ver-1c

Diffusion Problem with Partially Preloaded Slab

This verification problem is taken from Longhurst et al. (1992) and Ambrosek and Longhurst (2008), and it has been updated and extended in Simon et al. (2025). Diffusion of tritium through a semi-infinite SiC layer is modeled with an initial loading of 1 atom/m in the first 10 m of a 100 m slab. TMAP4 uses a slab length of 2275 m (the slab length is not specified in the TMAP7 document Ambrosek and Longhurst (2008)); however, using a smaller slab length was found not to change the results. Additionally, the smaller domain size allows getting a finer simulation mesh for the same computational cost, which improves the agreement between the TMAP8 and analytical calculations.

Diffusivity is set to 1 m/s and no trapping is included. The boundary condition on the left-hand side of the slab (at m) is different for the TMAP4 and TMAP7 cases. For TMAP4, an insulating boundary condition is assumed, and the analytical solution is given by (Carslaw and Jaeger, 1959):

(1)

but for TMAP7, which specifies , the analytical solution is (Carslaw and Jaeger, 1959)

(2)

where m is the thickness of the pre-loaded portion of the layer, is the initial concentration in the pre-loaded section of the slab, is the error function, is the position along the slab, and is the diffusivity.

warningwarning:Typo in Longhurst et al. (1992)

The value of found in Longhurst et al. (1992) has a typographical error, should be at the denominator. Eq. (1) follows the form of Carslaw and Jaeger (1959).

TMAP4 and TMAP7 verification cases are also evaluated at slightly different locations: TMAP4 verifies the mobile species concentration at three points:

  1. a point at the free surface ( m)

  2. a point at the end of the pre-loaded region ( m)

  3. a point in the initially unloaded region ( m)

The comparison of the values calculated with TMAP8 and analytically for the TMAP4 cases is shown in Figure 1. TMAP7 verifies the mobile species concentration at similar points

  1. a point near the free surface ( m), as the concentration at the surface is specified

  2. a point at the end of the pre-loaded region ( m)

  3. a point in the initially unloaded region ( m)

The comparison of the values calculated with TMAP8 and analytically for the TMAP7 cases is shown in Figure 2. In all cases, the TMAP8 calculations are found to be in good agreement with the analytical solution.

Figure 1: Comparison of concentration as a function of time at m, 10 m, and 12 m calculated with TMAP8 and analytically (TMAP4 cases)

Figure 2: Comparison of concentration as a function of time at m, 10 m, and 12 m calculated with TMAP8 and analytically (TMAP7 cases)

Input files

The input file for this case can be found at (test/tests/ver-1c/ver-1c.i), which is also used as test in TMAP8 at (test/tests/ver-1c/tests). The TMAP4 and TMAP7 verification tests use the same input file, but different command line arguments for TMAP4.

    cli_args = 'BCs/lhs/type=NeumannBC Outputs/file_base=ver-1c_tmap4'
(test/tests/ver-1c/tests)

and TMAP7

    cli_args = 'BCs/lhs/type=DirichletBC Outputs/file_base=ver-1c_tmap7'
(test/tests/ver-1c/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]
  2. H. S. Carslaw and J. C. Jaeger. Conduction of Heat in Solids. 2nd Edition, Oxford University Press, 1959.[BibTeX]
  3. GR Longhurst, SL Harms, ES Marwil, and BG Miller. Verification and Validation of TMAP4. Technical Report EGG-FSP-10347, Idaho National Engineering Laboratory, Idaho Falls, ID (United States), 1992.[BibTeX]
  4. Pierre-Clément A. Simon, Casey T. Icenhour, Gyanender Singh, Alexander D Lindsay, Chaitanya Vivek Bhave, Lin Yang, Adriaan Anthony Riet, Yifeng Che, Paul Humrickhouse, Masashi Shimada, and Pattrick Calderoni. MOOSE-based tritium migration analysis program, version 8 (TMAP8) for advanced open-source tritium transport and fuel cycle modeling. Fusion Engineering and Design, 214:114874, May 2025. doi:10.1016/j.fusengdes.2025.114874.[BibTeX]