Problem set up

This verification problem models two species 'T' (gaseous tritium) and 'D' (gaseous deuterium) equilibrating in pressure between two enclosures. It recreates the ver-1hb case in Ambrosek and Longhurst (2008). Enclosure 1 is pre-charged with tritium and enclosure 2 is pre-charged with deuterium. The gases flow between the two enclosures until the T and D partial pressures in the two enclosures equilibrate. The pressure change rates for this system for gas T are given by (1) (2) and for gas are given by (3) (4) where is the volumetric flow rate (ms), is the volume (m), is the pressure in enclosure of gas species ( = or ).

We solve these time evolution equations for the T and D pressures in the two enclosures using TMAP8 with the time and with the initial condition set to Pa, and Pa. We use m and ms.

Analytical solution

Mass balance between the two enclosures gives the following relationship between the pressure of species in the two enclosures (5) where is time, and is the initial pressure of the gas in enclosure . By substituting Eq. (5) into Eq. (1) and Eq. (2), we get (6) where . Similarly for D we get (7) where .

Note that the TMAP7 verification case in Ambrosek and Longhurst (2008) provides the initial values of gas pressures, and compares plots of gas pressure, but the analytical solutions use concentration variables. We used pressure variables throughout the derivation here to keep consistent units, but the pressure can be converted to concentration if needed using the ideal gas law as discussed in ver-1ha.

Results and comparison against analytical solution

The comparison of TMAP8 results against the analytical solution is shown in Figure 1 and Figure 2. The match between TMAP8's predictions and the analytical solution of T pressure is satisfactory, with root mean square percentage errors (RMSPE) of 0.1 and 0.13 % for the first and second enclosures respectively. The RMSPE values are flipped for the D calculations, since the initial pressure conditions are also flipped for D compared to T.

Figure 1: Comparison of pressure of species T for the first and second enclosures equilibrating predicted by TMAP8 and provided by the analytical solution. The RMSPE is the root mean square percent error between the analytical solution and TMAP8 predictions. This recreates the verification figure from TMAP7 Ambrosek and Longhurst (2008).

Figure 2: Comparison of pressure of species D for the first and second enclosures equilibrating predicted by TMAP8 and provided by the analytical solution. The RMSPE is the root mean square percent error between the analytical solution and TMAP8 predictions. This recreates the verification figure from TMAP7 Ambrosek and Longhurst (2008).

Input files

References