Case Description

This validation problem is taken from Ambrosek and Longhurst (2008), which utilized experimental data from Kizu et al. (2001). Kizu et al. describe permeation experiments in which H and D permeate through thin Pd membranes. This case is part of the validation suite of TMAP7 as val-2e (Ambrosek and Longhurst, 2008).

The experimental equipment has two vacuum chambers separated by a Pd membrane. The membrane area is 1.8 m and has thicknesses of either 0.025 mm or 0.05 mm, depending on the test. Copper gaskets hold the membrane in place and it is reasonably assumed that gas can only move from one chamber to another by diffusing through the membrane. The membrane's temperature is controlled between 820 and 870 K using an electric heater and a thermocouple. Gas is supplied to one chamber (the upstream chamber) from regulated bottles at different compositions and pressures. The base pressure on both upstream and downstream chambers is maintained below 1 Pa using turbomolecular and rotary pumps. Pressure is measured by an ion gage on each side, and downstream gas composition is measured with a quadrupole mass spectrometer. Flow rates through the membrane are determined by pressure increases in the downstream chamber at a fixed pumping rate of 0.1 m/s.

The structure of the system is shown in Figure 1. Enclosure 1 is the source of background pressure to the experimental system. Enclosure 4 is the vacuum pumping system that provides a sink for all system flows. Enclosure 5 is the gas feed to the upstream experimental chamber, enclosure 2. The pressures in enclosures 1, 4, and 5 are held constant or as a functional value while the pressures in enclosures 2 and 3 evolve with the flow and diffusion. Depending on the experiment, the feed pressure of H is 0, 0.14, or 0.063 Pa. Combined with the evacuation to enclosure 4, this provides the upstream H pressure for permeation. The D pressure is a stepped function of time, one step corresponding to each of the data points from Kizu et al. (2001). Steps are arbitrarily set at 100 s, but equilibrium is achieved in times much shorter than that. Effectively no HD is fed into the upstream experimental chamber, in keeping with the experimental setup given by Kizu et al. (2001). Rather, with either solution-law or recombination limited-boundary conditions for diffusion, HD is formed in accordance with the laws of chemical equilibrium. Likewise in the downstream chamber, enclosure 3, HD is formed together with H and D in chemical equilibrium from diffusing H and D.

Figure 1: Schematic of the modeled enclosures for val-2e.

The first three test cases are permeation tests of D alone through membranes of each thickness. For the 0.025 membrane, tests are conducted at both 825 K and 865 K whereas the 0.05 mm membrane is tested only at 825 K.

Table 1 described the different cases subcases being modeled in this validation effort.

Model Description

In the first three cases (val-2ea, val-2eb, and val-2ec), TMAP8 simulates a one-dimensional domain to represent the deuterium diffusion and gas flowing in the enclosures. Note that this case can easily be extended to a two- or three-dimensional case.

This model considers the diffusion of deuterium on both sides of the Pd membrane. First, the diffusion of deuterium in the membrane is described as:

(1)

where is the concentration of hydrogen isotope in membrane, represents hydrogen isotope H or D, is the time, is the diffusivity of hydrogen isotope .

The flow of gas into enclosures 2 and 3 can be given by

(2)

where is the volumetric flow rate of always 0.1 m/s, is the volume of current enclosure , and are the pressure of gas molecules in the previous enclosure and current enclosure , gas molecules represents H, or D, or HD.

The hydrogen isotopes recombine into gas on both sides of the membrane, and there are several ways to model this process, two of which are used in this validation case: (1) Assuming steady state at the surface, also called lawdep boundary conditions in TMAP4 and TMAP7 (Ambrosek and Longhurst, 2008), and (2) capturing the kinetics of dissociation and recombination, also called ratedep conditions in TMAP4 and TMAP7 (Ambrosek and Longhurst, 2008).

By assuming that the kinetics of the dissociation and recombination processes are faster than diffusion, surface reactions can be assumed to be at steady state. The concentrations on both sides can then be described as

(3)

where is the solubility of hydrogen isotope in the membrane, and is the exponent for the relation of pressure and concentration.

However, for the kinetics of surface reactions to be captured, the boundary conditions is set as

(4)

where is the recombination flux of hydrogen isotope out of the sample sides in enclosure , is the area on either side of the membrane, and and are the recombination and dissociation coefficients, respectively. The coefficient of 2 accounts for the fact that 2 atoms (D and H) form one isotope molecule (HD).

The two co-permeation simulations, i.e., val-2ed and val-2ec, also include the chemical reactions in the upstream and downstream enclosure. The reaction rates for H, D, and HD are described by

(5)

for lawdep boundary condition and

(6)

for ratedep boundary condition, where is the Boltzmann’s constant, is the surface area of the membrane, are the temperature.

Case and Model Parameters

The pressure history of deuterium on the enclosure 5 for first three simulations (val-2ea, val-2eb, and val-2ec) is presented in Table 2, as shown in Figure 2. The initial pressures on other enclosures are presented in Table 3.

Figure 2: The pressure history of D in enclosure 5 for simulations val-2ea, val-2eb, and val-2ec.

The pressure history of D on the enclosure 5 for next two simulations (val-2ed and val-2ee) is presented in Table 4, as shown in Figure 3. The initial pressures on other enclosures are presented in Table 5.

Figure 3: The pressure history of D in enclosure 5 for simulations val-2ed and val-2ee.

Other case and model parameters used in TMAP8 are listed in Table 6:

Results

Figure 4 shows the comparison of the TMAP8 calculation and the experimental data from Kizu et al. (2001), including simulations val-2ea, val-2eb, and val-2ec. There are reasonable agreements between the TMAP predictions and the experimental data with root mean square percentage errors (RMSPE) of RMSPE = 23.51 %, 30.80 %, and 47.70 % for val-2ea, val-2eb, and val-2ec, respectively.

Figure 4: Comparison of the D flux versus D pressure on the upstream side of the membrane from TMAP8 calculation and the experimental data.

Figure 5 shows the comparison of the D, H, and HD flux versus effective D pressure on the upstream side of the membrane from val-2ed using TMAP8 assuming steady state for surface reactions (i.e., lawdep condition) and the experimental data. The effective D pressure is calculated using . Although the total gas release is well captured, the release of individual gases are not well predicted, except for H at low pressure, and D at high pressure. In this scenario, RMSPE = 76.26 %, 100.83 %, 115.79 %, 10.77 % for H, D, HD, and the summation of all gases, respectively.

Figure 5: Comparison of the D, H, and HD flux versus effective D pressure on the upstream side of the membrane from TMAP8 calculation using steady state surface condition (i.e., lawdep) for case val-2ed and the experimental data.

Figure 6 shows the comparison of the D, H, and HD flux versus effective D pressure on the upstream side of the membrane from val-2ee using TMAP8 accounting for dissociation and recombination at the membrane's surface (i.e., ratedep condition) and the experimental data. There is a reasonable agreement between the TMAP8 predictions and the experimental data, except for the high pressure conditions greater than 0.1 Pa and for HD, especially at higher pressures. In this scenario, RMSPE = 11.34 %, 30.70 %, 79.72 %, 43.99 % for H, D, HD, and the summation of all gases, respectively. Note that the agreement could be improved by adjusting the model parameters. It is also possible to perform this optimization with MOOSE's stochastic tools module.

Figure 6: Comparison of the D, H, and HD flux versus effective D pressure on the upstream side of the membrane from TMAP8 calculation capturing dissociation and recombination at the surface (i.e., ratedep condition) for case val-2ee and the experimental data.

Input files

References