Untitled - Aerobib - Universidad Politécnica de Madrid

196 CHAPTER 6. LAMINAR FLAMES
This can be done by means of the diffusion relations given in Eqs. (6.264) and (6.265),
the steady-state Eq. (6.246), and Eq. (6.255).
expressed as
Let X 1f and X 3f be the final values for X 1 and X 3 . These variables can be
X 1 = X 1f − α 1 , (6.285)
X 3 = X 3f + α 3 , (6.286)
where α 1 and α 3 are functions of θ which approach zero when θ approaches one.
When Eqs. (6.246), (6.285) and (6.286) are combined with Eq. (6.255), the
following equation is obtained for the determination of X 2
√
k 1 RT f
X 1f − α 1 + X 3f + α 3 + X 2 +
k 5 p θX 2 = 1. (6.287)
The computation that follows holds for hydrogen-rich flames where X 2f =
X 4f = 0 and the following condition is consequently satisfied
X 1f + X 3f = 1. (6.288)
Equation (6.287) can now be written as
√
k 1 RT f
X 2 +
k 5 p θX 2 − (α 1 − α 3 ) = 0. (6.289)
On solving Eq. (6.289), one obtains for X 2
(√
√ )2
k 1
X 2 = θ RT f
+ α 1 − α 3 − 1 k 1
θ RT f
. (6.290)
4k 5 p
2 k 5 p
Von Kármán and Penner [40] give for
where
√
k1
k 5
the expression
√
k1
k 5
= 1.676 e − θ r
θ , (6.291)
θ r =
Finally, combining Eqs. (6.291) and (6.290) it is obtained
22 605
RT f
. (6.292)
X 2 =
(√
g(θ)2 + α 1 − α 3 − g(θ)) 2
, (6.293)