In some applications of solute transport, the user may want to model solute transport from a well-stirred bath into a tissue. One way to solve this problem is to assume that the bath has an infinite solute content, in which case the bath solute concentration may be prescribed as a boundary condition on the tissue. In that case, the bath does not need to be modeled explicitly.
However, there are times where the user may want to account for the fact that the bath has a finite amount of solute that may get depleted as solute transports into the tissue. In that case, the bath solution must be modeled explicitly and the solute content in the bath should be prescribed as an initial condition for the bath (not a boundary condition on the tissue). If the bath is well stirred (e.g., if there is a stir bar in the bath, or if the tissue and bath are set on an orbital shaker), we expect that the solute concentration in the bath should stay uniform.
To simulate this problem of a well-stirred bath with uniform (but time-varying) solute content, it suffices to set the solute diffusivity in the bath much higher than the diffusivity in the tissue (e.g., 1000x higher), and then analyze the problem on a time-scale relevant to the solute transport in the tissue.
In the attached example (WellStirredBath.feb), a tissue sample (8 mm x 3 mm) is placed in a slightly larger bath (12 mm x 5 mm). Due to symmetry, only one-half of the actual geometry is modeled. The initial solute concentration in the bath is set to 1 mM, whereas the solute concentration in the tissue is initially zero. The solute diffusivity in the tissue is D=0.5e-3 mm^2/s and the free diffusivity is D0=1.0e-3 mm^2/s (this accounts for hindrance by the porous solid matrix). The solute diffusivity in the bath is set to D=D0=1 mm^2/s to simulate well-stirred condition.
The attached movie shows the solute concentration contours as a function of time. Note the following:
1) The bath concentration remains essentially uniform over the entire temporal response.
2) The bath concentration decreases over time as solute transports into the tissue, thereby getting partially depleted from the bath.
3) The solute concentration in the tissue diffuses much more slowly as expected, initially exhibiting a narrow boundary layer near the tissue periphery.
This example illustrates how easy it is to simulate a well-stirred bath containing a finite amount of solute by simply picking a much higher diffusivity than in the tissue. The same principle may be applied for related analyses, such as solute desorption from the tissue into a well-stirred bath (same analysis as above, just invert the initial conditions by setting the tissue initial concentration to 1 mM and the bath initial concentration to 0 mM). If the bath is not stirred, all that needs to be done is set D=D0=1e-3 mm^2/s in the bath (i.e., the true free diffusivity of the solute).
However, there are times where the user may want to account for the fact that the bath has a finite amount of solute that may get depleted as solute transports into the tissue. In that case, the bath solution must be modeled explicitly and the solute content in the bath should be prescribed as an initial condition for the bath (not a boundary condition on the tissue). If the bath is well stirred (e.g., if there is a stir bar in the bath, or if the tissue and bath are set on an orbital shaker), we expect that the solute concentration in the bath should stay uniform.
To simulate this problem of a well-stirred bath with uniform (but time-varying) solute content, it suffices to set the solute diffusivity in the bath much higher than the diffusivity in the tissue (e.g., 1000x higher), and then analyze the problem on a time-scale relevant to the solute transport in the tissue.
In the attached example (WellStirredBath.feb), a tissue sample (8 mm x 3 mm) is placed in a slightly larger bath (12 mm x 5 mm). Due to symmetry, only one-half of the actual geometry is modeled. The initial solute concentration in the bath is set to 1 mM, whereas the solute concentration in the tissue is initially zero. The solute diffusivity in the tissue is D=0.5e-3 mm^2/s and the free diffusivity is D0=1.0e-3 mm^2/s (this accounts for hindrance by the porous solid matrix). The solute diffusivity in the bath is set to D=D0=1 mm^2/s to simulate well-stirred condition.
The attached movie shows the solute concentration contours as a function of time. Note the following:
1) The bath concentration remains essentially uniform over the entire temporal response.
2) The bath concentration decreases over time as solute transports into the tissue, thereby getting partially depleted from the bath.
3) The solute concentration in the tissue diffuses much more slowly as expected, initially exhibiting a narrow boundary layer near the tissue periphery.
This example illustrates how easy it is to simulate a well-stirred bath containing a finite amount of solute by simply picking a much higher diffusivity than in the tissue. The same principle may be applied for related analyses, such as solute desorption from the tissue into a well-stirred bath (same analysis as above, just invert the initial conditions by setting the tissue initial concentration to 1 mM and the bath initial concentration to 0 mM). If the bath is not stirred, all that needs to be done is set D=D0=1e-3 mm^2/s in the bath (i.e., the true free diffusivity of the solute).
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