Page 352 - Advanced thermodynamics for engineers

15.4 FLAMES 341

Heikal et al. (1979) applied a similar approach to engine calculations, and defined the ratio of turbulent
to laminar flame speed, often called the flame speed factor,as

1a 1=2
2 3
0 d
r u [ þ cV
u t bn p
6 7
f f ¼ ¼ 41 þ @ A 5 (15.20)
u [ aP r n
where
a ¼ molecular thermal diffusivity;
P r ¼ Prandtl number;
V p ¼ mean piston speed;
a, b, c, d are all empirical constants.
A flame speed model by Brehod and Newman (1992) is of the type in Eqn (15.18) and gives the
flame speed, u t ,as

1=2
u t r u u 0
¼ 1 þ C 1 e r f =r c (15.21)
u [ r b u [
where
r f ¼ instantaneous flame radius;
r c ¼ a term of the order of the turbulence integral length scale, [;
C ¼ coefficient of the order of unity.
Such an equation contains a number of parameters which should be important in evaluating the
flame speed in the engine, but it also has a number of degrees of freedom in matching the predictions to
experiment. It is difficult to know how ‘universal’ the prediction is likely to be. One of the benefits of
this model is that it accounts for the enhancement in flame speed that is likely to occur as the flame
grows from a small kernel, when r f =r c < 1, to a developed flame, when r f =r c > 1. In the case of the
small flame the exponential term ð1 e r f =r c Þ is approximately zero, resulting in the flame speed being
almost equal to the laminar flame speed. When the flame radius is larger than the length scale the
exponential term approaches unity, and Eqn (15.21) becomes similar to Eqn (15.18).
A very comprehensive model has been proposed by Herweg and Maly (1992), and this is given
below:
1=2
2
h i
U þ u 1=2
( 02 )
u t 1=2 r f
¼ I 0 þ I 0 1 exp
u [ h 2 02 i 1=2 [ I
U þ u
|fflfflfflfflffl{zfflfflfflfflffl} þ u l |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
I: strain
III: size dependent integral
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
II: effective turbulence length scale
factor
0
1=2 5=6
t u
1 exp (15.22)
s 0 u [
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl}
IV: size dependent integral V: fully developed
time scale turbulent
combustion