Untitled - Aerobib - Universidad Politécnica de Madrid

192 CHAPTER 6. LAMINAR FLAMES
Selection is made of the two equations corresponding to X 1 and X 3 because that
corresponding to X 2 presents some difficulties, as will be seen later on. Thus
and
dX 1
dx
dX 3
dx
= RT 4∑
(
)
f
p
θ 1 ε j ε 1
X 1 − X j , (6.261)
D
j=1 1j M j M 1
= RT 4∑
(
)
f
p
θ 1 ε j ε 3
X 3 − X j . (6.262)
D
j=1 3j M j M 3
Hence the system of the flame equations has been reduced to the five relations
given in Eqs. (6.229), (6.246), (6.251), (6.254) and (6.255) and to the four
differential equations (6.250), (6.256), (6.261) and (6.262) for the nine unknowns ε i
(i = 1, 2, 3, 4), X i (i = 1, 2, 3, 4) and θ. If, furthermore, the differential equations
(6.250), (6.261) and (6.262) are divided by Eq. (6.256), the coordinate x is eliminated
and θ is introduced as independent variable. The results are
dε 1
dθ = λ ( ) 3/2
√
p
m 2 2M 1 θ −3/2 k1
k 2
c p RT f k 5
(
X 3 X 3/2
2 X 2 + k )
4
X 1 − 1
k 3
·
θ − 1 + q 1 (ε 1f − ε 1 ) − q 4 (ε 4f − ε 4 ) . (6.263)
dX 1
dθ
dX 3
dθ
(
)
4∑ 1 ε j ε 1
= λ
X 1 − X j
RT f
mc p p θ j=1D 1j M j M 1
θ − 1 + q 1 (ε 1f − ε 1 ) − q 4 (ε 4f − ε 4 ) , (6.264)
(
)
4∑ 1 ε j ε 3
= λ
X 3 − X j
RT f
mc p p θ j=1D 3j M j M 3
θ − 1 + q 1 (ε 1f − ε 1 ) − q 4 (ε 4f − ε 4 ) . (6.265)
Solution of the reaction equations
Von Kármán and Penner [40] give the following expressions for k 2
√
k1
k 5
and k 4
k 3
and
where
k 2
√
k1
k 5
= 0.8 × 10 14 e −θ 1/θ , (6.266)
k 4
= 1 = 0.119, (6.267)
k 3 8.4
θ a =
40 200
RT f
(6.268)