Page 42 - gas transport in porous media

Chapter 3: Vapor Transport Processes
35
where
C(r = r o ) = C o (3.17)
and
(3.18)
C(r →∞) = C ∞
where r o is the radius of the evaporating droplet or the radial extent of the evaporating
plume. The solution is given as follows:
r o
C(r) = (C o − C ∞ ) + C ∞ (3.19)
r
The evaporation rate is determined by applying Fick’s law at the edge of the
evaporating surface:

dC
˙ m e =−DA = 4πDr o (C o − C ∞ ) (3.20)
dr
r=r o
The vapor-phase concentrations can be determined from the vapor pressures using
appropriate equations of state, and it should be noted that the vapor concentration
at the edge of the evaporating liquid surface may be greater than or less than the
saturated vapor concentration above a flat liquid surface depending on the radii of
curvature of the evaporating liquid.

3.4.3 Transient One-Dimensional Diffusion-Limited Evaporation
In contrast to the steady solutions provided in the previous sections, transient solutions
are presented here for cases where the boundary of the evaporating source is moving
or when the distance between the source and the ambient concentration boundary is
infinite.
Thegoverningequationfortransientone-dimensionalgasdiffusioninporousmedia
is given as follows:

2
∂C ∂ C
= D (3.21)
∂t y
3
where C is the gas concentration [kg/m ], D is the effective diffusivity that accounts
2
for gas saturation and porosity [m /s], t is time [s], and y is distance [m]. If C(y →
∞, t) = C(y,0) = C ∞ and C(0, t) = C s , the transient concentration profile can be
written as follows (adapted from Carslaw and Jaeger, 1959):

C(y, t) − C s y
= erf √ (3.22)
C ∞ − C s 2 Dt
The evaporation rate is calculated using Fick’s law at y = 0:
DA (C s − C ∞ )
˙ m e (t) = √ (3.23)
πDt