On the selection of metal foam volume fraction for hydriding time minimization of metal hydride reactors

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On the selection of metal foam volume fraction for hydriding time minimization of metal hydride reactors Meng-Lung Tsai, Tian-Shiang Yang* Department of Mechanical Engineering, National Cheng Kung University, 1 University Road, Tainan 701, Taiwan

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abstract

Article history:

Here we examine how the hydriding time of a metal hydride reactor (MHR) varies with the

Received 3 June 2010

volume fraction, 4mf, of a metal foam installed in the reactor. Technically, an experimen-

Received in revised form

tally validated mathematical model accounting for the hydrogen absorption kinetics of

16 July 2010

LaNi5 is used to compute the heat and mass transport in a cylindrical MHR. We then

Accepted 16 July 2010

demonstrate that, with a fixed amount of metal hydride powder sealed in the reactor,

Available online 15 August 2010

saving a relatively small fraction (say, 1%) of the MHR internal volume to accommodate a metal foam usually suffices to substantially facilitate heat removal from the reactor,

Keywords:

thereby greatly shortening the MHR hydriding time. However, for a metal foam of fixed

Metal hydride reactor

apparent size, increasing 4mf would reduce the metal hydride content, and hence the

Heat conduction augmentation

maximum hydrogen storage capacity, of the MHR. Consequently, if a prescribed amount of

Metal foam

hydrogen is to be stored in the MHR, the hydriding time would decrease with increasing 4mf

Hydriding time

at first (due to heat conduction augmentation), reach a minimum at an “optimal” 4mf value,

Hydrogen storage capacity

and then increase drastically due to metal hydride underpacking. ª 2010 Professor T. Nejat Veziroglu. Published by Elsevier Ltd. All rights reserved.

1.

Introduction

In the development of hydrogen power systems, fuel (hydrogen) storage and transportation are an important issue. Currently, because of its relatively high hydrogen storage capacity and safety implications, hydrogen storage in metal hydrides appears to be a promising option [1]. In practice, a hydriding intermetallic compound is pulverized into powder form (to increase the reaction surface area per unit apparent volume), and then sealed in a metal hydride reactor (MHR), which usually takes the shape of a cylindrical container [2,3]. As hydrogen absorption and desorption of metal hydrides involve heat release and consumption, respectively, thermal management of an MHR strongly affects its hydriding/dehydriding time, and therefore is an important issue in MHR design and optimization [4,5]. Meanwhile, the hydriding/ dehydriding process in an MHR is further complicated by the

gaseous hydrogen flow through the porous metal hydride bed. (Other key issues in the technical design of MHRs are discussed recently by Yang et al. [6]). Due to the synergetic interactions of the aforementioned factors, the performance of an MHR is highly sensitive to its operating conditions (such as the heat transfer fluid temperature [7e9] and reactor inlet/exit pressure [9e11]). Also, thermal management of MHRs presents some practical challenges, since metal hydride powders typically have low effective thermal conductivities (on the order of 0.1 W/m-K [12]). Various methods of heat conduction augmentation therefore have been proposed for MHRs. Basically, such methods can be classified into two categories [13], namely using extended surfaces (in the forms of fins, foams, or meshes; see [13e15] for example) and binding metal hydrides into a solid matrix formed by a high-conductivity material (such as copper, aluminum, or nickel [14,16e18]). Recently, to

* Corresponding author. Tel.: þ886 6 2757575x62112; fax: þ886 6 2352973. E-mail address: [email protected] (T.-S. Yang). 0360-3199/$ e see front matter ª 2010 Professor T. Nejat Veziroglu. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijhydene.2010.07.081

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further facilitate MHR heat removal, Mellouli et al. [19] inserted a spiral heat exchanger (and a finned spiral heat exchanger instead in a follow-up work [20]) in an MHR, and demonstrated that the hydriding time can be significantly reduced. (Note, however, that when the hydriding and dehydriding processes are mass transfer controlled, instead of heat transfer controlled, the MHR performance may not be improved as significantly by sophisticated thermal management strategies [21]). The present work is motivated by recent studies on the use of metal foam in MHRs. Specifically, on the basis of a oneedimensional (1-D) model, Laurencelle and Goyette [22] demonstrated that the hydriding and dehydriding times of an MHR can be substantially reduced by installing an aluminum foam in it. (In the 1-D model, all system state variables depend on time and the radial coordinate only.) A particularly impressive finding was that, when an aluminum foam of 9% volume fraction (or 91% porosity) is used, the reactor diameter can be increased by a factor of 7.5 while maintaining a reaction rate similar to that of a smaller reference reactor (which does not contain an aluminum foam). In other words, it is possible to achieve a 50-fold increase in hydrogen storage capacity (per unit reactor height) by installing an aluminum foam in a larger reactor (instead of using the smaller plain reactor) without increasing the hydriding and dehydriding times. Some of their model predictions were validated by comparison with experimental data. Meanwhile, Mellouli et al. [23] extended the 2-D model of Jemni et al. [24] to account for the presence of a metal foam in an MHR. They then studied numerically how the thermophysical properties, pore density, and pore size of a metal foam (with the technical data for Duocel aluminum foam [25]) affect the hydriding time of an MHR containing the metal foam.

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In the works cited above, the effectiveness of using metal foam for improving the MHR performance through heat conduction augmentation has been clearly demonstrated. However, it appears that such works did not examine the effects of the metal foam volume fraction (4mf, which is the ratio of the net volume occupied by the metal material to the apparent volume of the foam as a whole) on the overall MHR performance. This issue is worth addressing for the reason that, while the presence of a metal foam improves heat conduction in an MHR, it also reduces the amount of metal hydride powder that can be sealed in the MHR to store hydrogen. So, for an MHR to achieve a certain hydrogen storage capacity, there may exist an “optimal” choice of 4mf value, and here we wish to explore such a possibility. To serve this purpose, however, a slightly more comprehensive theoretical model than that cited above is needed. Briefly, the spatial variation of hydrogen pressure in the MHR was taken into account by Mellouli et al. [23], but was neglected in the model of Laurencelle and Goyette [22]. Meanwhile, both models did not explicitly consider the dependence of the hydrogen flow resistance on the metal foam volume fraction. As the spatial non-uniformity of hydrogen pressure distribution in larger MHRs may be significant enough to affect the hydriding/dehydriding kinetics, here we shall basically follow the model of Mellouli et al. [23], but attach to it certain additional components to allow for a tunable metal foam volume fraction. Meanwhile, to expedite computations, we shall also make the 1-D approximation like that in Laurencelle and Goyette [22]. The modified model will be explained in Section 2, and the numerical method for its solution will be discussed in Section 3. After validating our modified model by comparing its numerical results with the experimental data of Laurencelle and Goyette [22] in Section 4.1, we shall discuss a series of

Fig. 1 e (a) Schematic of a cylindrical MHR within which a metal foam is installed; (b) the simplified 1-D computational domain.

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parameter studies for a cylindrical MHR containing an annular metal foam of fixed apparent outer diameter (which equals the MHR internal diameter; see Fig. 1). The apparent inner radius (Ri) and volume fraction (4mf) of the metal foam are the design parameters. First, keeping constant the amount of metal hydride powder sealed in the reactor, the effects of varying 4mf on the MHR hydriding time are examined. (Note that the apparent inner radius Ri then varies with 4mf in order to conserve the metal hydride amount in the MHR.) Our numerical results suggest that, with a properly chosen 4mf value, the metal foam’s presence can reduce the time required to charge the MHR to 95% of its maximum hydrogen storage capacity by a factor of about 40 (under certain conditions; see Section 4.2). Also, a rather small 4mf value (say, 0.01) usually suffices to reduce the MHR hydriding time substantially. Next, in Section 4.3, we consider the scenario that the apparent inner radius of the metal form is fixed, while its volume fraction 4mf is being varied. It is important to note that, as the apparent size of the metal foam is fixed accordingly, increasing 4mf would reduce the amount of metal hydride sealed in the reactor, thereby reducing the maximum hydrogen storage capacity of the MHR. So, to store a specified amount of hydrogen in the MHR, there is a maximum admissible value of 4mf. When 4mf becomes too close to such a maximum admissible value, the MHR hydriding time (i.e., the time it takes for the MHR to store the specified hydrogen mass) would increase drastically due to metal hydride underpacking in the reactor. On the other hand, for smaller values of 4mf, the effects of heat conduction augmentation in the MHR may not be fully exploited. An “optimal” value of 4mf for a specified hydrogen storage capacity of the MHR therefore can be found, which minimizes the MHR hydriding time (see Section 4.3). These findings are expected to be useful for optimizing the selection of metal foam volume fraction for MHRs. Some additional remarks will be given in Section 5 to conclude this paper.

2.

Mathematical model

As explained in the previous section, to allow for a tunable metal foam volume fraction, the model of Mellouli et al. [23] needs some modifications, which will be detailed below. Other components of the model, however, have been used and explained by many authors (see, for example, [11,24,26,27]), and therefore will only be briefly outlined. First, as shown in Fig. 1, here we consider a cylindrical MHR containing an annular metal foam. The apparent outer radius of the metal foam, Ro, is assumed to be the same as the MHR internal radius. (Symbols for all variables are listed in the nomenclature.) In the MHR, hydrogen permeates through a composite porous medium formed by the metal hydride powder and the metal foam. During the charging process, hydrogen is supplied at the apparent inner radius of the metal foam, where r ¼ Ri. Suppose also that the height of the MHR is much larger than its diameter, so that the longitudinal temperature and pressure variations in the MHR can be neglected. Due to axisymmetry as well, all the state variables of the systemdsuch as the hydrogen temperature and pressure, and the local hydrogen content of the metal hydride powderdtherefore depend upon time (t) and the radial coordinate

(r) only, and so it is appropriate to calculate the heat and mass transfer in the MHR using a 1-D mathematical model. It is also important to note that, since a 4mf fraction of the MHR interior volume has been occupied by the metal foam, the apparent volume fraction of the metal hydride powder is (1  4mf). Accordingly, if the metal hydride powder has a porosity of e, the gaseous hydrogen would occupy a (1  4mf)e fraction of the MHR interior volume, and the actual volume fraction of the metal hydride powder would be (1  4mf)(1  e). Meanwhile, although the volume of metal hydride powder generally would change during the hydriding/dehydriding process [28], for simplicity here we neglect this volume change and take the porosity e of the metal hydride powder to be constant. In particular, following Laurencelle and Goyette [22], we set e ¼ 0.55. (All parameter values are summarized in Table 1).

2.1.

Mass conservation and Darcy’s law

Hydrogen mass balance for the metal hydride powder can be written as


 
vr _ 1  4mf ð1  3Þ s ¼ 1  4mf m: vt

(1)

Table 1 e List of parameter values. MHR Ambient fluid temperature, Tf Charging pressure, pch Effective heat transfer coefficient, h Initial pressure, p0 Initial temperature, T0 Internal radius, Ro

298 K 10 bar 1652 W/m2-K 0.1 bar 298 K 5 cm

Hydrogen Atomic weight, MH Ideal gas constant, Rg Kinetic viscosity, ng Specific heat capacity, cp,g Thermal conductivity, kg

1 kg/kmol 4.16 kJ/kg-K 1.05  104 m2/s 14.27 kJ/kg-K 0.190 W/m-K

Metal hydride (LaNi5) Density (bulk), rM Molecular weight, MM Particle diameter, ds Porosity, e Saturated density, rss Specific heat capacity, cp,s Thermal conductivity, ks

8310 kg/m3 432.4 kg/kmol 30 mm 0.55 8416 kg/m3 0.355 kJ/kg-K 0.222 W/m-K

Metal foam (aluminum) Density (bulk), rmf Specific heat capacity, cp,mf Thermal conductivity, kmf

2700 kg/m3 0.963 kJ/kg-K 121.1 W/m-K

Hydrogen absorption kinetics Activation energy, Ea  Heat release, DH Reaction rate constant, Ca Reference temperature, Tref Scaled enthalpy difference, DH/Rg

21.18 kJ/kg 1.5  104 kJ/kg 59.19 s1 303 K 3594 K

Coefficients in polynomial function f(H/M ) a0, a 1 a2, a 3 a4, a 5 a6, a 7

0.34863, 10.1059 14.2442, 10.3535 4.20646, 0.962371 0.115468, 0.00563776

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here the mass density rs(r, t) is based on the actual volume, instead of the apparent volume of the metal hydride powder. _ is defined to be the rate of hydrogen Note also that m absorption per unit apparent volume of the metal hydride powder. Accordingly, while the left-hand side of Eq. (1) is multiplied by a factor of (1  4mf)(1  e), the right-hand side is multiplied by (1  4mf) only. It is assumed here that hydrogen behaves like an ideal gas in the MHR, so that the hydrogen mass density rg ¼ p/RgT. Moreover, the average hydrogen permeation velocity through the composite porous medium (which only has a radial component, ug, in the present model) is assumed to be linearly related to the hydrogen pressure gradient by Darcy’s law: ke vp ug ¼  : mg vr

(2)

To derive the continuity equation for hydrogen mass conservation in the MHR, one may follow the arguments in [11], while taking into account the various volume fractions specified above. The result can be written as  3 v
p  k v  r vp 

e _  ¼  1  4mf m: 1  4mf r vr ng vr Rg vt T

(3)

In Eq. (3), the first and second terms on the left-hand side are the local and convective rates of change, respectively, of the hydrogen mass density rg (¼p/RgT ). While rg is a function of hydrogen pressure p and temperature T, here the kinematic viscosity ng ¼ mg/rg is taken to be constant (which is a good approximation; see Fig. B.2 in [29]). Meanwhile, the effective permeability of the composite porous medium, ke, is taken to be a function of 4mf, so that the dependence of the hydrogen flow resistance on the metal foam volume fraction can be taken into account. Specifically, in terms of 4mf and the permeability ks of plain metal hydride powder (i.e., that without metal foam), we have ke ¼ ks $


1  4mf  ; 1 þ 24mf =3

(4)

provided that the ligament diameter dmf of the metal foam is sufficiently large such that ks =d2mf  1 [30]. With the values of the metal hydride powder diameter (ds) and porosity (e) specified in Table 1, the value of ks is estimated from the KozenyeCarman equation to be ks ¼ d2s e3 /180(1  e)2 ¼ 4.1 mm2, while dmf usually is greater than 76.2 mm in practice [25]; hence the use of Eq. (4) is justified.

2.2.

Energy balance

In the experiments of Jemni et al. [24], it was verified that convective heat transfer and the local temperature difference between the hydrogen and metal hydride powder in an MHR generally can be neglected. Intuitively, when a small volume fraction of the MHR is occupied by a metal foam having a somewhat higher thermal conductivity than that of the metal hydride powder, the assumptions of negligible convective heat transfer and local thermal equilibrium would still be valid. In fact, such assumptions were made in the numerical simulations of Laurencelle and Goyette [22] and Mellouli et al. [23], and will be used here as well. Accordingly,

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for the equivalent homogenized continuum in the MHR consisting of the hydrogen, metal hydride powder, and metal foam, the energy balance can be expressed as follows:  
 v   1 v vT 0 _ : (5) rcp e T ¼ ke r þ 1  4mf mDH vt r vr vr in Eq. (5), (rcp)e ¼ (1  4mf)[ergcp,g þ (1  e)rscp,s] þ 4mfrmfcp,mf is the equivalent heat capacity of the homogenized continuum. Similarly, the equivalent thermal conductivity of the homogenized continuum is ke ¼ (1  4mf)[ekg þ (1  e)ks] þ 4mfkmf. In the work of Laurencelle and Goyette [22], the metal foam is made of aluminum, while the metal hydride powder is LaNi5. Here the parameter values for hydrogen, bulk aluminum, and bulk LaNi5 will be chosen to be the same as that in [22]; see Table 1. Also, following MacDonald and Rowe [15], the unitmass heat release of hydriding reaction is set to be  DH ¼ 1.5  104 kJ/kg (appropriate for LaNi5).

2.3.

Hydriding kinetics model

We now discuss the hydriding kinetics model that relates the _ to the density rs of the metal hydrogen absorption rate m hydride powder, and the hydrogen pressure p and tempera_ Eqs. (1), (3) ture T. In view of this parameter dependence of m, and (5) are three coupled equations for the three state variables rs, p and T. Following Jemni and Ben Nasrallah [26], here the hydrogen absorption rate is given by !   p Ea _ $exp  ; (6) m ¼ Ca $ðrss  rs Þ$ln Rg T peq where Ca is a reaction rate constant, and Ea is the activation energy (with values specified in Table 1). Meanwhile, rss is the mass density of the metal hydride powder with saturated hydrogen absorption (to be explained below). Moreover, the equilibrium pressure peq is given by 

 DH 1 1 ; peq ¼ f ðH=MÞexp  Rg T Tref

(7)

where H/M is the hydrogen-to-metal atomic ratio, i.e., the number of adsorbed hydrogen atoms per metal molecule. With the constant volume approximation explained in Section 2.1, one has H ðrs  rM Þ=MH : ¼ rM =MM M

(8)

the function f(H/M ) will be specified below. In [26], the enthalpy difference DH associated with the hydrogen phase change is related to the ideal gas constant for hydrogen, Rg, by DH/Rg ¼ 3594 K. Now, for LaNi5, f(H/M ) is a polynomial function obtained by fitting experimental data [31]: f ðH=MÞ ¼ pð0Þ eq

7 X

an ðH=MÞn :

(9)

n¼0

5 for the reference equilibrium pressure p(0) eq ¼ 1.01  10 Pa (¼1 bar) and reference temperature Tref ¼ 303 K, the values of the dimensionless coefficients a0, 1, $$$, 7 are given in Table 1. Note that enough significant digits of the coefficients a0, 1, $$$, 7 have to be retained in order for Eq. (9) to give accurate values

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of f(H/M ). As shown in Fig. 2, the value of f(H/M ) generally increases with H/M. However, when LaNi5 undergoes a phase change, H/M varies between about 1 and 5, while f stays relatively constant (at about 2.5e3 bar). Note also that, for LaNi5, the saturated mass density rss ¼ 8.416  103 kg/m3 [22]. Accordingly, the maximum admissible value of H/M can be calculated from Eq. (8) to be 5.54; see Fig. 2.

2.4.

Initial and boundary conditions

In our computations, the initial temperature and pressure in the MHR are set at T0 ¼ 298 K and p0 ¼ 10.1 kPa (¼0.1 bar), respectively. Suppose also that, before charging, the gaseous hydrogen existing in the reactor is in equilibrium with the metal hydride. The corresponding equilibrium pressure peq therefore is 10.1 kPa as well. Using then Eqs. (7) and (9), it is readily calculated that the hydrogen-to-metal atomic ratio has an initial value of H/M ¼ 0.05 throughout the MHR (indicated by point “I” in Fig. 2). Moreover, with rM ¼ 8.310  103 kg/ m3 for the bulk density of the metal hydride (i.e., LaNi5 here), Eq. (8) gives the initial density of the metal hydride powder rs(r, 0) ¼ 8.311  103 kg/m3 for c0  r  R. As for thermal boundary conditions, since the hydrogen flow entering the MHR at r ¼ Ri has a relatively small mass flow rate, the heat transfer there generally is negligible compared with that at the side wall of the MHR (r ¼ Ro). For this reason, here we set (for ct > 0) vT ðRi ; tÞ ¼ 0; vr

ke

 vT ðRo ; tÞ ¼ h TðRo ; tÞ  Tf ; vr

(10)

where h is the effective heat transfer coefficient on the side wall of the MHR. In our computations, the ambient fluid temperature is taken to be Tf ¼ 298 K. Also, using the value determined in the experiments of Jemni et al. [24], the heat transfer coefficient is chosen to be h ¼ 1652 W/m2 K. For mechanical boundary conditions, since the side wall of the MHR is impermeable, the hydrogen velocity there must be zero. Also, since the hydrogen flow enters the MHR at r ¼ Ri, the hydrogen pressure there is set at the charging pressure pch. Accordingly, we have the following pressure boundary conditions for ct > 0:

pðRi ; tÞ ¼ pch ;

3.

vp ðRo ; tÞ ¼ 0: vr

(11)

Numerical method

The above governing equations and the associated initial and boundary conditions are discretized by use of the finitevolume method. Also, second-order central difference formulas are used to discretize the spatial derivatives of the state variables of the system, while first-order implicit difference formulas are used to discretize the temporal derivatives. As for the numerical solution procedure, the initial metal hydride density (rs), hydrogen pressure ( p) and temperature (T ) distributions (or that at the current time step) are used as a first estimate for that at the next time step. Based upon such an estimate, and the equilibrium pressure ( peq) distribution calculated accordingly by use of Eqs. (7) and (8), Eqs. (1) and (6) then give the corresponding distributions of the _ and updated metal hydride local hydrogen absorption rate m density rs at the next time step. Moreover, with the estimated _ distribution, the discretized versions of Eqs. (3) and (5) are m solved to update the pressure and temperature distributions at the next time step. The procedures then are repeated until the metal hydride density, hydrogen pressure and temperature distributions converge. In our computations, uniform grid points are deployed in the interior of the MHR, with 50 points in the r-direction. The time step is taken to be 0.1 s so that the fastest variation of the system state in the MHR can be well resolved. When the relative variations of all state variables between two consecutive time steps are less than 108 times their current values at all points in the computational domain, the solution is taken to be converged. We have checked that the computational grid and convergence criterion specified above produce grid-independent numerical results.

4.

Results and discussion

For convenience in the following discussion, here we define a few useful capacity measures of MHRs. First, the mass of metal hydride powder sealed in an MHR is   
(12) m0M ¼ rM $p R2o  R2i $ 1  4mf $ð1  3Þ per unit reactor height, and the instantaneous hydrogen mass stored in the MHR (again, per unit reactor height) is ZRo
 m0H ¼ 1  4mf $ð1  3Þ$ ðrs  rM Þ$ð2prÞdr:

(13)

Ri

The maximum hydrogen storage capacity of the MHR (containing a 4mf volume fraction of metal foam) is reached when rs ¼ rss throughout the MHR, and therefore is   
m0H;max ¼ ðrss  rM Þ$p R2o  R2i $ 1  4mf $ð1  3Þ;

Fig. 2 e The polynomial function f(H/M ) given by Eq. (9) for calculating the hydrogen equilibrium pressure peq.

(14)

as can be readily calculated from Eq. (13). Meanwhile, the nominal weight percentage of hydrogen stored in an MHR with a metal foam in it is

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m0H $100%
 i: wt:% ¼   h 2 2 p Ro  Ri $ 4mf rmf þ 1  4mf ð1  3ÞrM

(15)

Finally, by substituting m0 H,max given by Eq. (14) for m0 H in Eq. (15), one obtains the maximum hydrogen-to-metal weight percentage
 ðrss  rM Þ$ 1  4mf $ð1  3Þ$100%
 : ðwt:%Þmax ¼ 4mf rmf þ 1  4mf ð1  3ÞrM

4.1.

(16)

Model validation

In order to validate the mathematical model outlined in Section 2, here we compare its numerical results with the experimental data of Laurencelle and Goyette [22]. Note, however, that some parameter values for their experiments differ from that specified in Table 1, and therefore are detailed below. Specifically, in their experiments two MHRs were tested. A metal foam having a volume fraction of 4mf ¼ 0.09 was installed in the “large reactor” of radius Ro ¼ 6.350 mm to enhance heat conduction, but not in the “small reactor” with Ro ¼ 3.175 mm. Note that, since the metal foam filled up the whole internal volume of the large reactor in the experiments of Laurencelle and Goyette [22], its apparent outer diameter also was the internal diameter of the reactor, and its apparent inner diameter Ri ¼ 0. Moreover, the large and small reactors were charged at pch ¼ 12.73 bar and 5.96 bar, respectively, while the ambient fluid (air) temperature was set at Tf ¼ 298 K. Note also that the initial MHR temperature and pressure were not explicitly specified in [22], and in our computations the reasonably estimated values of T0 ¼ Tf and p0 ¼ 0.1 bar are chosen for both reactors. For the large reactor, the maximum hydrogen storage capacity (per unit rector height) and nominal weight percentage are calculated from Eqs. (14) and (16) to be m0 H,max ¼ 55.2 mg/cm and (wt.%)max ¼ 1.19%, respectively. Similarly, for the small reactor m0 H,max ¼ 15.2 mg/cm and (wt.%)max ¼ 1.28%. Note that the maximum hydrogen-to-metal weight percentage, (wt.%)max, is independent of the reactor radius; see Eq. (16). Moreover, while the presence of the metal foam reduces the maximum amount of hydrogen that can be stored in the large reactor, since the metal foam material (i.e., aluminum here) generally has a lower mass density than that of the metal hydride powder (LaNi5 here), the maximum hydrogen-to-metal weight percentage of the large reactor is only slightly lower than that of the small reactor. Note also that the large and small MHRs in the experiments of Laurencelle and Goyette [22] actually have a 2-D (axisymmetric) geometry, and hydrogen was supplied at the top surface of the MHRs. However, in our 1-D model, hydrogen has to be supplied at a nonzero Ri so as to have finite hydrogen flow rate and permeation velocity there. To circumvent this minor problem, in our computations the apparent inner radius Ri of the metal foam is set at a small value of Dr/2, where Dr is the grid size. Now, the temporal variations of the hydrogen-to-metal weight percentage in the small and large reactors of Laurencelle and Goyette [22] (Figs. 4 and 7 in their paper) are reproduced in Fig. 3(a) and (b), respectively. (No error bars were included in the original figures, and therefore cannot be shown here.) It is seen in both figures that, with a properly tuned

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effective heat transfer coefficient of h ¼ 500 W/m2-K, the numerical results of the present model agree reasonably well with the experimental data. Of course, here the heat transfer coefficient has a somewhat lower value than that specified in Table 1 (1652 W/m2-K). This can be understood on physical grounds. Specifically, the higher value specified in Table 1 is a reasonable estimate when the MHR is heated in a water bath [24]. However, in the experiments of Laurencelle and Goyette [22], the MHR was heated in an oven through an air layer, and so the effective heat transfer coefficient would have a lower value. This also suggests that, as the heat transfer coefficient is readily tunable, variations in MHR heating condition would not be a threat for the applicability of the present model. An additional interesting observation can be made by comparing the charging times (or, hydriding times) of the two reactors. Specifically, while the large reactor has a larger hydrogen storage capacity than that of the small reactor (as calculated above), the charging time of the large reactor is about 2.5 min [see Fig. 3(b)], which is much shorter than the charging time of about 7.5 min [see Fig. 3(a)] of the small reactor. Of course, this substantial reduction in hydrogen charging time results from the enhanced heat conduction in the large reactor by use of the metal foam. However, as is

Fig. 3 e Comparison of the experimental data of Laurencelle and Goyette [22] with the numerical predictions of the present model, for (a) the “small reactor” and (b) the “large reactor” studied in [22].

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intuitively clear, the presence of the metal foam to some extent would reduce the interior space available for the metal hydride powder that can absorb hydrogen. So, in the following subsections, we shall carefully examine this effect (using the parameter values specified in Table 1), and see if the MHR and metal foam parameters can be optimized in certain ways.

as specified here). To seal in the MHR an amount of metal hydride powder that is close to the maximum admissible amount (m0 M,max ¼ 293.7 g/cm), say m0 M ¼ 290 g/cm, it would take about 570 min to charge the MHR to 95% of its full

4.2. MHR charging time for a specified amount of metal hydride First, suppose that a prescribed amount of metal hydride powder is sealed in an MHR of fixed internal diameter (Ro ¼ 5 cm). A metal foam will also be installed in the MHR, with its volume fraction 4mf being the design parameter, and here we wish to examine how the hydrogen charging time of the MHR depends on 4mf. It should be noted, however, that as 4mf varies, the MHR internal space available for the metal hydride powder and hydrogen would vary accordingly. In order to accommodate the prescribed amount of metal hydride powder, the apparent inner radius of the metal foam, Ri, would have to vary as well. Specifically, assuming that the porosity e of the metal hydride powder is independent of 4mf, for a given m0 M the apparent inner radius of the metal foam is calculated from Eq. (12) to be " Ri ¼

R2o

#1=2 m0H  : prM ð1  fmf Þð1  3Þ

(17)

since Ri  0, it can be readily deduced from Eq. (17) that for a given m0 M the maximum admissible value of 4mf is 1  m0 M/ [pR2o rM(1  e)]. Moreover, since the maximum admissible value of 4mf must also have a nonnegative value, the maximum admissible value of m0 M is m0 M,max ¼ pR2o rM(1  e), corresponding to 4mf ¼ 0 (i.e., an MHR without metal foam). With the parameter values specified in Table 1, we have m0 M,max ¼ 293.7 g/cm. Note also that here we define the hydrogen charging time, tch, to be the time it takes for the MHR to store 0.95 m0 H,max of hydrogen (for the specified m0 M) at the specified inlet pressure ( pch ¼ 10 bar). For various values of m0 M, the calculated hydrogen charging times tch are plotted in Fig. 4(a) against the metal foam volume fraction 4mf. From a practical standpoint, it is unlikely to grossly under-exploit the hydrogen storage capacity of an MHR, and so here we do not consider values of m0 M that are less than one half of m0 M,max. Note also that the tch curves terminate at their corresponding maximum admissible values of 4mf. As seen in Fig. 4 (a), for a given 4mf it would take a longer time to charge the reactor to a higher capacity and so tch increases with m0 M. Meanwhile, for a specified m0 M, a mere 1% of metal foam volume fraction (i.e., 4mf ¼ 0.01) is sufficient to reduce the hydrogen charging time tch from that with 4mf ¼ 0 (i.e., without metal foam) by a factor of about six. For smaller values of m0 M (say, 170 g/cm), which allow for larger values of 4mf, the hydrogen charging time can even be reduced by a factor of about forty! Of course, the substantial reduction in tch results from the heat conduction augmentation by use of the metal foam. A somewhat more interesting use of the data presented in Fig. 4(a) is explained as follows. Suppose that a certain space is allocated for an MHR to be installed on a hydrogen power system. The reactor radius Ro then is determined (say, at 5 cm

Fig. 4 e Dependence of (a) the hydrogen charging time, tch, (b) thermal resistance, Rth, and (c) maximum hydrogen-tometal (including foam and hydride) weight percentage, (wt.%)max, of an MHR on the volume fraction 4mf of a metal foam installed in the MHR, for various values of the mass of metal hydride powder, m0 M per unit reactor height, sealed in the MHR. See Section 4.2 for the definitions of tch and Rth.

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capacity, as indicated by point “A” in Fig. 4(a). One way to reduce the charging time of the MHR is to sacrifice its hydrogen storage capacity to some extent, and seal less metal hydride powder in it. As an example, suppose that only m0 M ¼ 170 g/cm of metal hydride powder is sealed in the MHR, as indicated by point “B” in Fig. 4(a). Accordingly, the hydrogen charging time (tch) would be reduced to about 144 min. But then a significant internal space (about 40%) of the MHR would be wasted, and there is a better alternative for reducing tch. Specifically, without reducing the metal hydride mass m0 M, one may install a metal foam of suitable volume fraction 4mf (which then determines the apparent inner radius Ri of the metal foam). As indicated by point “A0 ” in Fig. 4(a), the same reduced charging time of tch z 144 min can be achieved for the original metal hydride mass of m0 M ¼ 290 g/cm when a metal foam with 4mf z 0.5% is used. In fact, if 4mf is further increased, an even shorter tch can be achieved; see Fig. 4(a). It is also important to note, however, that the hydrogen charging time tch does not always decrease monotonically in the admissible range of 4mf. Taking m0 M ¼ 170 g/cm for example, tch reaches a minimum of about 3.82 min at 4mf z 0.38. To understand the appearance of this minimum charging time, recall that the apparent inner radius Ri of the metal foam decreases with increasing 4mf, and hence the apparent thickness of the metal foam increases with 4mf. Meanwhile, the effective thermal conductivity ke of the equivalent homogenized continuum in the MHR increases with 4mf. As the thermal resistance of the MHR intuitively would decrease with ke but increase with the apparent thickness of the metal foam, under suitable conditions there would exist a particular 4mf value that minimizes the thermal resistance, and hence the hydrogen charging time, of the MHR. To put the above explanation in more quantitative terms, let us calculate the thermal resistance of the MHR as follows. First, while the hydriding reaction taking place in the MHR generally is non-uniform both temporally and spatially, to simplify matters here we consider steady heat conduction with uniform heat generation in the MHR. Accordingly, the temperature distribution in the MHR can be calculated from r1d[r(dT/dr)]/dr þ q000 /k ¼ 0 [cf. Eq. (5)], with q000 being the rate of heat generation per unit volume. Using also Eq. (10), it can be readily calculated that the difference between the MHR interior temperature at Ri and the ambient fluid temperature Ti  Tf ¼ q000 Rth, where the “thermal resistance”

     1 ke 1  2 Ro : (18) Rth ¼ þ Ro  R2i  R2i ln Ri 2ke hRo 2 for the same parameter ranges as that shown in Fig. 4(a), the thermal resistances Rth calculated from Eq. (18) are plotted against 4mf in Fig. 4(b). It is clearly seen that the curves in Fig. 4 (b) are highly correlated to the corresponding curves in Fig. 4 (a). In particular, for a given m0 M, the minimum of Rth occurs at a particular 4mf value that is close to the maximum admissible value of 4mf (corresponding to zero apparent inner radius of the metal foam, Ri ¼ 0). The corresponding tch curves in Fig. 4(a) exhibit a similar trend, and so the interpretation given above for the existence of a minimum MHR charging time for a given m0 M is justified. An additional important capacity measure of MHRs is the maximum weight percentage of hydrogen-to-metal,

Fig. 5 e Dependence of (a) the hydrogen charging time, tch, (b) thermal resistance, Rth, and (c) hydrogen-to-metal (including foam and hydride) weight percentage, (wt.%), of an MHR on the volume fraction 4mf of a metal foam installed in the MHR, for various values of the mass of hydrogen, m0 H per unit reactor height, stored in the MHR. In (c), the maximum hydrogen-to-metal weight percentage, (wt.%)max, is shown as a function of 4mf for reference. Also, the particular 4mf value producing the minimum tch for a given m0 H is indicated by the symbol on the corresponding curve.

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(wt.%)max, given by Eq. (16). Note that (wt.%)max does not depend explicitly on m0 M, and that for a given m0 M the maximum admissible value of 4mf is 1  m0 M/[pR2o rM(1  e)]. So, for the values of m0 M shown in Fig. 4(a), the corresponding (wt.%)max values are plotted against 4mf as the single curve in Fig. 4(c), but with different ending 4mf values. Note also that, for a given m0 M, both the mass of the metal hydride powder and the maximum hydrogen storage capacity are specified. So, increasing the metal foam volume fraction 4mf increases the total metal mass, and therefore decreases (wt.%)max. However, for 4mf values less than about 0.01, the reduction in (wt.%)max is rather insignificant, as is clearly seen in Fig. 4(c). Recall also that a small 4mf value of 0.01 is sufficient to bring about a significant reduction in the hydrogen charging time tch of the MHR (by a factor of about six).

4.3. MHR charging time for a specified amount of hydrogen storage From a practical standpoint, it perhaps is more important to specify the amount of hydrogen, m0 H, to be stored in an MHR. The design task then is to identify the proper choices of m0 M and 4mf that would allow the MHR to reach the capacity of m0 H within a satisfactory time. Meanwhile, in order to facilitate hydrogen flow into the MHR, the apparent inner radius Ri of the metal foam also should exceed a certain minimum value. In view of these practical considerations, here we fix Ri at 2 cm in our computations. Also, the time it takes for the MHR to reach the capacity of m0 H is taken to be the hydrogen charging time tch. [Consistent with this definition, the charging time tch for a given m0 M defined in Section 4.2 is that for m0 H ¼ 0.95 m0 M$(rss  rM)/rM]. First let us examine how tch varies with 4mf for various values of m0 H; see Fig. 5(a). To understand the basic trend of the curves there, note that for given Ri, o the amount of metal hydride powder sealed in the MHR, m0 M, is a function of the metal foam volume fraction 4mf; see Eq. (12). Also, for a given 4mf (and m0 M), it is impossible for m0 H to exceed the corresponding maximum hydrogen storage capacity of the MHR, m0 H,max, which can be calculated from Eq. (14). Accordingly, as is clearly seen in Fig. 5(a), for each value of 4mf there is an upper limit for m0 H. Of course, the truly maximum hydrogen storage capacity of the MHR is obtained when 4mf ¼ 0, with m0 H,max ¼ 3.14 g/cm for the present parameter setting. The specified values of m0 H therefore could not exceed such a value. Also, here we do not choose m0 H below one half of this maximum capacity, because in such cases the hydrogen storage capacity of the MHR clearly is highly under-exploited.

The curves in Fig. 5(a) also can be read in an alternative way. Specifically, for a given m0 H, the maximum admissible value of 4mf corresponds to that with m0 H ¼ m0 H,max, and is readily deduced from Eq. (14) to be 1  m0 H/[p(R2o  R2i )$(1  e)$ (rss  rM)]. At such a maximum admissible value, the hydrogen charging time tch theoretically would become infinite, as illustrated in Fig. 5(a). On the other hand, for smaller 4mf values, due to heat conduction augmentation tch would decrease with increasing 4mf. There therefore exists an “optimal” value of 4mf for a given m0 H, which minimizes the hydrogen charging time tch; see Fig. 5(a). In Table 2, the optimal values of the metal foam volume fraction, denoted by 4mf*, that result in the shortest hydrogen charging time, tch*, for various m0 H values are summarized. It is seen that, as m0 H decreases, the maximum admissible value of 4mf would increase accordingly, since there would be more spare room in the MHR to accommodate the metal foam. The optimal values 4mf* therefore follow the same trend. Meanwhile, as is intuitively clear, the minimum hydrogen charging time tch* decreases with decreasing m0 H. From a practical standpoint, it is also important to calculate the mass of metal hydride powder sealed in the MHR, (m0 M)*, maximum hydrogen storage capacity, (m0 H,max)*, and m0 H/ (m0 H,max)* ratio corresponding to the optimal 4mf value. Such results are included in Table 2 as well, and it is observed that (m0 H,max)* decreases with decreasing m0 H as expected. A less obvious conclusion that can be drawn from Table 2 is that the m0 H/(m0 H,max)* ratio decreases with decreasing m0 H. To understand such a trend, we plot the thermal resistance Rth defined by Eq. (18) in Fig. 5(b). Note that for a given 4mf value the dimensions of the MHR and the metal foam installed in it are completely determined, and so Rth does not depend on m0 H. [Recall, however, that for a given m0 H there exists a maximum admissible value of 4mf, which is indicated by a symbol in Fig. 5 (b)] It can be seen in Fig. 5(b) that Rth decreases monotonically with increasing 4mf. So, from the viewpoint of heat conduction augmentation, it would be advisable to choose a largest possible 4mf value for a given m0 H. Recall also that decreasing Rth generally would decrease the hydrogen charging time tch as well. However, for a specified m0 H, this ceases to be true when 4mf becomes close to its maximum admissible value, because the metal hydride powder then has to be nearly fully charged to meet the specification of m0 H, therefore requiring an extremely long tch. To avoid such a “saturation effect,” the optimal choice of 4mf, 4mf*, would have to be somewhat less than its maximum admissible value. Moreover, since the saturation effect is a property of the metal hydride (rather than that of the metal

Table 2 e Optimal values of the metal foam volume fraction, 4mf*, resulting in the shortest hydrogen charging time, tch*, for various masses of hydrogen stored in the MHR (m0 H per unit reactor height). The corresponding mass of metal hydride powder sealed in the MHR, (m0 M)*, maximum hydrogen storage capacity, (m0 H,max)*, and ratio of m0 H to (m0 H,max)* also are listed. m0 H (g/cm)

4mf*

tch* (min)

(m0 M)* (g/cm)

(m0 H,max)* (g/cm)

m0 H/(m0 H,max)*

3.0 2.7 2.4 2.1 1.8

0.048 0.130 0.195 0.245 0.290

18.18 7.81 5.21 3.80 2.83

234.8 214.6 198.6 186.2 175.2

3.01 2.75 2.54 2.38 2.24

0.998 0.983 0.944 0.881 0.803

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foam), the necessary increase in Rth to avoid it perhaps is not sensitive to the specific value of m0 H. Now, as seen in Fig. 5(b) (which is a double logarithmic plot), the magnitude of dRth/d4mf decreases monotonically with increasing 4mf. Therefore, for an increase in Rth of similar magnitude, it suffices to move 4mf slightly below its maximum admissible value to reach 4mf* for larger values of m0 H (which have smaller maximum admissible values of 4mf), while a more significant decrease in 4mf would be necessary for smaller values of m0 H. This interpretation is consistent with the dependence of the m0 H/(m0 H,max)* ratio on m0 H observed in Table 2. The variations of the hydrogen-to-metal weight percentage, wt.%, with 4mf for the same m0 H values used in Fig. 5(a) are plotted in Fig. 5(c). It is seen that, for each m0 H, wt.% increases with increasing 4mf. This results from the fact that, as 4mf increases, more and more of the lighter metal foam material (aluminum here) replaces the heavier metal hydride powder (LaNi5) in the MHR of fixed internal volume. Meanwhile, as explained above, for a given m0 H there exists a maximum admissible value of the metal foam volume fraction 4mf, which gives the corresponding maximum wt.%, (wt.%)max. In Fig. 5(c), (wt.%)max is plotted against 4mf, and the optimal 4mf values (which demand the shortest tch for each value of m0 H) also are indicated by symbols. It is also clearly seen there that the optimal value of 4mf is closer to its maximum admissible value when the specified value of m0 H is larger. Finally, to conclude this discussion, let us make an additional interesting observation in Fig. 5(a) and (c). Specifically, suppose again that a certain space is allocated for the MHR on a hydrogen power system. The maximum hydrogen storage capacity therefore is 3.14 g/cm (corresponding to 4mf ¼ 0) as calculated above. Since the maximum capacity is relatively close to 3.0 g/ cm, it is indicated approximately by point “A” in Fig. 5(a). Suppose also that it is not necessary to use up this maximum hydrogen storage capacity of the available MHR space. As a specific example, consider the case that only m0 H ¼ 1.8 g/cm is required for the targeted application of the power system. Then, without using a metal foam, one may charge the MHR to point “B” in Fig. 5(a), and reduce the hydrogen charging time tch substantially (from 356 min to 89 min) at the same time. The tch curve for m0 H ¼ 1.8 g/cm in Fig. 5(a), however, suggests that with a metal foam of suitable volume fraction (4mf ¼ 0.290; see Table 2) installed in the MHR, an even more profound reduction in tch to 2.8 min can be obtained. Meanwhile, as shown in Fig. 5(c), such a strategy also has the advantage of increasing the hydrogen-to-metal weight percentage slightly from 0.73% to 0.79%. The above observation suggests that installing a properly designed metal foam in an MHR not necessarily would demand more space for MHR on a hydrogen power system. Moreover, such a strategy would substantially reduce the hydrogen charging time, and increase the hydrogen-to-metal weight percentage of the MHR simultaneously.

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affected by the volume fraction (4mf) of a metal foam installed in the MHR to enhance heat conduction. After validating the theoretical model by comparing its numerical results with existing experimental data, two cases were studied numerically. First, in Section 4.2, the amount of metal hydride powder contained in the MHR (m0 M per unit reactor height) was specified, and it was found that the metal foam’s presence indeed can substantially reduce the time (tch) required to charge the MHR to 95% of its maximum hydrogen storage capacity, with relatively insignificant reduction in wt.%. It was also shown that the effective thermal resistance of the MHR, Rth, is highly correlated with the hydrogen charging time of the MHR. Moreover, for a given m0 M there exists a particular 4mf value that would minimize Rth, which is rather close to the value of 4mf that minimizes tch. A specific example also was discussed in Section 4.2 to demonstrate how the numerical results of the present model can be used to help identify the design parameters of the MHR and metal foam that would meet the specified hydrogen storage capacity and charging time of the MHR. Meanwhile, specifying the amount of hydrogen to be stored in an MHR (m0 H per unit reactor height) instead, the effects of varying 4mf on tch and wt.% were examined in Section 4.3. As the apparent size of the metal foam was fixed in the computations, increasing 4mf would reduce the amount of metal hydride sealed in the MHR, thereby reducing the maximum hydrogen storage capacity of the MHR. For a specified m0 H, there therefore exists a maximum admissible value of 4mf that could meet that specification. Indeed, the numerical results discussed in Section 4.3 indicated that if 4mf becomes too close to such a maximum admissible value, the hydrogen charging time tch of the MHR would increase drastically due to metal hydride underpacking. On the other hand, for smaller values of 4mf, tch would decrease with increasing 4mf because of heat conduction augmentation. Consequently, for a specified m0 H there exists an optimal value of 4mf that would minimize tch of the MHR. In Section 4.3, the optimal values of 4mf were found for various m0 H values, and the corresponding MHR thermal resistance and wt.% of the MHR were discussed. Moreover, a specific example was discussed to illustrate that, in practical applications, installing a metal foam of suitable volume fraction in an MHR not only would substantially reduce the hydrogen charging time, but also would slightly increase the hydrogen-to-metal weight percentage of the MHR. To sum up, in the present work we have clearly demonstrated the effectiveness of installing a suitably designed metal foam in an MHR to minimize its hydrogen charging time, without necessarily sacrificing its hydrogen storage capacity. Also, the theoretical model and the computational and analysis methodologies proposed in this work are useful for providing important insights into the optimization of MHR and metal foam parameter design.

Acknowledgments 5.

Summary and concluding remarks

Here, using a 1-D theoretical model that accounts for the mass and energy balance and the hydrogen absorption kinetics in a cylindrical MHR, we have examined how the charing time (tch) and hydrogen-to-metal weight percentage (wt.%) of an MHR are

The authors gratefully acknowledge the Taiwan National Science Council for supporting this work through grant No.NSC98-2221-E-006-170-MY2. They would also like to thank Professors C.-J. Ho and C.-D. Wen of NCKU, and Dr. B.-H. Chen of ITRI, for a number of fruitful discussions on this work and other related topics.

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references [22] [1] Zhang J, Fisher TS, Ramachandran PV, Gore JP, Mudawar I. A review of heat transfer issues in hydrogen storage technologies. J Heat Transf 2005;127:1391e9. [2] Zu¨ttel A. Materials for hydrogen storage. Mater Today 2003;6: 24e33. [3] Sakintuna B, Lamari-Darkrim F, Hirscher M. Metal hydride materials for solid hydrogen storage: a review. Int J Hydrogen Energy 2007;32:1121e40. [4] Choi H, Mills AF. Heat and mass transfer in metal hydride beds for heat pump applications. Int J Heat Mass Transf 1990; 33:1281e8. [5] Georgiadis MC, Kikkinides ES, Makridis SS, Kouramas K, Pistikopoulos EN. Design and optimization of advanced materials and processes for efficient hydrogen storage. Comput Chem Eng 2009;33:1377e90. [6] Yang FS, Wang GX, Zhang ZX, Meng XY, Rudolph V. Design of the metal hydride reactorsea review on the key technical issues. Int J Hydrogen Energy 2010;35:3832e40. [7] Gopal MR, Murthy SS. Prediction of heat and mass transfer in annular cylindrical metal hydride beds. Int J Hydrogen Energy 1992;17:795e805. [8] Gopal MR, Murthy SS. Studies on heat and mass transfer in metal hydride beds. Int J Hydrogen Energy 1995;20: 911e7. [9] Ye J, Jiang L, Li Z, Liu X, Wang S, Li X. Numerical analysis of heat and mass transfer during absorption of hydrogen in metal hydride based hydrogen storage tanks. Int J Hydrogen Energy 2010;35:8216e24. [10] Kikkinides ES, Georgiadis MC, Stubos AK. On the optimization of hydrogen storage in metal hydride beds. Int J Hydrogen Energy 2006;31:737e51. [11] Yang T-S, Tsai M-L, Ju D-S. Effects of exit-pressure variation on the hydrogen supply characteristics of metal hydride reactors. Int J Hydrogen Energy 2010;35:8597e608. [12] Dehouche Z, Grimard N, Laurencelle F, Goyette J, Bose TK. Hydride alloys properties investigations for hydrogen sorption compressor. J Alloys Compd 2005;399:224e36. [13] Oi T, Maki K, Sakaki Y. Heat transfer characteristics of the metal hydride vessel based on the plate-fin type heat exchanger. J Power Sourc 2004;125:52e61. [14] Chen Y, Sequeira CAC, Chen C, Wang X, Wang Q. Metal hydride beds and hydrogen supply tanks as minitype PEMFC hydrogen sources. Int J Hydrogen Energy 2003;28: 329e33. [15] MacDonald BD, Rowe AM. Impacts of external heat transfer enhancements on metal hydride storage tanks. Int J Hydrogen Energy 2006;31:1721e31. [16] Nagel M, Komazaki Y, Suda S. Effective thermal conductivity of a metal hydride bed augmented with a copper wire matrix. J Less-Common Met 1986;120:35e43. [17] Kim KJ, Montoya B, Razani A, Lee K-H. Metal hydride compacts of improved thermal conductivity. Int J Hydrogen Energy 2001;26:609e13. [18] Kim KJ, Lloyd G, Razani A, Feldman Jr KT. Development of LaNi5/Cu/Sn metal hydride powder composites. Power Technol 1998;99:40e5. [19] Mellouli S, Askri F, Dhaou H, Jemni A, Ben Nasrallah S. A novel design of a heat exchanger for a metal-hydrogen reactor. Int J Hydrogen Energy 2007;32:3501e7. [20] Dhaou H, Souahlia A, Mellouli S, Askri F, Jemni A, Ben Nasrallah S. Experimental study of a metal hydride vessel based on a finned spiral heat exchanger. Int J Hydrogen Energy 2010;35:1674e80. [21] Yang F, Meng X, Deng J, Wang Y, Zhang Z. Identifying heat and mass transfer characteristics of metal hydride reactor

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Nomenclature Ca: hydrogen absorption rate constant, s1 cp: specific heat capacity, kJ/kg-K dmf: ligament diameter of metal foam, mm ds: particle diameter of metal hydride powder, mm Ea: activation energy for hydrogen absorption, kJ/kg h: effective heat transfer coefficient on MHR wall, W/m2-K H/M: hydrogen-to-metal atomic ratio k: thermal conductivity, W/m-K M: molecular weight, kg/kmol _ hydrogen absorption rate per unit volume, kg/m3-s m: m0 H: mass of hydrogen stored in an MHR per unit reactor height, g/cm m0 M: mass of metal hydride sealed in an MHR per unit reactor height, g/cm p: pressure, Pa (¼ 0.99  105 bar) r: radial coordinate, cm Rg: ideal gas constant of hydrogen, kJ/kg-K Ri: apparent inner radius of metal foam, cm Ro: apparent outer radius of metal foam, cm T: temperature, K t: time, s tch: MHR charging (or hydriding) time, s u: radial component of hydrogen permeation velocity, m/s wt.%: hydrogen-to-metal (including foam and hydride) weight percentage Greek letters DH: enthalpy difference of hydrogen phase change, kJ/kg  DH : heat release of hydriding reaction, kJ/kg

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e: porosity of metal hydride powder k: permeability of metal hydride powder, m2 m: dynamic viscosity, N-s/m2 n: kinetic viscosity, m2/s r: density, kg/m3 4mf: volume fraction of metal foam Subscripts 0: initial ch: charging (or hydriding)

e: equivalent continuum in MHR eq: equilibrium f: ambient fluid g: gas (hydrogen) H: hydrogen M: metal hydride max: maximum s: solid (metal hydride powder) ss: solid, with saturated hydrogen absorption ref: reference

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