < l, which makes it possible to reduce the number of
variables of the problem in l − l ′ .
Equation of Motion
Equation (3.52) reduces to
m dv
dx = − dp
dx + 4 3
(
d
µ dv )
, (3.71)
dx dx
20 As it can easily be proved, the difference l − g ′ between the number of species and the number
of components is the maximum number of independent reactions equations which can exist among the l
species. This property simplifies the interpretation of the conclusion concerning the number of independent
w i . In fact, if l − g ′ < r, the r chemical reactions are not linearly independent from each other.

3.9. THE CASE OF ONLY TWO CHEMICAL SPECIES 77
which by integration gives
where i is an integration constant.
p + mv − 4 3 µ dv = i, (3.72)
dx
Energy Equation
In this case it is convenient to make use of the total energy equation (3.54) which by
integration gives
mh + ρ ∑ i
Y i h i v di + 1 2 mv2 − λ dT
dx − 4 dv
µv = e, (3.73)
3 dx
where e is an integration constant.
The two first terms of this equation give the enthalpy flux m ∑ ε i h i of the
i
mixture per unit surface normal to x, as can easily be proved by keeping in mind
(3.61) and (3.68). Therefore Eq. (3.73) may also be written in the form
(
1
m
2 v2 + ∑ )
ε i h i − λ dT
dx − 4 dv
µv = e, (3.74)
3 dx
i
whose physical interpretation is obvious.
Diffusion Equations
Equations (3.56) remain valid
∑
j
Y j
M j
[
vdi − v dj
D ij
+
( 1 dY i
Y i dx − 1 dY j
Y j dx
( 1 dp
p dx
+ M j − M i
M m
)
)]
= 0, (i = 1, 2, . . . , l),
∑
Y i v di = 0.
i
(3.75)
3.9 The case of only two chemical species
Let us now assume that in the preceding case the number of chemical species of the
mixture reduces to two, A 1 and A 2 . Then, a single chemical variable Y 1 , denoted Y ,
is enough to define the composition of the mixture, since Y 2 = 1 − Y . Similarly the
flux fraction ε i of A 1 will be denoted ε and one has ε 2 = 1 − ε.

78 CHAPTER 3. GENERAL EQUATIONS
reduces to
Y v d1
D 12
System (3.69) reduces in this case to the single equation
m = dε = w. (3.76)
dx
Equation of motion (3.72) remains the same and the energy equation (3.74)
m
(h 2 + (h 1 − h 2 ) ε + 1 )
2 v2 − λ dT
dx − 4 dv
µv = e. (3.77)
3 dx
Finally, the system of diffusion equations (3.75) reduces to the single equation
(
+ dY
dx + (M2 − M 1 ) [ ])
(M 2 − M 1 ) Y + M 1 Y (1 − Y ) dp
= 0, (3.78)
M 1 M 2 p dx
which making use of Eq. (3.68) that in this case reduces to
can be written as
mε = ρY (v + v d1 ) = mY + ρY v d1 , (3.79)
dY
dx + (M 2 − M 1 ) [ ]
(M 2 − M 1 ) Y + M 1 Y (1 − Y ) dp
M 1 M 2 p dx = m (Y − ε). (3.80)
ρD 12
The unknown of the problem are in this case v, p, ρ, T , Y and ε. The six equation
which determine their values are the following: continuity (3.65) and (3.76), motion
(3.72), energy (3.77), diffusion (3.80) and state (3.39). The system thus formed
will be studied in detail in chapters 5 and 6 in discussing the structure of the combustion
waves, specially under the simplified form studied in the following. Let us
assume, in particular, the following two conditions:
1) The heat capacities at constant pressure of both species are equal
c p1 = c p2 = c p . (3.81)
2) The molar masses of both species are equal, or else the gradient of pressure is
very small.
By virtue of the first assumption, and taking into account (3.60), one has
h 2 − h 1 = h 20 − h 10 = q, (3.82)
where q is the heat of reaction per unit mass of the mixture.
Due to (3.82), equation (3.77) reduces to
m
(h 2 − qε + 1 )
2 v2 − λ dT
dx − 4 dv
µv = e. (3.83)
3 dx

3.10. STATIONARY, ONE-DIMENSIONAL MOTION OF IDEAL GASES WITH HEAT ADDITION 79
By virtue of the second assumption, the diffusion equation (3.80) reduces to
which is the expression of Fick’s Law.
dY
dx = m (Y − ε), (3.84)
ρD 12
If, furthermore, the heat capacity c p is independent from temperature, then
equation (3.83), considering (3.60), takes the form
m
(c p T − qε + 1 )
2 v2 − λ dT
dx − 4 dv
µv = e, (3.85)
3 dx
where e must now include the constant terms that come from (3.60).
3.10 Stationary, one-dimensional motion of ideal gases
with heat addition
Let us assume in the above case that the action of viscosity and thermal conductivity
can be neglected in the equation of motion (3.72) and energy (3.85). This is justified if
the gradients of velocity and temperature are not very ”large”. 21 Then (3.72) reduces
to
p + mv = i. (3.86)
Likewise (3.85) reduces to
m
(c p T − qε + 1 )
2 v2 = e. (3.87)
These two equations together with (3.65)
ρv = m (3.88)
and the equation of state (3.39)
p
ρ = R mT, (3.89)
determine the values for p, ρ, T and v corresponding to each value of ε, if the values of
such variables corresponding to a given value of ε are known, for example, to ε = 0.
This enables the determination of the constants of Eqs. (3.86), (3.87) and (3.88). In
such case, in Eq. (3.87) the term Q = qε represents the heat added to the gas per unit
mass from the initial state ε = 0. The result is independent from the way in which the
heat is added; be it by chemical reactions or transmitted from external sources.
21 See chapter 5 for the exact meaning of this expression.

80 CHAPTER 3. GENERAL EQUATIONS
In the following, R m is assumed to be constant as occurs if the molar masses
of both species are equal.
Taking in abscissas the heat added to the gas and in ordinates the values for
v and T 22 a diagram can be drawn which clearly shows the transformations taking
place within the gas. This has been done in Fig. 3.3 for the case where the relation
γ of heat capacities is 1.4. The dimensionless variables which will be defined in the
following are introduce in order to have an universal diagram.
1) Let
b = γ + 1
γ
ma s0
, (3.90)
i
where a s0 = √ γR m T s0 is the velocity of sound at the stagnation condition T s0
corresponding to the initial state Q = 0.
A simple calculation shows that b can be expressed as a function of the Mach
number M 0 in the form:
b = (γ + 1) M 0
1 + γM 2 0
√
1 + γ − 1 M0 2 2
. (3.91)
Similarly, let M ∗ 0 be ratio of v 0 to the critical velocity v cr of the gas at the initial
state. b can also be expressed as a function of M ∗ 0 in the form
b = √ 2(γ + 1)
M ∗ 0
1 + M ∗2
0
. (3.92)
b and M 0 are represented in Fig. 3.2 as functions of M ∗ 0 for γ = 1.4.
2) Let T s be the stagnation temperature of the gas when the added heat is Q, and
let n be the ratio of T s to T s0 ,
n = T s
T s0
. (3.93)
The relation between T s and T s0 is obtained when expressing e in Eq. (3.87) by
means of T s and T s0 . Thus resulting
n = 1 +
Q
c p T s0
. (3.94)
The added heat is expressed by means of the variable x defined by
3) The velocity is expressed in dimensionless form by
22 See von Kármán, Ref. [7].
x = b 2 n. (3.95)
v ∗ = b v
a s0
. (3.96)

3.10. STATIONARY, ONE-DIMENSIONAL MOTION OF IDEAL GASES WITH HEAT ADDITION 81
1.2
6
1.0
0.8
b
5
4
b
0.6
3
M 0
0.4
2
0.2
M 0
1
0
0
0.0 0.5 1.0 1.5 2.0 2.5
*
M 0
Figure 3.2: M 0 and b (Eq. (3.92)) as a function of M0 ∗ .
4) Likewise, the temperature is expressed in dimensionless form by
θ = b 2 T
T s0
. (3.97)
If these variables are taken into the previous system and this system is solved
for v ∗ and θ, one obtains
(v ∗ − 1) 2 = 1 − 2x
γ + 1 , (3.98)
θ = x − γ − 1 v ∗2 . (3.99)
2
The values of v ∗ and θ are represented in the diagram Fig. 3.3 where the law
of variation of the Mach number M of the motion is also included. M is given by the
expression
M = v∗
√
θ
. (3.100)
It can easily be verified that for v ∗ and θ to be real, x cannot exceed the value
x max = γ + 1 , (3.101)
2
to which corresponds a maximum value n max of n given by the expression
n max = γ + 1
2b 2 . (3.102)

82 CHAPTER 3. GENERAL EQUATIONS
2.5
1.25
v * , M
2.0
1.5
1.0
θ −
B
v +
*
θ +
M +
A
1.00
0.75
0.50
θ
0.5
M − v−
*
0.25
O
0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.00
1.4
x
Figure 3.3: Mach number (M), nondimensional velocity (v ∗ ) and temperature (θ) as a function
of the nondimensional heat added to the gas (x).
To this value corresponds a maximum value Q max of Q given by the expression
( ) γ + 1
Q max =
2b 2 − 1 c p T s0 . (3.103)
This value is determined by the initial condition through b 2 and c p T s0 . Q max
is the maximum heat that can be added to the flow consistent with the given initial
conditions.
The values of v ∗ 1, θ 1 and M 1 of v ∗ , θ and M corresponding to x = x max are
v ∗ 1 = 1, θ 1 = 1, M 1 = 1. (3.104)
Therefore the velocity of the gas at this point is the velocity of sound. In the
curve of velocities point A, corresponding to x = x max , divides this curve into two
branches OA subsonic and AB supersonic. Fig. 3.4 shows that if the initial velocity
is subsonic the gas accelerates as the heat is added whilst if the initial velocity is
supersonic the gas decelerates. In both cases the maximum heat that can be added
is given by (3.102) or (3.103). Conversely, the maximum initial velocity M0,max ∗ for
each value of n is obtained when Eq. (3.102) is solved for M 0 . Thus results
M0,max ∗ = √ n ± √ n − 1 . (3.105)
Sign plus corresponds to supersonic velocities since M ∗ 0,max > 1. Sign minus

3.10. STATIONARY, ONE-DIMENSIONAL MOTION OF IDEAL GASES WITH HEAT ADDITION 83
corresponds to subsonic velocities. This case has great importance in technical applications
as it determines, for example, the maximum velocity permissible at the inlet
of a constant cross-section combustion chamber as a function of the heat released in
the same. 23 The value of the corresponding Mach number M 0,max is given by
√
2M0,max
∗2
M 0,max =
(γ + 1) − (γ − 1) M0,max
∗2 . (3.106)
The values of M 0,max and M ∗ 0,max as a function of n are represented in Fig. 3.4.
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1
14
subsonic
supersonic
n (supersonic)
12
M * 0,max , M 0,max (subsonic)
1.0
0.8
0.6
0.4
M 0,max
M * 0,max
10
8
6
4
M * 0,max , M 0,max (supersonic)
0.2
2
M 0,max
0.0
0
1 3 5 7 9 11 13 15 17 19 21 23
n (subsonic)
Figure 3.4: Values of M 0,max and M ∗ 0,max as a function of n.
It is easily seen that this choking effect imposes an important limitation to the Mach
number at the inlet of the combustion chamber.
With respect to the law of variation of temperature as a function of Q it is
interesting to remark that the maximum temperature has the value
θ max =
(γ + 1)2
, (3.107)
4γ
and it is reached for a value of n slightly smaller that its maximum value as well as
for a subsonic velocity. This is due to the fact that for velocities slightly subsonic the
acceleration produced by the added heat is so large that the heat received is unable to
balance the cooling produced by the expansion of the accelerating gas. In chapter 5,
where these results are applied, it is seen that this effect produces in detonation.
23 See chapter 10.

84 CHAPTER 3. GENERAL EQUATIONS
3.11 Appendix: Notation and repertoire of vectorial
and tensorial formulae used in the present
chapter
Scalar: a
Vector: ā (components a 1 , a 2 and a 3 )
Scalar product: ā · ¯b (scalar)
Vectorial product: ā × ¯b (vector)
The Hamilton ”nabla”:
Gradient of a scalar:
Divergence of a vector:
∂ (·) ∂ (·) ∂ (·)
∇ (·) = ī 1 + ī 2 + ī 3 (vectorial operator)
∂x 1 ∂x 2 ∂x 3
ī 1 , ī 2 and ī 3 are unit vectors parallel to the coordinate axes.
∇a = ī 1
∂a
∂x 1
+ ī 2
∂a
∂x 2
+ ī 3
∂a
∂x 3
(vector)
∇ · ā = ∂a 1
∂x 1
+ ∂a 2
∂x 2
+ ∂a 3
∂x 3
(scalar)
Curl of a vector:
( ∂a3
∇ × ā = ī 1 − ∂a ) (
2 ∂a1
+ ī 2 − ∂a ) (
3 ∂a2
+ ī 3 − ∂a )
1
∂x 2 ∂x 3 ∂x 3 ∂x 1 ∂x 1 ∂x 2
(vector)
Convective derivative:
¯v · ∇ (·) = v 1
∂ (·)
∂x 1
+ v 2
∂ (·)
∂x 2
+ v 3
∂ (·)
∂x 3
∇ · (a¯b ) = ¯b · ∇a + a ( ∇ · ¯b ) .
(scalar operator)
Tensor:
τ =
⎛
⎜
⎝
⎞
τ 11 τ 12 τ 13
⎟
τ 21 τ 22 τ 23 ⎠
τ 31 τ 32 τ 33
Transposed ¯τ of tensor τ (permutation of rows by columns):
⎛
⎞
τ 11 τ 21 τ 31
⎜
⎟
¯τ = ⎝ τ 12 τ 22 τ 32 ⎠
τ 13 τ 23 τ 33

3.11. APPENDIX: NOTATION AND REPERTOIRE OF VECTORIAL AND TENSORIAL FORMULAE 85
Symmetric tensor: τ = ¯τ .
If τ is symmetric:
τ ij = τ ji (i, j = 1, 2, 3)
Product of a vector by a tensor:
ā · τ = ī 1 (a 1 τ 11 + a 2 τ 21 + a 3 τ 31 )
+ ī 2 (a 1 τ 12 + a 2 τ 22 + a 3 τ 32 )
+ ī 3 (a 1 τ 13 + a 2 τ 23 + a 3 τ 33 ) (vector)
Divergence of a tensor:
∇ · τ =
(
∂τ11
ī 1 + ∂τ 21
+ ∂τ )
31
∂x 1 ∂x 2 ∂x 3
(
∂τ12
+ ī 2 + ∂τ 22
+ ∂τ )
32
∂x 1 ∂x 2 ∂x 3
(
∂τ13
+ ī 3 + ∂τ 23
+ ∂τ )
33
∂x 1 ∂x 2 ∂x 3
(vector)
Gradient of a vector:
⎛
∇ ā =
⎜
⎝
∂a 1
∂x 1
∂a 2
∂x 1
∂a 3
∂x 1
∂a 1
∂x 2
∂a 2
∂x 2
∂a 3
∂x 2
∂a 1
∂x 3
∂a 2
∂x 3
∂a 3
∂x 3
⎞
⎟
⎠
(tensor)
Double product:
ā · (¯b · τ
)
(scalar)
If τ is symmetric, then:
ā · (¯b · τ
)
= ¯b ·
(ā
· τ
)
Contraction of two tensors:
3∑ 3∑
τ : τ ′ = τ ij τ ji
′
i=1 j=1
(scalar)
Therefore
and
τ : τ ′ = τ ′ : τ
∇ · (ā · τ) = ā · (∇ · τ) + (∇ā) : τ (scalar)

86 CHAPTER 3. GENERAL EQUATIONS
If τ is symmetric
∇ · (ā · τ) = ā · (∇ · τ) + (∇ā) : τ = ā · (∇ · τ) + 1 (∇ā + ∇ā) : τ
2
The Gauss formula:
In a domain V enclosed by surface S:
∫∫
∫∫∫
(¯n ◦ ϕ) dσ =
S
V
(∇ ◦ ϕ) dV
ϕ can be a scalar, vector or tensor. Symbol ◦ represents any of the several types of
products previously defined.
Derivative of an integral with changing boundary:
∫∫∫
∫∫∫
∫∫
d
∂ϕ (t, ¯x)
ϕ (t, ¯x) dV =
dV +
dt
∂t
V (t)
V (t)
Σ(t)
(¯n · ¯v) ϕ dσ
where ¯v is the velocity of Σ at dσ and ¯n is the outwards normal to Σ at dσ.
References
[1] Kuethe, A. N. and Schetzer, J. D.: Foundations of Aerodynamics. John Wiley
and Sons, Inc. New York. 1950.
[2] Green, R. S.: The Molecular Theory of Fluids. Interscience Publishers, Inc,
New York, 1952.
[3] Richardson, J. M. and Brinkley, S. R.: Mechanics of Reacting Continua. Combustion
Processes, Sec. F, Vol. II of High Speed Aerodynamics and Jet Propulsion,
Princeton University Press, 1956, p. 203.
[4] Hirschfelder, J. O., Curtiss, C. F. and Bird, R. B.: Molecular Theory of Liquid
and Gases. John Wiley and Sons, Inc., New York, 1954.
[5] De Groot, S. R.: Thermodynamics of Irreversible Processes. North Holland
Publishing Comp., Amsterdam, 1951.
[6] Prigogine, I. and Defay, R.: Chemical Thermodynamics. Longmans Green and
C., New York, 1954.
[7] von Kármán, Th.: The Theory of Shock Waves and the Second Law of Thermodynamics.
L’Aerotecnica, Febr. 15, 1951, pp. 82-3.
[8] Chambré, P. and Lin, C. C.: On the Steady Flow of a Gas through a Tube with
Heat Exchange or Chemical Reaction. The Journal of Aeronautical Sciences,
Oct. 1946, pp. 537-42.

3.11. APPENDIX: NOTATION AND REPERTOIRE OF VECTORIAL AND TENSORIAL FORMULAE 87
[9] Shapiro, A. H. and Hawthorne, W. R.: The Mechanics and Thermodynamics
of Steady One-Dimensional Gas Flows. Journal of Applied Mechanics, Trans.
A.S.M.E., Vol. 14, No. 4, 1947, p. 317.
[10] Foa, J. V. and Rudinger, G.: On the Addition of Heat to a Gas Flowing in a
Pipe at Subsonic Speed. The Journal of Aeronautical Sciences, Febr. 1949,
pp. 84-94.

88 CHAPTER 3. GENERAL EQUATIONS

Chapter 4
Combustion Waves
4.1 Detonation and deflagration
When an explosive mixture is ignited, a wave or reaction front originates which propagates
the combustion throughout the mixture. The wave thickness is normally very
small. By assuming this thickness to be zero, that is, assuming that gases burn instantaneously
as they cross the wave, the state of the mixture and the state of the burn
gases can be compared, in a way similar to that used when studying shock waves; 1
that is, by applying the laws of the conservation mass, momentum and energy across
the front, without taking into consideration the intermediate transformations between
the initial and final states Thus, the following system of three equations is obtained
ρ 1 v 1 = ρ 2 v 2 , (4.1)
ρ 1 v1 2 + p 1 = ρ 2 v2 2 + p 2 , (4.2)
1
2 v2 1 + h 1 = 1 2 v2 2 + h 2 , (4.3)
where ρ, p and h are the density, pressure and total enthalpy per unit mass and v the
normal velocity relative to the reaction front. Subscripts 1 and 2 refer to the unburnt
gases and to the combustion products respectively. In particular, v 1 is the propagation
velocity of the wave, that is, the velocity at which the wave moves through the unburnt
gases, Fig. 4.1.
For a given composition, the specific enthalpy h 1 of the unburnt gases is a
function of the pressure p 1 and the density ρ 1 of the mixture
h 1 = h 1 (p 1 , ρ 1 ) . (4.4)
1 See i.e. Courant-Friedrichs: Supersonic Flow and Shock Waves. Intersciences Pub. Inc., New York,
1948, pp. 121 and following.
89

function of p 2 and ρ 2
h 2 = h 2 (p 2 , ρ 2 ) . (4.5)
90 CHAPTER 4. COMBUSTION WAVES
Unburned gases
Burned gases
V
1
V 2
p ρ 1 T p ρ T
1 1
2 2 2
Combustion wave
Figure 4.1: Schematic diagram of a combustion wave to obtain the relations between initial
and final states.
Similarly, if the burnt gases are in thermodynamic equilibrium, h 2 is only a
When equations (4.4) and (4.5) are substituted into (4.3), this equation together
with Eq. (4.1) and (4.2) form a system of three equations with six unknowns: p 1 , ρ 1 ,
v 1 , p 2 , ρ 2 and v 2 . If three of these values are known, the system determines the values
for the other three. As said before, the state of the unburnt gases is defined by the
values p 1 and ρ 1 . Therefore, in order to determine the propagation, another of these
values must be known, for instance that of the propagation velocity v 1 . The analysis
of the propagation velocity shows the existence of two types of essentially different
waves. In fact, the elimination of v 2 between Eq. (4.1) and (4.2), gives for v 1 ,
v 1 = 1 ρ 1
√ √√√ p 2 − p 1
1
ρ 1
− 1 ρ 2
. (4.6)
The study is simplified by introducing in the above system the specific volume
τ, which is related to the density ρ by
τ = 1 ρ , (4.7)
thus obtaining for v 1
v 1 = τ 1
√
p2 − p 1
τ 1 − τ 2
. (4.8)
satisfied
Since the propagation velocity must be real, the following condition must be
p 2 − p 1
τ 1 − τ 2
> 0. (4.9)

4.2. KINDS OF DETONATIONS AND DEFLAGRATIONS 91
Consequently, if p 2 > p 1 , then τ 1 > τ 2 , and if p 2 < p 1 , then τ 1 < τ 2 .
Therefore, when the pressure increases across the wave the gases contract, and if the
pressure decreases, the gases expand. Both types of waves are physically observed.
The first type is called detonation and the second deflagration. The propagation velocity
of a detonation wave in an explosive mixture is of the order of several thousand
meters per second. The propagation velocity of a deflagration wave is normally of the
order of 1 m/s.
4.2 Kinds of detonations and deflagrations
The study can be simplified by representing the state of unburnt and burnt gases in the
diagram (p, τ), Fig. 4.2, proposed by Grussard [1].
p
III
H
I
p 2
A
p 1
II
P
B
IV
0 τ0 τ1 τ2 τ3
Figure 4.2: Sketch of the Hugoniot curve for the final states.
D
τ
In this diagram, let P (p 1 , τ 1 ) be the representative point of the initial state.
Condition (4.9) excludes regions I and II from this diagram, since in these regions
(p 2 − p 1 ) / (τ 1 − τ 2 ) is smaller than zero. The representative points of the final states
must be, consequently, within regions III and IV. The points in region III correspond to
detonations, since, in this region p 2 > p 1 and τ 2 < τ 1 . Points in region IV correspond
to deflagrations.
The possible final states corresponding to the initial state (p 1 , τ 1 ) and compatible
with conditions (4.1), (4.2) and (4.3), lie on a curve H, called the Hugoniot curve.

92 CHAPTER 4. COMBUSTION WAVES
This curve is obtained from the elimination of v 1 and v 2 between Eqs. (4.1), (4.2) and
(4.3). There results for H, the following equation
h 2 − h 1 − 1 2 (τ 1 + τ 2 ) (p 2 − p 1 ) = 0. (4.10)
This curve has a detonation branch and a deflagration branch.
Let p 2 be the final pressure corresponding to an adiabatic combustion at constant
volume τ 1 . Since the reaction is exothermic p 2 > p 1 and the representative point
of the state (p 2 , τ 1 ), will be, for instance, point A. The detonation branch starts at this
point. The propagation velocity corresponding to this detonation at constant volume
is, according to Eq. (4.8), infinite. Hence, at this point the combustion propagates
instantaneously throughout the mass.
Similarly, let τ 2 be the specific volume corresponding to an adiabatic combustion
at constant pressure p 1 . Here τ 2 > τ 1 , and point B, corresponding to the state p 1 ,
τ 2 , is the starting point of the deflagration branch. The propagation velocity of this
limit constant pressure deflagration is, according to (4.8), zero.
Hereinafter, it is assumed that the Hugoniot curve satisfies the following conditions
( ) ( ∂p
∂ 2 p
< 0 ,
∂τ
H
∂τ
)H
2 > 0, (4.11)
where subscript H indicates differentiation along the Hugoniot curve. These conditions
mean that H is monotonically decreasing and turns its concavity to the axis
p > 0. Both conditions correspond to the curves generally observed in practice. In
general, curve H has an asymptote, parallel to the pressure axis, for τ = τ 0 > 0, and
cuts the specific volume axis at point τ 3 . Therefore, the general form of H is the one
shown in Fig. 4.2.
Let us consider a straight line that starts from point P , corresponding to the
initial state, and enters in region III of detonations. Let α, Fig. 4.3(a), be the angle
between this line and the negative direction of the axis. Depending on the values of α,
the three following cases are possible:
1) If α is smaller than a certain limit value α min which corresponds to the tangent
from P to H, the straight line corresponding to α cannot cut the Hugoniot curve.
2) If α = α min , the corresponding line is, as aforesaid, tangent to H at point J.
3) If α > α min , the corresponding line cuts the Hugoniot curve at two points E
and E ′ at different sides of point J.

4.2. KINDS OF DETONATIONS AND DEFLAGRATIONS 93
p
C
E’
p
1
p
p 1
P
B
J
α
α min
E
A
P
τ 0 1
(a) Detonation branch
τ
τ
J’
D
τ 1 τ 3
τ
(b) Deflagration branch
Figure 4.3: Chapman-Jouguet points in the detonation and deflagration branches of the
Hugoniot curve.
These properties can immediately be translated into properties for the propagation
velocity of the detonation. In fact, we have 2
tan α = p 2 − p 1
τ 1 − τ 2
. (4.12)
Taking this value into Eq. (4.8), the following expression for the propagation velocity
is obtained
v 1 = τ 1
√
tan α. (4.13)
Therefore, it results that the propagation velocity of a detonation is minimum
at point J, where the straight line P J is tangent to H. This property was observed
by Chapman in 1899 [2].
Point J is called the Chapman-Jouguet point and the
corresponding detonation the Chapman-Jouguet detonation. The Chapman-Jouguet
detonation is important since it is the one usually observed. Hence, one concludes
that of all the possible detonations, compatible with the given initial conditions, the
Chapman-Jouguet detonation is the one which propagates at a minimum velocity.
2 To account for dimensions a constant should be included. Hereinafter it is omitted for simplicity.

94 CHAPTER 4. COMBUSTION WAVES
The detonations corresponding to the upper part JC of H, for which the pressure
of the burnt gases is greater than the pressure of the Chapman-Jouguet detonation,
have been called by Courant and Friedrichs strong detonations and those corresponding
to the lower part AJ, weak detonations.
In the same manner it can be proven, Fig. 4.3(b), that in the deflagration branch
there exists a point J ′ in which H and P J ′ are tangent and the corresponding deflagration,
also called Chapman-Jouguet, propagates with a maximum velocity. Point J ′
divides the deflagration branch in two parts, one J ′ D of strong deflagrations and the
other BJ ′ of weak deflagrations.
4.3 Velocity of the burnt gases
It is interesting to determine the subsonic or supersonic character of the velocity v 2
of the burnt gases with respect to the wave front, since the stability of the detonation
wave depends on the character of this velocity, that is, the possibility for this wave to
propagate unchanged through the gas. In order to determine the character of v 2 it is
necessary to compare this velocity with the sound velocity a 2 in the burnt gases. Let
us see how this can be attained.
Gas Dynamics shows 3 that the sound velocity a in a gas is given by the expression
a = τ
√
−
( ) ∂p
, (4.14)
∂τ
S
where subscripts S means differentiation along the isentropic that passes through the
representative point of the gas state.
Let α S , Fig. 4.4, be the angle between the tangent to the isentropic that passes
through the point (p 2 , τ 2 ) of the detonation branch, and the negative direction of x
axis. From Eq. (4.14) we have
a 2 = τ 2
√ tan αS . (4.15)
On the other hand, the velocity v 2 of the burnt gases with respect to the flame front,
in virtue of Eq. (4.1), is v 2 = τ 2
τ 1
v 1 . Then, because of Eq. (4.13), this velocity can be
expressed as
v 2 = τ 2
√
tan α , (4.16)
where α is the previously defined angle. If α is larger, equal to or smaller than α s ,
then v 2 will be larger, equal to or smaller than a 2 ; and the flow of the burnt gases
3 See Courant-Friedrichs: Supersonic Flow and Shock Waves. Intersciences Pub. Inc., New York, 1948.

4.3. VELOCITY OF THE BURNT GASES 95
α S
α > : supersonic α = : sonic α < : subsonic
α S
α S
τ ,
α
α S
( τ , )
( τ , )
( τ , )
2 p 2
S
α
2 p 2
α
α S
S
2 p 2
αS
P( τ 1,
p 1 )
P( τ 1,
p 1 )
P( 1 p 1 )
S
Figure 4.4: Schematic diagram showing the line connecting the initial and final states and
the corresponding isentropic curve.
relative to the wave will be supersonic, sonic or subsonic. Hence, the problem reduces
to a comparison between the slope of the isentropic that passes through each point of
H, and that of the radius vector that joins this point with point P of the initial state.
The three possible cases appear schematically in Fig. 4.4.
To perform this comparison let us start by studying the variation of the entropy
s, along the Hugoniot curve. The elemental entropy variation ds corresponding to the
variations dh and dp of enthalpy h and pressure p, is
T ds = dh − τ dp. (4.17)
Now, by differentiating (4.10), keeping p 1 and τ 1 constant and taking the result into
(4.17), the following expression for the entropy variation along H, is obtained
( ∂s2
T 2 =
∂τ 2
)H
τ [ ( ]
1 − τ 2 p2 − p 1 ∂p2
+
2 τ 1 − τ 2 ∂τ 2
)H
(4.18)
= (τ 1 − τ 2 ) (tan α − tan α H )
,
2
where α H is the angle between the tangent to H at point p 2 , τ 2 , and the negative
direction of τ axis.
Fig. 4.3 shows that in the strong detonation branch α H
virtue of (4.18), in this branch
( ∂s2
∂τ 2
)H
> α. Therefore, in
< 0. (4.19)
On the contrary, in the weak detonation branch α H < α, that is
( ∂s2
> 0. (4.20)
∂τ 2
)H

96 CHAPTER 4. COMBUSTION WAVES
Finally, at the Chapman-Jouguet point α H = α, that is
( ∂s2
= 0. (4.21)
∂τ 2
)H
From these three inequalities it results that the entropy of the burnt gases is
minimum at the Chapman-Jouguet point and increases from there on, both in strong
and weak detonation branches.
Since at point J condition (4.21) is satisfied, the isentropic passing through this
point is tangent to the Hugoniot curve. Therefore, at point J the following conditions
are satisfied
α S = α H = α = α min . (4.22)
These values, when taken into Eqs. (4.15) and (4.16), give at point J
a 2 = v 2 , (4.23)
thus resulting the following important property: in a Chapman-Jouguet detonation
the velocity of the burnt gases with respect to the wave is sonic. This property is
the one that best characterizes a Chapman-Jouguet detonation. As will presently be
seen, owing to this property, the detonations physically observed are of this same type.
Jouguet stated this property in 1905 [3].
For the purpose of determining the character of v 2 in strong and weak detonations,
it is necessary, as previously seen in Fig. 4.3, to compare α and α s . This
means that it is necessary to determine the relative position of the isentropic that passes
through each point of H, with respect to the straight line that joins P to that point. For
the determination of this relative position it is not sufficient to know the entropy variation
along H, but it is also necessary to know the entropy variation along another
direction, for instance, that of the radius vector with its origin at point P . This can
easily be attained by considering the function
F (p 1 , τ 1 ; p 2 , τ 2 ) ≡ h 2 − h 1 − 1 2 (τ 1 + τ 2 ) (p 2 − p 1 ) . (4.24)
On the Hugoniot curve H the value of this function is zero, as results from (4.10).
This curve divides the plane in two regions: the lower region, that contains point P ,
representative of the initial state, and the upper region. F takes opposite signs in both
regions. As can be easily be verified, in the lower region F < 0 and in the upper
region F > 0. For this purpose, it is enough to study, for instance, the sign of F in P .
In B we have (Fig. 4.2)
F (p 1 , τ 1 ; p 1 , τ 2 ) = h 2 (p 1 , τ 2 ) − h 1 (p 1 , τ 1 ) = 0. (4.25)

4.3. VELOCITY OF THE BURNT GASES 97
Similarly, in P
F (p 1 , τ 1 ; p 1 , τ 1 ) = h 2 (p 1 , τ 1 ) − h 1 (p 1 , τ 1 ) . (4.26)
Now subtracting (4.25) from (4.26), results
F (p 1 , τ 1 ; p 1 , τ 1 ) = h 2 (p 1 , τ 1 ) − h 2 (p 1 , τ 2 ) . (4.27)
But, τ 1 is smaller than τ 2 , and, since to reduce the volume of the burnt gases from τ 2
to τ 1 without varying their pressure p 1 it is necessary to cool the burnt gases, that is to
reduce their enthalpy, then
h 2 (p 1 , τ 1 ) < h 1 (p 1 , τ 2 ) , (4.28)
that is
F (p 1 , τ 1 ; p 1 , τ 1 ) < 0. (4.29)
Therefore F is negative in the lower region and positive in the upper region.
By means of the function F , the entropy variation can be expressed in a simple
manner by differentiating Eq. (4.24), keeping p 1 and τ 1 constant and then combining
the result with Eq. (4.17), thus, obtaining
T 2 ds 2 = dF + 1 2 (p 2 − p 1 ) dτ 2 + 1 2 (τ 1 − τ 2 ) dp 2 . (4.30)
But the variations dp 2 and dτ 2 corresponding to the straight line that joins P with
(p 2 , τ 2 ) must satisfy condition
dp 2
dτ 2
= p 2 − p 1
τ 2 − τ 1
, (4.31)
this expression, when taken into Eq. (4.30), gives for entropy variation along a radius
vector
T 2 ds 2 = dF. (4.32)
Therefore, along a radius vector, F and s 2 increase and decrease in the same direction.
Now, at point E (Fig. 4.5), F increases from E to E ′ . Therefore, s 2 also
increases from E to E ′ . In the same manner, at point E ′ , F and s 2 increase in the
direction E ′ E.
Furthermore, s 2 decrease along H in the directions E ′ J and EJ. Thereby it is
deduced that the isentropic curves that pass by E and E ′ must, necessarily, have the
positions shown in Fig. 4.5. There results that:
a) In E, α S < α and the velocity of the burnt gases is supersonic.
b) In E ′ , α S > α and the velocity of the burnt gases is subsonic.

98 CHAPTER 4. COMBUSTION WAVES
p
C
E’
S 2
S 2
J
E
A
p
1
P
τ 0 1
Figure 4.5: Position of the isentropic curves in the detonation branch of the Hugoniot curve.
τ
τ
Hence the velocity of the burnt gases is supersonic in the weak detonations and subsonic
in the strong detonations.
Similarly, it can be demonstrated that the entropy of the burn gases in the deflagration
branch is maximum at J ′ . It can also be demonstrated that the velocity of
the unburnt gases is subsonic in the weak deflagrations, sonic in the Chapman-Jouguet
deflagration and supersonic in the strong deflagrations.
4.4 Propagation velocity
In order to complete the study, we shall analyze the character of the propagation velocity
v 1 , by comparing its values with the value of the velocity a 1 of the sound propagation
in the state (p 1 , τ 1 ). The study can be made in an identical manner to that
previously used for v 2 , that is by considering all the states (p 1 , τ 1 ) of the unburnt gases
that are compatible with a given state (p 2 , τ 2 ) of the burnt gases. All these states lie
on a Hugoniot curve H ′ which is obtained by fixing in Eq. (4.10) the values of p 2 and
τ 2 and varying p 1 and τ 1 . This curve, like curve H, has two branches (Fig. 4.6): a
lower detonation branch and an upper deflagration branch. It can easily be proved that
in the former, the entropy of the unburnt gases decreases starting from A, and in the
latter increases starting from B. Furthermore by studying the variation of function F ,
which is defined by (4.24) fixing the values of p 2 and τ 2 and varying p 1 and τ 1 , we
obtain that H ′ divides the plane (p, τ) in two regions. The upper region containing

4.4. PROPAGATION VELOCITY 99
p H’
S 2
B ( τ 2
, p 2
)
p 2
S 2
A
H’
Figure 4.6: Position of the isentropic curves in the Hugoniot curve of the admissible initial
states.
τ 2
τ
point (p 2 , τ 2 ), in which F < 0, and the lower region, in which F > 0. Finally, it
is also verified that along the radius vectors that pass through the point (p 2 , τ 2 ) the
variations of F and s 1 are opposite. Now, by combining these properties as previously
done for v 2 , we can determine the relative position of the isentropic that passes
through each point H ′ with respect to the straight line that joins this point to the point
(p 2 , τ 2 ). It results that this position is the one indicated in Fig. 4.6. Therefore, in the
detonation branch α < α S and in the deflagration branch α > α S . Therefore, one
concludes that the propagation velocity of a detonation is always supersonic, and the
propagation velocity of a deflagration is always subsonic.
As a result of the preceding study, the Table 4.1 can be constructed. This
table contains the results of Jouguet’s studies and defines the characteristics of the
combustion waves.
Detonations
Weak Chapman-Jouguet Strong
Velocity before
Supersonic
Velocity after Supersonic Sonic Subsonic
Deflagrations
Weak Chapman-Jouguet Strong
Velocity before
Subsonic
Velocity after Subsonic Sonic Supersonic
Table 4.1: Characteristics of the combustion waves.

100 CHAPTER 4. COMBUSTION WAVES
4.5 Applications
As an application of the previous study we shall consider the propagation of a combustion
wave, assuming that both the unburnt and burnt gases are perfect gases and
that the chemical composition of the burnt gases is independent from their state. In
this case, the enthalpy of the burnt gases can be expressed in the form
∫ T2
h 2 = h 21 + c p2 dT, (4.33)
T 1
where h 21 is their enthalpy at the temperature T 1 . Moreover, it is assumed that the
heat capacity c p2 is independent from the temperature in the interval (T 1 , T 2 ). Then
Eq. (4.33) reduces to
h 2 = h 21 + c p2 (T 2 − T 1 ) . (4.34)
Now, by taking this expression into the Hugoniot equation (4.10), we have
c p2 T 2 = (h 11 − h 21 ) + c p2 T 1 + 1 2 (τ 1 + τ 2 ) (p 2 − p 1 ) , (4.35)
where h 11 − h 21 is the difference between the formation enthalpies of the burnt and
unburnt gases at the temperature T 1 of the unburnt gases. Let Q be this value. Then
Eq. (4.35) can be written in the form
c p2 T 2 = Q + c p2 T 1 + 1 2 (τ 1 + τ 2 ) (p 2 − p 1 ) . (4.36)
Let
p 2 τ 2 = R 2 T 2 (4.37)
be the state equation of the burn gases, and γ 2 = c p2 /c v2 the ratio of their capacities.
Furthermore, let x = τ 2 /τ 1 and y = p 2 /p 1 the specific volume and pressure of the
burnt gases, referred to the corresponding values for the unburnt gases, and
q = 2
( ) ( )
γ2 − 1 Q
+ 2γ 2ν
γ 2 + 1 p 1 τ 1 γ 2 + 1 − γ 2 − 1
γ 2 + 1 , (4.38)
where ν = τ ′ 1/τ 1 and τ ′ 1 is the specific volume that would correspond to the burnt
gases at the pressure p 1 and temperature T 1 of the unburnt gases, that is,
ν = R 2
R 1
= M u
M b
, (4.39)
where M u and M b are the average mole masses for the unburnt gases and for the
combustion products respectively.

4.5. APPLICATIONS 101
By substituting these values and that of the temperature given by Eq. (4.37)
into Eq. (4.36), the following expression, for the Hugoniot curve, is obtained
xy = q + γ 2 − 1
(y − x) . (4.40)
γ 2 + 1
Therefore, the Hugoniot curve is an equilateral hyperbola, with an asymptote parallel
to the axis y for x 0 = (γ 2 − 1) / (γ 2 + 1). This curve cuts axis x at the point x 1 =
q (γ 2 + 1) / (γ 2 − 1).
Let us see how the Chapman-Jouguet points are determined. The equation of
an isentropic for the burnt gases is
yx γ2 = const. (4.41)
Therefore, the Chapman-Jouguet points are obtained by solving the system formed
by Eq. (4.40) and by the equations that express the tangency condition of the curves
Eq. (4.40) and Eq. (4.41), namely,
(γ 2 + 1) y + (γ 2 − 1)
(γ 2 + 1) x − (γ 2 − 1) = γ y
2
x . (4.42)
The solution of Eq. (4.40) and Eq. (4.42) gives for x and y the following equations
y 2 − ( (γ 2 + 1)q − (γ 2 − 1) ) y + q = 0, (4.43)
x 2 − 1 γ 2
(
(γ2 + 1)q + (γ 2 − 1) ) x + q = 0. (4.44)
When solving these equations the root x 1 < 1 must be assign to the root y 1 > 1 and
root x 2 > 1 to the root y 2 < 1.
In virtue of Eq. (4.8), the propagation velocity is
v 1
= 1
√
y − 1
√
a 1 γ1 1 − x , (4.45)
where a 1 is the sound velocity of the unburnt gases, and γ 1 the ratio of their heat
capacities.
Similarly, the velocity of the burnt gases with respect to the wave front is
v 2
=
x
√
y − 1
√
a 1 γ1 1 − x . (4.46)
The corresponding Mach number M 2 is
√
M 2 = √ 1 ( )
x y − 1
. (4.47)
γ2 y 1 − x

102 CHAPTER 4. COMBUSTION WAVES
By subtracting from the left hand side of Eq. (4.42) the right hand side multiplied
by two, we obtain
y
γ 2
x = y − 1
1 − x , (4.48)
which, together with Eq. (4.47), show that at the Chapman-Jouguet points, the velocity
of the burnt gases with respect to the combustion point is equal to the sound velocity.
For example, by taking the typical values p 1 = 1 atm, τ 1 = 1000 cm 3 /gr,
T 1 = 300 K, γ 1 = γ 2 = 1.4, Q = 1000 cal gr −1 , c p2 = 0.46 cal gr −1 K −1 and ν = 1
the following is obtained for the Hugoniot curve
xy = 15.23 + 1 (y − x) . (4.49)
6
This curve has been represented in Fig. 4.7. The branch of detonations (Fig. 4.7(a))
corresponds to the values of x in the interval
1
6 ≤ x ≤ 1 . (4.50)
The branch of deflagrations (Fig. 4.7(b)) corresponds to the interval
13.2 ≤ x ≤ 91.4 . (4.51)
The Chapman-Jouguet points for detonations and deflagrations are, respectively,
Detonations: x 1 = 0.58, y 1 = 35.7 .
Deflagrations: x 2 = 24.82, y 2 = 0.45 .
The corresponding propagation velocities are
Chapman-Jouguet detonation:
Chapman-Jouguet deflagration:
v 1
= 7.68 .
a 1
v 1
= 0.126 .
a 1
4.6 Remarks
If the variation of the chemical composition along the Hugoniot curve is taken into
account, that is the influence of the dissociation of the combustion products on the
characteristics of the wave, the problem is far more complicated. To attain a solution
one must resort to the equations of thermodynamic equilibrium, which must be combined
with the state equations and the invariants across the wave. Then, the Hugoniot
curve must be drawn point by point. For additional information see Ref. [4].

4.7. INDETERMINACY OF THE SOLUTION 103
y
125
100
x =1/6 0
(a) Detonation branch
75
50
J
25
0
P A
0.0 0.2 0.4 0.6 0.8 1.0
y
1.0
0.8
P
B
(b) Deflagration branch
x
0.6
0.4
J’
0.2
0.0
0
20
40
60
80
100
x
Figure 4.7: Hugoniot curve (x = τ/τ 1, y = p/p 1) corresponding to p 1 = 1 atm, τ 1 =
1000 cm 3 /gr, T 1 = 300 K, γ 1 = γ 2 = 1.4, Q = 1000 cal gr −1 , c p2 = 0.46
cal gr −1 K −1 , ν = 1.
4.7 Indeterminacy of the solution
The study performed in this chapter allows the establishment of the different types of
combustion waves that are compatible with the principles of the conservation of mass,
momentum and energy. This has been attained by comparison between the initial and
final state of the gases. In order to obtain this result it has not been necessary to take
into account the intermediate states undergone by the gas, within the wave. Thus, the
problem is essentially simplified. However, if the influence of the intermediate states
is neglected the result obtained is incomplete, since the propagation velocity of the
combustion wave remains undetermined, whilst experience shows that it takes a well
defined value for each different case. In order to eliminate the said indeterminacy and
select among the combustion waves obtained herein those that will actually occur, it
is necessary to analyze the internal structure of the wave and see the mechanism that

104 CHAPTER 4. COMBUSTION WAVES
propagates the reaction to the unburnt gases. This study will be the subject of the
following chapter. Therein, it will be seen that:
1) The weak detonations and strong deflagrations are impossible.
2) The detonations physically observed must be of the Chapman-Jouguet type.
3) Deflagrations propagate with a well defined velocity.
References
[1] Crussard, L.: Ondes de choc et onde explosive. Bulletin de la Société de
l’Industrie Minérale, Vol. VI, 1907.
[2] Chapman, D. L.: On the rate of explosion in gases. Phylosophical Magazine,
Jan. 1899.
[3] Jouguet, E.: Sur la propagation des réactions chimiques dans les gaz. Journal
de Mathématiques Pures et Appliquées, 1905-6.
[4] Taylor, J.: Detonation in Condensed Explosives. Oxford University Press, 1952.

Chapter 5
Structure of the combustion
waves
5.1 Introduction
The different types of combustion waves compatible with the principles of the conservation
of mass, momentum and energy throughout the wave have been previously
established in chapter 4. Table 4.1 of that chapter summarizes the results of the said
analysis. These results where attained simply by comparing the initial and final state
of the gases. The conclusions are correct provided that the wave thickness is small
enough, so that its geometrical shape has no influence upon the transformations that
occur within the wave.
Owing to its nature, the aforementioned analysis is necessary incomplete, due
to the fact that the possible influence of the intermediate states is ignored. Therefore,
it is not to be expected that all the types of combustion waves compatible with the said
principles of conservation will actually occur. On the contrary, when analyzing the
internal mechanism of the propagation of the process, the existence of new limitations
is to be expected. It should also be expected that such limitations will enable us to
eliminate the indeterminacy, mentioned at the end of the preceding chapter, and to
select from all the waves compatible with the laws of conservation the one which will
actually occur in each case.
To make this point clear, it is necessary to take into consideration the internal
structure of the wave. This is done in the present chapter, following a similar procedure
to the one used in Gas Dynamics [1] when studying the internal structure of
shock waves.
105

106 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
For simplicity, the study herein will be restricted to the case of a perfect gas
with heat capacity and number of moles constant throughout the wave. The limiting
solutions, obtained by assuming that a characteristic time of the thermodynamic
transformations is very small compared to a characteristic time of the chemical transformations
(as explained in §3), will be carefully analyzed. This plausible assumption
enables an essential simplification of the equations, favouring the study of the properties
of the corresponding solutions. This analysis will follow, mainly, the line of von
Kármán’s reasoning [2]. More detailed studies, for example those of Friedrichs [3] and
Hirschfelder [4], which take into account the influence of the terms neglected herein,
show that the discrepancy between both cases is very small. Therefore, the method
followed in this study is well justified. Furthermore, these studies allow one to derive
conclusions for the cases in which, due to the existence of ”abnormal” reaction rates,
the simplified treatment developed herein is not applicable.
In the following analysis special attention is given to the study of detonations
(see §5), since deflagrations are the subject of a detailed analysis in the following
chapter.
5.2 Wave equations
Let us consider an indefinite plane wave which propagates in undisturbed uniform
combustible mixture. We shall adopt a reference system fixed to the wave were the x
axis is parallel to the propagation direction and positive towards the burnt gases. With
respect to this reference system, the process is stationary and x is the only independent
variable. The values of x vary from x = −∞ for the unburnt gases to x = +∞ for
the burnt gases.
For simplification it will be assumed that the following considerations are satisfied
throughout the wave:
1) The mixture behaves as a perfect gas.
2) The heat capacity at constant pressure c p is independent from the temperature
and composition of the mixture.
3) The ratio γ of the heat capacities is independent from the composition of the
mixture.
4) The chemical composition of the mixture at each point of the wave is determined
by only one chemical parameter. We shall adopt as chemical parameter the degree
of advancement of the combustion, measured by the mass fraction ε(x) of
the gas that has burnt when the point x of the wave is reached. Due to the action

5.2. WAVE EQUATIONS 107
of diffusion, this degree of advancement of the combustion differs from the mass
fraction Y (x) of the burnt gases that defines the mixture composition at point x,
that is, the fraction Y (x) of burnt gases that would be obtained by analyzing the
composition of a gas sample taken from the said point.
The simplifying assumptions previously stated do not limit the extent of the
study that follows, which is of a qualitative nature. On the other hand, these assumptions
simplify calculations.
The wave equations are obtained by particularizing the general equations of
continuity, momentum and energy 1 to the case of a one-dimensional stationary motion
(∂/∂y = ∂/∂z = ∂/∂t = 0). When this is done, and in addition to the preceding
assumptions are taken into account, the following system of equations is obtained:
a) Continuity equation.
that is
d (ρv)
dx
ρv = m,
= 0, (5.1)
(5.1.a)
where m is the mass flow normal to the wave, per unit surface.
b) Continuity equation for the burnt gases. Since the mass flow per unit surface is
ρv and the burnt fraction is ε, the mass flow of the burnt gases at section x is
ρvε. Its variation with x is due to chemical reactions. The equation that gives
this variation is
d (ρvε)
= w, (5.2)
dx
where w is the reaction rate. In virtue of (5.1.a), equation (5.2) can also be
written in the form
c) Momentum equation.
m dε
dx = w.
ρv dv
dx = − dp
dx + 4 3
(5.2.a)
(
d
µ dv )
. (5.3)
dx dx
This equation can be immediately integrated by taking into account (5.1.a), giving
p + mv − 4 3 µ dv
dx = i,
(5.3.a)
where i is an integration constant that must be determined by the boundary conditions.
2
1 See chapter 3.
2 In equations (5.3) and (5.4) it has been assumed that the second viscosity coefficient of the mixture
(see chapter 2) is zero. When not, it is sufficient to substitute µ for `µ + 3 4 µ′´ in (5.3) and (5.4).

108 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
d) Energy equation.
ρv d ( 1
dx 2 v2 + c p T − qε)
= d (
λ dT )
+ 4 dx dx 3
(
d
µv dv )
, (5.4)
dx dx
which can be immediately integrated, giving
( )
1
m
2 v2 + c p T − qε − λ dT
dx − 4 dv
µv
3 dx = me,
(5.4.a)
where e is an integration constant which, like i, must be determined by the
boundary conditions.
e) Diffusion equations.
which can be integrated and gives
(
d
ρD dY ) ( dY
= ρv
dx dx dx − dε )
, (5.5)
dx
ρD
m
dY
dx − Y + ε = f,
(5.5.a)
where f is an integration constant.
The previous system must be completed with the state equation. It can easily be
proven that assumptions 2) and 3) imply that the average mole masses of the unburnt
and burnt gases are equal. Therefore, the particular constant R m = R/M m of the gas
is independent from its composition, and the state equation valid throughout the wave
is
where R m is independent from Y .
p
ρ = R mT, (5.6)
Equations (5.1.a), (5.2.a), (5.3.a), (5.4.a), (5.5.a) and (5.6) form a system of
six equations for the six unknowns p, ρ, T , v, ε and Y . With the adequate boundary
conditions, this system determines the possible solutions of the problem.
Let p 0 , ρ 0 , T 0 and v 0 , be the values for p, ρ, T and v in the unburnt gases, and
p f , ρ f , T f and v f the corresponding values for the burnt gases. Furthermore, since in
the unburnt gases there are no combustion products, in them Y = ε = 0. Similarly,
in the burn gases, Y = ε = 1. Therefore, the looked-for solutions must satisfy the
following boundary conditions:
1) Unburnt gases,
x → −∞ : p → p 0 , ρ → ρ 0 , T → T 0 , v → v 0 , Y → 0, ε → 0. (5.7)

5.2. WAVE EQUATIONS 109
2) Burnt gases,
x → +∞ : p → p f , ρ → ρ f , T → T f , v → v f , Y → 1, ε → 1. (5.8)
Such conditions imply, naturally, the following ones,
x → ±∞ :
dp
dx → 0, dρ
dx → 0, dT
dx → 0, dY
dx → 0,
dε
→ 0. (5.9)
dx
Herein, the possibility of satisfying the previous boundary conditions, will not
be discussed since it is subjected to a careful study in the chapter dedicated to deflagrations.
Therein, it will be seen that the boundary conditions relative to the cold
boundary of the flame give rise to a problem which, so far, has not been satisfactorily
solved.
In the present study it will be assumed that the boundary conditions can be
satisfied and, therefore, that the problem is completely determined.
By taking the boundary conditions (5.7), (5.8) and (5.9) into the system of
equations (5.1.a), (5.4.a)) and (5.4.a), the following laws of conservation are obtained,
which agree with the invariants deduced in the preceding chapter
ρ 0 v 0 = ρ f v f , (5.10)
p 0 + ρ 0 v 2 0 = p f + ρ f v 2 f , (5.11)
1
2 v2 0 + c p T 0 + q = 1 2 v2 f + c p T f . (5.12)
The boundary conditions relative, for instance, to the burnt gases, make possible
the elimination of constants i, e and f from equations (5.3.a), (5.4.a) and (5.5.a),
thus obtaining
p + mv − 4 3 µ dv
dx = p f + mv f , (5.13)
1
)
m(
2 v2 + c p T − qε − λ dT
dx − 4 dv
( 1
)
µv
3 dx = m 2 v2 f + c p T f − q , (5.14)
ρD dY
− Y + ε = 0. (5.15)
m dx
The discussion and analysis of the possible solutions of the previous system are
rather complicated. An example of a discussion for a similar system (but in which the
diffusion neglected) can be found in Friedrichs’s work [3]. As a result of his analysis,
Friedrichs concludes that except in the case where the reaction rate w is exceptionally
high (and takes a well defined value) weak detonations are impossible. Consequently,
the only possible detonations with a normal reaction rate are strong detonations and

110 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
Chapman-Jouguet detonations. As will presently be seen, the type of detonation that
occurs in each particular case depends on the boundary conditions on the burnt side of
the detonation wave. As for the deflagrations, Friedrichs obtained the conclusion that
strong deflagrations are impossible and that weak deflagrations are possible only for a
well defined value of the propagation velocity. This value depends on the state of the
unburnt gases, on the reaction rate and on the transport coefficients of the mixture.
The discussion of the differential system of combustion waves is considerably
simplified by analyzing, as done by von Kármán [2], the limiting system deduced from
the previous one by assuming that a characteristic time of the thermodynamic transformations,
for example the average time between two molecular collisions, is very small
compared to a characteristic time of the chemical transformations, for example the average
time between two collisions of molecules of different species under the required
circumstances for these molecules to react to one another. This assumption appears
justified if one considers that, in a mixture of reactants, of all the molecular collisions
only a few are accompanied by chemical transformations. In fact, for this to happen
a number of favourable circumstances must concur: for example, the molecules must
be of the adequate species, the orientation of the collision and the energy in certain
degrees of freedom must be the proper ones, etc. 3
In the following, the solutions of the aforementioned differential system will
be discussed mainly from this stand-point, following von Kármán’s reasoning.
5.3 Characteristic times
As a characteristic τ t of the thermodynamic transformations the following ratio is
adopted
τ t = µ p . (5.16)
This ratio is proportional to the average time between molecular collisions, which is
given by ratio l/¯v of the mean free path l of the molecules to the average velocity ¯v
of the molecular motion. In fact, the Kinetic Theory of Gases shows that µ is of the
form
µ ∼ ρ¯vl, (5.17)
and that ¯v is of the form
√ p
¯v ∼
ρ . (5.18)
3 See chapter 1.

5.4. LIMITING FORM OF THE WAVE EQUATIONS 111
Now, by substituting the expressions of ¯v and l deduced from (5.17) and (5.18) into
the ratio l/¯v, it results that this ratio is proportional to τ t as defined by (5.16).
As a characteristic time τ c of the chemical transformations, the following ratio
is adopted
τ c = ρ w , (5.19)
that measures the average time between those molecular collisions successful in producing
chemical reaction. 4
Let
α = τ t
= µw
τ c ρp
(5.20)
be the ratio of the thermodynamic time τ t to the chemical time τ c . The assumption
that chemical transformations are very slow when compared to the thermodynamic
transformations is expressed by the condition that parameter α is much smaller than
unity
α ≪ 1. (5.21)
The limiting form of the wave equations, obtained when taking assumption (5.21) into
the system deduced in the preceding paragraph, is given in the paragraph that follows.
5.4 Limiting form of the wave equations
In order to discuss the relative values of the various terms of the differential system
of the wave, we shall start by showing the influence of α and the flow Mach number
M = v/a = v/ √ γp/ρ.
For this purpose, let us first eliminate x from (5.3.a), (5.4.a) and (5.5.a), making
use of equation (5.2.a), then
p + mv − 4 µw dv
= i,
3 m dε
(5.22)
1
2 v2 + c p T − qε − λw dT
m 2 dε − 4 µw
3 m 2 v dv = e,
dε
(5.23)
ρDw
m 2
dY
dε
− Y + ε = 0. (5.24)
4 See chapter 1.

112 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
Now, the system of equations (5.22), (5.23) and (5.24) can be written in the
following way, which evidences the influence of α and M,
(
p 1 + γM 2 − 4 3 α 1 )
dv
= i, (5.25)
v dε
(
c p T 1 + γ − 1 M 2 −
q
2 c p T ε − 1 α 1 dT
γP r M 2 T dε − 4 γ − 1
α 1 )
dv
= e, (5.26)
3 γ v dε
1 α dY
γS c M 2 − Y + ε = 0, (5.27)
dε
where P r = µc p /λ and S c = µ/ (ρD) are the Prandtl and Schmidt numbers, respectively,
for the mixture.
Since α ≪ 1, the last term of the left hand side of equation (5.25) can be
neglected when compared to unity, thus obtaining, instead of equation (5.25)
p ( 1 + γM 2) = i. (5.28)
Similarly, in equation (5.26) one can neglect the last term of the left hand side. Since
the Prandtl P r and the Schmidt S c numbers are of order unity, 5 the order of magnitude
of terms 1 α 1 dT
P r M 2 T dε of equation (5.26), and 1 α dY
S c M 2 of equation (5.27) depends
dε
on the values of the ratio α/M 2 . Here, there are two possible cases:
a) If the Mach number M of the flow is of the order of magnitude of unity or larger,
then α/M 2 ≪ 1, and the said terms can also be neglected.
b) Whereas if α/M 2 are of the order one, then said terms must be preserved. In
this case, however, M 2 ≪ 1, and γM 2 can be neglected in equation (5.28)
and γ − 1 M 2 can be neglected in equation (5.26). Therefore the two following
2
cases are obtained:
1) α ≪ 1, M 2 ∼ 1.
Will be designated case A. For this case, Eq. (5.28) is valid and will be
written in the form
p + ρv 2 = i.
(5.28.a)
In Eq. (5.26) the fourth and fifth terms of the left hand side disappear,
obtaining the following simplified equation
(
c p T 1 + γ − 1 M 2 −
q )
2 c p T ε = e, (5.29)
that can be written
c p T + 1 2 v2 − qε = e.
(5.29.a)
5 See Hirschfelder, Curtiss & Bird: Molecular Theory of Gases and Liquids, John Wiley & Sons Inc.,
New York, 1954, p. 16.

5.4. LIMITING FORM OF THE WAVE EQUATIONS 113
Equation (5.27) reduces to
Y = ε. (5.30)
Therefore in this case, the effects of viscosity, thermal conductivity and
diffusion can be neglected, and the mixture can be considered as an ideal
gas whose composition varies due to chemical reactions.
2) α ≪ 1, α/M 2 ∼ 1.
Will be designated case B. As aforesaid, in this case M 2
(5.28) is reduced to
≪ 1, and Eq
p = i, (5.31)
that is to say, the combustion occurs in first approximation at constant pressure.
In Eq. (5.26) the second and last terms of the left hand side can be
neglected. Then, the following simplified equation is obtained
(
c p T 1 − q
c p T ε − 1 )
α 1 dT
P r M 2 = e, (5.32)
T dε
which can be written in the following form, returning to the old variable x
by means of equation (5.2.a)
c p T − qε − λ m
dT
dx = e.
(5.32.a)
All the terms of equation (5.27) must be preserved. By returning to variable
x, equation (5.15) is obtained.
Of the two limiting cases determined herein, the former is represented by the
system of equations (5.1.a), (5.2.a), (5.28.a) and (5.30), that is, by
Case A: α ≪ 1, M 2 ∼ 1 .
ρv = m,
m dε
dx = w,
p + ρv 2 = i,
c p T + 1 2 v2 − qε = e,
(5.1.a)
(5.2.a)
(5.28.a)
(5.29.a)
Y = ε. (5.30)
As aforesaid, in this system, the influence of viscosity, thermal conductivity
and diffusion has disappeared. As will presently be seen, the solutions corresponding
to this system represent detonation waves.

114 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
The second case is represented by the system of equations (5.1a), (5.2a), (5.31)
and (5.32a), that is, by
Case B: α ≪ 1, α/M 2 ∼ 1 .
ρD
m
dY
dx
c p T − qε − λ m
ρv = m,
m dε
dx = w,
(5.1.a)
(5.2.a)
− Y + ε = 0, (5.15)
p = i, (5.31)
dT
dx = e,
(5.32.a)
in which the influence of the kinetic energy and viscosity has disappeared. This system
represents a combustion at constant pressure, as limiting case of the weak deflagrations
that are physically observed.
In the following paragraphs both cases will be studied separately.
5.5 Detonations
Let us first discuss the solutions represented by system A).
The three equations (5.1.a), (5.28.a) and (5.29.a), together with the state equation
(5.6), allows one to express the variation laws of p, T and v as function of the
degree of advancement ε of the combustion, or of the mass fraction Y = ε of the
burnt gases in the mixture. This is the problem of heat addition in the ideal gas in
one-dimensional and stationary motion. 6 In particular, the following equation for v is
obtained
v 2 γ i − 1)
− 2 v + 2(γ (e + qε) = 0, (5.33)
(γ + 1) m γ + 1
which shows that to each value of ε satisfying the condition
(
0 ≤ ε ≤ 1 γ 2 ( ) 2 i
q 2(γ 2 − e)
, (5.34)
− 1) m
corresponds two different real values of v, given by the expression
( √
)
γ i
v =
1 ± 1 − 2 (γ2 − 1)
( m
) 2
(e + qε)
(γ + 1) m
γ 2
. (5.35)
i
6 See chapter 3, paragraph 9.

5.5. DETONATIONS 115
Of these two velocities, one is subsonic and the other supersonic. In fact, the
critical velocity v cr , corresponding to the point defined by the value ε of the degree of
advancement of the combustion, is given, in virtue of (5.29a), by
v 2 cr =
2 (γ − 1)
γ + 1
(e + qε) . (5.36)
This value, however, is exactly the product of roots v 1 and v 2 of equation (5.33).
Consequently
v 1 v 2 = v 2 cr, (5.37)
that is, of both velocities one is subcritical, that is to say subsonic, and the other supercritical,
that is to say, supersonic. Furthermore, the two values v 1 and v 2 correspond to
the velocities before and after a normal shock wave, since (5.37) is the Prandtl relation
for shock waves. 7
A
F
v
C
E
B
ε
D
Figure 5.1: Schematic diagram showing the two possible velocities resulting from the heat
addition.
Figure 5.1 represents the pair of values corresponding to Eq. (5.33) for variable
ε. Their corresponding curve is a parabola. The upper branch of this parabola, AC,
corresponds to the supersonic velocities, and the lower branch, BC, corresponds to the
subsonic velocities. At point C, where both branches join, the velocity of the gases
with respect to the wave is equal to the sound velocity.
7 See R. Courant and K. O. Friedrichs: Supersonic Flow and Shock Waves. Interscience Pub. Inc., New
York, 1948, p. 147.

116 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
Let A and B be the two representative points of the two possible initial states
(ε = 0), compatible with the assumed values for m, i and e. Velocity is supersonic
in A and subsonic in B. The jump from A to B occurs through a shock wave, as
aforesaid. Since, as previously seen in the preceding chapter, the propagation velocity
of a detonation is supersonic, the representative point for the initial state of the mixture
is A. When ε increases, that is, when the reaction occurs, two alternatives are possible
either the representative point of the intermediate states of the mixture throughout the
wave moves along the upper branch AC (see Fig. 5.1), or else a shock wave produces
first which changes the velocity from A to B and then, as the combustion progresses,
the representative point moves along the lower branch BC. A decision between both
alternatives can not be taken from purely hydrodynamic considerations. Let us analyze
both cases separately.
Suppose that the point moves on the upper branch, starting from A. This means
that the combustion initiates in the state of the unburnt gases. But in this state, the
temperature is small and the reaction velocity cannot be sufficiently large to burn
the gases with the speed required by the detonation wave. Therefore, the detonation
must be initiated by a shock wave which, by making the gases pass from the state
represented by point A to the one represented by point B, compresses and heats the
gases, taking them to a state in which the reaction velocity can be sufficiently large to
burn them with the required speed.
The ratio of the thickness of the shock wave to the thickness of the detonation
wave is measured by the ratio of the thermodynamic characteristic time τ t to the chemical
characteristic time τ c . Therefore, the thickness of the shock wave is very small
with respect to the thickness of the detonation wave. 8 This means that while the gases
pass through the shock wave, the burnt fraction is insignificant. Thus the detonation
wave appears as formed by the shock wave, followed by a combustion wave.
The need of a shock wave to initiate the chemical reaction in the detonation
wave has always been acknowledged. This has been the stand-point, for instance, for
Vieille [5] and Jouguet [6] in 1900. 9 These authors, however, have consider the detonation
always as a discontinuity, in which not only the shock wave is instantaneously
produced, but also the subsequent reaction. The ideas developed in the present study,
concerning the structure of the wave, belong to the modern theories on detonation
8 The study of the dynamic transformations within the shock wave cannot be performed with the system
obtained by neglecting the action of viscosity and thermal conductivity, whose action is essential within the
wave. See for example M. Roy: Structure de l’onde de choc et des flammes deflagrantes. ONERA, Paris,
1952.
9 In the fundamental work of Jouguet [6] data can be found concerning the historical evolution of the
ideas relative to the classical theory of the detonation waves.

5.5. DETONATIONS 117
waves. Such theories were initiated by the works of Taylor in England, von Neumann
[7] in the U.S.A., Zeldovich [8] in the U.R.S.S. and Döring [9] in Germany.
Within the combustion zone that follows the initial shock wave, the gases move
along the lower branch of the curve shown in Fig. 5.1, starting from point B. The point
ε = 1, at which the combustion ends and the thermodynamic equilibrium is reached
after the wave, cannot lie at the right hand side of D. Therefore, the three following
cases can occur:
1) The combustion ends at a point such as E of the lower branch, in which the
velocity is subsonic. This case represents a strong detonation.
2) The combustion ends at the point C that separates the subsonic from the supersonic
branch. In C the velocity of the burnt gases with respect to the detonation
wave is sonic. The corresponding detonation for this case is of the Chapman-
Jouguet type.
3) The representative point of the final state is point F in the supersonic branch.
Consequently, the final velocity is supersonic and the corresponding detonation
is weak.
We shall presently see that the last case cannot possibly occur. In fact, in order
to reach point F either one has to pass by point C, in which case the reaction between
C and F would be endothermic and therefore in contradiction with what happen in the
combustion, or else one must pass from B to D and from D to F by means of a jump
or expansion shock which is also impossible. Consequently, weak detonations are not
possible.
On the other hand, nothing opposes a strong detonation or a detonation of the
Chapman-Jouguet type. For a given case the occurrence of one or the other depends
on the boundary conditions after the wave. Let us see the influence of these conditions.
When the detonation is strong, the velocity of the burnt gases, with respect to
the wave is subsonic. Any perturbation produced in the burnt gases will propagate
throughout them with sound velocity and can therefore reach the wave and alter its
nature. For example, an expansion of the burnt gases will reach the wave, reducing
its strength down to the point where the velocity of the burnt gases with respect to the
wave equals the sound velocity. In such a case the wave becomes insensible to the
perturbations that occur in the burnt gases. Such is the situation that normally occurs
in practice; for example, when the detonation propagates along a tube, starting either
at an open or closed end. Thus the detonations normally observed in practice are of the
Chapman-Jouguet type, due to the fact that these are the only stable ones with respect
to perturbations coming from the burnt gases.

118 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
The study of the aerodynamic field subsequent to a detonation wave can be performed
by applying the Gas Dynamic methods. 10 For instance, Zeldovich [10] has calculated
the aerodynamic field following a detonation wave that propagates unchanged
from the closed end of a tube, initially full of detonant gas at rest. By neglecting the
friction and the heat lost through the walls of the tube, a solution is obtained which
propagates with the Chapman-Jouguet velocity. The detonation wave is followed by
an expansion region with an approximate length of 50% of the total length covered
by the detonation wave. The remaining burnt gases (approximately 40% of the burnt
mass) are at rest. By including the effect of friction and heat losses through the walls
of the tube, a stable Chapman-Jouguet wave is obtained, followed by an expansion
in which the direction of the motion of the burnt gases in the tube is reversed. Following
the expansion there also exists in this case a region at rest in which the burnt
gases have cooled down to ambient temperature. There is photographic evidence of
the existence of this reversal motion in the expansion region.
The aerodynamic field that follows a spherical detonation wave has been studied
by G.J. Taylor [11] in England, and by Zeldovich in U.S.S.R. [12]. The calculations
also demonstrate the existence of a solution which propagates unchanged with
the Chapman-Jouguet velocity. Immediately after the detonation wave there is a strong
expansion region, followed by a central nucleus at rest. Within the expansion region
more than 90% of the burnt mass is in motion. The spherical detonation waves have
been experimentally observed, for example, by Manson and Ferrié [13]. A strong detonation
can occur, for example, when the detonation propagates inside a tube, starting
from an end closed by a piston, that moves after the wave with a subsonic velocity
with respect to the wave. In such a case the intensity of the strong detonation would
be determined by the compatibility condition obtained by expressing that the velocity
of the burnt gases with respect to the wave must equal the velocity of the piston.
Then, when friction and heat losses through the walls of the tube are neglected, it
results that the strong detonation wave propagates unchanged throughout the mass of
unburnt gases.
The relation between pressure and density within the wave can be obtained
from Eqs. (5.1.a) and (5.28.a) by elimination of v. Thus resulting
p + m2
ρ
= i, (5.38)
or else, introducing the state (p 1 , ρ 1 ) of the unburnt gases
( 1
p − p 1 = m 2 − 1 )
. (5.39)
ρ 1 ρ
10 See Courant and Friedrichs, ib., pp. 218 and 416 for plane and spherical waves, respectively.

5.5. DETONATIONS 119
This relation shows that the pressure at each point of the wave is determined
only by the corresponding value of the density. This relation plays here the same part
as, for example, in Gas Dynamics the relation p = Cρ γ for the isentropic motions of
non-reacting gases.
In order to represent the result in the diagram (p, τ), as done in chapter 3, the
specific volume τ = 1/ρ must be introduced in place of the density in Eq. (5.39),
which then takes the form
p − p 1 = m 2 (τ 1 − τ). (5.40)
This equation is represented in diagram (p, τ), see Fig. 5.2, by a straight line joining
the representative points of the initial and final states. On this line also lie all the
representative points of the intermediate states corresponding to the reaction zone. 11
70
60
D
p/p 1
E
50
C
Trayectory through shock wave
40
30
ε=1.0
J
20
E‘
ε=0.3 ε=0.7
10
ε=0
1
P
0.0 0.2 0.4 0.6 0.8 1.0
τ /τ 1
Figure 5.2: Hugoniot curves from different values of ε.
Bringing forth in Eqs. (5.1.a), (5.28.a) and (5.29.a) the specific volume and the
initial conditions corresponding to the unburnt gases, there results
v
τ = v 1
τ 1
, (5.41)
p + v2
τ = p 1 + v2 1
τ 1
, (5.42)
c p T + 1 2 v2 − qε = c p T 1 + 1 2 v2 1. (5.43)
11 W. Michelson was the first to postulate that within the reaction zone of the detonation wave the linear
relation (40) is verified.

120 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
The elimination of T , v, T 1 and v 1 between these equations and the state equations,
pτ = R m T and p 1 τ 1 = R m T 1 , (5.44)
gives the following Hugoniot relation for each value of the degree of advancement ε
of the combustion
γ + 1
2(γ − 1) (pτ − p 1τ 1 ) + 1 2 (p 1τ − pτ 1 ) = qε. (5.45)
This equation represents a family of hyperbolas. For each value of ε a hyperbola
is obtained, which represents the locus of the possible states of the gas when
burnt fraction is ε and the initial state is (p 1 , τ 1 ). The Hugoniot curve E’JE is obtained
for the particular case ε = 1. This curve corresponds to all the possible states of the
burnt gases, see Fig. 5.2. Similarly, for ε = 0 the Hugoniot curve PD is obtained,
which corresponds to all the shock waves that can form in the unburnt gases at the
initial state represented by point P . Between both curves lie those corresponding to
the intermediate states. Two of them are shown in Fig. 5.2, corresponding to ε = 0.3
and ε = 0.7.
Now, let us consider, for example, a Chapman-Jouguet detonation. Equation
(5.40) corresponding to this detonation is represented by the straight line PJC. The
transformations that occur throughout the shock wave, preceding the combustion,
carry the gas from the initial state P to the state C after the shock wave with no combustion.
However, the trajectory of the gases through the shock wave is not represented
by the straight line PC. In fact, we have seen that the thickness of the shock wave
is very small and, as aforesaid, the thermal conductivity and viscosity therein cannot
be neglected. In particular, Eq. (5.40) must be substituted by the following relation,
deduced from (5.3.a), which takes into account the viscosity, 12
p − p 1 = m 2 (τ 1 − τ) + 4 3 µ dv
dx . (5.46)
But since, through the shock wave the velocity decreases, dv/ dx is negative. Therefore,
the representative points of Eq. (5.46) lie below the straight line PC, as indicated
in Fig. 5.2. 13
The representative states of the combustion that follow the shock wave, that
is, those corresponding to section BE, Fig. 5.1, are obtained by traversal segment CJ,
starting from C. The points at which this segment intersects the successive Hugoniot
12 Assuming that the transformations through the shock wave can be described by variables corresponding
to a continuum. See chapter 3, §1.
13 Hirschfelder et al. ib. page 810.

5.5. DETONATIONS 121
curves, represent the successive states of the mixture for the corresponding values of
the degree of advancement of the combustion.
Now, let us consider a strong detonation, which in Fig. 5.2 would be represented
by a straight line such as PED. The initial shock wave produces a jump from
state P to state D. From D to E the reaction takes place and is accompanied by an
expansion and acceleration of the gases. The intermediate states are represented by
the points of segment DE.
Figure 5.2 also shows that weak detonations are not possible. In fact, once
point E is reached, point E’ corresponding to a weak detonation can only be reached
by means of an expansion wave between E and E’, which is impossible, or else by
means of an endothermic reaction, corresponding to segment EE’.
The temperature variation throughout the wave can be analyzed in the same
manner as that used for the pressure variations. For this purpose it is sufficient to
eliminate p and p 1 from Eq. (5.40), by making use of (5.44). Thus, we obtain
T
= (1 + γM1 2 ) τ ( ) 2 τ
− γM1
2 , (5.47)
T 1 τ 1 τ 1
in which the value of the Mach number M 1 of the unburnt gases has been made explicit.
To each value of the Mach number M 1 corresponds a different parabola, on
which lie the representative points of the successive states of the mixture within the
combustion zone of the detonation wave. In Fig. 5.3 two parabolas are shown. One
corresponds to the Chapman-Jouguet detonation and the other to a strong detonation.
The same letters are used to designate homologous points in Figs. 5.1 and 5.2.
The Hugoniot curves are obtained from Eq. (5.45) eliminating p and p 1 by
means of Eq. (5.44), as has been done to obtain Eq. (5.47). There results
( ) 2 τ
+ γ + 1 T τ
− T ( 2γ qε
−
+ γ + 1 ) τ
= 0. (5.48)
τ 1 γ − 1 T 1 τ 1 T 1 γ − 1 c p T 1 γ − 1 τ 1
These curves are a family of hyperbolas. A hyperbola is obtained for each value of
ε. In particular, for ε = 1 one obtains the Hugoniot curve of the burnt gases and for
ε = 0 the curve of the shock waves of the unburnt gases. Both have been taken into
Fig. 5.3. Their intersection with curve of Eq. (5.47), representative of the intermediate
states, determines section CJ or DE, corresponding to the said intermediate states. The
state corresponding to a given fraction of burnt gases is given by the intersection of
curve of Eq. (5.48), corresponding to this fraction, with section CJ, or DE, as shown
in Fig. 5.3 for the values ε = 0.3 and ε = 0.7.
The preceding considerations give an idea of the structure of a detonation wave.
The initial shock wave compresses, decelerates and heats suddenly the gasses, within a

122 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
zone of negligible thickness compared to that of the detonation wave, where no appreciable
chemical reaction occurs. After the shock wave the combustion is initiated. The
gases expand and accelerate and the pressure decreases. Temperature rises due to the
reaction. For Chapman-Jouguet detonation and slightly strong detonations, the maxi-
20
18
T/T 1
16
14
12
10
8
6
D
C
ε =0.3
E
ε =1
J
K
ε =0.7
E‘
4
2
0
ε =0
Trayectory through shock wave
P
0.2 0.4 0.6 0.8 1.0
τ /τ 1
Figure 5.3: Hugoniot diagram in variables (T, τ).
15
50
1.0
13
40
C
P/P 1
K
J
0.8
11
30
J
0.6
T/T 1
9
P/P 1
20
T/T 1
J
0.4
τ/τ 1
7
10
C
C
τ/τ 1
0.2
5
0.0 0.2 0.4 0.6 0.8 1.0
Figure 5.4: Typical values of T , p and τ in a Chapman-Jouguet detonation as a function of
the fraction of burnt gases.
ε

5.6. DEFLAGRATIONS 123
mum temperature is reached shortly before the combustion ends (as is clearly shown
in Fig. 5.3), after which a slight temperature drop (100 to 200 ◦ C for typical cases)
takes place due to the fast expansion of the combustion products. For the very strong
detonations, like DE represented in Fig. 5.3, the maximum temperature is reached at
the end of the combustion. These results appear in Fig. 5.4, which shows the values
corresponding to a typical case for a Chapman-Jouguet detonation.
5.6 Deflagrations
We have said that the deflagrations, also known as flames, will be the subject of a
careful study in the following chapter. Consequently, herein we shall only include a
few brief considerations concerning their possible existence, structure and propagation
velocity.
In the preceding chapter we have seen that the propagation velocity of a deflagration
wave is always subsonic. Thereby it cannot be ascertained that in the deflagration
waves the condition α ≪ M 2 will be satisfied, and the system that must be used
for the study of its structure is B, at least within the region of the flame close to the
cold boundary. 14 Let us consider separately the weak and strong deflagrations, and let
us start by demonstrating that the latter cannot exist.
In the strong deflagration waves, the final velocity of the gases is supersonic.
Consequently, at least in the final zone of such waves, the condition M 2 ∼ 1 is satisfied,
and system A can be used. In particular in this zone, the variation law for the
velocity of the gases with respect to the wave is given in Fig. 5.1. However, due to
the influence of thermal conductivity and diffusion, this law can differ within the zone
close to the cold boundary. Hence, in a strong deflagration wave, the variation of the
velocity of the gases through the wave must follow a law as the one represented in
Fig. 5.5, where the curve in Fig. 5.1 representing the limiting solution of system A has
also been represented by a dash line. The velocity of the unburnt gases is represented
by point P . Starting from this point, the gases accelerate. When their Mach number
is such that condition α ≪ M 2 is satisfied, the curve overlaps the dash line of the
limiting solution (in the figure this happens at point B). If now one must reach point
D, corresponding to the final state where the velocity is supersonic, this can only be
achieved by passing through point C where the velocity is sonic. But the jump from C
14 It has been shown in the preceding chapter that the propagation velocity of a deflagration is always equal
to or smaller than that corresponding to the Chapman-Jouguet deflagration. In this case the Mach number
corresponding to the propagation velocity is always much smaller than unity. Thereby, the propagation
Mach number of the deflagrations observed is always much smaller than unity. In practice, of the order of
10 −3 .

124 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
to D could only be attained through an endothermic reaction, and this kind of reaction
is impossible in a combustion process. As a consequence, the non-existence of the
strong deflagrations is concluded.
D
v
C
B
P
ε
Figure 5.5: Schematic diagram showing the variation of the velocity of gases in a deflagration.
On the other hand, since for a weak deflagration the final velocity is subsonic,
nothing is opposed to its existence.
In the weak deflagrations diffusion and heat conduction are important and the
system to be used is system B. Diffusion and heat conduction are responsible for
the propagation of the combustion to the unburnt gases, activating their reaction by
diffusion of the active centers (atoms, radicals, etc.) and by heating. As they heat, the
gases expand and accelerate, producing a slight pressure drop throughout the wave.
Fig. 5.6 shows qualitatively the structure of a deflagration wave bringing forth, by
comparison with Fig. 5.3, the difference in the mechanism that propagate the process
in each case. The system B of differential equations that must be integrated in order
to obtain the structure of the deflagration wave can be simplified by eliminating x, as
done in §4. Thus obtaining
p Dw
R m T m 2
dY
dε
= Y − ε, (5.49)
λw dT
m 2 dε = c pT − qε + e. (5.50)
In the deduction of this system, the state equation p = ρR m T has been used in order
to eliminate the density from the diffusion equation.

5.6. DEFLAGRATIONS 125
8
7
v/v 1
=T/T 1
6
5
4
3
−∆ P/(1/2)ρ 1
v 1
2
T 2
/T 1
=8
ρ D c p
/λ=1.5
2
1
Y
0
0 0.2 0.4 0.6 0.8 1
ε
Figure 5.6: Typical values of T , v, Y and ∆p in a deflagration as a function of the fraction
of burnt gases .
In this system the reaction rate w is a known function of Y and T . The pressure
p is constant in virtue of Eq. (5.31). One must look for a solution of this system, in
the interval 0 ≤ ε ≤ 1, satisfying the following boundary conditions
ε = 0 : Y = 0, T = T 0 , (5.51)
ε = 1 : Y = 1, T = T f . (5.52)
Since this is a system of two first order equations, the four conditions (5.51)
and (5.52) cannot be satisfied unless parameters e and m take well defined values.
Parameter e is given by the expression: e = q − c p T f , which results from expressing
the condition w = 0 for ε = 1, that is, at the end of the combustion. As for m it must
taken well defined value, so that the previous differential system to be compatible.
This means that weak deflagrations can only propagate with a well defined velocity
which depends on the state of the mixture, its reaction rate and its coefficients of thermal
conductivity and diffusion. The problem of determining the propagation velocity
of the flame through a combustible mixture appears, thus, as an “eigenvalue” problem.
The solution to this problems will be studied in the following chapter.
The same conclusions are reached when taking into account the influence of all
the terms in the differential equations of the combustion wave, instead of considering
a limiting solution as done herein.

126 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
5.7 Transition from deflagration to detonation
The preceding study presents deflagrations and detonations as independent phenomena.
Both, deflagrations and detonations can be initiated by means of various procedures.
For example, deflagrations can be initiated by means of an electric spark within
a combustible mixture, and detonations by making a shock wave cross the detonant
mixture. Such procedure was applied, for example, by Fay [14] using a shock tube.
However, a detonation can also be produced starting from a deflagration by compression
of the unburnt gases and acceleration of the combustion wave. This can occur, for
example, when a detonant mixture filling a tube is ignited at the closed end. In such
a case a deflagration wave initiates at the ignition point. This deflagration accelerates
as it propagates along the tube, producing a succession of intermediate states, nonstationary,
which end with the establishment of a stable Chapman-Jouguet detonation,
Hereinafter, we shall restrict ourselves to a qualitative description of the process, following
the lines of Zeldovich’s work [8], who carefully studied this phenomenon. For
this purpose the flame front will be considered as a discontinuity that travels along the
tube, as done in the preceding chapter.
Let ϕ be the propagation velocity of the flame through the unburnt gases, and
S the area of the cross-section of the tube. If the flame front is plane and normal to the
tube axis, the mass of the gases burnt per unit time is ρ 1 ϕS, where ρ 1 is the density
of the unburnt gases before the flame. Before burning, this mass occupies a volume
ϕS. Since the gases expand as they burn, the volume that must occupy the said mass
at the same pressure 15 is nϕS. Here n is the ratio of the temperature of the burnt gases
to temperature of the unburnt gases. Consequently, the increase in volume due to the
combustion is (n − 1)ϕS. Such an increase in volume sets the unburnt gases into
motion in front of the combustion wave to empty the necessary space. The total mass
of unburnt gases is not set into motion simultaneously, but progressively by a pressure
wave that travels through the mass with a velocity close to the velocity of sound. The
situation is shown schematically in Fig. 5.7. The gases are at rest within the region
between O and the flame front A. Between the front A and the pressure wave B, lie the
unburnt gases, compressed and moving towards the right-hand side. Starting from the
pressure wave B and towards tho right-hand side, lie the unburnt gases in the initial
state and at rest. The intensity of the pressure jump across the pressure wave is such
that the burnt gases are at rest. Let v p1 and v p2 be the velocities of the unburnt gases
before and after the pressure wave, measured with respect to this wave. The velocity
at which the pressure wave travels along the tube is v p1 . The velocity of the unburnt
15 The pressure drop across the flame is negligible.

5.7. TRANSITION FROM DEFLAGRATION TO DETONATION 127
O
A
B
P P
n ϕ
2
1
P v p − v p = (n−1)ϕ v p1
3 1 2
P 3
P 2
P, v
(n−1)ϕ
P 1
Figure 5.7: Schematic diagram of the flame propagation in a closed tube.
gases after the wave, with respect to the tube, is v p1 −v p2 . The condition that the burnt
gases must be at rest is
v p1 − v p2 = (n − 1)ϕ, (5.53)
as it can easily be verified. The velocity v a at which the flame travels along the tube is
v a = nϕ, (5.54)
where ϕ is the propagation velocity of the flame throughout the gases in their state
after the pressure wave. Such propagation velocity differs from that corresponding
to the state of the gases before the flame. This state of affairs can be maintained
unchanged along the tube. For the detonation to occur it is necessary to activate the
combustion, increasing the burnt mass per second. Such an increase is necessary in
order to increase the intensity of the pressure jump across the pressure wave, up to the
point of self-ignition of the mixture after the wave, needed to maintain the detonation.
The slight compression of the mixture, produced by the compression wave, is not
sufficient to increase, appreciably, the propagation velocity of the flame. Therefore,
the increase of burnt mass per second must be due to other effects. The explanation
can be found in the influence of the tube walls. In fact, due to friction, the distribution
of velocities in the cross-section of a tube is not uniform but maximum at the axis
and zero on the walls. Due to this non-uniformity of the velocity distribution, the
flame front curves increasing its surface. Due to this surface increase, the burnt mass
per second increases and therewith the velocity of the gases and the intensity of the
pressure jump, producing an interaction of both effects such that when the tube is
sufficiently long it leads to the establishment of the detonation wave. This standpoint
is corroborated by the fact that for tubes with rough walls or small diameter the

128 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
detonation is soon establish because both effects increase the influence of friction. On
the other hand, if the mixture is ignited at the open end of the tube, the detonation is
established with greater difficulty as the burnt gases can escape through the open end
and the pressure wave that precedes the combustion wave is much weaker. In this case,
a longer tube is necessary so that the pressure wave may reach the required intensity
for the detonation to occur.
For the case previously analyzed, A.K. Oppenhein [15] has studied the succession
of intermediate states that must be produced in the transition from deflagration to
detonation by acceleration of the deflagration wave up to the point where it overtakes
the shock wave. He demonstrated his results by means of a hydraulic analogy.
References
[1] Backer, R.: Stosswelle und Detonation. Zeitschrift für Physic, Vol. 8, 1922.
[2] von Kármán, Th.: Aerothermodynamics and Combustion Theory. L’Aerotecnica,
Vol. XXXIII, Fasc. 1st., 1953, pp. 80-86.
[3] Friedrichs, K. O.: On the Mathematical Theory of Deflagrations and Detonations.
NAVORD Report 79-46, Institute for Mathematics and Mechanics, New
York University, 1946.
[4] Hirschfelder, J. O., Curtiss, C. F. and Campbell, D.E.: The Theory of Flames
and Detonations. Fourth Symposium (International) on Combustion, Williams
and Wilkins Co., Baltimore, 1953, pp. 190-211.
[5] Vieille, P.: Rôle des Discontinuites dans la Propagation des Phénomènès Explosifs.
Comptes Rendus des Séances de 1’Académie des Sciences, Vol. CXXXI,
1900, p. 413.
[6] Jouguet, E.: Mecanique des Explosifs, Paris, 1917, p. 325.
[7] Neumann, J. von: Progress Report on the Theory of Detonation Waves. N D R
C, Division B, 0 S R D, No. 549, 1942.
[8] Zeldovich, Y. B.: On the Theory of Propagation of Detonation. Journal of
Experimental and Theoretical Physics, Vol. 10, 1940, p. 542, Translated as
NACA Technical Memorandum No. 1261, 1950.
[9] Döring, W.: Annalen der Physic, Vol. 43, 1943, p. 421.
[10] Zeldovich, Y. B.: Theory of Combustion and Detonation of Gases. A9-T-45,
Air Documents Division, Air Material Command, U S A F .

5.7. TRANSITION FROM DEFLAGRATION TO DETONATION 129
[11] Taylor, G. I.: Detonation Waves. Ministry of Supply, Explosives Research Committee,
R.C. 178, A.C. 639, 1941.
[12] Zeldovich, Y. B.: The Distribution of Pressure and Velocity in the Products of
a Detonation Explosion. Journal of Experimental and Theoretical Physics, Vol.
12, 1942, p. 389. Also ref. [8].
[13] Manson, N. and Ferrié, P.: Contribution to the Study of Spherical Detonation
Waves. Fourth Symposium (International) on Combustion, Williams and
Wilkins Co., Baltimore, 1953, pp. 486-94.
[14] Fay, J. A.: Some Experiments on the Initiation of Detonation in 2H 2 -O 2 Mixtures
by Uniform Shock Waves. Fourth Symposium (International) on Combustion,
Williams and Wilkins Co., Baltimore, 1953, pp. 501-507.
[15] Oppenheim, A. K.: Gasdynamics Analysis of the Development of Gaseous Detonation
and its Hydraulic Analogy. Fourth Symposium (International) on Combustion,
Williams and Wilkins Co., Baltimore, 1953, pp. 471-480.

130 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES

Chapter 6
Laminar flames
6.1 Introduction
It was demonstrated in chapter 4 that through a combustible mixture of gases two
types of combustion waves may propagate: detonation waves and deflagration waves
or flames. Chapter 5 includes the analysis of the nature of such waves, their structure
and the conditions which determine their propagation velocity. This analysis proved
that while in detonation waves the propagation velocity is determined only by a condition
of mechanical stability within the zone of burnt gases, which simplifies extremely
its determination, when dealing with flames their propagation velocity is determined
by an internal equilibrium between the processes of chemical reaction, heat transfer
(conductivity) and transport of the chemical species (diffusion), which makes the
study specially difficult.
The initial attempt to establish a theory on these bases, and in particular the
deduction of the first formula for the computation of the propagation velocity of a
flame through a given mixture, which is the fundamental problem of combustion, is
own to Mallard and Le Chatelier [1]. Their theory is purely thermal and it establishes
a primitive balance between the heat received by the gases through conduction, when
an assumed ignition temperature is reached, and the heat content of these gases. The
concept of reaction velocity is not applied in this theory.
The work by Mallard and Le Chatelier was the starting point for the development
of the “thermal theories of the flame”, so called because they did not include the
diffusion of species.
The idea of chemical reaction velocity, incorporated by Crussard [2] to this
thermal model, was a definite step forward in the progress of the theory. From there
131

132 CHAPTER 6. LAMINAR FLAMES
on, the interest of the later works was centered both on the use of more realistic expressions
for the reaction velocity which considered the influence of the concentration
of species and of the temperature of the mixture, as well as on the development of
approximate methods, analytical and numerical, for the integration of the flame equations.
A critical discussion on thermal theories, whose development and application
have been pursued up to our days, may be found in the work by Goward and Payman
[3]. Frank-Kamenetskii-Semenov [4], Boys-Corner [5] and von Kármán [6] have specially
contributed to the development of analytical methods and Hirschfelder and his
collaborators [7] to the expansion of numerical ones.
Initially these studies were performed with a very simplified kinetic model
which considered only two different chemical species (reactants and products) and
they applied a law of reaction which reflected approximately the relationship between
the reaction velocity of the mixture and its temperature and composition at each point.
Later on these models were improved by taking into account the influence of the different
species and chemical radicals which act in the process.
The first attempt to include the influence of both diffusion and chemical radicals
on the velocity of the flame is own to Lewis and von Elbe [8] in their study of
ozone decomposition flame, which will be given further consideration in §13. This
study established the bases for the development of a complete theory on flames. Other
authors followed their steps, trying to find a more accurate and complete formulation
for the problem. At the same time the analytical and numerical methods of the thermal
theory were extended to the new formulation. A very complete analysis of the different
theories and methods available, as well as of the assumptions used in expanding
them, may be found in the work by Evans [9].
In the last years a great effort has been applied to the development of the theory
of the flame, which has resulted in the obtaining of a complete formulation of the
problem and its boundary conditions. At the same time a great amount of attention was
given to the development of analytical and numerical methods, very approximated,
for the solution of the resulting equations, while the theory was applied to a certain
number of practical cases, although very few due to the fact that it is very difficult to
know the physico-chemical constants of the process and of the chemical scheme of
the reactions. It is practically impossible to try to give a complete bibliography of the
work achieved in these last years since it is very abundant and scattered, specially in
the United States, United Kingdom, and Russia. Therefore, it is far more practical
to refer the reader to the Proceedings of the International Symposia on Combustion,
in special starting from the third one which took place in 1948, as well as to the
volumes of the Selected Combustion Problems published by AGARD, containing the

6.1. INTRODUCTION 133
papers presented at the two International Colloquia sponsored by this organization in
1954 and 1956. The work by von Kármán [10] is a complete review of the state of
knowledge on this problem up to 1956.
At the same time that this theory which considers the flame as a wave propagating
through a continuous medium was developed, other theories have attempted to
emphasize only partial aspects of the phenomenon. From them the most representative
is the one proposed by Tanford and Pease on the diffusion of radicals from the hot
boundary [11]. It has also been objected that flames cannot be treated by the methods
of the Mechanics of Continua. This objection has originated such theories as the
one proposed by van Tiggelen [12], who applies a method analogous to that used by
Semenov in studying chain reactions.
At present, however, it seems clearly established and widely accepted that the
correct method which should be applied to this problem is the one expanded in the following
paragraphs, and that its difficulty reduces to the use of the adequate chemical
kinetic scheme for each case; the knowledge of the corresponding physico-chemical
constants and of the transport coefficients of the mixtures; and finally to the obtaining
of satisfactory methods of solution of the difficult mathematical problem of the
integration of the flame equations.
In addition to this progress of theory, a substantial improvement has been
achieved in the field of experimental techniques to measure the propagation velocity
of the flame. An increasing number of data is now available on the influence of
several parameters such as the composition, pressure, initial temperature of the mixture,
etc., on the flame propagation velocity. As a bibliography for the study of these
techniques, their limitations and applicability we recommend the works by Jost [13],
Gaydon and Wolfhard [14] and specially the one of Ref. [15]. Linnet [16] has published
a review on these methods, in which he classifies the experimental techniques
most commonly used into the following five groups:
1) Method of the bunsen burner (Gouy and Michelson). The most universal, of
which a great number of variations are used.
2) Method of the tube (Coward-Hartwell, Gerstein, etc). Not considered reliable
enough.
3) Method of the spherical flame in a constant volume bomb (Fick, Lewis-von
Elbe). Not frequently used and difficult to observe.
4) Method of the soap film or spherical flame at constant pressure. Not frequently
used and limited to mixtures which are not sensitive to the influence of water
vapor.

134 CHAPTER 6. LAMINAR FLAMES
5) Method of the flat burner (Egerton-Pauling, Spalding-Botha). Indicated for mixtures
with a low flame velocity.
At the same time, several attempts have been made, with limited success, to
measure the distribution of certain variables through the flame in order to analyze
its structure. Information of this matter may be found in the above mentioned work
by Linnet and in the Proceedings of the 6th International Symposium on Combustion.
Initially, attempts were made to obtain temperature profiles whose results show
a fair agreement with theoretical predictions. Later on, attempts were made to obtain
distributions of the concentration of the main species, and finally, of the radicals.
These observations are difficult due to the small thickness of the flame. Although it
may be increased by reducing pressure, then the perturbation effects of the measuring
instruments and of radiation become more important, and it is difficult to know the
magnitude of the corrections which must be introduced into the results obtained so
that the values be exact.
In the following paragraphs of this chapter, we shall deduce and discuss the
flame equations, its boundary conditions and the methods of integration most commonly
used. The application of the theory to some practical cases will be performed
in §13, 14 and 15.
6.2 Equation for the combustion wave in the case of
two chemical species
The equations for the propagation of a plane stationary combustion wave were deduced
in the preceding chapter, system B, under the following assumptions:
1) There are only two different chemical species, reactants and products.
2) The mixture behaves like a perfect gas.
3) The specific heat of the mixture is independent from its composition and temperature.
The system obtained under these conditions, referred to the axis advancing with the
wave, and preserving the notation previously used, is as follows:
a) Continuity equation.
ρv = m (6.1)
b) Chemical reaction equation.
m dε = w(Y, p, T ) (6.2)
dx

6.3. BOUNDARY CONDITIONS 135
c) Diffusion equation.
d) Momentum equation.
e) Energy equation.
f) State equation.
ρD dY
dx
= m(Y − ε) (6.3)
p = const. (6.4)
c p T − qε − λ m
dT
dx = e (6.5)
p
ρ = R gT (6.6)
In this system, m and e are two constants, whose values will result from the
boundary conditions. The elimination of ρ through (6.6) reduces the system to three
first order differential equations (6.2), (6.3) and (6.5) with three unknown ε, Y and T .
6.3 Boundary conditions
In order for the solution of this system to represent a stationary wave, it is necessary
that it satisfies the following boundary conditions, which insures the transition from an
uniform state of the unburnt gases before the wave to an uniform state of the products
in chemical equilibrium after it,
Unburnt gases, x → −∞ : Y → 0, ε → 0, T → T 0 . (6.7)
Products, x → +∞ : Y → 1, ε → 1, T → T f . (6.8)
These conditions imply, as well, the following
x → ±∞ :
dY
dx → 0,
dε
dx → 0,
dT
dx
→ 0. (6.9)
The first of conditions (6.9) is satisfied by virtue of (6.3), since ε and Y take
the same values both in x = +∞ and in x = −∞, by virtue of (6.7) and (6.8).
The third condition (6.9) is also satisfied by virtue of (6.5) when the following
values are assigned to q and e
q = c p (T f − T 0 ),
e = c p T 0 ,
(6.10)

136 CHAPTER 6. LAMINAR FLAMES
which, furthermore, enables (6.5) to be written as
λ dT
dx = mc (
p (T − Tf ) + (T f − T 0 )(1 − ε) ) , (6.5.a)
in which form it will be used further on.
As for the second condition (6.9) it is also satisfied for x = +∞, since the
condition of chemical equilibrium for the products may be expressed as
w(1, ρ f , T f ) = 0. (6.11)
To the contrary, condition dε → 0 for x → −∞, will not be satisfied unless
dx
w(0, ρ 0 , T 0 ) = 0, (6.12)
which, generally, will be in contradiction with the laws of Chemical Kinetics.
This circumstance implies a fundamental difficulty, whose origin lies upon the
fact that it is impossible to maintain invariable the composition of the unburnt gases
located before the wave, as imposed by boundary conditions (6.7) and (6.9), when the
reaction velocity of such gases is not zero.
The existence of stationary combustion waves observed in practice may be
explained by the fact that such waves do not correspond exactly to the one-dimensional
model proposed in this work, since the mixing of reactant species forms, in general,
shortly before they reach the wave, which always looses a certain amount of heat
through the walls of the flame holder, for example, through the walls of the burner.
Furthermore, these walls act as chain breakers for the chemical reactions of the wave.
There is a possibility of eluding the above mentioned difficulty by incorporating
to the one-dimensional model the effect of all these complex circumstances, which
prevent the reaction of the unburnt gases in a real flame, through a slight modification
of the boundary conditions at the neighborhood of the “cold limit” of the flame, which
may be attained in several different ways. The practical interest of these solutions
is based on the fact that for reactions with an appreciable activation energy, as those
expected in combustion, the structure and propagation velocity of the flame are insensible
to a modification of the said boundary conditions, which makes it unnecessary to
define them exactly, within a wide range of values of the defining parameters. Moreover,
the proposed solutions are equivalent since they lead to the same fundamental
results. The following paragraph is devoted to a discussion of these solutions.

6.4. MODIFICATION OF THE CONDITIONS AT THE “COLD BOUNDARY” 137
6.4 Modification of the conditions at the “cold
boundary”
There are several solutions available in order to elude the difficulty of the cold boundary,
which has passed unnoticed until recently. The history of the evolution of thought
on this subject may be followed by consulting references [17] through [19]. Herein
we will only study the two solutions currently used, which consist in introducing an
ignition temperature T i and a flame holder.
Ignition temperature
By adopting the same assumption used in the classical theories of Combustion, that
is, by assuming that there is an ignition temperature T i , such that reaction velocity of
the mixture is zero for T < T i , this temperature divides the combustion wave into
two zones: a “heating and diffusion zone”, corresponding to temperatures under T i , at
which no chemical reaction takes place, and a “reaction zone”, which corresponds to
temperatures over T i .
The differential equations for the reaction zone are the same given in the preceding
paragraph, which are summarized in the following, as well as the boundary
conditions, assuming that the origin of coordinates x = 0 is located at ignition point
T = T i .
Reaction zone, x > 0.
m dε = w(Y, ρ, T ), (6.2)
dx
ρD dY
dx
= m(Y − ε), (6.3)
λ dT
dx = mc p
The boundary conditions for this system will be
(
(T − Tf ) + (T f − T 0 )(1 − ε) ) . (6.5.a)
x = 0 : ε = 0, T = T i , (6.13)
x → +∞ : ε = 1, T = T f , Y = 1. (6.14)

138 CHAPTER 6. LAMINAR FLAMES
The equations for the heating and diffusion zone, x < 0, may be obtained
from those given §2 by introducing in them the conditions expressing the absence of
chemical reaction, that is: w = ε = 0. There resulting
Heating and diffusion zone, x < 0.
ρD dY
dx
The corresponding boundary conditions are
= mY, (6.15)
λ dT
dx = mc p(T − T 0 ). (6.16)
x = −∞ : T → T 0 , Y → 0, (6.17)
x = 0 : T = T i . (6.18)
Since for x → −∞ both dY / dx and dT / dx tend to zero, as it can be verified
by taking (6.17) into (6.15) and (6.16), the introduction of an ignition temperature
eliminates the difficulty at the cold boundary. Furthermore by comparing the equations
for the heating and diffusion zone with those for the reaction zone, given in the
preceding paragraph, and the boundary conditions (6.13) and (6.18), it may be readily
verified that the transition from one zone to another at point x = 0 is continuous for
all variables when adopting the additional condition that the value for Y at x = 0,
which is undetermined, be the same for both solutions
x = 0 : Y (0 − ) = Y (0 + ). (6.19)
The objection to this way of dealing with the difficulty at the cold boundary
states that the solution obtained, and in special the propagation velocity for the flame,
will depend on the value adopted for T i , which does not actually exist. Nevertheless, it
will be proved further on, when studying the solutions for the combustion wave, that as
long as the velocity of the chemical reaction depends substantially on the temperature
of the mixture, it will happen that the solution obtained will be independent from the
values of T i , except for values of this temperature very close to T 0 or to T f . Under
such conditions the influence of the ignition temperature will vanish completely from
the result, when calculating the propagation velocity of the flame.
The presence of a flame holder
Another solution proposed to the problem of the cold boundary consists in imagining
the presence of a flame holder placed in front of the wave which has the double mission

6.4. MODIFICATION OF THE CONDITIONS AT THE “COLD BOUNDARY” 139
of absorbing a certain amount of heat Q from the flame and of acting as a filter which
allows the unburnt gases to reach the flame but prevents the combustion products from
diffusing towards the unburnt gases. By assuming that the holder is located at point
x = 0, the system of Eqs. (6.2), (6.3) and (6.5a) will still hold, but conditions (6.13)
at the cold boundary will have to be substituted by the following
x = 0 : T = T 0 , ε = 0, λ dT
dx
≠ 0. (6.20)
To the contrary, for x < 0 the composition and state of the mixture are uniform
x < 0 : Y = ε = 0, T = T 0 . (6.21)
The value of Y at x = 0 remains undetermined, but different from zero. Therefore
through x = 0 there exists a discontinuity in the composition of the mixture which
becomes possible due to the existence of the filter.
It happens here, as in the case of ignition temperature, that the structure and
propagation velocity of the flame are independent from the value of parameter Q,
provided it is not close to heat of reaction q or very close to zero, which justifies the
practical value of the proposed model.
T f
T f
T T 0 i 1−ε T 0
1−Y
Karman ´ ´ boundary condition
1−Y
Hirschfelder boundary condition
1−ε
Figure 6.1: Boundary conditions for ignition temperature and flame holder models.
Figure 6.1 summarizes and compares the conditions at the cold boundary for
the two solutions proposed. Since both lead to the same results, in the following we
will use only the assumption that an ignition temperature exists.
Figure 6.2 shows in a qualitatively way the form of the solutions obtained for
the propagation velocity of the flame in a given mixture when the ignition temperature,
assumed for it, is changed. It is seen that there is a wide interval AB of ignition
temperatures within which the propagation velocity of the flame is independent from
T i . The quantitative solutions for some typical cases will be included further on.

140 CHAPTER 6. LAMINAR FLAMES
1
0.8
0.6
S b
0.4
A
B
0.2
0
0 0.2 0.4 0.6 0.8 1
T 0
/T f T /T i f
Figure 6.2: Schematic diagram showing the flame propagation velocity vs ignition temperature.
6.5 Propagation velocity of the flame
A simple check of the boundary conditions at the “reaction zone” shows that they
are superabundant. In fact, since we are dealing with a system of three equations of
first-order, the solution of the system is determined for each value of m by boundary
conditions (6.14), except for a translation, which is determined by the additional condition
that for x = 0 be T = T i as imposed by (6.13). In order to satisfy as well the
additional condition x = 0 : ε = 0, it is necessary that m takes a particular value: the
“eigenvalue” of the system which makes compatible boundary conditions (6.13) and
(6.14).
This eigenvalue will determine the propagation velocity of the flame, which by
virtue of (6.1) is given by
u 0 = m . (6.22)
ρ 0
6.6 Example
Even in the case of only two chemical species as considered in the preceding paragraphs,
the non-linear character of the flame equations and in particular the shape of
the reaction velocity add difficulties to the problem to such an extent that it becomes

6.6. EXAMPLE 141
generally impossible to obtain explicit solutions for the equations. Hence, it is necessary
to resort to cumbersome numerical integrations or to analytical methods which
will be discussed further on. However, in order to illustrate the nature of the solutions
and the way in which the eigenvalue appears determined, the present paragraph offers
an example in which through an adequate simplification of both transport coefficients
and chemical reaction velocity, it is possible to obtain an explicit solution of the problem
giving correct qualitative predictions. A possible objection to this solution could
be that the results obtained depend on the ignition temperature of the mixture, which,
as before said, does not exist. But this is due to the fact that the simplification assumed
for the reaction velocity is excessive and consequently the inconvenient will vanish for
more realistic cases, as will be proven in the following.
For this example the following assumptions will be adopted:
1) The coefficient of thermal conductivity λ and the product ρD are constant.
2) The reaction is of first-order and its velocity is independent from temperature, in
which case it has the form 1 w = ρ 0 k(1 − Y ). (6.23)
It is precisely this simplified form of the reaction velocity and, in special, the fact
that in it the activation energy is assumed to be zero, an indispensable condition
for the integrability of the system, which makes the solution depends on the
ignition temperature, as aforesaid.
When expression (6.23) is taken into (6.2), it gives
The diffusion equation subsists in the form (6.3)
m dε
dx = ρ 0k(1 − Y ). (6.24)
ρD dY
dx
= m(Y − ε). (6.25)
Finally, when the dimensionless temperature θ = T/T f is introduced, the energy
equation (6.5.a) may be written
where is θ 0 = T 0 /T f .
λ dθ
mc p dx = θ − 1 + (1 − θ 0)(1 − ε), (6.26)
1 See §8 of Chap. 1.

142 CHAPTER 6. LAMINAR FLAMES
In order to find the solution, it is convenient to eliminate x from the preceding
system. This is done by dividing (6.24) and (6.25) by (6.26), thus resulting
In this system
dY
dθ = L Y − ε
θ − 1 + (1 − θ 0 )(1 − ε) , (6.27)
dε
dθ = Λ 1 − Y
θ − 1 + (1 − θ 0 )(1 − ε) . (6.28)
L =
λ
ρDc p
(6.29)
is the Lewis-Semenov number of the mixture, which is constant.
Λ is a dimensionless parameter, also constant, defined by
Λ = λρ 0k
m 2 c p
. (6.30)
The problem lies, precisely, in determining the eigenvalue of this parameter. Once it is
known, the velocity u 0 of the flame propagation may be derived from it and Eq. (6.22),
thus obtaining
√
λk
u 0 = Λ −1/2 . (6.31)
ρ 0 c p
The system of equations (6.27) and (6.26) holds only within the reaction zone, where
boundary conditions are
where we have written θ i = T i /T f .
θ = θ i : ε = 0, (6.32)
θ = 1 : ε = θ = 1, (6.33)
Precisely because these three conditions exist for a system of second-order, it
is necessary to look for the value of Λ which makes them compatible.
The system is readily integrated by testing lineal solutions of the form
1 − ε = α(1 − θ), (6.34)
1 − Y = β(1 − θ), (6.35)
which satisfy boundary conditions (6.33) at the hot boundary.
Condition (6.32) at the cold boundary gives the following relation
1 = α(1 − θ i ). (6.36)

6.6. EXAMPLE 143
Two more relations may be obtained by taking (6.34) and (6.35) into (6.27) and
(6.28), there resulting
α =
β =
Λβ
−1 + (1 − θ 0 )α , (6.37)
α − β
−1 + (1 − θ 0 )α . (6.38)
The three Eqs. (6.36), (6.37) and (6.38) become a determined system for the
computation of α, β and Λ. The solution is
α = 1
1 − θ i
, (6.39)
β = 1 (1 + 1 ) −1
θ i − θ 0
, (6.40)
1 − θ i L 1 − θ i
Λ = θ i − θ 0
(1 + 1 )
θ i − θ 0
. (6.41)
1 − θ i L 1 − θ i
Once Λ is known, then Eq. (6.31) provides the following dimensionless expression
for the propagation velocity of the flame
θ = θ i ,
u 0
√
ρ0 c p
λk = Λ−1/2 . (6.42)
Finally, since we know β, Eq. (6.35) gives the value Y i for Y corresponding to
Y i = 1 − β(1 − θ i ). (6.43)
The solution for the heating and diffusion zone, x < 0, will be obtained by
integrating the following equation
dY
dθ = L Y , (6.44)
θ − θ 0
which results from (6.27) when making ε = 0. The solution to this equation is
Y = Y i
( θ − θ0
θ i − θ 0
) L
. (6.45)
Once the preceding solutions are known, one may pass to the physical space
by making use of solutions (6.34) and (6.35) and Eqs. (6.24), (6.25) and (6.26), as
follows. 2
2 This far, the problem could have been solved under less drastic assumptions, i.e., not being constant λ
nor ρD, but the Lewis-Semenov number.

144 CHAPTER 6. LAMINAR FLAMES
a) Reaction zone.
After substituting (6.34) into (6.26), we have
dθ
dξ = θ i − θ 0
1 − θ 0
(1 − θ), (6.46)
where the following dimensionless distance was introduced
being
ξ = x l , (6.47)
l =
λ
mc p
(6.48)
a characteristic length of the wave. The integration of (6.46), once it is assumed that
reaction starts at point x = 0, gives
Likewise, one obtains for ε and Y
1 − θ = (1 − θ i ) e −θ i − θ 0
1 − θ i
ξ
. (6.49)
1 − ε = e −θ i − θ 0
1 − θ i
ξ
, (6.50)
1 − Y =
(
1 + 1 ) −1
θ i − θ 0
e −θ i − θ 0
ξ
1 − θ i . (6.51)
L 1 − θ i
1.0
0.8
Y(L=0.5)
0.6
ε, Y
0.4
Y(L=1)
ε
0.2
Y(L=2)
0.2 θ 0.6 0.8 1.0
i
=0.4
θ =0.125 0 θ
Figure 6.3: Distributions of Y and ε as a function of θ when the activation energy is zero.

6.7. REACTION VELOCITY 145
1.0
0.8
Y(L=2)
θ, ε, Y
0.6
0.4
0.2
θ 0
θ
Y(L=0.5)
Y(L=1)
ε
0.0
−10 −8 −6 −4 −2 0 2 4 6 8 10
ξ
Figure 6.4: Distributions of Y , ε and θ as functions of ξ, when the activation energy is zero.
b) Heating zone.
Likewise, in this zone one obtains
θ = θ 0 + (θ i − θ 0 ) e ξ , (6.52)
Y = 1 L
θ i − θ 0
(1 + 1 ) −1
θ i − θ 0
e Lξ , (6.53)
1 − θ i L 1 − θ i
ε = 0. (6.54)
Figures 6.3 and 6.4 represent the solutions computed for the following typical
values: θ 0 = 0.125, θ i = 0.4 and L = 0.5, 1 and 2.
6.7 Reaction velocity
The correct expression for the reaction velocity w is, in accordance with the laws of
Chemical Kinetics, 3 w = ρ n k(1 − Y ) n . (6.55)
3 See Chap. 1, §8.

146 CHAPTER 6. LAMINAR FLAMES
In this expression, n is the order of reaction and k a function of temperature, which,
generally, is of the form
k = A
( T
T f
) δ
e −E/RT , (6.56)
where A is a constant and E is the activation energy of the reaction.
It is convenient to eliminate the density from Eq. (6.55) so that only two variables
appear explicitly: the temperature and the mass fraction of products.
Let M r and M p be the molar masses of reactants and products, respectively,
and a the relation
Then, the mean molar mass of the mixture is 4
a = M r
M p
− 1. (6.57)
M =
M r
1 + aY , (6.58)
and the density of the mixture may be expressed as a function of pressure and temperature
in the form
By taking now Eq. (6.56) and (6.59) into (6.55), we finally obtain for the reaction
velocity
ρ = pM r
RT
1
1 + aY . (6.59)
w = A
( ) n ( ) n ( ) δ pMr 1 − Y T
e −E/RT . (6.60)
RT 1 + aY T f
It is advantageous to bring forth in Eq. (6.60) the dimensionless temperature θ =
T/T f as defined in the preceding paragraph, and to introduce dimensionless activation
temperature θ a as defined by
Now, Eq. (6.60) may be written
θ a =
E
RT f
. (6.61)
( ) n 1 − Y
w = Bθ δ−n e −θ a/θ , (6.62)
1 + aY
where
( ) n pMr
B = A
(6.63)
RT f
is a constant of the process. Eq. (6.62) is the form of the reaction velocity that will be
used in the following.
4 See Chap. 1, Eq. (1.36).

6.8. FLAME EQUATIONS 147
6.8 Flame equations
When (6.62) is taken into (6.2) , the later takes the form
m dε ( ) n 1 − Y
dx = Bθδ−n e −θ a/θ . (6.64)
1 + aY
Equation (6.3) is still written
ρD dY
dx
= m(Y − ε). (6.65)
If we substitute into Eq. (6.5.a) the dimensionless temperature, we have for it
λ dθ
dx = mc (
p θ − 1 + (1 − θ0 )(1 − ε) ) . (6.66)
In order to solve the above system it is convenient to divide (6.64) and (6.65)
by (6.66), as was done with the example treated in §6, thus obtaining
In this system,
(
λ 1 − Y
θ δ−n 1 − θ
dε
dθ = Λ(1 − θ λ f 1 + aY
0)
θ − 1 + (1 − ε)(1 − θ 0 ) e−θ a
θ , (6.67)
dY
dθ = L Y − ε
θ − 1 + (1 − θ 0 )(1 − ε) . (6.68)
Λ =
Bλ f e −θ a
m 2 c p (1 − θ 0 )
) n
(6.69)
is an undetermined parameter of the problem and L is the previously defined Lewis-
Semenov number of the mixture. 5
The boundary conditions of system (6.67), (6.68) are the same given in (6.32)
and (6.33), namely,
Hot boundary: θ = 1, Y = 1, (6.70)
Cold boundary: θ = θ i , ε = 0. (6.71)
The problem lies in determining the value of Λ which makes compatible these
three conditions in the system formed by the two first-order equations (6.67) and (6.68).
5 In the numerator of (6.67) we bring forth factor e −θ a 1−θ
θ , in lieu of e − θa θ in order to simplify
calculations since θ a is generally much larger than unity and otherwise, we would have to deal with very
small numbers instead of doing it with numbers of the order of unity.

148 CHAPTER 6. LAMINAR FLAMES
Once Λ is known then the velocity u 0 of the flame is given by
√
Bλ f e
u 0 =
−θa
ρ 2 0 c p(1 − θ 0 ) Λ−1/2 . (6.72)
6.9 Solution of the flame equations
As before said, the preceding system cannot by integrated analytically. Hence, it
becomes necessary to resort either to numerical solutions or else to semi-analytical
approximation methods.
The numerical methods are very cumbersome since we are dealing with an
eigenvalue problem with given boundary conditions at both extremes of the interval,
and, therefore, we would have to make numerous tentatives before reaching the exact
solution. Generally, the use of electronic computers is required and in particular they
were widely utilized by Hirschfelder and his collaborators.
In recent years, several approximate methods have been proposed generally
based in the unusual behavior of the solutions due to the presence of factor e −θa/θ in
the reaction velocity. In fact, when the reduced activation temperature θ a is large, as
should be expected in combustion reactions, due to this factor the result is determined,
essentially, by the form of the solution close to the hot boundary and this enables
the development of methods, as those proposed by Zeldovich-Frank-Kamenetskii-
Semenov [4] and the one by Boys-Corner [5], Adams [20], Wilde [21] and von Kármán
[6], with different variations and approximations. Only very recently a comparative
study of these methods has been performed. Information on the subject will be found
in the work by von Kármán [10] and, specially, in the more complete study carried
out by Millán, Sendagorta and Da Riva [22]. Figures 6.5 and 6.6, taken from this
study show a comparison between the approximate values for Λ −1/2 ( to which the
flame velocity is proportional by virtue of (6.72)), obtained through several of those
methods, with the exact value, given by a numerical integration. This comparison was
performed for several values of the temperature of the unburnt gases and of the activation
energy of the reaction. All the results shown in these figures correspond to a
Lewis-Semenov number equal to unity and to a constant mean molar mass of the mixture.
However, the unpublished results of the calculations performed for more general
cases indicate that the same conclusions are still valid.
Figures 6.5 and 6.6 show that the Boys-Corner method in its first iteration
gives defect values with an important deviation. The same happens with the method
by Zeldovich et al., but giving excess values. The approximation improves with the

6.9. SOLUTION OF THE FLAME EQUATIONS 149
second iteration of the Boys-Corner method but it is very arduous to perform and,
furthermore, like the first one it does not converge towards the correct solution for
increasing activation temperatures.
1.5
1.4
1.3
ZELDOVICH
1.2
Λ −1/2 / Λ −1/2
exact
1.1
1.0
0.9
0.8
WILDE
KARMAN 3
SENDAGORTA
BOYS−CORNER 2nd IT.
KARMAN 2
KARMAN 1
0.7
BOYS−CORNER
0.6
2 4 6 8 10 12 14 16
θ a
Figure 6.5: Comparison between the flame propagation problem eigenvalue obtained by
different approximated methods and the exact value for θ 0 = 0.125.
1.3
Λ −1/2 / Λ −1/2
exact
1.2
1.1
1.0
0.9
0.8
0.7
ZELDOVICH
WILDE SENDAGORTA
KARMAN 3
BOYS−CORNER 2nd IT.
KARMAN 2
KARMAN 1
BOYS−CORNER
0.6
2 4 6 8 10 12 14 16
θ a
Figure 6.6: Comparison between the flame propagation problem eigenvalue obtained by
different approximated methods and the exact value for θ 0 = 0.250.

150 CHAPTER 6. LAMINAR FLAMES
The other methods are considerably better mainly when applied to normal activation
temperatures. This is specially true about Wilde’s method, the second and third
alternatives of von Kármán’s one and that proposed by Sendagorta [23], which is the
best and does not require more work than the others.
1.0
0.8
BOYS−CORNER
0.6
EXACT
θ
0.4
0.2
KARMAN 3
0.0
−0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0
ε
Figure 6.7: Comparison of the distributions of ɛ vs θ obtained by two approximate methods
with the exact result.
Figure 6.7 shows, for one of the case calculated, the law of the variation of ε
with θ of the Boys-Corner approximation, Kármán’s third alternative and the exact
solutions. It is evident that the approximation reached with Kármán’s method is very
satisfactory. The following studies the nature of this method after referring the reader
to the above references for a more detailed analysis of the other methods mentioned.
If in Eq. (6.67) we bring the denominator of the right hand side into the left one,
and this equation is integrated between the cold boundary (6.71) and the hot boundary
(6.70), we obtain
1 − θ 0
2
−
∫ 1
0
∫ 1
( ) n
λ 1 − Y
(1 − θ) dε = Λ(1 − θ 0 ) θ δ−n e −θ 1 − θ
a
θ dθ.
θ i
λ f 1 + aY
(6.73)
Consequently, in order to determine Λ, the problem reduces now to obtaining
an approximation of θ vs ε on the integral of the left side of Eq. (6.73) and an approximation
of Y vs θ on the integral of the right side of the same equation. These integrals

6.9. SOLUTION OF THE FLAME EQUATIONS 151
will be written for shortness as
∫ 1
( ) n
λ 1 − Y
I =(1 − θ 0 ) θ δ−n e −θ 1 − θ
a
θ dθ, (6.74)
λ f 1 + aY
J =
∫ 1
0
θ i
(1 − θ) dε. (6.75)
With this notation, the fundamental Eq. (6.73) may be written
1 − θ 0
2
− J = ΛI.
The following is a separate study of both approximations.
(6.73.a)
Approximation of integral I
Heat transfer λ is a function of temperature θ of the mixture and its composition
defined by the value of Y . 6 Consequently, in order to calculate I the problem reduces
to finding an approximation for Y as a function of θ. The solution depends on the
value of the Lewis-Semenov number.
Lewis-Semenov number equal to unity
If L = 1, then Y is a lineal function of θ in the following form
In fact, when condition
Y = θ − θ 0
1 − θ 0
. (6.76)
L ≡
is satisfied, the diffusion Eq. (6.65) may be written in the form
λ
ρDc p
= 1 (6.77)
λ dY
dx = mc p(Y − ε). (6.78)
By adding to it the energy Eq. (6.66), multiplied by an arbitrary constant C,
λ d
(
dx (Y + Cθ) = mc p Y + C(θ − θ 0 ) − ( C(1 − θ 0 ) + 1 ) )
ε . (6.79)
This equation is satisfied identically with the following two conditions
6 See Chap. 2, §4.
C(1 − θ 0 ) + 1 = 0, (6.80)
Y + C(θ − θ 0 ) = 0, (6.81)

152 CHAPTER 6. LAMINAR FLAMES
the first is satisfied by adopting the following value for C
C = − 1
1 − θ 0
. (6.82)
This value when taken into Eq. (6.41) gives for Y the expression (6.76) which is, in
fact, the desired solution valid throughout the interval of temperatures as the boundary
conditions are met at both extremes.
θ = θ 0 : Y = 0, θ = 1 : Y = 1. (6.83)
0.40
θ a
=4
0.35
θ 0
=0.125
Λ −1/2
0.30
0.25
0.20
0.5 1.0 1.5 2.0
L
Figure 6.8: Dimensionless flame velocity as a function of the Lewis number for θ a = 4 and
θ 0 = 0.125.
Lewis-Semenov number different from unity
Actually the Lewis-Semenov number is only identical to unity when dealing with pure
mono-atomic gases (in which case D is the coefficient of autodiffusion). Generally
in more complex cases L differs from unity and it varies with the composition and
temperature of the mixture. However, it is frequently very close to unity and the
preceding approximation is satisfactory. Moreover, the influence of the value of the
Lewis-Semenov number on the flame velocity is, generally speaking, quite small. This
is verified in Fig. 6.8 taken from Ref. [22] which shows, as an example, that when
L doubles its value, the flame velocity increases less than 20 per cent, in this case
corresponding to a first-order reaction with an activation temperature θ a = 4, which

6.9. SOLUTION OF THE FLAME EQUATIONS 153
is small, and therefore the influence of L is more obvious than it would be for larger
values of θ a .
On the other hand, due to the presence of factor e −θ 1 − θ
a
θ
in the integral of
the right hand side of Eq. (6.73) it occurs that for large values of θ a , as being those
normally appearing in combustion, the value of the integral is influenced only by the
value taken at the neighborhood of θ = 1 by the quantity under the integral.
1.0
0.8
θ a
=4
L=0.5
ε
0.6
θ
Y
0.4
0.2
θ 0
=0.125
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
ε, Y
Figure 6.9: Solutions of the flame equations 6.67 and 6.68 for L = 0.5, θ a = 4, θ 0 =
0.125, θ i = 0.4, n = 1, a = 0 and δ = 1.
In order to obtain the desired approximation Y vs θ it is sufficient to observe
Fig. 6.3 corresponding to the example treated in §7, since this figure shows that even
when the activation energy θ a is zero it happens that near θ = 1 is 1 − ε ≫ 1 − Y ,
and also 1 − ε ≫ 1 − θ. This fact appears more evident for reactions with a high
activation temperature as shown in Fig. 6.9 which was calculated for a more realistic
case. Eq. (6.68) may be written
dY
dθ = L (1 − ε) − (1 − Y )
1 − θ 0
(1 − ε) − 1 − θ , (6.84)
1 − θ 0
and, in accordance with the preceding considerations, both 1 − Y and 1 − θ
1 − θ 0
are negligible
close to θ = 1 and the following approximation may be obtained for Eq. (6.84)
θ ≃ 1 :
dY
dθ ≃ L , (6.85)
1 − θ 0

154 CHAPTER 6. LAMINAR FLAMES
whose solution is
θ ≃ 1 : 1 − Y ≃ L
1 − θ 0
(1 − θ), (6.86)
this approximation must replace Eq. (6.76) when L is different from unity and furthermore
the value taken for L must be that corresponding to the temperature and
composition of the hot boundary conditions of the flame, T = T f .
If the activation energy has a small value and an approximation of Y vs θ that
will be valid within a wider range of temperatures is desired, then the linear approximation
(6.86) may be substituted by the following parabolic
Y ≃
( θ − θ0
1 − θ 0
) L
, (6.87)
which for θ = 1 coincides with (6.86) and for θ ≃ θ 0 it behaves like the solution
corresponding to the heating zone (ε ≡ 0). This can readily be verified. Fig. (6.10)
shows some of the curves (6.86) and (6.87).
1.0
0.8
L=0.5
L=0.75
0.6
L=1.5
L=2.0
Y
0.4
L=1.0
0.2
0.0
0.125 0.2 0.4 0.6 0.8 1.0
θ
Figure 6.10: Linear and parabolic approximations of Y vs θ given by Eqs. 6.86 and 6.87
for θ 0 = 0.125.
The preceding considerations prove that it is generally sufficient, when calculating
the integral to take for λ/λ f the value unity which corresponds to the hot
boundary, or some other approximation at the neighborhood of this point. Frequently
the following approximations are used
λ
λ f
≃ θ,
λ
λ f
≃ √ θ. (6.88)

6.9. SOLUTION OF THE FLAME EQUATIONS 155
1.4
1.3
Λ −1/2 / Λ −1/2
λ=λ λ=λ θ
f f
1.2
1.1
θ 0
=0.250
θ 0
=0.125
1.0
2 4 6 8 10 12 14 16
θ a
Figure 6.11: Ratio of flame velocities corresponding to λ = λ f and λ = λ f θ as a function
of the activation temperature θ a for two different values of the initial temperature
θ 0.
Figure 6.11, taken from Ref. [22], sets forth the scarce influence of the variation
of λ with temperature on the flame velocity, specially for high activation temperature.
Two different solution are compared in this figure, one assuming that λ is
constant through the flame λ = λ f ; and another where λ varies linearly with temperature
λ = λ f θ, in accordance with (6.88). The solution was computed for two different
values of the temperature of unburnt gases The deviations are under 10 per cent.
Once established that the preceding considerations solve the problem of the
calculation of integral I we shall proceed with the study of J.
Approximation of integral J
In order to approximate integral J, we must first consider that in the normal cases, this
is to say when θ a ≫ 1, this integral is much smaller than the other term of the left
hand side of Eq. (6.73), that is
J ≪ 1 − θ 0
. (6.89)
2
Such a property is illustrated in Fig. 6.12 which corresponds to a typical case
where the value of J is shown by a dot-area while the value for (1 − θ 0 )/2 is represented
by the area of triangle ABC.

156 CHAPTER 6. LAMINAR FLAMES
1.0
A
1
∫ (1−θ) dθ
θ0
0.8
0.6
θ
0.4
(1−θ 0
)/2
0.2
θ 0
=0.125
C
B
0.0
0.0 0.2 0.4 0.6 0.8 1.0
ε
Figure 6.12: Areas representing the values of J = R 1
θ 0
(1 − θ)dε and (1 − θ 0)/2 in a typical
case.
Hence, it results that a first approximation is obtained when J is neglected
respect to (1 − θ 0 )/2, in which case the value of Λ is given by expression
Λ = 1 − θ 0
. (6.90)
2I
Actually, this approximation was introduced by Zeldovich et al., although by
means of a more complicated justification. Furthermore, the authors applied an inadequate
approximation for the value of integral I which is the main reason for the
error resulting from their method as it was shown in Figs. 6.5 and 6.6. Professor
von Kármán, after a more accurate calculation of integral I, as before said, has improved
the approximation obtained by Zeldovich et al. This can be seen in Figs. 6.5
and 6.6 where the curves named Kármán correspond to (6.90) when using for I the
approximation developed in the preceding paragraph.
Von Kármán improved the approximation for J, through an approximation of
1 − θ vs 1 − ε, which is obtained by studying the behavior of the differential equation
(6.67) at the neighborhod of the hot boundary in the following way.
Close to point θ = 1, difference 1 − ε is an infinitesimal, whose order depends
only on the order of 1 − θ and it can be easily derived from (6.67) by introducing into
it the following approximations:

6.9. SOLUTION OF THE FLAME EQUATIONS 157
1) We neglect 1 − θ respect to (1 − θ 0 ) (1 − ε), in agreement with the procedure
used in obtaining (6.85).
2) We substitute (1 − Y ) by its approximation as a function of (1 − θ), given by
Eq. (6.86).
3) The remaining terms of (6.67) are substituted by the corresponding values at the
hot boundary.
Once these approximation are introduced into (6.67) one obtains the following
approximation for this equations, which is valid at the neighborhood of the hot
boundary
1 − θ ≪ (1 − ε)(1 − θ 0 ) :
dε
dθ ≃
Λ (
) n
L (1 − θ) n
1 − θ 0 (1 − θ 0 )(1 + a) 1 − ε . (6.91)
After integration the following approximation 1 − θ vs 1 − ε, is obtained
( n + 1
1 − θ ≃
2
) 1
1 − θ 0
Λ
n + 1 ( (1 − θ 0 )(1 + a)
L
which when taken into (6.75) and integrated, gives for J
J = n + 1 ( n + 1
n + 3 2
) 1
1 − θ 0
Λ
n + 1 ( (1 − θ 0 )(1 + a)
L
) n
n + 1 (1 − ε)
2
n + 1 , (6.92)
) n
n + 1 . (6.93)
By taking this value of J and the one for I given by Eq. (6.74) into Eq. (6.73), the
following equation results for the calculation of Λ
1 − θ 0
2
− n + 1 ( n + 1
n + 3 (1 − θ 0)
2
) 1
n + 1 ( 1 + a
L
) n
n + 1 Λ
− 1
n + 1 = ΛI, (6.94)
which may be readily solved, either with a graphical method or by iteration, using for
this last case as a first approximation the one given by (6.90) and as the correcting
term for successive iterations the right hand side of Eq. (6.94).
The result reached with (6.94) are represented in Figs. 6.5 and 6.6 where the
corresponding curves are named Kármán 2.
The approximation for J may still be improved by writing it in the form
J =
∫ 1
θ
(1 − θ) dε dθ, (6.95)
dθ
and using for dε an approximation of (6.67) derived as follows
dθ
(1 − ε) dε
dθ ≃
Λ (
L
1 − θ 0 (1 − θ 0 )(1 + a)
) n
λ
λ f
θ δ−n e −θ a
1 − θ
θ (1 − θ) n , (6.96)

158 CHAPTER 6. LAMINAR FLAMES
which through integration, gives
√ ( 2Λ L
1 − ε ≃
1 − θ 0 (1 − θ 0 )(1 + a)
) n 2
√ ∫ 1
λ (1 − θ) n
λ f θ n−δ e −θ 1 − θ
a
θ dθ. (6.97)
This expression, when taken into the denominator of (6.67) after we have neglected
1 − θ, supplies an approximation for dε/ dθ which introduced in (6.95) gives
the desired value for J. The result obtained from this approximation corresponds to
the curves named Kármán 3 of Fig. 6.5 and 6.6.
Tho approximation for J is even better when, after Sendagorta, we calculate
dε/ dθ in (6.95) by using for ε vs θ an approximation of the form
where P is a polynomial of the form
θ
ε = P e −θ 1 − θ
a
θ , (6.98)
P = 1 + a 1 (1 − θ) + a 2 (1 − θ) 2 + · · · + a m (1 − θ) m , (6.99)
whose coefficients must be determined by studying the behavior ε close to θ = 1 in
the differential equation (6.67).
For example, in the case of a first-order reaction (n = 1), it gives for J
J = H 0 + a 1 H 1 + a 2 H 2 + · · · + a m H m , (6.100)
where
H j =
∫ 1
θ i
(1 − θ) j e −θ a
1 − θ
θ dθ, (j = 1, 2, . . . m). (6.101)
This integral may be expressed by a law of recurrences starting from H 0 , which
in turn is expressed through the exponential integral in the following way
being
H 0 =
∫ 1
θ i
e −θ 1 − θ
a
θ dθ
= 1 − θ a e θ a
E 1 (θ a ) − θ i e −θ a
E 1 (θ) =
∫ ∞
θ
1 − θ i
θ i
e −z
z
+ θ a e θ a
E 1
(
θa
θ i
)
,
(6.102)
dz. (6.103)
For example, if in the preceding case one limits approximation (6.99) to the first two
terms,
P = 1 + a 1 (1 − θ), (6.104)

6.10. STRUCTURE OF THE COMBUSTION WAVE 159
in which case the only undetermined parameter would be a 1 to be obtained when
comparing the value for ( dε/ dθ) θ=1 given by (6.98),
( ) dε
= θ a − a 1 , (6.105)
dθ
θ=1
with the one resulting from (6.67)
( ) dε
(1 − θ 0 = ( )
dθ
θ=1
dε
(1 − θ 0 )
dθ
θ=1
The following second-degree equation will be obtained for a 1
. (6.106)
− 1
1 − θ 0
θ a − a 1 = Λ
(1 − θ 0 )(θ a − a 1 ) − 1 , (6.107)
which when solved gives the value of the undetermined parameter, which in turn,
when taken into (6.100) provides the desired approximation for J.
Therefore, this approximation of J depends on Λ, and when introduced into
(6.73.a), it supplies an equation for Λ, whose solution gives the eigenvalue which
solves the problem.
6.10 Structure of the combustion wave
Once determined the value of Λ which makes compatible the boundary conditions of
the flame equations, a simple numerical integration of such equations, through any of
the methods available, will give the distribution of temperature, concentrations, etc.
through the combustion wave. If the integration is performed by using system (6.67),
(6.68), the curves obtained will be as those shown in Fig. 6.9. When the structure is
desired on a physical plane it is sufficient to integrate the system of equations (6.64),
(6.65) and (6.66). However for this case one must select the position of origin of coordinates
which is undetermined. Fig. 6.13 shows the solution corresponding to the case
represented in Fig. 6.9. It is seen that the structure is analogous to the one represented
in Fig. 6.4, corresponding to the simplified case discussed in §6. Fig. 6.13, shows a
dot-line representation of the distribution of temperatures corresponding to the case of
pure thermal conduction, where chemical reaction is absent, which practically coincides
with the wave up to a θ = 0.5, proving that up this point the action of chemical
reaction is extremely small.
the same
As a characteristic length of the phenomenon it was adopted, for its solution,
l = λ f
mc p
(6.108)

160 CHAPTER 6. LAMINAR FLAMES
applied before in §6. This is the characteristic length of the process and it is a measure
of the thickness of the flame, since although theoretically it is infinite, almost the
entire variation of the temperature and composition of the gases takes place within a
narrow zone which is a few times the value of l, as shown in the solution computed in
Fig. 6.13.
By calculating the value of l corresponding to a typical case it may be verified
that it is a small fraction of a millimetre and, consequently, the thickness of the flame,
under normal conditions, has the same order of magnitude, as proven by experimental
observation.
This circumstance increases considerably the difficulties of the experimental
observation of the structure of the flame, since the recording of the distribution of
temperatures, for instance, must be performed within a very narrow zone and when
attempting to measure the composition the difficulties encountered are even greater.
Several attempts have been made of measurements of this nature, mainly by the research
team of the Applied-Physics Laboratory of the Johns Hopkins University. A
description of the present state of this problem may be found in Ref. [24]. The measurements
performed agree completely with theoretical predictions.
1.0
0.9
0.8
0.7
0.6
Y
ε
Y, θ, ε
0.5
0.4
0.3
0.2
0.1
θ
L=0.5
θ =4 a
θ 0
=0.125
θ =0.4 i
0
−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5
ξ=mc p
x/λ
Figure 6.13: Structure of the combustion wave represented in Fig. 6.9. The origin of coordinates
corresponds to the ignition temperature, where chemical reaction starts.

6.11. IGNITION TEMPERATURE 161
6.11 Ignition temperature
We have established that, except when activation temperature θ a is zero or very small,
the ignition temperature θ i bears no influence on the flame velocity unless it has a
value close to the temperature of the burnt gases θ = 1, or very close to initial temperature
θ 0 .
1.6
1.2
θ a
=0
Λ −1/2
0.8
θ a
=2
0.4
θ a
=4
θ a
=8
0.0
0.0 0.2 0.4 0.6 0.8 1.0
θ =0.125 0
θ i
Figure 6.14: Flame velocity as a function of the ignition temperature θ i for different values
of the activation temperature θ a.
This property is illustrated by Fig. 6.14, where the law of variation of Λ −1/2
(to which u 0 is proportional as before said), as a function of θ i is shown for a typical
case, corresponding to a first-order reaction, with a Lewis-Semenov number equal to
unity and a temperature of the unburnt gases θ 0 = 0.125. The figure represents the
solutions corresponding to values of θ a comprised between 0 and 8 and it is seen that
except for the first case (θ a = 0), there exists a wide range on values of θ i in which
Λ −1/2 is practically constant, as it was previously announced. 7
This property is of a general nature and it justifies the assumption of a ignition
temperature which eliminate the difficulty of the cold boundary.
7 For θ a ≫ 1 and θ i very close to θ 0 , it can be shown by means of asymptotic techniques that Λ −1/2 =
„
(1 − θ 0 ) ln` 1 ´ 1 − θ 0 ´«1/2
exp`−θa , so that when θ i → θ 0 there exists an exponentially
θ a(θ i − θ 0 )
θ 0
thin layer in which the transition for Λ −1/2 = constant to Λ −1/2 = ∞ occurs, Ed.

162 CHAPTER 6. LAMINAR FLAMES
0.05
0.04
θ a
=8
(1−θ)e −θ a (1−θ)/θ
0.03
0.02
0.01
θ l
0.00
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
θ
Figure 6.15: Curve showing the typical values of the integrand of I as a function of θ.
The reason for the insensibility of the solutions to θ i appears clearly in Fig. 6.15
where it is seen that the values of θ i comprised between θ 0 and θ l do not influence in
the value of integral I, since in this zone the integrand is practically zero. To the
contrary, if θ i is comprised between θ l and 1, then one disregards in the solution the
contribution of the part corresponding to θ l < θ < θ i . Since Λ −1/2 is proportional to
√
I, this explains the fact that Λ −1/2 should decrease and tend to zero when is θ i → 1.
The fact that the combustion velocity tends to ∞ when θ i → θ 0 may be physically
explained since then all the mass of gas burns simultaneously giving infinite
wave velocity. In order to give a theoretical explanation it would be necessary to
analyze in detail the differential system.
The fact that Λ −1/2 is independent from θ i enables the elimination of this value
in all the equations of the preceding paragraphs, in special in the lower limit of integral
I (Eq. 6.74), by substituting it for zero without changing the result.
6.12 General equations for the combustion wave in the
case of more than two chemical species
Hirschfelder and his collaborators were the first to give the general equations for the
combustion wave in the case of more than two chemical species [7]. A derivation of
these equations may be found in a paper by Kármán and Penner [6].

6.12. GENERAL EQUATIONS FOR THE COMBUSTION WAVE 163
Such equations are a particular case of those corresponding to the one-dimensional
stationary flow of a mixture of gases which were deduced in §8, chapter 3 of this book
and will be used as a starting point for the present study, although certain modifications
and simplifications will be required in order to adapt these equations to the
problem in which we are now interested.
Reaction equations
The reaction equation which corresponds to species i is the Eq. (3.63) of the above
mentioned chapter
m dε i
dx = w i, (i = 1, 2, . . . , l). (6.109)
where l is the number of different chemical species. In this expression w i is the
reaction velocity of species i, which, in the case of more than one chemical reaction,
is the resultant of the contributions of each one of them
r∑
w i = w ij . (6.110)
j=1
Here, w ij is the mass of species i produced per volume unit and time unit due
to reaction j, and r is the number of different reactions, counting as such, the two
opposite ones which take place in each reaction.
Velocity w ij corresponding to reaction j is given by expression (1.119) of
chapter 1 which will be written herein in the following way by bringing forth in it
the molar fractions X of the species in lieu of the mass fractions which were applied
in the case of two chemical species (this change is done for the reasons stated further
on the diffusion equations)
w ij = M i (ν ′′
ij − ν ′ ij)k j
( p
RT
) nj
l∏
s=1
X ν′ sj
s . (6.111)
In this expression M i is the molar mass of species i, ν ij ′ and ν′′ ij are the stoichiometric
coefficients of this species in the left and right sides of reaction equation j, n j =
l∑
ν sj ′ is the order of the reaction and k j is a function of the temperature of the mixture
s=1
which, as in the case of two species, is generally of the form
k j = A j T δ j
e −E j/RT . (6.112)
When (6.112) is taken into (6.111), the equation may be written for shortness
w ij = B j T δ j−n j
e −E j/RT
l ∏
s=1
X ν′ sj
s , (6.111.a)

164 CHAPTER 6. LAMINAR FLAMES
which is the form that will be used is the following. In the equation it has been taken
B j = A j
( p
R
) nj
. (6.113)
As aforesaid said in chapter 3, §8, even when there exist l reaction equations (6.109)
corresponding to i chemical species they are not all independent from one another.
In fact, the number of independent equations is the smallest of the following two: 1)
number of chemical reactions; 2) number of independent components of the mixture,
in the sense of the rule of phases.
If we take (6.111.a) into (6.109), the latter may be written in the form
m dε i
dx =
r∑
j=1
M i B j (ν ′′
ij − ν ′ ij)T δj−nj e −E j/RT
which is the form that will be used in the following.
l ∏
s=1
X ν′ sj
s , (6.109.a)
Diffusion equations
Since it is evident that
ρv = m, (6.114)
equation (3.68) of Chap. 3 may be written
ρY j v dj = m(ε j − Y j ), (j = 1, 2, . . . , l). (6.115)
If we now take this expression of the diffusion velocities into the equations of system
(3.75) (which in this case still holds if we nullify the term corresponding to pressure
diffusion since it is neglectable), we shall obtain the desired system of diffusion equations.
However, when written in this form, the said system has the inconvenient that
the derivatives of the mass fractions dY j / dx do not appear explicit as it would be
necessary in order to obtain the system of equations for the flame in the canonical
form. Such an inconvenience can be easily avoided by simply expressing the result
as a function of molar fractions X i in lieu of the mass fractions. In fact, in this case
the system of diffusion equations to be used is the one given by Eq. (2.28), which
after applied to the one-dimensional flow and neglecting the pressure and temperature
diffusions takes the form
dX i
dx =
l∑
j=1
X i X j
D ij
(v dj − v di ). (6.116)

6.12. GENERAL EQUATIONS FOR THE COMBUSTION WAVE 165
On the other hand, considering that 8
and
Y j = M j
M m
X j (6.117)
ρ = pM m
RT , (6.118)
the elimination of v di , v dj , Y j and ρ between these three last equations and (6.115),
finally gives the following system of diffusion equations in the form which will be
applied to the solution of the flame equations
dX i
dx = mRT
p
l∑
j=1
(
)
1 ε j ε i
X i − X j , (i = 1, 2, . . . , l). (6.119)
D ij M j M i
Energy equation
The energy equation is the one give in Eq. (3.74), when we disregard in it the terms
due to kinetic energy and viscosity, thus reducing it to the following
m
l∑
j=1
ε j h j − λ dT
dx
where e is a constant defined by the initial or final conditions.
= e, (6.120)
Before transforming this equations into the form in which it will be applied we
shall first consider the boundary conditions.
Boundary conditions
These conditions are analogous to those applied in the case of two chemical species,
namely,
Cold boundary, x → −∞:
T → T 0 , ε i → ε i0 , X i → X i0 ,
dT
dx → 0, dε i
dx → 0, dX i
→ 0. (6.121)
dx
Hot boundary, x → +∞:
T → T f , ε i → ε if , X i → X if ,
dT
dx → 0, dε i
dx → 0, dX i
→ 0. (6.122)
dx
8 See Chap. 1, Eq.1.34

166 CHAPTER 6. LAMINAR FLAMES
In (6.121), T 0 , ε i0 = Y i0 and X i0 = Y i0 M m0 /M i are the values of T , ε i and
X i corresponding to the temperature and composition of the unburnt gases. M m0 is
the value of M m under these conditions.
Likewise, in (6.122), T f , ε if = Y if and X if = Y if M mf /M i are the temperature
and composition corresponding to the adiabatic combustion products in chemical
equilibrium.
A recount of conditions, similar to the one performed in §6 for the case of two
chemical species, shows that conditions are superabundant and that their compatibility
imposes that m takes an “eigenvalue” which determines the propagation velocity of
the flame.
Boundary conditions (6.121) and (6.122) determine the value of constant e in
(6.120), when it is particularized for both extremes, thus obtaining
l∑
l∑
e = m ε j0 h j0 = m ε jf h jf . (6.123)
j=1
j=1
In this expression the equality of the last two terms expresses simply that the combustion
is adiabatic and at constant pressure.
form
Substituting (6.123) into (6.120) the latter may be written as
λ dT
dx = m
l∑
(ε j h j − ε jf h jf ). (6.124)
j=1
In Chap. 1, §3, it was established that the enthalpy of a diluted gas is of the
∫ T
h j = h 0 j + c pj dT, (6.125)
T 0
where h 0 j is the formation enthalpy9 of species j at temperature T 0 and c pj is the
specific heat of the same at constant pressure.
When taking (6.125) into (6.124) the latter is written
⎛
⎞
λ dT
dx = m ⎝ ∑ ∫ T ∑
h 0 jε j + c pj ε j dT ⎠
j
T 0 j
⎛
⎞
− m ⎝ ∑ ∫ Tf ∑
h 0 jε jf + c pj ε jf dT ⎠ .
j
T 0
j
(6.126)
9 To avoid confusion with h j0 , total enthalpy of species j at T 0 , the notation is slightly different from
that used in chapter 1. Ed.

6.12. GENERAL EQUATIONS FOR THE COMBUSTION WAVE 167
In this equation, expression ∑ c pj ε j depends on temperature T and on the values for
j
mass fluxes ε j of the species. In order to simplify this expression we adopt a mean
value c p independent from temperature and the composition of the mixture so that
Eq. (6.126) may be written in the form
λ dT (∑
)
dx = m h 0 j(ε j − ε jf ) + c p (T − T f ) . (6.127)
j
The value for c p is derived from (6.127) when expressing the condition that
(6.127) vanishes at the cold boundary in agreement with (6.121) obtaining
∑
j
c p =
h0 j (ε j0 − ε jf )
, (6.128)
T f − T 0
or else since, as before said, ε jf = Y jf and ε j0 = Y j0 due to the fact that diffusion is
absent in both limits
c p =
∑
j h0 j (Y j0 − Y jf )
T f − T 0
. (6.129)
If, as before done, we introduce dimensionless temperature θ = T/T f into Eq. (6.127),
this may be written
λ dθ (
dx = mc p θ − 1 +
l∑
j=1
h 0 )
j
(ε j − ε jf ) . (6.130)
c p T f
It is also convenient here to eliminate variable x, by dividing the reaction equation
(6.109.a) and diffusion Eq. (6.119) by (6.130), as it was done for the case of two
chemical species. Thus the system of the flame equations reduces to the following
Reaction equations
dε i
dθ = λ λ f
r∑
j=1
(ν ′′
ij − ν ′ ij)Λ ij θ δ j−n j
e −θ aj/θ
θ − 1 +
l∑
q j (ε j − ε jf )
j=1
l ∏
s=1
X s ν ′ sj
, (i = 1, 2, . . . , l), (6.131)
where, by analogy with the case of two chemical species, we have taken
Λ ij = M iλ f B j T δ j−n j
f
m 2 , (6.132)
c p
θ aj =
E j
RT f
, (6.133)

168 CHAPTER 6. LAMINAR FLAMES
and
q j =
h j
c p T f
. (6.134)
In comparison with the case of only two chemical species, we observe here the appearing
of a set of parameters Λ ij . However, all these parameters have only one unknown
quantity, the value of m, this is to say the flame velocity, therefore the set Λ ij may be
expressed as a function of just one of the parameters.
Diffusion equations
l∑
( )
Mi
L ij X i ε j − X j ε i
dX
M
i
dθ = j=1
j
, (i = 1, 2, . . . , l), (6.135)
l∑
θ − 1 + q j (ε j − ε jf )
j=1
where
L ij =
λRT
M i c p pD ij
, (6.136)
Boundary conditions
The boundary conditions reduce to the following
θ → θ 0 : ε i → ε i0 , X i → X i0 , (6.137)
θ → 1 : ε i → ε if , X i → X if . (6.138)
Like in the case of two chemical species the difficulty at the cold boundary may be
eliminated by introducing an ignition temperature θ i whose value does not influence
the result when the reduced activation energies θ aj , or at least some of them, are large
as normally happens in practical cases for the same reasons stated in detail in the two
species case.
A recount of the number of boundary conditions when compared to the number
of equations proves that the conditions are superabundant and hence, an eigenvalue of
m, this is, one of the values of Λ ij , must exist which makes compatible the said
conditions and therefore the problem is essentially the same than in the case of two
species only.

6.13. OZONE DECOMPOSITION FLAME 169
Be Λ ij the parameter chosen to express the rest of them as functions of it. Once
its value is known, the value for velocity u 0 of the flame, by virtue of Eqs. (6.22) and
(6.132), is
√
u 0 =
M iλ f B j T δj−nj
f
ρ 2 0 c p
Λ −1/2
ij . (6.139)
6.13 Ozone decomposition flame
The ozone decomposition flame in a mixture of this gas with oxygen is the first example
we have chosen among the few cases of flame propagation which have been
calculated. Its first theoretical study was due to Lewis and von Elbe [8] who thus became
the first to analyze flame propagation considering, in addition to the effects of
heat transfer, those of the diffusion of reactants and products, as well as the influence
of the radicals (oxygen atoms) in the process. For this purpose they applied the law of
chemical reaction which includes the influence of temperature and the concentrations
of the various species on the reaction rate. The three chemical species in the process
are ozone, molecular oxygen and atomic oxygen. Lewis and von Elbe simplified the
problem by adopting the following assumptions:
1) The Lewis-Semenov number for the mixture O 2 - O 3 is equal to unity. This
assumption enables the expression of the concentrations of ozone and molecular
oxygen as functions of temperature.
2) The chemical processes take place in accordance with the following kinetic
scheme
O 3 ⇆ O 2 + O, (6.140)
O + O 3 → 2 O 2 . (6.141)
The first of these two reaction equations expresses that the concentration of
atoms of oxygen at each point of the flame is the one that would correspond
to a mixture of O, O 2 and O 3 in chemical equilibrium at the temperature and
composition of the point.
Having stated the problem, and if, furthermore, we take into account that the
molar fraction of atomic oxygen is much smaller than unity, its solution is straightforward.
In fact, the first of the above assumptions allows the expression of the molar
fractions of O 2 and O 3 as functions of temperature, as shown in §9 of this chapter,
and equation (6.140) allows the same for the molar fraction of O.

170 CHAPTER 6. LAMINAR FLAMES
Thus, the problem reduces to the integration of only one reaction equation
(that corresponding to O 2 or to O 3 ), which may be performed by means of one of the
methods enumerated in the aforesaid.
Lewis and von Elbe carried out a more arduous calculation, through the numerical
integration of the resulting equation. A detailed information on their calculations
may be found in the above reference or in [9]. Even when the values of the flame velocity
obtained from this calculation are several times larger than those experimentally
observed, considering the state of knowledge at the time, their work was published,
the fact that predicted and experimental values were of the same order of magnitude
was considered an important success.
Recently, von Kármán and Penner [10] have recomputed the flame velocity for
a set of mixtures of O 2 and O 3 , using the kinetic scheme proposed by Lewis and von
Elbe, but applying the semi-analytical methods developed in §9. The results agree
with those obtained by Lewis and von Elbe and they shown that the influence of the
composition of the mixture on velocity of the flame obtained through this procedure,
differ essentially from those observed by experimentation. This discrepancy is basically
due to the fact that the set of chemical reactions proposed by Lewis and von Elbe
and, in particular, the distribution of oxygen atoms within the flame, are different from
the actual ones.
Hirschfelder and his associates [25] have recently calculated the same flame,
applying the following complete kinetic scheme
O 3 + G → O + O 2 + G 1)
O + O 2 + G → O 3 + G 2)
O + O 3 → 2 O 2 3)
2 O 2 + G → O + O 3 4)
O 2 + G → 2 O + G 5)
(6.142)
2 O + G → O 2 + G 6)
This computation was carried out by means of an arduous numerical integration
for which electronic computers were used.
Later on, von Kármán and Penner [6] performed a simplified analysis of the
problem, utilizing the same kinetic scheme and physico-chemical constants as Hirschfelder
but applying Kármán’s analytical method of integration, and postulating a distribution
of oxygen atoms within the flame determined through the extension of the
steady state assumption which is widely used in classical Chemical Kinetics.

6.13. OZONE DECOMPOSITION FLAME 171
The following is a summary of the von Kármán-Penner study which can be
found fully described in the papers of Refs. [6] and [10].
If we assign subscripts 1, 2 and 3 to species O, O 2 and O 3 respectively, and
making use of the flame equations derived in §11 of this chapter, we obtain the following
system.
Reaction equation
It is sufficient to write two equations corresponding, for example, to species O 2 and
O 3 thus obtaining after Eq. (6.131)
dε 1
dθ = λ λ f
[
Λ 11 θ −3/2 e −θ a1/θ X 3 − Λ 12 θ −5/2 e −θ a2/θ X 1 X 2
− Λ 13 θ −3/2 e −θa3/θ X 1 X 3 + Λ 14 θ −3/2 e −θa4/θ X2
2 ]
(6.143)
+ 2Λ 15 θ −3/2 e −θa5/θ X 2 − 2Λ 16 θ −5/2 e −θa6/θ X1
2
[
] −1,
· θ − 1 + q 1 (ε 1 − ε 1f ) + q 2 (ε 2 − ε 2f ) + q 3 (ε 3 − ε 3f )
dε 3
dθ = λ [
−Λ 31 θ −3/2 e −θa1/θ X 3 + Λ 32 θ −5/2 e −θa2/θ X 1 X 2
λ f
]
− Λ 33 θ −3/3 e −θa3/θ X 1 X 3 + Λ 34 θ −3/2 e −θa4/θ X2
2 (6.144)
[
] −1.
· θ − 1 + q 1 (ε 1 − ε 1f ) + q 2 (ε 2 − ε 2f ) + q 3 (ε 3 − ε 3f )
Diffusion equations
Likewise, the diffusion equations corresponding to O and O 3 are, according to (6.135)
dX 1
dθ = L ( 1
12 2 X ) (
1ε 1 − X 2 ε 1 + 1 L13
3 X )
1ε 1 − X 3 ε 1
θ − 1 + q 1 (ε 1 − ε 1f ) + q 2 (ε 2 − ε 2f ) + q 3 (ε 3 − ε 3f ) , (6.145)
dX 3
dθ = L 31 (3X 1 ε 3 − X 3 ε 1 ) + L 32
( 3
2 X 3ε 2 − X 2 ε 3
)
θ − 1 + q 1 (ε 1 − ε 1f ) + q 2 (ε 2 − ε 2f ) + q 3 (ε 3 − ε 3f ) . (6.146)
The following Table 6.1shows the values of the physico-chemical constants
corresponding to the six reactions of Eq. (6.142).

172 CHAPTER 6. LAMINAR FLAMES
Reaction
Order
n j
δ j
E j
cal/mol
A j
1 2 1/2 24.14 10.56 × 10 12
2 3 1/2 0.00 0.230 × 10 12
3 2 1/2 6.00 7.15 × 10 12
4 2 1/2 99.21 2.93 × 10 12
5 2 1/2 117.34 8.92 × 10 12
6 3 1/2 0.00 0.482 × 10 12
Table 6.1: Physico-chemical parameters for reactions 6.142.
The boundary conditions for this system are as follows:
a) Hot boundary.
At the hot boundary (θ = 1) the concentration of ozone is zero and the concentration
of atoms of oxygen is very small. Therefore it can be written
θ = 1 : ε 1 ≡ ε 1f ≃ 0, ε 3 ≡ ε 3f = 0, (6.147)
X 1 ≡ X 1f ≃ 0, X 3 ≡ X 3f = 0. (6.148)
b) Cold boundary.
At the cold boundary since no chemical reactions has been yet produced, the
mass flux of ozone is the one corresponding to the initial mixture and the one of
atomic oxygen is zero
θ = θ i : ε 1 ≡ ε 10 = 0, ε 3 = Y 30 (datum). (6.149)
Simplification of the equations
Before proceeding to integrate the system, we will simplify it, following von Kármán
and Penner. For the purpose, we shall begin by analyzing the values of the physicochemical
constants of the reactions which appear summarized in the above Table 6.1,
starting by the activation energies.
We see in this table that for reactions 4 and 5 in (6.142), this is, those producing
atomic oxygen from molecular oxygen, the activation energies are much larger than
for the rest, and consequently the reaction velocities in which they participate will be
very small and may be neglected. Hence, we can eliminate the terms corresponding
to these reactions from equations (6.143) and (6.144).
Let us now consider, coefficients Λ ij which are the ones that may also influence
the order of magnitude of the reaction velocities. In order to simplify this comparison,

6.13. OZONE DECOMPOSITION FLAME 173
in the following table we give their values, referred to the value of Λ 31 which we are
adopting as a reference according to the reasons stated in the foregoing.
Λ 11
Λ 31
= 0.333
Λ 14
Λ 31
= 0.925
Λ 32
= 3.486 × 10 −3 T −1
f
Λ 31
Λ 12
= 1.162 × 10 −3 T −1
f
Λ 31
Λ 15
Λ 31
= 0.282
Λ 33
Λ 31
= 0.677
Λ 13
Λ 31
= 0.226
Λ 16
= 2.435 × 10 −3 T −1
f
Λ 31
Λ 34
Λ 31
= 0.277
Table 6.2: Values of the ratios Λ ij/Λ 31 appearing in Eqs. (6.145) and (6.146).
It appears here that Λ 12 , Λ 16 and Λ 32 , which correspond to reactions 2 and 6
in (6.142), are much smaller than the others, due to the following two reasons. First,
because the corresponding coefficients A j are smaller, as shown by Table (6.1), and
second because they are reactions requiring triple collision, as shown by Eq. (6.142),
which explains the presence of term T −1
f
in these coefficients.
Hence, it results that we can also eliminate the corresponding terms in reaction
equations (6.143) and (6.144).
Finally, with reference to the denominator of these equations, it can also be
simplified, keeping in mind that:
1) In accordance with boundary conditions (6.147) it is ε 1f = ε 3f = 0.
2) Moreover, ε 1 is very small compared to unity throughout the flame, and therefore,
the corresponding term may be neglected.
3) Finally, in accordance with Eq. (6.124) and considering that h 0 2 is zero, it is
q 2 = 0.
following
Taking into account these considerations, the said denominator reduces to the
θ − 1 + q 1 (ε 1 − ε 1f ) + q 2 (ε 2 − ε 2f ) + q 3 (ε 3 − ε 3f )1 ≃ θ − 1 + q 3 ε 3 . (6.150)
Consequently if we introduce the above simplifications into the system of reaction
equations (6.143) and (6.144), this system reduces to
dε 1
dθ = Λ λ θ ( )
−3/2 e −θa1/θ X 3 − e −θa3/θ X 1 X 3
, (6.151)
λ f θ − 1 + q 3 ε 3
dε 3
dθ = −Λ λ λ f
θ −3/2 ( e −θ a1/θ X 2 + e −θ a3/θ X 1 X 3
)
θ − 1 + q 3 ε 3
. (6.152)

174 CHAPTER 6. LAMINAR FLAMES
Steady state assumption
In spite of the simplification introduced into the system, it is still very difficult to
perform its integration by semi-analytical methods.
In an effort to find further simplifications, von Kármán and Penner checked the
applicability of the steady state assumption for the distribution of atomic oxygen. This
assumption is expressed through nullifying the reaction velocity of the oxygen atoms,
thus obtaining
e −θ a1/θ X 3 − e −θ a3/θ X 1 X 3 = 0, (6.153)
from which we derive the following distribution of atomic oxygen throughout the
flame
X 1 = e − θ a1 − θ a3
θ . (6.154)
10
9
8
7
Hirschfelder
Karman−Penner
6
10 4 X 1
5
4
3
2
1
0
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
θ
Figure 6.16: Mole fraction of atomic oxygen as a function of the temperature for the ozone
decomposition flame.
Figure 6.16 taken from [6] compares the distribution of X 1 corresponding to
this assumption, with that obtained by Hirschfelder through numerical integration of
the flame equations, without making use of the assumption. It is seen that the result is
completely satisfactory and it plainly justifies the assumption of Kármán and Penner,
which is not applicable just within a narrow zone very close to the maximum temperature,
at which, since the concentration of ozone is practically reduced to zero there
are other reactions, different from 1 and 3 in (6.142) controlling the formation of O.

6.13. OZONE DECOMPOSITION FLAME 175
But this zone is unimportant when studying the flame structure and in particular its
propagation velocity.
The applicability of the steady state assumption is an important contribution
to the theory of flames which essentially simplifies. The success reached with ozone
flames has encouraged the idea of applying this assumption to the study of similar
practical cases, as will be seen later on.
Recently, however, the applicability of such an assumption has been discussed,
among others, by Campbell [26], Giddings and Hirschfelder [27] and Spalding [28].
They based their objections first on the fact that the crossing of the flame is so rapid
that there is not sufficient time for the formation of radicals to reach the level required
by the condition of equilibrium established by the said assumption. On the other hand,
even if such level were reached the diffusion of radicals would change substantially
the distribution. Although other studies by Gilbert and Altman [29] and by Millán
and Sanz [30] tend to confirm the applicability of the steady state assumption for the
cases studied in the foregoing paragraphs, however, the problem cannot be considered
as solved, until a general study enables the establishment of the conditions that must
be satisfied in a flame so that this assumption be valid.
Let us proceed with the study of the ozone flame. The application of the steady
state assumption eliminates Eq. (6.151) and reduces (6.152) to the following
where, to simplify notation, we have written
dε 3
dθ = −2Λ λ λ f
θ −3/2 e −θa1/θ X 3
θ − 1 + q 3 ε 3
, (6.155)
Λ 13 = Λ = M 1λ f B 3
. (6.156)
m 2 c p T 3/2
f
Taking into account the preceding considerations, the diffusion equation reduces to
where also, for shortness, we write
dX 3
dθ = L 3
2 X 3 − ε 3 − 1 2 X 3ε 3
θ − 1 + q 3 ε 3
, (6.157)
L 13 = L =
λRT
M 1 c p pD 13
. (6.158)
The problem reduces now to the integration of these two equations with the following
boundary conditions
θ = θ i : ε 3 = 0, (6.159)
θ = 1 : ε 3 = X 3 = 0. (6.160)

176 CHAPTER 6. LAMINAR FLAMES
The compatibility of these three conditions determines, the “eigenvalue” of Λ, as established
in §9, and once it is determined,we obtain with it the value for the flame
velocity by virtue of Eq. (6.139).
The integration of the system is very simple using Kármán’s method, in the
following way. In the zone of interest, this is for θ close to 1, it is ε 3 ≫ X 3 and
ε 3 ≫ 1 − θ, therefore a first approximation of Eq. (6.157) is the following
where considering that for θ = 1 it is X 3 = 0, we obtain
dX 3
dθ ≃ − L q 3
, (6.161)
X 3 ≃ − L q 3
(1 − θ). (6.162)
If this value is taken into (6.155) and integrating between θ = 1 and θ = θ i ≃ 0 as
expressed in §9 of this chapter, we obtain the desired value of Λ.
Von Kármán and Penner have calculated the result for X 30 = 0.25, 0.40, 0.50,
0.75, and 1.00. Furthermore, these authors calculated as well the flame velocity for
the same values by means of the Lewis-von Elbe chemical scheme.
A comparison of results with those obtained through experimentation by Grosse
[31] proves that the approximation of the theoretical values reached by Kármán and
Penner is satisfactory since the deviations, in both cases, are under 25 per cent.
6.14 Hydrazine decomposition flame
Hydrazine combustion has been experimentally studied by Murray and Hall [32].
Tests were performed by burning mixtures of hydrazine and water vapours in a Bunsen
burner at ambient pressure. The flame propagation velocity was obtained by measuring
the area of the luminous cone of the flame.
The analysis of the combustion products shown that they correspond to the
overall reaction
N 2 H 4 ⇄ 1 2 H 2 + 1 2 N 2 + NH 3 . (6.163)
Murray and Hall also calculated the theoretical propagation velocity of the
flame for one of the mixtures experimentally analyzed (97.2% hydrazine). They assumed
that such velocity was determined by the unimolecular decomposition reaction
N 2 H 4 → 2NH 2 , (6.164)

6.14. HYDRAZINE DECOMPOSITION FLAME 177
which is the initial reaction in the complex process that is stoichiometrically represented
by (6.163).
Making use of the thermal theory of Zeldovich and Frank-Kamenetskii [33]
and using the specific reaction rate
k = 4 × 10 12 e −60 000/RT , (6.165)
proposed by Szwarc [34], they obtained a theoretical propagation velocity of the flame
of 110 cm/s compared to the 185 cm/s obtained from experiments.
More precise calculations were performed by Hirschfelder [35] and by Kármán
and Penner [6], making use of Murray and Hall’s kinetic scheme with results that
confirm those obtained by Murray and Hall. Hirschfelder and Kármán and Penner
also studied the influence over the propagation velocity of the flame of the diffusion
of the different species. In this case they obtain values considerably smaller than
the experimental ones since diffusion reduces substantially the value of the flame’s
propagation velocity.
The following table summarizes the results of the works by Murray-Hall and
Kármán-Penner.
Experimental value
Theoretical values
Refs. Murray-Hall Murray-Hall Kármán-Penner
Zero diffusion Zero diffusion Non-zero diffusion 10
u 0 185 cm/s 110 cm/s 95 cm/s 42 cm/s
Table 6.3: Theoretical and experimental propagation velocities of the flame in a mixture of
vapour of hydrazine and water. Mass fraction of hydrazine = 0.972. Pressure =
1 atm. Initial temperature of the mixture = 150 ◦ C.
If the reaction controlling the process is the one given by Eq. (6.164) with the
specific reaction rate (6.165), the following two conditions must be satisfied:
1) Since chemical reaction (6.164) is of the first order, the propagation velocity of
the flame must be inversely proportional to the square root of pressure.
2) Since the activation energy E of reaction (6.164) is 60 000 cal/mol and the flame
propagation velocity is approximately proportional to exp (−E/2RT f ) according
to Eq. (6.72) where T f is the flame temperature, when such velocity is represented
as a function of 1/T f in the logarithmic scale one must evidently obtain
a slope equal to −E/2R.
10 For a Lewis number equal to unity.

178 CHAPTER 6. LAMINAR FLAMES
With the experiments made by Murray and Hall it is not possible to verify the
first conclusion since they were all performed at ambient pressure, and the second
conclusion was not checked by the authors. In order to verify both, Adams and Stock
[36] measured the combustion velocity of hydrazine at different pressures. For this
purpose, they stabilized the flame over a column of liquid hydrazine contained in a
capillary tube and measured the recession velocity of the liquid level as its vapour
burnt. This method is not practical to measure the absolute propagation velocity of
the flame due to the irregularity of its front but is very convenient for the comparative
study intended by these authors. The results of their experiments show that the influence
of pressure is approximately that corresponding to a first order reaction. The
discrepancies could be due to the experimental method used.
As for the apparent activation energy it decreases with the drop in flame temperature
and it is in all cases way under the 60 000 cal/mol that correspond to the
process proposed by Szwarc and applied to the theoretical studies mentioned herein.
Such result led Adams and Stocks to conclude that the decomposition of hydrazine
should occur through a complicated system of chain reactions of which (6.164)
is only the initial one. As a result of a detailed discussion of all possible reactions the
above mentioned authors proposed a simplified kinetic scheme [36], which summarizes
the actual process. This proposed scheme considers only three chemical species:
reacting species A (hydrazine), stable products C of the combustion (mixture of NH 3 ,
N 2 , H 2 , etc.), and one radical B, that propagates the chain (which could be NH 2 , H,
etc). The Adams and Stock simplified scheme is a follows
A → 2 B, (6.166)
B + A → B + 2 C, (6.167)
B + B + X → 2 C + X. (6.168)
Of the three reactions, the first initiates the chain, the second is the propagation reaction
and the third the chain breaking reaction.
Lately, Spalding [37] has calculated the propagation velocity of the flame with
the scheme proposed by Adams and Stocks and he applied an interesting method of
numerical integration developed by him. Spalding calculates two cases corresponding,
respectively, to a cold flame and to one of the hot flames experimented by Murray
and Hall. In both cases he gets results in close agreement with those experimentally
obtained by Murray and Hall. These results are summarized in Table 6.4.

6.14. HYDRAZINE DECOMPOSITION FLAME 179
Theoretical
(Spalding)
Experimental
Hot flame 190 cm/s 185 cm/s (Murray-Hall)
Cold flame 12 cm/s 10 cm/s (Adams-Stocks)
Table 6.4: Propagation velocity of the flame in cm/s.
As Spalding points out the excellent numerical agreement is casual to a certain
extent since there are doubts respect to the correct values of the transport coefficients.
Moreover, Spalding did not include the influence of the variation of such coefficients
with the temperature across the flame. More interesting is the fact that Spalding’s
results predict correctly the influence of the combustion temperature on the velocity
of the flame.
Spalding also concludes that the propagation velocity of the flame that would
be obtained when calculating the radical distributions by means of the steady state
assumption would be much too large. He bases this conclusion upon the fact that
the distribution of radicals obtained with the steady state assumption is considerably
larger, at temperatures close to combustion temperature, than the distribution given by
correct calculation.
Millán and Sanz [30] calculated the propagation velocity of hydrazine decomposition
flame applying the same simplified model proposed by Adams and Stocks.
Two cases were considered, the first assuming steady state for concentration of radicals
and solving the flame equations through Kármán’s method; the second, for which
an approximate calculation method was developed to obtain flame velocity without
considering the steady state assumption. The authors concluded that the results obtained
applying the steady state assumption to the radical concentrations are satisfactory
when compared to those corresponding to a more correct distribution for the
same.
Simultaneously, Gilbert and Altman [38] obtained the flame propagation velocity
corresponding to complete kinetic model proposed by Adams and Stocks [36]
through the application of the Boys-Corner method in finding the solution of the flame
equations and calculating the concentration of radicals throughout the flame under the
steady state assumption introduced by Kármán and Penner [6].
Recently [39] a new analysis of the problem was conducted using the complete
kinetic model of Adams and Stocks, and adopting the steady state assumption for the
determination of the concentration of radicals, with Kármán’s method in calculating
the solution.

180 CHAPTER 6. LAMINAR FLAMES
Kinetic model of the reaction of hydrazine decomposition
The mechanism of decomposition proposed by Adams and Stocks is as follows
N 2 H 4 → 2NH 2 1)
N 2 H 4 + NH 2 → NH 3 + N 2 H 3 2)
N 2 H 3 → N 2 + H 2 + H 3)
H + N 2 H 4 → 2NH 3 + NH 2 4)
H + NH 2 + X → NH 3 + G 5 c)
(6.169)
H + H + X → H 2 + G 5 d)
In these equations, G represents any one of the particles of the mixture.
In order to identify the molecules of the components of the mixture the following
subscripts will be used
1 ↔ N 2 H 4 reactant
2 ↔ NH 3 product
3 ↔ N 2 product
4 ↔ H 2 product
(6.169.a)
5 ↔ NH 2 radical
6 ↔ N 2 H 3 radical
7 ↔ H radical
According to the law of the mass action, the velocity reaction w i of the radicals
must be expressed as a function of the mole concentration in the following way.
For NH 2 :
For N 2 H 3 :
For H :
w 5
M 5
= 2k 1 cX 1 − k 2 c 2 X 1 X 5 + k 4 c 2 X 1 X 7 − k 5c c 3 X 5 X 7 , (6.170)
w 6
M 6
= k 2 c 2 X 1 X 5 − k 3 cX 6 , (6.171)
w 7
M 7
= k 3 cX 6 − k 4 c 2 X 1 X 7 − k 5c c 3 X 5 X 7 − 2k 5d c 3 X 2 7 . (6.172)
Likewise, the hydrazine decomposition velocity is
− w 1
M 1
= k 1 cX 1 + k 2 c 2 X 1 X 5 + k 4 c 2 X 1 X 7 . (6.173)

6.14. HYDRAZINE DECOMPOSITION FLAME 181
Here, X i are the mole fractions of the different species, k i the specific reaction velocities
given by Eq. (6.112) (subscript of k indicates the corresponding reaction in accordance
with the number assigned to them in the preceding paragraph) and c = ρ/M
is the mole concentration per cm 3 .
The following values, also applied by Gilbert and Altman [38], are adopted for
specific velocities
k 1 = 4 × 10 12 e −60 000/RT s −1 (6.174)
k 2 = 10 13 e −4 600/RT cm 3 mol −1 s −1 (6.175)
k 4 = 10 13 e −7 000/RT cm 3 mol −1 s −1 (6.176)
k 5c = 5 × 10 15 cm 6 mol −2 s −1 (6.177)
k 5d = 10 16 cm 6 mol −2 s −1 (6.178)
Reaction velocity of hydrazine under to steady state assumption for
radicals
The steady state assumption is expressed by making to reaction velocities of the radicals
referred by 5, 6 and 7 in (6.187.a) equal to zero. Thereby
k 2 cX 1 X 5 = k 3 X 6 , (6.179)
cX 1 (k 2 X 5 − k 4 X 7 ) = 2k 1 X 1 − k 5c c 2 X 5 X 7 , (6.180)
cX 1 (k 2 X 5 − k 4 X 7 ) = k 5c c 2 X 5 X 7 + 2k 5d c 2 X7 2 . (6.181)
Since mole fraction X 7 of radical H is small throughout the reaction with
respect to reactant X 1 , it occurs that k 5c cX 7
is negligible when compared to 2k 1
.
k 2 X 1 k 2 cX 5
Hence Eqs. (6.180) and (6.181) may be written
k 2 X 5 − k 4 X 7 = 2k 1
c , (6.182)
k 2 X 5 − k 4 X 7 = 2k 5dcX 2 7
X 1
. (6.183)
Consequently, the concentration of radicals H is supplied by
√
k1 √
X 7 = X1
k 5d c 2 . (6.184)

182 CHAPTER 6. LAMINAR FLAMES
From (6.182) and (6.179) the following expressions may be deduced for mole
fractions X 5 of NH 2 and X 6 of N 2 H 3 :
X 5 = 2k 1 + k 4 cX 7
, (6.185)
k 2 c
X 6 = 2k 1 + k 4 cX 7
k 3
X 1 . (6.186)
If the values for X 7 and X 5 given by (6.184) and (6.185) are substituted into
the hydrazine reaction velocity given by (6.173), the following is obtained
− w 1
M 1
= cX 1 (3k 1 + kX 1/2
1 ), (6.187)
where
( ) 4k
2 1/2
k = 4 k 1
= 4 × 10 11 e −37 000/RT . (6.188)
k 5d
It is easily seen that the first term in the parenthesis of Eq. (6.187) may be neglected
respect to the second. Hence it finally results
− w 1
M 1
= kcX 3/2
1 . (6.189)
Let us assume that the proportion between concentrations of products NH 3 , N 2 and
H 2 is the one corresponding to the complete reaction, and the concentrations of radicals
are very small, then the following relations will be obtained between the mole
fractions and the mass fractions of the main species
Furthermore, the mean mole mass of the mixture will be
X 1 + 2X 2 = 1, (6.190)
Y 1
= X 1
M 1 M , (6.191)
Y 2
= X 2
M 2 M . (6.192)
M = M 1 X 1 + X 2
(M 2 + 1 2 M 3 + 1 2 M 4
From the system of equations (6.190) to (6.193) it results
)
. (6.193)
1 − 32
X 1 = 17 Y 2
1 + 32 . (6.194)
17 Y 2
The reaction velocity of ammonia results to be
⎛
w 2
= − w 1 − 32 ⎞3/2
1 ⎜
= kc ⎝
17 Y 2
⎟
M 2 M 1
1 + 32 ⎠ . (6.195)
17 Y 2

6.14. HYDRAZINE DECOMPOSITION FLAME 183
Energy equation
When mean specific heat is assumed to be constant and the influence of radicals on
the energy of the mixture is neglected, the energy equation may be written
( )
λ dT 17
m dx = q r
32 − ε 2 − c p (T f − T ), (6.196)
where q r is the reaction heat per gram of ammonia produced. If reduced temperatures
θ = T T f
and θ 0 = T 0
T f
(6.197)
are introduced into Eq. (6.196) and taking into consideration the conditions at the cold
boundary, θ = θ 0 , ε 2 = 0, this equation may be written
where
λ
mc p
Chemical reaction and diffusion equations
The reaction equation for ammonia is
dθ
dx = (1 − θ 0)(1 − ε) − (1 − θ), (6.198)
ε = 32
17 ε 2. (6.199)
m dε 2
dx = w 2. (6.200)
The diffusion equation corresponding to the same is given by Fick’s law and written
dY 2
dx = m (Y 2 − ε 2 ). (6.201)
ρD 2m
Coordinate x is eliminated from the above two equations through Eq. (6.198), and the
following system results
where
and
dε
dθ = 32
17
is the Lewis-Semenov number.
λ
m 2 c p
w 2
(1 − θ 0 )(1 − ε) − (1 − θ) , (6.202)
dY
dθ = L Y − ε
(1 − θ 0 )(1 − ε) − (1 − θ) , (6.203)
Y = 32
17 Y 2, (6.204)
L =
λ
ρDc p
(6.205)

184 CHAPTER 6. LAMINAR FLAMES
where
Taking (6.195) into (6.202)
( ) 3/2 1 − Y
e −θ 1 − θ
a
θ
dε
dθ = Λ 1 + Y
(1 − θ 0 )(1 − ε) − (1 − θ) , (6.206)
Λ = 128 × 10 11 e −θ a
If a linear variation of λ with temperature is assumed
we obtain from Eq. (6.207).
Λ = 128 × 10 11 e −θ a
( p
) λ
RT m 2 . (6.207)
c p
λ = λ f θ, (6.208)
( ) p λf
RT f m 2 . (6.209)
c p
Then the problem reduces to the integration of the system of Eqs. (6.203) and (6.206),
with the following boundary conditions
θ = θ 0 : ε = 0, (6.210)
θ = 1 : ε = 1, Y = 1. (6.211)
The value of Λ which makes compatible both boundary condition will be the
eigenvalue for the system, and it determines the flame propagation velocity.
Flame velocity
The flame velocity is given by
u 0 = m ρ 0
= 1 ρ 0
(
128 × 10 11 e −θ a
p
RT f
where Λ is the eigenvalue referred to.
When Eq. (6.206) is written
1 − θ 0
2
−
∫ 1
0
λ f
c p
) 1/2
Λ −1/2 , (6.212)
∫ 1
( ) 3/2 1 − Y
(1 − θ) dε = Λ
e −θ 1 − θ
a
θ dθ, (6.213)
θ 0
1 + Y
the problem reduces (following the idea of Karman’s method) to finding an approximation
of θ vs ε for the calculation of integral
∫ 1
Y vs θ for the computation of the integral of the right hand side.
0
(1 − θ) dε, and an approximation of

6.14. HYDRAZINE DECOMPOSITION FLAME 185
The parabolic approximation given by Eq. (6.87) is adopted for Y vs θ
( ) L θ − θ0
Y =
, (6.214)
1 − θ 0
which satisfies Eq. (6.203) for ε = 0, and for ε = 1 is tangent to the straight line
1 − Y = L 1 − θ
1 − θ 0
, (6.215)
which, in turn, is also tangent to the solution of Eq. (6.216) as it can easily be verified
when considering that for θ ≃ 1 is
and
1 − ε ≫ 1 − Y, (6.216)
1 − ε ≫ 1 − θ . (6.217)
1 − θ 0
When approximation (6.214) is taken into Eq. (6.213), it results
where
1 − θ 0
2
−
∫ 1
∫ ⎡ ( ) 1 θ − L ⎤3/2
θ0
1 −
⎢ 1 − θ
I K = ⎣
0 ⎥
( ) θ − L ⎦
θ0
θ 0 1 +
1 − θ 0
The approximation of θ vs ε for the computation of
0
(1 − θ) dε = ΛI K , (6.213.a)
e −θ 1 − θ
a
θ dθ. (6.218)
∫ 1
θ
(1 − θ) dε can be obtained
from the solution of Eq. (6.206) for values of ε close to 1. Through the procedure
applied in Ref. [30] , it results 11
Therefore
∫ 1
0
[ ( ) ] 3/2 −2/5
1 − θ
= (1 − ε) 4/5 4 L
1 − θ 0 5 Λ . (6.219)
2
[
(1 − θ) dε ≃ 5 ( ) ] 3/2 −2/5
9 (1 − θ 4 L
0)
Λ −2/5 . (6.220)
5 2
If Eqs. (6.218) and (6.220) are taken into Eq. (6.213) the following equation will be
obtained for the eigenvalue Λ
⎛
Λ = 1 − θ 0
2I K
⎝1 − 10
9
which is solved through iteration or else graphically.
[ ( ) ] ⎞
3/2
−2/5
4 L
Λ −2/5 ⎠ , (6.221)
5 2
11 Essentially, it reduces to neglecting (1 − θ) respect to (1 − ε) in the denominator of Eq. (6.206) and to
apply the linear approximation given by Eq. (6.215) in order to eliminate Y in the numerator of the same.

186 CHAPTER 6. LAMINAR FLAMES
In order to perform the numerical calculation of the flame velocity the following
values given by Hirschfelder were applied. These same values were also used by
Gilbert and Altman in his work:
ρ 0 = 9.17 × 10 −4 gr/cm 3 ,
T 0 = 423 K,
T f = 1933 K,
λ = 0.00067
cal
cm s K ,
c p = 0.6623 cal
gr K ,
L = 1.33.
(6.222)
With them, the reduced activation temperature results to be
θ a =
The numerical integration of Eq. (6.218) gives
When the precedent values are taken into (6.221)
which, when taken into (6.212), finally results
37 000
= 9 632. (6.223)
1.933 × 1.9872
I K
1 − θ 0
= 270.9 × 10 −5 . (6.224)
Λ −1/2 = 8.2962 × 10 −2 , (6.225)
u 0 = Λ−1/2
3.97
≃ 209 cm/s. (6.226)
The agreement between this value and the 200 cm/s given by experimental results is
better than it could be expected considering the dubiousness of many of the numerical
values used.
Influence of pressure
As shown in Eq. (6.212) the flame velocity varies with pressure in accordance with
the following law
u 0 ∼ p1/2
ρ 0
∼ p −1/2 , (6.227)
in agreement with the predictions of theory for first-order reaction flames Eq. (6.75)
and as verified by the experiments carried-out by Adams and Stocks [36].

6.15. FLAME PROPAGATION IN HYDROGEN-BROMINE MIXTURES 187
6.15 Flame propagation in Hydrogen-Bromine mixtures
The present example contains a calculation of the velocity of the hydrogen-bromine
flame performed with a variation of the method developed by von Kármán and Penner
[40]. The method presented here differs from earlier calculations because the
influence of dissociation of bromine on the flame velocity is evaluated properly. Von
Kármán and Penner included the influence of the atoms of bromine on the reaction
rate but when calculating the distribution of the mole fraction of Br 2 , H 2 and HBr,
they neglected the mole fraction of Br atoms. However, for temperatures close to the
maximum flame temperature the dissociation of Br 2 reduces substantially the value
for the mole fraction of Br 2 , which has an important influence on the flame velocity.
Hence it seemed desirable to compute the burning velocity correctly, especially
in hydrogen-rich flames where the influence of dissociation is most important. Von
Kármán and Penner also neglect the influence of the energy of dissociation of Br 2 ,
which should be taken into account since it may be of the same order of magnitude as
the energy transferred by convection through the flame.
In the following, a method is developed for the correct calculation of the mole
fraction of Br 2 and of the influence of its dissociation energy on the velocity of the
flame, within the limitations of the steady-state assumption for the distribution of
bromine and hydrogen atoms. This method will be applied to computation for the
four hydrogen-rich flames previously considered by von Kármán and Penner. The results
are compared with those obtained by these authors and are shown to differ by not
more than 25% even for the hottest flame.
A detailed study of the structure of the flame is also presented. It will be shown
that dissociation of Br 2 reduces the flame velocity because of a decrease in the number
of molecules of bromine which exist near the hot flame boundary. On the other hand,
it will be seen that the influence of the dissociation energy can be neglected if thermal
convection is ignored because these two effects tend to cancel.
The procedure developed here follows closely the work performed in Ref. [41]
and could also be useful for computations on other types of flames.
Flame Equations
Since there are only five different chemical components, namely, HBr, Br 2 , H 2 , Br
and H, the unknowns are:
1) Five flux fractions, ε i .

188 CHAPTER 6. LAMINAR FLAMES
2)

6.15. FLAME PROPAGATION IN HYDROGEN-BROMINE MIXTURES 189
Expressions for the reaction rates w i are obtained from chemical kinetics. Between
the five chemical components, the following independent chemical reactions
exist [42]
Br 2 + X −→ k1
2Br + X, (6.233)
k
Br + H
2
2 −→ HBr + H, (6.234)
k
H + Br
3
2 −→ HBr + Br, (6.235)
HBr + H k 4
−→ H 2 + Br, (6.236)
2Br + X k 5
−→ Br 2 + X. (6.237)
These equations give the following expressions for the reaction rates w 1 , w 4
and w 5
( ) 2 p
w 1 = M 1 θ −2 (k 2 X 3 X 4 + k 3 X 2 X 5 − k 4 X 1 X 5 ), (6.238)
RT f
( ) p
w 4 = M 4 θ −1 p
(2k 1 X 2 − 2k 5 θ −1 X4 2 ) − M 4
w 5 , (6.239)
RT f RT f M 5
( ) 2 p
w 5 = M 5 θ −2 (k 2 X 3 X 4 − k 3 X 2 X 5 − k 4 X 1 X 5 ). (6.240)
RT f
Energy Equation
The energy equation is
λ dθ (
dx = mc p (θ − 1) +
5∑
i=1
)
(ε i − ε if ) h0 i
. (6.241)
c p T f
Diffusion Equations
Between mole fractions X i the following relation exists
X 1 + X 2 + X 3 + X 4 + X 5 = 1. (6.242)
Consequently, four additional equations are needed in order to complete the determination
of all of the mole fractions. The remaining equations are those for the diffusion
of four of the components, and they are obtained from Eq. (6.119)
dX i
5∑
(
)
dx = mRT f
p
θ 1 ε j ε i
X i − X j , (i = 1, 2, 3, 4). (6.243)
D ij M j M i
j=1

190 CHAPTER 6. LAMINAR FLAMES
Equations (6.228) through (6.232), (6.241), (6.242) and the four expressions
contained in Eq. (6.243) form a system of eleven equations for the computation of
the eleven unknowns of the flame. In the following section we simplify the problem
by introducing the steady-state approximation for the concentration of bromine and
hydrogen atoms.
The steady-state approximation
The following is the expression of the steady-state assumption for bromine and hydrogen
atoms, respectively
w 4 = 0, (6.244)
w 5 = 0. (6.245)
If Eqs. (6.239) and (6.240) are combined with Eqs. (6.244) and (6.245), we obtain
for the mole fractions of Br and H the following relations in terms of the principal
components and of the temperature
X 4 =
√
k1
k 5
√
RT f
√
θX2 , (6.246)
p
and
X 5 = k √ √
2 k1 RT f
k 3 k 5 p
X 3
√ θX2
X 2 + k 4
k 3
X 1
. (6.247)
From Eqs. (6.238), (6.242) and (6.246) the following expression is deduced for w 1
( ) 2 p
w 1 = 2M 1 θ −2 k 3 X 2 X 5 . (6.248)
RT f
Using in Eq. (6.248) the value for X 5 given in Eq. (6.247), we obtain
( ) 3/2
√
p
k1
w 1 = 2M 1 k 2 θ −3/2 X 3 X 3/2
2
RT f k 5
X 2 + k . (6.249)
4
X 1
k 3
Equation (6.249) expresses the rate of formation of HBr as a function of the mole
fractions of the principal components and of temperature. Introduction of Eq. (6.249)
into Eq. (6.230) leads to the result
m dε ( ) 3/2
√
1
p
dx = 2M k1
1
k 2
RT f
θ −3/2 X 3 X 3/2
2
k 5
X 2 + k 4
k 3
X 1
. (6.250)

6.15. FLAME PROPAGATION IN HYDROGEN-BROMINE MIXTURES 191
The steady-state assumption implies the hypothesis that the diffusion of radicals may
be neglected. Therefore, the flux fractions of Br and H must be proportional to their
corresponding mole fractions, viz.,
ε 4 = M 4
M m
X 4 , (6.251)
ε 5 = M 5
M m
X 5 . (6.252)
The mole fraction of hydrogen, X 5 , is always negligibly small. Therefore, in
view of Eq. (6.252), the corresponding flux fraction may also be neglected.
The preceding assumption and considerations allow a considerable simplification
of the flame equations. Since
Eqs. (6.228) and (6.229) reduce to
X 5 ≪ 1 and ε 5 ≪ 1, (6.253)
ε 1 + ε 2 + ε 3 + ε 4 = 1, (6.254)
X 1 + X 2 + X 3 + X 4 = 1. (6.255)
The mole fraction of Br, X 4 , is expressed as a function of X 2 through. Eq. (6.246).
The rate of formation of HBr is given by Eq. (6.250).
Eq. (6.254), now reduces to
where
and
The energy equation, see
λ dθ (
)
dx = mc p θ − 1 + q 1 (ε 1f − ε 1 ) − q 4 (ε 4f − ε 4 ) , (6.256)
q 1 =
(
M4
h 0 2 + M )
5
1
h 0 3 − h 0 1
(6.257)
M 1 M 1 c p T f
q 4 = (h 0 4 − h 0 1
2)
(6.258)
c p T f
are, respectively, the reduced reaction heats per unit mass corresponding to reactions
1
2 H 2 + 1 2 Br 2 −→ HBr + M 1 q 1 c p T f , (6.259)
and
Br + Br −→ Br 2 + M 2 q 4 c p T f . (6.260)
Finally, by virtue of Eqs. (6.251) and (6.252), only two of the four diffusion equations,
see Eqs. (6.243), remain. Moreover, the terms involving X 5 and ε 5 may be ignored.

192 CHAPTER 6. LAMINAR FLAMES
Selection is made of the two equations corresponding to X 1 and X 3 because that
corresponding to X 2 presents some difficulties, as will be seen later on. Thus
and
dX 1
dx
dX 3
dx
= RT 4∑
(
)
f
p
θ 1 ε j ε 1
X 1 − X j , (6.261)
D
j=1 1j M j M 1
= RT 4∑
(
)
f
p
θ 1 ε j ε 3
X 3 − X j . (6.262)
D
j=1 3j M j M 3
Hence the system of the flame equations has been reduced to the five relations
given in Eqs. (6.229), (6.246), (6.251), (6.254) and (6.255) and to the four
differential equations (6.250), (6.256), (6.261) and (6.262) for the nine unknowns ε i
(i = 1, 2, 3, 4), X i (i = 1, 2, 3, 4) and θ. If, furthermore, the differential equations
(6.250), (6.261) and (6.262) are divided by Eq. (6.256), the coordinate x is eliminated
and θ is introduced as independent variable. The results are
dε 1
dθ = λ ( ) 3/2
√
p
m 2 2M 1 θ −3/2 k1
k 2
c p RT f k 5
(
X 3 X 3/2
2 X 2 + k )
4
X 1 − 1
k 3
·
θ − 1 + q 1 (ε 1f − ε 1 ) − q 4 (ε 4f − ε 4 ) . (6.263)
dX 1
dθ
dX 3
dθ
(
)
4∑ 1 ε j ε 1
= λ
X 1 − X j
RT f
mc p p θ j=1D 1j M j M 1
θ − 1 + q 1 (ε 1f − ε 1 ) − q 4 (ε 4f − ε 4 ) , (6.264)
(
)
4∑ 1 ε j ε 3
= λ
X 3 − X j
RT f
mc p p θ j=1D 3j M j M 3
θ − 1 + q 1 (ε 1f − ε 1 ) − q 4 (ε 4f − ε 4 ) . (6.265)
Solution of the reaction equations
Von Kármán and Penner [40] give the following expressions for k 2
√
k1
k 5
and k 4
k 3
and
where
k 2
√
k1
k 5
= 0.8 × 10 14 e −θ 1/θ , (6.266)
k 4
= 1 = 0.119, (6.267)
k 3 8.4
θ a =
40 200
RT f
(6.268)

6.15. FLAME PROPAGATION IN HYDROGEN-BROMINE MIXTURES 193
is the reduced activation temperature for the formation of HBr. These authors assume
also that the coefficient of thermal conductivity λ is independent of the composition
of the mixture and that it varies proportionally with the square root of the temperature,
viz.,
λ = λ f
√
θ. (6.269)
If Eqs. (6.266), (6.267), and (6.269) are introduced into Eq. (6.263), we obtain
where
dε 1
dθ = X 3 X 3/2
Λθ−1 2 (X 2 + 0.119X 1 ) −1 e −θ 1 − θ
a
θ
, (6.270)
θ − 1 + q 1 (ε 1f − ε 1 ) − q 4 (ε 4f − ε 4 )
Λ = 1.6 × 10 14 λ ( ) 3/2
f p
m 2 M 1 e −θ a (6.271)
c p RT f
is the eigenvalue which determines the propagation velocity u 0 of the flame through
the relation
u 0 = 1.265 × 10 7 1 ρ 0
√
λ f M 1
c p
( p
RT f
) 3/4
e −θ a/2 Λ −1/2 . (6.272)
The eigenvalue Λ can be obtained by applying the method of von Kármán
and Penner [6] which involves integration of Eq. (6.270) between the cold and hot
boundaries of the flame in the following form
ε 2 ∫ ε1f (
) ∫ 1
1f
q 1
2 − X 3 X 3/2
2 e −θ 1 − θ
a
θ
1 − θ + q 4 (ε 4f − ε 4 ) dε 1 = Λ
0
θ 0
θ(X 2 + 0.119X 1 )
The problem lies now in obtaining:
dθ. (6.273)
1) Approximate expressions for (1 − θ) and (ε 4f − ε 4 ) as function of (ε 1f − ε 1 )
for the computation on the integral on the left hand of Eq. (6.273).
2) Approximate expressions for X 1 , X 2 , and X 3 as functions of θ, for the computation
of the integral appearing on the right hand side.
Von Kármán and Penner [40] performed this computation by assuming that
X 4 as well as ε 4 may be neglected (compared with unity). In this case, the diffusion
equations (for Lewis number close to unity) show that X 1 , X 2 , and X 3 are linear
functions of 1 − θ near θ = 1; a study of Eq. (6.270) shows that ε 1f − ε 1 is a function
of θ of the form
ε 1f − ε 1 = (1 − θ) n (6.274)

194 CHAPTER 6. LAMINAR FLAMES
near θ = 1, where the exponent n takes the following values
X 3,0 < 0.5 : n = 1,
X 3,0 = 0.5 : n = 7 4 ,
(6.275)
X 3,0 > 0.5 : n = 5 4 .
The approximation given in Eq. (6.274) holds only for values of θ very close
to unity. In fact, the values for ε 1f − ε 1 given by such an approximation decrease very
rapidly with 1 − θ, reaching the value zero for values of θ very close to unity. A better
approximation for ε 1f − ε 1 can be obtained as follows. In Eq. (6.270), (1 − θ) as well
as q 4 (ε 4f − ε 4 ) are much smaller than q 1 (ε 1f − ε 1 ) for the interesting range of values
of θ. 13 Therefore, a good approximation for Eq. (6.270) can be obtained if these terms
are neglected, thus yielding
(ε 1f − ε 1 ) dε 1
dθ = Λ X 3 X 3/2
2 e −θ 1 − θ
a
θ
q 1 θ(X 2 + 0.119X 1 ) . (6.276)
Equation (6.276) may be integrated from the hot boundary forward. The result is
√ ∫ 1
ε 1f − ε 1 = √ 2Λ q 1
θ
f(θ ′ ) dθ ′ , (6.277)
where
f(θ) = X 3X 3/2
2 e −θ 1 − θ
a
θ
θ(X 2 + 0.119X 1 ) . (6.278)
It can be seen that Eq. (6.277) is an approximation for (ε 1f −ε 1 ) which is valid
for a range of temperatures far more extensive than that covered by Eq. (6.274).
The improved approximation is necessary when the velocity of the flame is
computed, as it is done here, without neglecting the influence of ε 4 and X 4 . In fact, if
an approximation similar to Eq. (6.274) is used in this case, when computing the integral
on the left-hand side of Eq. (6.273), the contribution of this integral would be considerably
overestimated, as can be verified easily. The approximation of Eq. (6.277)
gives
√
dε 1
dθ = Λ
2q 1
13 However a very small interval near θ = 1 must be excluded.
f(θ)
√ ∫ . (6.279)
1
θ f(θ′ ) dθ ′

6.15. FLAME PROPAGATION IN HYDROGEN-BROMINE MIXTURES 195
This expression permits the integral of the left-hand side of Eq. (6.273) to be written
as follows
∫ ε1f
(
1 − θ+q4 (ε 4f − ε 4 ) ) dε 1
0
√
∫
Λ 1 (
)
f(θ)
=
1 − θ + q 4 (ε 4f − ε 4 ) √
2q 1 ∫ dθ.
θ 1
0
θ f(θ′ ) dθ ′
(6.280)
By using these expressions in Eq. (6.273), the following equation for Λ is obtained
ε 2 ∫ 1
1f
q 1
2 − Λ f(θ) dθ
θ
√
0
∫
Λ 1 (
M
)
4
f(θ)
−
1 − θ + q 4 (X 4f − X 4 ) √
2q 1 θ 0
M m ∫ dθ = 0.
1
θ f(θ′ ) dθ ′
Here Eq. (6.251) was used in order to eliminate ε 4 . 14
notation will be used
and
I =
∫ 1
θ 0
f(θ) dθ =
∫ 1
(6.281)
For simplicity the following
X 3 X 3/2
2 e −θ 1 − θ
a
θ
dθ, (6.282)
θ 0
θ(X 2 + 0.119X 1 )
∫ 1 (
M
)
4
f(θ)
J = (1 − θ) + q 4 (X 4f + X 4 ) √
θ 0
M m ∫ dθ. (6.283)
1
θ f(θ′ ) dθ ′
If these expressions are used in Eq. (6.281) and the resulting equation is solved, the
following is obtained for √ Λ
√
Λ =
√q 1 ε 2 1f
2I
+ 1 ( ) 2 J
+ 1
8q 1 I 2 √ J
2q 1 I . (6.284)
This value, when substituted in Eq. (6.272), gives the corresponding flame velocity.
Computation of X 2 and X 4
Equation (6.284) shows that when computing Λ it is only necessary to know the values
for I and J. In order to compute I, one must know f(θ) whereas for the calculation
of J one must know f(θ) and X 4 as functions of θ. Finally, Eq. (6.278) shows that
to obtain f(θ), one must know X 1 , X 2 , and X 3 as function of θ. Consequently, the
problem reduces now to the computation of X 1 , X 2 , X 3 , and X 4 as functions of θ.
14 In Eq. (6.281) when changing from ε 4 to X 4 , it has also been assumed that M m is constant through
the flame. Actually its variation is very small.

196 CHAPTER 6. LAMINAR FLAMES
This can be done by means of the diffusion relations given in Eqs. (6.264) and (6.265),
the steady-state Eq. (6.246), and Eq. (6.255).
expressed as
Let X 1f and X 3f be the final values for X 1 and X 3 . These variables can be
X 1 = X 1f − α 1 , (6.285)
X 3 = X 3f + α 3 , (6.286)
where α 1 and α 3 are functions of θ which approach zero when θ approaches one.
When Eqs. (6.246), (6.285) and (6.286) are combined with Eq. (6.255), the
following equation is obtained for the determination of X 2
√
k 1 RT f
X 1f − α 1 + X 3f + α 3 + X 2 +
k 5 p θX 2 = 1. (6.287)
The computation that follows holds for hydrogen-rich flames where X 2f =
X 4f = 0 and the following condition is consequently satisfied
X 1f + X 3f = 1. (6.288)
Equation (6.287) can now be written as
√
k 1 RT f
X 2 +
k 5 p θX 2 − (α 1 − α 3 ) = 0. (6.289)
On solving Eq. (6.289), one obtains for X 2
(√
√ )2
k 1
X 2 = θ RT f
+ α 1 − α 3 − 1 k 1
θ RT f
. (6.290)
4k 5 p
2 k 5 p
Von Kármán and Penner [40] give for
where
√
k1
k 5
the expression
√
k1
k 5
= 1.676 e − θ r
θ , (6.291)
θ r =
Finally, combining Eqs. (6.291) and (6.290) it is obtained
22 605
RT f
. (6.292)
X 2 =
(√
g(θ)2 + α 1 − α 3 − g(θ)) 2
, (6.293)

6.15. FLAME PROPAGATION IN HYDROGEN-BROMINE MIXTURES 197
where
θ √
g(θ) = 0.838 e − r
θ θ RT f
p . (6.294)
Likewise, when Eq. (6.291) is substituted into Eq. (6.246), one finds
√
X 4 = 1.676 e − θr
θ θ RT √
f X2 . (6.295)
p
Equations (6.293) and (6.295) permit the computation of X 2 and X 4 as soon as α 1
and α 3 are known. Nevertheless, before initiating the calculation of these quantities
let us study the behavior of X 2 as a function of θ, since this relation influences more
than any other the value of the flame velocity, as shown by Eq. (6.282).
For θ → 1, the following conditions are satisfied
√
g(θ) → 0.838 e −θ RT
r f
≠ 0,
p
α 1 − α 3 → 0.
(6.296)
Therefore, for θ near 1
α 1 − α 3
g(θ) 2 ≪ 1, (6.297)
it is now simple to show from Eq. (6.293) that, near θ = 1, X 2 behaves as follows
X 2 ≃ 1
2θ r
p
θ −1 e θ (α 1 − α 3 ) 2 . (6.298)
2.808 RT f
√
On the other hand, when the difference 1−θ increases, the term 0.838 e −θr/θ θ RT f
p
decreases very rapidly since θ r ≫ 1, whilst the term α 1 − α 3 increases. Therefore,
when the difference 1 − θ is not very small compared to unity, X 2 behaves as follows
X 2 ≃ α 1 − α 3 . (6.299)
In the different behaviors of Eqs. (6.298) and (6.299) lie essentially the influence
of the dissociation of bromine on the propagation velocity of the flame. In fact,
if the term X 4 is neglected in Eq. (6.255) when computing X 2 , as was done in [40],
then the approximation given in Eq. (6.299) holds for all temperatures. On the contrary,
when the influence of X 4 is taken into account, the values for X 2 near θ = 1
are far smaller for than those given by Eq. (6.299), since then X 2 varies as the square
of (α 1 − α 3 ), as shown by Eq. (6.298). Since the values for X 2 are smaller, the value
for the integral I is also smaller, as shown by Eq. (6.28). Finally, if I decreases, √ Λ
increases as shown by Eq. (6.284). Therefore, according to Eq. (6.272), u 0 decreases.

198 CHAPTER 6. LAMINAR FLAMES
Hence, dissociation of bromine reduces the flame propagation velocity basically because
such dissociation reduces the molar fractions of Br 2 at temperatures close to
T f . Dissociation also influences the value for J, as shown by Eq. (6.283). However,
this influence is small and it acts opposite to the influence of thermal conductivity; the
value for J is very small and its influence is negligible. This last conclusion will be
verified when numerical calculations are performed later on.
Introducing Eq. (6.298) into Eq. (6.295) we find, for θ ≃ 1, the following
behavior of X 4
X 4 ≃ 1.676 e −θ r
√
RT f
p (α 1 − α 3 ), (6.300)
thus X 4 is a linear function of (α 1 − α 3 ) and, for θ ≃ 1, X 4 ≫ X 2 .
Evaluation of X 1 and X 3
The evaluation of X 1 and X 3 can be easily performed from the diffusion relations
Eqs. (6.264) and (6.265).
Equations (6.229) and (6.254) permit us to express ε 2 and ε 3 as functions of ε 1
and ε 4 as follows
and
ε 2 = M 4
M 1
(ε 1f − ε 1 ) − ε 4 , (6.301)
ε 3 = ε 3f + M 1 − M 4
M 1
(ε 1f − ε 1 ). (6.302)
By introducing these expressions, together with Eqs. (6.285) and (6.286) , into
Eq. (6.264) and (6.265), and if ε 4 , X 2 and X 4 are expressed as functions of (α 1 − α 3 )
by use of the relations given in Eqs. (6.251), (6.298) and (6.300), two equations are
obtained which depend only on θ, (ε 1f − ε 1 ), α 1 and α 3 . These expressions can be
developed in series of these variables near θ = 1. The results are
dX 1
dθ
dX 3
dθ
[( =λ f RT f M4 X 1f
+ M 1 − M 4
pc p q 1 M 1 M 2 D 12 M 1 M 3
(
α1 α 3
+O , ,
ε 1f − ε 1 ε 1f − ε 1
[( =λ f RT f M4 X 3f
− M 1 − M 4
pc p q 1 M 1 M 2 D 23 M 1 M 3
(
α1 α 3
+O , ,
ε 1f − ε 1 ε 1f − ε 1
X 1f
D 13
+
1 )
X 3f
M 1 D 13
)
1 − θ
+ higher order terms
ε 1f − ε 1
X 1f
D 13
−
1 )
X 3f
M 1 D 13
)
1 − θ
+ higher order terms
ε 1f − ε 1
]
, (6.303)
]
. (6.304)

6.15. FLAME PROPAGATION IN HYDROGEN-BROMINE MIXTURES 199
dX 1
dθ
Hence, for θ ≃ 1, we find
≃ λ (
f RT f M4 X 1f
+ M 1 − M 4 X 1f
+
pc p q 1 M 1 M 2 D 12 M 1 M 3 D 13
1 )
X 3f
≡ β 1 , (6.305)
M 1 D 13
dX 3
dθ
≃ λ (
f RT f M4 X 3f
− M 1 − M 4 X 1f
−
pc p q 1 M 1 M 2 D 23 M 1 M 3 D 13
1 )
X 3f
≡ −β 3 , (6.306)
M 1 D 13
which leads to the following approximations for X 1 and X 3
X 1 ≃ X 1f − β 1 (1 − θ), (6.307)
X 3 ≃ X 3f + β 3 (1 − θ). (6.308)
Thus
where
α 1 − α 3 = (β 1 − β 3 )(1 − θ) = β 2 (1 − θ), (6.309)
β 2 = β 1 − β 3 . (6.310)
near θ = 1
In short, the following approximations for X 1 , X 2 , X 3 and X 4 are obtained
X 1 = X 1f − β 1 (1 − θ), (6.311)
X 2 =
(√
g(θ)2 + β 2 (1 − θ) − g(θ)) 2
, (6.312)
X 3 = X 3f − β 3 (1 − θ), (6.313)
(√ )
X 4 = 2g(θ) g(θ)2 + β 2 (1 − θ) − g(θ) . (6.314)
Evaluation of f(θ), I and J
When computing f(θ), the only values of X 1 , X 2 and X 3 needed are those for θ near
1, since the reduced activation temperature θ a is much larger than unity. The relevant
expressions are given in Eqs. (6.311) through (6.313). Once f(θ) is known, the
integral I can be computed by numerical of graphical integration. Likewise, knowing
f(θ) and X 4 , the integral J can be obtained either numerically or graphically.
Computation of the flame velocity
The preceding results may be summarized by the following method for the computation
of the velocity of the flame.

200 CHAPTER 6. LAMINAR FLAMES
1) Values for β 1 , β 3 and β 2 are obtained from Eqs. (6.305), (6.306) and (6.310).
2) Introducing these values into Eqs. (6.307), (6.308) and (6.312) one obtains X 1 ,
X 2 and X 3 as functions of θ.
3) Substituting the results into Eq. (6.278), the value of f(θ) is obtained.
4) From the numerical or graphical integration of f(θ) between θ 0 and 1, the value
of the integral I, defined in Eq. (6.282), is obtained.
5) The numerical or graphical integration of f(θ) between θ and 1 gives the value
of
∫ 1
θ
f(θ ′ ) dθ ′ .
6) We introduce this value of
∫ 1
θ
f(θ) dθ, the value of f(θ), computed in (6.230),
and the value for X 4 , given by Eq. (6.314), into Eq. (6.283). Then we integrate
Eq. (6.283), either numerically or graphically, between θ 0 and 1 in order to
obtain the value of J.
7) When both the values for I and J are substituted into Eq. (6.284) the value for
√
Λ is obtained.
8) Finally, u 0 is computed from Eq. (6.272).
It has been stated that the influence of J on the value of u 0 is negligible. This
fact allows a considerable simplification of the method since it eliminates steps 5) and
6). By neglecting J in Eq. (6.284), the following simplified equation results
√
Λ =
√q 1 ε 2 1f
2I . (6.315)
Results
The method developed herein has been applied to the computation of the four cases
calculated by von Kármán and Penner for hydrogen-rich flames. The results obtained
are given in Table (6.5), in which are also listed the values obtained by von Kármán
and Penner.
X 3,0 0.55 0.60 0.65 0.70
Von Kármán and Penner 30.6 42.3 34.2 23.6
u 0 calculated from Eq. (6.315) 22.9 33.7 30.5 21.9
u 0 calculated from Eq. (6.284) 23.3 34.2 – –
Table 6.5: Flame velocity (cm/s) for hydrogen-rich H 2-Br 2 mixtures.

6.15. FLAME PROPAGATION IN HYDROGEN-BROMINE MIXTURES 201
The physico-chemical constants given by von Kármán and Penner were used
in our calculations. The values of these constants, as well as those for the integrals
and parameters needed for evaluation of u 0 , are summarized in Table 6.6.
In order to analyze the influence of convection and of dissociation of bromine
upon the value of u 0 , the flame velocity was computed for two cases, using the correct
relation given in Eq. (6.284) for √ Λ. The results are listed in Table 6.5 and 6.6, which
show that the correction is negligible small due to the cancellation of convection and
dissociation energy effects.
X 3,0 0.55 0.60 0.65 0.70
X 2,0 0.45 0.40 0.35 0.30
X 1,f 0.90 0.80 0.70 0.60
X 3,f 0.10 0.20 0.30 0.40
ε 1f 0.997 0.994 0.990 0.984
T f K 1660 1585 1460 1324
10 5 λ f (cal/cm-s-K) 19.8 25.0 29.4 33.1
c p (cal/g-K) 0.105 0.116 0.132 0.151
M (g/mole) 73.1 65.2 57.3 49.4
θ a 12.2 12.75 13.9 15.28
θ r 6.81 7.13 7.83 8.55
θ 0 0.1945 0.204 0.221 0.244
β 1 1.532 1.745 1.762 1.684
β 2 1.380 1.560 1.560 1.470
β 3 0.152 0.185 0.202 0.214
q 1 0.807 0.800 0.787 0.768
q 4 1.63 1.55 1.475 1.42
10 3 I 0.667 1.601 2.639 3.395
√
Λ 24.54 13.82 12.09 10.47
u 0
√
Λ (cm/s) 551 529 369 228
u 0 (cm/s) 22.9 33.7 30.5 21.8
10 3 J -0.87 -0.94 — —
√
Λ from Eq. (6.284) 24.03 15.49 — —
u 0 (cm/s) from Eq. (6.284) 23.34 34.15 —
Table 6.6: Parameters for the computation of flame velocities.
The results given in Table 6.5 show that dissociation of bromine reduces the
flame velocity considerably. Values for u 0 have been plotted in Fig. 6.17 which also
shows the results calculated by von Kármán and Penner [40] and by Gilbert and Altman
[29] . Also listed are experimental data obtained by Anderson and his collaborators
[43]. It will be seen that the influence of dissociation of bromine on u 0 predicted
by Gilbert and Altman is exaggerated. This conclusion is not surprising if one considers
that these authors estimated the influence of dissociation from the results obtained
through a thermal theory. Then they applied the reduction factor obtained from a ther-

202 CHAPTER 6. LAMINAR FLAMES
S b
(cm/s)
60
50
40
30
20
von Kármán − Penner
von Kármán − Millán
Gilbert − Altman (Dissoc. Negl.)
Gilbert − Altman (Dissoc. Incl.)
Boys − Corner (Diss. Negl.)
(1) Flame cone area method
(2) Flame cone angle method
(3) Tube method
(1)
(2)
(3)
10
0
0.50 0.55 0.60 0.65 0.70
X 3,0
Figure 6.17: The quantity S b as a function of X 3,0 for hydrogen-rich H 2 − Br 2 mixtures.
mal theory to a diffusion theory in which the influence of dissociation was neglected.
On the other hand, the excellent agreement between their results and the experimental
ones is not too relevant since the former were obtained by applying the method of Boys
and Corner, which gives considerably smaller values for u 0 than those obtained from
the application of a more correct method, as will now be verified. With the purpose of
estimating the uncertainty due to the use of a poor method of computation, the value
of u 0 for X 3,0 = 0.70 was computed with the method of Boys and Corner, as applied
by Gilbert and Altman, and with the method of von Kármán and Penner. In both cases
the influence of the dissociation of bromine was neglected and the physico-chemical
constants of von Kármán and Penner were used, so that the difference in numbers was
only due to the difference in method. The following are the results obtained.
Boys-Corner method, as applied by Gilbert and Altman:
Von Kármán and Penner method:
u 0 = 13.8 cm/s,
u 0 = 23.6 cm/s.
Therefore, if Gilbert and Altman had applied a better method than that of Boys and
Corner, they would have obtained considerably larger values for u 0 . The agreement
between their values and those obtained from experiments must be considered to be
largely fortuitous.

6.15. FLAME PROPAGATION IN HYDROGEN-BROMINE MIXTURES 203
1.00
0.75
X 1
X 2
0.50
X 2
(dissociation neglected)
X 3
ε 1f
− ε 1
X 4
0.25
0.0 0.1 0.2 0.3
1 − θ
Figure 6.18: Composition and flux profiles for X 3,0 = 0.65.
Structure of the Flame
The distributions of X 1 , X 2 , X 3 , X 4 and ε 1f − ε 1 corresponding to one of the cases
considered are shown in Fig. 6.18.
In order to show the influence of X 4 on the distribution of X 2 , the figure also
indicates the values for X 2 obtained when X 4 is neglected. Obviously, the influence
of X 4 is very important, especially for values of θ near 1. On the other hand, this is
the most important influence of neglecting X 4 . It may be seen that small deviations of
X 1 and X 3 from the linear law have little influence upon the value of u 0 , since X 1f
and X 3f are different from zero,
References
[1] Mallard, E. and Le Chatelier, H.: Recherches sur la Combustion des Melanges
Gateaux Explosives Ann. Mines, Vol. 8 series 4, 1883, p. 274.
[2] Crussard, L.: Les Deflagrations on Régime Permanent dans les Mileux Conducteurs.
Compt. Rend., Vol. 158, 1914, pp. 125, 340.
[3] Coward H. F. and Payman W.: Chem. Rev., Vol. 21, 1937, p. 359.
[4] Zeldovich, Y. B. and Semenov, N.: Kinetics of Chemical Reactions in Flames.
NACA Tech. Memo. No. 1084, 1946.

204 CHAPTER 6. LAMINAR FLAMES
[5] Boys, S. F. and Corner, J.: The Structure of the Reaction Zone in a Flame. J.
Proc. Roy Soc. London, 1949, A1-97,90.
[6] von Kármán, Th. and Penner, S. S.: Fundamental Approach to Laminar Flame
Propagation. Selected Combustion Problems, Vol. I, AGARD, 1954, pp. 5-41.
[7] Hirschfelder, J. O., Curtiss, C. F. and Campbell, E.: The Theory of Flames and
Detonations. Fourth Symposium (International) on Combustion, Williams and
Wilkins Co., Baltimore, 1953.
[8] Lewis, B. and von Elbe, G.: On the Theory of Flame Propagation. Phys., Vol.
2, 1934, pp. 537-546.
[9] Evans, N. W.: Current Theoretical Concepts of Steady-State Flame Propagation.
Chem. Rev., Vol. 51, 1952.
[10] von Kármán, Th.: The Present Status of the Theory of Laminar Flame Propagation.
Sixth Symposium (International) on Combustion, Reinhold Publishing
Corp., New York, 1957.
[11] Tanford, C. and Pease, R.: Equilibrium Atom and Free Radical Concentrations
in Carbon Monoxide Flames and Correlation with Burning Velocities. J. Chem.
Phys., Vol. 15, 1947, pp. 431-433.
[12] van Tiggelen, A.: Chemical Theory of the Speed of Flame Propagation. Bull.
Soc. Chim. Belg., Vol. 58, 1949, p. 259.
[13] Jost, W.: Explosion and Combustion Processes in Gases. McGraw-Hill, New-
York-London, 1946.
[14] Gaydon A. G. and Wolfhard, H. G.: Flames, Their Structure, Radiation and
Temperature. Chapman Hall, London, 1953.
[15] Ladenburg, R. W., Lewis, B., Pease, R. N. and Taylor, H. S.: Physical Measurements
in Gas Dynamics and Combustion. Vol. IX of High Speed Aerodynamics
and Jet Propulsion, Princeton University Press, 1954.
[16] Linnet, J. W.: Methods of Measuring Burning Velocities. Fourth Symposium
(International) on Combustion, Williams and Wilkins Co., Baltimore, 1953,
pp. 20-35.
[17] Emmons, H. W., Harr, J. A. and Strong, P.: Harvard University, Contract No.
AT (30-1)-497, 1950.
[18] Adamson, T. C.: On the Theory of One Dimensional Flame Propagation. Jet
Propulsion, January-February 1952.
[19] von Kármán Th. and Millán, G.: The Thermal Theory of Constant Pressure
Deflagration. Biezeno’s Anniversary Volume, Delft, Holland, 1953.

6.15. FLAME PROPAGATION IN HYDROGEN-BROMINE MIXTURES 205
[20] Adams E. N., quoted by Henkel, M. J., Spaulding, W. P. and Hirschfelder J. O.:
Theory of Propagation of Flames, Part II Approximate Solutions. Third Symposium
(International) on Combustion, Williams and Wilkins Co., Baltimore,
1949, pp. 127-135.
[21] Wilde, K. A.: J. Chem. Phys., Vol. 22, 1954, p. 1788.
[22] Millán G., Sendagorta, J. M. and Da Riva, I.: Comparison of Analytical Methods
for the Calculation of Laminar Flame Velocity. ARDC Contract AF 61
(514), 997, 1957.
[23] Sendagorta, J. M.: Method for the Computation of the Propagation Velocity of
a Plane Laminar Flame. ARDC Contract AF 61 (514), 997, 1957.
[24] Fristrom, R. M.: The Structure of Laminar Flame Fronts. Sixth Symposium
(International) on Combustion, Reinhold Publishing Corp., New York, 1957.
[25] Hirschfelder, J. O., Curtis, J. P. and Campbell, D. E.: The Theory of Flame
Propagation. IV Rep. No. OM-756. Univ. Wisconsin, 1952.
[26] Campbell, E. S.: Theoretical Study of the Hydrogen Bromine Flame. Sixth
Symposium (International) on Combustion, Reinhold Publishing Corp., New
York, 1957.
[27] Giddings, J. C. and Hirschfelder J. O.: Flame Properties and the Kinetics of
Chain-Branching Reactions. Sixth Symposium (International) on Combustion,
Reinhold Publishing Corp., New York, 1957.
[28] Spalding, D. B.: The Theory of Flame Phenomena with a Chain Reaction. Phil.
Trans. Roy. Soc., London 1956.
[29] Gilbert, M. and Altman, D.: The Chemical Steady State in HBr Flames. Sixth
Symposium (International) on Combustion, Reinhold Publishing Corp., New
York, 1957.
[30] Millán, G. and Sanz, S.: Hydrazine Decomposition Flame. ARDC Contract No.
AF 61 (514)-734-O, 1956.
[31] Strong, A. G. and Grosse, A. V.: The Ozone to Oxygen Flame. Sixth Symposium
(International) on Combustion, Reinhold Publishing Corp., New York,
1957.
[32] Murray, R. O. and Hall. A. R.: Flame Speeds in Hydrazine Vapour and in
Mixtures of Hydrazine and Ammonia with Oxygen. Trans. Faraday Soc., Vol.
57, 1951, pp. 743-751.
[33] Zeldovich, Y. B.: Theory of Flame Propagation. NACA Tech. Memo. No.
1282, 1951.
[34] Szwarc M. J.: Chem. Phys. Vol. 17, 1949, p. 505.

206 CHAPTER 6. LAMINAR FLAMES
[35] Hirschfelder, J. O. and Curtiss O. F.: J. Phys Chem. Vol. 57.
[36] Adams, G. K. and Stocks, G. W.: The Combustion of Hydrazine. Fourth Symposium
(International) on Combustion, Williams and Wilkins Co., Baltimore,
1953, pp. 239-248.
[37] Spalding, D. B.: The Theory of Flame Phenomenon with a Chain Reaction.
Phil. Trans. Roy. Soc. London, 1956.
[38] Gilbert M. and Altman D.: Hydrazine Decomposition Flame. ORDCIT Project
Contract No. DA-04-495, Ord. 18 Progress Report No. 20-278, 1956.
[39] Millán, G. and Sendagorta J. M.: Hydrazine Decomposition Flame. ARDC
Contract No. AF 61-(514)-997, 1957.
[40] von Kármán, Th. and Penner S. S.: The Theory of One-Dimensional Laminar
Flame Propagation for Hydrogen-Bromine Mixtures. Part I. Dissociation
Neglected. Tech. Rep. No. 16, Contract No. DA 04-495-0rd. 446, 1956.
[41] von Kármán Th. and Millán G.: The Theory of One-Dimensional Laminar
Flame Propagation for H 2 -Br 2 Mixtures. Part II. Dissociation Included. Tech.
Rep. No. 16, Contract No. DA-04-495-Ord.-446, 1956.
[42] Mileson, D. F.: The Thermal Theory of Laminar Flame Propagation for Hydrogen-
Bromine Mixtures. Tech. Rep. No. 6, Contract No. DA 04-495-Ord.-446, 1954.
[43] Anderson, R. C.: Hydrogen-Halogen Flames. AGARD Combustion Panel Meeting,
Oslo, 1956.

Chapter 7
Turbulent flames
7.1 Introduction
There is experimental evidence that turbulence increases the propagation velocity of
the flame through a combustible mixture. This is a very important fact in technical
applications since it allows a considerable reduction of the space and time required to
burn a given mass. The influence of turbulence over combustion was first recognized
by Mallard and Le Chatelier in 1883 [1]. However the attempts made to determine the
causes of this influence as well as to estimate quantitatively its value are quite recent.
The basic problem lies in determining the propagation velocity of the flame through a
combustible mixture in turbulent motion knowing the characteristics of the turbulence
and the state and composition of the mixture.
From the experimental stand-point, several techniques are available for the determination
of the flame velocity. However the basis for such techniques are not as
solid as those applied to the case of laminar flames and the measurements taken are
not as numerous or systematic. Among them, the technique more commonly used
consists in photographing a flame, obtained for example in a bunsen burner, then measuring
the area of the combustion front and dividing it by the flow rate of the burner
as is done for the case of laminar flames. One of the difficulties of such techniques
is the fact that the combustion front of a turbulence flame is not well defined since
the long exposure photographs show a thick luminous zone (see Fig. 7.1a) whereas
a short exposure “Schlieren” photograph reveals a very irregular and wrinkled structure
(see Fig. 7.1b). These circumstances impose the introduction of an arbitrariness
in estimating the flame area. Damköhler [2], for instance, adopted as surface of the
turbulent combustion front the one that limits internally the luminous zone. Later on,
207

208 CHAPTER 7. TURBULENT FLAMES
(a) Time exposure photograph
(b) Instantaneous Schlieren photograph
Figure 7.1: Stoichiometric natural gas-air flame, Re = 25 000, burner tube diameter =
5.08 cm (by courtesy of Bureau of Mines).
preference was given to the use of the mean surface between those internally and externally
limiting the luminous zone [3] or else to the surface where luminosity reaches
its maximum intensity [4].
Additional difficulty lies in the fact that the paths of the gas particles are not
known. Therefore it is risky to identify the fraction of unburnt mixture corresponding
to each element of the flame front. Such difficulty has often been eluded by measuring
the mean velocity obtained when the total flow rate of the mixture is divided by the
area of the flame front. The propagation velocity of turbulent flames has been measured
with these or similar techniques by Damköhler [2] , Bollinger and Williams [3],
Scurlock [4], Williams, Hottel and Scurlock [5], Karlovitz, Denniston and Wells [6],
Wohl, Shore, von Rosemberg and Weil [7], Leason [8], Bowditch [9], Wohl and Shore
[10], Mickelsein and Ernstein [11], etc.
Figure 7.2 taken from Ref. [3] shows, as an example, the results of the measurements
performed by Bollinger and Williams in mixtures of several hydrocarbons
and air. For each case the mixture giving the maximum velocity was used. This figure
also shows the laminar velocities corresponding to the same mixtures. It is seen that
the effect is maximum in the acetylene flame where the value of the laminar flame ve-

7.1. INTRODUCTION 209
300
250
TURBULENT FLAME SPEED, cm/s
200
150
100
50
0
0
10 20 30 40x10 3
REYNOLDS NUMBER OF PIPE FLOW
Figure 7.2: Variation of flame speed with Reynolds number of flow.
locity is half of that for a Reynolds number 3 × 10 4 . Other measurements have given
turbulent velocities considerably higher than these observed here.
From the theoretical stand-point several attempts have been made to explain the
activation of combustion due to turbulence. The first attempt was made by Damköhler
[2] who pointed-out two causes for the action of turbulence. One is the increase in the
transport coefficients (conductivity and diffusion) due to turbulent diffusivity. The second
cause is the distortion of the laminar flame front due to the turbulent oscillations
of the velocity which would increase, considerably, the effective surface of the combustion
front. The importance of either one of these two factors would depend in each
case on the relation between the scale of turbulence and the thickness of the laminar

210 CHAPTER 7. TURBULENT FLAMES
flame. In technical applications this ratio is generally very large, hence the main cause
for the activation of combustion would be in this case the distortion of the flame front.
Figure 7.1b seems to confirm this point of view. The major part of the later works on
the subject have attempted to estimate the importance of either one of these causes,
preserving Damköhler’s model. In 1952 von Kármán and Marble [12] proposed a different
model which considered the turbulent flame as a zone whose structure should
be defined by mean values of the characteristic variables (temperature, reaction rate,
etc.) to which were superimposed static oscillations of turbulent nature. Later, this
model was adopted and developed by Sommerfield and his collaborators [13], who
obtained some experimental evidence (not sufficient) that the approximation is correct.
The comparison between theoretical and experimental results is not conclusive
enough in any case to establish either one of the proposed theories, which are actually
in a preliminary phase.
Consequently the present study will only be a brief exposure of the basic principles
and fundamental results of these theories.
7.2 Turbulent combustion theories
So far, the only type of turbulence whose influence on combustion has been studied is
the isotropic. This turbulence is characterized by two magnitudes: its scale l and its
intensity v ′ . 1 Damköhler studies the influence of turbulence on the flame by comparing
l and v ′ with the corresponding magnitudes of the laminar combustion wave of the
mixture; these being thickness d l of the flame and the laminar propagation velocity
u l . When performing this comparison the following cases arise.
flame
1) The turbulence scale is small when compared to the thickness of the laminar
l
d l
≪ l (7.1)
In this case the action of turbulence is reduced to an activation of the transport coefficients
conductivity and diffusion at the flame). Let
x =
λ
ρc p
(7.2)
and
ε ∼ lv ′ , (7.3)
1 For an exposure of the principles of turbulence, see, i.e., H.L. Dryden: A review of the Statistical Theory
of Turbulence. Quart. Applied Math., Vol. 1, 1943, pp. 7-42.

7.2. TURBULENT COMBUSTION THEORIES 211
be the laminar and turbulent thermal diffusivities, respectively. The turbulent propagation
velocity u t of the flame is obtained from u l when x is substituted by ε. Since
u l is proportional to √ x, it results for u t
u t
u l
=
√ ε
x
(7.4)
Shelkin [14] also considers the influence of laminar diffusivity on the turbulent flame.
In this case, ε must be substituted in (7.3) by x + ε, thus resulting
√
u t
= 1 + ε u l x
(7.5)
This expression has the advantage over (7.4) that when turbulence approaches zero,
u t approaches u l .
Case 1) very seldom arises since under normal conditions the thickness of the
laminar flame is only of some tenths of a millimeter.
2) Scale of turbulence and flame thickness are of the same order of magnitude
l
d l
≃ 1. (7.6)
Damköhler does not consider this case. However Shelkin has proven that the
action of turbulence does not reduce then to an activation of the transport coefficients
but it also influences the combustion time of the mixture, which not only depends on
the reaction velocity of the species but on the rapidity of the turbulent mixing. To date,
no formula is available for this case.
3) Scale of turbulence is large when compared to flame thickness
l
d l
≫ 1. (7.7)
In this case combustion takes place through a laminar front but turbulence wrinkles
the laminar flame front thus increasing its surface. Thereby, the effective combustion
surface of the mixture is increased. Two possibilities should be considered
depending on the intensity of the turbulence.
3.a) The intensity of turbulence is very large with respect to the laminar propagation
velocity of the flame
v ′
u l
≫ 1. (7.8)
This case was not analyzed by Damköhler. The following consideration are
owned to Shelkin. When condition (7.8) takes place the structure of the combustion

212 CHAPTER 7. TURBULENT FLAMES
B
A
BURNED
UNBURNED
GASES
GASES
B’
A’
Figure 7.3: Structure of the turbulent flame in the case l ≫ d l and v ′ ≫ u l , according to
Shelkin.
zone shows islands of unburnt gas which have been dragged by the strong turbulent oscillations
towards the region of combustion products without time to burn, see Fig. 7.3.
In this case, the zone between AA ′ and BB ′ can be considered as a combustion wave
which advances with a propagation velocity u t . The propagation velocity of a combustion
wave is determined by an equilibrium between transport processes, which are
characterized by combustion time τ. Independently from the mechanism which determines
the values for ε and τ, a dimensionless analysis shows that u t must be of the
form
u t ∼
√ ε
τ . (7.9)
For the case shown in Fig. 7.3, ε is clearly the turbulent diffusivity. Combustion
velocity is determined by the mixing rapidity and therefore τ is proportional to the
time of turbulent mixing
τ ∼ l v ′ . (7.10)
When the expressions for ε and τ given by (7.2) and (7.10) are taken into (7.9)
u t ∼ v ′ , (7.11)
Hence, the propagation velocity is proportional to the intensity and independent from
the scale of turbulence. Furthermore, u t results to be independent from laminar velocity
u l . Such conclusion contradicts experimental evidence.
3.b) The intensity of turbulence is of the same order of magnitude or smaller
than the propagation velocity of the laminar flame
v ′
u l
1. (7.12)
In this case the islands disappear and the action of turbulence reduces to deforming

7.2. TURBULENT COMBUSTION THEORIES 213
BURNED
σ
l
UNBURNED
σ
GASES
GASES
Figure 7.4: Structure of the turbulent flame in the case l ≫ d l and v ′ u l , according to
Shelkin.
the laminar flame front thus increasing its surface as shown in Fig. 7.4. Let σ l be the
effective surface of combustion and σ the surface of access to the flame of the unburnt
gases. Velocity u t is given in this case by expression
u t
u l
= σ l
σ . (7.13)
The problem lies then in estimating the value for σ l , for which several models have
been proposed.
Shelkin assumes that σ l is formed by a system of cones whose base is proportional
to the square of the scale of turbulence l 2 and whose height is proportional to
distance v ′ l/u l travelled by a mass of gas with a diameter l due to turbulent oscillations
during the time l/u l needed by the laminar flame to cross this mass. Here, σ l /σ
is the ratio of the lateral to the base area of the cones, and from (7.13), it results
√
( )
u t
v
′ 2
= 1 + k , (7.14)
u l u l
where k is a numerical coefficient of the order of magnitude unity. Therefore, u t is
also independent from the scale of turbulence. When turbulence is weak, Eq. (7.14)
shows that its effect is of the second order, whilst for very intense turbulence Eq. (7.14)
reduces to (7.11). Through a different analysis M. Tucker [15] obtains the following
expression for the case of weak turbulence
( )
u t
v
′ 2
= 1 + k(θ) , (7.15)
u l u l
which is valid if v ′ /u l ≪ l. In this formula, k(θ) is a coefficient depending on ratio
θ of temperature T f of the burnt gases to temperature T 0 of unburnt gases. It is seen
that when v ′ /u l ≪ l. Eqs. (7.14) and (7.15) agree.
Karlovitz [16] reasons as follows. Let ¯X =
√ ¯X2 be the root mean square
displacement of a particle due to turbulent oscillations. ¯X is a increasing function of

214 CHAPTER 7. TURBULENT FLAMES
time counted from the point at which measurements were initiated. Let t = l/u l be
the time taken by the laminar flame to cross a turbulent vortex. The total path travelled
by the flame in this time is ¯X + l and when this expression is divided by t one obtains
for u t
That is to say
u t = u l
l
The value for ¯X is given by Taylor’s formula
d ¯X 2
dt
¯X + u l , (7.16)
u t
= 1 + ¯X
u l l . (7.17)
= 2v ′ 2
∫ t
0
R t dt, (7.18)
where R t is the correlation coefficient between the velocities of the same particle at
two different instants. Karlovitz uses from R t the following expression
where
R t = e −tv′ l ′ , (7.19)
∫ ∞
l ′ = v ′ R t dt (7.20)
0
is the Lagrangian scale of turbulence. By substituting (7.19) into (7.18) and with
t = l/v ′ it results for ¯X
√
¯X = l
(
2a v′
1 − au [
l
u l v ′ 1 − exp
(
− 1 )])
v ′
, (7.21)
a u l
Here a is the ratio from the Lagrangian to the Eulerian scales of turbulence. Karlovitz
assigns to a a value 1 while Scurlock and Grover [17] and Wohl [18] choose 1/2.
Taking (7.21) into (7.17) one obtains for u t
√ (
u t
= 1 + 2a v′
1 − au [ (
l
u l u l v ′ 1 − exp − 1 a
)])
v ′
. (7.22)
u l
For very intense turbulence (7.22) reduces to
√
u t
≃ 1 + 2a v′
, (7.23)
u l u l
which is valid provided that
v ′
≫ 1.
u l
Eq. (7.23) is in contradiction with (7.12) and for weak turbulence is reduces to
u t
u l
≃ 1 + v′
u l
, (7.24)

7.2. TURBULENT COMBUSTION THEORIES 215
valid for
which also contradicts (7.16).
v ′
u l
≪ 1,
Wohl has shown that (7.22) can also be deduced from (7.13) through purely
geometric considerations by assuming that the distortion of the flame front consists in
the formation of a system of prisms instead of the cones proposed by Shelkin.
Scurlock and Graver [17] calculate σ l by assuming with Shelkin that the laminar
surface is formed by a set of cones whose base is proportional to the square l 2
of the scale of turbulence and whose height is proportional to the root mean square
displacement ¯X of the turbulent oscillations suffered by a flame element. Therefore,
as in Eq. (7.14), u t is given by expression
√
u t
= 1 + k ¯X 2
u l l 2 . (7.25)
The problem lies in computing ¯X. Its magnitude, according to Scurlock, is
governed by three different mechanisms:
1) Turbulent diffusivity tends to increase indefinitely the value of ¯X as time elapses.
Figure 7.5 shows two consecutive positions of the flame front in Scurlock’s
model after the instant at which the flame was plane.
Initial flat flame
Flame after passage
of short time
Flame after passage
of longer time
Figure 7.5: Schematic diagram showing wrinkling with passage of time of an initially flat
flame element exposed to turbulence.
2) Laminar combustion tends to absorb oscillations reducing the value of ¯X. Karlovitz
and his collaborators [19] were the first to acknowledge this fact which is
represented in Fig. 7.6
3) The flame generates turbulence 2 and this tends to increase the value of ¯X.
2 See §4.

216 CHAPTER 7. TURBULENT FLAMES
ARBITRARY WAVY FLAME FRONT
BURNED GAS
FRESH GAS
FLAME FRONT AFTER ∆t TIME
Figure 7.6: The effect of laminar flame propagation on the evolution of a turbulent flame
front.
Since the locus of an element of the flame at two different instants are related
to two different particles of the mixture, when calculating the influence of turbulent
diffusivity on ¯X, a combined time-space coefficient of correlation R tx should be used
substituting coefficient time-correlation R t in Taylor’s formula [17].
The influence of the laminar propagation of the flame in the value of ¯X may
be obtained through purely geometric considerations. Finally, the influence of the
turbulence generated by the flame is taken into consideration by substituting into the
expression of ¯X, the intensity v ′ of turbulence of the unburnt gases flow by the resultant
of it, plus the intensity of the turbulence originated by the flame.
The combination of these three effects gives a differential equation which in
turn supplies the law of variation for ¯X as a function of time t during which the
particle has been exposed to turbulent oscillations. This differential equation replaces
Taylor’s equation (7.18) for this case. Through integration of this equation the value
for ¯X is obtained, and when taken into Eq. (7.22) it gives a formula for the propagation
velocity of the turbulent flame.
Now we must determine the time t that each element of the flame has been
exposed to the action of turbulence. Scurlock considers only flames inclined respect
to the flow such as the one obtained with a flame-holder or from the ring of a bunsen
burner. He identifies t with the time taken by a gas particle to cross the flame front
from the holders section to the point under consideration. Let v t by the component
tangential to the flame front of the velocity of the unburnt gases and s the distance to
the flame-holder measured along the flame front. The following expression is obtained
for t
t =
∫ s
0
ds
v t
. (7.26)

7.2. TURBULENT COMBUSTION THEORIES 217
The selection of t, which is arbitrary to a certain extent, is justified by the
theoretical and experimental studies carried out by Markstein [20] on the propagation
of disturbance along inclined flame fronts. These studies show that such disturbances
propagate with the tangential velocity of the flow as they become amplified.
BURNED
GASES
BURNED
GASES
BURNED
GASES
MEAN POSITION
OF FLAME
MEAN POSITION
OF FLAME
MEAN POSITION
OF FLAME
β
INSTANTANEOUS
FLAME FRONT
INSTANTANEOUS
FLAME FRONT
INSTANTANEOUS
FLAME FRONT
UNBURNED
GASES
UNBURNED
GASES
UNBURNED
GASES
U U U
BURNER
RIM
BURNER
RIM
BURNER
RIM
Figure 7.7: Cross section of stabilized unconfined Bunsen flames constructed on basis of
theory to demonstrate under typical conditions the predicted individual effects
of eddy diffusion, flame propagation and flame generated turbulence.
Figure 7.7 taken from the work by Scurlock and Grover, shows the influence of
the above mentioned facts for the case of an open flame stabilized in a bunsen burner.
The following values were adopted for computations: diameter of the burner = 5 cm,
velocity on the gases in the burner= 200 cm/s, u 1 = 10 cm/s, v ′ = 10 cm/s, intensity
of turbulence 5%, scale of turbulence= 0.25 cm, and width of the flame= 2 ¯X.

218 CHAPTER 7. TURBULENT FLAMES
7.3 Turbulence generated by the flame
Markstein [21] has shown that the disturbances originated at a certain point of an incline
flame propagate and amplify along the same and eventually generate turbulence.
In their studies of turbulent flames stabilized in tubes, Willians, Hottel and Scurlock
[5] recognized the existence of turbulence originated by the flame. They considered
this turbulence as a consequence of the strong gradients of velocity which originate
through the flame in this case. 3
Scurlock and Grove have given a formula for the
maximum intensity v ′ of the possible turbulence. Such formula was deduced from elementary
considerations on mass and momentum conservation across the flame before
and after the mixing of burnt gases. This formula is
√
( )
v m ′ K m ρ u
= (vu 3 ρ 2 − u 2 l ) ρu
− 1 . (7.27)
b ρ b
Here, K m = ∆p ∆p
−1, where is the relation between pressure jumps across
∆p
′
∆p
′
the flames before and after the turbulent mixing, v u is the velocity of the unburnt gases
normal to the mean flame front and ρ u /ρ b is the ratio between densities of unburnt and
burnt gases. Formula (7.27) gives a maximum limit to the turbulence that could be
originated in the flame but does not allow the computation of the possible turbulence
for each case. This arises the problem of introducing an additional undetermined
element in the theory which adds difficulties to the comparison between theoretical
and experimental results. In practice v ′ m can be considerably larger than the turbulence
of the incident stream.
Karlovitz [16] when comparing the values for u t predicted by his theory with
those obtained from experimenting with flames stabilized in bunsen burner verified
that experimental velocities were much larger than those calculated and he considered
this discrepancy to be due to the influence of the turbulence originated by the flame.
Karlovitz explains the production of turbulence as follows: through the laminar front
an increase in velocity takes place of the order of magnitude of ρ u
ρ b
− 1. In a laminar
flame this increase has a constant direction but in a turbulent flame it oscillates since
it must be normal to the temporary position of the flame front at all instants which is
variable. The statistic oscillations of this increase are the source of the turbulence.
Through a not too legitimate calculation Karlovitz deduces the following expression
for the maximum intensity of turbulence produced by the flame
v m ′ = √ 1 ( )
ρu
− 1 u l . (7.28)
3 ρ b
3 See chapter 10.

7.4. COMPARISON WITH EXPERIMENTAL RESULTS 219
It is also impossible here to estimate which fraction of the intensity given by (7.28)
will actually turn into turbulence in each case.
7.4 Comparison with experimental results
As aforesaid the available experimental evidence is not sufficient to establish either
one of the proposed theories in a definite way precluding the others. We owe the most
complete set of experiments to Bollinger and Williams [2] and to Wohl and Shores [7]
referred to herein. Some of the results obtained show laws of variation of u t which
approach those predicted by Karlovitz and by Scurlock and Grover. However, their
effects appear which are either not predicted by theory or in contradiction with it. An
example of effects not predicted is the influence of the mixture’s composition, since
it is verified that the effect of turbulence is not the same for rich mixtures than for
poor ones, even when operating under identical conditions and with the same laminar
propagation velocity. This is especially true for certain combustibles (like butane) and
this fact cannot be explained with any of the proposed theories. Wohl believes this
effect to be due to the different laminar stability in poor and rich mixtures. An example
of effects appearing in contradiction with theory is the influence of the scale of
turbulence, which after Scurlock’s predictions, should be considerable and which according
to experimental results is very small. Consequently it is necessary to increase
experimental evidence before final decision may be reached.
References
[1] Mallard, F. and Le Chatelier, H.: Ann. de Mines. Vol. 8, Sev. 4, 1884, p. 274.
[2] Damköhler, G.: The Effect of Turbulence on the Flame Velocity in Gas Mixtures.
NACA Tech. Mem. No. 1112, 1947.
[3] Bollinger L. M. and Williams, D. T.: Effect of Reynolds Number in the Turbulent-
Flow Range on Flame Speeds of Bunsen-Burner Flames. NACA Tech. Note
No. 1707, Sept. 1948.
[4] Scurlock, A. C.: Flame Stabilization and Propagation in High Velocity Gas
Streams. Meteor Report No. 19, July 1948.
[5] Williams, G. C., Hottel, H. C. and Scurlock, A. C.: Flame Stabilization and
Propagation in High Velocity Gas Streams. Third Symposium (International)
on Combustion, Williams and Wilkins Co., Baltimore, 1949, pp. 21-40.
[6] Karlovitz, B. Denniston, D. W. and Wells, F. E.: Investigation of Turbulent
Flames. Journal of Chemical Physics, Vol. 19, No. 5, May 1951.

220 CHAPTER 7. TURBULENT FLAMES
[7] Wohl, K., Shore, L., von Rosemberg, H. and Weil, C. M.: The Burning Velocity
of Turbulent Flames. Fourth Symposium (International) on Combustion,
Williams and Wilkins Co., Baltimore, 1953.
[8] Leason, D. B.: Turbulence and Flame Propagation in Premixed Gases. Fuel,
Vol. XXX, No. 10, Oct. 1951.
[9] Bowditch, F.: Some Effects of Turbulence on Combustion. Fourth Symposium
(International) on Combustion, Williams and Wilkins Co., Baltimore, 1953,
pp. 620-635.
[10] Wohl, K. and Shore, L.: Experiments with Butane-Air and Methane-Air Flames.
Industrial and Engineering Chemistry, April 1955.
[11] Mickelsein, W. R. and Ernstein, N. E.: Propagation of a Free Flame in a Turbulent
Gas Stream. NACA Tech. Note No. 3456, 1955.
[12] von Kármán, Th. and Marble, F.: Combustion in Turbulent Flames. Fourth
Symposium (International) on Combustion, Williams and Wilkins Co., Baltimore,
1953, p. 923.
[13] Sommerfield, M., Reiter, S. H., Kebely, V. and Mascolo, R.W.: The Structure
and Propagation Mechanism of Turbulent Flames in High Speed Flow. Jet
Propulsion, August 1955.
[14] Shelkin, F. I.: On Combustion in a Turbulent Flow. NACA Tech. Mem. No.
1110, Febr. 1947.
[15] Tucker, E.: Interaction of a Free Flame Front with a Turbulence Field. NACA
Tech. Note No. 3407, 1955.
[16] Karlovitz, B.: A Turbulent Flame Theory Derived From Experiments. Selected
Combustion Problems, Vol.I, AGARD, 1954, pp. 248-262.
[17] Scurlock, A. C. and Grover, J. F.: Experimental Studies on Turbulent Flames.
Selected Combustion Problems, Vol.I, AGARD, 1954, pp. 215-247.
[18] Wohl, K.: Burning Velocity of Unconfined Turbulent Flames Industrial Engineering
Chemistry, April 1955, p. 825.
[19] Karlovitz, B.: Open Turbulent Flames. Fourth Symposium (International) on
Combustion, Williams and Wilkins Co., Baltimore, 1953, pp. 60-67.
[20] Markstein, G. H.: Discussion on Turbulent Flames. Selected Combustion Problems,
Vol.I, AGARD, 1954, pp. 263-265.
[21] Markstein, G. H.: Interaction of Flow Propagation and Flame Disturbances.
Third Symposium (International) on Combustion, Williams and Wilkins Co.,
Baltimore, 1949, pp. 162-167.

Chapter 8
Ignition, flammability and
quenching
8.1 Introduction
In the preceding chapter we have analyzed the properties of a plane wave propagating
in a stationary regime, through an indefinite combustible mixture whose state and
composition are uniform. Although the problem is far from being completely solved,
the essential characteristic of those waves are well understood, and a precise formulation
of the same is available as well as several methods to solve the resulting equations,
including some practical solutions, which may be considered as fairly acceptable when
compared to experimental results.
However, it must be said that the stage of knowledge is not as favorable for
other fundamental questions relative to flames such as: the problem of ignition of
a combustible mixture; the possibility of a flame to propagate through a mixture of
given state and composition; and the influence of the vicinity of the walls on the
characteristics of the flame. Even when a great amount of attention has been applied
during recent years to the study of these problems, the progress achieved, specially
from the theoretical stand-point, has been considerably less than for the case of an
indefinite plane wave, to the extreme that the causes of some of the phenomena are
still unknown as well as their governing laws.
The present chapter contains a brief description of each of the above mentioned
problems, accompanied by a selected bibliography which will enable a more detailed
study on each one.
221

222 CHAPTER 8. IGNITION, FLAMMABILITY AND QUENCHING
8.2 Ignition
Considering a combustible mixture capable of propagating a flame, in order for this
to happen it is necessary to supply it with a certain amount of energy, located within a
small volume and reduced interval of time, to initiate the wave. The study of this process,
both theoretically and experimentally, has been the subject of numerous reports.
Among the various sources of energy that may be utilized, the most adequate is the
electrical spark because it is capable of steering great amounts of energy in reduced
spaces and times. For this reason it is the most widely used and has been the subject
of more systematic and complete experimental studies, of which a description may be
found in Refs. [1] and [2].
The fundamental problem of ignition lies on determining the minimum energy
required to ignite a mixture of given composition, at known pressure and temperature.
The results of experiments have pointed out that to each mixture it corresponds a well
defined minimum ignition energy, which increases very rapidly with the decrease in
pressure. Such energy results to be independent from the distance between electrodes,
provided this distance exceeds a given minimum value under which it grows very
rapidly and finally the flame cannot propagate.
Several theories have been attempted for the calculation of this minimum energy
with fair success. The present state of knowledge on this problem may be considered
as comparable to the situation existing 20 years ago respect to theories on
flames.
All the theories developed are based on the following model. Let us consider
a small volume V of gas to which energy H is instantaneously communicated, rising
its temperature from T 0 to T f . Starting from this instant, the gas contained in V
tends to cool, by thermal conductivity, heating the gas layers surrounding V . On the
other hand, the chemical reaction which produces in the heated mass releases heat,
which tends to compensate the cooling effect. From the balance between this two
phenomena, it will depend that the flame may progress towards a wave of the type
described in Chap. 6, or, to the contrary, that it extinguishes.
The available theories differ in the development of this idea and on the conditions
applied to determine the minimum volume V and the necessary energy H. For
instance, Lewis and von Elbe [3] assume that volume V is the one corresponding to
the quenching 1 distance and that the energy is determined by the excess enthalpy of
the flame. The validity of this concept does not appear clearly justified, although the
1 See §4.

8.2. IGNITION 223
comparison between the minimum values of the calculated energy and those measured
experimentally show a surprising agreement [3].
Fenn [4] performs an elementary computation which is actually more of a dimensionless
analysis of the problem. He reasons as follows: be r the radius of volume
V and let us assume that combustion takes place through a second-order reaction of
the form
w = Aρ 2 Y (1 − Y )e −E/RT . (8.1)
Then, the heat released per unit time in volume V due to the combustion of the mixture
will be
Q = 4 3 πr3 qAρ 2 Y (1 − Y )e −E/RT f , (8.2)
where T f is the temperature of gases, and q the heat of reaction.
In turn, the heat lost by conductivity through the surface limiting V , will be
Q ′ = 4πr2 λ(T f − T 0 )
, (8.3)
cr
where cr is the thickness of the combustion wave, which separates hot from cold
gases.
The condition of minimum ignition energy for propagation, will be
Q = Q ′ , (8.4)
since if it is Q < Q ′ the mass cools and the flame extinguishes.
radius of V
By taking (8.2) and (8.3) into (8.4) we obtain the following expression for the
√
3λ(T f − T 0 )
r =
cqAρ 2 Y (1 − Y )e −E/RT . (8.5)
f
Finally, the minimum energy H is the one needed to rise the mass contained in V from
temperature T 0 to T f which is
where for r one must use expression (8.5).
H = 4 3 πr3 c p ρ(T f − T 0 ), (8.6)
Recently Swott [5] has extended this theory to the study of the problem of
the ignition of flowing mixtures including as well the effects of turbulence. Other
studies on the matter, specially directed to the ignition in combustion chambers, with
extensive bibliography, will be found in Ref. [6].
The preceding theories are of the thermal type, since they ignore the influence
of diffusion, which could be important, specially the diffusion of radicals.

224 CHAPTER 8. IGNITION, FLAMMABILITY AND QUENCHING
A correct stating of the problem would require first an adequate formulation of
the same, in an analogous way to the one used in Chap. 6 for the stationary wave. In
the second place it would be necessary to perform an analysis of the type of solutions
for the system of equations obtained under the set of initial and boundary conditions
corresponding to the model previously described.
If we consider the more simple case of an indefinite plane wave corresponding
to a one-dimensional problem, in which case the initial energy must be stored between
two parallel layers, it may be easily verified that the system of equations of Chap. 6,
corresponding to a stationary wave, keeping the same notation, must be substituted by
the following:
a) Continuity equation.
b) Energy equation.
c) Diffusion equation.
∂ρ
∂t + ∂(ρv) = 0. (8.7)
∂x
( ∂T
ρc p
∂t + v ∂T )
= ∂ (
λ ∂T )
+ qw. (8.8)
∂x ∂x ∂x
ρ
( ∂Y
∂t + v ∂Y )
= ∂ (
ρD ∂Y )
+ w. (8.9)
∂x ∂x ∂x
Thus we obtain a system of three equations for the three unknowns T , v and Y .
Since the process takes place at constant pressure, ρ is already determined, as function
of T .
With reference to initial and boundary conditions corresponding to the model
under consideration, if 2d is the width of the heated slab and the origin of coordinates
is fixed at the central point of the slab, we will have:
a) Initial conditions (t = 0).
0 < x < d : T = T f ,
d < x < ∞ : T = T 0 ,
(8.10)
b) Boundary conditions (by symmetry).
0 < x < ∞ : v = Y = 0.
x = 0 :
∂T
∂x = v = ∂Y
∂x = 0.
Furthermore an additional condition must be introduced expressing that w must
be zero for T = T 0 , as it was done for the case of a stationary wave.

8.3. FLAMMABILITY LIMITS 225
The preceding system of equations allows an easy simplification, by using the
stream function ψ, frequently applied to the problem of Fluid Mechanics.
The said function is defined by the following conditions
ρ = ∂ψ
∂x ,
ρv = −∂ψ
∂t , (8.11)
through which Eq. (8.7) is identically satisfied. When taking Eq. (8.11) into Eqs. (8.8)
and (8.9), this system reduces to the following
∂T
∂t = ∂ ( ρλ ∂T
∂ψ c p ∂ψ
∂Y
∂t = ∂ ( ρ 2 D
∂ψ
Thus the variable v has been eliminated.
c p
∂Y
∂ψ
)
+ q c p
w
ρ , (8.12)
)
+ w ρ . (8.13)
The integration of the above system has not yet been performed. However,
Spalding [7] has obtained graphical solutions of Eq. (8.12) disregarding diffusion.
These solutions show the existence of a critical value d cr for d under which the wave
extinguishes, while for d > d cr it propagates indefinitely.
There are some experimental observations on the evolution followed by an
ignited mass before the flame establishes [8].
8.3 Flammability limits
The theoretical studies performed so far have shown that the combustible mixture is
always capable of maintaining a flame. Experimentally, however, it has been verified
that it is not so, since there are numerous mixtures which, in practise, are not able to
propagate a flame, even when theory predicts otherwise.
In fact, experiments disclose that when analyzing the behavior of a combustible
mixture under pressure and temperature constant, but changing its composition, for instance
a mixture of fuel and air, the flame velocity is maximum for a definite value
of the composition which is normally very close to the stoichiometric one, and it decreases
for mixtures either rich or loan, up to values known as inflammability limits of
the mixture, beyond which it cannot propagate a flame. Moreover the flame velocity
does not vanish at the inflammability limits but it generally reaches values of a few
centimeters per second. Inflammability limits depend also on pressure and they are
difficult to determine experimentally because it is necessary to operate under marginal
conditions, on which the experimental techniques might have an influence. Tables

226 CHAPTER 8. IGNITION, FLAMMABILITY AND QUENCHING
and graphics may be found in Ref. [1], giving the inflammability limits of several
mixtures of hydrocarbons with air, under different pressures. Ref. [3] supplies an
abridged review of the problem. The Proceedings of the Fourth International Symposium
on Combustion include, as well, several papers on the subject, Coward and
Jones, [9], reviewed the state of knowledge up to 1952, in a report including an extensive
experimental and bibliographic material. Two recent, very interesting, reviews
are those by Egorton [10] and Linnot-Simpson [11].
At present, the situation of the problem is such, that the causes for the existence
of inflammability limits are still unknown and, even more, it is questioned
whether they actually exit as quantities governed only by the state and composition
of the mixture or, to the contrary, they depend upon the characteristics of the experimental
device used. Lewis and von Elbe [3] suppose that the existence of such limits
may be due to the internal stability of the combustion wave, which then would be unstable
beyond the limits. Several attempts to study the problem from this stand-point
have been made, [12] and [13], in addition to the one by Lewis and von Elbe. Of
all, the most satisfactory is own to Werner and Rosen [14], who analyze the internal
stability of the wave with respect to small disturbances of temperature by means of a
linearization of the system of Eqs. (8.7), (8.8) and (8.9) around their stationary solution.
These authors solved the resulting system after introducing several simplifying
assumptions regarding the disturbances of the composition of the mixture and of the
mass flux. They reach the conclusion that the stability of the wave depends on the
behavior of the disturbances of Y with respect to those of T . Although this study
must be considered as uncompleted it suggests the possibility for some waves to be
internally unstable, which could explain the existence of inflammability limits. An
application of this theory to the ozone-oxygen flame is given in Ref. [15], the result
of which shows that the flame is stable for ozone rich mixtures and unstable for loan
ones. However, the experiments carried out by Streng and Grosse [16] disclose that
the flame is stable even for loan mixtures.
Recently Spalding [17] has analyzed the possible influence of heat losses of
the flame on its behavior. He reaches the conclusion that there exist two different
flame velocities, and that the smaller is unstable. When heat losses augment the velocities
approach one another and finally coincide. Spalding identifies the coincidence
of velocities with the inflammability limits. Although this work does represent an important
contribution, yet an unpublished work, being carried out at the Combustion
Laboratory of the INTA, reveals that his conclusions are not free from criticism.

8.4. QUENCHING 227
8.4 Quenching
So far studies have dealt with flames propagating through unlimited mixtures, not
taking into account the influence of the proximity of walls,
A set of phenomena exist, of great importance, relative to the behavior of
flames close to walls which have been the subject of such attention in recent years.
Among those phenomena are quenching, blow-off and flash-back.
Experiments prove that when the diameter of a tube full of a combustible gas
is sufficiently small, it is impossible for a flame to propagate through it. This phenomenon,
which has been recognized for a long time and which has been applied to
the prevention of explosions, is known as “quenching”.
The way in which the walls of the tube act to prevent flame propagation is
not well understood at present, The most direct explanation is to attach the effect to
the cooling of the flame due to the proximity of the walls. However, it is though
that chemical action of the wall when acting as chain breaker may also be significant.
Analogously to what happened for some time with flame theory, if one considers either
cooling or diffusion as the only acting effect, a “thermal” or “diffusive” theory will
be obtained. Examples of thermal theories are those by Lewis-von Elbe [1] and by
Kármán-Millán [18], and on diffusion theories the one by Simon-Bellos [19].
The difference between this case and the unlimited flame theory lies on the fact
that both the formulation of the equation and boundary conditions and their solution
are far more difficult. Hence, the solutions may only be attempted through drastic
simplifications.
A great number of measurements have been carried out with the purpose of
determining the quenching distance in ducts of different cross-sections and the influence
on it, of the mixture. For an additional study of this question as well as on the
phenomena of blow-off and flash-back we refer the reader to Refs. [20] through [23].
In particular references [20] and [23] include an extensive bibliography.
References
[1] Lewis, B. and von Elbe, G.: Combustion, Flames and Explosions of Gases.
Academic Press, New York, 1951.
[2] Lewis, B, Pease, R. N. and Taylor, H,S.: Combustion Processes. Vol. II of High
Speed Aerodynamics and Jet Propulsion, Princeton University Press, 1956.

228 CHAPTER 8. IGNITION, FLAMMABILITY AND QUENCHING
[3] Lewis, B. and von Elbe, G.: Fundamental Principles of Flammability and Ignition.
Selected Combustion Problems, Vol. II, AGARD, 1956, pp. 63-72.
[4] Fenn, J. B.: Lean Inflammability Limit and Minimum Spark Ignition Energy.
Industrial and Engineering Chemistry. Vol. Dec., 1951, pp. 2865-68.
[5] Swott, C. C.: Spark Ignition of Flowing Gases. IV. Theory of Ignition in Nonturbulent
and Turbulent Flow Using Long Duration Discharges. NACA Research
Memorandum No. R M E54F29a, 1954.
[6] Wigs. L. D.: The Ignition of Flowing Gases. Selected Combustion Problems,
Vol. II, AGARD, 1956, pp. 73-82.
[7] Spalding, D. B.: Some Fundamentals of Combustion. Butterworths Scientific
Publications, 1955.
[8] Olsen, E. L., Gayhart, E. L. and Edmoneon, R. B.: Propagation of Incipient
Spark–Ignited Flames in Hydrogen–Air and Propane–Air Mixtures. Fourth
Symposium (International) on Combustion, Williams and Wilkins Co., Baltimore,
1953, pp. 144-148.
[9] Coward, H. F. and Jons, G. W.: Limits of Flammability of Gases and Vapors.
U.S. Bureau of Mines. Bull., 503, 1952.
[10] Egerton, A. C.: Limits of Inflammability. Fourth Symposium (International) on
Combustion, Williams and Wilkins Co., Baltimore, 1953, pp. 4-13.
[11] Linnet, J. W. and Simpson, O. J. S.: Limits of Inflammability. Sixth Symposium
(International) on Combustion, Reinhold Publishing Corp., New York, 1957.
[12] Richardson, J. M.: The Existence and Stability of Simple One–Dimensional,
Steady–State Combustion Waves. Fourth Symposium (International) on Combustion,
Williams and Wilkins Co., Baltimore, 1953, pp. 182-189.
[13] Layzer, D.: Journal of Chemical Physics, Vol. 22, 1954, p. 222.
[14] Wehner, J. F. and Rosen, J. B.: Temperature Stability of the Laminar Combustion
Wave. Combustion and Flame, Sept. 1957, pp. 339-345 .
[15] Rosen, J. B.: Stability of the Ozone Flame Propagation. Sixth Symposium
(International) on Combustion, Reinhold Publishing Corp., New York, 1957.
[16] Streng, A. G. and Grosse, A. V.: The Ozone to Oxygen Flame. Sixth Symposium
(International) on Combustion, Reinhold Publishing Corp., New York, 1957.
[17] Spalding, D. B.: A Theory of Inflammability Units and Flame–Quenching .
Proc. Roy, Soc. London, Ser. A, Vol. 240, No. 1220, 1957, pp. 83-100.
[18] von Kármán, Th. and Millán, G.: Thermal Theory of Laminar Flame Front near
a Cold Wall. Fourth Symposium (International) on Combustion, Williams and
Wilkins Co., Baltimore, 1953, pp. 173-177.

8.4. QUENCHING 229
[19] Simon, D. M. and Belles, F. E.: An Active Particle Diffusion Theory of Flame
Quenching for Laminar Flames. NACA Research Memorandum No. RM
E51L18, 1952.
[20] Wohl, K.: Quenching Flash–Back, Blow–Off. Theory and Experiment. Fourth
Symposium (International) on Combustion, Williams and Wilkins Co., Baltimore,
1953, pp. 68-89.
[21] Berlad, A. L. and Potter, A. B.: Effect of Channel Geometry on the Quenching
of Laminar Flames. NACA Research Memorandum RM E54C05, 1954.
[22] Berlad, A. L.: Flame Quenching by a Variable Width Rectangular–Channel
Burner as a Function of Pressure for Various Propane–Oxygen–Nitrogen Mixtures.
Journal of Physical Chemistry, Vol. 58, 1954, pp. 1023-1026.
[23] Massey, B. S. and Lindley, B. O.: Flame Quenching. Journal of the Royal
Aeronautical Society, Jan. 1958, pp. 32-42.

230 CHAPTER 8. IGNITION, FLAMMABILITY AND QUENCHING

Chapter 9
Flows with combustion waves
9.1 Introduction
As previously seen in chapter 6, in a combustible mixture at ambient of higher pressure,
the thickness of the flame is of the order of a fraction of millimeter. This length
is generally “small” when compared to those of interest in aerothermodynamic problems.
Therefore, when studying these problems it is justified to assume that the flame
thickness is zero. In such a case the flame may be considered as a surface of discontinuity
of the pressure, temperature and velocity. The same idea is applied, for example,
in the study of gas dynamic problems involving shock waves. The problem has been
studied from this view point mainly by Emmons and his co-operators [1], [2], [3], [4].
In this chapter the general lines of Emmons’s work are followed. First the conditions
that must be satisfied across the flame are established and then some general properties
deduced. In the chapter that follows, the method is applied to the study of the
aerothermodynamic field originated by a flame stabilized in a combustion chamber.
This problem has been studied by Scurlock, Tsien, and Fabri-Siestrunck-Fouré.
9.2 Conditions that must be satisfied by the jump across
a flame front.
As aforesaid, the relations deduced herein are applicable only if the thickness of the
flame front is small when compared to a characteristic length of the phenomenon under
study. 1 The flame thickness must also be small compared to the radius of curvature
1 For instance, this does not occur in rarefied mixtures where the thickness of the flame can be large.
231

232 CHAPTER 9. FLOWS WITH COMBUSTION WAVES
of the flame. In fact, if the radius of curvature and the flame thickness are of the
same order of magnitude the flame speed depends not only on the thermodynamic
conditions of the unburnt gases, but also on the shape of the flame in the neighborhood
of the point under consideration. Such is the case in the flame tip of a Bunsen burner.
Here, the flame speed is increased due to the favorable effect of the curvature on heat
transfer and diffusion of the active particles towards the unburnt gases.
V n1
V t1
p , T ,
ρ
1 1 1
V t2
V
n2
p , T ,
2 2
ρ
2
Figure 9.1: Schematic diagram of a surface element of a flame.
Thus, let it be a flame front of small thickness and small curvature as indicated
in Fig. 9.1, and let us consider a surface element of this front. By applying the continuity,
momentum, and energy equations to the gases at both sides of this element, as
it is done in the study of the invariants across a shock wave 2 , the following conditions
are obtained which relate the state of the unburnt and burnt gases at both sides of the
elements:
1) Continuity equation.
ρ 1 v n1 = ρ 2 v n2 . (9.1)
2) Momentum equation.
a) Normal component.
ρ 1 v 2 n1 + p 1 = ρ 2 v 2 n2 + p 2 . (9.2)
b) Tangential component.
v t1 = v t2 . (9.3)
3) Energy equation.
1
2 v2 1 + h 1 = 1 2 v2 2 + h 2 . (9.4)
Subscripts 1 and 2 refer to conditions before and after the flame, respectively,
v is the velocity of the gases relative to the flame front, assumed stationary, and v n
2 See chapter 6.

9.2. CONDITIONS THAT MUST BE SATISFIED BY THE JUMP ACROSS A FLAME FRONT. 233
and v t are the normal and tangential components of this velocity. Moreover ρ and p
are, as usual, the density and pressure of the gases, and h is the total specific enthalpy,
that is, (as seen in chapter 1), the thermal enthalpy h T plus the formation enthalpy h f ,
h = h T + h f , (9.5)
where h T is expressed, in terms of the heat capacity c p at constant pressure and the
absolute temperature T , as follows
h T =
∫ T
0
which, for the special case c p = constant reduces to
c p dT, (9.6)
h T = c p T. (9.7)
At each side of the flame front, the pressure, density and temperature are related
by the state equation, which is assumed to be that of the perfect gases, that is
p 1
ρ 1
= R 1 T 1
and
p 2
ρ 2
= R 2 T 2 , (9.8)
where
R 1 = R M u
and R 2 = R M b
(9.9)
are the specific constants of the unburnt and burnt gases, respectively, and M u and M b
their respective average molecular masses.
Since the tangential component of the velocity is continuous throughout the
flame, there results that the incoming and outgoing velocities and the vector normal to
the combustion front are coplanar.
When the state of the unburnt gases and the incline α 1 of the incident flow
relative to the flame front, see Fig. 9.2, are known, the system of equations (9.1),
(9.2), (9.3), (9.4) and (9.8) determines the state of the burnt gases.
Let ϕ be the flame propagation velocity corresponding to the composition of
the unburnt gas mixture in the thermodynamic state defined by the values p 1 and ρ 1
of pressure and density. Obviously we have
ϕ = v n1 , (9.10)
and for the incline α 1 of the flame front relative to the incident flow
sin α 1 = v n1
v 1
= ϕ v 1
. (9.11)

234 CHAPTER 9. FLOWS WITH COMBUSTION WAVES
V t1
V n1
α 2
ϕ = −
V n1
α 1
V 1
α 1
δ
V n2
V 2
V t2 =
Vt1
Figure 9.2: Velocity components at both sides of the front.
Therefore, α 1 is determined by the velocity v 1 of the incident flow relative to the
front and by the burning velocity ϕ of the mixture. This velocity depends only on the
composition, pressure and density or temperature, and must be considered as a datum,
determined either theoretically or experimentally (see chapter 6).
In order to simplify calculations, it is assumed in the following that the thermal
enthalpy can be expressed as indicated in (9.7). Let
q = h f1 − h f2 (9.12)
be the difference between the formation enthalpies of the unburnt and burnt gases. By
substituting equations (9.7) and (9.12) into (9.4), this equation can be written
1
2 v2 1 + c p1 T 1 + q = 1 2 v2 2 + c p2 T 2 . (9.13)
Let T s1 and T s2 be the stagnation temperatures of the unburnt and burnt gases
respectively. By virtue of equation (9.13) they are related by
c p1 T s1 + q = c p2 T s2 , (9.14)
and their ratio n is given by
For the particular case c p1 = c p2 , we have
n = T (
s2
= 1 + q )
cp1
. (9.15)
T s1 c p1 T s1 c p2
n = 1 +
q
c p1 T s1
. (9.16)

9.3. NORMAL FLAME FRONT 235
9.3 Normal flame front
Let us consider a normal flame front as indicated in Fig. 9.3. In such a case, the
equations that relate the conditions before and after the flame reduce to
ρ 1 v 1 =ρ 2 v 2 , (9.17)
ρ 1 v 2 1 + p 1 =ρ 2 v 2 2 + p 2 , (9.18)
1
2 v2 1 + c p1 T 1 + q = 1 2 v2 2 + c p2 T 2 . (9.19)
V
V
1 2
ϕ
p , T ,
ρ
1 1 1
p , T ,
2 2
ρ
2
Figure 9.3: Schematic diagram of a normal flame front.
Since, v 1 is equal to the flame propagation velocity in the unburnt gases, (which
is normally of the order of 1 m/s) and v 2 is only a few times larger, both v 1 and v 2
are very “small” when compared to the sound speed in the unburnt and burnt gases
respectively. Therefore, in equation (9.19) the terms containing the kinetic energy of
the unburnt and burnt gases can be neglected. There results
c p1 T 1 + q ≃ c p1 T s1 + q = c p2 T s2 ≃ c p2 T 2 . (9.20)
That is
T 1 ≃ T s1 , T 2 ≃ T s2 . (9.21)
Therefore, by virtue of Eq. (9.15)
T 2
T 1
≃ T s2
T s1
= n. (9.22)
Furthermore, as results from Eq. (9.18) the pressure drop p 2 − p 1
is very small
compared to unity, and, consequently, for the calculation of the thermodynamic state
of the burnt gases it may be assumed to be zero. The density ratio is then given by
ρ 2
ρ 1
= T 1
T 2
= 1 n . (9.23)
p 1

236 CHAPTER 9. FLOWS WITH COMBUSTION WAVES
The ratio of velocities is
obtained from Eqs. (9.23) and (9.17).
v 2
v 1
= n, (9.24)
Therefore, once the density and temperature of the unburnt gases and the value
of n are known, the temperature of the burnt gases is determined by equation (9.22),
its density by equation (9.23), and the pressure drop across the flame by
p 2 − p 1
p 1
= γ 1 M 2 1 (1 − n), (9.25)
where γ 1 is the ratio of the heat capacity at constant pressure to the heat capacity at
constant volume of the unburnt gases, and
M 1 = v 1
a 1
(9.26)
is the flow Mach number for the unburnt gases, where
√
p 1
a 1 = γ 1 (9.27)
ρ 1
is the sound speed in these same gases.
9.4 Inclined flame front
Here, v n1 and v n2 are still small compared to the sound speed of the unburnt and
burnt gases respectively. Therefore, the pressure drop across the flame front is still
small, see Eq. (9.2), and the ratio between densities on each side of the flame front is
determined by the ratio of temperature T 2 to temperature T 1 . Let λ be this ratio, there
results
Let
λ = T 2
T 1
= ρ 1
ρ 2
= v n2
v n1
. (9.28)
δ = α 2 − α 1 (9.29)
be the velocity deviation across the flame front (see Fig. 9.2) . A simple calculation
gives for δ
tan δ = (λ − 1) tan α 1
1 + λ tan 2 α 1
. (9.30)
This ratio has been taken into Fig. 9.4 for different values of λ. As it can easily be
seen, the maximum deviation δ max corresponds to
tan α 1 = 1 √
λ
. (9.31)

9.4. INCLINED FLAME FRONT 237
60
50
40
λ = 12
8
6
5
4
δ ( o )
30
3
20
λ = 2
10
δ max
0
10 30 50 70 90
α 1
( o )
Figure 9.4: Velocity deviation across the flame front as a function of α 1.
Its value is given by
tan δ max = 1 2
( )
√λ 1 − √λ . (9.32)
Even though the normal velocity is always “small”, the tangential velocity v t , which
by virtue of equation (9.3) is the same at both sides of the front, can be “large”. Therefore,
it must be taken into account when writing the energy equation. As a result, the
temperature at each side of the front can differ, considerably, from the corresponding
stagnation temperature. This is exactly the opposite to what occurs in the case of a
normal front.
Therefore, two different cases are to be considered, either
v t ≃ O(v n ), (9.33)
in which case
or
in which case
T 1 ≃ T s1 , T 2 ≃ T s2 , λ ≃ n, (9.34)
v t ≫ v n , (9.35)
v 2 1 ≃ v 2 2 ≃ v 2 t , (9.36)
and equation (9.13) reduces to
c p1 T 1 + q = c p2 T 2 . (9.37)

238 CHAPTER 9. FLOWS WITH COMBUSTION WAVES
By virtue of equation (9.15) and of the relation
the following expression is obtained for λ
T s1
= 1 + γ 1 − 1
M1 2 , (9.38)
T 1 2
λ = n + γ 1 − 1
2
(
n − c )
p1
M1 2 , (9.39)
c p2
which shows that for large values of M 1 , that is, when the flame front is very inclined
to the incident flow, λ can appreciably differ from n.
The values of λ/n as a function of M 1 , for c p2 = c p1 and different values of
χ = γ 1 − 1 n − 1
, have been represented in Fig. 9.5.
2 n
λ / n
1.20
1.18
1.16
1.14
1.12
1.10
1.06
1.06
1.04
1.02
χ=(γ 1
−1)(n−1)/2n
χ=0.20
χ=0.18
χ=0.16
χ=0.14
χ=0.12
χ=0.10
1.00
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
M 1
Figure 9.5: Values of the ratio λ/n as a function of incident Mach number M 1.
Relation (9.39) is only valid for very inclined flame front, that is, for not too
small values of M 1 . For values of M 1 close to zero, it must be substituted by
λ ≃ n. (9.40)
The first case, that is to say, the case of slow flows (M 1 ≪ 1) with flame
fronts, has been examined in detail by Gross and Esch [4], reaching the following
conclusions:
1) If ϕ, λ and q are constant and the motion is irrotational before the flame, after
the flame it continues being irrotational.

9.5. ENTROPY JUMP ACROSS THE FLAME FRONT 239
PRODUCTS
FLAME FRONT
REACTANTS
Figure 9.6: Straight-line flame flow field according to Gross and Esch for ϕ/v 1 = 0.1 and
λ = 7.
2) Opposite to what occurs with fast flows, if the flow is slow pressure drop across
the flame front cannot be neglected.
In order to analyze the motion, the flame front can be considered as a surface with a
distribution of sources. For example, in the case of a plane motion, the strength of the
(λ − 1)ϕ
source per unit length of the front is . Now by expressing the condition that
2π
in the unburnt gases the velocity normal to the front must be ϕ, an integral equation
is obtained which must ϕ satisfied at the front. This integral equation determines the
flame shape which ”a priori” is unknown. Gross and Esch give approximate solutions
for certain cases. For example, for the case in which the flame front reduces to a
straight segment inclined to the incident flow. Fig. 9.6, taken from the said work,
shows the shape of the streamlines in this case.
9.5 Entropy jump across the flame front
The entropy of the unburnt gases is given by the expression 3
3 See chapter 1.
S 1 = S 01 + c p1 ln p1/γ 1
1
, (9.41)
ρ 1

240 CHAPTER 9. FLOWS WITH COMBUSTION WAVES
where S 01 depends only on the composition of the mixture. Likewise, the entropy of
the burnt gases is
S 2 = S 02 + c p2 ln p1/γ2 2
, (9.42)
ρ 2
Therefore, the entropy jump ∆S across the flame, is
∆S = S 2 − S 1 = (S 02 − S 01 ) + c p2 ln p1/γ 2
2
− c p1 ln p1/γ 1
1
. (9.43)
ρ 2 ρ 1
As previously seen, the pressure drop across the flame is very small. Therefore, in
(9.43) we can take
p 2 ≃ p 1 = p. (9.44)
Making use of this simplification and of the relation (9.28) between p 1 and p 2 , the
following expression is obtained for ∆S
being
∆S = S 2 − S 1 = (S 02 − S 01 ) + c p2 ln λ + (c p2 − c p1 ) ln p1/γ 12
ρ 1
, (9.45)
γ 12 = c p2 − c p1
c v2 − c v1
, (9.46)
where c v1 and c v2 are the heat capacities at constant volume of the unburnt and burnt
gases respectively.
In the particular case
c p2 = c p1 = c p , (9.47)
the entropy jump across the flame reduces to
∆S = (S 02 − S 01 ) + c p ln λ. (9.48)
Equation (9.39) shows that λ can vary along the flame front when M 1 varies. Therefore,
if the local conditions of the flow vary, the entropy jump across the flame front
can vary considerably from one point to another. Due to this variation of the entropy
jump along the flame front, the motion after the front can be rotational, even
if the motion before the front is potential. This is similar to what occurs in the case
of supersonic motions with shock waves, when their incline varies from one point to
another. 4
If the motion before the flame is isentropic, the value λ depends only on the
incline α 1 of the flame front. In fact, by using the relation,
57.
tan α 1 = v n1
v t
= ϕ v t
, (9.49)
4 See A. Ferri: Elements of Aerodynamic of Supersonic Flows. Mac Millan Comp., New York, 1949, p.

9.5. ENTROPY JUMP ACROSS THE FLAME FRONT 241
that gives the incline of the flame front, and the relation
M 1 ≃ v t
a , (9.50)
which, as seen in the preceding paragraph, is valid for the very inclined flame fronts,
Bernoulli equation for unburnt gases can be written
1 + γ 1 − 1
M1 2 =
2
(
a01
ϕ
) 2
M 2 1 tan 2 α 1 , (9.51)
where a 01 is the sound speed at the stagnation point of the unburnt gases.
By eliminating M 1 between this equation and equation (9.39), λ can be expressed
as a function of the flame incline, as follows
λ = n +
2
γ 1 − 1
n − c p1
(
a01
ϕ
c p2
) 2
tan 2 α 1 − 1. (9.52)
When the flame incline is increased, α 1 decreases, and relation (9.52) show that λ
increases. Therefore, the entropy jump ∆S increases when the flame incline is increased.
Such behavior is the opposite to that of a shock wave where the entropy jump
decreases when the wave incline is increased. See Fig. 9.7 [5] . The influence of
this variation of the entropy jump on the flow will be studied in the following paragraph.
The simplified expression (9.48) for the entropy jump will be used, and since
the constant S 02 − S 01 has no influence on the flow, we shall express in short
∆S = c p ln λ. (9.53)
∆ S
ω 2 ∆ S
ω 2
(a) Flame
(b) Shock wave
Figure 9.7: Entropy jump across a flame front and a shock wave.

242 CHAPTER 9. FLOWS WITH COMBUSTION WAVES
9.6 Vorticity across the flame
In the flow of a perfect gas, where viscosity and mass forces are neglected, the variation
of the rotational ¯ω = ∇ × ¯v as a function of the stagnation temperature T s and
specific entropy S can be expressed as follows
∂ ¯ω
∂t + ¯v × ¯ω = c p∇T s − T ∇S. (9.54)
For isoenergetic (T s = const.) and stationary (∂/∂t = 0) flows, only this
case will be considered in the present study, 5 equation (9.54) reduces to the following
(Crocco’s Theorem)
¯v × ¯ω = −T ∇S. (9.55)
From this equation, the jump of the rotational, across the flame, can be computed
when the values of ¯v, T and S are known. The jump of the rotational is obtained by
(9.55) to both sides of the flame. For the calculation, we shall adopt on each point
of the flame front a cartesian rectangular coordinate system, defined in the following
way:
Axis n: Normal to the flame front.
Axis t: Intersection of the plane tangent to the flame with the plane of the incident
and emergent velocities ¯v 1 and ¯v 2 .
Axis τ: Normal to the (n, t) plane forming a positive trihedron.
The velocity components relative to this system, will be v n , v t and 0. The
vorticity components ω n , ω t and ω τ . Therefore, those of the vector product ¯v × ¯ω will
be v t ω τ , −v n ω τ and (v n ω t − v t ω n ). Consequently, equation (9.55) breaks down into
the following three equations 6
v t ω τ = −T ∂S
∂n , (9.56)
v n ω τ =
T ∂S
∂t , (9.57)
v n ω t − v t ω n = −T ∂S
∂τ . (9.58)
The jump of the normal component of the vorticity ω n can be computed directly.
The above equations are not necessary for the computation. In fact, since v t
5 If the flow before the flame is isoenergetic and if the heat q released in the combustion is constant, as
will be assumed hereinafter, the flow after the flame is also isoenergetic, as results from (9.14).
6 The derivative ∂S/∂t in the tangential direction must not be confused with a time derivatives.

9.6. VORTICITY ACROSS THE FLAME 243
is continuous across the flame, by applying Stokes theorem to both faces of a surface
element of the flame, the following is obtained
ω n1 = ω n2 , (9.59)
that is, the normal component of the vorticity is continuous across the flame.
The jump of ω τ can then be obtained from (9.57) by writing this equation for
each side of the flame and forming the difference. Thus, when taking into account that
v n1 = ϕ, v n2 = λϕ, T 2 = λT 1 , S 2 = S 1 + ∆S there results
ω τ2 = ω τ2
λ + T [
1 λ − 1 ∂S 1
ϕ λ ∂t + ∂(∆S) ]
. (9.60)
∂t
Likewise by using (9.58) we obtain for ω t2
ω t2 = ω t1
λ − T [
1 λ − 1
ϕ λ
∂S 1
∂τ + ∂(∆S) ]
. (9.61)
∂τ
On the other hand, equations (9.57) and (9.58) , when written for conditions before
the flame, give
ϕω τ1 = T 1
∂S 1
∂t , (9.62)
ϕω t1 − v t ω n1 = − T 1
∂S 1
∂τ . (9.63)
By eliminating ∂S 1
and ∂S 1
between these two equations and (9.60) and (9.61), we
∂t ∂τ
finally obtain for ω τ2 and ω t2 ,
ω τ2 = ω τ1 + T 1
ϕ
∂(∆S)
, (9.64)
∂t
ω t2 = ω t1 − λ − 1
λ ω n1 cot α 1 − T 1
ϕ
∂(∆S)
. (9.65)
∂τ
In particular, if the motion before the flame is irrotational (ω n1 = ω t1 = ω τ1 =
0), the vorticity produced by the flame results
ω n2 = 0, (9.66)
ω t2 = − T 1
ϕ
ω τ2 = T 1
ϕ
∂(∆S)
,
∂τ
(9.67)
∂(∆S)
.
∂t
(9.68)
Furthermore, if the entropy jump can be expressed as (9.53), we have
ω t2 = − c pT 1
ϕ
ω τ2 = c pT 1
ϕ
1 ∂λ
λ ∂τ , (9.69)
1 ∂λ
λ ∂t , (9.70)

244 CHAPTER 9. FLOWS WITH COMBUSTION WAVES
or else, if the motion is rotational, the equations resulting from (9.64) and (9.65) when
∆S is substituted therein for its value (9.53).
In the special case of a plane motion, the only component different from zero
of the vorticity is ω τ , and its value after the flame is given by (9.64) if the motion
before the flame is rotational, and by (9.68) if it is potential. In Fig. 9.7.a, the rotation
sense of the vorticity generated by the flame has bean indicated, in the case of a plane
motion, assuming that the motion before the flame is potential. In Fig. 9.7.b, the sense
corresponding to a shock wave is shown for the same case.
References
[1] Emmons, H. W., Ball, G. A. and Maier, A. D.: Development of a Combustion
Tunnel. Army Ordenance Project Report, Harvard University, Cambridge
Mass., 1954.
[2] Emmons, H. W.: Fundamentals of Gas Dynamics. Sec. E, Vol. III of High
Speed Aerodynamics and Jet Propulsion. Princeton University Press. 1958.
[3] Gross, R. A.: Combustion Tunnel Laboratory Interim Technical Report No. 2.
Harvard University, June 1952.
[4] Gross, R. A. and Esch, R.: Low Speed Combustion Aerodynamics. Jet Propulsion,
March-April 1954, pp. 95-101.
[5] von Kármán, Th.: Aerothermodynamics and Combustion Theory. L’Aerotecnica,
Vol. XXXIII, Fasc. 1st., 1953, pp. 80-86.

Chapter 10
Aerothermodynamic field of a
stabilized flame
10.1 Introduction
As an example of the application of the aerothermodynamic method that was outlined
in the preceding chapter for the study of gas flows, in the present chapter we shall
study the characteristics of the flow in a combustion chamber with a stabilized flame.
y
+h
F
u 0
0
x
−h
F’
Figure 10.1: Schematic diagram of a stabilized flame front in a two-dimensional combustion
chamber.
The problem is outlined in Fig. 10.1. To simplify calculation we shall consider
the case of a two dimensional chamber of constant width 2h and unit thickness normal
to the plane of the figure. An uniform flow of homogeneous pre-mixed combustible
enters the combustion chamber. The velocity, pressure, density and temperature of
the mixture are u 0 , p 0 , ρ 0 and T 0 , respectively. The flame front FOF’ is stabilized at
point O by one of the methods studied in the next chapter. As expressed in the pre-
245

246 CHAPTER 10. AEROTHERMODYNAMIC FIELD OF A STABILIZED FLAME
ceding chapter, this flame front can be considered as a discontinuity surface between
unburnt and burnt gases.
The problem lies in determining:
a) The width of the flame as a function of the fraction of gas burnt.
b) The pressure drop along the chamber.
c) The velocity distribution in the different sections of the chamber.
d) The shape of the flame.
The following simplifying assumptions are introduced:
1) Combustion efficiency is the same at all points, that is, the heat released in the
combustion per unit mass of fuel is independent from the state of the gas before
the flame.
2) Heat capacity at constant pressure c p is independent from temperature, and has
the same value for the unburnt and burnt gases.
3) Unburnt and burnt gases behave as perfect gases. The constant R g of their state
equation has the same value for both gases
p
ρ = R gT. (10.1)
4) The flame propagation velocity is constant along the front and very small when
compared to the gas velocity.
5) Unburnt and burnt gases are ideal fluids, their viscosity and thermal conductivity
are negligible.
Furthermore, in this study only the case of stationary flow will be considered.
Thus stated, this problem has been studied by A.C. Scurlock [1] and [2], who introduced
further simplification by assuming both unburnt and burnt gases to be incompressible
fluids.
Since the flame speed is small compared to the gas speed, the angle between the
front and the streamlines is very small. Therefore, the streamlines are approximately
parallel to the chamber axis. Furthermore, due to the negligibility of the pressure drop
across the flame front, it can be assumed that pressure is constant at each cross section
of the chamber. Moreover, gas speed can be substituted by its component parallel
to the chamber axis. Therefore, from these assumptions an almost one-dimensional
theory can be worked out.
By using the aforementioned simplifying assumptions Scurlock obtained numerical
solutions for the equations of motion in some typical cases. In order to per-

10.1. INTRODUCTION 247
form numerical computations, Scurlock substitutes the differential equations for finite
differences.
Some of the results, taken from Ref. [1], are given in Figs. 10.2 to 10.5. These
results correspond to the case where the density ratio of the unburnt gases to the burnt
gases is 6, and the ratio of the flame speed to the gas velocity at the chamber’s inlet is
λ = 0.10.
y/h
A B C
1
0
x/h
−1
0
1
2
3
4
5
6
7
8
9
10
11
∆ P / ∆ P t
1
0 1 2 3 4 5 6 7 8 9 10 11 x/h
Figure 10.2: Streamlines of flow through a flame front in a two-dimensional chamber and
pressure drop along it.
Fig. 10.2 shows the shape of the flame front and of the streamlines, as well
as the pressure drop ∆p along the chamber referred to the total pressure drop ∆p t ,
between the inlet and outlet sections.
Fig. 10.3 shows the velocity distribution for the cross-sections A, B and C
indicated in Fig. 10.2. These sections correspond to the point at which combustion
starts (section A), to the point where the burnt fraction is 0.25 (section B) and to the
point where this fraction is 0.75 (section C). In this figure all velocities are referred to
the velocity u 0 of the gas at the inlet of the chamber. Fig. 10.3 shows that:
1) While the velocity of the unburnt gases is constant at each cross section, the
velocity of the burnt gases increases from the flame front towards the chamber
axis where it is maximum.

248 CHAPTER 10. AEROTHERMODYNAMIC FIELD OF A STABILIZED FLAME
y/h
1
0
A
B
C
U/U 0
−1
0
1 2 3 4 5 6 7 8 9 10
Figure 10.3: Velocity profiles for three cross-sections of the two-dimensional chamber (see
Fig. 10.2).
1.0
0.9
0.8
0.7
0.6
η / h
0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
f
Figure 10.4: Flame width as a function of the burnt fraction for incompressible flow and
λ = 6.
2) Due to the pressure drop produced by combustion, the mass of gas accelerates
downstream of the chamber.
Fig. 10.4 shows the variation law of the flame width, referred to that of the
chamber, as a function of the burnt fraction.
Fig. 10.5 shows the fraction of pressure drop along the chamber, referred to
the total pressure drop between the inlet and outlet sections, as a function of the burnt
fraction.

10.1. INTRODUCTION 249
∆ P /∆ P t
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
f
Figure 10.5: Pressure drop as a function of the burnt fraction for incompressible flow and
λ = 6.
G.A. Ball, [3] and [4], has studied the same problem by applying the relaxation
method.
The assumption that density, for both unburnt and burnt gases, is constant,
is permissible only when the variations of the Mach number along the chamber are
small. Due to the acceleration produced by the pressure drop, the local Mach number
close to the walls and near the end of the chamber can be large, even for small entrance
Mach numbers. As a consequence a choking of the chamber can occur preventing the
complete combustion of the gas, as will be shown later. This choking phenomenon
cannot be predicted by Scurlock’s hydrodynamic theory.
H.S. Tsien in the U.S.A. [5], and J. Fabri, R. Siestrunck and C. Fouré in France,
[6], [7] and [8], have studied (simultaneous but independently) the influence of gas
compressibility on the problem.
By assuming at each section a linear velocity distribution for the burnt gases
(see Fig. 10.3), H. S. Tsien has demonstrated that an approximate analytical solution
of Scurlock’s problem can be easily obtained. The results are very close to those
obtained by Scurlock. The linear velocity a distribution is a good approximation to the
profile calculated by Scurlock. Moreover, Tsien has extended the analysis, taking into
account compressibility effects, by applying the same idea of a quasi one-dimensional
theory. He also assumes a linear velocity distribution for the burnt gases, showing that
in this case choking of the chamber can occur.

250 CHAPTER 10. AEROTHERMODYNAMIC FIELD OF A STABILIZED FLAME
Fabri, Siestrunck and Fouré have developed a similar theory. They also include
the effect of density variations, but do not postulate a linear velocity profile for burnt
gases. Their stating of the problem leads then to an integral equation for the stream
function. For some typical cases they have integrated this equation numerically. Furthermore,
they extended the analysis to other types of chambers. For instance, the
cylindrical chamber with circular cross section and different arrangements of the flame
stabilizer.
As a result of their studies Fabri, Siestrunck and Fouré also predict the occurrence
of choking when the velocity at the inlet section is larger than a critical velocity.
In the following sections the approximate method of Tsien will be described
first and then the method developed by Fabri-Siestrunck-Fouré.
10.2 Tsien method
h
p’ p
B
ρ’
1
D
ρ’
2
C
y
u 0
ρ
1
u e
ρ
1
y
u
1
u
0
A
Figure 10.6: Notation for the Tsien method.
Figure 10.6 contains the necessary elements for Tsien’s approximate calculation.
obtained
By applying the continuity equation to section AB, the following equation is
∫ y1
ρ 1 u 1 (h − y 1 ) + ρu dy = ρ 0 u 0 h. (10.2)
0
By introducing η = y 1 /h and U = u 1 /u 0 it can be written in the following dimensionless
form
∫
ρ η
1
ρ u
( y
)
U(1 − η) + d = 1. (10.2.a)
ρ 0 0 ρ 0 u 0 h

10.2. TSIEN METHOD 251
As aforesaid, Tsien assumes that the velocity distribution of the burnt gases
between A and C is linear, that is, u (Fig. 10.6) is given by the expression
u = u 1 + (u e − u 1 )
(1 − y )
, (10.3)
y 1
where u e is the velocity on the axis of the chamber.
Furthermore, Tsien assumes that the ratio λ of the density of the unburnt gases
to that of the burnt gases at each point of the flame front is constant. 1 Then, it can be
easily shown that the density of the burnt gases is constant between A and C and equal
to ρ 1 /λ
In fact, in D we have
ρ = ρ 1
λ . (10.4)
ρ ′ 1
ρ ′ 2
= λ. (10.5)
But due to the fact that the expansions of both unburnt and burnt gases between
pressure p ′ in D and p in C are isentropic, there result
and
ρ ′ 1
ρ 1
=
( ) 1 p
′
γ
p
(10.6)
The combination of (10.5), (10.6) and (10.7) leads to (10.4).
ρ ′ ( ) 1
2 p
′
ρ = γ . (10.7)
p
Since pressure and density of the burnt gases are constant between A and C, the
temperature T of the burnt gases must also be constant between A and C, and equal to
T 2
T = T 2 = λT 1 = T e , (10.8)
where T e is the temperature at A.
The Bernoulli equation, when applied to the unburnt gas between u 0 and u 1 ,
gives
1
2 u2 0 + c p T 0 = 1 2 u2 1 + c p T 1 . (10.9)
If T 1 is eliminated by combining this equation with the equation
( ) 1
ρ 1 T1 γ − 1
=
ρ 0 T 0
(10.10)
1 Relation (9.36) in the preceding chapter proves that this ratio actually varies with the flame front incline.

252 CHAPTER 10. AEROTHERMODYNAMIC FIELD OF A STABILIZED FLAME
of the reversible adiabatics, it results for ρ 1 /ρ 0
(
ρ 1
= 1 − γ − 1 M0 2 (U 2 − 1)
ρ 0 2
) 1
γ − 1 = λ
ρ
ρ 0
, (10.11)
where M 0 = u 0 / √ (γ − 1)c p T 0 is the Mach number at the inlet section of the combustion
chamber.
On the other hand, Bernoulli’s equation applied to the burnt gases between O
and A, taking (10.8) into account, gives
1
2 u2 0 + λc p T 0 = 1 2 u2 e + λc p T 1 . (10.12)
Finally, the elimination of T 0 and T 1 between (10.9) and (10.12) gives for u e
u e
u 0
= √ 1 + λ(U 2 − 1). (10.13)
When (10.3), (10.11) and (10.13) are substituted into (10.2.a) and the integration
performed, the following expression for η as a function of U is obtained
η =
(
U −
1
2λ
1 − γ − 1 M0 2 (U 2 − 1)
2
) 1 −
γ − 1
(
(2λ − 1)U − √ 1 + λ(U 2 − 1)
The fraction f of the burnt gases is given by the expression
f = ρ 0u 0 h − ρ 1 u 1 (h 1 − y 1 )
ρ 0 u 0 h
=1 − ρ 1u 1
ρ 0 u 0
(1 − η)
) . (10.14)
(
=1 − U(1 − η) 1 − γ − 1
) 1
M0 2 (U 2 γ − 1
− 1) . (10.15)
2
System (10.14) and (10.15) allow computation of the fraction f of burnt gas as a
function of the flame width η, through parameter U. Fig. 10.7 shows several of the
results given by Tsien in his work for the case λ = 6.
The main result of Tsien’s work is the following: for each value of λ there is a
critical Mach number M cr for the flow at the inlet section, such that when M 0 < M cr
the combustion is complete and the flame reaches the chamber walls. On the other
hand, if M 0 > M cr the maximum burnt fraction f max is smaller than unity and the
flame cannot reach the walls. This is known as the choking phenomenon. It shows

10.2. TSIEN METHOD 253
1.0
λ=6.0
M 0
=0.2
M 0
=0, 0.1
0.8
η
0.6
M 0
=0.4
0.50
M 0
=0.4
0.4
0.25
M 0
=0.6
0.2
M 0
=0.6
0.00
0.0 0.1 0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
f
Figure 10.7: Flame width as a function of the burnt fraction for density ratio λ = 6. For
each value of the initial Mach number M 0, a dot indicates the maximum values
of η and f.
that, aside from the difficulties arising from the problem of flame stabilization at high
speed, 2 other difficulties are to be expected when trying to burn gas at high speed, due
to the interaction between aerodynamic and combustion processes.
Choking occurs due to the fact that the contraction of the unburnt gases originated
by downstream acceleration is not sufficient to compensate the expansion of
the gases as they burn. Compressibility acts because the contraction corresponding
to a given acceleration is smaller in a compressible fluid than in an incompressible
one. This is the reason why the choking phenomenon does not exist in Scurlock’s
hydrodynamic theory.
The critical Mach number that corresponds to each value of λ can be obtained
by expressing the condition that the curve of f as a function of U, obtained by eliminating
η between (10.14) and (10.15), must have a maximum at the point f = 1. That
is, when the conditions
or their equivalents
are satisfied.
df
dU
dη
dU
= 0 and f = 1, (10.16)
= 0 and η = 1, (10.17)
2 The problem of stabilizing a flame at high speed is studied in the next chapter.

254 CHAPTER 10. AEROTHERMODYNAMIC FIELD OF A STABILIZED FLAME
U η=1
10
8
6
4
2
M η=1
M cr
U η=1
1.0
0.8
0.6
0.4
0.2
M cr
, M η=1
0
0.0
0 2 4 6 8 10 12
λ
Figure 10.8: Critical conditions for the complete combustion as a function of the density
ratio λ.
Figure 10.8, in which the values of M cr thus calculated have been taken from
[5], shows that the choking effects impose an important limitation to the velocity at
which gases enter the combustion chamber. Choking effects can be avoided by using
an expanding combustion chamber. It is interesting to point out that when the maximum
Mach number is calculated at critical conditions, that is, at the initiation of the
choking phenomenon, this number is slightly larger than one. This value corresponds
to the unburnt gases at the point where the flame reaches the chamber wall. Therefore,
in the neighbourhood of this point, there exists a small area where the flow is slightly
supersonic. Details of the calculation can be found in Tsien’s work. Furthermore,
Tsien’s analysis shows that Scurlock’s hydrodynamic theory gives satisfactory results
when the entry Mach number is not too close to the critical.
Recently, G. Ernst [9] has applied Tsien’s method to the same problem as well
as to the study of other types of combustion chambers and other arrangements of the
stabilizer. Ernst substitutes Tsien’s linear velocity profile for a parabolic one with
an exponent 1.33, which is the one that best fits the profiles obtained by Scurlock.
Moreover, instead of assuming a constant density for the burnt gases at each cross
section of the chamber, he includes the effect of the slight density variation of the burnt
gases. However, Ernst’s results only differ slightly from those obtained by Tsien.

10.3. METHOD OF FABRI-SIESTRUNCK-FOURÉ 255
10.3 Method of Fabri-Siestrunck-Fouré
The flow in the combustion chamber is two dimensional and stationary. Therefore, a
stream function ψ exists. This function is defined by the following relations
ρu =ρ 0 u 0 h ∂ψ
∂y , (10.18)
ρv =ρ 0 u 0 h ∂ψ
∂x . (10.19)
Thus defined, ψ is non-dimensional.
The x axis is a streamline to which the value ψ = 0 is assigned. The upper
wall of the chamber, y = h, is also a streamline for which ψ = 1, as results from the
integration of (10.18). Therefore, in the upper half of the chamber, ψ varies from the
value zero, at the axis, to the value one, at the wall.
A linearization of the problem, by assuming that all velocities are small compared
to u, leads to the conclusion that pressure is constant at each cross section of
the chamber. Therefore, since the flow of unburnt gases is isentropic, their density,
temperature and velocities are equally constant at each cross section. Moreover, the
value of either one of these magnitudes determines the values for the others.
ψ
ψ’
D u’
ρ
1
B
C
ρ’’
E
u
u’’
ψ
ψ’
0
x
A
Figure 10.9: Notation for the method of Fabri-Siestrunck-Fouré.
Let us consider a cross section AB, at a distance x from the stabilizer, as shown
in Fig. 10.9. Here, evidently
∫
1 h
dy = 1, (10.20)
h 0
which, by virtue of (10.18), can be written in the following form
∫ 1
0
ρ 0 u 0
ρu
dψ = 1. (10.21)

256 CHAPTER 10. AEROTHERMODYNAMIC FIELD OF A STABILIZED FLAME
Let us consider the streamline ψ that crosses the flame front at section AB
(Fig. 10.9) at point C. At this section, the streamline separates the unburnt from the
burnt gases. To the first correspond values ψ ′ of the stream function between ψ and 1.
To the latter correspond values ψ ′ between 0 and ψ. By separating in (10.21) both
intervals, the following is obtained;
∫ ψ
0
∫
ρ 0 u 1
0 ρ 0 u 0
ρ ′′ u ′′ dψ′ +
ψ ρu dψ′ = 1, (10.22)
where ρ ′′ and u ′′ are the corresponding values of ρ and u, on the streamline ψ ′ at point
E on section AB.
As previously said, both density and velocity of the unburnt gas are constant
at each cross section of the chamber. If ρ 1 and u are their values at section BC, the
second integral in (10.22) can be integrated, obtaining
∫ 1
This value, when carried into (10.21), gives
∫ ψ
This equation can be written
0
ψ
ρ 0 u 0
ρ 1 u dψ′ = ρ 0u 0
(1 − ψ). (10.23)
ρ 1 u
ρ 0 u 0
ρ ′′ u ′′ dψ′ = 1 − ρ 0u 0
(1 − ψ). (10.24)
ρ 1 u
∫
ρ 1 u
ψ
ρ 1 u
− 1 + ψ =
ρ 0 u 0 0 ρ ′′ u ′′ dψ′ . (10.25)
Hereinafter all velocities will be referred to the maximum velocity u max =
√
2cp T s,1 of the unburnt gases, and the ratio named U. Equation (10.25) is then
written as follows
∫
ρ 1 U
ψ
ρ 1 U
− 1 + ψ =
ρ 0 U 0 0 ρ ′′ U ′′ dψ′ . (10.26)
The problem lies now in expressing ρ 1 U
ρ ′′ U ′′ as a function of U and of the dimensionless
velocity U ′ at the point D where the streamline ψ ′ crosses the flame,
Fig. 10.9. Let us see how it can be done.
Proceeding in the same manner as done for the deduction of equations (10.4)
and (10.8) in Tsien’s method, that is, by comparing the expansions along ψ and ψ ′ ,
the following is obtained
λ ′ = ρ ′′
1 T
= . (10.27)
ρ
′′
T 1

10.3. METHOD OF FABRI-SIESTRUNCK-FOURÉ 257
Moreover, it is immediately obtained that
λ ′ = n − U ′ 2
1 − U ′ 2 , (10.28)
from which results
′′
ρ 1 T n − U ′ 2
=
ρ
′′
T 1 1 − U ′ 2 . (10.29)
Bernoulli’s equation when applied to E, gives
U ′′ 2 +
T ′′
T s,1
= n. (10.30)
Likewise, Bernoulli’s equation when applied to C for the unburnt gases, gives
U 2 + T 1
T s,1
= 1. (10.31)
By combining (10.29), (10.30) and (10.31), the following is obtained for U ′′
√
U ′′ nU
=
2 − U ′2 (n − 1 + U 2 )
1 − U ′ 2
. (10.32)
Substituting (10.29) and (10.32) into (10.26), and making use of relation
ρ 1
ρ 0
=
the following equation is finally obtained
( 1 − U
2
1 − U 2 0
) 1
γ − 1 , (10.33)
( ) 1
U 1 − U
2
γ − 1 − 1 + ψ
U 0 1 − U0
2 ∫ ψ
= √ U (n − U ′2 )
√ n
0
(1 − U ′2 )
(U 2 − U ′ 2 n − 1 + U 2 ) dψ′ .
n
(10.34)
This expression is an integral equation which determines ψ as a function of U. Once
the solution is known, the value of ψ gives the burnt fraction as a function of the
velocity U of the gases at the point where the streamline ψ crosses the flame front.
The flame width η = y/h is given by the expression
η = 1 − ρ 0U 0
(1 − ψ). (10.35)
ρ 1 U
To determine the shape of the flame front, the abscissa ξ = x/h corresponding to ψ
must also be known. This abscissa is given by the following expression
ξ =
∫ ψ
0
ρ 0 U 0
ρ 1 ψ ′ dψ′ , (10.36)

258 CHAPTER 10. AEROTHERMODYNAMIC FIELD OF A STABILIZED FLAME
which is deduced from (10.19) by making v ≃ −ψ. When computing this expression
the following relation must be used
ρ ′ 1
ρ 0
=
(
) 1
1 − U ′ 2 γ − 1
1 − U0
2
(10.37)
as well as the values ψ ′ = ψ(U ′ ) given by the solution of (10.34).
Equation (10.34) can be integrated numerically by substituting the integral for
a summation with a finite number of terms. Fabri, Siestrunck and Fouré have, thus,
calculated the solutions corresponding to several typical cases. Some of their results
can be found in the references included herein.
When calculating the solution it is found that for each value of n there is a
corresponding value U 0,cr of U 0 such that if U 0 > U 0,cr the curve of ψ = ψ(U)
presents an horizontal tangent for a value of ψ smaller than one. This means that in
such a case the burnt fraction must be smaller than unity. Therefore, the existence
of a critical Mach number for the inlet flow is also obtained here, thereupon choking
occurs. Furthermore, it can be proved that the value of U 0,cr is equal to the value
U 0,cr = (√ n − √ n − 1 ) √ γ − 1
γ + 1 , (10.38)
even by the one-dimensional theory that results from the assumption that at each crosssection
of the chamber the distribution of velocities is uniform. For such a case the
value of the final Mach number of the burnt gases is unity. 3
Fig. 10.10, taken from Ref. [8], shows a solution of (10.34) that corresponds
to the case n = 6 for the three following values of U 0 : subcritical value U 0 = 0.05,
critical value U 0 = U 0,cr = 0.087, and supercritical value U 0 = 0.100. In the latter,
the burnt fraction would be only ψ max = 0.8.
The experimental evidence available is not enough to judge the good approximation
of these methods.
10.4 Cylindrical chambers
As Fabri, Siestrunck and Fouré have demonstrated [6] the case of a cylindrical chamber
of circular cross-section reduces to the two-dimensional problem studied in the
preceding paragraph. In fact, in such case equation (10.18), which defines the stream
3 See chapter 3, §9.

10.5. CHAMBER WITH SLOWLY VARYING CROSS-SECTION 259
0.5
0.4
0.3
U 0
= 0.100
U
0.2
U 0
= 0.087
0.1
U 0
= 0.050
0
0.2 0.4 0.6 0.8 1.0
ψ
Figure 10.10: Velocity profiles as a function of the burnt mass fraction for λ = 6 and three
values of the initial velocity (subcritical, critical and supercritical).
function, must be substituted by
ρu = 1 2 ρ 0u 0
R 2
r
∂ψ
∂r , (10.39)
where R is the radius of the chamber and r is the distance to its axis. Coefficient 1/2
is introduced so that stream function ψ changes from value zero at the axis to value 1
at the chamber wall.
The change of variable
( r
) 2 y =
R h , (10.40)
transforms (10.39) into (10.18) keeping conditions ψ = 0 at y = 0 and ψ = 1 at
y = 1. The remaining calculations are identical in both cases. Hence, the problem
reduces to the two-dimensional case. Fabri, Siestrunck and Fouré have also studied the
case where stabilizer is at the chamber wall. The annular stabilizer has been studied
by Ernst [9]. All these cases reduce to the two-dimensional problem by an adequate
change of variables.
10.5 Chamber with slowly varying cross-section
The preceding method can be applied to an approximate study by substituting equations
(10.18) or (10.39) for an equation that includes the variation of the cross-section

260 CHAPTER 10. AEROTHERMODYNAMIC FIELD OF A STABILIZED FLAME
along the chamber axis. The same is done for the approximate study of gas motions
within ducts with slowly varying cross-section.
References
[1] Scurlock, A. C.: Flame Stabilization and Propagation in High-Velocity Gas
Streams. Meteor Report No. 19, Fuels Research Laboratory, M.I.T., May 1948.
[2] Williams, G. C., Hottel, H.C. and Scurlock, A. C.: Flame Stabilization and
Propagation in High Velocity Gas Streams. Third Symposium (International)
on Combustion, Williams and Wilkins Co., Baltimore, 1949, pp. 21-40.
[3] Ball, G.: A Study of a Two-Dimensional Flame. Combustion Tunnel Laboratory
Report, Harvard University, Cambridge Mass., 1951.
[4] Emmons, H. W.: Fundamentals of Gas Dynamics. Section F, vol. III of High
Speed Aerodynamics and Jet Propulsion, Princeton University Press, 1956
[5] Tsien, H. S.: Influence of Flame Front on the Flow Field. Journal of Applied
Mechanics, June 1951, pp. 188-194.
[6] Fabri, J., Siestrunck, R. and Fouré, C.: Sur le Calcul du Champ Aérodynamique
des Flammes Stabilisées. Comtes Rendus des Séances de l’Académie des Sciences,
Paris, 1951, p. 1263.
[7] Fabri, J., Siestrunk, R. and Fouré, C.: Phénomène d’Obstruction Intervenant
dans la Stabilisation des Flammes. La Recherche Aéronautique, No. 25, 1952,
pp. 21-27.
[8] Fabri, J., Siestrunk, R. and Fouré, C.: On the Aerodynamic Field of Stabilized
Flames. Fourth Symposium (International) on Combustion, Williams and
Wilkins Co., Baltimore, 1953, pp. 443-450.
[9] Ernst, G.: Propagation a Faible Vitesse d’une Flamme dans un Ecoulement
Compressible. Technique et Science Aéronautiques, 1955, pp. 1-12.

Chapter 11
Similarity in combustion.
Applications
11.1 Introduction
The methods of dimensional analysis and physical similarity have been applied with
very good results to the study, both theoretical and experimental, of classical Aerothermodynamics.
Recently, numerous attempts have been made to extend these methods
to the study of Aerothermochemistry. In some instances, like in the work by Schultz-
Grunow listed in Ref. [1], such an extension was intended in order to facilitate the
establishment of a correlation between several experimental results. Other attempts
tried to find a rational basis for the experimentation with models as well as rules to
apply the results obtained through such models to the processes at normal scale.
Before treating a problem of this nature, it is necessary to determine the dimensionless
parameters that characterize the process. In the phenomena whose study
is the purpose of Aerothermochemistry, such parameters are the same studied by classical
Aerothermodynamics, but increased by the chemical parameters introduced by
Damköhler [2] when he applied physical similarity to the study of chemical reactors.
We owe to Penner [3] the first study on the application of the laws of physical similarity
to Aerothermochemistry. A concise work on the subject, analyzing the origin
and significance of these parameters, is own to von Kármán and listed in Ref. [4].
Recently, Weller [5] has performed a review of the practical applications. Several applications
of this theory to the study of technological problem of combustion will be
found in Penner’s work and in the corresponding sections of the Proceedings of Sixth
International Symposium on Combustion and of Second Colloquium on Combustion,
AGARD.
261

262 CHAPTER 11. SIMILARITY IN COMBUSTION. APPLICATIONS
In the following we will first deduce the dimensionless parameters of Aerothermochemistry
starting from the fundamental equations governing motion. Thereafter
two practical application will be performed, one to the rule of the scaling of rockets
and another to the problem of flame stabilization.
11.2 Dimensionless parameters of Aerothermochemistry
The physical similarity between two analogous processes consists in the proportionality
between corresponding magnitudes (velocities, temperatures, etc.) through space
and time. In the first place, it implies a geometrical similarity and, moreover, proportionality
of times.
The origin of this similarity and the required conditions for it to occur may be
conceived as follows. Let us consider the equations of the process and write them in
dimensionless form, referring each one of the variable quantities comprised in them
(length, pressure, velocity, temperature, etc.) to a characteristic value of the same.
Thus, we shall obtain a system of equations between dimensionless variables, with
coefficients that will also be dimensionless combinations of such characteristic values.
All the combinations of these values for which the coefficients are invariable will
correspond to processes represented by identical systems of equations. If, furthermore,
we keep invariable the dimensionless expressions of the initial and boundary
conditions, then the solution of the system thus obtained will correspond to a physically
similar set of processes. We will pass from one another of these processes
by introducing changes into the characteristic values which keep invariable the said
coefficients of the equations and their initial and boundary conditions. Hence, these
coefficients are the dimensionless parameters of the physical similarity. Therefore, the
problem reduces to finding, among them, those independent from one another, and to
select the most simple expressions possible for the same.
Let us see now in what way this can be attained, utilizing, for simplicity and
clearness, the equations corresponding to an one-dimensional stationary flow with
only two chemical species, since the generalization to other cases is straightforward
and it only requires multiplying the number of parameters.
The system of equations that we must apply to this study was deduced in §2 of
Chap. 5, but it is reproduced here in order to assist the reader:
a) Equation of mass conservation.
ρv = m. (11.1)

11.2. DIMENSIONLESS PARAMETERS OF AEROTHERMOCHEMISTRY 263
b) Continuity equation.
c) Momentum equation.
d) Energy equation.
( )
1
ρv
2 v2 + c p T − qε
e) Diffusion equation.
ρv dε = w. (11.2)
dx
p + ρv 2 − 4 3 µ dv = i. (11.3)
dx
D
v
dY
dx
− λ dT
dx − 4 dv
µv = me. (11.4)
3 dx
− Y + ε = 0. (11.5)
Be l 0 , ρ 0 , v 0 , w 0 , p 0 , µ 0 , c p0 , T 0 , λ 0 and D 0 characteristic values of the corresponding
quantities and let us change the variables of the preceding system as follows
x = l 0 x ′ , ρ = ρ 0 ρ ′ , v = v 0 v ′ , w = w 0 w ′ , p = p 0 p ′ ,
µ = µ 0 µ ′ , c p = c p0 c ′ p, T = T 0 T ′ , λ = λ 0 λ ′ , D = D 0 D ′ .
(11.6)
Thus all the “prime” quantities are dimensionless. For ε and Y this change is
not necessary since both are already dimensionless and their variation field has been
selected from zero to one.
When these new variables (11.6) are introduced into the preceding system it
may be written as follows:
a) Equation of mass conservation.
ρ ′ v ′ =
( m
ρ 0 v 0
)
. (11.7)
b) Continuity equation. ( )
ρ0 v 0
ρ ′ w ′ dε
l 0 w 0 dx = w′ . (11.8)
c) Momentum equation.
( )
p0
ρ 0 v0
2 p ′ + ρ ′ v ′ 2 4 −
3
d) Energy equation.
ρ ′ v ′ [ 1
2
( v
2
0
)
v ′ 2 + c
′
c p0 T p T ′ −
0
(
− 4 3
(
µ0
) (
µ ′ dv′
ρ 0 v 0 l 0 dx ′ =
( ) ] (
q
ε −
c p0 T 0
µ 0 v0
2 )
µ ′ v ′ dv′
l 0 ρ 0 v 0 c p0 T 0
i
ρ 0 v 2 0
)
. (11.9)
)
λ 0 T 0
λ ′ dT ′
l 0 ρ 0 v 0 c p0 T 0 dx ′
( )
dx = me
. (11.10)
ρ 0 v 0 c p0 T 0

264 CHAPTER 11. SIMILARITY IN COMBUSTION. APPLICATIONS
e) Diffusion equation.
( )
D0 D
′
v 0 l 0 v ′
dY
dx
− Y + ε = 0. (11.11)
The set of dimensionless coefficients P i characteristic of the system is
P 1 = ρ 0v 0
, P 2 = p 0
l 0 w 0 ρ 0 v0
2
P 5 =
q
c p0 T 0
, P 6 =
, P 3 = µ 0
,
ρ 0 v 0 l 0
P 4 = v2 0
,
c p0 T 0
λ 0
, P 7 = µ 0v 0
,
l 0 ρ 0 v 0 c p0 l 0 ρ 0 c p0 T 0
P 8 = D 0
.
v 0 l 0
(11.12)
The physical similarity requires that the values of all these coefficients be equal for
the two processes compared, when using as characteristic values of the variables for
their calculations those at corresponding points and instants.
Such equality guarentees the identity of the left-hand sides of Eqs. (11.7) to
(11.11) in both phenomena. If, furthermore, we guarentee as well the equality of the
right-hand sides, which correspond to the boundary conditions of the problem, we will
have insured the identity of the solution, thus reaching the physical similarity of the
phenomena. Let us proceed to discuss the set of parameter of Eq. (11.12).
We readily observe that P 7 is equal to the product of P 3 by P 4
P 7 = P 3 · P 4 , (11.13)
while the remaining parameters are independent from one another. Consequently, the
number of independent dimensionless parameters is seven. Let us see which are these
seven.
by it
It is clear that P 3 is reciprocal to the Reynolds number and may be substituted
1) Re = ρ 0v 0 l 0
µ 0
. (11.14)
The combination of P 2 and P 4 gives the following parameter P ′ 4 which may
substitute P 4
P ′ 4 =
p 0
ρ 0 c p0 T 0
. (11.15)
Yet, the state equation p 0
ρ 0
= R g T 0 allows P ′ 4 to be written
P ′ 4 = R g
c p0
= c p0 − c v0
c p0
= 1 − 1 γ . (11.16)
Consequently, the constancy of P ′ 4 is equivalent to the constantcy of relation (11.16)
between specific heats, which supplies the second characteristic dimensionless parameter

11.2. DIMENSIONLESS PARAMETERS OF AEROTHERMOCHEMISTRY 265
2) γ = c p0
c v0
. (11.17)
Keeping in mind that velocity a 0 of sound in the mixture, at pressure p 0 and
with density ρ 0 , is a 0 = √ γp 0 /ρ 0 , we can promptly see that P 2 may be written in the
form
being
P 2 = 1
γM0
2 , (11.18)
3) M 0 = v 0
a 0
, (11.19)
the Mach number of the flow, which will be used as a third characteristic dimensionless
parameter in lieu of P 2 .
The combination of P 3 and P 6 gives the following value
4) Pr = P 3
P 6
= µ 0c p0
λ 0
, (11.20)
which is the Prandtl number, fourth dimensionless parameter the substitutes P 6 .
Likewise, the combination of P 3 and P 8 gives
5) Sc = P 3
P 8
= µ 0
ρ 0 D 0
, (11.21)
which is the Schmidt number, fifth dimensionlees parameter in lieu of P 8 .
The aforegoing parameters are the same used in classical Aerothermodynamics
and the characteristic constants of the chemical reaction have no bearing on them. This
constants act through two other parameter, P 1 and P 5 . The first may be written
where
P 1 = v 0τ ch0
l 0
, (11.22)
τ ch0 = ρ 0
w 0
(11.23)
is a characteristic chemical time. The reciprocal of (11.22) is the first parameter of
Damköhler,
6) Da 1 = l 0
v 0 τ ch0
, (11.24)
and it is the sixth dimensionless parameter of the process.
Finally, P 5 is the second parameter of Damköhler
7) Da 2 = q
c p0 T 0
, (11.25)
this being the seventh dimensionless parameter of the process.

266 CHAPTER 11. SIMILARITY IN COMBUSTION. APPLICATIONS
The interpretation of the meaning of Damköhler two parameters is simple. The
first is a measurement of the relationship between the mean mechanical time required
by a particle to travel the characteristic lenght l 0 at the characteristic velocity v 0 and
the time required by the chemical reaction of the mixture to take place. The second
parameter of Damköhler is a measure of the relationship between the heat produced
by unity of mixture when burning at constant pressure and the heat contained by the
same at characteristic temperature T 0 .
It should be noted that in the non-stationary processes a additional parameter
appears, Strouhal number, which originates from the time derivatives ∂/∂t of the
equations of motions. Likewise, if the mass forces are present as occurs in the phenomena
of free convection, then another dimensionless parameter appears, due to the
gravity terms of the equations, which originates the Froude number.
Finally, when the number of species and chemical reaction is increased, the
number of the Damköhler parameters (of both kinds time and heat) increases as well.
In technical literature we frequently find other parameter which are combinations
of the above mentioned. Por example, occasionally Prandtl number is substituted
by Peclet number, which is a product of the numbers of Reynolds and Prandtl
Pe = Pr · Re = ρ 0v 0 l 0 c p0
λ 0
. (11.26)
Likewise, the Schmidt number is replaced by the Lewis-Semenov number which is
obtained as follows
L = Pr
Sc = ρ 0D 0 c p0
, (11.27)
λ 0
used specially in combustion processes as seen in Chap. 6.
The advantage of utilizing the Peclet and Lewis-Semenov numbers in combustion
phenomena is based on the fact that the conductivity and diffusion coefficients
appear explicity in their expressions. Such coefficients are the ones of interest in these
processes in which the influence of the viscosity coefficient is negligible in many
cases, as said in Chap. 5.
We shall now see some practical applications of this theory.
11.3 Scaling of rockets
The problem of the scaling of rockets has a great practical interest since its solutions
would allow the prediction of the behavior of the rockets after tests made with reduced
scale models.

11.3. SCALING OF ROCKETS 267
Furthermore, both the scaling under steady operation as well as under low and
high frequency oscillations are important in practice. This problem was recently studied
by Penner-Tsien [6], Rose [7], Crocco [8], Barrère [9] and Penner-Fuhs [10].
We have seen in §2 that physical similarity imposses that the seven characteristic
parameters be equal, when passing from the model to the rocket under steady state
conditions and some additional ones under non-steady conditions.
We shall now study the scaling rules imposed by these equalities. If we assume
that we are working with the same mixture of gases and at the same temperature, the
equality of γ, as well as that of the Prandtl and Schmidt numbers and of Da 2 , is insured.
Consequently we only have to insure the equality of the other three parameters.
Disregarding subscript zero and using subscripts 1 for the model and 2 for the rocket,
we will have:
a) Equality of the Reynolds Number. The equality of temperatures makes µ 1 = µ 2 .
Furthermore, ρ may be substituted by p, to which it is proportional. Therefore,
this condition reduces to the following
p 1 v 1 l 1 = p 2 v 2 l 2 . (11.28)
b) Equality of the Mach Number. The equality of temperature insures the equality
of the velocity of sound a 1 = a 2 . Hence, this condition reduces to the following
v 1 = v 2 , (11.29)
this is to say that the velocity must be the same in both the model and rocket.
c) Equality of Da 1 . This condition gives
l 1
v 1 τ ch1
= l 2
v 2 τ ch2
. (11.30)
Since l is the length of the combustion chamber and v the gas velocity through it,
relation
τ r = l v
(11.31)
is the residence time of the gases in the chamber. τ ch is a physico-chemical time, characteristic
of the process, the so called time lag or delay from the instant at which the
fuel enters the chamber to the instant at which it transforms into products. Therefore,
it accounts for the time needed by the fuel to form droplets, to evaporate, to mix with
the gases and burn. Hence, Da 1 can be written in the form
Da 1 = τ r
τ ch
. (11.32)

268 CHAPTER 11. SIMILARITY IN COMBUSTION. APPLICATIONS
Let
K = l 1
l 2
(11.33)
be the linear relation of sizes. Then from (11.28), (11.29), (11.30) and (11.33) one
derives
p 1
= 1 p 2 K , (11.34)
τ ch1
= K.
τ ch2
(11.35)
The first condition imposses that the variation of pressure be inversely proportional
to size. The second one cannot be insured since in a rocket, as we have seen,
τ ch is the resultant of a complicated physico-chemical process. Crocco, by means of
a detailed analysis [11], has reached the conclusion that τ ch may be represented, in
many cases, through a decreasing function of pressure in the form
where n is an exponent differing slightly from unity.
τ ch ∼ p −n , (11.36)
If n = 1, by taking (11.36) into (11.34) and comparing with (11.35), we see
that the later is satisfied, thus obtaining a complete physical similarity.
If n is different from unity the complete similarity is not possible. In such case,
if we disregard the condition of equality of the Mach numbers, which is actually not
too important under steady-state conditions, since velocities at the combustion chamber
are very small and compressibility effects can be neglected, the following conditions
are obtained, instead of (11.29). By eliminating v 1 and v 2 between Eqs. (11.28)
and (11.26), which remain valid, and taking into account Eq. (11.33), it results
p 1
p 2
= τ ch1
τ ch2
K −2 . (11.37)
If expression (11.36) is assumed for the time lag, the following condition is obtained
for the ratio of pressures
p 2
1
= K − 1 + n , (11.38)
p 2
whereas, from here and Eqs. (11.28) and (11.33), it results for the ratio at velocities
1 − n
v 1
= K 1 + n . (11.39)
v 2
Penner and Tsien [6] adopt a different view point. They also neglect the equality
of Mach numbers, but assume that both rockets operate at the same pressure
p 1 = p 2 . (11.40)

11.3. SCALING OF ROCKETS 269
From here and Eq. (11.37), one obtains
τ ch1
τ ch2
= K 2 . (11.41)
The authors propose that these condition be satisfied by proper control of the
time lag, which may be reached, for instance, if the mean size of the droplets is conveniently
varied, since the evaporation time of a droplet depends on its size as will be
shown in chapter 13.
Now let us see the scaling rules for the thrust and injector.
Physical similarity implies that the number of injectors be the same in both
rockets and identically distributed.
Be e the thrust, g the flow rate of fuel, v 1 its velocity through the injector and
d its diameter. It is promptly verified that the following system of relations is valid
provided the temperature is preserved.
e ∼ g ∼ v i d 2 ∼ ρvl 2 ∼ pvl 2 , (11.42)
From here, the following system of scaling conditions is derived
e 1
= g 1
= v i1 d 2 1
e 2 g 2 v i2 d 2 = p 1v 1 l1
2
2 p 2 v 2 l2
2 . (11.43)
Moreover, since the velocity fields must be similar, it results
v i1
= v 1
. (11.44)
v i2 v 2
When Eqs. (11.28) and (11.33), which in all cases remain valid, are taken into
Eq. (11.43), one obtains for the ratio of thrusts
e 1
e 2
= K. (11.45)
The combination of this relation with Eqs. (11.43) and (11.44) gives for the ratio of
injector diameters
√
d 1
= K v 2
. (11.46)
d 2 v 1
Hence, the ratio of diameters depends on the scaling law for the velocities at
the chamber, which, as we have seen, depends on the rule applied. For example, for
the Penner-Tsien rule it results, by virtue of Eq. (11.28),
d 1
d 2
= K, (11.47)
whereas for Crocco’s rule, from Eq. (11.39) it is obtained
n
d 1
= K 1 + n . (11.48)
d 2

270 CHAPTER 11. SIMILARITY IN COMBUSTION. APPLICATIONS
11.4 Scaling of rockets for non-steady conditions
The motion on gases through the combustion chamber of a rocket is always strongly
turbulent with oscillations of pressure, velocities, etc., distributed at random in time
and space, which does not cause important disturbances in the normal operation of the
rocket. Under these circumstances the combustion is called steady.
However, there is also the case, frequently observed, where combustion is unstable
with strong self-excited periodic oscillations of pressure which in some case
destroy the rocket in a few seconds. This phenomenon is at present one of the main
difficulties for the development of rockets, specially large ones. It was observed for
the first time in the United States in 1941 by the research staff of the Jet Propulsion
Laboratory of the California Institute of Technology under the guidance of Professor
von Kármán. Thereafter the great effort applied to the study of this problem has
consderably increased the technical literature on the subject.
Ross-Datner [11] and Crocco-Gray-Grey [12] have written two excellent reviews
on this problem describing the kinds of instabilities observed, their causes, experimental
techniques and the results of the measurements performed, in addition to
the fundamentals of the theories available. These reviews include as well an extensive
bibliography. The most up to date and complete work on the subject is the one
performed by Crocco and Cheng [13] which was recently published by AGARD, including
an abundant bibliography.
Initially, due to limitations of the instruments available for observation, it was
only possible to isolate one type of low-frequency oscillations of the order of about
100 cycles per second, which is normally known as chugging. Later on, as the instrumentation
was improved, it became feasible to isolate other high-frequency oscillations,
over 1000 c.p.s., generally called screaming.
Soon, the origine of the low-frequency oscillations was known and it was possible
to derive practical rules to prevent them which resulted efficient when applied to
practice. Summerfield [14] stated the fundamentals of the theory and the preventive
rules born from it. In low-frequency oscillations, the combustion chamber behaves
like a resonating cavity whose oscillations of pressure act on the feeding system,
changing the rate of fuel injected. The influence of this variation reflects on the chamber
with a certain delay which is due partially to the relaxation times of the chamber
and of the feeding system, and partially to the physico-chemical time lag described
in the preceding paragraph. If this delay is the adequate one, the oscillations result
self-excited. Consequently, the unit combustion chamber-feeding system behaves as a
dynamic system with a characteristic time lag, able of producing unstable oscillations.

11.4. SCALING OF ROCKETS FOR NON-STEADY CONDITIONS 271
2
0
Chamber pressure
t
−2
−4
−6
Injection pressure drop
Injection rate
τ i
t
t
−8
−10
Burning rate
τ ch
τ c
t
−12
−14
Effect of burning rate
Half period
t
Figure 11.1: Time delays leading to self-exciting oscillationes.
Figure 11.1, taken from Ref. [13], clearly illustrates the mechanisn of the process
capable of producing self-excited oscillations. In this figure, it is assumed that the
feeding system is of the constant pressure type. Therefore, the pressure drop through
the injector, and with it the rate of fuel, decreases as the pressure of the chamber is
increased. However, the variation in rate of fuel follows the drop in pressure with a
certain delay τ i , determined by the dynamic characteristics of the feeding system. The
fuel injected turns into combustion products with a delay τ ch , which is the time lag,
and such products act in turn on the pressure of the chamber with a delay τ c , which
depends on the dynamic characteristics of the same. If the sum τ i + τ ch + τ c is equal
to half the period of the pressure oscillations, then there is a phase agreement between
cause an effect and the oscillations amplify with the only limitation of damping effects.
The above analysis shows that in order to eliminate oscillations it is necessary
to reduce τ i , τ ch and τ c . The measures taken to this effect confirm theoretical prediction.
Two efficient ways, for example, are to reduce the relation between pressure drop
through the injector and the pressure in the chamber, or to reduce τ ch by eliminating
the propellants having an excessive time lag [15].
Aside from the aforegoin causes, Crocco [15] has pointed out an intrinsic cause
for instability, so called because it depends only on the conditions in the combustion
chamber and not on the interaction between it and the feeding system, indispensable,
as it was established, for the previously analyzed case. The intrinsic instability of
Crocco shows the possible existence of low-frequency oscillations in a rocket with a
constant volume feeding system, which is impossible with the preceding model.

272 CHAPTER 11. SIMILARITY IN COMBUSTION. APPLICATIONS
We have said in the preceding paragraph of this chapter, that Crocco has proven
that time lag τ ch is not constant but a function of the conditions in the chamber. In
his study he schematizes this by making τ ch depending on the pressure, as shown in
Eq. (11.36). In intrinsic instability the variation of τ ch with p plays the same part than
the consumption of fuel in Figure 11.1, since the element acting on the pressure of
the chamber is not the fuel rate injected into it, but the rate of fuel transformed into
products, which may vary from one instant to another, even if the former is constant,
when τ ch varies with pressure. Under such conditions, if a coupling of phases is
reached we will have a self-excited oscillation as in the preceding case.
Let us now see the rules obtained for the scaling of rockets, to maintain physical
similarity, so that both rockets be equaly stable with respect to the low-frequency
oscillations.
In such case, aside from the conditions derived in the preceding paragraph, it
is necessary that some additional ones be satisfied relative to feeding system, since its
influence is important.
In particular, we must keep the equality of the ratio of the pressure drop ∆p
through the injector to the pressure in the chamber, that is
the injector
∆p 1
p 1
= ∆p 2
p 2
. (11.49)
From here, one obtains the following scaling law for the pressure drop through
∆p 1
∆p 2
= p 1
p 2
. (11.50)
However, we must keep in mind that the pressure drop through the injector
determines velocity v i of the fuel when passing through the same, and that the ratio
of such velocities has been determined before by the condition of physical similarity
between velocity fields in Eq. (11.44). Consequently we cannot insure the simultaneous
satisfaction of conditions (11.50) and (11.44), except for certain particular cases
or by adopting special precautions. For instance, if the injectors have similar friction
characteristics, then the following ratio would exist between ∆p and v i
and we have
∆p ∼ v 2 i , (11.51)
( ) 2
∆p 1 vi1
= , (11.52)
∆p 2 v i2
which is not compatible with Eq. (11.50), except for n = 2, as it can readily be
verified.

11.4. SCALING OF ROCKETS FOR NON-STEADY CONDITIONS 273
The similarity conditions that still need to be satisfied refer to the dynamic
characteristics of the feeding system which determine the value for τ i in Fig. 11.1.
For their study, we refer the reader to the papers by Crocco [8] and Penner-Fuhs [10]
previously mentioned.
Low-frequency oscillations may be analyzed by adopting the assumption that
the state of the gases in the chamber is constant, because their period is very long compared
to the propagation time of a wave through the chamber. To the contrary, when
both the period of the oscillations and the propagation time of the wave are of the
same order of magnitude, it is necessary to consider the differences in state between
different points of the chamber. Such is the situation in the case of high-frequency
oscillations, where, as shown by experimentation, the chamber oscillates like an organ
pipe with longitudinal and transversal oscillations. The existence of a time lag
makes possible the maintenance of these oscillations in a similar way as it happens
for low-frequency oscillations. High-frequency oscillations are particularly dangerous
because, in addition to pressure oscillations being very large, the transmision of
heat to the walls or injectors increases drastically and they are destroyed within a few
seconds. The study of these oscillations is less advanced than for the low-frequency
ones which are easier to control.
As for the scaling law, it appears evident that the residence time τ r should be
substituted by propagation time τ p of a pressure wave through the chamber. Therefore,
in order to mantain the same level of stability for high-frequency oscillation when
passing from the model to the rocket, the following condition must be satisfied
τ ch1
τ ch2
= τ p1
τ p2
. (11.53)
However, since the velocity of sound is the same for both rockets, between τ p1 and
τ p2 the following ratio exists
τ p1
τ p2
= K. (11.54)
Consequently we obtain
τ ch1
τ ch2
= K. (11.55)
It happens also as for low-frequency oscillations, that this condition can not always be
satisfied. In particular, it can be verified that for n = 1 and applying Crocco’s rule the
above condition may be attained.

274 CHAPTER 11. SIMILARITY IN COMBUSTION. APPLICATIONS
11.5 Flame stabilization
In the preceding chapter we have studied the aerodynamic field of a flow with a flame
stabilized by a holder. We verified that the expansion of the burnt gases may originate
a chocking effect which imposses a limitation to the maximum velocity of the flow
entering the chamber.
Another limitation to this velocity, bearing a great practical interest, is that impossed
by the need to fix the flame to the holder in order to maintain combustion.
In fact, it has been experimentally observed that when the velocity of the incoming
flow exceeds a certain value, the flame blows out. The blowing velocity of the flame
is determined by the composition and state of the combustible mixture and by the
characteristics of the holder. The problem arising from this circunstance holds a great
practical interest in the new propulsion systems, specially in ramjets and after-burners,
in which their limits of practical application are often determined by this effect. Consequently,
this phenomenon has been the subject of a particular interest during the last
years and abundant experimental data have been gathered on a great variety of types of
holder and on the influence of the parameters of the mixture on the blowing velocity
of the flame. An extensive bibliography on the matter can be found in the proceedings
of the congresses and colloquia on combustion celebrated in recent years.
The initial study on this problem was carried out by Scurlock [16], who pointed
out the significance of the recirculation zone that forms in the wake of the holder. Figure
11.2 taken from Ref. [17] illustrates the phenomenon. Behind the holder a recirculation
zone forms in which the flame stabilizes. The boundary of this zone consists
in a mixing zone formed by a free boundary layer which separates the unburnt gases
from the combustion products. Within this mixing zone is where the combustible
gases ignite through a process of heat and mass transport, whose characteristics are
not yet well known.
Since the presence of a recirculation zone is essential for the existence of the
flame, the holders used for this purpose are given a shape which produces a large
aerodynamic wake.
In many of the tests conducted, attempts were made to establish a correlation
between the blowing velocity of the flame, in a mixture of given composition and
temperature, and the size of the holder, characterized by a linear dimension of the
same, for example by its diameter in the case of a holder of circular section as the one
shown in Fig. 11.2. A summary of the works performed from this stand-point, up to
1953, and of the state or knowledge at the time may be found in Ref. [18].

11.5. FLAME STABILIZATION 275
Flame front
End of recirculation zone
Recirculation zone
Flame holder
Mixing zone
Propagating zone
Figure 11.2: Zones of a stabilized flame.
The theoretical study of this problem is very difficult. Several theories are
available whose development may be found in the works listed under Refs. [19]
through [21], but none of them has been experimentally confirmed. On the other
hand, the correlation between experimental results as presented by Longwell is very
difficult. At this stage it appears justified to perform the study by means of a dimensionless
analysis, searching for the significant variables and the relation existing
between them.
Inasmuch as we are far from the actual regime in which the compressibility
effect would appear, the dimensionless variable characteristic of the motion is the
Reynolds number referred to the velocity and state of the gases before the holder and
to a linear dimension of the same. The composition of the mixture is characterized by
the relationship between the fuel fraction and that corresponding to the stoichiometric
one. Finally, the variable we are trying to analyze is the blowing velocity or, if written
in dimensionless form although it is not generally necesary, this velocity referred to
the propagation velocity of the flame.
Zukoski and Marble [22] have performed a very interesting systematic analysis
on the experimental data available. A summary of this analysis appears in Figs. 11.3
and 11.4
Figure 11.3, corresponding to a circular flame holder as the one shown in
Fig. 11.2, gives the following: a) the influence of the Reynolds number on the maximum
blowing velocity of the flame, and b) the composition of the mixture for which
such velocity is reached, for a given Reynolds number.
Figure 11.4 shows the law of variation of the maximum blowing velocity as a
function of the Reynolds number for a set of experiments performed by other authors
with different types of flame holders.

276 CHAPTER 11. SIMILARITY IN COMBUSTION. APPLICATIONS
Maximum blowoff
velocity (ft/s) φ m
2.0
1.5
1.0
600
400
200
100
10
2 4 6 8 10 2 4 6 8 10
Reynolds number
3 4 5
Figure 11.3: Maximum blowoff velocity and corresponding equivalence ratio for a cylindrical
flame holder.
Maximum blowoff velocity (ft/s)
1000
800
600
400
300
200
150
100
4
2.5 4 6 8 10 1.5 2 3 4 6 8 10 5 1.5
Reynolds number
Figure 11.4: Maximum blowoff velocity for different types of flame holders (see [22] for
more details).
Both figures show clearly the existence of a transition Reynolds number, approximately
equal to 10 4 , which separates two different regions. For Re < 10 4 , the
maximum blowing velocity corresponds to a composition different from the stoichiometric
one, which furthermore varies with the Reynolds number, whilst if Re exceeds
the transition number the composition remains constant. Likewise, for Re > 10 4
the blowing velocity varies with the square root of the Reynolds number, while for
Re < 10 4 it follows a different law.
Zukoski and Marble, Ref. [17], have demonstrated that such a change is due to
the fact that the free boundary layer of the mixing zone changes from laminar to turbulent.
This transition is important because the transfer of heat and chemical species between
unburnt gases and combustion products, within the mixing zone, occurs through
laminar diffusion under subcritical conditions and through turbulence in the opposite
case.

11.5. FLAME STABILIZATION 277
Flame holder
geometry
D
D
1/8 inch
3/16
1/4
τ i
3.09 × 10 −4 s
2.85
2.80
D
1/4 3.00
mesh screen
D
D
1/4
3/8
1/2
3/4
1/4
3/8
1/2
3/4
2.38
2.70
2.65
2.58
3.46
3.12
3.05
3.03
D
3/4 3.05
D
3/4 2.70
Table 11.1: Ignition time for different types and dimensions of flame holders.
Since the existence of the flame is governed by the possibility of igniting the
mixture, while it travels along the mixing zone, the residence time τ r of the mixture
within this zone must be an important factor of the phenomenon. The residence time
may be expressed by the ratio
τ r = L (11.56)
V
of length L of the mixing zone to the velocity V of the gases when travelling along
it, which is approximately the velocity of the free flow. Consequently, if this interpretation
of the phenomenon is correct, the blowing velocity may be reached when τ r
reaches a limiting value τ i , below which there is no time for ignition. Furthermore,
τ i must depend exclusively on the composition and state of the mixture and on the
conditions at the free boundary layer, but not on the shape of the holder or any other
mechanical factor.
In order to check the validity of this assumption, Marble and Zukoski computed
the value of τ i for a set of quite different cases. Results are summarized in Table

278 CHAPTER 11. SIMILARITY IN COMBUSTION. APPLICATIONS
11.1 in which the constancy of the ignition time may be clearly seen. Hence, the
characteristic dimensionless group determining the blowing velocity of the flame is
Damköhler’s first parameter
and blowing velocity is reached for Da 1 = 1.
Da 1 = L/V
τ i
, (11.57)
The advantage of this study lies on the fact that it separates two variables. One,
being mechanical, depends only on the conditions of motion, whereas the other, being
physico-chemical, depends exclusively on the composition and state of the mixture.
Fig. 11.5, from Ref. [17], shows the law of variation for τ i as a function of the mixture
composition for a typical case.
1.8
1.6
Characteristic ignition time (ms)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
φ, Equivalence ratio
Figure 11.5: Ignition time as a function of the composition of the mixture (see [17] for more
details).
With reference to the mechanical time, the problem reduces to a study of the
relationship between the length L of the wake and the characteristics of the holder.
For a great number of holders it has been proved that, in the supercritical regime, this
relation is of the form
L ∼ D 1/2 , (11.58)
which explains the law of variation of V with D observed by experimentation.
There exist, however, other cases for which the following condition is satisfied
L ∼ D, (11.59)

11.5. FLAME STABILIZATION 279
which is the reason for the special behavior of the holders observed by Longwell [18].
In this formulae, D is a linear dimension of the holder.
The problem which remains to be solved is the process in the mixing zone
which determines the value for τ i .
References
[1] Schultz-Grunow, F.: Similarity Laws of Deflagration. Fourth Symposium (International)
on Combustion, Williams and Wilkins Co., Baltimore, 1953, pp. 439-
443.
[2] Damköhler, G.: Einflüsse der Strömung, Diffusion and des Warmeliberganges
auf die Leistung von Reaktionsofen. Z. Elektrochem., Vol. 42, 1936, pp. 846-
862.
[3] Penner, S. S.: Similarity Analysis for Chemical Reactors and the Scaling of
Liquid Fuel Rocket Engines. Combustion Researches and Reviews, AGARD,
1955.
[4] von Kármán, Th.: Dimensionslose Grösen in Grenzgebieten der Aerodynamik.
Z. F. W., Jan-Febr. 1956, pp. 3-5.
[5] Weller, A. E.: Similarities in Combustion, a Review. Selected Combustion Problems,
Vol. II, AGARD, 1956, pp. 371-383.
[6] Penner, S. S.: Models in Aerothermochemistry. International Symposium on
Models in Engineering, Venice, Italy, 1955.
[7] Ross, Ch. C.: Scaling of Liquid Fuel Rocket Combustion Chambers. Selected
Combustion Problems, Vol. II, AGARD, 1956, pp. 444-456.
[8] Crooco, L.: Considerations on the Problem of Scaling Rocket Motors. Selected
Combustion Problems, Vol. II, AGARD, 1956, pp. 457-468.
[9] Barrère, M.: Similarity of Liquid Fuel Rocket Combustion Chambers. ONERA,
Paris, 1956.
[10] Penner, S. S. and Fuhs, A. E.: On Generalized Scaling Procedures for Liquid-
Fuel Rocket Engines. Combustion and Flame, Vol. I, June 1957, pp. 229-240.
[11] Ross, Ch. C. and Datner, P. P.: Combustion Instability in Liquid Propellant
Rocket Motors. A Survey. Selected Combustion Problems, Vol I, AGARD,
1954, pp. 352-380.

280 CHAPTER 11. SIMILARITY IN COMBUSTION. APPLICATIONS
[12] Crocco, L. and Grey, J.: Combustion Instability in Liquid Propellant Rocket
Motors. Proceedings of the Gas Dynamic Symposium on Aerothermochemistry,
Northwestern Univ., Evanston Illinois, 1956.
[13] Crocco, L. and Cheng, S. I.: Theory of Combustion Instability in Liquid Propellant
Rocket Motors. Butterworths Scientific Publications, 1956.
[14] Summerfield, M.: A Theory of Unstable Combustion in Liquid Propellant Rocket
Systems. Journal of the American Rocket Society, Sept. 1951, pp. 108-114.
[15] Crocco, L.: Aspects of Combustion Stability in Liquid Propellant Rooket Motors.
Journal of the American Rocket Society, Nov. 1951, pp. 163-178.
[16] Scurlock, A. C.: Flame Stabilization and Propagation in High Velocity Gas
Streams. Meteor Report No. 19, Mass. Inst. Tech., May 1948.
[17] Zukoski, E. E. and Marble, F. E.: Experiments Concerning the Mechanism of
Flame Blowoff from Bluff Bodies. Proceedings of the Gas Dynamic Symposium
on Aerothermochemistry, Northwestern Univ., Evanston Illinois, 1956.
[18] Longwell, J. P.: Flame Stabilization by Bluff Bodies and Turbulent Flame in
Ducts. Fourth Symposium (International) on Combustion, Williams and Wilkins
Co., Baltimore, 1953, pp. 90-97.
[19] Longwell, J. P., Frost, E.E. and Weis, M.A.: Flame Stability in Bluff Body Recirculation
Zones. Industrial and Engineering Chemistry, Vol. 45, August 1953,
pp. 1629-1633.
[20] Spalding, D. B.: Theoretical Aspects of Flame Stabilization. Aircarft Engineering,
Sep. 1953, pp. 264-268,276.
[21] Lees, L.: Fluid Mechanical Aspects of Plane Stabilization. Jet Propulsion, July-
August 1954, pp. 234-236.
[22] Zukoski, E. E. and Marble, F. E.: The Role of Wake Transition in the Process
of Flame Stabilization on Bluff Bodies. Combustion Researches and Reviews,
AGARD, 1955, pp. 167-180.

Chapter 12
Diffusion flames
12.1 Introduction
So far we have considered combustion problems dealing with premixed reactant gases.
Yet, it can also happen that species are initially separated, and mixing and combustion
take place simultaneously. Thus, a new type of flame is obtained which Burke and
Schumann [1] called diffusion flame. In these flames the reaction zone separates the
two reacting species which diffuse through inert gases and combustion products from
each side towards the flame. A flame of this type is obtained, for example, when a jet
of combustible gas discharges from a burner into the open atmosphere or into an air
stream parallel to the jet burning at the outlet of the burner, see Fig. 12.1. Such flames,
are largely used in industrial applications. The flame length needed to burn a given
x
Flame
Air
Air
o
r
Fuel
Figure 12.1: An open diffusion flame.
281

282 CHAPTER 12. DIFFUSION FLAME
quantity of fuel per unit time under given conditions is highly interesting in these applications,
since the dimensions of burners and furnaces depend on it. The reacting
species burn very rapidly as they reach the reaction zone. Thereby, the combustion
velocity is generally conditioned by the accessibility of the species to the flame. In
other words, by their facility to diffuse across the inert gases and combustion products.
Diffusion can be either laminar or turbulent depending on the conditions of the
phenomenon, originating the flames accordingly.
The actual state of knowledge on diffusion flames is by far less advanced than
on premixed flames The fundamental work by Burke and Schumann [1] has been the
starting point for practically all the later studies. Even though considerably ahead the
publication of this work a correct qualitative idea on diffusion flames was held, 1 Burke
and Schumann were the first to achieve a quantitative study of the phenomenon. For
the purpose they used a highly simplified model that enabled an analytical solution
of the problem. This model retains the essential factors of the process and can be
summarized in the following two assumptions:
1) The thickness of the reaction zone is zero.
2) The reacting species burn instantaneously upon reach the flame.
This simple model eliminates chemical kinetics from the process thus making
available its computation. This simplification is justified when the reaction rate is very
large compared to the diffusion velocities of the species towards the flame. Thereby it
is not applicable, for example, to the study of flames rarefied or very diluted by inert
gases where the time of reaction can appreciably influence the process. The model
disregards the structure of the reaction zone as was done in Chap. 6 for premixed
flames. The difference between both cases lies upon the fact that while in the case
of premixed flames the combustion front is a discontinuity surface for velocity, temperature,
composition, etc., in diffusion flames such magnitudes change continuously
across the front.
The first assumption reduces the reaction zone to a surface called the flame
front, which acts as a sink for reacting species and as a source for combustion products.
The second assumption implies the following consequences:
a) Concentrations of reactants at the flame must be zero.
b) Reactants must diffuse towards the flame in the stoichiometric ratio corresponding
to the produced reactions.
1 See, i.e., Barr’s review paper [2] where data and bibliography on the subject can be found.

12.1. INTRODUCTION 283
In fact, otherwise, some of the species would diffuse across the flame and react with
those on the other side.
In order to judge the validity of this model Hottel and Hawthorne [3] performed
some experiments to determine the composition of the gases in a diffusion flame of
hydrogen burning in the open atmosphere, taking samples of the gas at different points
of the flame by means of a special sound. Details of these experiments can be found
in the said work. Figure 12.2, taken from it, shows the results of the sounding made
in three cross sections of the flame. The results of such measurements prove the
correctness of Burke and Schumann’s model.
100
75
50
25
H 2
flame front
distance from port= 30.5 cm
Percentage in dry sample
0
100
75
50
25
0
100
O 2
H 2
N 2
N 2
axis of flame
flame front
O 2
distance from port= 22.85 cm
75
50
25
H 2
N 2
flame front
distance from port= 15.25 cm
0
O 2
0 10 20 30 40
Radial distance (mm)
Figure 12.2: Gas composition in a hydrogen diffusion flame
Burke and Schumann have successfully applied their model to the calculation
of the shape and height of laminar diffusion flames for the case where the fuel jet
discharges within a tube where an air stream moves with the same velocity than the
fuel jet, Fig. 12.3. This device eliminates the difficulties originating from momentum
transfer between the fuel and the surrounding air when their velocities are not equal.
Furthermore, for this reason it is easier to obtain a stable flame. By introducing several
drastic simplifications, to be considered further on, Burke and Schumann were able
to calculate the shape of the flame. Some of their results are outlined in Fig. 12.3.
Fig. 12.3 (a) corresponds to an overventilated flame, which closes at the axis of the

284 CHAPTER 12. DIFFUSION FLAME
air fuel air air fuel air
(a) overventilated
(b) underventilated
Figure 12.3: Confined laminar diffusion flames.
tube, and Fig. 12.3 (b) to an underventilated flame, which ends at the wall of the
exterior tube. The fuel-air ratio can be changed by varying the diameters ratio of both
tubes. Both types of flames were experimentally observed by Burke and Schumann.
The flame length is proportional to the amount of fuel burnt per second and does not
depend on the diameter of the burner.
Burke and Schumann’s analysis on confined flames has been extended by J.
Barr, [4] and [5], to the case where air and fuel velocities are different. He has experimentally
analyzed the appearance of the flames that form when the streams of air
and fuel change within limits far more extended than those covered by any previous
investigation. As a result he concludes that the length of a laminar diffusion flame is
proportional to the fuel consumption not only for the case of short flames, as stated
by Lewis and von Elbe [6], but also for flames of a length as much as a hundred times
the diameter of the burner. The results of his work are summarized in Fig. 12.4, taken
from Ref. [5], which shows the different types of flames observed when the flow rates
of air and fuel are changed. In the same figure, regions 1 and 2 correspond to flames
with excess of air and fuel respectively, of the type studied by Burke and Schumann.
In between these regions a zone of carbon formation exists. Region 3 corresponds to
meniscus flames for which diffusion in the axial direction is important. A meniscus
flame blows-out when the fuel flow rate is reduced. Region 4 corresponds to the socalled
convective or Lambent flames. Within this region, where the fuel and air ratios
are small, buoyancy forces are important and oscillating flames are obtained. Region
5, where fuel flow rate is very large, corresponds to tilted flames preceding blow-off.
Region 6, where the fuel and air streams are very strong, corresponds to lifted flames

12.1. INTRODUCTION 285
which preceded blow-off. Region 7 corresponds to vortex flames. Finally, under given
conditions, flames similar to the manometric and singing flames were also observed
by Barr. This incomplete enumeration of the types of flames observed gives idea on
the complexity of the phenomenon. Further data can be found in Ref. [4].
100
10
Extinction
2
Smoke
6
Lifted flames
Butane flow (cm /s)
3
1
0.1
5
4
Smoke point
1
Laminar
diffusion
flames
7
Extinction
Meniscus flames
3
Vortex flames
Convective flames
Extinction
0.01
1 10 100
1000 10000
3
Air flow (cm /s)
Figure 12.4: Flow limits for formation of enclosed diffusion flames.
The case of an open diffusion flame where the fuel jet discharges and burns in
an atmosphere at rest has been experimentally analyzed, among others, by Rembert
and Haslam [7] , Cuthbertson [8] , Gaunce [9], Hottel and Hawthorne [3], Wohl,
Gazley and Kapp [10], Barr and Mullins [11], Parker and Wolfhard [12] and Yagi and
Saji [13]. The fundamental results of their experiments are as follows:
1) When the Reynolds number of the jet at the mouth of a burner is small, a laminar
flame establishes.
2) The length of the flame increases when the flow rate of the jet is increased.
3) When Reynolds number is larger than a given value a turbulent region initiates

286 CHAPTER 12. DIFFUSION FLAME
at the flame tip. When Reynolds number of the jet is increased this region propagates
towards the flame base.
4) When the tip of the flame becomes turbulent its length decreases as Reynolds
number is increased until turbulence reaches its base. There on it remains constant
independently from the discharge velocity of the jet.
5) Within the laminar region the length of the flame is independent from the burner’s
diameter and for a given fuel it depends only on its flow rate. Such result agrees
with that obtained by Burke and Schumann for confined flames.
6) Within the turbulent region the length of the flame is proportional to the diameter
of the tube and independent from the fuel velocity.
Diffusion flames
Transition region
Fully developed turbulent flames
Envelope of flame heights
Height
Increasing nozzle velocity
Envelope of
break points
Figure 12.5: Change in flame type with increasing nozzle velocity.
Fig. 12.5 taken from the work by Hottel and Hawthorne summarizes the experiments
performed by these authors. This figure shows the law of variation of the height of
the flame with the velocity of the jet, and the extension of the turbulent zone. The
following states are observed:
a) Laminar. When the velocity of the jet is small. Here the length of the flame
increases, at first proportionally to the jet velocity and then slows down up to the
maximum height that limits the laminar zone.
b) Transition. The turbulent zone of the flame initiates close to the tip and when
the velocity of the jet increases it propagates towards its base. The total length
of the flame decreases when the extension of the turbulent zone increases. This
is such because the diffusivity of the reactant gases is much larger in turbulent
than in laminar state.
c) Turbulent. The whole flame is turbulent throughout its extension except within a
small zone close to the mouth of the burner. In this state the length of the flame

12.1. INTRODUCTION 287
remains appreciably constant as the velocity of the jet increases. Fig. 12.6 gives
some of the results of the measurements done by Gaunce [9] with city gas in a 3
mm diameter burner. The three aforementioned zones are clearly shown in this
figure.
24
20
Distance from nozzle (inch)
16
12
8
4
Total length, on−port flames
Break point length
Flame separation begins here
Upper part of flame blows−off
0
0 50 100 150 200 250 300
Nozzle velocity (ft/s)
Figure 12.6: Effect of nozzle velocity on flame length.
Hottel and Hawthorne [3], Wohl, Gazley and Kapp [10], Yagi and Saji [13]
and Barr [5] have attempted an extension of Burke and Schumann’s method to the
prediction of the length of open flames both laminar and turbulent. Through rudimentary
approximations they obtain an expression for the flame length containing an
unknown function which they determine empirically from the results of their experiments.
Fig. 12.7 taken from Ref. [5] shows the theoretical and experimental lengths
of some laminar open flames as functions of the fuel flow rate. This figure gives an
idea on the agreement to be expected between experimental measurements and those
predicted by semi-empirical formulae. The extrapolation to values not included in
this figure is risky. A summary on the state of knowledge regarding this matter can
be found in the work by Hottel listed in Ref. [14] where additional bibliography is
included.
Lately, J. A. Fay [15] has calculated the shape and characteristics of the laminar
diffusion flame obtained when a fuel jet discharges into the open atmosphere for the
two-dimensional case and for the case with axial symmetry. Fay has taken into account
the influence of the variation of the velocity in cross direction to the jet, computing
the cross distributions of the velocities, concentrations and temperatures. The model

288 CHAPTER 12. DIFFUSION FLAME
120
100
Butane tube
A
B
Flame length (cm)
80
60
40
City gas nozzle
C
E
City gas tube
D
B
Theory
Experiments
A & E: Wohl et al. Butane, Wohl et al.
20
B, D & F: Barr City gas, Wohl et al.
C: Hottel et al. City gas, Rembert & Haslem
Gaunce
0
0 50 100 150 200 250
Fuel flow (cm 3 /s)
Figure 12.7: Length of different open flames.
proposed by Burke and Schumann is also used by Fay by reducing the reaction zone
to a surface. This work represents the first serious attempt to analyze the velocity field
induced by a diffusion flame.
So far no attempt has been made for the study of the influence of free convection
on the phenomenon which can be very important specially when the discharge
velocity of the jet is small.
The structure of the reaction zone of a diffusion flame has experimentally been
analyzed by Wolfhard and Parker [12] by means of a two-dimensional laminar diffusion
flame obtained with two parallel jets, combustible and oxidizer respectively.
This type of flame is specially suitable for spectroscopic studies. Wolfhard and Parker
have mainly experimented with ammonia-oxygen and ethylene-oxygen flames. Their
studies confirm Burke and Schumann’s stand-point. The reaction zone is in thermal
and chemical equilibrium and therein temperature is practically constant. The concentrations
of reacting species are very small within this zone and, except when dealing
under special conditions that reduce the reaction rate, the process is governed by the
diffusivity of the species. The reacting species generally dissociate before reaching
the reaction zone due to temperature. In the case of hydrocarbons such cracking leads
to carbon formation. Temperatures observed agree fairly with those predicted by theory.
2 Zeldovich [16] has taken into consideration the finite thickness of the reaction
2 See further on §3.

12.2. GENERAL EQUATIONS FOR LAMINAR DIFFUSION FLAMES 289
zone to explain the blowing-off phenomenon. A similar study has been performed
by Spalding to explain the extinction phenomenon in the combustion of fuel droplets
with convection. 3
The following paragraphs are devoted to the deduction of the general equations
for diffusion flames as well as to the simplified equations on which their study is based.
12.2 General equations for laminar diffusion flames
The general equations given in chapter 3 will be applied here to the study of laminar
diffusion flames by assuming that Burke and Schumann’s conditions stated in §1 are
satisfied. The study limits to the case of steady motions. In the deduction that follows
the notation in chapter 3 will apply.
Continuity equations
1) For the mixture.
Since the motion is steady, equation (3.6) reduces to
∇ · (ρ¯v) = 0. (12.1)
2) For the species.
As per Burke and Schumann’s first condition, the reaction rate outside the flame’s
surface Σ f is zero. Therefore, system (3.7) reduces to
ρ(¯v · ∇)Y i + ∇ · (ρY i¯v di ) = 0, i = 1, 2, . . . , l. (12.2)
Equation of motion
Equation of motion (3.18) takes the form
ρ(¯v · ∇)¯v = −∇p + ∇ · τ ev + ρ ¯F . (12.3)
The following cases are specially interesting:
1) Pressure is constant and the influence of mass forces negligible. Then Eq. (12.3)
takes the form
ρ(¯v · ∇)¯v = ∇ · τ ev . (12.4)
3 See §13 of Chap. 13.

290 CHAPTER 12. DIFFUSION FLAME
Moreover, as done in the theory of free jets, here only some terms of the viscosity
forces need to be preserved. 4
2) The influence of mass forces is important. Hydrostatic equilibrium in the fluid
undisturbed by the flame determines the pressure field. If ρ 0 is the density of the
undisturbed fluid
−∇p + ρ 0 ¯F = ¯0, (12.5)
which gives for Eq. (12.3)
ρ(¯v · ∇)¯v = ∇ · τ ev + (ρ − ρ 0 ) ¯F . (12.6)
Furthermore, in each case only the significant terms of ∇ · τ ev will be retained. So
far a solution that includes the influence of the last term of Eq. (12.6) has not been
obtained.
Energy equation
Kinetic energy of motion, energy dissipated by viscosity and work done by mass
forces are negligible. Hence, equation (3.38) reduces to
(
ρ(¯v · ∇)h − ∇ · (λ∇T ) + ∇ · ρ ∑ )
Y i h i¯v di = 0. (12.7)
i
Diffusion equations
Such equations are given by system (3.8) which takes the form
∑
(
Y j ¯vdi − ¯v dj
+ ∇Y i
− ∇Y )
j
= ¯0, (i = 1, 2, . . . , l)
M
j j D ij Y i Y j
∑
Y j ¯v dj = ¯0,
j
(12.8)
where pressure and thermal diffusion are not included since they are negligible.
State equation
This is equation (3.39)
p
ρ = R mT. (12.9)
4 See Fay, Ref. [15].

12.3. BOUNDARY CONDITIONS ON THE FLAME 291
The previous system together with the adequate boundary conditions determine
the values for p, ρ, T , ¯v and for the mass fractions Y i at each point as well as the shape
of the flame which so far is unknown.
12.3 Boundary conditions on the flame
Chemical species A i forming the mixture can be classified into three different groups:
a) Reacting species A r i , which diffuses towards the flame where they burn. These
species are unable to cross Σ f .
b) Reaction products A p i , which produce at the flame and diffuse from it towards
both sides.
c) Inert species A d i , which dilute reacting species. These species can cross Σ f .
According to Burke and Schumann’s model mass fractions Yi
r
species must be zero on the flame surface, that is
of the reacting
Y r
1 = Y r
2 = · · · = Y r
l r
= 0 (12.10)
on Σ f , where l r is the number of reacting species. 5
position of the flame surface.
These conditions determine the
Let m r i be the mass of species Ar i that reaches Σ f per unit surface and per unit
time. m r i is given by m r i = −ρY i¯v i · ¯n i , (12.11)
where ¯n i is the unit vector normal to Σ f at A r i side. mr i is consumed by the chemical
reactions taking place at Σ f . Let
∑
ν ijA ′ r i → ∑ ν ijA ′′ p i , (j = 1, 2, . . . , r), (12.12)
i
i
be one of these reactions, which transforms reacting species A r i into reaction products
A p i . Since the thickness of the reaction zone is assumed to be zero, it becomes necessary
to define a surface reaction rate for each reaction. Such rate gives the masses of
the species consumed or produced by the reaction per unit surface and per unit time.
Be r j the value of such surface reaction rate for reaction j. The fraction m r ij of mr i
consumed by this reaction is
where M r i is the molar mass of species Ar i .
m r ij = M r i ν ′ ijr j , (j = 1, 2, . . . , r), (12.13)
5 Actually, these mass fractions are very small but not strictly zero. Their values are practically those of
the equilibrium of species under the prevailing conditions at the flame.

292 CHAPTER 12. DIFFUSION FLAME
Therefore m r i
is given by
∑
m r i = Mi
r ν ijr ′ j , (i = 1, 2, . . . , l r ). (12.14)
j
Likewise if m p i is the mass of species Ap i produced per unit surface and per
unit time one has
m p i = M p i
∑
j
ν ′′
ijr j , (i = l r + 1, l r + 2, . . . , l a ), (12.15)
where l a = l r + l p is the number of active species (reactants plus products) and l p is
the number of products.
Reaction rates r j are unknown “a priori”. Elimination of the same between
equations (12.14) and (12.15) gives l a −r relations between Yi
r and Y p
i which enables
to express all mass fractions of active species as functions of r selected from them.
Thus the variables of the problem reduce in l a − r.
So far no solutions have been obtained for this general system of equations.
In order to make computation available, additional assumptions must be introduced
to simplify considerably the problem. Such simplifications will be performed in the
following paragraph.
12.4 Simplified equations
In this approximate study the number of species is assumed to be three. Namely, fuel
A 1 , oxidizer A 3 and products A 2 . Products include not only species resulting from
reactions, but also diluents of fuel and oxidizer initially mixed with them. 6 Flame surface
divides space into two regions: an interior region at the fuel side and an exterior
one at the oxidizer side. Only A 1 and A 2 species exist within the interior region. In
the exterior one only A 2 and A 3 . Therefore, the composition of the mixture within the
interior region is determined by the value of Y 1 which must satisfy equation (12.2)
ρ(¯v · ∇)Y 1 + ∇ · (ρY 1¯v d1 ) = 0. (12.16)
Similarly in the exterior region one has for Y 3
ρ(¯v · ∇)Y 3 + ∇ · (ρY 3¯v d3 ) = 0. (12.17)
6 A larger number of species could easily be included by distinguishing, for example , between diluents
and products. Such is done, among others, by Zeldovich [17] and Fay [15]. However, this does not represent
any fundamental advantage and computations become more elaborate. A differentiation between products
and diluent can be done once the problem is solved.

12.4. SIMPLIFIED EQUATIONS 293
As for Y 2 , its value is given by expressions
Interior region: Y 2 = 1 − Y 1 . (12.18)
Exterior region: Y 2 = 1 − Y 3 . (12.19)
Diffusion velocities ¯v d1 and ¯v d3 are given by Fick’s law 7
Y 1¯v d1 = −D 12 ∇Y 1 , (12.20)
Y 3¯v d3 = −D 23 ∇Y 3 . (12.21)
On surface Σ f
Y 1 = Y 3 = 0; Y 2 = 1. (12.22)
Furthermore, since Y 1 and Y 3 must diffuse towards the flame in the stoichiometric
ratio from (12.11), (12.20) and (12.21) we have on Σ f
νD 12
∂Y 1
∂¯n i
= D 23
∂Y 3
∂¯n e
. (12.23)
Here ν is the ratio between the masses of oxidizer and fuel needed for complete combustion.
¯n e and ¯n i are normals to Σ f towards the exterior and interior regions respectively.
System of equations (12.16) and (12.17) can be substituted by a single equation
for a new variable Y defined as follows
⎧
⎪⎨ Y 1 in the interior region,
Y =
⎪⎩ − 1 ν Y 3 in the exterior region.
Y must satisfy the following differential equation
(12.24)
ρ(¯v · ∇)Y − ∇ · (ρD∇Ȳ ) = 0, (12.25)
where D takes the value
⎧
⎨
D =
⎩
D 12
D 23
in the interior region,
in the exterior region.
(12.26)
Furthermore, Y is zero on the flame and takes values of opposite sign at both sides of
it. The derivatives of Y at both sides of the flame in normal direction to it must satisfy
condition
7 See Chap. 2.
D 12
∂Y
∂¯n i
= −D 23
∂Y
∂¯n e
. (12.27)

294 CHAPTER 12. DIFFUSION FLAME
In particular, if condition
D 12 = D 23 = D (12.28)
is satisfied, Y as well as its derivatives are continuous even at the flame. Continuity
Eq. (12.1) for the mixture and Eq. (12.3) for motion still hold unchanged.
As for the energy equation we shall assume that the specific heats at constant
pressure of the three species are constant and equal
c p1 = c p2 = c p3 = c p . (12.29)
In such case, one can immediately check that Eq. (12.7) reduces to the following
which hold throughout space.
ρc p¯v · ∇T − ∇ · (λ∇T ) = 0, (12.30)
Let us assume, that besides condition (12.28) the following is satisfied
λ
ρDc p
= 1, (12.31)
in accordance with the result of the elementary Kinetic Theory of Gases. Then, comparison
between (12.25) and (12.30) suggests the existence of solutions of the form
T = aY + b, (12.32)
where a and b are constant but can be different for the interior and exterior regions.
These solutions are available for the study of diffusion flames if boundary conditions
can be satisfy through them. In particular, since on the flame Y = 0, temperature T b
of the flame will be constant for the cases where Eq. (12.32) is valid. The study of
diffusion flames reduces, hence, to a computation of the values for ρ , T , ¯v and Y . For
this purpose, system of Eqs. (12.1) and (12.3) is available which must be completed
with the boundary conditions adequate for each case. These refer in general to the
conditions at the discharge sections of fuel and oxidizer and at great distance from the
flame. Let us assume, for example, that in the fuel’s discharge section
and in the oxidizer’s discharge section
Y = Y 10 , T = T 0 , (12.33)
Y = Y 30 , T = T 0 , (12.34)
Then Eq. (12.32) holds and one obtains in the interior region,
T = T b − (T b − T 0 ) Y
Y 10
= T b − (T b − T 0 ) Y 1
Y 10
, (12.35)

12.4. SIMPLIFIED EQUATIONS 295
while in the exterior region
T = T b + ν(T b − T 0 ) Y
Y 30
= T b − (T b − T 0 ) Y 3
Y 30
. (12.36)
In these expressions T b is still undetermined. Its value can be obtained by considering
an element of the flame as the one in Fig. 12.8 and applying to it the principle of
conservation of energy. One immediately obtains
m 2 h 2 − (m 1 h 1 + m 3 h 3 ) = λ
( ∂T
∂¯n i
+ ∂T
∂¯n e
)
. (12.37)
Here, m 1 and m 3 are the masses of fuel and oxidizer that reach ∑ f
per unit surface
ν 1
m 2,i + m 2,e = =(1+ ) m
m 2
m 2,i
m 1
m 2,e
m 3 = νm 1
n i
λ T n
’
i
n e
λT’ n e
Flame
Figure 12.8: Schematic diagram of an element of a diffusion flame.
and per unit time and m 2 is the mass of products that emerges from it. But
m 3 = νm 1 , m 2 = (1 + ν)m 1 . (12.38)
Furthermore, from Eqs. (12.11), (12.20) and (12.28)
From Eqs. (12.35) and (12.36)
m 1 = ρD ∂Y 1
∂¯n i
. (12.39)
∂T
= −(T b − T 0 ) 1 ∂Y 1
, (12.40)
∂¯n i Y 10 ∂¯n i
∂T
= −(T b − T 0 ) 1 ∂Y 3
. (12.41)
∂¯n e Y 30 ∂¯n e
When expressions for m 1 , m 2 , m 3 , ∂T /∂¯n i and ∂T /∂¯n e are taken into Eq. (12.37)
keeping in mind Eqs. (12.23), (12.29) and (12.31), one obtains for T b
T b − T 0 = q r
c p
Y 10 Y 30
Y 30 + νY 10
, (12.42)
where
q r = h 1 + νh 3 − (1 + ν)h 2 (12.43)
is the heat of reaction per unit mass of fuel.

296 CHAPTER 12. DIFFUSION FLAME
Formula (12.42) shows that T b is the temperature that would correspond to a
premixed flame where fuel and its diluent were mixed with oxidizer and its diluent in
the stoichiometric ratio. In fact, in such mixture, mass fraction of fuel is
Y 10 Y 30
Y 30 + νY 10
and the corresponding temperature of the flame is given by (12.42).
12.5 Solutions of the simplified system
Some approximate solutions have been obtained for the case of two-dimensional and
axis symmetrical flames either confined or open. A detailed study of such flames is
available in the works listed in Refs. [1], [3], [6], [10], [15] and [17]. The following
chapter studies in detail the diffusion flame that forms surrounding a fuel droplet
which evaporates and burns in an oxidizing atmosphere. The problem has great importance
in jet propulsion.
The following simplifying assumptions have generally been adopted:
1) Stream velocity is throughout space parallel to the flame axis and uniform throughout
space or at least at each cross-section of the same.
2) All transport coefficients are independent from the composition and temperature
of the mixture.
3) Diffusion produces only in cross direction to the flame axis.
These assumptions reduce the problem to the integration of the one-dimensional
diffusion equation for the computation of the fuel, oxidizer and temperature distributions.
For example, in the case of an axis symmetrical flame, Fig. 12.1, the preceding
assumptions reduce Eq. (12.25) to
v ∂Y
∂x = D 1 (
∂
r ∂Y )
. (12.44)
r ∂r ∂r
Here, D and v are constant or at least independent from r. In the first case
the integration of Eq. (12.44) is straightforward. This also occurs in the second case
through the previous change of variable
x =
∫ τ
0
v
dτ, (12.45)
D
which to be determined would require knowing v/D as a function of distance x to
the flame base, which has been introduced by Hottel [3], Wohl [10] and Barr [5] as

12.5. SOLUTIONS OF THE SIMPLIFIED SYSTEM 297
an unknown function empirically determined. The aforementioned simplifying assumptions
only represent a poor approximation to reality. In fact, with respect to the
velocity field it can easily be verified that it is only constant in open flames where fuel
and oxidizer move at equal velocity within the discharge section and, furthermore,
where pressure is uniform throughout space. In such case the gas expansion produces
entirely in cross direction to the flame. As for transport coefficients they can change
in the ratio ten to one between flame surface and unburnt gases. While the assumption
that diffusion produces only in cross direction is well justified, except for special cases
like that of meniscus flames which so far have not been theoretically analyzed.
Before ending this brief review on diffusion flames we shall develop a simple
argument by W. Jost [19] which enables an easy establishment of the influence on the
length of a diffusion flame of some of the fundamental variables of the process. In
view of the rudimentary approximations needed in Burke-Schumann’s calculations in
order to obtain usable expressions, Jost assumes the same validity for his argument.
R
z
L
vt
z
v
fuel
v
oxigen
burner axis
burner wall
Figure 12.9: Schematic diagram of a overventilated diffusion showing the elements of the
Jost’s criterion.
Let us consider, for example, a confined flame with an excess of air or an
open flame, Fig. 12.9. According to Jost, the length of the flame is determined by the
condition that the oxygen of the atmosphere surrounding the jet can reach the jet’s axis
through diffusion. Let D be the oxygen diffusion coefficient and z its mean quadratic
displacement due to diffusion at instant t. Diffusion theory gives for z as function of t
z 2 = 2Dt. (12.46)
But, at the point where the flame closes
z = R, (12.47)

298 CHAPTER 12. DIFFUSION FLAME
where R is the burner’s radius, and
t = L v , (12.48)
where v is the mean velocity of the jet between the base and the tip of the flame and L
the length of the same. Through elimination of t and z between (12.46), (12.47) and
(12.48) one obtains
L ∼ R2 v
2D . (12.49)
In laminar flames D depends only on the properties of the gases. Furthermore
R 2 v is proportional to the fuel flow rate G. Consequently, we have
in accordance with the result obtained by Burke-Schumann.
L ∼ G
2D , (12.50)
In a turbulent flame D is the turbulent diffusivity which has the form
When this value is substituted into (12.49) it results
D ∼ vR. (12.51)
L ∼ R, (12.52)
which as previously seen also agrees with experimental results.
References
[1] Burke, S. F. and Schumann, T. E. W.: Diffusion Flames. Industrial and Engineering
Chemistry, Vol. 20, October 1928, pp. 998-1004.
[2] Barr, J.: Diffusion Flames in the Laboratory. AGARD Memorandum No.
AG11/M7, 1954.
[3] Hottel, H. C. and Hawthorne, W.R.: Diffusion in Laminar Flame Jets. Third
Symposium (International) on Combustion, Williams and Wilkins Co., Baltimore,
1949, pp. 254-266.
[4] Barr, J.: Diffusion Flames. Fourth Symposium (International) on Combustion,
Williams and Wilkins Co., Baltimore, 1953, pp. 765-771.
[5] Barr, J.: Length of Cylindrical Laminar Diffusion Flames. Fuel, Jan. 1954,
pp. 51-59.
[6] Lewis, B. and von Elbe, G.: Combustion, Flames and Explosions of Gases.
Academic Press Inc. Publishers., New York, 1951.

12.5. SOLUTIONS OF THE SIMPLIFIED SYSTEM 299
[7] Rembert, E. W. and Haslam, R. T.: Factors Influencing the Length of a Gas
Flame Burning in Secondary Air. Industrial and Engineering Chemistry, Dec.
1925, pp. 1236-1238.
[8] Cuthbertson, J.: Journal Society of Chemical Industries, Transactions, Vol. 50,
1931, pp. 451-457.
[9] Gaunce, H.: Unpublished Research on Flames, M.I.T., 1937.
[10] Whol, K., Gazley, C. and Kapp, N. M.: Diffusion Flames. Third Symposium
(International) on Combustion, Williams and Wilkins Co., Baltimore, 1949,
pp. 288-301.
[11] Barr, J. and Mullins, B. P.: Combustion in Vitiated Atmospheres. Fuel, Vol. 28,
1949, pp. 131, 200, 205, 225.
[12] Parker, W. G. and Wolfhard, H. G.: Journal of the Chemical Society, 1950,
p. 2038.
[13] Yagi, S. and Saji, K.: Problems of Turbulent Diffusion and Flame Jets. Fourth
Symposium (International) on Combustion, Williams and Wilkins Co., Baltimore,
1953, pp. 771-781.
[14] Hottel, H. C.: Burning in Laminar and Turbulent Fuel Jets. Fourth Symposium
(International) on Combustion, Williams and Wilkins Co., Baltimore, 1953,
pp. 97-113.
[15] Fay, J. A.: The Distributions of Concentration and Temperature in a Laminar Jet
Diffusion Flame. Journal of the Aeronautical Sciences, October 1954, pp. 581-
589.
[16] Gaydon, A. G. and Wolfhard, H. G.: Flames, their Structure, Radiation and
Temperature. Chapman and Hall Ltd., London, 1953, p. 135.
[17] Zeldovich, Y. B.: On the Theory of Combustion of Initially Unmixed Gases.
N.A.C.A. Tech. Memo. No. 1296, June 1951.
[18] Spalding, D. B. : The Combustion of Liquid-Fuels. Fourth Symposium (International)
on Combustion, Williams and Wilkins Co., Baltimore, 1953, pp. 847-
864.
[19] Jost, W.: Explosion and Combustion Processes in Gases. McGraw-Hill Book
Comp., New York, 1946, p. 210.

300 CHAPTER 12. DIFFUSION FLAME

Chapter 13
Combustion of liquid fuels
13.1 Introduction
The present chapter is devoted to the study of some of the problems of the combustion
of liquid fuels. In certain cases surface reactions can take place in the liquid phase, for
example, in the combustion of hypergoles. On the contrary, when a fuel burns in an
oxidizing atmosphere the reaction takes place in the gaseous phase. Thereby, the fuel
evaporation must preceded combustion. In such case, if the mixing of the fuel vapours
and the oxidizing gas takes place before the chemical reaction, a flame produces of
the type studied in chapter 6. On the other hand, if mixing and combustion take place
simultaneously, a diffusion flame is obtained, similar to those studied in chapter 12.
This is the type of combustion considered in the present chapter. Khudiakov [1], [2]
has studied the characteristics of the diffusion flame that forms on the free surface of
a fuel burning in the atmosphere. Spalding [3] has also studied this problem taking
into account the influence of free and forced convection. He made this study as an
application of his uniform method [4] for the study of all the mass transport processes
(absorption, evaporation, condensation and combustion).
Of further technical interest is the case where fuel forms droplets of small
diameter. Fuel mists formed by droplets of very small diameter (smaller than a few
hundredths of a millimeter) burn forming a flame similar to the premixed gas flames
[5], [6]. In particular, in the said mists, like in the premixed gas flames, the propagation
velocity of the flame, its flammability limits, etc., are well defined magnitudes. Their
values are close, but not equal, to those corresponding to premixed gas flames [7].
The difference is probably due to the droplets evaporation effect. The diameter of the
droplets is so small that the evaporation produces within the heating zone of the flame,
so that the fuel reaches the reaction zone in gaseous phase.
301

302 CHAPTER 13. COMBUSTION OF LIQUID FUELS
If the diameter of the droplet is larger than, for example, a few tenths of a
millimeter the phenomenon becomes more complicated. Such size is too large to
allow evaporation across the heating zone of a flame. Therefore, in this case individual
diffusion flames surrounding the droplets are produced.
For technical applications the fuel is normally introduced in the combustion
chamber by means of an atomizer producing droplets of different sizes. These droplets
spray and evaporate in contact with the atmosphere of the chamber, which is strongly
turbulent. Furthermore, in the combustors of jet engines, the gases flow at high velocity
through the combustion chamber. Generally, in such cases not enough space
is available for the complete evaporation of the fuel before it reaches the combustion
zone. Consequently when the fuel reaches the said zone it is only partially evaporated
and the mixing with the oxidizing atmosphere is incomplete. The entire process starts
with the entrance of the fuel in the combustion chamber and ends when the combustion
products are formed. The complete study of this process includes the following stages:
atomizing, spraying, mixing, evaporation and combustion of the fuel jet. Due to the
complexity of the process a study in detail can not be attempted with any probabilities
of success. Therefore, we must resort to empiricism and to the study of simplified
physical models which emphasize some of the features of the phenomenon.
Hereinafter we shall first briefly refer to the atomization, spraying and mixing
of fuel jets and then, more closely, to the combustion of isolated droplets. Finally the
evaporation of droplets with no combustion will be considered.
13.2 Atomization
A great effort has been made both theoretically and experimentally for the study of
the atomization of fuel jets [8], [9]. The process depends on:
1) The geometrical characteristics of the atomizer.
2) The physical properties of the liquid and the atmosphere in which it discharges.
3) The working conditions.
Depending on the discharge velocity of the liquid and on the conditions of the surrounding
atmosphere, several states of atomization can be observed [9]. In the discharge
at high velocity, normally produced in burners, the atomization starts at the
nozzle of the atomizer. Atomization is produced by the turbulence of the liquid, whose
radial velocities tend to brake the jet, and by the action of the atmosphere at the outlet.
Surface tension and viscosity of the liquid oppose to the jet atomization. The prob-

13.2. ATOMIZATION 303
lem has been reviewed by Heinze [10], who studied in detail the action of its various
factors.
Atomization is characterized by the average size of the droplets and by their
distribution in sizes. The average size of the spray droplets is characterized by the
Sauter diameter d s [11]. This is the diameter that would be obtained in an ideal atomization
where all the droplets were of equal size and where the surface and total
volume of the fuel were equal to those of actual atomization. The value d s depends
on the variables that characterize the particular conditions of atomization, according
to rules empirically determined (see Ref. [9], p. 117 and f.).
The size distribution in characterized by the mass fraction R (d/d s ) of the fuel
corresponding to droplets of a diameter larger than d. Therefore, the size distribution
dR
function is
d (d/d s ) . Several empirical formulas have been proposed for R (d/d s) or
dR
, [12]. The two formulas generally used are the Rosin-Rammler formula [13]
d (d/d s )
and the Nukiyama-Tanasawa formula [14]. The results given by these two formulas
are in fair agreement with the distributions experimentally observed [15], [16]. These
formulas are as follows.
Rosin-Rammler:
or else
( ) δ−1
dR d
d (d/d m ) = δ e −(d/d m) δ , (13.1)
d m
R = 1 − e −(d/d m) δ , (13.2)
In these formulas d m is an average diameter, characteristic of the size of atomization,
which is related to the Sauter diameter through formula
d m
d s
(
= Γ 1 − 1 )
, (13.3)
δ
where Γ is the factorial function, 1 and δ is a characteristic parameter of size uniformity.
When δ increases the distribution becomes more uniform.
Nukiyama-Tanasawa:
( ) 5
dR
d (d/d m ) = δ d
e −(d/d m) δ (13.4)
Γ (6/δ) d m
1 Also called gamma function, defined as Γ(a) = R ∞
0 ta−1 e −t dt, Ed.

304 CHAPTER 13. COMBUSTION OF LIQUID FUELS
or else
R =
γ
(6/δ, (d/d m ) δ)
Γ (6/δ)
(13.5)
Here the parameters have the same meaning that in the Rosin–Rammler formula and γ
is the incomplete gamma function. 2 The Sauter diameter is related with d m by means
of
(
ds
d m
)
= Γ (6/δ)
Γ (5/δ) . (13.6)
13.3 Mixing
Once the fuel is atomized it mixes with the surrounding atmosphere due to the action
of turbulence. Lately, Longwell and Weiss [17] have studied the problem for the case
where the fuel atomizes in a turbulent gas stream flowing with uniform velocity. Let
f be the ratio of the fuel mass to the air mass. Assuming that:
1) Turbulent diffusion produces only transversely to the main flow.
2) Mean motion is stationary.
3) The process has axial symmetry.
The following approximate equation for f is obtained
∂f
∂x = E ( )
1 ∂f
v r ∂r + ∂2 f
∂r 2 . (13.7)
Here v is the motion velocity, E is the coefficient of turbulent diffusion of the
fuel and x and r are the cylindrical coordinates of the system. For the derivation of this
equation E is assumed to be constant. Equation (13.7) is identical to the molecular
diffusion equation under similar conditions.
If the fuel is vaporized, E is the coefficient of turbulent diffusion defined by
Taylor [18]. In this case, E equals the product of intensity by scale of turbulence.
If the fuel is in the liquid state, E is appreciably smaller due to the inertia of the
droplets which are unable to follow the air fluctuations. In a typical case calculated by
Longwell and Weiss, E was only 35% of the value corresponding to the diffusion of
the vapour.
If the boundary conditions in the atomizing section are known, equation (13.7)
can be integrated. For example, if the mixing starts from the origin of coordinates,
which acts as a source of fuel of strength G c , and if the duct radius is large, the
2 Defined as γ(a, x) = R x
0 ta−1 e −t dt, Ed.

13.4. COMBUSTION 305
solution corresponding to Eq. (13.7) is
f = G vr2
c v
e− 4Ex , (13.8)
G a 4πE
where G a is the air mass flux in the duct. This solution corresponds, for example, to
the case of a cylindrical atomizer discharging downstream of the gas flow. Formula
(13.8) can be used to obtained the value for E.
From this fundamental solution, the solutions corresponding to different distributions
of fuel sources in the atomizing section can be found by superposition. Such
as discs, rings, etc. These solutions represent with fair approximation the actual mixing
processes for several practical cases, such as the upstream atomization or the case
of a swirl atomizer, etc. Some of these solutions can be found in the work by Longwell
and Weiss.
13.4 Combustion
The study under given conditions of the combustion of a fuel spray, for example, under
the prevailing conditions in the combustion chamber of a jet engine, is very arduous.
At present it can only be attempted empirically. 3 Therefore it is justified as a preliminary
phase to analyze the combustion of isolate droplets under laboratory conditions.
This problem has lately been studied successfully both theoretically and experimentally
by several investigators. Such problem is the subject of this and the following
paragraphs where it will he considered in detail. Further on a complementary study
will be made on the evaporation of the droplet when combustion is absent.
Experimental evidence shows that combustion takes place within a region of
small thickness surrounding the droplet. From each side towards this region diffuse
fuel vapours and oxygen. Burnt gases diffuse from this region towards the exterior.
Therefore we have a diffusion flame of the type considered in chapter 12. Thus it
will be assumed that the flame thickness is zero and that the mass fractions of fuel
and oxidizer at the flame are also zero. Moreover, the reacting species must diffuse
towards the flame in the stoichiometric ratio.
The flame must supply the necessary heat for the previous evaporation of the
fuel. This heat is transferred to the droplet surface partially through conduction and
partially through radiation. The energy transmitted through radiation is only a small
fraction of the heat absorbed by the droplet through conduction. Therefore, in the
3 See §14 of the present chapter.

306 CHAPTER 13. COMBUSTION OF LIQUID FUELS
present approximate study its influence is neglected. Otherwise, the analysis of the
problem becomes far more complicated. By doing so the results obtained do not
substantially varies.
The heat received by the droplet is partially used for its heating and partially for
the evaporation of the fuel. The first fraction increases the temperature of the droplet
up to the boiling point T s of the fuel. Thereon it remains constant and equal to T s . All
the heat received is used up in evaporating the fuel. Hereinafter, the transient initial
period will be neglected by assuming that the droplet temperature is initially T s and
that no thermal gradients exist therein.
Strictly the process is non-stationary since the size of the droplet decreases as
combustion progresses. However, the recession velocity of the droplet surface is very
small compared to the velocity of the burnt gases. Thereby, an excellent approximation
is obtained by assuming that the phenomenon is stationary. That is to say by
neglecting the local variations with time of temperature and mass fractions produced
by the droplet reduction in size. This is an important simplification of the problem
since it eliminates an independent variable from the equations.
The existence of free and forced convection introduces a privileged direction
destroying the spherical symmetry of the phenomenon. Even in the case of free convection
with approximately round droplets experimental results show that the shape
of the flame differs appreciably from the spherical within the upper region. A study of
the problem, taking into account these convection effects is very arduous and has not
yet been achieved. Nevertheless, when assuming that both the droplet and the flame
front are spherical, that is when neglecting these effects, the results obtained show a
good agreement with the experimental results as for the overall magnitudes such as the
droplet burning velocity and its law of variation with size. Thereby, neglect of convection
effects is justified. With this assumption and the assumption of quasi-stationary
state previously stated, there is only one independent variable, this is the distance r to
the center of the droplet.
The composition of the fuel vapours, oxidizing atmosphere and burnt gases is
very complex. However, as done in the study of diffusion flames, we shall assume for
simplicity that only three different chemical species exist, namely: fuel, oxidizer and
inert gases, which include these in the atmosphere surrounding the droplet as well as
the combustion products.
Summarizing, the following assumptions are adopted:
1) The phenomenon has spherical symmetry.

13.5. NOTATION 307
2) The phenomenon is quasi-stationary.
3) Combustion takes place at constant pressure.
4) The droplet temperature is uniform and equal to the boiling temperature of the
fuel at ambient pressure.
5) The chemical reaction takes place upon a spherical surface, named the flame
front. Fuel vapours and oxygen diffuse in the stoichiometric ratio towards this
flame front from which the burnt gases flow. The gases that reach the flame front
react instantaneously. Thereby, on the flame front both the mass fractions of fuel
vapours and of oxygen are zero.
6) Only three chemical species exist, namely: fuel, oxygen and inert gases.
These assumptions allow a simple analysis of the problem and lead to results
which have been experimentally confirmed. Such a treatment of the problem has been
applied by Godsave [19] and Spalding [3] in England, and by Penner and Goldsmith
[20] in the U.S.A. Aside from the aforementioned assumptions Godsave and Spalding
also assume that transport coefficients are independent from temperature by adopting
a mean value for these coefficients. This rather arbitrary assumption reduces appreciably
the suitability of the method. In fact, for the existing range of temperatures, these
coefficients can vary in a ratio of ten to one. Goldsmith and Penner take into account
this variation as well as that of heat capacities. The following study is mainly based
on the work of these two authors but it assumes that heat capacities are independent
from temperature since their variation has no substantial influence on the results.
13.5 Notation
In the present study the notation used in the preceding chapter will be completed with
the following (see Fig. 13.1):
M = Droplet mass
m = Mass flow across a closed surface surrounding the droplet.
q l = Latent heat of evaporation per unit mass of fuel.
r = Radial distance to the center of the droplet.
v = Radial velocity of the mixture.
v j = Radial velocity of species A j .
v jd = Radial diffusion velocity of species A j .
Y j = Mass fraction of species A j .
ν = Stoichiometric ratio of oxygen mass to fuel mass.

308 CHAPTER 13. COMBUSTION OF LIQUID FUELS
Subscripts:
1 = Fuel
2 = Inert gases
3 = Oxygen
e = Exterior to the flame front
i = Interior to the flame front
l = On the flame
s = On the droplet surface
∞ = At great distance from the droplet
Y
Y 2
r l
Y 1
Y 3
Y 2 oo
oo
r s
T s
r
l
T
r
Y 3
oo T
T
T l
Figure 13.1: Schematic diagram of the droplet combustion problem.
13.6 Continuity equations
Since the process is stationary, the fluid mass that crosses any spherical surface concentric
to the droplet must be constant. Due to the spherical symmetry of the problem
such condition can be expressed in the form
4πr 2 ρv = const. (13.9)
The constant can be determined by particularizing this equation on the droplet surface
r = r s . This surface acts as a fuel source. The strong of this source is the mass m of
vapour produced per unit time. Therefore, we have
4πr 2 ρv = m. (13.10)
This equation is valid throughout the fluid space surrounding the droplet.

13.6. CONTINUITY EQUATIONS 309
Chemical reactions take place only on the flame surface r = r l , which acts as
a sink for fuel and oxidizer and as a source for the combustion products. Therefore
one obtains for each different chemical species an equation similar to Eq. (13.10). In
this equation ρ must be substituted by the partial density ρ i = ρY i corresponding to
species A i , v by velocity v i = v + v di , where v di is the radial diffusion velocity of the
species, and m must be substituted by the partial flow m i of the species. Thus
4πr 2 ρY i v i = m i , (i = 1, 2, 3). (13.11)
Moreover, the following evident conditions must be satisfied
∑
Y i v i = v, (13.12)
i
∑
m i = m. (13.13)
i
Equation (13.11) is valid for each species A i , as long as r does not cross the flame.
Let us now apply these equations separately to the interior and exterior regions
of the flame.
Interior region r s ≤ r ≤ r l
In this region only fuel vapours and inert gases exist, since the incoming oxygen is
entirely consumed as it reaches the flame without crossing it.
Equation (13.11) applied to the fuel vapours A 1 gives
4πr 2 ρY 1 v 1 = m 1 = m, (13.14)
since on the droplet surface the mass flow m is the mass flow m 1 of the fuel produced
by evaporation.
By comparing (13.9) and (13.14)
Y 1 v 1 = v. (13.15)
Equation (13.12) gives
Y 1 v 1 + Y 2 v 2 = v. (13.16)
From Eqs. (13.15) and (13.16) results
v 2 = 0, (13.17)
that is to say within the interior region the inert gases are at rest and the fuel vapours
diffuse through them towards the flame.

310 CHAPTER 13. COMBUSTION OF LIQUID FUELS
Exterior region r ≥ r l
In this region only oxygen and inert gases exist. Equation (13.11) applied to each of
them gives
Inert gases: 4πr 2 ρY 2 v 2 =m 2 . (13.18)
Oxygen: 4πr 2 ρY 3 v 3 =m 3 . (13.19)
Similarly, Eqs. (13.12) and (13.14) give
Y 2 v 2 + Y 3 v 3 = v, (13.20)
m 2 + m 3 = m. (13.21)
Let ν be the stoichiometric ratio of oxygen mass to fuel mass. Since, the fuel
mass reaching the flame per unit time is m, the oxygen mass that reaches the flame
per unit time must be νm. And since the oxygen moves towards the flame, v 3 must be
negative, that is
m 3 = −νm. (13.22)
From here and (13.21) there results for m 2
m 2 = (1 + ν)m. (13.23)
Equations (13.22) and (13.23) determine the constants for the continuity equations
(13.18) and (13.19) as functions of m, thus obtaining
4πr 2 ρY 2 v 2 = (1 + ν)m, (13.24)
4πr 2 ρY 3 v 3 = −νm. (13.25)
Therefore, within the exterior region the flame acts as a sink of strength νm for the
oxygen and as a source of strength (1 + ν)m for the combustion products.
13.7 Energy equation
The process is stationary and it takes place at constant pressure. Moreover the kinetic
energy of the motion and the work of the viscous forces can be neglected. Therefore
the summation of the heat and enthalpy fluxes through any spherical surface concentric
to the droplet must be constant. Due to the spherical symmetry, this condition can be
expressed in the form
∑
i
m i h i − 4πr 2 λ dT
dr
= const. (13.26)

13.7. ENERGY EQUATION 311
The value for the constant can be obtained by applying this equation to the droplet
surface r = r s . Since, here m 1 = m and m 2 = m 3 = 0, we have
(
mh 1s − 4πrs
2 λ dT )
= const., (13.27)
dr
(
where 4πrs
2 λ dT )
is the heat received by the droplet surface per unit time. Since
dr
s
the droplet temperature is assumed to be constant and equal to the boiling temperature
T s of the fuel, this heat must be used in evaporating the combustible. Furthermore
this heat is the only source of energy for the evaporation, since the energy received
from the flame by heat radiation has been neglected. 4
s
Let q l be the latent heat of
evaporation per unit mass at temperature T s . Since the evaporated mass per second is
m, the following condition is obtained
(
4πrs
2 λ dT )
= mq l . (13.28)
dr
s
Consequently, the value for the constant is m(h 1s − q l ), which taken into
Eq. (13.26) gives
∑
i
m i h i − 4πr 2 λ dT
dr = m(h 1s − q l ). (13.29)
The specific enthalpy h i of species A i can be expressed in the form
∫ T
h i = h 0i + c pi dT. (13.30)
T 0
When this expression is substituted into (13.29) we obtain
4πr 2 λ dT ∫ ( )
T ∑
dr − m i c pi dT = m(q l − h 1s ) + ∑
T 0 i
i
m i h 0i . (13.31)
The form taken by this expression within the interior and exterior regions will
be studied separately.
Interior region r s ≤ r ≤ r l
As aforesaid in this region only fuel vapours ant inert gases exist. Furthermore due
to Eq. (13.17) the flow of inert gases throughout the spherical control surface is zero.
4 This matter has been considered by G.A. Godsave who concludes that the radiation energy received
by the droplet per gram of evaporated fuel is approximately proportional to the droplet radius. For large
droplets (r s ≃ 1 mm) it can become a 20% of the energy needed to produce evaporation.

312 CHAPTER 13. COMBUSTION OF LIQUID FUELS
Therefore, there only remains the enthalpy flux of the fuel vapours and Eq. (13.31)
reduces to
4πr 2 λ dT
dr − m ∫ T
T 0
c p1 dT = m
(
q l −
∫ Ts
T 0
c p1 dT
)
, (13.32)
which determines the distribution of temperature between the droplet surface and the
flame front.
The integration constant of Eq. (13.32) is determined by expressing that the
value of the temperature on the droplet surface must be T s , that is
r = r s : T = T s . (13.33)
Exterior region r ≥ r l
Here, only oxygen and inert gases exist and their flows are given by Eqs. (13.22) and
(13.23) respectively. Therefore, the energy equation (13.31) takes the form
4πr 2 λ dT ∫ T
dr − m ( )
(1 + ν)cp2 − νc p3 dT =
T
(
0
= m q l − ( ∫ )
) Ts
h 01 + νh 03 − (1 + ν)h 02 − c p1 dT
T 0
(13.34)
Let q r be the heat of reaction per gram of fuel in gas state at the temperature
T 0 . This heat is
q r = h 01 + νh 02 − (1 + ν)h 03 . (13.35)
Substituting Eq. (13.35) into (13.34), the latter takes the form
4πr 2 λ dT ∫ (
T
∫ )
Ts
dr − m c p dT = m q l − q r − c p1 dT , (13.36)
T 0
T 0
where, in short
c p = (1 + ν)c p2 − νc p3 . (13.37)
The constant resulting from the integration of this equation is determined by
expressing that the temperature at great distance from the droplet has the value T ∞
r → r ∞ : T → T ∞ . (13.38)
The continuity of the temperature at the flame imposes an additional condition
T (r − l ) = T (r+ l
), (13.39)
which will be used in determining the position of the flame.

13.8. DIFFUSION EQUATIONS 313
13.8 Diffusion equations
To compute the distribution of the mass fractions of fuel vapours and oxygen, the
equations (13.14) and (13.25) must be integrated. For this v 1 and v 3 must be expressed
as functions of the mixture velocity v and of the mass fractions Y 1 and Y 3 by using the
diffusion equations. Both the interior and exterior region will be studied separately.
Interior region r s ≤ r ≤ r l
The continuity equation 13.14) for the fuel vapour can be written in the form
4πr 2 ρY 1 (v + v d1 ) = m, (13.40)
where v d1 is the diffusion velocity of the fuel vapours through the atmosphere of inert
gases. Fick’s law, 5 gives for this velocity
v d1 = − 1 Y 1
D 12
dY 1
dr , (13.41)
where D 12 is the diffusion coefficient for the fuel vapours and the inert gases.
When Eq. (13.41) is substituted into Eq. (13.40) and Eq. (13.10) is taken into
account, one obtains
4πr 2 ρD 12
dY 1
dr = −m(1 − Y 2) (13.42)
for the determination of Y 1 . The integration constant for this equation is determined
by expressing that the mass fraction of the fuel vapours at the flame is zero
r = r l : Y 1 = 0, (13.43)
Exterior region r ≥ r l
This procedure applied to Eq. (13.25) gives the following equation for the distribution
of oxygen in the exterior region of the flame
4πr 2 ρD 23
dY 3
dr = m(ν + Y 3). (13.44)
The solution to this equation must satisfy the condition that on the flame front mass
fraction of oxygen must be zero,
r = r l : Y 3 = 0, (13.45)
which determines the value for the corresponding integration constant.
5 See chapter 2.

314 CHAPTER 13. COMBUSTION OF LIQUID FUELS
The solution of Eq. (13.44) must satisfy another additional condition. In fact,
the mass fraction of oxygen must tend to the value Y 3∞ corresponding to the composition
of the atmosphere surrounding the droplet at great distance from the same
r → ∞ : Y 3 → Y 3∞ . (13.46)
This equation will be used in determining the burning velocity m of the droplet.
13.9 Combustion velocity of the droplet. Temperature
and position of the flame
From the preceding paragraph it results that the solution of the combustion problem
of a droplet reduces to the following:
1) The integration of the system of differential equations
4πr 2 λ dT ∫ (
T
∫ )
Ts
dr − m c p1 dT =m q l − c p1 dT , (13.32)
T 0
T 0
4πr 2 ρD 12
dY 1
dr = − m(1 − Y 1), (13.42)
for the determination of temperature T and mass fraction Y 1 of the fuel vapours
within the interior region r s ≤ r ≤ r l , with the two boundary conditions
r = r s : T = T s , (13.33)
r = r − l
: Y 1 = 0. (13.43)
2) The integration of the differential equations
4πr 2 λ dT ∫ (
T
dr − m c p dT =m q l − q r −
T 0
with boundary conditions
∫ Ts
T 0
c p1 dT
)
, (13.36)
4πr 2 ρD 23
dY 3
dr =m(ν + Y 3), (13.44)
r →∞ : T →T ∞ , (13.38)
r =r + l
: Y 3 =0, (13.45)
for the determination of temperature and mass fraction Y 3 of oxygen within the
exterior region r ≥ r l .

13.9. COMBUSTION VELOCITY OF THE DROPLET 315
When Y 1 and Y 3 are known, the distribution of inert gases Y 2 is determined by
the following equations
r s ≤ r ≤ r l : Y 2 = 1 − Y 1 ,
r ≥ r 1 : Y 2 = 1 − Y 3 .
(13.47)
The solution of the system must also satisfy conditions
r = r l : T (r − l ) = T (r+ l
), (13.39)
r → ∞ : Y 3 → Y 3∞ . (13.46)
These two additional conditions determine, as aforesaid, the burnt burning velocity m
of the droplet and the position r l of the flame front.
The flame temperature T l is the value for T given by the solution of Eq. (13.32),
or (13.36) since both values are equal for r = r l .
For the integration of these systems the specific enthalpies of the various species
must be known. Therefore, the laws of variation with temperature of the heat capacities
at constant pressure for the said species must be known. Furthermore one must
also know the coefficients of thermal conductivity and diffusion as functions of the
temperature and composition of the mixture. As well as the heat of reaction q r and the
latent heat of evaporation q l of the fuel. Hereinafter it will be assumed that the heat
capacity at constant pressure does not vary with temperature, within the range under
consideration. 6 In such case Eq. (13.32) takes the form
and Eqs. (13.32) and (13.36) reduce respectively to
and
h i = h 0i + c pi (T − T 0 ), (13.48)
4πr 2 λ dT
dr − mc p1T = m(q l − c p1 T s ) (13.49)
4πr 2 λ dT
dr − mc pT = m ( q l − q r − c p T 0 − c p1 (T s − T 0 ) ) . (13.50)
Such is the form in which these equations are used in the following calculations.
As for the transport coefficients we shall only consider the case where λ is
independent from the mixture composition but varies proportionally to the absolute
temperature. That is
6 Goldsmith and Penner in [20] take into consideration the variation of the heat capacity with temperature.
λ
T
= const., (13.51)

316 CHAPTER 13. COMBUSTION OF LIQUID FUELS
where, in general, the value for the constant will vary when passing from the interior
to the exterior region.
Moreover, it is assumed that the following condition is satisfied
λ
ρD ij c pi
= δ i = const., (13.52)
in accordance with the results of they elementary theory of diffusion. Experimental
evidence shows that this condition is approximately satisfied.
13.10 Integration of the equations
Under the assumptions stated in §9 the system of equations (13.49) and (13.42), corresponding
to the interior region, takes the form
4πr 2 λ s
T
T s
dT
dr − mc p1T = m(q l − c p1 T s ), (13.53)
where the values of λ and ρD 12 are expressed as
4πr 2 (ρD 12 ) s
T
T s
dY l
dr = m(1 − Y 1), (13.54)
λ = λ s
T
T s
, (13.55)
ρD 12 = (ρD 12 ) s
T
T s
, (13.56)
as functions of the corresponding values on the droplet surface and of the ratio of absolute
temperatures T/T s . The integration of Eq. (13.53) is straightforward. Making
use of condition (13.33) one obtains
(
1
− 1 r s r = 4πλ (
s T
− 1 + 1 − q ) [
l
ln 1 + c ( )] )
p1T s T
− 1 . (13.57)
mc p1 T s c p1 T s q l T s
The integration of (13.54) is simplified by previously eliminating r between
(13.53) and (13.54). Thus one obtains for as a function of T
( )δ cp1 (T − T s ) + q 1 l
Y 1 = 1 −
, (13.58)
c p1 (T l − T s ) + q l
where conditions (13.43) has been used and according to (13.44) and δ 1 is given by
δ 1 =
λ
ρD 12 c p1
. (13.59)

13.10. INTEGRATION OF THE EQUATIONS 317
Similarly, the system of Eqs. (13.50) and (13.44) corresponding to the exterior
region takes the form
4πr 2 λ ∞
T
T ∞
dT
dr − mc pT = m ( q l − q r − c p T 0 − c p1 (T s − T 0 ) ) , (13.60)
4πr 2 (ρD 23 ) ∞
T
T ∞
dY 3
dr = m(ν + Y 3), (13.61)
where λ and ρD 23 are expressed as functions of ratio T/T ∞ and of their values at
infinity.
Proceeding as for the interior region and with boundary conditions (13.38) and
(13.45), the following solution is obtained for this system
1
r = 4πλ (
∞
1 − T +
q ln q − c )
pT ∞
, (13.62)
mc p T ∞ c p T ∞ q − c p T
( ( ) ) δ q − cp T
Y 3 = ν
− 1 , (13.63)
q − c p T l
where the following abbreviations
q = q r − q l + (c p − c p1 )T 0 + c p1 T s , (13.64)
δ =
are used for simplicity.
λ
ρD 23 c p
=
λ
ρD 23 c p3
c p3
c p
=
δ 3
(1 + ν) c p2
c p3
ν , (13.65)
Taking Eq. (13.46) into (13.63) one obtains
( (q ) ) δ − cp T ∞
Y 3∞ = ν
− 1 . (13.66)
q − c p T l
This expression gives for T l
(
T l = T ∞ 1 + Y ) (
−1/δ
3∞
+ q (
1 − 1 + Y ) ) −1/δ
3∞
. (13.67)
ν c p ν
Consequently, the temperature of the flame is independent from its position
and from the size of the droplet.
When condition (13.39) is expressed making use of Eqs. (13.57) and (13.62)
one obtains
(
1
= 4πλ s
r s mc p1
T l
T s
− 1 +
+ 4πλ (
∞
1 − T l
+
mc p T ∞
(
1 − q ) [
l
ln 1 + c ( )] )
p1T s Tl
− 1
c p1 T s q l T s
q ln q − c )
pT ∞
.
c p T ∞ q − c p T l
(13.68)

318 CHAPTER 13. COMBUSTION OF LIQUID FUELS
From which results for m
m
= 4πλ (
∞
1 − T l
+
q ln q − c )
pT ∞
r s c p T ∞ c p T ∞ q − c p T l
(
+ 4πλ (
s T l
− 1 + 1 − q ) [
l
ln 1 + c ( )] )
p1T s Tl
− 1 .
c p1 T s c p1 T s q l T s
(13.69)
Since the right-hand side of this equation depends only on the physicochemical characteristics
of the process, it results that the burning velocity of the droplet is directly
proportional to its radius. There is a simple explanation to this fact. The evaporated
mass is proportional to the droplet surface, Σ s , and to the heat Q received per unit
surface and per unit time, m ∼ Σ s Q. But, Σ s ∼ rs. 2 Furthermore, Q ∼ 1/r s , since
Q is proportional to the temperature gradient and this gradient is inversely proportional
to the distance from the flame to the droplet surface, which is proportional to
r s . Therefore it results m ∼ rs(1/r 2 s ) = r s .
Let ρ c be the density of the fuel. The droplet mass M is
M = 4 3 πr3 sρ e (13.70)
and its variation with time is
− dM dt
= −4πr 2 sρ e
dr s
dt . (13.71)
When this expression is combined with Eq. (13.69) the following law is obtained
for the time variation of the droplet radius
2r s
dr s
dt
= −k, (13.72)
where
k =
m = 2λ (
∞
1 − T l
+
2πρ e r s ρ e c p T ∞
(
+ 2λ s T l
− 1 +
ρ e c p1 T s
q ln q − c )
pT ∞
c p T ∞ q − c p T l
(
1 − q l
c p1 T s
)
ln
[
1 + c p1T s
q l
is a constant of the process named evaporation constant. 7
( )] ) (13.73)
Tl
− 1
T s
When (13.72) is integrated the following expression is obtained for the law of
variation of the droplet radius as a function of time
7 The evaporation constant used by Godsave is equal to 4k.
r 2 s = r 2 si − kt, (13.74)

13.11. NUMERICAL APPLICATION 319
where r si is the initial radius. This formula is the fundamental result of the present
theory.
The combustion time t c of a droplet of radius r si is, obviously,
t c = r2 si
k . (13.75)
The position r l /r s of the flame front is obtained by writing T = T l in Eq. (13.62).
Furthermore if m/r s is eliminated from the result by means of Eq. (13.73), one obtains
r l
r s
=
1 − T l
T ∞
+
kρ e c p
2λ ∞
q ln q − c . (13.76)
pT ∞
c p T ∞ q − c p T l
Therefore, the radius of the flame and the droplet are proportional.
Formula (13.73) enables the study of the influence of the physical constants of
the fuel and the state of the surrounding atmosphere on the burning rate of the droplet. 8
In particular, the said formula shows that the heat transmitted from the flame to the
droplet surface is the determining factor for its burning velocity. This is due to the
fact that the burning rate of the droplet is determined by the heat received by the same.
Therefore, the essential factor in the combustion process is not the volatility of the fuel
but its heat of evaporation. Experiments results confirm this conclusion. In fuels at the
high temperature reached when combustion occurs, the evaporation mechanism differ
essentially from the evaporation when the temperature of the surrounding atmosphere
is normal. In the last case the evaporation is maintained by the diffusion of the vapors
from the droplet towards the exterior. The evaporation velocity is determined by the
difference between the vapour pressures on the droplet surface and at infinity. 9
13.11 Numerical application
The above formulas are applied here to the study of the combustion of a gasoline
droplet in the air. The following typical values are adopted
T ∞ = 300 K, T s = 355 K, ρ c = 0.85 gr/cm 3 ,
λ ∞ = λ s = 55 × 10 −6 cal/cm s K, q l = 95 cal/gr, q = 10 000 cal/gr,
c p1 = 0.60 cal/gr K, c p2 = 0.35 cal/gr K, c p3 = 0.26 cal/gr K,
Y 3∞ = 0.23, ν = 3, δ 1 = δ 2 = 0.9.
8 Information on the subject can be found in Godsave’s works [19].
9 See §15, Eq. (13.103).

320 CHAPTER 13. COMBUSTION OF LIQUID FUELS
obtained
Taking these values into Eq. (13.67) for the temperature of the flame, it is
The evaporation constant is obtained from Eq. (13.73)
The position of the flame is given by (13.76)
T l = 3112 K. (13.77)
k = 2.26 × 10 −3 cm 2 /s. (13.78)
r l
r s
= 9.48. (13.79)
The distributions of temperature and mass fractions are given by Eqs. (13.57) and
(13.58) for the interior region and by Eqs. (13.62) and (13.63) for the exterior region
Interior region
r
r s
=
Y 1 = 1 −
12.42
13.60 − T − 0.66 ln
T ∞
(
1.89 T
T ∞
− 1.24
), (13.80)
( ( ) ) 0.9
1 T
− 0.56 . (13.81)
9.72 T ∞
Exterior region
r
r s
=
Y 3 = 3 ⎣
10.84
1 − T + 53.76 ln
T ∞
⎡(
1
43.38
(
53.76 − T
T ∞
) ) 0.38
52.76
53.76 − T
T ∞
, (13.82)
⎤
− 1⎦ . (13.83)
The distribution of Y 2 is given in the interior region by
Y 2 = 1 − Y 1 , (13.84)
and in the exterior region by
Y 2 = 1 − Y 3 . (13.85)
The distribution of T/T ∞ , Y 1 , Y 2 and Y 3 have been taken into Figs. 13.2 and 13.3.

13.11. NUMERICAL APPLICATION 321
10
8
T/T ∞
6
4
2
T s
/T ∞
T l
/T ∞
r l
/r s
=9.48
0
0 5 10 15 20 25 30
r/r s
Figure 13.2: Temperature profile given by Eqs. (13.80) and (13.82).
1.0
Y 1s
Y 2
Y 2
0.8
Y 2∞
Y 1
, Y 2
, Y 3
0.6
0.4
Y 1
Y 3
0.2
Y 3∞
Y 2s
r l
/r s
=9.48
0.0
0 5 10 15 20 25 30
r/r s
Figure 13.3: Mass fraction profiles given by Eqs. (13.81) and (13.83)-(13.85).
The combustion time for a droplet of a radius r si millimeters is given by
Eq. (13.75). We obtain
t c = 4.42 rsi 2 s. (13.86)
In Fig. 13.4 the variation law of t c as a function of rsi 2 is represented.

322 CHAPTER 13. COMBUSTION OF LIQUID FUELS
5
4
3
t c
(s)
2
1
0
0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000
r (mm) si
Figure 13.4: Droplet combustion time as a function of the initial size.
13.12 Comparison with experimental results and limitations
of the theory
Godsave [19], [21], Topps [22], Hall and Diederichsen [23], Spalding [24], Hottel
et al.
[25], Wise et al [26], Kobayasi [27], Nishiwaki [28] and others have published
experimental results on the combustion of droplets. Such results have been
obtained from different technical procedures and they confirm the main conclusions
of the present theory. For example, Fig. 13.5 obtained in the I.N.T.A Combustion
Laboratory at Madrid 10 confirms the linear law (13.74) of variation of the square of
the radius with time. The following table 13.1, taken from the work by Goldsmith and
Penner [20], compares theoretical values of k with these experimentally obtained by
Godsave [21] for different fuels. As seen the agreement is generally excellent.
Fuels k × 10 3 cm 2 /s
Theoretical
Experimental
Benzene 2.5 2.43
Ethyl alcohol 1.98 2.03
Ethyl Benzene 2.13 2.15
N-heptane 2.15 2.43
Table 13.1: Comparison between theoretical and experimental results for k.
10 The results have been obtained by applying Godsave’s technique, that is to say, by suspension of a
droplet from a quartz filament and photographing the variation of radius as a function of time.

13.12. COMPARISON WITH EXPERIMENTAL RESULTS AND LIMITATIONS OF THE THEORY 323
0.5
0.4
r 2 s × 102 (cm 2 )
0.3
0.2
0.1
0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
t (s)
Figure 13.5: Experimental results showing the linear dependence between r 2 s and time, obtained
for isooctane in air at ambient pressure.
On the other hand a comparison of the theoretical and experimental radius
of the flame [20] shows that the former is two or three times larger than the latter.
Although formula (13.76) shows that ratio r l /r s is practically independent from
pressure, Wise, Lovell and Wood [26] have experimentally observed that this ratio
increases as pressure decreases. The authors explain theoretically this effects as a
consequence of free convection. Photographic evidence shows that combustion is accompanied
by strong free convection flows due to which the spherical flame appears
only in the lower half of the droplet whilst in the upper half it takes a considerably
elongate shape. The intense formation of carbon observed makes difficult the optical
study of this region. This convection effect may account for the difference between
the theoretical and experimental values of the flame radius making the values of k
predicted by theory agree with the experimental values as previously seen.
Formula (13.73) shows that the burning velocity of a droplet is practically independent
from pressure. In fact, its only influence shows small reduction of latent
heat of evaporation that accompanies and increase in pressure. However, Hall and
Diederichsen [23] have experimentally checked that the burning velocity of a droplet
is approximately proportional to the 4th root of pressure. It has been suggested [20]
that the following factors could account for this effect, aside from the aforementioned
decreases in evaporation heat:

324 CHAPTER 13. COMBUSTION OF LIQUID FUELS
1) Augmentation of the energy transmitted by radiation from the flame to the droplet
which increases very rapidly with pressure.
2) Augmentation of the rate of chemical reactions which could be very significant
if the burning velocity of the droplet would depend, even if only slightly, on the
said reaction rate, contrary to what is postulated in the present theory.
3) Augmentation of the free convection effects due to the increase of the Grashof
number with pressure.
The theoretical study of such effects is very arduous as it implies taking into
account the influence of radiation, finite thickness of the flame and free convection.
So far this study has not been satisfactorily accomplished.
13.13 Influence of convection
The present theory excludes convection effects. Experiments with suspended droplets
show that the influence of free convection is not significant. Less experimental information
is available for the case of forced convection. Spalding [24] has performed
some measurements for very large Reynolds numbers (from 400 to 4 000). His experiments
show that for this range the burning velocity of droplets with forced convection
can be computed by applying the Frössling formula 11 for the evaporation with no
combustion. Spalding’s experiments bring forth the fact that, at least in the analyzed
range, two different combustion states can exist. For convection velocities lower than
a given critical value, a semi-spherical flame surrounds the front part of the droplet,
whilst in the opposite side a long wake forms with a strong formation of carbon. If the
convection velocity is larger than the critical value the front flame extinguishes and
the wake flame remains. It is questionable whether this second state also produces in
the case of very small droplets.
Spalding explains this extinction as follows. Even in the present theory the
thickness of the flame is assumed to be zero, actually it is finite. Now, the convection
activates evaporation and increases the flame thickness in order to burnt the largest
quantity of fuel that must be consumed per second. But such an increase in thickness
is accompanied by a decrease in the maximum temperature of the flame. Since reaction
rate changes very rapidly with temperature if the said decrease is large enough
the reaction is incomplete and combustion extinguishes. The same occurs in diffusion
flames. 12
11 See Eq. (13.106)
12 See chapter 12.
When assuming a reaction rate of the Arrhenius type, Spalding’s calcula-

13.14. COMBUSTION OF FUEL SPRAYS 325
tions demonstrate that the extinction velocity is proportional to the droplet diameter.
Experimental results confirm such prediction. Spalding also arrives to the conclusion
that, even when no convection exists, the flame extinguishes if the diameter of the
droplet is very small.
13.14 Combustion of fuel sprays
The present theory enables the calculation with good approximation of the burning
velocity of an isolated droplet under laboratory conditions. In particular, this theory
allows the study of the influence of the physical characteristics of the fuel and the
state of the surrounding atmosphere on the burning velocity of the droplet. Now these
results must be extended to the study of the combustion of fuel sprays of the type
existing in the combustion chambers of jet engines [29]. Such an extension has not
yet been achieved. In the actual state of knowledge this seems impossible due to the
interaction of the many factors of the process. In fact, for this extension to become
possible it is necessary to know in advance the distribution of droplets in the spray
and the characteristics of the surrounding atmosphere, in a system in which the fuel
is partially in the liquid phase and partially in the gaseous phase as it enters the combustion
zone. It is also necessary to take into account the interaction of droplets and
the influence of the highly turbulent motion of the gas. All this makes the theoretical
study of the problem extremely difficult. For this reason the studies made up to date
are of a highly empirical character. These studies limit themselves either to an analysis
of the influence of some fundamental parameters on the combustion efficiency of a
burner [30], or to obtain general conclusions as for the way in which several variables
can influence the process [24].
13.15 Droplet evaporation
When combustion is absent the evaporation process of a fuel spray is also a very
complicated phenomenon. This problem has only been studied empirically for some
typical cases [16]. Bahr [30], for example, has given an empirical formula to express
under given conditions the evaporated fraction as a function of the distance to the
atomizer. The Bahr formula represents a good approximation to the results obtained
from his own experiments and those performed by Ingebo [16].
As a first step towards the solution of the problem, the evaporation of isolated
droplets can be studied by the same procedure applied in the preceding paragraphs

326 CHAPTER 13. COMBUSTION OF LIQUID FUELS
to the study of combustion. The extrapolation of the obtained values to the case of a
spray is difficult. In fact, it is necessary to know the size distribution of the droplets
and the composition of the surrounding atmosphere. Furthermore, in order to account
for the convection effects, which are very important, the motion of the droplets relative
to the atmosphere must also be known. The available information is not sufficient to
show the way in which this extrapolation can be performed [34].
The evaporation velocity of a droplet when convection is absent can be easily
obtained from the formulas deduced in the preceding paragraphs. In fact, assuming
that the droplet is isothermal, it is enough to make use of the equations corresponding
to the interior region, enlarged to infinity. Therein the following boundary conditions
must be satisfied which determine the integration constants
r → ∞ : T → T ∞ , Y 1 → Y 1∞ , (13.87)
where Y 1∞ is the mass fraction of the fuel vapours at great distance from the droplet.
Moreover, if one keeps the assumptions previously established with respect to the law
of variation of the values of the transport coefficients as functions of temperature, the
two equations for T and Y 1 are Eqs. (13.53) and (13.54), that is
The last equation when applied to the droplet surface r = r s gives the following
relation
4πr 2 λ ∞
T
T ∞
dT
dr − mc p1T = m(q l − c p1 T s ), (13.88)
4πr 2 (ρD 12 ) ∞
T
T ∞
dY 1
dr = −m(1 − Y 1). (13.89)
The solution of this system that satisfies conditions (13.87) is
1
r = 4πλ (
∞
1 − T + q l − c p1 T s
ln c )
p1(T − T s ) + q l
, (13.90)
mc p1 T ∞ c p1 T ∞ c p1 (T ∞ − T s ) + q l
( ) δ
1 − Y 1 cp1 (T − T s ) + q l
=
. (13.91)
1 − Y 1∞ c p1 (T ∞ − T s ) + q l
(
) δ
1 − Y 1s
q l
=
. (13.92)
1 − Y 1∞ c p1 (T ∞ − T s ) + q l
Let p 1s be the partial pressure of the fuel vapour on the droplet surface. Thermodynamics
shows 13 that p 1s is determined by the temperature T s on the droplet surface.
13 See Prigogine, I. and Defay, R.: Chemical Thermodynamics. Longmans Green & Co., 1954, pp. 332
and f.

13.15. DROPLET EVAPORATION 327
4
0.25
P 1
(kg/cm 2 )
3
2
P 1
(kg/cm 2 )
0.20
n−heptane
0.15
0.10
n−exane
0.05
n−octane
0.00
250 275 300 325 350
T (K)
n−heptane
n−octane
n−exane
1
n−decane
n−dodecane
0
250 300 350 400 450 500
T (K)
Figure 13.6: Partial pressure of fuel vapour as a function of temperature.
Therefore a relation exists between p 1s and T s of the form
Figure 13.6 gives Eq. (13.93) for some typical fuels.
f(p 1s , T s ) = 0. (13.93)
Furthermore 14 p 1
p = M a Y
( 1
) , (13.94)
M c Ma
1 + − 1 Y 1
M c
where M c and M a are, respectively, the molar masses of the fuel vapour and of the
gas through which it diffuses.
When (13.94) is particularized on the droplet surface, taking the result into
(13.93), the following relation between Y 1s and T s is obtained
This relation and (13.92) determine Y 1s and T s .
F (Y 1s , T s ) = 0. (13.95)
Once T s is known the evaporation velocity m of the droplet is deduced from
(13.90) by making r = r s and T = T s . Thus obtaining
m = 4πλ (
∞r s
1 − T s
+ q )
l − c p1 T s q l
ln
. (13.96)
c p1 T ∞ c p1 T ∞ c p1 (T ∞ − T s ) + q l
14 See chapter 1.

328 CHAPTER 13. COMBUSTION OF LIQUID FUELS
The parenthesis on the right-hand side of this equation is independent from
r s . Therefore, evaporation velocity, like combustion velocity, is proportional to the
droplet radius.
Making use of Eq. (13.71) a relation similar to (13.74) is obtained for the variation
of r s
r 2 s = r 2 si − k v t, (13.97)
where k v is the evaporation constant, which from (13.96) is given by the expression
k v = 2λ (
∞
1 − T s
+ q )
l − c p1 T s q l
ln
. (13.98)
ρ c c p1 T ∞ c p1 T ∞ c p1 (T ∞ − T s ) + q l
If the following condition is satisfied
a linearization of (13.96) gives for m
A similar linearization of (13.92) gives
c p1 (T s − T ∞ )
q l
≪ 1, (13.99)
m = 4πλ ∞r s
q l
(T ∞ − T s ). (13.100)
c p1 (T ∞ − T s )
q l
= 1 δ
By taking this expression into (13.100), it results for m
or else, as a function of p 1s and p 1∞ by virtue of (13.94),
Y 1s Y 1∞
1 − Y 1∞
. (13.101)
m = 4πr s ρD 12
Y 1s Y 1∞
1 − Y 1∞
, (13.102)
m = 4πr sD 12
R c T ∞
p 1s − p 1∞
p − p 1∞
p, (13.103)
where R c = R/M c is the gas constant for the fuel vapour. Eq. (13.103) is Langmuir’s
evaporation formula valid for small evaporation velocities.
Let
x = c p1(T ∞ − T s )
q l
. (13.104)
If this value is taken into (13.96) and the result divided by (13.100), the following
relation between m and m 0 is obtained
m
=
q [ (
l
1 − 1 − c ) ]
p1T ∞ ln(1 + x)
+ x
m 0 c p1 T ∞ q l
x
(13.105)
where m is the actual evaporation velocity and m 0 is the velocity that would be obtained
if Langmuir’s formula would apply to all cases.

13.15. DROPLET EVAPORATION 329
1
0.9
0.8
0.7
0.6
m/m 0
0.5
0.4
0.3
0.2
0.1
c T /q =2
p1 ∞ l 4 6 8 10
0
0 1 2 3 4 5 6 7 8 9 10
x
Figure 13.7: The ratio m/m 0 as a function of x = c p1(T ∞ − T s)/q l for some values of
c p1T ∞/q l .
Figure 13.7 gives m/m 0 as a function of x, for some values of c p1 T ∞ /q 1 .
When the evaporation velocity is high the fast decrease of the evaporation constant
can be seen due to the transport of enthalpy done by the vapour motion.
The previous formulae do not take into account the influence of convection.
Such effect has been studied by Frössling [32], who obtained the following relation
between the evaporation velocity m c with convection and the evaporation velocity at
rest
m c
m = (
1 + 0.276 Sc 1/3 Re 1/2) . (13.106)
Here Sc = µ/ρD 12 and Re = ρvd s /µ are, respectively, the Schmidt number
and the Reynolds number of the motion, µ is the gas viscosity coefficient, d s the
droplet diameter and v the motion velocity. Ranz and Marshall [33] have experimentally
verified this formula. Fig. 13.8 gives m c /m as a function of Sc 2/3 Re.
Formula (13.105) was obtained from experiments at ambient temperature. Ingebo
[34] gives the following empirical formula which represents with good approximation
the experimental results obtained by him from 30 ◦ C to 500 ◦ C
m c
m = ( 1 + 0.151(Sc Re) 0.6) . (13.107)

330 CHAPTER 13. COMBUSTION OF LIQUID FUELS
24
20
Ingebo (S c
=0.65)
m c
/ m
16
12
Frössling
8
4
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
2/3
S Re × 10
−3
c
Figure 13.8: Experimental relations of Frössling and Ingebo, giving m c/m as a function of
Sc 2/3 Re.
This formula is represented by the dash line in Fig. 13.8. It is seen that the Frössling
formula underestimates the evaporation velocity for the large Reynolds numbers.
Miesse [35] has lately calculated the motion of an evaporating droplet by assuming:
1) The coefficient of the aerodynamic drag of the droplet is inversely proportional
to the Reynolds number (laminar flow).
2) The evaporation velocity of the droplet follows the Frössling law.
Miesse has obtained solutions for the case where the air velocity changes linearly with
distance. Such solutions allow the calculation of the distance covered by the droplet
as a function of the reduction of its diameter and the distance needed for complete
evaporation, etc. His work contains interesting practical applications. However, when
trying to extend his conclusions to the case of sprays the remarks made at the beginning
of this paragraph should be taken into account. On the other hand the possible
interaction of evaporation and aerodynamic drag of the droplet is neglected.
Penner [36] has calculated the evaporation time of a propellant droplet within
the combustion chamber of a rocket by assuming that the droplet is isothermal but its
temperature changes with time. He has estimated the possible influence of the radiation
energy on the evaporation process reaching the conclusion that such influence is
of no significance.

13.16. APPENDIX: APPLICATION OF PROBERT’S METHOD 331
13.16 Appendix: Application of Probert’s method for
the combustion of fuel sprays
Probert 15 has developed a theoretical method for the study of evaporation or combustion
of fuel sprays, under the assumption that evaporation constant is the same for all
droplets. The justification of this assumption and the value that should assigned to the
constant if it is valid, depend on the results the experimental measurements.
After chapter 13 was written, some theoretical works on the application of
Probert’s method have been performed at the I.N.T.A. 16 corresponding to steady burning
as well as to transition from ignition to steady burning and periodic combustion.
In these computations the size distribution function of Mugele-Evans was used with
preference to those of Rosin-Rammler and Nukiyama-Tanasawa, since it allows the
prediction of the sizes with a very good approximation taking into account the maximum
size of the droplets. Let F be the mass fraction of fuel corresponding to droplets
with a diameter smaller than d. Mugele-Evans’ formula gives for F the following
expression
F = 1 2
( (
))
θd
1 + erf ε ln
. (13.108)
d max − d
Here erf is the error integral, d max is the droplet’s maximum diameter and ε and θ
are two parameters characteristic of the distribution. Fig. 13.9 shows some of the
distributions corresponding to typical values for these parameters. It is seen that the
increment of ε increases the uniformity of the spray, whilst when θ increases the mean
diameter of the droplets reduces.
Let G be the volume of fuel, injected to the burner per unit time, and g the
volume of the droplets existing in the burner. It can be verified that g is expressed as a
function of G through formula
where I is given by expression
I =
∫ 1
0
x 4 dx
∫ 1 − x
0
g = εI √ π
G t v , (13.109)
−
e
(
p ) 2
x2 + y
ε ln
1 − p x 2 + y
and t v is the life time of the largest droplets of the spray.
(x 2 + y)
(1 5/2 − √ ) dy, (13.110)
x 2 + y
15 Probert. R.P.: The Influence of Spray Particle Size and Distribution in the Combustion of Oil Droplets.
Philosophical Magazine, February 1946.
16 Millán, G., and Sanz, S.: Analysis of the Combustion Processes in Gas Turbines. Fourth International
Congress of Combustion Engines, Zurich, 1957.

332 CHAPTER 13. COMBUSTION OF LIQUID FUELS
1.0
F
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(0.5,1.5)
(1.5,1.0)
(1.0,1.0)
(1.5,1.5)
(1.0,1.5)
(1.0,0.5)
(0.5,0.5)
(0.5,1.0)
(1.5,0.5)
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
d / d max
Figure 13.9: Droplet size distribution without combustion according to Mugele and Evans
(fraction F of droplets with a diameter smaller than d/d max), for several values
of the parameters (ε , θ).
Magnitude (13.109) is interesting since the combustion intensity of the burner
should be considered inversely proportional to it.
Table 13.2 gives the values for g/G t v for three typical cases corresponding to
θ = 1.
ε 0.5 1 1.5
g
0.129 0.110 0.105
G t v
( )
ds
0.269 0.438 0.895
d max
spray
(
ds
d max
)flame
0.517 0.454 0.401
Table 13.2: Values of g/Gt v for θ = 1 and ε = 0.5, 1, 1.5.
It is seen that when ε increases, that is when the uniformity of the spray increases,
the mass fraction of fuel in the burner decreases. Consequently, it is advantageous
to work with spray as uniform as possible in order to decrease the volume of
the primary zone of the burner.

13.16. APPENDIX: APPLICATION OF PROBERT’S METHOD 333
Combustion changes the droplet size distribution, preserving, obviously the
maximum diameter. Figure 13.10 shows the distribution functions at the flame corresponding
to the three cases studied. For comparison this figure includes the distributions
corresponding to the spray. Table 13.2 also gives Sauter’s diameters for both the
spray and the flame.
1.0
0.9
0.8
F
0.7
0.6
0.5
0.4
0.3
0.2
0.1
ε = 1.5
ε = 0.5
ε = 1.0
Spray
Combustion
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
d / d max
Figure 13.10: Effect of combustion on the droplet size distribution for some values of ε.
References
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USER, Div. Chem. Sci., Nos. 10-11, 1945.
[2] Khudyakov, G. N.: Distribution of Temperature within a Liquid Burning from
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[3] Spalding, B. D.: The Combustion of Liquid Fuels. Fourth Symposium (International)
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[4] Spalding B. D.: The Calculation of Mass Transfer Rates in Absorption, Vaporization,
Condensation and Combustion Processes. Proceedings of the Institution
of Mechanical Engineers, Vol. 168, No. 19, 1954.

334 CHAPTER 13. COMBUSTION OF LIQUID FUELS
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[6] Jones, G. W. et al.: Research on the Flammability Characteristics of Aircraft
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Oxidizing Atmosphere. Jet Propulsion, July-August 1954, pp. 245-251.

13.16. APPENDIX: APPLICATION OF PROBERT’S METHOD 335
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Progress, 1952, pp. 141-46 and 173-180.

336 CHAPTER 13. COMBUSTION OF LIQUID FUELS
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Drops. NACA Tech. Note No. 2368, 1951.
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Motors. Journal of the American Rocket Society, March-April 1953, pp. 85-88.