s ti an Jiaotong University, 28 Xianning Road, Xi’an 710049, Shaanxi hwest U The metal hydride (MH for short) reactor is widely used for proposed, as was summarized in a former paper [7]. Among these candidates, the tubular reactor (TR for short) is the earliest one developed and has been investigated extensively by researchers [8–12]. In general, the tubular reactor has the features of good sealing, high bearing pressure and be subdivided into two kinds, ‘‘inside the tube type’’ and simulation. In the 1980s, a 1-D mathematical model consid- ering heat conduction and reaction kinetics was used widely in describing the adsorption/ desorption process in the MH reactor [13–16]. Later Choi and Mills [17] used Darcy’s law to calculate hydrogen flow in the reactor; this treatment also * Corresponding author. Fax: þ86 29 8266 0689. Avai lab le at www.sc iencedi rect .com w. i n t e r n a t i on a l j o u r n a l o f h y d r o g e n en e r g y 3 4 ( 2 0 0 9 ) 1 8 5 2 – 1 8 6 1 E-mail address:
[email protected] (Z. Zhang). many industrial purposes, such as hydrogen storage, heat pump, thermal compression, etc. Therefore its design and performance analysis are important issues and have attracted many authors’ attentions [1–7]. Configuration design is a crucial part in the research of MH reactor. Until now many types of reactors have been ‘‘outside the tube type’’, the former beingmore popular. TR in this article refers to ‘‘inside the tube type’’. For certain types of MH reactor, performance analysis and optimization were carried out by many researchers, and average reaction ratewas often used as themain performance index. Generally these studies were completed by numerical Reaction rate ª 2009 Published by Elsevier Ltd on behalf of International Association for Hydrogen Energy. 1. Introduction guaranteed heat and mass transfer effect by using thin tubes. According towhere the alloy is packed, this type of reactor can a r t i c l e i n f o Article history: Received 20 April 2008 Received in revised form 15 August 2008 Accepted 24 November 2008 Available online 24 January 2009 Keywords: Metal hydride Reactor Heat transfer Mass transfer 0360-3199/$ – see front matter ª 2009 Publis doi:10.1016/j.ijhydene.2008.11.119 a b s t r a c t Transport process affects the performance of a metal hydride reactor significantly. Therefore in a former paper presented by the same authors, two parameters, which are known as heat transfer controlled reaction rate and mass transfer controlled reaction rate, were defined to account for this effect and assist the design of the reactors. However, a few simplifications were adopted in that article, which may result in some errors. In order to achieve better accuracy and clarity, more factors such as the external convection heat transfer and propagation of reaction front were considered here in the formulation of the parameters. Then numerical simulations for the adsorption in a tubular reactor were carried out and the situation under which parameter analysis can be applied was dis- cussed. More characteristics in the process were revealed by the newly formulated parameters, which could be seen from the comparison of the results by parameter analysis and numerical simulation. bSchool of Chemical Engineering, Nort niversity, Taibai Road, Xi’an 710069, PR China State Key Laboratory of Multiphase Flow in Power Engineering, Xi’ Province, PR China Fusheng Yang , Xiangyu Meng , Jianqiang Deng , Yuqi Wang , Zaoxiao Zhang * a Identifying heat and mass tran hydride reactor during adsorp about parameter analysis a a journa l homepage : ww hed by Elsevier Ltd on be fer characteristics of metal on: Improved formulation a b a, e lsev ie r . com/ loca te /he half of International Association for Hydrogen Energy. ib inertia boundary sf between solid and gas phase i n t e r n a t i o n a l j o u rn a l o f h y d r o g e n en e r g y 3 4 ( 2 0 0 9 ) 1 8 5 2 – 1 8 6 1 1853 introduced the notion that the process in a MH reactor can be considered as flow and transport phenomena in porous Nomenclature a coefficient (–) A area (m2) As specific surface area (m 2/m3) c1 constant heat flow (W) c2 constant heat generation rate (W/m 3) Cp specific heat capacity (J/(kg K)) E activation energy (J/mol) h heat transfer coefficient (W/(m2 K)) DH reaction heat (J/mol H2) k reaction rate constant (s�1) K permeability (m2) _m source term of reaction (kg/m3 s) MH molar mass of hydrogen (kg/mol) P pressure (bar ¼ 105 Pa) Q heat flow or hydrogen flow (W or mol/s) r reaction rate (mol H2/(m 3 s) R radial length in cylindrical coordinate (m) Rr universal gas constant (J/(mol K)) t time (s) T temperature (K) U ! gas velocity (m/s) V volume (m3) x length in the concerning direction (m) X reacted fraction (–) media. Then the concept was further developed by Jemni and his co-workers [18–20]. They formulated 2-D mathematical models based on a volume averaging method, which is a kind of classicalmethod in the study of porousmedia. In a different way following this notion, Lloyd et al. derived the governing equations of representative element volume from the basic conservation law [21]. Henceforward the basic procedure of numerical simulation in a MH reactor has been established, with more details being taken into account in recent studies [22–25]. Usually modeling of the process in a MH reactor is more andmore intricate, yet the conclusions of the studies by numerical simulation do not changemuch: transport process, especially heat transfer, affects the average reaction rate greatly, thus measures for enhancement of heat transfer should be undertaken to improve the performance of the MH reactor, such as inserting Al/ Cu matrix, decreasing the bed thickness, enhancement of external convection, and so on [26–30]. Considering the requirements for convenience in engi- neering practice, it seems necessary to introduce some tool other than numerical simulation to assist in the design and analysis of a MH reactor, which is easier to use yet capable of reflecting the characteristics of the dynamic process in the reactor. In an attempt to do so, the authors defined two key parameters, heat transfer controlled reaction rate and mass transfer controlled reaction rate, in a former article [7], and their validity were proved by comparison with the results of a numerical simulation. However, some simplifications in formulating the parameters resulted in some deviation of the theoretical description from the realistic case, thus a corrected formulation of the param- eters is presented in this article, towards improving their N heat transfer fluid in inner inl inlet m mass transfer mb mass transfer boundary n normal direction out outer s solid phase Z axial length in cylindrical coordinate (m) a modification coefficient (–) d thickness of reaction completion layer (m) 3 porosity (–) l thermal conductivity (W/(m K)) m dynamic viscosity (Pa s) r density (kg/m3) Subscripts a adsorption e equilibrium eff effective f gas phase h heat transfer hb heat transfer boundary accuracy and extending their use. Furthermore some discussion about the application of parameter analysis is given. 2. Improved definition of the parameters Some notations in using the two parameters were explained [7], including the sources of the errors. Here a few simplifica- tions adopted before are removed or clarified to improve the usability of the parameters. 2.1. Redefinition of heat transfer controlled reaction rate rh In the former paper [7], rh was defined by comparing Qh1 and Qh2, namely the heat flow determined by heat transfer conditions and the heat flow generated by the reaction. Two approximations were made for the formulation of Qh1: Qh1 ¼ leffVT,Ahzleff Tib � Thb Dxh Ahbzleff Tib � TN Dxh Ahb (1) Either ‘‘z’’ stands for one approximation. A modification coefficient ah is introduced here to compensate for the first approximation, which accounts for the heat flow accumula- tion and variation of geometry conditions; therefore Qh1 is rewritten as, Qh1 ¼ ahleff Tib � Thb Dxh Ahb (2) i n t e r n a t i on a l j o u r n a l o f h y d r o g e n en e r g y 3 4 ( 2 0 0 9 ) 1 8 5 2 – 1 8 6 11854 Next we can remove the second approximation in equation (1). On both sides of the heat transfer boundary the heat flow keeps constant, thus the heat conduction in the reaction bed and the convection outside the bed can be correlated as below: Qh1 ¼ ahleff Tib � Thb Dxh Ahb ¼ hNðThb � TNÞAhb (3) By simple manipulation the convective heat transfer is taken into account in calculating Qh1, Qh1 ¼ Tib � TNDxh ahleff þ 1 hN Ahb (4) If other thermal resistances are also to be considered, they can be dealt with in much the same way and can be added to the denominator of equation (4). Thus the new expression of parameter rh is: rh ¼ Tib � TNDxh ahleff þ 1 hN Ahb DH,V (5) 2.2. Redefinition of mass transfer controlled reaction rate rm Similarly, rm is defined by comparing Qm1 and Qm2, namely the hydrogen flow determined by mass transfer conditions and the hydrogen flow generated by reaction. Qm1 was written as, Qm1 ¼ K m VPfAm,Pf= � RrTf � z K m Pmb � Pib Dxm Amb Pmb RrTmb (6) Here the linear approximation of pressure distribution was made. Before further deduction we need to formulate Qm1 in another way, that is Qm1 ¼ K m vPf vxm Am Pf RrTf ¼ K 2mRrTf vP2f vxm Am (7) Thus the hydrogen flow through the mass transfer boundary can be approximated as: Qm1z K 2mRrTmb P2mb � P2ib Dxm Amb (8) Then another modification coefficient am is introduced to correct the approximation in equation (8), Qm1 ¼ amK2mRrTmb P2mb � P2ib Dxm Amb (9) Finally parameter rm is expressed as follows: rm ¼ amK2mRrTmb P2mb � P2ib Dxm Amb V (10) The purpose of redefiningQm1 in equation (7) is explained: The initial formulations of Qh1 and Qm1 resemble each other in that Qh1 ¼ c1 (11) i. If heat transfer proceeds in the axial direction, xh is Z and Ah is constant, then the gradient of T is uniform every- where in the reaction bed and equals to the lumped difference, therefore ah ¼ 1. ii. If heat transfer proceeds in the radial direction, xh turns out to be R. Then the following equation holds: Qh1 ¼ leff vT vxh Ah ¼ leffvT vR 2pRZ (12) SinceQh1 is constant, the logarithmic distribution of T in radial direction is known by integration of the equation (12). Therefore both the local gradient and lumped difference of T can be expressed by functions of R, and then ah is derived according to where the heat transfer between the reaction bed and the fluid takes place. ah ¼ Rout � Rin ðlnRout � lnRinÞRout : on the outer boundary Rout � Rin� lnReff � lnRin � Rin : on the inner boundary 8>>< >>: (13) 2. Constant heat generation rate and steady state condition were written as, dQh1 ¼ c2,dV ¼ c2,Ah,dxh (14) i. If heat transfer proceeds in the axial direction, xh is Z and Ah is constant. According to equation (14), Qh1 increases linearly along xh. Therefore, according to equation (1), the gradient of T also grows linearly from zero to the local gradient on the heat transfer boundary. Knowing the quadratic distribution of T, we can find that ah ¼ 2. ii. If heat transfer proceeds in the radial direction, xh is R. Then the heat flow Qh1 can be calculated according to In Sections 2.1 and 2.2 two coefficients ah and am are used to correct the expressions of rh and rm respectively. Nextwewant to know the values of the coefficients. Both ah and am are defined to formulate the actual heat or hydrogen flowby lumped difference of T or Pf, which should be calculated based on the local gradient. Therefore the values of ah and am should reflect this deviation, which are determined by the 1-D distribution of T and Pf. Generally speaking, the detailed distribution information about T and Pf can hardly be obtained in advance, yet analysis under certain simplified conditions may help to specify the range of ah and am. Here ah in 2-D cylindrical coordinate is taken for example, suppose the inner and outer radiuses of reaction bed are respectively Rin and Rin, the length of reaction bed is Z. 1. Constant heat flow condition was written as follows: they are mainly determined by the gradient of a certain vari- able (T or Pf 2), and the linear approximation of the variables can be discussed in a similar way, thus the conclusions reached for ah are applied to am, which could simplify the study. 2.3. Values of the modification coefficients ah and am where the heat transfer between reaction bed and fluid takes place. The distribution of T in radial direction is obtained after combine equations (12) and (15), and ah is derived as, ah ¼ 2 � R2out � R2in � RoutðRout þ RinÞ � 2R2in lnðRout=RinÞ,RoutRout�Rin : on the outer boundary 2 � R2out � R2in � 2R2out lnðRout=RinÞ,Rin Rout�Rin � RinðRout þ RinÞ : on the inner boundary 8>>>>< >>>>: (16) According to equations (13) and (16), ah depends on Rin/Rin if heat transfer proceeds in the radial direction. Suppose Rin/Rin is between 1.1 and 10, the values of ah in the above two cases are calculated and shown in Figs. 1 and 2. On one hand, a is static. Take the heat transfer for example: as the reaction proceeds, the reaction front propagates from the boundary towards the inside. The bed near the heat transfer boundary tends to be saturated by hydrogen first, and then no accu- mulation of heat flow takes place here, thus a different rh should be determined. In this case, assuming the thickness of the reactor bed saturated by hydrogen is dh, this part can be simply taken as a thermal resistance and combined with 1/hN, while the remaining part of the bed is still dealt with following the procedure in Section 2.1. For heat transfer proceeding in the axial direction, rh can be easily written as, rh ¼ Tib � TN Ahb (17) Qh1 ¼ ( R R Rin dQh1 ¼ c2 R R Rin dV ¼ c2,p � R2 � R2in � Z : on the outer boundaryR Rout R dQh1 ¼ c2 R Rout R dV ¼ c2,p � R2out � R2 � Z : on the inner boundary (15) i n t e r n a t i o n a l j o u rn a l o f h y d r o g e n en e r g y 3 4 ( 2 0 0 9 ) 1 8 5 2 – 1 8 6 1 1855 h larger when the heat transfer takes place on the inner boundary in both cases, because the shrinkage of section area towards the boundary results in a higher local gradient, which implies a larger ah. On the other hand, heat flow accumulation also favors a higher local gradient on the boundary, therefore ah in the second case (constant heat generation rate) is larger than that in the first one (constant heat flow). Finally we turn to the actual case: reaction tends to be faster near the trans- port boundary, so a more significant heat flow accumulation will be observed over there. As a result, it is reasonable to deduce that the value of ah in the actual reaction should be even greater than the one calculated in the second case. Since constant generation rate condition is a better approximation of the actual case, the following parameter analysis will be carried out based on the assumption. 2.4. Dynamic analysis The formulations in Sections 2.1 and 2.2 were completed for the initial state of adsorption process and thus somehow Fig. 1 – The variation of modification coefficient ah (am) with geometry in the first case (constant heat flow). Dxh � dh ahleff þ dh leff þ 1 hN DH,V From the discussions in Section 2.3, it is shown that actual value of ah is larger than that is calculated under constant heat generation assumption, which is always larger than 1, there- fore equation (17) implies a decreasing rate as the reaction front propagates. The formulation of rh for radial heat transfer is complicated for the variation of Ahb, yet the same conclu- sion holds. Although the discussions in Section 2.3 and herein are conducted on ah, if we replace Qh1 with Qm1, T with Pf 2 and Dxh with Dxm, we can see that am can be dealt with in a similar way. Under constant mass flow or constant mass adsorption rate conditions, rm varies with geometric condi- tions in the same manner as rh. Ignoring the flow resistance Fig. 2 – The variation of modification coefficient ah (am) with geometry in the second case (constant heat generation rate). vTs vxn ¼ 0; vTf vxn ¼ 0; vPf vxn ¼ 0 (26a) Heat transfer wall: ð1� 3ÞlsvTs vxn ¼ hsðTs � TNÞ; 3lf vTf vxn ¼ hf � Tf � TN � ; vPf vxn ¼ 0 (26b) Mass transfer boundary: vTs vxn ¼ 0; Tf ¼ Tinl; Pf ¼ Pinl (26c) A fully implicit scheme based on control volume method was used to discretize the governing equations, and then the resulting algebraic equations were solved by ADI method. Converge was assumed when error of Ts, Tf and Pf were respectively lower than 10�3 K, 10�3 K and 10�6 bar. 3.2. Model validation Jemni et al. have carried out some experimental studies on a cylindrical MH reactor in 1999 [32]. The schematic of the reactor is shown in Fig. 3, and the temperatures of three locations(R ¼ 15 mm and Z ¼ 25, 35, 45 mm) were measured and recorded during adsorption. LaNi5 powder was packed in the reactor, the inlet pressure of hydrogen was 8 bar and the temperature of heat transfer medium was 293 K. Under the i n t e r n a t i on a l j o u r n a l o f h y d r o g e n en e r g y 3 4 ( 2 0 0 9 ) 1 8 5 2 – 1 8 6 11856 other than that of the bed, the dynamic rm is derived as follows: rm ¼ K2mRrTf P2mb � P2ib Dxm � dm am þ dm Amb V (18) 3. Dynamic process modeling As is pointed out in Section 1, the flow and heat transfer theory of porous media can be applied to describe the dynamic process in the reactor. With the differences between two phases considered, the non-local thermal equilibrium model for adsorption is as follows. 3.1. Governing equations and boundary conditions Mass equation for solid phase: vðð1� 3ÞrsÞ vt ¼ _m (19) Mass equation for gas phase: v3rf vt þ V � rf U !� ¼ � _m (20) Energy equation for solid phase: v �ð1� 3ÞrsCpsTs� vt ¼ Vðð1� 3ÞlsVTsÞ þ hsfAs � Tf � Ts �þ _m�DH=MH þ CpfTf � (21) Energy equation for gas phase: v � 3rf CpfTf � vt þ V � rf Cpf U ! Tf � ¼ V�3lfVTf �þ hsfAs�Ts � Tf � � _mCpfTf (22) Momentum equation for gas phase takes the form of Darcy’s law: U !¼ K m VPf (23) Reaction kinetic equations for either adsorption or desorption vary with the type of alloy used, the widely applied alloy- LaNi5 was taken for discussion in our article here. Kinetic equation for adsorption is as follows [18]: _m ¼ kaexp � � Ea RrTs � ln � Pf Pe � ð1� XÞ (24) P-C-T state equation is the one recommended by Dhaou et al.[31]: Pe ¼ X7 i¼0 aiX iexp � DH Rr � 1 Ts � 1 303 �� (25) The boundary conditions of MH reactors can be classified into three types: adiabatic wall (or symmetry boundary), heat transfer wall and mass transfer boundary, their specifications are: Adiabatic wall (or symmetry boundary): same conditions, numerical simulation on 60 � 25 grids was conducted. The temperature histories of the three locations predicted by our model are shownwith the experimental data presented by Jemni et al. in Fig. 4. It is seen that the simulation results agree well with the experimental results, thus the validity of the model is proved. Fig. 3 – The schematic of a cylindrical MH reactor. Fig. 4 – Comparison of the bed temperature between d e si g n d im e n si o n s a n d o p e ra ti o n co n d it io n s. D x h (m m ) P in (b a r) T N (K ) 2 0 3 0 6 8 1 0 2 8 3 2 93 3 0 3 7 1 7 1 3 .3 3 6 6 .4 4 9 1 1 .0 9 9 1 3 .3 3 6 1 5 .1 5 8 1 5 .7 1 8 1 3 .3 3 6 1 0 .9 5 4 6 6 7 6 5 0 2 7 3 9 2 0 3 6 9 9 7 0 6 6 7 6 5 0 1 0 5 0 4 0 0 6 7 5 0 9 0 6 6 7 6 5 0 6 5 1 5 0 0 i n t e r n a t i o n a l j o u rn a l o f h y d r o g e n en e r g y 3 4 ( 2 0 0 9 ) 1 8 5 2 – 1 8 6 1 1857 4. Results and discussion The tubular reactor (TR) is taken for discussion hereafter. The length, inner diameter and bed thickness of TR are respectively 60 mm, 40 mm and 20 mm. The inlet pressure and temperature are 8 bar and 293 K. The temperature and reacted fraction of reaction bed are 293 K and 0.1, and the equilibrium is assumed initially. The temperature of the heat transfer fluid is 293 K and the convection heat transfer coefficient is 1500 W/m2 K. The boundary conditions of TR in 2-D cylindrical coordinates are the same as has been described in Ref. [6]. 30 � 10 grids were used for simulation, and the time step was chosen to be 0.01 s. Average reaction rate is the main index used to describe the performance of the MH reactor, which can be calculated according to the simulation results of adsorption by tracking simulation and experiment. the temporal changes of average reacted fraction in the bed, aswasdefined in the formerpaper [7]. Thephysical properties of the material and data in simulation are listed in Table 1. 4.1. Re-examination of the former results In the former paper [7] the validity of parameter analysis has been proved by several examples. Here the parameters are Table 1 – Physical properties of the material and data in simulation. Metal (LaNi5) Hydrogen Density, rs/kg m �3 8400 – Specific heat, Cp/J kg �1 K�1 419 14890 Thermal conductivity, l/Wm�1 K�1 2.4 0.16 Permeability, K/m2 1.11 � 10�12 – Reaction enthalpy, DH/J mol�1 3.1 � 104 – Porosity, 3 0.50 – Universal gas constant, Rr/J mol �1 K�1 – 8.314 T a b le 2 – T h e ca lc u la ti o n re su lt s o f r h a n d r m fo r T R a t d if fe re n t R in (m m ) 1 0 2 0 3 0 1 0 r h (m o lH 2 /m 3 s) 1 4 .8 4 6 1 3 .3 3 6 1 2 .6 3 9 4 5 . r m (m o l H 2 /m 3 s) 5 7 7 2 5 0 6 6 7 6 5 0 7 1 2 3 9 0 2 9 5 7 4 0 0 former study can be drawn, that is smaller Rin, Dxh (denoted by Fig. 7 – Variation of average reaction rate with pre- i n t e r n a t i on a l j o u r n a l o f h y d r o g e n en e r g y 3 4 ( 2 0 0 9 ) 1 8 5 2 – 1 8 6 11858 d in the former paper) and TN, and larger Pin favor the adsorption process, while the order of their influence is Dxh > Pinz TN > Rin. 4.2. The characteristics of the average reaction rate curve and comparison with the results of the parameter analysis The average reaction rate curve is plotted using the calcula- redefined, thus we need to check about the calculation results again to guarantee the consistence of the method. The example of TR for validation is examined and the results calculated by newly defined parameters are shown in Table 2. As can be seen, the same analytical conclusion as that of the Fig. 5 – The comparison of average reaction rate with that calculated by parameter analysis. tion results by numerical simulation, as is shown in Fig. 5. Obviously, after a very fast kinetics during the initial period until around 14 s, the reaction slows down. The phenomenon Fig. 6 – The distribution of temperature in the reaction bed at 0, 14 and 50 s. can be explained as follows. When the process starts, the temperature is low and a driving force large enough is guar- anteed for the reaction throughout the bed. Basically, essen- tial reaction kinetics dominates the whole process in this stage and the average reaction rate is high. As the reaction proceeds, the heat is released and the temperature of reaction bed rises; later a relatively steady temperature distribution (as is shown in Fig. 6) will be built based on the heat transfer conditions. It is known that temperaturewill affect the driving force of the reaction, thus the average reaction rate drops for the rise of temperature and the distribution of reaction rate reflects the distribution of the temperature, so the heat transfer takes control of the process hereafter. It is also at this stage that the parameter analysis proposed by the authors works. The initial value of rh is calculated to be 10.56 mol H2/ (m3 s) and a corresponding reaction rate curve is plotted exponential factor of reaction kinetics. against the actual one. It is seen that the average reaction rate in the actual case is higher than rh, which also proves that ah Fig. 8 – Variation of average reaction rate with convection heat transfer coefficient (Dxh[ 1 mm). Fig. 9 – Variation of average reaction rate with convection heat transfer coefficient (Dx [ 2 mm). h e a t tr a n sf e r co n d it io n s. 2 0 3 0 1 .8 5 9 1 .8 4 4 6 0 0 9 0 0 1 2 0 0 1 5 0 0 6 0 0 9 0 0 1 2 0 0 1 5 0 0 9 .5 0 9 1 0 .0 6 4 1 0 .3 6 6 1 0 .5 5 7 4 .7 5 7 4 .9 4 8 5 .0 4 9 5 .1 1 2 i n t e r n a t i o n a l j o u rn a l o f h y d r o g e n en e r g y 3 4 ( 2 0 0 9 ) 1 8 5 2 – 1 8 6 1 1859 calculated in the assumed case of constant heat generation rate is underestimated. 4.3. The effect of the essential reaction kinetics on the average reaction rate The discussion is rather complicated, because the terms including Tor Pf in equation (24)may also be affectedbyheat or mass transfer conditions, under which situation the shear effect of reaction kinetics can hardly be identified. Therefore, only the pre-exponential factor kawas varied (the actual value of ka is around 60 s �1) and corresponding reaction rates are compared in Fig. 7. In the first stage that reaction kinetics dominates, it is clearly seen that reaction rate increases with the increase of k . While in the second stage that heat transfer h a dominates, the effect of ka is much attenuated and the corre- sponding reaction rates show little difference. Apparently, this Fig. 10 – Variation of average reaction rate with convection heat transfer coefficient (Dxh[ 3 mm). T a b le 3 – T h e ca lc u la ti o n re su lt s o f r h fo r T R a t v a ri e d co n v e ct io n D x h (m m ) 1 0 a h (– ) 1 .8 9 8 h (W /m 2 K ) 6 0 0 9 0 0 1 2 0 0 1 5 0 0 r h (m o l H 2 /m 3 s) 2 9 .8 1 3 2 .9 8 3 4 .8 3 3 6 .0 4 Convection heat transfer coefficient h is taken into consider- Refrigeration 1999;22:137–49. [10] Park JG, Jang KJ, Lee PS, Lee JY. The operating characteristics i n t e r n a t i on a l j o u r n a l o f h y d r o g e n en e r g y 3 4 ( 2 0 0 9 ) 1 8 5 2 – 1 8 6 11860 ation in the formulation of rh, so the effect of h on the reaction rate can be determined by both numerical simulation and parameter analysis. For a certain bed thickness Dxh (1, 2, 3 mm), hN was varied from 600 to 1500W/(m 2 K) and the corresponding reaction rates are compared in Figs. 8–10. Obviously, in the first stage that the reaction kinetics domi- nates, the reaction rates for different h are almost the same. However, in the second stage that the heat transfer domi- nates, the reaction rate increases with the increase of hN, yet the effect is less significant for higher h. Moreover, the effect of hN decreases with the increase of Dxh, the reason being that the heat conduction in the reaction bed will play a more important role for larger Dxh. These conclusions can also be drawn qualitatively from the parameter analysis, as is shown in Table 3. Discussions above are carried out for TR, in which heat transfer is the controlling step after the initial period. It was shown thatmass transfer can be the controlling step too [7]. In that case, essential reaction kinetics dominates in the initial stage and then mass transfer takes control. 4.5. The application of parameter analysis method Through the comparisonwith the simulation results in several cases, the validity of the method in characterizing the actual reaction process has been proved, thus the method can be used effectively in the analysis of theMH reactor and assist its primary design. For example, the performance of a MH reactor can be easily assessed by the method for different schemes, including the variations of alloys, operation conditions and configuration dimensions. This is particularly meaningful in the initial stage of a reactor’s design: after comparing the performances of reactors for different schemes, we can quickly concentrate on the superior schemes before more intensive studies of numerical simulation or experiments are carried out, which provides much convenience. 5. Conclusions An improved formulation of the two parameters character- izing the dynamic process in a MH reactor was proposed, in which more realistic factors such as the convection heat transfer and thepropagationof reaction frontwereconsidered. Then a 2-Dmathematicalmodel of the adsorption processwas built and solved numerically. TR was taken for discussion and the average reaction rate curve was examined. It was found coincideswith the conclusion obtained by parameter analysis: even the average reaction rate determined by the slowest kinetics (ka ¼ 30 s�1) is still as high as 102.71 mol H2/(m3 s), which is much larger than rh, thus heat transfer rather than reaction kinetics controls the actual process and thedifference in reaction kinetics only plays a minor role in this case. 4.4. 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Int J Hydrogen Energy 1999;24:631–44. i n t e r n a t i o n a l j o u rn a l o f h y d r o g e n en e r g y 3 4 ( 2 0 0 9 ) 1 8 5 2 – 1 8 6 1 1861 Identifying heat and mass transfer characteristics of metal hydride reactor during adsorption: Improved formulation about parameter analysis Introduction Improved definition of the parameters Redefinition of heat transfer controlled reaction rate rh Redefinition of mass transfer controlled reaction rate rm Values of the modification coefficients alphah and alpham Dynamic analysis Dynamic process modeling Governing equations and boundary conditions Model validation Results and discussion Re-examination of the former results The characteristics of the average reaction rate curve and comparison with the results of the parameter analysis The effect of the essential reaction kinetics on the average reaction rate The effect of the convection heat transfer on the average reaction rate The application of parameter analysis method Conclusions Acknowledgment References