Page 350 - Advanced Thermodynamics for Engineers, Second Edition

340 CHAPTER 15 COMBUSTION AND FLAMES

15.4.6.1 Flame speed models
The simplest flame speed model, which is used in many simulations, is based on a spherical flame front
propagating from an initiation point. This gives the equation:

dm b
¼ r F f S f (15.15)
u
dt
where dm b /dt is the rate of production of products from the reactants, r u is the density of the un-
burned reactants, F f is the surface area of the flame and S f is theflamespeed. S f may be either the
laminar or turbulent flame speed. This model will result in a flame volume which is a portion of a
sphere travelling out from the spark plug. This modelisconsideredinmoredetailinthe section on
two-zone models in Chapter 16.
15.4.6.1.1 Laminar flame speed
Laminar flame speed can be measured in a number of ways by considering adiabatic combustion of an
air–fuel mixture. Laminar flame speed has the form

a
b
T u p
u [ ¼ u [;0 (15.16)
T 0 p 0
where u [,0 , a and b are constants for a particular fuel and air–fuel ratio. Metgalchi and Keck (1980,
1982) have shown the variation of u [ with equivalence ratio and fuel for a number of
common fuels, see Fig. 15.6. The variation of laminar flame speed with equivalence ratio has the
general form
0:77
u l ðx b Þ¼ u l;x b ¼0 1 2:06x b (15.17)
where x b is the mole fraction of burned gas diluting the charge. Hence, this can be used to estimate the
effect of lean mixtures, and also exhaust gas recirculation.
15.4.6.1.2 Turbulent flame speed
There are a large number of models for relating turbulent flame speed to the laminar one. The simplest
ones are simply the addition of an effect due to turbulence, or an enhancement (multiplying) of the
effect due to turbulence. The former models have the form

u t u 0
¼ 1 þ c (15.18)
u [ u [
whereas the latter are

u t u 0
¼ c : (15.19)
u [ u [
In the first case if the enhancement due to turbulence is higher relative to the laminar flame speed
then the intrinsic effects of the laminar flame speed can be swamped by the turbulence. This means that
it is necessary to re-evaluate the enhancement at different equivalence ratios. The second approach
does retain the intrinsic effects of the actual mixture.