Untitled - Aerobib - Universidad Politécnica de Madrid

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)