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 & 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.  doi: 10.1006/jfan.1996.3040.  Google Scholar

[2]

E. Bonetti, P. Colli and P. Laurencot, Global existence for a hydrogen storage model with full energy balance,, Nonlinear Analysis: Theory, 75 (2012), 3558.  doi: 10.1016/j.na.2012.01.015.  Google Scholar

[3]

E. Bonetti, M. Fremond and C. Lexcellent, Hydrogen storage: Modeling and analytical results,, Applied Mathematics and Optimization, 55 (2007), 31.  doi: 10.1007/s00245-006-0862-5.  Google Scholar

[4]

R. Bowen, Continuum Physics, vol. 3,, Acad. Press, (1976).   Google Scholar

[5]

E. Chiodaroli, A dissipative model for hydrogen storage: Existence and regularity results,, Mathematical Methods in the Applied Sciences, 34 (2011), 642.  doi: 10.1002/mma.1390.  Google Scholar

[6]

B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity,, Arch. Rational Mech. Anal., 13 (1963), 167.  doi: 10.1007/BF01262690.  Google Scholar

[7]

S. De Groot and P. Mazur, Non-Equilibrium Thermodynamics,, Dover Publ., (1984).   Google Scholar

[8]

J. Divisek, J. Fuhrmann, K. Gärtner and R. Jung, Performance modelling of a direct methanol fuel cell,, J. Electrochem. Soc., 150 (2003).  doi: 10.1149/1.1572150.  Google Scholar

[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.  doi: 10.1098/rspa.2007.0205.  Google Scholar

[10]

W. Dreyer, C. Guhlke and R. Huth, Hysteresis in the context of hydrogen storage and lithium-ion batteries,, Preprint WIAS No. 1410, (1410).   Google Scholar

[11]

P. Edwards, V. Kuznetsov, W. David and N. Brandon, Hydrogen and fuel cells: Towards a sustainable energy future,, Energy Policy, 36 (2008), 4356.  doi: 10.1016/j.enpol.2008.09.036.  Google Scholar

[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.  doi: 10.1002/mma.1089.  Google Scholar

[13]

J. Fuhrmann, Mathematical and numerical modeling of flow, transport and reactions in porous structures of electrochemical devices,, Simulation of flow in porous media, 12 (2013), 139.   Google Scholar

[14]

D. Jou, J. Casas-Vázquez and G. Lebon, Extended Irreversible Thermodynamics,, 4th edition, (2010).  doi: 10.1007/978-90-481-3074-0.  Google Scholar

[15]

S. Kjelstrup and D. Bedeaux, Non-equilibrium Thermodynamics of Heterogeneous Systems,, World Scientific, (2008).  doi: 10.1142/9789812779144.  Google Scholar

[16]

A. Kulikovsky, Analytical Modelling of Fuel Cells,, Elsevier, (2010).   Google Scholar

[17]

M. Latroche, Structural and thermodynamic properties of metallic hydrides used for energy storage,, J. Physics & Chemistry Solids, 65 (2004), 517.  doi: 10.1016/j.jpcs.2003.08.037.  Google Scholar

[18]

G. Libowitz, Metallic hydrides; fundamental properties and applications,, J. Physics & Chemistry Solids, 55 (1994), 1461.  doi: 10.1016/0022-3697(94)90571-1.  Google Scholar

[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.   Google Scholar

[20]

I. Müller, Thermodynamics,, Pitman, (1985).   Google Scholar

[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.  doi: 10.1007/s00205-010-0349-z.  Google Scholar

[22]

K. Promislow and B. Wetton, PEM fuel cells: a mathematical overview,, SIAM J. Appl. Math., 70 (2009), 369.  doi: 10.1137/080720802.  Google Scholar

[23]

T. Roubíček, Incompressible ionized non-Newtonean fluid mixtures,, SIAM J. Math. Anal., 39 (2007), 863.  doi: 10.1137/060667335.  Google Scholar

[24]

T. Roubíček, Nonlinear Partial Differential Equations with Applications,, 2nd edition, (2013).  doi: 10.1007/978-3-0348-0513-1.  Google Scholar

[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.  doi: 10.1142/S0218202511004976.  Google Scholar

[26]

T. Roubíček and G. Tomassetti, Thermodynamics of shape-memory alloys under electric current,, Z. Angew. Math. Mech., 61 (2010), 1.  doi: 10.1007/s00033-009-0007-1.  Google Scholar

[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.  doi: 10.1007/s00205-013-0648-2.  Google Scholar

[28]

W. van Roosbroeck, Theory of flow of electrons and holes in germanium and other semiconductors,, Bell System Tech. J., 29 (1950), 560.   Google Scholar

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.  doi: 10.1006/jfan.1996.3040.  Google Scholar

[2]

E. Bonetti, P. Colli and P. Laurencot, Global existence for a hydrogen storage model with full energy balance,, Nonlinear Analysis: Theory, 75 (2012), 3558.  doi: 10.1016/j.na.2012.01.015.  Google Scholar

[3]

E. Bonetti, M. Fremond and C. Lexcellent, Hydrogen storage: Modeling and analytical results,, Applied Mathematics and Optimization, 55 (2007), 31.  doi: 10.1007/s00245-006-0862-5.  Google Scholar

[4]

R. Bowen, Continuum Physics, vol. 3,, Acad. Press, (1976).   Google Scholar

[5]

E. Chiodaroli, A dissipative model for hydrogen storage: Existence and regularity results,, Mathematical Methods in the Applied Sciences, 34 (2011), 642.  doi: 10.1002/mma.1390.  Google Scholar

[6]

B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity,, Arch. Rational Mech. Anal., 13 (1963), 167.  doi: 10.1007/BF01262690.  Google Scholar

[7]

S. De Groot and P. Mazur, Non-Equilibrium Thermodynamics,, Dover Publ., (1984).   Google Scholar

[8]

J. Divisek, J. Fuhrmann, K. Gärtner and R. Jung, Performance modelling of a direct methanol fuel cell,, J. Electrochem. Soc., 150 (2003).  doi: 10.1149/1.1572150.  Google Scholar

[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.  doi: 10.1098/rspa.2007.0205.  Google Scholar

[10]

W. Dreyer, C. Guhlke and R. Huth, Hysteresis in the context of hydrogen storage and lithium-ion batteries,, Preprint WIAS No. 1410, (1410).   Google Scholar

[11]

P. Edwards, V. Kuznetsov, W. David and N. Brandon, Hydrogen and fuel cells: Towards a sustainable energy future,, Energy Policy, 36 (2008), 4356.  doi: 10.1016/j.enpol.2008.09.036.  Google Scholar

[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.  doi: 10.1002/mma.1089.  Google Scholar

[13]

J. Fuhrmann, Mathematical and numerical modeling of flow, transport and reactions in porous structures of electrochemical devices,, Simulation of flow in porous media, 12 (2013), 139.   Google Scholar

[14]

D. Jou, J. Casas-Vázquez and G. Lebon, Extended Irreversible Thermodynamics,, 4th edition, (2010).  doi: 10.1007/978-90-481-3074-0.  Google Scholar

[15]

S. Kjelstrup and D. Bedeaux, Non-equilibrium Thermodynamics of Heterogeneous Systems,, World Scientific, (2008).  doi: 10.1142/9789812779144.  Google Scholar

[16]

A. Kulikovsky, Analytical Modelling of Fuel Cells,, Elsevier, (2010).   Google Scholar

[17]

M. Latroche, Structural and thermodynamic properties of metallic hydrides used for energy storage,, J. Physics & Chemistry Solids, 65 (2004), 517.  doi: 10.1016/j.jpcs.2003.08.037.  Google Scholar

[18]

G. Libowitz, Metallic hydrides; fundamental properties and applications,, J. Physics & Chemistry Solids, 55 (1994), 1461.  doi: 10.1016/0022-3697(94)90571-1.  Google Scholar

[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.   Google Scholar

[20]

I. Müller, Thermodynamics,, Pitman, (1985).   Google Scholar

[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.  doi: 10.1007/s00205-010-0349-z.  Google Scholar

[22]

K. Promislow and B. Wetton, PEM fuel cells: a mathematical overview,, SIAM J. Appl. Math., 70 (2009), 369.  doi: 10.1137/080720802.  Google Scholar

[23]

T. Roubíček, Incompressible ionized non-Newtonean fluid mixtures,, SIAM J. Math. Anal., 39 (2007), 863.  doi: 10.1137/060667335.  Google Scholar

[24]

T. Roubíček, Nonlinear Partial Differential Equations with Applications,, 2nd edition, (2013).  doi: 10.1007/978-3-0348-0513-1.  Google Scholar

[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.  doi: 10.1142/S0218202511004976.  Google Scholar

[26]

T. Roubíček and G. Tomassetti, Thermodynamics of shape-memory alloys under electric current,, Z. Angew. Math. Mech., 61 (2010), 1.  doi: 10.1007/s00033-009-0007-1.  Google Scholar

[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.  doi: 10.1007/s00205-013-0648-2.  Google Scholar

[28]

W. van Roosbroeck, Theory of flow of electrons and holes in germanium and other semiconductors,, Bell System Tech. J., 29 (1950), 560.   Google Scholar

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