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Homogenization method for effective thermal conductivity of metal hydride bed
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International Journal of Hydrogen Energy 29 (2004) 209 – 216
www.elsevier.com/locate/ijhydene
Homogenization method for e%ective thermal conductivity of metal hydride bed Yusuke Asakuma∗ , Shinsuke Miyauchi, Tsuyoshi Yamamoto, Hideyuki Aoki, Takatoshi Miura Department of Chemical Engineering, Tohoku University, 07 Aoba Aramaki, Aoba-ku, Sendai 980-8579, Japan Accepted 26 March 2003
Abstract The e%ective thermal conductivity of the metal hydride bed is analyzed by the homogenization method. This method can represent the microstructure in the bed precisely. The referenced material is LaNi4:7 Al0:3 , which is practically used. Temperature and pressure ranges investigated here are from −80◦ C to +140◦ C and from 10−6 to 100 bar, respectively. Hydrogen, helium, nitrogen and argon are chosen as the 6lling gas. The validity of the homogenization method as the multi-scale analysis is con6rmed by a good correspondence with the experimental data that the e%ective thermal conductivity depends on the pressure of the 6lling gas under these various conditions (Int. J. Hydrogen Energy 23(2) (1998) 107). The homogenization method can become a powerful tool for the estimation of the e%ective thermal conductivity of the metal hydride bed by considering the microscopic behavior such as the pulverization and the change of the contact area. ? 2003 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. Keywords: E%ective thermal conductivity; Homogenization method; Multiscale analysis; Microstructure
1. Introduction The metal hydride is promising for the storage method of hydrogen in the point that it has the advantage of a lower pressure operation and a reversible reaction. However, heat conduction in the hydride bed becomes very complicated because the metal hydride has various characteristics of the packed structure such as the pulverization, the change of the void fraction and the contact area due to the expansion and the contraction of hydride particles. When the metal hydride absorbs and desorbs hydrogen, heat transfer in the packed bed becomes very complicated. The e%ective thermal conductivity is very important because heat of reaction has to be removed for the exothermic formation and added for the endothermic decomposition [1–9]. Recently, the hydriding and dehydriding kinetics are measured and heat and mass transfer characteristic in the metal hydride bed is ∗ Corresponding author. Tel.: +81-22-217-7252; fax: +81-22217-6165. E-mail address: [email protected] (Y. Asakuma).
investigated [10,11]. From these analytical and experimental results, we have proposed a numerical model, which can predict the various conditions of reaction pro6les. Absorption and desorption rates depend on heat transfer rate because the metal hydride is used in a powdery form in order to provide a large surface area for the reaction. Therefore, thermal and structural analyses have to be conducted simultaneously and precisely. However, there is less research, which focuses on the microstructure of the hydride particle. Here, the homogenization method is introduced as the multi-scale analysis, which can reCect the microstructure (particle diameter is 10−5 –10−6 m) in the bed on the macroproperty such as the thermal conductivity. Although this method is applied to the stress and the fracture analysis frequently [12,13], it is developing in thermal analysis recently [14–18]. The conventional models for heat transfer in the metal hydride bed [2,8] are not perfectly adequate because they are simpli6ed and treated as two-dimensional system. Therefore, we think that the homogenization method is very useful for the metal hydride in the point that it can consider the change of the microstructure after several cycles
0360-3199/03/$ 30.00 ? 2003 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. doi:10.1016/S0360-3199(03)00106-X
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Y. Asakuma et al. / International Journal of Hydrogen Energy 29 (2004) 209 – 216
Nomenclature a A A∗ b Bi CA CB g G h IH Kn l L n p p0 Peq R IS T IT v x xH x∗
contact area ratio surface area of cell (=l2 ) contact area constant Biot number gas speci6c constant in Eq. (28) gas speci6c constant in Eq. (28) volumetric rate of heat generation non-dimensional heat generation number interfacial thermal conductance enthalpy Knudsen number characteristic microscopic length characteristic macroscopic length unit normal to pressure reference pressure equilibrium pressure gas constant entropy temperature imposed temperature di%erence weight function non-dimensional macroscale variable hydrogen concentration dimensional macroscale variable
precisely by using three-dimensional 6nite element method. The e%ective thermal conductivity is calculated to study the mechanism of heat transfer and the e%ect of the characteristic behavior including the pulverization and the change of the contact area is investigated. 2. Model For the analysis of the metal hydride packed bed in Fig. 1(a), a periodic composite is considered simply and illustrated in Fig. 1(b). A periodic structure consists of two domains, solid phase, &s and gas phase, &g , as shown in Fig. 1(c). In the following equations, the subscripts s and g mean solid and gas, respectively. denotes the interface between the components. The periodic domain & is small compared to the characteristic length, L at the macroscopic scale: l = 1; (1) L where is a scale parameter. l and L can be assimilated to the characteristic sizes of the microscopic and the macroscopic sample, respectively. In this analysis, l presents the particle diameter of the metal hydride and is about 10−4 –10−6 .
xH max y ’0 ! " "m "0 # $ % &
hydrogen saturation concentration non-dimensional microscale variable ratio of thermal conductivity hysteresis particular solution for T identity matrix small parameter (l=L) accommodation factor for plateau inclination common boundary of the two media Porosity non-dimensional thermal conductivity dimensional thermal conductivity mean free path dimensional thermal conductivity at p = 1 weight function temperature density domain
Subscripts e% g p q s 0,1,2
e%ective gas phase number of spatial dimension number of spatial dimension solid phase asymptotic expansion indices
The multi-scale periodic heat conduction problem in the medium described above, under the steady condition, can be mathematically expressed by @ @Ts − ∗ "s ∗ = gs in &s ; (2) @xj @xj −
@ @xj∗
"g
@Tg @xj∗
= gg
in &g ;
(3)
− "s
@Ts @Tg nj = −"g ∗ nj @xj∗ @xj
on ;
(4)
− "s
@Ts nj = h(Ts − Tg ) @xj∗
on ;
(5)
where ", T and g are the thermal conductivity, the temperature 6eld and the volumetric rate of heat generation at microscale, respectively. n is the unit vector locally normal to the boundary and pointing to the outside of &s . Also h is interfacial thermal conductance. The above equations are general expressions, gs and gg become zero in the case of the metal hydride bed. Here, the meaning of the symbol ‘in x’ is dimensional parameter.
Y. Asakuma et al. / International Journal of Hydrogen Energy 29 (2004) 209 – 216
211
problem. The method proceeds by using the non-dimensional temperature 6eld as a function of two space variables, $(x; y), where x is given by x≡
x∗ : L
(15)
Multi-scale asymptotic expansions are introduced:
Fig. 1. Schematic diagram of homogenization method: (a) packed bed, (b) periodic structure, (c) unit cell.
De6nitions of the non-dimensional quantities are y≡
x∗ ; l
$≡
T ; IT
!≡
"g : "s
(6)
Here, IT is an external temperature di%erence at macroscale. We can rewrite Eqs. (2)–(5) as @$s @ = Gs in &s ; (7) − @yj @yj @$g @ ! = Gg in &g ; (8) − @yj @yj −
@$s @$g nj = −! nj @yj @yj
−
@$s nj = Bi($s − $g ) @yj
on ;
(9)
on ;
(10)
where the non-dimensional heat generation number and Biot number are as follows: l2 G s ≡ gs ; "s IT
l2 G g ≡ gg ; "g IT
hl Bi ≡ : "s
(11)
By means of multiplying Eqs. (7) and (8) by the weight function, #, integrating over & and applying the 6rst form of Green’s theorem, we obtained @#s @$s @$s dy − #s nj ds = Gs #s dy; (12) @yj &s @yj @yj & @$g @#g @$g ! dy + !#g nj ds = Gg #g dy: @yj @yj @yj &g &g (13) By applying Eqs. (9) and (10) to Eqs. (12) and (13), we obtained @# @$ dy − Bi #$ ds = G# dy; (14) @yj @yj & & where = 1 if y ∈ &s , and = ! if y ∈ &g . The homogenization method is applied to the variational weak form Eq. (14) of the multi-scale heat conduction
$(x; y) = $0 (x; y) + $1 (x; y) + 2 $2 (x; y) + · · · ;
(16)
#(x; y) = #0 (x; y) + #1 (x; y) + 2 #2 (x; y) + · · · ;
(17)
where $k (x; y) and #k (x; y) are periodic functions in y. In this computation, we must take into account the fact that x and y should be considered as independent variables and the derivation operator is expressed by @ @ @ = + : @yj @yj @xj
(18)
The homogenization process, → 0, produces a set of equations satis6ed by $0 , which presents the macroscopic behavior of the heat transfer in this bed. See Ref. [19] for details. By using the chain rule of Eq. (18), we insert Eqs. (16) and (17) into Eq. (14) and obtain @#0 @#0 @#1 2 @#1 2 @#2 + + + + @yj @xj @yj @xj @yj & @$1 @$2 @$0 @$0 @$1 × dy + + + 2 + 2 @yj @xj @yj @xj @yj + Bi(#0 + #1 + 2 #2 )($0 + $1 + 2 $2 ) ds
=
&
G(#0 + #1 + 2 #2 ) dy:
(19)
The last step of homogenization process is to collect the same powers of , leading to two boundary value problems, one in a homogenized macroscopic region, and the other is in a periodic cell [20]. By collecting the terms of order 0 , we obtain that the temperature 6eld, $0 does not vary on the macroscale. In addition, we assumed that Bi = O( 0 ) (i.e., Bi1= ) and G = O( 2 ), and by collecting the terms of order 2 , we obtained @#1 @$0 @$1 dy + + Bi #1 $1 ds = 0: (20) @yj @xj @yj & We introduce an arbitrary additive y independent function, (y), as follows: $1 (x; y) = −p (y)
@$0 (x) : @xp
(21)
This characteristic function, p (y) is a periodic solution to Eq. (20) and corresponding to a unit temperature gradient.
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Y. Asakuma et al. / International Journal of Hydrogen Energy 29 (2004) 209 – 216 Table 1 Physical properties of LaNi4:7 Al0:3 Reaction enthalpy IH (J=mol H2 ) Reaction entropy IS (J=mol H2 K) Accommodation factor for plateau inclination (dimensionless), 0 (dimensionless) Hysteresis Density %s (g=cm3 ) Thermal conductivity "s (W=mK) Porosity when 6rst 6lled in ’0 (dimensionless)
Fig. 2. Finite element mesh of the unit cell: (a) particle, (b) external view.
−33820 −107:4 0.3, 0.005 0.098 7.44 12.5 0.531
through both the 6lling gas and the hydride particle, that is, the conduction against the neighboring cells is considered by using periodic boundary condition [14]. 3. Materials 3.1. Metal hydride The e%ective thermal conductivity from the homogenization method were compared with the experimental data by Hahne and Kallweit [1]. The kind of the metal hydride was LaNi4:7 Al0:3 HxH (xH max = 6). The important physical properties are given in Table 1. The steady-state concentration/pressure isotherms (PCT) are shown in the following equation: the Van’t Ho% equation: IH IS ln peq = − RT R
Fig. 3. De6nition of contact area ratio.
By inserting Eq. (21) into Eq. (20), we obtained @$0 @p @#1 @$0 jp − dy = Bi #1 p ds; @yj @yj @xp @xp &
+ ( ± 0 ) tan[.((xH =xH max ) − 0:5)] ± =2; (25) (22)
where is Kronecker delta. By simplifying Eq. (22), we got @#1 @p @#1 dy + Bi #1 p ds = dy: (23) @y @y @y j j p & & This equation can become the cell problem for the characteristic function, p (y), which is solvable by the 6nite element method. Here, the e%ective thermal conductivity is obtained as the homogenized property, as follows: 1 @q "e% = dy: (24) pq − & & @yp The 6nite element meshes of the unit cell are shown in Fig. 2, where the part of the hydride particle and the external view are presented. The number of nodes and elements were 4880 and 4374, respectively. Here, the validity of this model and the mesh number were con6rmed by comparing with the solution of Rocha and Cruz [14]. Biot number and contact area ratio, a were changed as a parameter for the e%ective thermal conductivity, "e% . De6nition of contact area ratio is shown in Fig. 3. Heat conduction
where peq , IH , IS, R, and are the equilibrium pressure, the enthalpy, the entropy changes, the gas constant, the accommodation factor for the plateau inclination and the hysteresis and these parameters are deduced from 6tting of the experimental results. 3.2. Filling gas H2 , He, N2 and Ar were considered as the 6lling gases. When the pressure is fairly low, the frequency of collisions between molecules in a microregion a%ects on the thermal conductivity of the gas. Actually, with the decrease in the pressure, the thermal conductivity of the gas decreases because of the increase in the mean free path of the gas molecules (Smoluchowski e%ect [2]). The thermal conductivity of the 6lling gas, "g is given as "0 "g = ; (26) 1 + 2b Kn "m ; (27) Kn = where "0 is the thermal conductivity of the gas at p = 1 bar and b is constant and Kn is Knudsen number. Here, is
Y. Asakuma et al. / International Journal of Hydrogen Energy 29 (2004) 209 – 216
213
Fig. 5. E%ect of Biot number on the e%ective thermal conductivity. Fig. 4. E%ect of the pressure on the thermal conductivity.
de6ned by the mean diameter of the pore. The mean free path, "m is p 0 CA "m = ; (28) p(1 + CB =T ) where CA and CB are gas speci6c constants, p0 is 0:00133 bar and T is temperature. The e%ect of the pressure on the thermal conductivity is shown in Fig. 4 when the 6lling gas are H2 , He, N2 and Ar and T is 20◦ C. For the high pressure (p ¿ 1 bar or Kn ¡ 0:01), the conductivities are constant. This area is de6ned as the gas continuum region. On the other hand, for the low pressure (p ¡ 1 bar), the conductivities are inCuenced strongly by the pressure. The upper and lower parts of this area consist of the Smoluchowski region (10−4 ¡ p ¡ 1 bar or 0:01 ¡ Kn ¡ 10) and the free molecular region (p ¡ 10−4 bar or Kn ¿ 10), respectively. 3.3. Validity of this model From the experimental data of the transient hot wire method by Hahne and Kallweit et al. [1], the e%ect of particle decay on the e%ective thermal conductivity was considered for LaNi4:7 Al0:3 in the case of the non-activated state (the 6lling gas helium) after several cycles. There are three kinds of characteristic behavior as follows. (1) The decrease in the conductivity at the high pressure (p ¿ 0:01 bar). (2) The shift of the conductivity forward the side of the high pressure. (3) The increase in the conductivity at the low pressure (p ¡ 0:01 bar). Hahne and Kallweit [1] explained such behaviors on the grounds of that both the particle diameter, l and the mean diameter of the pore, became smaller by absorption and desorption cycle of hydrogen. This particle decay in this analysis corresponds to the following two points, the
Fig. 6. E%ect of particle diameter on the e%ective thermal conductivity.
decrease in Biot number (≡ hl="s ) as shown in Eq. (11) and the increase in Knudsen number (≡ "m = ) as shown in Eq. (27). First, Fig. 5 shows the e%ect of Biot number on the e%ective thermal conductivity, "e% without considering Smoluchowski e%ect by the decrease in the mean diameter of the pore when the contact area ratio and temperature are constant (a = 0:0009; T = 20◦ C; = 36:1 m). These results represent the decrease in the conductivity only at the high pressure (p ¿ 0:01 bar) by the decreasing in Biot number, as shown in their experimental data [1]. When Biot number is small, the results at the high pressure accord with the values of H2 in Fig. 4. Also, it is predicted that Biot number becomes more than 10 in the comparison with their experimental data. Secondly, the e%ect on the e%ective thermal conductivity is investigated, considering Smoluchowski e%ect by the decrease in the mean diameter of the pore ( = 36:1; 14:5; 7:5 m) on condition that Biot number, the contact area ratio and temperature are constant (Bi = 100; a = 0:0009; T = 20◦ C). This e%ect is shown in Fig. 6. The e%ective thermal conductivity shifts forward the side of the high pressure by Smoluchowski e%ect, that is, the increase in Kn number by the decrease in the mean diameter of the pore. These behaviors are caused by the decrease in
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Y. Asakuma et al. / International Journal of Hydrogen Energy 29 (2004) 209 – 216
Fig. 8. E%ect of particle decay on the e%ective thermal conductivity for LaNi4:7 Al0:3 . Fig. 7. E%ect of contact area ratio on the e%ective thermal conductivity.
apparent conductivity of the 6lling gas, "g as shown in Eq. (26), however maximum and minimum values are constant. For the increase in the conductivity at the low pressure (p ¡ 0:01 bar) after several hydriding cycles, Hahne and Kallweit [1] has explained the reason by the change of poly-dispersed bed structure towards the mono-dispersed one on condition that the void fraction is constant. This change of the dispersed structure means the change of the contact area between particles. Therefore, the e%ect of particle decay on the e%ective thermal conductivity, "e% was analyzed against the contact area ratio between particles, a in unit cell in Fig. 2. This e%ect of the contact area is shown in Fig. 7 when Biot number, temperature and the mean diameter of the pore are constant (Bi = 100; T = 20◦ C; = 36:1 m). By the change of the contact area ratio in this analysis, these results could express the increase in the conductivity only at the low pressure (p ¡ 0:01 bar), at the free molecular region due to the increase in cycle number. Although it is important to consider the contact area between particles by the expansion or the contraction at the low pressure (p ¡ 0:01 bar), it is not necessary to consider the contact area at the high pressure (p ¿ 0:1 bar), actual operating pressure. Also, it is predicted that the contact area ratio becomes about 0.0009 in the comparison with their experimental data. From our consideration of Figs. 5, 6 and 7, the change of Biot, Knudsen number and the contact area of hydride particles must be considered to express the behaviors from the experimental data of Hahne and Kallweit [1] precisely and simultaneously. Fig. 8 shows the e%ect of both Biot number, the contact area ratio and the mean diameter of the pore on the e%ective conductivity, "e% . When the particle diameter after cycles is the half of initial diameter [1], the initial condition is Bi = 10, a = 0:0009, = 14:9 m and the condition after several hydriding cycles is Bi = 5, a = 0:0016, = 7:5 m, respectively. These results accord with their experimental data [1] qualitatively. Therefore, this analysis
Fig. 9. E%ect of the pressure on the e%ective thermal conductivity.
could express their experimental data and prove their consideration and the validity of this model. 4. Results For the metal hydride bed of LaNi4:7 Al0:3 HxH and four kinds of the 6lling gas (H2 , He, N2 and Ar), the e%ective thermal conductivity, "e% was calculated by the homogenization method at their delivery, i.e. in the non-activated state. The results are shown in Fig. 9 when Biot number is 100 from Fig. 5, the mean diameter of the pore, is 14:9 m from Fig. 6 and the contact area ratio between the particles, a is constant (a = 0:0009) from Fig. 7. The other detail conditions are described in Ref. [1]. The curves are the calculations by this method and plots are the experimental data of Hahne and Kallweit [1]. Analytical results by the homogenization can represent the typical S-curves of evaluated beds [2] and the inCuences of the 6lling gas, and show good agreements with experimental results. Di%erences between their experimental data and these results at high pressure are caused by the assumption that the hydride bed is expressed by the simple cell model (BCC; body-centered-cubic), which has the minimum contact points between particles and the minimum surface
Y. Asakuma et al. / International Journal of Hydrogen Energy 29 (2004) 209 – 216
215
Fig. 11. The e%ect of the metal hydride conductivity on the e%ective thermal conductivity. Fig. 10. Relation between the e%ective thermal conductivity and hydrogen concentration for LaNi4:7 Al0:3 : (a) experimental data [1], (b) homogenization method.
area between particles and gas as shown in Fig. 2. Better accordance with their experimental data will be obtained if more complex particle shape, which has larger surface area and more contact points, can be represented by 6nite element mesh. However, it is hard to make 6nite element mesh of face-center-cubic (FCC) model as unit cell. Also, it is assumed that the bed structure is periodic in this analysis. Actually, because the metal hydride bed consists of particles with various diameters, the bed structure becomes more complex. To resolve this problem, unit cell, which represents the complex and dispersed structure, must be considered by 6nite element mesh. However, it is diQcult for three-dimensional 6nite element method analysis to use the unit cell with more meshes than this cell as shown in Fig. 2 because of the computational capacity. From the hydrogen concentration/pressure isotherm (PCT: Eq. (25)) by the experimental data, the relation between the hydrogen concentration and the e%ective thermal conductivity, "e% was evaluated for LaNi4:7 Al0:3 HxH in various temperatures assuming that the contact area, Biot number and the mean diameter of the pore are constant (a = 0:0009, Bi = 100, = 14:9 m) from Figs. 5, 6 and 7. Their experimental data [1] and our results are shown in Figs. 10(a) and (b), respectively. These results could express the characteristic behaviors that the conductivities at high hydrogen concentration (-phase) approach limit values and the conductivities at low hydrogen concentration (-phase) increase rapidly. When temperature is high (T ¿ 20◦ C), that is, the equilibrium pressure is high (p ¿ 0:1 bar), good agreements are obtained each other. However, there are di%erences in the case of low temperature (T ¡ − 20◦ C). Two following points, the change of the contact area by the expansion and the change of the thermal conductivity of the metal hydride, "s by temper-
ature, explain these di%erences. First, in the case of the high equilibrium pressure (p ¿ 0:1 bar; T ¿ 20◦ C), at the gas continuum region, there is little inCuence of the contact area due to the expansion by absorbing hydrogen as shown in Fig. 7. However, in the case of the low equilibrium pressure (p ¡ 0:1 bar; T ¡ − 20◦ C), at the free molecular region or the Smoluchowski region, the e%ect of the contact area by absorbing hydrogen must be considered from Fig. 7. In this analysis, the e%ect of the contact area cannot be considered due to less research of correlation between the contact area and the hydrogen concentration. If the e%ect of the contact area by the expansion is considered, the conductivity at plateau region increases with the hydrogen concentration at the low equilibrium pressure region (p ¡ 0:1 bar; T ¡ − 20◦ C). Second, the e%ect of temperature on the thermal conductivity of metal hydride "s was analyzed. In general the conductivity of the metal hydride becomes large as temperature becomes low. The e%ect of the conductivity of the metal hydride on the e%ective thermal conductivity, "e% are shown in Fig. 11. When the conductivity of the metal hydride becomes large, the e%ective conductivity becomes large mainly at low equilibrium pressure region. Judging from the behavior that the di%erences at low temperature (T = −60◦ C) are caused from xH = 3 in Fig. 10, the e%ect of this thermal dependence becomes small. As a result, because actual operating temperature is high (T ¿ 20◦ C), the application of the homogenization method is useful for heat transfer in the hydride bed analysis. For the microstructure such as the metal hydride, heat generation must be considered as shown in right term of Eq. (19). However, we assume that temperature distribution in the microstructure, which is dependent on scale ratio, 2 does not exist. In future, the inCuence of heat generation must be investigated by using new characteristic function.
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5. Conclusions The homogenization method was introduced as new multi-scale model against heat transfer analysis of the metal hydride bed. This homogenization method could become a powerful tool for the e%ective thermal conductivity of the metal hydride considering microscopic behaviors such as the pulverization, the changes of the void fraction and the contact area. By considering the behaviors such as the pulverization, the changes of the contact area, the discussion and the consideration of Hahne and Kallweit [1] could be proved from these calculations that the conductivity increases with Biot number at the high pressure region and increases with the contact area ratio at the low pressure region. Furthermore, the validity of this model was con6rmed from the e%ect of the parameter, such as Biot and Knudsen number on the effective thermal conductivity. The three regions of the e%ective thermal conductivity could be expressed against the pressure and the various 6lling gases with Smoluchowski e%ect at the non-active conditions. Here, three regions consist of free molecular region, the Smoluchowski region and the gas continuum region. Also, the e%ective thermal conductivity was analyzed for the H2 concentration of the metal hydride. From good agreements with the experimental data [1] at high temperature, it was indicated that the homogenization method was useful for heat transfer analysis. References [1] Hahne D, Kallweit J. Thermal conductivity of metal hydride materials for storage of hydrogen: experimental investigation. Int J Hydrogen Energy 1998;23(2):107–14. [2] Griesinger A, Spindler K, Hahne E. Measurement and theoretical modelling of the e%ective thermal conductivity of zeolites. Int J Heat Mass Transfer 1999;42:4363–74. [3] Kapischke J, Hapke J. Measurement of the e%ective thermal conductivity of a Mg-MgH2 packed with oscillating heating. Exp Thermal Fluid Sci 1998;17:347–55. [4] Suda S, Komazaki Y. The e%ective thermal conductivity of a metal hydride bed packed in a multiple-waved sheet metal structure. J Less-Common Met 1991;172–174:1130–7. [5] Pons M, Dantzer P. Determination of thermal conductivity and wall heat transfer coeQcient of hydrogen storage materials. Int J Hydrogen Energy 1994;19:611–6.
[6] Pons M, Dantzer P. The e%ective thermal conductivity in hydride packed beds I. Study of basic mechanism with help of the Bauer and Schlunder Model. J Less-Common Met 1991;172–174:1147–56. [7] Sun DW, Deng SJ. Theoretical description and experimental measurements on the e%ective thermal conductivity in metal hydride powder beds. J Less-Common Met 1990;160:387–95. [8] Sun DW, Deng SJ. A theoretical model predicting the e%ective thermal conductivity in powdered metal hydride beds. Int J Hydrogen Energy 1990;15(5):331–6. [9] Guo Z, Sung HJ. Technical note conjugate heat and mass transfer in metal hydride beds in the hydriding process. Int J Heat Mass Transfer 1999;42:379–82. [10] Inomata A, Aoki H, Miura T. Measurement and modeling of hydriding and dehydriding kinetics. J Alloy Compound 1998;278:103–9. [11] Nakagawa T, Inomata A, Aoki H, Miura T. Numerical analysis of heat and mass transfer characteristics in the metal hydride bed. Int J Hydrogen Energy 2000;25:339–50. [12] Sun H, Di S, Zhang N, Changchun W. Micromechanics of composite materials using multivariable 6nite element method and homogenization theory. Int J Solids Struct 2001;38(17):3007–20. [13] Terada K, Ito T, Kikuchi N. Characterization of the mechanical behaviors of solid–Cuid mixture by the homogenization method. Comput Methods Appl Mech Eng 1998;153:223–57. [14] Rocha RPA, Cruz ME. Computation of the e%ective conductivity of unidirectional 6brous composites with an interfacial thermal resistance. Numer Heat Transfer Part A 2001;39:179–203. [15] Auriault JL, Ene HI. Macroscopic modeling of heat transfer in composites with thermal barrier. Int J Heat Mass Transfer 1994;38(18):2885–92. [16] Chung PW, Tamma KK. Homogenization of temperature-dependent thermal conductivity in composite materials. J Thermophys Heat Transfer 2001;15(1):10–7. [17] Taghite MB, Rohmattulla A, Lanchon-Ducaquis H, Taous K. Homogenization of thermal problem in the plate of a heat exchanger. Comput Methods Appl Mech Eng 1997;145: 381–402. [18] Bouddour A, Auriault JL, Mhamdi-Alaoui M. Heat and mass transfer in wet porous media in presence of evaporation-condensation. Int J Heat Mass Transfer 1998;41(11):2263–77. [19] Nguetseng G. Asymptotic analysis for a sti% variational problem arising in mechanics. SISM J Math Anal 1989;21:1394–414. [20] Auriault JL. Heterogeneous medium. Is an equivalent macroscopic description possible? Int J Eng Sci 1991;29: 785–95.
International Journal of Hydrogen Energy 29 (2004) 209 – 216
www.elsevier.com/locate/ijhydene
Homogenization method for e%ective thermal conductivity of metal hydride bed Yusuke Asakuma∗ , Shinsuke Miyauchi, Tsuyoshi Yamamoto, Hideyuki Aoki, Takatoshi Miura Department of Chemical Engineering, Tohoku University, 07 Aoba Aramaki, Aoba-ku, Sendai 980-8579, Japan Accepted 26 March 2003
Abstract The e%ective thermal conductivity of the metal hydride bed is analyzed by the homogenization method. This method can represent the microstructure in the bed precisely. The referenced material is LaNi4:7 Al0:3 , which is practically used. Temperature and pressure ranges investigated here are from −80◦ C to +140◦ C and from 10−6 to 100 bar, respectively. Hydrogen, helium, nitrogen and argon are chosen as the 6lling gas. The validity of the homogenization method as the multi-scale analysis is con6rmed by a good correspondence with the experimental data that the e%ective thermal conductivity depends on the pressure of the 6lling gas under these various conditions (Int. J. Hydrogen Energy 23(2) (1998) 107). The homogenization method can become a powerful tool for the estimation of the e%ective thermal conductivity of the metal hydride bed by considering the microscopic behavior such as the pulverization and the change of the contact area. ? 2003 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. Keywords: E%ective thermal conductivity; Homogenization method; Multiscale analysis; Microstructure
1. Introduction The metal hydride is promising for the storage method of hydrogen in the point that it has the advantage of a lower pressure operation and a reversible reaction. However, heat conduction in the hydride bed becomes very complicated because the metal hydride has various characteristics of the packed structure such as the pulverization, the change of the void fraction and the contact area due to the expansion and the contraction of hydride particles. When the metal hydride absorbs and desorbs hydrogen, heat transfer in the packed bed becomes very complicated. The e%ective thermal conductivity is very important because heat of reaction has to be removed for the exothermic formation and added for the endothermic decomposition [1–9]. Recently, the hydriding and dehydriding kinetics are measured and heat and mass transfer characteristic in the metal hydride bed is ∗ Corresponding author. Tel.: +81-22-217-7252; fax: +81-22217-6165. E-mail address: [email protected] (Y. Asakuma).
investigated [10,11]. From these analytical and experimental results, we have proposed a numerical model, which can predict the various conditions of reaction pro6les. Absorption and desorption rates depend on heat transfer rate because the metal hydride is used in a powdery form in order to provide a large surface area for the reaction. Therefore, thermal and structural analyses have to be conducted simultaneously and precisely. However, there is less research, which focuses on the microstructure of the hydride particle. Here, the homogenization method is introduced as the multi-scale analysis, which can reCect the microstructure (particle diameter is 10−5 –10−6 m) in the bed on the macroproperty such as the thermal conductivity. Although this method is applied to the stress and the fracture analysis frequently [12,13], it is developing in thermal analysis recently [14–18]. The conventional models for heat transfer in the metal hydride bed [2,8] are not perfectly adequate because they are simpli6ed and treated as two-dimensional system. Therefore, we think that the homogenization method is very useful for the metal hydride in the point that it can consider the change of the microstructure after several cycles
0360-3199/03/$ 30.00 ? 2003 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. doi:10.1016/S0360-3199(03)00106-X
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Nomenclature a A A∗ b Bi CA CB g G h IH Kn l L n p p0 Peq R IS T IT v x xH x∗
contact area ratio surface area of cell (=l2 ) contact area constant Biot number gas speci6c constant in Eq. (28) gas speci6c constant in Eq. (28) volumetric rate of heat generation non-dimensional heat generation number interfacial thermal conductance enthalpy Knudsen number characteristic microscopic length characteristic macroscopic length unit normal to pressure reference pressure equilibrium pressure gas constant entropy temperature imposed temperature di%erence weight function non-dimensional macroscale variable hydrogen concentration dimensional macroscale variable
precisely by using three-dimensional 6nite element method. The e%ective thermal conductivity is calculated to study the mechanism of heat transfer and the e%ect of the characteristic behavior including the pulverization and the change of the contact area is investigated. 2. Model For the analysis of the metal hydride packed bed in Fig. 1(a), a periodic composite is considered simply and illustrated in Fig. 1(b). A periodic structure consists of two domains, solid phase, &s and gas phase, &g , as shown in Fig. 1(c). In the following equations, the subscripts s and g mean solid and gas, respectively. denotes the interface between the components. The periodic domain & is small compared to the characteristic length, L at the macroscopic scale: l = 1; (1) L where is a scale parameter. l and L can be assimilated to the characteristic sizes of the microscopic and the macroscopic sample, respectively. In this analysis, l presents the particle diameter of the metal hydride and is about 10−4 –10−6 .
xH max y ’0 ! " "m "0 # $ % &
hydrogen saturation concentration non-dimensional microscale variable ratio of thermal conductivity hysteresis particular solution for T identity matrix small parameter (l=L) accommodation factor for plateau inclination common boundary of the two media Porosity non-dimensional thermal conductivity dimensional thermal conductivity mean free path dimensional thermal conductivity at p = 1 weight function temperature density domain
Subscripts e% g p q s 0,1,2
e%ective gas phase number of spatial dimension number of spatial dimension solid phase asymptotic expansion indices
The multi-scale periodic heat conduction problem in the medium described above, under the steady condition, can be mathematically expressed by @ @Ts − ∗ "s ∗ = gs in &s ; (2) @xj @xj −
@ @xj∗
"g
@Tg @xj∗
= gg
in &g ;
(3)
− "s
@Ts @Tg nj = −"g ∗ nj @xj∗ @xj
on ;
(4)
− "s
@Ts nj = h(Ts − Tg ) @xj∗
on ;
(5)
where ", T and g are the thermal conductivity, the temperature 6eld and the volumetric rate of heat generation at microscale, respectively. n is the unit vector locally normal to the boundary and pointing to the outside of &s . Also h is interfacial thermal conductance. The above equations are general expressions, gs and gg become zero in the case of the metal hydride bed. Here, the meaning of the symbol ‘in x’ is dimensional parameter.
Y. Asakuma et al. / International Journal of Hydrogen Energy 29 (2004) 209 – 216
211
problem. The method proceeds by using the non-dimensional temperature 6eld as a function of two space variables, $(x; y), where x is given by x≡
x∗ : L
(15)
Multi-scale asymptotic expansions are introduced:
Fig. 1. Schematic diagram of homogenization method: (a) packed bed, (b) periodic structure, (c) unit cell.
De6nitions of the non-dimensional quantities are y≡
x∗ ; l
$≡
T ; IT
!≡
"g : "s
(6)
Here, IT is an external temperature di%erence at macroscale. We can rewrite Eqs. (2)–(5) as @$s @ = Gs in &s ; (7) − @yj @yj @$g @ ! = Gg in &g ; (8) − @yj @yj −
@$s @$g nj = −! nj @yj @yj
−
@$s nj = Bi($s − $g ) @yj
on ;
(9)
on ;
(10)
where the non-dimensional heat generation number and Biot number are as follows: l2 G s ≡ gs ; "s IT
l2 G g ≡ gg ; "g IT
hl Bi ≡ : "s
(11)
By means of multiplying Eqs. (7) and (8) by the weight function, #, integrating over & and applying the 6rst form of Green’s theorem, we obtained @#s @$s @$s dy − #s nj ds = Gs #s dy; (12) @yj &s @yj @yj & @$g @#g @$g ! dy + !#g nj ds = Gg #g dy: @yj @yj @yj &g &g (13) By applying Eqs. (9) and (10) to Eqs. (12) and (13), we obtained @# @$ dy − Bi #$ ds = G# dy; (14) @yj @yj & & where = 1 if y ∈ &s , and = ! if y ∈ &g . The homogenization method is applied to the variational weak form Eq. (14) of the multi-scale heat conduction
$(x; y) = $0 (x; y) + $1 (x; y) + 2 $2 (x; y) + · · · ;
(16)
#(x; y) = #0 (x; y) + #1 (x; y) + 2 #2 (x; y) + · · · ;
(17)
where $k (x; y) and #k (x; y) are periodic functions in y. In this computation, we must take into account the fact that x and y should be considered as independent variables and the derivation operator is expressed by @ @ @ = + : @yj @yj @xj
(18)
The homogenization process, → 0, produces a set of equations satis6ed by $0 , which presents the macroscopic behavior of the heat transfer in this bed. See Ref. [19] for details. By using the chain rule of Eq. (18), we insert Eqs. (16) and (17) into Eq. (14) and obtain @#0 @#0 @#1 2 @#1 2 @#2 + + + + @yj @xj @yj @xj @yj & @$1 @$2 @$0 @$0 @$1 × dy + + + 2 + 2 @yj @xj @yj @xj @yj + Bi(#0 + #1 + 2 #2 )($0 + $1 + 2 $2 ) ds
=
&
G(#0 + #1 + 2 #2 ) dy:
(19)
The last step of homogenization process is to collect the same powers of , leading to two boundary value problems, one in a homogenized macroscopic region, and the other is in a periodic cell [20]. By collecting the terms of order 0 , we obtain that the temperature 6eld, $0 does not vary on the macroscale. In addition, we assumed that Bi = O( 0 ) (i.e., Bi1= ) and G = O( 2 ), and by collecting the terms of order 2 , we obtained @#1 @$0 @$1 dy + + Bi #1 $1 ds = 0: (20) @yj @xj @yj & We introduce an arbitrary additive y independent function, (y), as follows: $1 (x; y) = −p (y)
@$0 (x) : @xp
(21)
This characteristic function, p (y) is a periodic solution to Eq. (20) and corresponding to a unit temperature gradient.
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Y. Asakuma et al. / International Journal of Hydrogen Energy 29 (2004) 209 – 216 Table 1 Physical properties of LaNi4:7 Al0:3 Reaction enthalpy IH (J=mol H2 ) Reaction entropy IS (J=mol H2 K) Accommodation factor for plateau inclination (dimensionless), 0 (dimensionless) Hysteresis Density %s (g=cm3 ) Thermal conductivity "s (W=mK) Porosity when 6rst 6lled in ’0 (dimensionless)
Fig. 2. Finite element mesh of the unit cell: (a) particle, (b) external view.
−33820 −107:4 0.3, 0.005 0.098 7.44 12.5 0.531
through both the 6lling gas and the hydride particle, that is, the conduction against the neighboring cells is considered by using periodic boundary condition [14]. 3. Materials 3.1. Metal hydride The e%ective thermal conductivity from the homogenization method were compared with the experimental data by Hahne and Kallweit [1]. The kind of the metal hydride was LaNi4:7 Al0:3 HxH (xH max = 6). The important physical properties are given in Table 1. The steady-state concentration/pressure isotherms (PCT) are shown in the following equation: the Van’t Ho% equation: IH IS ln peq = − RT R
Fig. 3. De6nition of contact area ratio.
By inserting Eq. (21) into Eq. (20), we obtained @$0 @p @#1 @$0 jp − dy = Bi #1 p ds; @yj @yj @xp @xp &
+ ( ± 0 ) tan[.((xH =xH max ) − 0:5)] ± =2; (25) (22)
where is Kronecker delta. By simplifying Eq. (22), we got @#1 @p @#1 dy + Bi #1 p ds = dy: (23) @y @y @y j j p & & This equation can become the cell problem for the characteristic function, p (y), which is solvable by the 6nite element method. Here, the e%ective thermal conductivity is obtained as the homogenized property, as follows: 1 @q "e% = dy: (24) pq − & & @yp The 6nite element meshes of the unit cell are shown in Fig. 2, where the part of the hydride particle and the external view are presented. The number of nodes and elements were 4880 and 4374, respectively. Here, the validity of this model and the mesh number were con6rmed by comparing with the solution of Rocha and Cruz [14]. Biot number and contact area ratio, a were changed as a parameter for the e%ective thermal conductivity, "e% . De6nition of contact area ratio is shown in Fig. 3. Heat conduction
where peq , IH , IS, R, and are the equilibrium pressure, the enthalpy, the entropy changes, the gas constant, the accommodation factor for the plateau inclination and the hysteresis and these parameters are deduced from 6tting of the experimental results. 3.2. Filling gas H2 , He, N2 and Ar were considered as the 6lling gases. When the pressure is fairly low, the frequency of collisions between molecules in a microregion a%ects on the thermal conductivity of the gas. Actually, with the decrease in the pressure, the thermal conductivity of the gas decreases because of the increase in the mean free path of the gas molecules (Smoluchowski e%ect [2]). The thermal conductivity of the 6lling gas, "g is given as "0 "g = ; (26) 1 + 2b Kn "m ; (27) Kn = where "0 is the thermal conductivity of the gas at p = 1 bar and b is constant and Kn is Knudsen number. Here, is
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213
Fig. 5. E%ect of Biot number on the e%ective thermal conductivity. Fig. 4. E%ect of the pressure on the thermal conductivity.
de6ned by the mean diameter of the pore. The mean free path, "m is p 0 CA "m = ; (28) p(1 + CB =T ) where CA and CB are gas speci6c constants, p0 is 0:00133 bar and T is temperature. The e%ect of the pressure on the thermal conductivity is shown in Fig. 4 when the 6lling gas are H2 , He, N2 and Ar and T is 20◦ C. For the high pressure (p ¿ 1 bar or Kn ¡ 0:01), the conductivities are constant. This area is de6ned as the gas continuum region. On the other hand, for the low pressure (p ¡ 1 bar), the conductivities are inCuenced strongly by the pressure. The upper and lower parts of this area consist of the Smoluchowski region (10−4 ¡ p ¡ 1 bar or 0:01 ¡ Kn ¡ 10) and the free molecular region (p ¡ 10−4 bar or Kn ¿ 10), respectively. 3.3. Validity of this model From the experimental data of the transient hot wire method by Hahne and Kallweit et al. [1], the e%ect of particle decay on the e%ective thermal conductivity was considered for LaNi4:7 Al0:3 in the case of the non-activated state (the 6lling gas helium) after several cycles. There are three kinds of characteristic behavior as follows. (1) The decrease in the conductivity at the high pressure (p ¿ 0:01 bar). (2) The shift of the conductivity forward the side of the high pressure. (3) The increase in the conductivity at the low pressure (p ¡ 0:01 bar). Hahne and Kallweit [1] explained such behaviors on the grounds of that both the particle diameter, l and the mean diameter of the pore, became smaller by absorption and desorption cycle of hydrogen. This particle decay in this analysis corresponds to the following two points, the
Fig. 6. E%ect of particle diameter on the e%ective thermal conductivity.
decrease in Biot number (≡ hl="s ) as shown in Eq. (11) and the increase in Knudsen number (≡ "m = ) as shown in Eq. (27). First, Fig. 5 shows the e%ect of Biot number on the e%ective thermal conductivity, "e% without considering Smoluchowski e%ect by the decrease in the mean diameter of the pore when the contact area ratio and temperature are constant (a = 0:0009; T = 20◦ C; = 36:1 m). These results represent the decrease in the conductivity only at the high pressure (p ¿ 0:01 bar) by the decreasing in Biot number, as shown in their experimental data [1]. When Biot number is small, the results at the high pressure accord with the values of H2 in Fig. 4. Also, it is predicted that Biot number becomes more than 10 in the comparison with their experimental data. Secondly, the e%ect on the e%ective thermal conductivity is investigated, considering Smoluchowski e%ect by the decrease in the mean diameter of the pore ( = 36:1; 14:5; 7:5 m) on condition that Biot number, the contact area ratio and temperature are constant (Bi = 100; a = 0:0009; T = 20◦ C). This e%ect is shown in Fig. 6. The e%ective thermal conductivity shifts forward the side of the high pressure by Smoluchowski e%ect, that is, the increase in Kn number by the decrease in the mean diameter of the pore. These behaviors are caused by the decrease in
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Y. Asakuma et al. / International Journal of Hydrogen Energy 29 (2004) 209 – 216
Fig. 8. E%ect of particle decay on the e%ective thermal conductivity for LaNi4:7 Al0:3 . Fig. 7. E%ect of contact area ratio on the e%ective thermal conductivity.
apparent conductivity of the 6lling gas, "g as shown in Eq. (26), however maximum and minimum values are constant. For the increase in the conductivity at the low pressure (p ¡ 0:01 bar) after several hydriding cycles, Hahne and Kallweit [1] has explained the reason by the change of poly-dispersed bed structure towards the mono-dispersed one on condition that the void fraction is constant. This change of the dispersed structure means the change of the contact area between particles. Therefore, the e%ect of particle decay on the e%ective thermal conductivity, "e% was analyzed against the contact area ratio between particles, a in unit cell in Fig. 2. This e%ect of the contact area is shown in Fig. 7 when Biot number, temperature and the mean diameter of the pore are constant (Bi = 100; T = 20◦ C; = 36:1 m). By the change of the contact area ratio in this analysis, these results could express the increase in the conductivity only at the low pressure (p ¡ 0:01 bar), at the free molecular region due to the increase in cycle number. Although it is important to consider the contact area between particles by the expansion or the contraction at the low pressure (p ¡ 0:01 bar), it is not necessary to consider the contact area at the high pressure (p ¿ 0:1 bar), actual operating pressure. Also, it is predicted that the contact area ratio becomes about 0.0009 in the comparison with their experimental data. From our consideration of Figs. 5, 6 and 7, the change of Biot, Knudsen number and the contact area of hydride particles must be considered to express the behaviors from the experimental data of Hahne and Kallweit [1] precisely and simultaneously. Fig. 8 shows the e%ect of both Biot number, the contact area ratio and the mean diameter of the pore on the e%ective conductivity, "e% . When the particle diameter after cycles is the half of initial diameter [1], the initial condition is Bi = 10, a = 0:0009, = 14:9 m and the condition after several hydriding cycles is Bi = 5, a = 0:0016, = 7:5 m, respectively. These results accord with their experimental data [1] qualitatively. Therefore, this analysis
Fig. 9. E%ect of the pressure on the e%ective thermal conductivity.
could express their experimental data and prove their consideration and the validity of this model. 4. Results For the metal hydride bed of LaNi4:7 Al0:3 HxH and four kinds of the 6lling gas (H2 , He, N2 and Ar), the e%ective thermal conductivity, "e% was calculated by the homogenization method at their delivery, i.e. in the non-activated state. The results are shown in Fig. 9 when Biot number is 100 from Fig. 5, the mean diameter of the pore, is 14:9 m from Fig. 6 and the contact area ratio between the particles, a is constant (a = 0:0009) from Fig. 7. The other detail conditions are described in Ref. [1]. The curves are the calculations by this method and plots are the experimental data of Hahne and Kallweit [1]. Analytical results by the homogenization can represent the typical S-curves of evaluated beds [2] and the inCuences of the 6lling gas, and show good agreements with experimental results. Di%erences between their experimental data and these results at high pressure are caused by the assumption that the hydride bed is expressed by the simple cell model (BCC; body-centered-cubic), which has the minimum contact points between particles and the minimum surface
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215
Fig. 11. The e%ect of the metal hydride conductivity on the e%ective thermal conductivity. Fig. 10. Relation between the e%ective thermal conductivity and hydrogen concentration for LaNi4:7 Al0:3 : (a) experimental data [1], (b) homogenization method.
area between particles and gas as shown in Fig. 2. Better accordance with their experimental data will be obtained if more complex particle shape, which has larger surface area and more contact points, can be represented by 6nite element mesh. However, it is hard to make 6nite element mesh of face-center-cubic (FCC) model as unit cell. Also, it is assumed that the bed structure is periodic in this analysis. Actually, because the metal hydride bed consists of particles with various diameters, the bed structure becomes more complex. To resolve this problem, unit cell, which represents the complex and dispersed structure, must be considered by 6nite element mesh. However, it is diQcult for three-dimensional 6nite element method analysis to use the unit cell with more meshes than this cell as shown in Fig. 2 because of the computational capacity. From the hydrogen concentration/pressure isotherm (PCT: Eq. (25)) by the experimental data, the relation between the hydrogen concentration and the e%ective thermal conductivity, "e% was evaluated for LaNi4:7 Al0:3 HxH in various temperatures assuming that the contact area, Biot number and the mean diameter of the pore are constant (a = 0:0009, Bi = 100, = 14:9 m) from Figs. 5, 6 and 7. Their experimental data [1] and our results are shown in Figs. 10(a) and (b), respectively. These results could express the characteristic behaviors that the conductivities at high hydrogen concentration (-phase) approach limit values and the conductivities at low hydrogen concentration (-phase) increase rapidly. When temperature is high (T ¿ 20◦ C), that is, the equilibrium pressure is high (p ¿ 0:1 bar), good agreements are obtained each other. However, there are di%erences in the case of low temperature (T ¡ − 20◦ C). Two following points, the change of the contact area by the expansion and the change of the thermal conductivity of the metal hydride, "s by temper-
ature, explain these di%erences. First, in the case of the high equilibrium pressure (p ¿ 0:1 bar; T ¿ 20◦ C), at the gas continuum region, there is little inCuence of the contact area due to the expansion by absorbing hydrogen as shown in Fig. 7. However, in the case of the low equilibrium pressure (p ¡ 0:1 bar; T ¡ − 20◦ C), at the free molecular region or the Smoluchowski region, the e%ect of the contact area by absorbing hydrogen must be considered from Fig. 7. In this analysis, the e%ect of the contact area cannot be considered due to less research of correlation between the contact area and the hydrogen concentration. If the e%ect of the contact area by the expansion is considered, the conductivity at plateau region increases with the hydrogen concentration at the low equilibrium pressure region (p ¡ 0:1 bar; T ¡ − 20◦ C). Second, the e%ect of temperature on the thermal conductivity of metal hydride "s was analyzed. In general the conductivity of the metal hydride becomes large as temperature becomes low. The e%ect of the conductivity of the metal hydride on the e%ective thermal conductivity, "e% are shown in Fig. 11. When the conductivity of the metal hydride becomes large, the e%ective conductivity becomes large mainly at low equilibrium pressure region. Judging from the behavior that the di%erences at low temperature (T = −60◦ C) are caused from xH = 3 in Fig. 10, the e%ect of this thermal dependence becomes small. As a result, because actual operating temperature is high (T ¿ 20◦ C), the application of the homogenization method is useful for heat transfer in the hydride bed analysis. For the microstructure such as the metal hydride, heat generation must be considered as shown in right term of Eq. (19). However, we assume that temperature distribution in the microstructure, which is dependent on scale ratio, 2 does not exist. In future, the inCuence of heat generation must be investigated by using new characteristic function.
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5. Conclusions The homogenization method was introduced as new multi-scale model against heat transfer analysis of the metal hydride bed. This homogenization method could become a powerful tool for the e%ective thermal conductivity of the metal hydride considering microscopic behaviors such as the pulverization, the changes of the void fraction and the contact area. By considering the behaviors such as the pulverization, the changes of the contact area, the discussion and the consideration of Hahne and Kallweit [1] could be proved from these calculations that the conductivity increases with Biot number at the high pressure region and increases with the contact area ratio at the low pressure region. Furthermore, the validity of this model was con6rmed from the e%ect of the parameter, such as Biot and Knudsen number on the effective thermal conductivity. The three regions of the e%ective thermal conductivity could be expressed against the pressure and the various 6lling gases with Smoluchowski e%ect at the non-active conditions. Here, three regions consist of free molecular region, the Smoluchowski region and the gas continuum region. Also, the e%ective thermal conductivity was analyzed for the H2 concentration of the metal hydride. From good agreements with the experimental data [1] at high temperature, it was indicated that the homogenization method was useful for heat transfer analysis. References [1] Hahne D, Kallweit J. Thermal conductivity of metal hydride materials for storage of hydrogen: experimental investigation. Int J Hydrogen Energy 1998;23(2):107–14. [2] Griesinger A, Spindler K, Hahne E. Measurement and theoretical modelling of the e%ective thermal conductivity of zeolites. Int J Heat Mass Transfer 1999;42:4363–74. [3] Kapischke J, Hapke J. Measurement of the e%ective thermal conductivity of a Mg-MgH2 packed with oscillating heating. Exp Thermal Fluid Sci 1998;17:347–55. [4] Suda S, Komazaki Y. The e%ective thermal conductivity of a metal hydride bed packed in a multiple-waved sheet metal structure. J Less-Common Met 1991;172–174:1130–7. [5] Pons M, Dantzer P. Determination of thermal conductivity and wall heat transfer coeQcient of hydrogen storage materials. Int J Hydrogen Energy 1994;19:611–6.
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