Untitled - Aerobib - Universidad Politécnica de Madrid

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)