Untitled - Aerobib - Universidad Politécnica de Madrid

12.4. SIMPLIFIED EQUATIONS 293
As for Y 2 , its value is given by expressions
Interior region: Y 2 = 1 − Y 1 . (12.18)
Exterior region: Y 2 = 1 − Y 3 . (12.19)
Diffusion velocities ¯v d1 and ¯v d3 are given by Fick’s law 7
Y 1¯v d1 = −D 12 ∇Y 1 , (12.20)
Y 3¯v d3 = −D 23 ∇Y 3 . (12.21)
On surface Σ f
Y 1 = Y 3 = 0; Y 2 = 1. (12.22)
Furthermore, since Y 1 and Y 3 must diffuse towards the flame in the stoichiometric
ratio from (12.11), (12.20) and (12.21) we have on Σ f
νD 12
∂Y 1
∂¯n i
= D 23
∂Y 3
∂¯n e
. (12.23)
Here ν is the ratio between the masses of oxidizer and fuel needed for complete combustion.
¯n e and ¯n i are normals to Σ f towards the exterior and interior regions respectively.
System of equations (12.16) and (12.17) can be substituted by a single equation
for a new variable Y defined as follows
⎧
⎪⎨ Y 1 in the interior region,
Y =
⎪⎩ − 1 ν Y 3 in the exterior region.
Y must satisfy the following differential equation
(12.24)
ρ(¯v · ∇)Y − ∇ · (ρD∇Ȳ ) = 0, (12.25)
where D takes the value
⎧
⎨
D =
⎩
D 12
D 23
in the interior region,
in the exterior region.
(12.26)
Furthermore, Y is zero on the flame and takes values of opposite sign at both sides of
it. The derivatives of Y at both sides of the flame in normal direction to it must satisfy
condition
7 See Chap. 2.
D 12
∂Y
∂¯n i
= −D 23
∂Y
∂¯n e
. (12.27)