Page 222 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

198 7 Combustion Process and Air Emission Formation

The fuel vapor resulted from the evaporation is then transported by diffusion and
convention from the surface of the droplet to the surrounding gas phase. The
corresponding convective diffusion can be described as

2
_ m v 1 f v Þ ¼ 4pr q D df v ð7:5Þ
ð
v
dr
where f v is the vapor mass fraction at the droplet surface. D is the diffusivity of the
vapor in the surrounding gas phase.
Integration of Eqs. (7.4) and (7.5), respectively, from the droplet surface r ¼ r s
to r !1 leads to

!

T s T v þ h fg c p c p _ m v
ln ¼ ð7:6Þ
T 1 T v þ h fg c p 4pk r s

1 f v;s 1 _ m v
ln ¼ ð7:7Þ
4pq D
1 f v;1 v r s
where subscript s is the for the droplet surface. T s is the surface temperature of the
droplet. Equation (7.6) gives the evaporation rate as

4pr s k c p T 1 T v Þ þ h fg
ð
_ m v ¼ ln : ð7:8Þ
c p c p T s T v Þ þ h fg
ð
At thermodynamic equilibrium state, the surface temperature is assumed to be
the same as the vaporization temperature, T s T v , then Eq. (7.8) is simplified as

4pr s k c p T 1 T s Þ
ð
_ m v ¼ ln 1 þ : ð7:9Þ
c p h fg
Conservation of mass leads to the relationship between the mass of the droplet
3
m l ¼ q 4p=3ð Þr and the vaporation rate can is _ m v ¼ dm l =dt
l s
d 4
dm l 3 2 dr s
_ m v ¼ ¼ pq r ¼ 4pq r : ð7:10Þ
l s
l s
dt dt 3 dt
Combination of Eqs. (7.9) and (7.10) leads to

ð
4pq r 2 dr s ¼ 4pr s k ln 1 þ c p T 1 T s Þ
l s
dt c p h fg

k c p T 1 T s Þ
ð
r s dr s ¼ ln 1 þ dt: ð7:11Þ
q c p h fg
l