Untitled - Aerobib - Universidad Politécnica de Madrid

200 CHAPTER 6. LAMINAR FLAMES
1) Values for β 1 , β 3 and β 2 are obtained from Eqs. (6.305), (6.306) and (6.310).
2) Introducing these values into Eqs. (6.307), (6.308) and (6.312) one obtains X 1 ,
X 2 and X 3 as functions of θ.
3) Substituting the results into Eq. (6.278), the value of f(θ) is obtained.
4) From the numerical or graphical integration of f(θ) between θ 0 and 1, the value
of the integral I, defined in Eq. (6.282), is obtained.
5) The numerical or graphical integration of f(θ) between θ and 1 gives the value
of
∫ 1
θ
f(θ ′ ) dθ ′ .
6) We introduce this value of
∫ 1
θ
f(θ) dθ, the value of f(θ), computed in (6.230),
and the value for X 4 , given by Eq. (6.314), into Eq. (6.283). Then we integrate
Eq. (6.283), either numerically or graphically, between θ 0 and 1 in order to
obtain the value of J.
7) When both the values for I and J are substituted into Eq. (6.284) the value for
√
Λ is obtained.
8) Finally, u 0 is computed from Eq. (6.272).
It has been stated that the influence of J on the value of u 0 is negligible. This
fact allows a considerable simplification of the method since it eliminates steps 5) and
6). By neglecting J in Eq. (6.284), the following simplified equation results
√
Λ =
√q 1 ε 2 1f
2I . (6.315)
Results
The method developed herein has been applied to the computation of the four cases
calculated by von Kármán and Penner for hydrogen-rich flames. The results obtained
are given in Table (6.5), in which are also listed the values obtained by von Kármán
and Penner.
X 3,0 0.55 0.60 0.65 0.70
Von Kármán and Penner 30.6 42.3 34.2 23.6
u 0 calculated from Eq. (6.315) 22.9 33.7 30.5 21.9
u 0 calculated from Eq. (6.284) 23.3 34.2 – –
Table 6.5: Flame velocity (cm/s) for hydrogen-rich H 2-Br 2 mixtures.