Untitled - Aerobib - Universidad Politécnica de Madrid

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)