# American Institute of Mathematical Sciences

September  2014, 19(7): 2313-2333. doi: 10.3934/dcdsb.2014.19.2313

## Thermomechanics of hydrogen storage in metallic hydrides: Modeling and analysis

 1 Mathematical Institute, Charles University, Sokolovská 83, CZ-186 75 Praha 8 2 Università degli Studi di Roma "Tor Vergata", Dipartimento di Ingegneria Civile e Ingegneria Informatica, Via Politecnico 1, 00133 Roma, Italy

Received  March 2013 Revised  May 2013 Published  August 2014

A thermodynamically consistent mathematical model for hydrogen adsorption in metal hydrides is proposed. Beside hydrogen diffusion, the model accounts for phase transformation accompanied by hysteresis, swelling, temperature and heat transfer, strain, and stress. We prove existence of solutions of the ensuing system of partial differential equations by a carefully-designed, semi-implicit approximation scheme. A generalization for a drift-diffusion of multi-component ionized gas'' is outlined, too.
Citation: Tomáš Roubíček, Giuseppe Tomassetti. Thermomechanics of hydrogen storage in metallic hydrides: Modeling and analysis. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2313-2333. doi: 10.3934/dcdsb.2014.19.2313
##### References:
 [1] L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), 237-258. doi: 10.1006/jfan.1996.3040. [2] E. Bonetti, P. Colli and P. Laurencot, Global existence for a hydrogen storage model with full energy balance, Nonlinear Analysis: Theory, Meth. & Appl., 75 (2012), 3558-3573, URL http://www.sciencedirect.com/science/article/pii/S0362546X120. 0020X. doi: 10.1016/j.na.2012.01.015. [3] E. Bonetti, M. Fremond and C. Lexcellent, Hydrogen storage: Modeling and analytical results, Applied Mathematics and Optimization, 55 (2007), 31-59. doi: 10.1007/s00245-006-0862-5. [4] R. Bowen, Continuum Physics, vol. 3, Acad. Press, New York, 1976. [5] E. Chiodaroli, A dissipative model for hydrogen storage: Existence and regularity results, Mathematical Methods in the Applied Sciences, 34 (2011), 642-669. doi: 10.1002/mma.1390. [6] B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal., 13 (1963), 167-178. doi: 10.1007/BF01262690. [7] S. De Groot and P. Mazur, Non-Equilibrium Thermodynamics, Dover Publ., New York, 1984. [8] J. Divisek, J. Fuhrmann, K. Gärtner and R. Jung, Performance modelling of a direct methanol fuel cell, J. Electrochem. Soc., 150 (2003), A811-A825. doi: 10.1149/1.1572150. [9] W. Dreyer and F. Duderstadt, On the modelling of semi-insulating GaAs including surface tension and bulk stresses, Proc. Royal Soc. A, 464 (2008), 2693-2720. doi: 10.1098/rspa.2007.0205. [10] W. Dreyer, C. Guhlke and R. Huth, Hysteresis in the context of hydrogen storage and lithium-ion batteries, Preprint WIAS No. 1410, Berlin, 2009. [11] P. Edwards, V. Kuznetsov, W. David and N. Brandon, Hydrogen and fuel cells: Towards a sustainable energy future, Energy Policy, 36 (2008), 4356-4362, URL http://www.sciencedirect.com/science/article/pii/S0301421508004503. doi: 10.1016/j.enpol.2008.09.036. [12] E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369. doi: 10.1002/mma.1089. [13] J. Fuhrmann, Mathematical and numerical modeling of flow, transport and reactions in porous structures of electrochemical devices, Simulation of flow in porous media, Radon Ser. Comput. Appl. Math., De Gruyter, Berlin, 12 (2013), 139-164. [14] D. Jou, J. Casas-Vázquez and G. Lebon, Extended Irreversible Thermodynamics, 4th edition, Springer, New York, 2010. doi: 10.1007/978-90-481-3074-0. [15] S. Kjelstrup and D. Bedeaux, Non-equilibrium Thermodynamics of Heterogeneous Systems, World Scientific, Hackensack, NJ, 2008. doi: 10.1142/9789812779144. [16] A. Kulikovsky, Analytical Modelling of Fuel Cells, Elsevier, Amsterdam, 2010. [17] M. Latroche, Structural and thermodynamic properties of metallic hydrides used for energy storage, J. Physics & Chemistry Solids, 65 (2004), 517-522, URL http://www.sciencedirect.com/science/article/pii/S0022369703004402. doi: 10.1016/j.jpcs.2003.08.037. [18] G. Libowitz, Metallic hydrides; fundamental properties and applications, J. Physics & Chemistry Solids, 55 (1994), 1461-1470, URL http://www.scopus.com/inward/record.url?eid=2-s2.0-0028741292 &partnerID=40&md5=d79dac704fa4eb54e7461cf24ad03a23. doi: 10.1016/0022-3697(94)90571-1. [19] F. Luterotti, G. Schimperna and U. Stefanelli, Global solution to a phase field model with irreversible and constrained phase evolution, Quart. Appl. Math., 60 (2002), 301-316. [20] I. Müller, Thermodynamics, Pitman, 1985. [21] P. Podio-Guidugli, T. Roubíček and G. Tomassetti, A thermodynamically consistent theory of the ferro/paramagnetic transition, Arch. Ration. Mech. Anal., 198 (2010), 1057-1094. doi: 10.1007/s00205-010-0349-z. [22] K. Promislow and B. Wetton, PEM fuel cells: a mathematical overview, SIAM J. Appl. Math., 70 (2009), 369-409. doi: 10.1137/080720802. [23] T. Roubíček, Incompressible ionized non-Newtonean fluid mixtures, SIAM J. Math. Anal., 39 (2007), 863-890. doi: 10.1137/060667335. [24] T. Roubíček, Nonlinear Partial Differential Equations with Applications, 2nd edition, Birkhäuser, Basel, 2013. doi: 10.1007/978-3-0348-0513-1. [25] T. Roubíček and G. Tomassetti, Ferromagnets with eddy currents and pinning effects: Their thermodynamics and analysis, Math. Models Methods in Appl. Sciences, 21 (2011), 29-55. doi: 10.1142/S0218202511004976. [26] T. Roubíček and G. Tomassetti, Thermodynamics of shape-memory alloys under electric current, Z. Angew. Math. Mech., 61 (2010), 1-20. doi: 10.1007/s00033-009-0007-1. [27] T. Roubíček and G. Tomassetti, Phase transformation in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis, Arch. Ration. Mech. Anal., 210 (2013), 1-43. doi: 10.1007/s00205-013-0648-2. [28] W. van Roosbroeck, Theory of flow of electrons and holes in germanium and other semiconductors, Bell System Tech. J., 29 (1950), 560-607.

show all references

##### References:
 [1] L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), 237-258. doi: 10.1006/jfan.1996.3040. [2] E. Bonetti, P. Colli and P. Laurencot, Global existence for a hydrogen storage model with full energy balance, Nonlinear Analysis: Theory, Meth. & Appl., 75 (2012), 3558-3573, URL http://www.sciencedirect.com/science/article/pii/S0362546X120. 0020X. doi: 10.1016/j.na.2012.01.015. [3] E. Bonetti, M. Fremond and C. Lexcellent, Hydrogen storage: Modeling and analytical results, Applied Mathematics and Optimization, 55 (2007), 31-59. doi: 10.1007/s00245-006-0862-5. [4] R. Bowen, Continuum Physics, vol. 3, Acad. Press, New York, 1976. [5] E. Chiodaroli, A dissipative model for hydrogen storage: Existence and regularity results, Mathematical Methods in the Applied Sciences, 34 (2011), 642-669. doi: 10.1002/mma.1390. [6] B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal., 13 (1963), 167-178. doi: 10.1007/BF01262690. [7] S. De Groot and P. Mazur, Non-Equilibrium Thermodynamics, Dover Publ., New York, 1984. [8] J. Divisek, J. Fuhrmann, K. Gärtner and R. Jung, Performance modelling of a direct methanol fuel cell, J. Electrochem. Soc., 150 (2003), A811-A825. doi: 10.1149/1.1572150. [9] W. Dreyer and F. Duderstadt, On the modelling of semi-insulating GaAs including surface tension and bulk stresses, Proc. Royal Soc. A, 464 (2008), 2693-2720. doi: 10.1098/rspa.2007.0205. [10] W. Dreyer, C. Guhlke and R. Huth, Hysteresis in the context of hydrogen storage and lithium-ion batteries, Preprint WIAS No. 1410, Berlin, 2009. [11] P. Edwards, V. Kuznetsov, W. David and N. Brandon, Hydrogen and fuel cells: Towards a sustainable energy future, Energy Policy, 36 (2008), 4356-4362, URL http://www.sciencedirect.com/science/article/pii/S0301421508004503. doi: 10.1016/j.enpol.2008.09.036. [12] E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369. doi: 10.1002/mma.1089. [13] J. Fuhrmann, Mathematical and numerical modeling of flow, transport and reactions in porous structures of electrochemical devices, Simulation of flow in porous media, Radon Ser. Comput. Appl. Math., De Gruyter, Berlin, 12 (2013), 139-164. [14] D. Jou, J. Casas-Vázquez and G. Lebon, Extended Irreversible Thermodynamics, 4th edition, Springer, New York, 2010. doi: 10.1007/978-90-481-3074-0. [15] S. Kjelstrup and D. Bedeaux, Non-equilibrium Thermodynamics of Heterogeneous Systems, World Scientific, Hackensack, NJ, 2008. doi: 10.1142/9789812779144. [16] A. Kulikovsky, Analytical Modelling of Fuel Cells, Elsevier, Amsterdam, 2010. [17] M. Latroche, Structural and thermodynamic properties of metallic hydrides used for energy storage, J. Physics & Chemistry Solids, 65 (2004), 517-522, URL http://www.sciencedirect.com/science/article/pii/S0022369703004402. doi: 10.1016/j.jpcs.2003.08.037. [18] G. Libowitz, Metallic hydrides; fundamental properties and applications, J. Physics & Chemistry Solids, 55 (1994), 1461-1470, URL http://www.scopus.com/inward/record.url?eid=2-s2.0-0028741292 &partnerID=40&md5=d79dac704fa4eb54e7461cf24ad03a23. doi: 10.1016/0022-3697(94)90571-1. [19] F. Luterotti, G. Schimperna and U. Stefanelli, Global solution to a phase field model with irreversible and constrained phase evolution, Quart. Appl. Math., 60 (2002), 301-316. [20] I. Müller, Thermodynamics, Pitman, 1985. [21] P. Podio-Guidugli, T. Roubíček and G. Tomassetti, A thermodynamically consistent theory of the ferro/paramagnetic transition, Arch. Ration. Mech. Anal., 198 (2010), 1057-1094. doi: 10.1007/s00205-010-0349-z. [22] K. Promislow and B. Wetton, PEM fuel cells: a mathematical overview, SIAM J. Appl. Math., 70 (2009), 369-409. doi: 10.1137/080720802. [23] T. Roubíček, Incompressible ionized non-Newtonean fluid mixtures, SIAM J. Math. Anal., 39 (2007), 863-890. doi: 10.1137/060667335. [24] T. Roubíček, Nonlinear Partial Differential Equations with Applications, 2nd edition, Birkhäuser, Basel, 2013. doi: 10.1007/978-3-0348-0513-1. [25] T. Roubíček and G. Tomassetti, Ferromagnets with eddy currents and pinning effects: Their thermodynamics and analysis, Math. Models Methods in Appl. Sciences, 21 (2011), 29-55. doi: 10.1142/S0218202511004976. [26] T. Roubíček and G. Tomassetti, Thermodynamics of shape-memory alloys under electric current, Z. Angew. Math. Mech., 61 (2010), 1-20. doi: 10.1007/s00033-009-0007-1. [27] T. Roubíček and G. Tomassetti, Phase transformation in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis, Arch. Ration. Mech. Anal., 210 (2013), 1-43. doi: 10.1007/s00205-013-0648-2. [28] W. van Roosbroeck, Theory of flow of electrons and holes in germanium and other semiconductors, Bell System Tech. J., 29 (1950), 560-607.
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