Untitled - Aerobib - Universidad Politécnica de Madrid

13.8. DIFFUSION EQUATIONS 313
13.8 Diffusion equations
To compute the distribution of the mass fractions of fuel vapours and oxygen, the
equations (13.14) and (13.25) must be integrated. For this v 1 and v 3 must be expressed
as functions of the mixture velocity v and of the mass fractions Y 1 and Y 3 by using the
diffusion equations. Both the interior and exterior region will be studied separately.
Interior region r s ≤ r ≤ r l
The continuity equation 13.14) for the fuel vapour can be written in the form
4πr 2 ρY 1 (v + v d1 ) = m, (13.40)
where v d1 is the diffusion velocity of the fuel vapours through the atmosphere of inert
gases. Fick’s law, 5 gives for this velocity
v d1 = − 1 Y 1
D 12
dY 1
dr , (13.41)
where D 12 is the diffusion coefficient for the fuel vapours and the inert gases.
When Eq. (13.41) is substituted into Eq. (13.40) and Eq. (13.10) is taken into
account, one obtains
4πr 2 ρD 12
dY 1
dr = −m(1 − Y 2) (13.42)
for the determination of Y 1 . The integration constant for this equation is determined
by expressing that the mass fraction of the fuel vapours at the flame is zero
r = r l : Y 1 = 0, (13.43)
Exterior region r ≥ r l
This procedure applied to Eq. (13.25) gives the following equation for the distribution
of oxygen in the exterior region of the flame
4πr 2 ρD 23
dY 3
dr = m(ν + Y 3). (13.44)
The solution to this equation must satisfy the condition that on the flame front mass
fraction of oxygen must be zero,
r = r l : Y 3 = 0, (13.45)
which determines the value for the corresponding integration constant.
5 See chapter 2.