Page 342 - Advanced thermodynamics for engineers

15.4 FLAMES 331

In obtaining Eqn (15.7) the assumption had been made that the Lewis number,

Le ¼ k rc p D ¼ 1;
where k ¼ thermal conductivity, r ¼ density, c p ¼ specific heat at constant pressure and D ¼ mass
diffusivity.
Hence, Le is the ratio between thermal and mass diffusivities, and this obviously has a major effect
on the transport of properties through the reaction zone. The assumption Le ¼ 1 can be removed to
give the following results for first- and second-order reactions.
For first-order reactions

8 9 1=2
! 2
Z T u n R A RT e
< 0 2 E=RT b =
2k b c p b b
u [ ¼ 2 2 (15.8a)
: r c T b n P B E
u p ðT b T u Þ ;
and for second-order reactions
2
2
2 0 2 2
( ! ) 1=2
2k b c Z c u T u n R A RT e E=RT b
p b b
u [ ¼ 3 3 (15.8b)
r c T b n P B E ðT b T u Þ
u p
where Z is the pre-exponential term in the Arrhenius equation and c u is the initial volumetric con-
0
centration of reactants.
Equations (15.8a) and (15.8b) can be simplified to
( 2 ) 1=2 1=2
kc k
u [ z p b c u Ze E=RT b z R z ðaRÞ 1=2 (15.9)
r c 3 r c p
u
u p
Hence the laminar flame speed is proportional to the square root of the product of thermal diffu-
sivity, a, and the rate of reaction, R. Glassman (1986) shows that the flame speed can be written as
( ) 1=2

k b T b T ig m w b
u [ ¼ R : (15.10)
r c p T ig T u r u
u
which is essentially the same as Eqn (15.9), where R ¼ Ze E=RT b . Obviously the laminar flame speed
is very dependent on the temperature of the products, T b , which appears in the rate equation. This
means that the laminar flame speed, u [ , will be higher if the reactants temperature is high, because the
products temperature will also be higher. It can also be shown that u [ fp ðn 2Þ=2 ; where n is the order of
the reaction, and n z 2 for a reaction of hydrocarbon with oxygen. This means that the effect of
pressure on u [ is small. Figure 15.5 (from Lewis and von Elbe (1961)) shows the variation of u [ with
reactant and mixture strength for a number of fundamental ‘fuels’. It can be seen that, in general, the
maximum value of u [ occurs at close to the stoichiometric ratio, except for hydrogen and carbon
monoxide which have slightly more complex reaction kinetics. It is also apparent that the laminar
flame speed is a function both of the reactant and the mixture strength. The effect of the reactant comes
through its molecular weight, m w . This appears in more than one term in Eqn (15.10) because density
and thermal conductivity are both functions of m w . The net effect is that u [ f1=m w : This explains the