ver-1kc-1

Sieverts’ Law Boundaries with No Volumetric Source

General Case Description

Two enclosures are separated by a membrane that allows diffusion according to Sievert's law, with no volumetric source present. Enclosure 2 has twice the volume of Enclosure 1.

Case Set Up

This verification problem is taken from Ambrosek and Longhurst (2008).

This setup describes a diffusion system in which tritium T is modeled across a one-dimensional domain split into two enclosures. The total system length is m, divided into 100 segments. The system operates at a constant temperature of 500 Kelvin. Initial tritium pressures are specified as Pa for Enclosure 1 and Pa for Enclosure 2.

The diffusion process in each of the two enclosures can be described by

and

where and represent the concentration fields in enclosures 1 and 2 respectively, is the time, and denotes the diffusivity.

This case is similar to the ver-1kb case, with the key difference being that sorption here follows Sieverts' law instead of Henry's law. The concentration in Enclosure 1 is related to the partial pressure and concentration in Enclosure 2 via the interface sorption law:

where is the ideal gas constant in J/mol/K, is the temperature in K, is the solubility, and is the exponent of the sorption law. For Sieverts' law, .

Results

We assume that , which is expected to lead to at equilibrium. As illustrated in Figure 1, the pressure jump maintains a ratio of , which is consistent with the relationship for and The concentration ratio between enclosures 1 and 2 in Figure 2 shows that the results obtained with TMAP8 are consistent with the analytical results derived from the sorption law for . As shown in Figure 3, mass is conserved between the two enclosures over time, with a variation in mass of only %. This variation in mass can be further minimized by refining the mesh, i.e., increasing the number of segments in the domain.

Evolution of species concentration over time governed by Sieverts' law with $K = \frac{10}{\sqrt{RT}}$.

Figure 1: Evolution of species concentration over time governed by Sieverts' law with .

Concentrations ratio between enclosures 1 and 2 at the interface for $K = \frac{10}{\sqrt{RT}}$. This verifies TMAP8's ability to apply Sieverts' law across the interface.

Figure 2: Concentrations ratio between enclosures 1 and 2 at the interface for . This verifies TMAP8's ability to apply Sieverts' law across the interface.

Total mass conservation across both enclosures over time for $K = \frac{10}{\sqrt{RT}}$.

Figure 3: Total mass conservation across both enclosures over time for .

Input files

The input file for this case can be found at (test/tests/ver-1kc-1/ver-1kc-1.i). To limit the computational costs of the test cases, the tests run a version of the file with a coarser mesh and less number of time steps. More information about the changes can be found in the test specification file for this case (test/tests/ver-1kc-1/tests).

References

  1. James Ambrosek and GR Longhurst. Verification and Validation of TMAP7. Technical Report INEEL/EXT-04-01657, Idaho National Engineering and Environmental Laboratory, December 2008.[BibTeX]