Untitled - Aerobib - Universidad Politécnica de Madrid

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.