Untitled - Aerobib - Universidad Politécnica de Madrid

158 CHAPTER 6. LAMINAR FLAMES
which through integration, gives
√ ( 2Λ L
1 − ε ≃
1 − θ 0 (1 − θ 0 )(1 + a)
) n 2
√ ∫ 1
λ (1 − θ) n
λ f θ n−δ e −θ 1 − θ
a
θ dθ. (6.97)
This expression, when taken into the denominator of (6.67) after we have neglected
1 − θ, supplies an approximation for dε/ dθ which introduced in (6.95) gives
the desired value for J. The result obtained from this approximation corresponds to
the curves named Kármán 3 of Fig. 6.5 and 6.6.
Tho approximation for J is even better when, after Sendagorta, we calculate
dε/ dθ in (6.95) by using for ε vs θ an approximation of the form
where P is a polynomial of the form
θ
ε = P e −θ 1 − θ
a
θ , (6.98)
P = 1 + a 1 (1 − θ) + a 2 (1 − θ) 2 + · · · + a m (1 − θ) m , (6.99)
whose coefficients must be determined by studying the behavior ε close to θ = 1 in
the differential equation (6.67).
For example, in the case of a first-order reaction (n = 1), it gives for J
J = H 0 + a 1 H 1 + a 2 H 2 + · · · + a m H m , (6.100)
where
H j =
∫ 1
θ i
(1 − θ) j e −θ a
1 − θ
θ dθ, (j = 1, 2, . . . m). (6.101)
This integral may be expressed by a law of recurrences starting from H 0 , which
in turn is expressed through the exponential integral in the following way
being
H 0 =
∫ 1
θ i
e −θ 1 − θ
a
θ dθ
= 1 − θ a e θ a
E 1 (θ a ) − θ i e −θ a
E 1 (θ) =
∫ ∞
θ
1 − θ i
θ i
e −z
z
+ θ a e θ a
E 1
(
θa
θ i
)
,
(6.102)
dz. (6.103)
For example, if in the preceding case one limits approximation (6.99) to the first two
terms,
P = 1 + a 1 (1 − θ), (6.104)