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Numerical investigation of heat and mass transfer in a metal hydride bed
Applied Mathematics and Computation 150 (2004) 169–180 www.elsevier.com/locate/amc
Numerical investigation of heat and mass transfer in a metal hydride bed Abdulkadir Dogan a,*, Yuksel Kaplan b, T. Nejat Veziroglu c a
Department of Mathematics, Nigde University, Fen-Edebiyat Fakultesi, 51100 Nigde, Turkey b Mechanical Engineering Department, Nigde University, 51100 Nigde, Turkey c Clean Energy Research Institute, University of Miami, Coral Gables, FL 33124, USA
Abstract This paper presents a mathematical model for hydrogen storage in a metal hydride bed. For this purpose, a two-dimensional mathematical model which considers complex heat and mass transfer during the hydriding process is developed. The coupled differential equations are solved with a numerical method based on integrations of governing equation over finite control volumes. The driving force is considered to be pressure difference because of the temperature distribution in the system. The numerical results showed that the hydriding performance depends on the temperature distribution in the hydride bed. Fluid flow enhances the hydriding rate in the system by driving the hot fluid to the colder regions. The numerical results were found to agree satisfactorily with the experimental data available in the literature. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Hydrogen storage; Mathematical modeling; Metal hydride; Chemical reaction; Numerical solution
1. Introduction Over the last decade world wide interest in the use of hydrogen has led to much research interest on its storage and usage. Recently, many
*
Corresponding author. E-mail address: [email protected] (A. Dogan).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00207-8
170
A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
Nomenclature Ca material constant (s1 ) Cp specific heat (J kg1 K1 ) Ea activation energy (J mol1 ) h heat transfer coefficient (W m2 K1 ) H/M hydrogen and metal atomic ratio m_ hydrogen mass absorbed (kg m3 s1 ) M molecular weight (kg mol1 ) P pressure (pa) r radial coordinate (m) R universal gas constant (J mol1 K1 ) t time (s) T temperature (K) z axial coordinate (m) Greek letters DH 0 reaction heat of formation (J kg1 ) e porosity k thermal conductivity (W m1 K1 ) q density (kg m3 ) / generic variable that representing the variable solved (i.e., u, v, T ) C exchange coefficient Subscripts e effective eq equilibrium f cooling fluid g gas 0 initial s solid ss saturated
mathematical and experimental works have concentrated on metal hydride beds to be used for hydrogen storage medium. Developments in metal hydride technology show that metal hydrides provide opportunity for hydrogen storage to a high standard of safety both for mobile and for stationary applications. Hydrogen is an interesting and very effective fuel as well as a notable medium for energy storage. It is a natural fuel and the operation of it has no serious problems. It is present in water virtually unlimited quantities and can be produced from plentiful raw materials without difficulty. Taking the world energy and environmental problems into consideration, use of hydrogen as an
A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
171
alternative fuel instead of fossil fuels seems to be best alternative for near future. The hydrogen energy meets all the requirements of an alternative energy such as plentiful availability, comparable cost, versatility of use and social acceptability. It is a reliable fuel that can be created in either from the electrolysis of water or closed chemical reactions. There are several methods for hydrogen storage. Recently, storing hydrogen in the metal hydride bed received many research interests due to superiorities of this method over the conventional ones. Nevertheless, the efficient use of metal hydride beds for proficient operations in both the charging and discharging requires analysis of the complex mass and heat transfer procedure inside the hydride bed. Simultaneous heat and mass transfer takes place with chemical reaction therefore hydriding or dehydriding technique is quite complicated. The gas motion and changing physical properties also adds additional difficulty to the problem. So a lot of experimental and numerical works exist on the diverse viewpoints of the metal hydrides. Most of numerical studies in the earlier work presume that the hydriding or dehyriding techniques are a transient heat conduction problem with an interior heat source and variable physical parameters. Fisher and Watson [1] developed mathematical models to predict the performance of metal hydride beds for hydrogen storage. The relative importance of heat and mass transfer, chemical kinetics and equilibrium have been calculated by comparing the models with experimental data. An equilibrium model without empirical parameters produces bed pressure and temperatures that show good agreement with experimental data. Lucas and Richards [2] developed a detailed mathematical model for metal hydride beds to describe its behavior under both hydriding and dehydring conditions. This model provides a convenient means of predicting the time taken to release or absorb given amounts of hydrogen. These are calculated from the heat transfer characteristics and diffusion properties of particular metal alloys. Selvam et al. [3] addressed the question of using magnesium and magnesium based alloys for rechargeable hydrogen storage media. The effect of addition of organic compounds as well as other metals on the sorption characteristics of magnesium is considered in detail. The current status of information in this field is reviewed considering the potential of these materials for use as hydrogen storage in vehicular applications. The energy transport equations were solved by Jemni and Ben Nasrallah [4] for gas and solid phase and made a comparison the results with one temperature model which considers the gas and solid phases a mixture. They have demonstrated that solving problem with the one temperature model does not significantly affect the result and also demonstrated that convection is not significant in the hydriding processes.
172
A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
Jemni and Nasrallah [5] presented an experimental and theoretical study of a metal hydrogen reactor. Aim of this study was to determine experimentally the effective thermal conductivity, and to test validity of the theoretical model comparing theoretical and experimental results. They have shown that the theoretical results are agreed satisfactorily with experimental data. Fedorov et al. [6] developed a mathematical model and set of software for metal hydride heat pump operations. The set of equations includes energy conservation in hydride sorbers considering heat source released during the chemical reaction. Nakagawa et al. [7] considered the hydriding techniques with a two dimensional model and obtained that the convection at first improved the hydriding process and exhibited an adverse effect at late times. More recently Mat and Kaplan [8] numerically examined the absorption in a metal hydride bed using a mathematical model similar to the Jemni et al.Õs [5] work. The aim of this work is to adapt Jemni and Ben NasrallahÕs mathematical model and put in application it to experimental studies usable in the earlier work and study the primary mechanisms of hydride formation and heat and mass transfer which take position in the hydride bed.
2. Mathematical model Metal hydride formation is considered in a two-dimensional metal hydride reactor shown in Fig. 1. The system is consisted of a cylindrical tank with cooling system. The hydrogen is charged from radially from the center and outside. The metal hydride system is sandwiched between two cooling regions. Due to the symmetric distribution of metal hydride regions only one of them is considered (Fig. 1b). The equations governing the heat and mass transfer and chemical reaction within the hydrogen storage beds are given as follows. 2.1. Energy equation Assuming thermal equilibrium between the storage bed and hydrogen a continuum energy equation is solved instead of separate equations for both solid and gaseous phases. oT 1 o oT o oT 0 _ ðqCpg Þ ¼ rke ke T ðCpg Cps Þ ð1Þ þ m½DH ot r or or oz oz where q, Cp , ke are effective density, specific heat and thermal conductivity respectively. ðqCp Þe is calculated as ðqCp Þe ¼ ðeqg Cpg þ ð1 eÞqs Cps Þ
ð2Þ
A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
173
Fig. 1. The schematic sketch of system considered: (a) metal hydride tank and (b) solution domain.
where, e is the porosity. The effective thermal conductivity is expressed as ke ¼ ekg þ ð1 eÞks
ð3Þ
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A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
Where kg and ks are the thermal effective conductivity of the gas and solid phases respectively. 2.2. Hydrogen mass balance
e
oðqg Þ ¼ m_ ot
ð4Þ
2.3. Mass balance for solid phase Similar to hydrogen mass balance equation excluding the advection term, mass balance for the solid phase can be expressed as: ð1 eÞ
oðqs Þ ¼ m_ ot
ð5Þ
2.4. Reaction kinetic The amount of hydrogen absorbed is directly linked to reaction-rate which is expressed as [4], Ea Pg m_ ¼ Ca exp ð6Þ ln ðqss qs Þ RTS Peq where Ca material is depended constant, qss is the density of the solid phase at saturation and Peq is the equilibrium pressure calculated using the vanÕt Hoff relationship; ln Peq ¼ A
B T
ð7Þ
where A and B are materials constants deduced from the experimental work of Jemni et al. [5] as A ¼ 100:75 and B ¼ 31 896:25. 2.5. Initial and boundary conditions Initially hydride bed is assumed to have constant temperature and pressure and hydrogen is assumed to be at rest. These conditions can be expressed mathematically as at t ¼ 0
P ¼ P0
T ¼ T0
ð8Þ
The boundary walls are assumed to be impermeable and no slip conditions are valid at the boundary walls. The reaction heat is removed from the boundary
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175
Table 1 Thermophysical properties of materials and data used in computations. Density, q (kg m3 ) Specific heat, Cp (J kg1 K1 ) Thermal conductivity, k (Wm1 K1 ) Inlet temperature of gas, T0 Initial temperature of bed and wall, Tf Heat transfer coefficient, h (Wm2 K1 )
Metal (LaNi5 )
Hydrogen
4200 419 1.2 283 K 283 K 1652
0.0838 14 890 0.12
walls with a cooling fluid whose temperature is Tf . Hydrogen is charged from sides with constant pressure, P0 and constant temperature, T0 . The boundary conditions can be expressed as oT ðri ; z; tÞ ¼ h1 ðT T0 Þ or
at r ¼ ri
k
at z ¼ 0
oT ðr; 0; tÞ ¼ 0 oz oT ðr0 ; z; tÞ ¼ h1 ðT T0 Þ or
at r ¼ r0
k
at z ¼ H
oT ðr; H ; tÞ ¼ h2 ðT Tf Þ oz
ð9Þ ð10Þ ð11Þ ð12Þ
Where h1 and h2 are the heat transfer coefficients between hydride bed and hydrogen gas and boundary walls and cooling fluid respectively. Thermophysical properties and the values of data used in the calculations are presented in Table 1.
3. Numerical method The partial differential equations are solved numerically with a fully implicit numerical scheme embodied in PHOENICS code [9]. This code solves following general differential equations, oðq/Þ þ rðqu/Þ ¼ rðCr/Þ þ Sh ot
ð13Þ
where / is a generic variable that representing the variable solved (i.e., u, v, T ), C is the exchange coefficient. Sh represents the source terms. An important advantage of PHOENICS code is that it allows the user to incorporate additional source term that not available in the main program. A 30 40 grid system is employed after a grid refinement test. A typical run until 3000 s takes about 8 h in a Pentium III PC.
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A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
4. Results The temperature distribution along the axis at a few locations in the system at t ¼ 500, 2000 and 4000 s is plotted in Fig. 2. At the beginning, the temperature of the bed is assumed to be the same as the cooling fluid. Fig. 2a indicates that in 500 s temperature in whole bed increases significantly. These results show that at initial times hydriding processes take place all over the bed and the heat released as a result of chemical reaction is used to heat up the bed. It is seen that temperature is slightly lower near the cooled boundary wall and inlet section. Fig. 2b denotes that in t ¼ 2000 s the temperature both walls are lower than Fig. 2a as time proceeds. It is clear that from Fig. 2b the temperature decreases at higher locations. Their temperatures for various z values remain constant except for both sides. The temperature profile at t ¼ 4000 s is presented in Fig. 2c. We observe that both sides have lower temperatures which are different from inner sections. Temperatures decrease at higher locations. It is clear that that from Fig. 2c the initially the temperature increases sharply, then it remains constant, then it suddenly decreases. Fig. 3 displays evolution of hydride formation in the reactor. It is seen that hydride formation raises little in vicinity of both walls but it remains almost zero at other sections at 500 s. At t ¼ 2000 s initially, hydride formation for z ¼ 1:00 cm and t ¼ 1:25 cm is higher but it suddenly decreases and it remains constant. Hydride formation decreases at z ¼ 0:25, z ¼ 0:50 and z ¼ 0:75. It is clear from Fig. 3b hydride formations are very similar and are nearly zero at lower sections. Fig. 3c indicates that the hydride formations increase at higher sections as time proceeds. For values of height z ¼ 0:25 cm and z ¼ 0:50 cm their curves are similar, when we look carefully we can see that there is a little change on the hydride formation. For z ¼ 0:75 cm, z ¼ 1:00 cm and z ¼ 1:25 cm hydride formations are higher at both walls. We observe that both a sharp increase and a sharp decrease at both sides. When we compare for z ¼ 0:75, z ¼ 1:00 and z ¼ 1:25 we see too much changes in hydride formations. The effect of hydrogen charge method is compared in Fig. 4. Two systems are mainly considered. In the first system hydrogen is charged only from one side which is called as a single entry system. In the second system hydrogen is charged from both wall and center section, this system is refereed as a double entry system. It can be seen that double entry increases sharply until about at t ¼ 1500 s then it remains constant. On the other hand, hydrogen formation in the single entry system increases smoothly, until about at t ¼ 3500 s then it remains constant. When we compare between two systems, we can see easily from Fig. 4 that hydrogen is produced quickly in the double
A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
177
Fig. 2. Temperature profile in the reactor at selected times. (a) t ¼ 500 s, (b) t ¼ 2000 s, (c) 4000 s.
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A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
Fig. 3. Evolution of hydride formation in the reactor. (a) t ¼ 500 s, (b) t ¼ 2000 s, (c) 4000 s.
A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
179
Fig. 4. The effect of hydrogen charge method.
entry system however in the single entry system charging time for hydrogen formation takes place a longer time. We can see that they intersect about at t ¼ 3500 s. 5. Conclusion In this study hydrogen storage in metal hydride bed is numerically investigated. Mathematical model includes complex heat and mass transfer take place during the storage process. Hydrogen storage in the metal hydride bed occurs with an exothermic reaction. Therefore, the temperature of the bed increases during the operation. Temperature increases adversely affect the formation rate. Therefore the system must be efficiently cooled for a rapid charge. The effect of two charge systems on filling time is investigated. It is found that hydrogen must be supplied radially from both sides for a rapid filling. Acknowledgements The useful financial support from Turkish State Planning Organization (DPT) under contract number 2002K120490 is gratefully acknowledged.
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A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
The authors would like to thank Dr. Mahmut D. Mat for the constructive suggestions which have been included in the paper.
References [1] P.H. Fisher, J.S. Watson, Modelling and evaluation of designs for solid hydrogen storage beds, International Journal of Hydrogen Energy 8 (1983) 109–119. [2] G.G. Lucas, W.L. Richards, Mathematical modelling of hydrogen storage systems, International Journal of Hydrogen Energy 9 (1984) 225–231. [3] P. Selvam, B. Vswanathan, C.S. Swamy, V. Srinivasan, Magnesium and magnesium alloy hydrides, International Journal of Hydrogen Energy 11 (1986) 169–192. [4] A. Jemni, S. Ben Nasrallah, Study of two-dimensional heat and mass transfer during absorption in a metal-Hydrogen rector, International Journal of Hydrogen Energy 20 (1995) 43–52. [5] A. Jemni, S. Ben Nasrallah, J. Lamloumi, Experimental and theoretical study of a metalhydrogen reactor, International Journal of Hydrogen Energy 24 (1999) 631–644. [6] E.M. Fedorov, Y.I. Shanin, L.A. Izhvanov, Simulation of hydride heat pump operation, International Journal of Hydrogen Energy 24 (1999) 1027–1032. [7] T. Nakagawa, A. Inomata, A. Aoki, T. Miura, Numerical analysis of heat and mass transfer characteristics in the metal hydride bed, International Journal of Hydrogen Energy 25 (2000) 339–350. [8] M. Mat, Y. Kaplan, Numerical study of hydrogen absorption in an Lm Ni5 hydride reactor, International Journal of Hydrogen Energy 26 (2001) 957–963. [9] H. Rosten, Spalding DB. PHOENICS Manual, CHAM. TR/1000, London, UK, 1986.
Numerical investigation of heat and mass transfer in a metal hydride bed Abdulkadir Dogan a,*, Yuksel Kaplan b, T. Nejat Veziroglu c a
Department of Mathematics, Nigde University, Fen-Edebiyat Fakultesi, 51100 Nigde, Turkey b Mechanical Engineering Department, Nigde University, 51100 Nigde, Turkey c Clean Energy Research Institute, University of Miami, Coral Gables, FL 33124, USA
Abstract This paper presents a mathematical model for hydrogen storage in a metal hydride bed. For this purpose, a two-dimensional mathematical model which considers complex heat and mass transfer during the hydriding process is developed. The coupled differential equations are solved with a numerical method based on integrations of governing equation over finite control volumes. The driving force is considered to be pressure difference because of the temperature distribution in the system. The numerical results showed that the hydriding performance depends on the temperature distribution in the hydride bed. Fluid flow enhances the hydriding rate in the system by driving the hot fluid to the colder regions. The numerical results were found to agree satisfactorily with the experimental data available in the literature. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Hydrogen storage; Mathematical modeling; Metal hydride; Chemical reaction; Numerical solution
1. Introduction Over the last decade world wide interest in the use of hydrogen has led to much research interest on its storage and usage. Recently, many
*
Corresponding author. E-mail address: [email protected] (A. Dogan).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00207-8
170
A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
Nomenclature Ca material constant (s1 ) Cp specific heat (J kg1 K1 ) Ea activation energy (J mol1 ) h heat transfer coefficient (W m2 K1 ) H/M hydrogen and metal atomic ratio m_ hydrogen mass absorbed (kg m3 s1 ) M molecular weight (kg mol1 ) P pressure (pa) r radial coordinate (m) R universal gas constant (J mol1 K1 ) t time (s) T temperature (K) z axial coordinate (m) Greek letters DH 0 reaction heat of formation (J kg1 ) e porosity k thermal conductivity (W m1 K1 ) q density (kg m3 ) / generic variable that representing the variable solved (i.e., u, v, T ) C exchange coefficient Subscripts e effective eq equilibrium f cooling fluid g gas 0 initial s solid ss saturated
mathematical and experimental works have concentrated on metal hydride beds to be used for hydrogen storage medium. Developments in metal hydride technology show that metal hydrides provide opportunity for hydrogen storage to a high standard of safety both for mobile and for stationary applications. Hydrogen is an interesting and very effective fuel as well as a notable medium for energy storage. It is a natural fuel and the operation of it has no serious problems. It is present in water virtually unlimited quantities and can be produced from plentiful raw materials without difficulty. Taking the world energy and environmental problems into consideration, use of hydrogen as an
A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
171
alternative fuel instead of fossil fuels seems to be best alternative for near future. The hydrogen energy meets all the requirements of an alternative energy such as plentiful availability, comparable cost, versatility of use and social acceptability. It is a reliable fuel that can be created in either from the electrolysis of water or closed chemical reactions. There are several methods for hydrogen storage. Recently, storing hydrogen in the metal hydride bed received many research interests due to superiorities of this method over the conventional ones. Nevertheless, the efficient use of metal hydride beds for proficient operations in both the charging and discharging requires analysis of the complex mass and heat transfer procedure inside the hydride bed. Simultaneous heat and mass transfer takes place with chemical reaction therefore hydriding or dehydriding technique is quite complicated. The gas motion and changing physical properties also adds additional difficulty to the problem. So a lot of experimental and numerical works exist on the diverse viewpoints of the metal hydrides. Most of numerical studies in the earlier work presume that the hydriding or dehyriding techniques are a transient heat conduction problem with an interior heat source and variable physical parameters. Fisher and Watson [1] developed mathematical models to predict the performance of metal hydride beds for hydrogen storage. The relative importance of heat and mass transfer, chemical kinetics and equilibrium have been calculated by comparing the models with experimental data. An equilibrium model without empirical parameters produces bed pressure and temperatures that show good agreement with experimental data. Lucas and Richards [2] developed a detailed mathematical model for metal hydride beds to describe its behavior under both hydriding and dehydring conditions. This model provides a convenient means of predicting the time taken to release or absorb given amounts of hydrogen. These are calculated from the heat transfer characteristics and diffusion properties of particular metal alloys. Selvam et al. [3] addressed the question of using magnesium and magnesium based alloys for rechargeable hydrogen storage media. The effect of addition of organic compounds as well as other metals on the sorption characteristics of magnesium is considered in detail. The current status of information in this field is reviewed considering the potential of these materials for use as hydrogen storage in vehicular applications. The energy transport equations were solved by Jemni and Ben Nasrallah [4] for gas and solid phase and made a comparison the results with one temperature model which considers the gas and solid phases a mixture. They have demonstrated that solving problem with the one temperature model does not significantly affect the result and also demonstrated that convection is not significant in the hydriding processes.
172
A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
Jemni and Nasrallah [5] presented an experimental and theoretical study of a metal hydrogen reactor. Aim of this study was to determine experimentally the effective thermal conductivity, and to test validity of the theoretical model comparing theoretical and experimental results. They have shown that the theoretical results are agreed satisfactorily with experimental data. Fedorov et al. [6] developed a mathematical model and set of software for metal hydride heat pump operations. The set of equations includes energy conservation in hydride sorbers considering heat source released during the chemical reaction. Nakagawa et al. [7] considered the hydriding techniques with a two dimensional model and obtained that the convection at first improved the hydriding process and exhibited an adverse effect at late times. More recently Mat and Kaplan [8] numerically examined the absorption in a metal hydride bed using a mathematical model similar to the Jemni et al.Õs [5] work. The aim of this work is to adapt Jemni and Ben NasrallahÕs mathematical model and put in application it to experimental studies usable in the earlier work and study the primary mechanisms of hydride formation and heat and mass transfer which take position in the hydride bed.
2. Mathematical model Metal hydride formation is considered in a two-dimensional metal hydride reactor shown in Fig. 1. The system is consisted of a cylindrical tank with cooling system. The hydrogen is charged from radially from the center and outside. The metal hydride system is sandwiched between two cooling regions. Due to the symmetric distribution of metal hydride regions only one of them is considered (Fig. 1b). The equations governing the heat and mass transfer and chemical reaction within the hydrogen storage beds are given as follows. 2.1. Energy equation Assuming thermal equilibrium between the storage bed and hydrogen a continuum energy equation is solved instead of separate equations for both solid and gaseous phases. oT 1 o oT o oT 0 _ ðqCpg Þ ¼ rke ke T ðCpg Cps Þ ð1Þ þ m½DH ot r or or oz oz where q, Cp , ke are effective density, specific heat and thermal conductivity respectively. ðqCp Þe is calculated as ðqCp Þe ¼ ðeqg Cpg þ ð1 eÞqs Cps Þ
ð2Þ
A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
173
Fig. 1. The schematic sketch of system considered: (a) metal hydride tank and (b) solution domain.
where, e is the porosity. The effective thermal conductivity is expressed as ke ¼ ekg þ ð1 eÞks
ð3Þ
174
A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
Where kg and ks are the thermal effective conductivity of the gas and solid phases respectively. 2.2. Hydrogen mass balance
e
oðqg Þ ¼ m_ ot
ð4Þ
2.3. Mass balance for solid phase Similar to hydrogen mass balance equation excluding the advection term, mass balance for the solid phase can be expressed as: ð1 eÞ
oðqs Þ ¼ m_ ot
ð5Þ
2.4. Reaction kinetic The amount of hydrogen absorbed is directly linked to reaction-rate which is expressed as [4], Ea Pg m_ ¼ Ca exp ð6Þ ln ðqss qs Þ RTS Peq where Ca material is depended constant, qss is the density of the solid phase at saturation and Peq is the equilibrium pressure calculated using the vanÕt Hoff relationship; ln Peq ¼ A
B T
ð7Þ
where A and B are materials constants deduced from the experimental work of Jemni et al. [5] as A ¼ 100:75 and B ¼ 31 896:25. 2.5. Initial and boundary conditions Initially hydride bed is assumed to have constant temperature and pressure and hydrogen is assumed to be at rest. These conditions can be expressed mathematically as at t ¼ 0
P ¼ P0
T ¼ T0
ð8Þ
The boundary walls are assumed to be impermeable and no slip conditions are valid at the boundary walls. The reaction heat is removed from the boundary
A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
175
Table 1 Thermophysical properties of materials and data used in computations. Density, q (kg m3 ) Specific heat, Cp (J kg1 K1 ) Thermal conductivity, k (Wm1 K1 ) Inlet temperature of gas, T0 Initial temperature of bed and wall, Tf Heat transfer coefficient, h (Wm2 K1 )
Metal (LaNi5 )
Hydrogen
4200 419 1.2 283 K 283 K 1652
0.0838 14 890 0.12
walls with a cooling fluid whose temperature is Tf . Hydrogen is charged from sides with constant pressure, P0 and constant temperature, T0 . The boundary conditions can be expressed as oT ðri ; z; tÞ ¼ h1 ðT T0 Þ or
at r ¼ ri
k
at z ¼ 0
oT ðr; 0; tÞ ¼ 0 oz oT ðr0 ; z; tÞ ¼ h1 ðT T0 Þ or
at r ¼ r0
k
at z ¼ H
oT ðr; H ; tÞ ¼ h2 ðT Tf Þ oz
ð9Þ ð10Þ ð11Þ ð12Þ
Where h1 and h2 are the heat transfer coefficients between hydride bed and hydrogen gas and boundary walls and cooling fluid respectively. Thermophysical properties and the values of data used in the calculations are presented in Table 1.
3. Numerical method The partial differential equations are solved numerically with a fully implicit numerical scheme embodied in PHOENICS code [9]. This code solves following general differential equations, oðq/Þ þ rðqu/Þ ¼ rðCr/Þ þ Sh ot
ð13Þ
where / is a generic variable that representing the variable solved (i.e., u, v, T ), C is the exchange coefficient. Sh represents the source terms. An important advantage of PHOENICS code is that it allows the user to incorporate additional source term that not available in the main program. A 30 40 grid system is employed after a grid refinement test. A typical run until 3000 s takes about 8 h in a Pentium III PC.
176
A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
4. Results The temperature distribution along the axis at a few locations in the system at t ¼ 500, 2000 and 4000 s is plotted in Fig. 2. At the beginning, the temperature of the bed is assumed to be the same as the cooling fluid. Fig. 2a indicates that in 500 s temperature in whole bed increases significantly. These results show that at initial times hydriding processes take place all over the bed and the heat released as a result of chemical reaction is used to heat up the bed. It is seen that temperature is slightly lower near the cooled boundary wall and inlet section. Fig. 2b denotes that in t ¼ 2000 s the temperature both walls are lower than Fig. 2a as time proceeds. It is clear that from Fig. 2b the temperature decreases at higher locations. Their temperatures for various z values remain constant except for both sides. The temperature profile at t ¼ 4000 s is presented in Fig. 2c. We observe that both sides have lower temperatures which are different from inner sections. Temperatures decrease at higher locations. It is clear that that from Fig. 2c the initially the temperature increases sharply, then it remains constant, then it suddenly decreases. Fig. 3 displays evolution of hydride formation in the reactor. It is seen that hydride formation raises little in vicinity of both walls but it remains almost zero at other sections at 500 s. At t ¼ 2000 s initially, hydride formation for z ¼ 1:00 cm and t ¼ 1:25 cm is higher but it suddenly decreases and it remains constant. Hydride formation decreases at z ¼ 0:25, z ¼ 0:50 and z ¼ 0:75. It is clear from Fig. 3b hydride formations are very similar and are nearly zero at lower sections. Fig. 3c indicates that the hydride formations increase at higher sections as time proceeds. For values of height z ¼ 0:25 cm and z ¼ 0:50 cm their curves are similar, when we look carefully we can see that there is a little change on the hydride formation. For z ¼ 0:75 cm, z ¼ 1:00 cm and z ¼ 1:25 cm hydride formations are higher at both walls. We observe that both a sharp increase and a sharp decrease at both sides. When we compare for z ¼ 0:75, z ¼ 1:00 and z ¼ 1:25 we see too much changes in hydride formations. The effect of hydrogen charge method is compared in Fig. 4. Two systems are mainly considered. In the first system hydrogen is charged only from one side which is called as a single entry system. In the second system hydrogen is charged from both wall and center section, this system is refereed as a double entry system. It can be seen that double entry increases sharply until about at t ¼ 1500 s then it remains constant. On the other hand, hydrogen formation in the single entry system increases smoothly, until about at t ¼ 3500 s then it remains constant. When we compare between two systems, we can see easily from Fig. 4 that hydrogen is produced quickly in the double
A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
177
Fig. 2. Temperature profile in the reactor at selected times. (a) t ¼ 500 s, (b) t ¼ 2000 s, (c) 4000 s.
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Fig. 3. Evolution of hydride formation in the reactor. (a) t ¼ 500 s, (b) t ¼ 2000 s, (c) 4000 s.
A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
179
Fig. 4. The effect of hydrogen charge method.
entry system however in the single entry system charging time for hydrogen formation takes place a longer time. We can see that they intersect about at t ¼ 3500 s. 5. Conclusion In this study hydrogen storage in metal hydride bed is numerically investigated. Mathematical model includes complex heat and mass transfer take place during the storage process. Hydrogen storage in the metal hydride bed occurs with an exothermic reaction. Therefore, the temperature of the bed increases during the operation. Temperature increases adversely affect the formation rate. Therefore the system must be efficiently cooled for a rapid charge. The effect of two charge systems on filling time is investigated. It is found that hydrogen must be supplied radially from both sides for a rapid filling. Acknowledgements The useful financial support from Turkish State Planning Organization (DPT) under contract number 2002K120490 is gratefully acknowledged.
180
A. Dogan et al. / Appl. Math. Comput. 150 (2004) 169–180
The authors would like to thank Dr. Mahmut D. Mat for the constructive suggestions which have been included in the paper.
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