Page 41 - gas transport in porous media

Ho
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Steady One-Dimensional Diffusion-Limited Evaporation
3.4.1
In the absence of externally induced flow (e.g., from soil–vapor extraction and baro-
metric pumping in the vicinity of wells), movement of gas in the subsurface can be
dominated by diffusive transport. Consider the case of one-dimensional evaporation
from the water table to the land surface. We assume that the concentration of water
3
vapor at the surface of the water table, C s [kg/m ], is constant and can be derived
from the ideal gas law:
P s
C s = (3.13)
RT
where P s is the vapor pressure at the liquid surface [Pa], R is the gas constant for the
compound of interest [J/kg · K], and T is the system temperature [K]. Note that the
vapor pressure, P s , depends on the curvature of the liquid–gas interface and can be
less than or greater than the saturated vapor pressure over a flat liquid surface (see
Section 3.3).
If the vapor concentration at the land surface, C ∞ , is also assumed constant, the
steady-state vapor concentration profile between the water table and the land surface
is linear and readily determined from the one-dimensional steady diffusion equation
with constant concentration boundary conditions as follows:
C ∞ − C s
C(y) = y + C s (3.14)
L
where L is the distance between the water table and the land surface. The rate of
evaporation, ˙m e [kg/s], is determined by applying Fick’s Law:
dC C s − C ∞
˙ m e =−DA = DA (3.15)
dy L
where D is the effective vapor diffusion coefficient that accounts for the effects of
liquid saturation, porosity, and tortuosity.

3.4.2 One-Dimensional Radial Solution in Spherical Coordinates
If we assume that the shape of the evaporating liquid is spherical, we can derive
a steady, one-dimensional solution for the radial concentration profile and evapora-
tion rate. Solutions of this form have been presented by Ho (1997) for evaporation
from a pendant water droplet. Analogous conditions can occur if evaporation occurs
spherically from a stagnant liquid zone in porous media surrounded by a region
with a constant initial vapor concentration. For example, a non-aqueous phase liq-
uid (NAPL) may be trapped in a low-permeability lens and is evaporating slowly
to its surroundings. For these scenarios, the steady one-dimensional radial diffusion
equation in spherical coordinates can be written as follows:

d 2 dC
r = 0 (3.16)
dr dr