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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.