ver-1kc-1.md

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 !cite.

This setup describes a diffusion system in which tritium T$_2$ is modeled across a one-dimensional domain split into two enclosures. The total system length is $2.5 \times 10^{-4}$ m, divided into 100 segments. The system operates at a constant temperature of 500 Kelvin. Initial tritium pressures are specified as $10^{5}$ Pa for Enclosure 1 and $10^{-10}$ Pa for Enclosure 2.

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

\begin{equation} \frac{\partial C_1}{\partial t} = \nabla D \nabla C_1, \end{equation} and \begin{equation} \frac{\partial C_2}{\partial t} = \nabla D \nabla C_2, \end{equation}

where $C_1$ and $C_2$ represent the concentration fields in enclosures 1 and 2 respectively, $t$ is the time, and $D$ 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:

\begin{equation} C_1 = K P_2^n = K \left( C_2 RT \right)^n \end{equation}

where $R$ is the ideal gas constant in J/mol/K, $T$ is the temperature in K, $K$ is the solubility, and $n$ is the exponent of the sorption law. For Sieverts' law, $n=0.5$.

Results

We assume that $K = \frac{10}{\sqrt{RT}}$, which is expected to lead to $C_1 = 10 \sqrt{C_2}$ at equilibrium. As illustrated in [ver-1kc-1_comparison_time_k10], the pressure jump maintains a ratio of $\frac{C_1}{\sqrt{C_2}} \approx 10$, which is consistent with the relationship $C_1 = K (RT C_2)^n$ for $K = \frac{10}{\sqrt{RT}}$ and $n=0.5$ The concentration ratio between enclosures 1 and 2 in [ver-1kc-1_concentration_ratio_k10] shows that the results obtained with TMAP8 are consistent with the analytical results derived from the sorption law for $K \sqrt{RT}=10$. As shown in [ver-1kc-1_mass_conservation_k10], mass is conserved between the two enclosures over time, with a variation in mass of only $0.4$ %. This variation in mass can be further minimized by refining the mesh, i.e., increasing the number of segments in the domain.

!media comparison_ver-1kc-1.py image_name=ver-1kc-1_comparison_time_k10.png style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto id=ver-1kc-1_comparison_time_k10 caption=Evolution of species concentration over time governed by Sieverts' law with $K = \frac{10}{\sqrt{RT}}$.

!media comparison_ver-1kc-1.py image_name=ver-1kc-1_concentration_ratio_k10.png style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto id=ver-1kc-1_concentration_ratio_k10 caption=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.

!media comparison_ver-1kc-1.py image_name=ver-1kc-1_mass_conservation_k10.png style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto id=ver-1kc-1_mass_conservation_k10 caption=Total mass conservation across both enclosures over time for $K = \frac{10}{\sqrt{RT}}$.

Input files

!style halign=left The input file for this case can be found at [/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 [/ver-1kc-1/tests].

!bibtex bibliography