To illustrate the use of diffusive flux links, we consider
a simple case involving intermedia diffusion across an air/water interface. This
particular example file, Cell3_Diffusion.gsm, can be found in the Contaminant
Transport Examples folder in your GoldSim directory (accessed by selecting
File | Open Example... from the main menu).
Suppose that a closed beaker contains 1 liter of water and
1 liter of air. The air-water interface has an area of 20
cm2. 1 mg of a volatile organic compound is injected into the
water, with a Henry's Law constant (the partition coefficient between air and
water) equal to 10 m3/m3. The water and air in the
beaker are gently stirred, so that you can assume that the water and air
portions are well-mixed. You wish to simulate the transfer of the compound
across the air-water interface. In particular, you want to know how long
it takes for the system to reach equilibrium.
Such a transfer process is often modeled by assuming that
the interface can be represented by two boundary layers: one on the water side,
and one on the air side. Because both the air and water are assumed to be
well-mixed, the concentration within the bulk of the air and water (outside of
the boundary layers) is assumed to be uniform. Therefore, the kinetics of
the transfer process are controlled by the diffusion of the organic from the
bulk water through the water boundary layer, the (instantaneous) partitioning
from water to air at the interface, and the diffusion for the organic through
the air boundary layer into the bulk air. For the purposes of this
example, we assume that the water boundary layer has a thickness of 0.1 mm, and
the air boundary layer has a thickness of 5 mm.
Note: Due to the nature of
turbulence, air boundary layer thickness are (perhaps somewhat
counter-intuitively) significantly larger than water boundary layer thicknesses.
Note, however, that due to the fact that the diffusivity of chemicals in air are
on the order of four orders of magnitude greater than in water, unless the
Henry’s Law constant is relatively small, the water boundary layer controls the
process (i.e., the resistance of the air boundary layer is negligible).
That is indeed the case in this example, in which the Henry’s Law constant is
quite large.
To simulate this system, you would do the following:
1. Create a fluid called Air and
assign it an appropriate reference diffusivity value (1E-5 m2/sec)
and define the Henry’s Law constant;
2. Define a Water_Side Cell containing 1 liter
of Water;
3. Define an Air_Side Cell
containing 1 liter of Air;
4. Create a diffusive flux
link between the Water_Side cell and the Air_Side cell.
5. Specify the simulation
settings (i.e., duration and timesteps), and run the model.
The output of this simulation, in the form of time
histories of the concentrations of the organic in both Cells, is shown
below:
Note that it takes about two days for the system to reach
equilibrium. At equilibrium, the concentration in the Air is ten times
higher than the concentration in Water (i.e., exactly equal to the Henry's Law
constant).
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