Numerical Solution Strategies in Permeation Processes

Abstract:

In this work, the strategy for numerical solutions in transport processes is investigated. Permeation problems can be solved analytically or numerically by means of the Finite Difference Method (FDM), while choosing the Euler forward explicit or Euler backwards implicit formalism. The first method is the easiest and most commonly used, while the Euler backwards implicit is not yet well established and needs further development. Hereafter, a possible solution of the Crank-Nicolson algorithm is presented, which makes use of matrix multiplication and inversion, instead of the step-by-step FDM formalism. If one considers the one-dimensional diffusion case, the concentration of the elements can be expressed as a time dependent vector, which also contains the boundary conditions. The numerically stable matrix inversion is performed by the Branch and Bound (B&B) algorithm [2]. Furthermore, the paper will investigate, whether a larger time step can be used for speeding up the simulations. The stability range is investigated by eigenvalue estimation of the Euler forward and Euler backward. In addition, a third solver is considered, referred to as Combined Solver, that is made up of the last two ones. Finally, the Crank-Nicolson solver [9] is investigated. All these results are compared with the analytical solution. The solver stability is analyzed by means of the Steady State Eigenvector (SSEV), a mathematical entity which was developed ad hoc in the present work. In addition, the obtained results will be compared with the analytical solution by Daynes [6,7].

• Periodical: Defect and Diffusion Forum (Volume 413)
• Pages: 29-46
• DOI: https://doi.org/10.4028/www.scientific.net/DDF.413.29
• Online since: December 2021
• Authors: Axel von der Weth*, Daniela Piccioni Koch, Frederik Arbeiter, Till Glage, Dmitry Klimenko, Georg Schlindwein, Volker Pasler, Norbert Prothmann, Kevin Zinn
• Keywords: Frankel Defects, Fusion Research, Gas Release Experiment, Hydrogen Diffusion, Matrix Solver, Partial Differential Equation (PDE)

References

[1] A. von der Weth et al., Numerical analysis of an isovolumetric thermal desorption experiment, Proceeding of Diffusion in Solids and Liquids (DSL) conference, Defect and Diffusion Forum., (Athens, 24.-28.June 2019).

[2] A. von der Weth et al., Permeation Data Analysis Considering a Nonzero Hydrogen Concentration on the Low Pressure Detector Side for a Purged Permeation Experiment, Defect and Diffusion Forum, 2019, 18-19, Vol 391, https://doi.org/10.4028/www.scientific.net/DDF.391.18.

DOI: 10.4028/www.scientific.net/ddf.391.18

[3] L. A. Sedano et al., Intrinsic hydrogen transport constants in the CFC matrix and fibres derived from isovolumetric desorption experiments, Journal of Nuclear Materials 273 (1999) 285±293.

DOI: 10.1016/s0022-3115(99)00055-0

[4] K. Nagatou, M. R. Schulz et al, ANALYTICAL SOLUTION OF A GAS RELEASE PROBLEM CONSIDERING PERMEATION WITH TIME DEPENDENT BOUNDARY CONDITIONS, ICTT (2019).

[5] bwUniCluster, https://www.scc.kit.edu/dienste/bwUniCluster.php.

[6] H. A. Daynes, The Process of Diffusion through a Rubber Membrane, Proc. Roy. Soc. (1920) A97, 286.

[7] J. Crank, The Mathematics of diffusion, Oxford 19752.

[8] H. R. Schwarz, Numerische Mathematik. Stuttgart (1983).

[9] Smith, Numerical Solution of Partial Differential Equation, London (1965).

[10] J. Ulm, Numerisch Lösung gewöhnlicher und partieller Differentialgleichungen, Renningen, (2018).

[11] R. M. Barer, Diffusion In And Through Solids, Cambridge (1941).

[12] W. Jost, Diffusion, Darmstadt, (1957).

[13] H. R. Schwarz, Numerische Mathematik. Stuttgart 20097.

[14] W. Dahmen, A. Reusken, Numerik für Ingenieure und Wissenschaftler, Berlin 20082.

[15] W. G. Bickley, R.S.H.G. Thomson, Matrices Their Meaning and Manipulation London (1964).

[16] T. Glage et al., On the Analytical Inversion of Solver Matrices used in num ericalApproximations for the Diffusion Equation, this conference.

[17] J. Akoka, Rounded [i. E. Bounded] branch and bound methode for mixed integer nonlinear preogramming problems, Masssachusettes Cambridge 1977 (reprint of a work from 1923).

[18] J. S. Wentzel, Elemente der dynamischen Optimierung, Moskau (1964).

[19] F. Weinberg, Branch and Bound: Eine Einführung, Berlin (1973).

[20] D. W. Marqardt et al. Least Squares Analysis of Electron Paramagnetic Resonance Spectra, Jornal of Molecular Spectroscopy 7, 269-279 (1961).

DOI: 10.1016/0022-2852(61)90360-5

[21] W. Hackbusch, Iterative Lösung großer schwachbesetzter Gleichungssysteme, Stuttgart (1993).

DOI: 10.1007/978-3-663-05633-1

[22] H. Wegener, Der Mössbauereffekt, Mannheim 1964².

[23] C. Reuter, Mößbauerspektroskopie an Eisen-Terbium-Schichten, Regensburg (1990).

[24] A. von der Weth, Berechnung statischer axialsymmetrischer Magnetfelder mit der Methode der Randelemente unter Berücksichtgug nichtlinearer Materisleigenschaften, pp.108-112, Frankfurt (1997).

[25] https://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm.