Untitled - Aerobib - Universidad Politécnica de Madrid

156 CHAPTER 6. LAMINAR FLAMES
1.0
A
1
∫ (1−θ) dθ
θ0
0.8
0.6
θ
0.4
(1−θ 0
)/2
0.2
θ 0
=0.125
C
B
0.0
0.0 0.2 0.4 0.6 0.8 1.0
ε
Figure 6.12: Areas representing the values of J = R 1
θ 0
(1 − θ)dε and (1 − θ 0)/2 in a typical
case.
Hence, it results that a first approximation is obtained when J is neglected
respect to (1 − θ 0 )/2, in which case the value of Λ is given by expression
Λ = 1 − θ 0
. (6.90)
2I
Actually, this approximation was introduced by Zeldovich et al., although by
means of a more complicated justification. Furthermore, the authors applied an inadequate
approximation for the value of integral I which is the main reason for the
error resulting from their method as it was shown in Figs. 6.5 and 6.6. Professor
von Kármán, after a more accurate calculation of integral I, as before said, has improved
the approximation obtained by Zeldovich et al. This can be seen in Figs. 6.5
and 6.6 where the curves named Kármán correspond to (6.90) when using for I the
approximation developed in the preceding paragraph.
Von Kármán improved the approximation for J, through an approximation of
1 − θ vs 1 − ε, which is obtained by studying the behavior of the differential equation
(6.67) at the neighborhod of the hot boundary in the following way.
Close to point θ = 1, difference 1 − ε is an infinitesimal, whose order depends
only on the order of 1 − θ and it can be easily derived from (6.67) by introducing into
it the following approximations: