Available online at www.sciencedirect.com International Journal of Hydrogen Energy 28 (2003) 537–557 www.elsevier.com/locate/ijhydene Study of two-dimensional and dynamic heat and mass transfer in a metal–hydrogen reactor Faouzi Askri, Abdelmajid Jemni∗, Sassi Ben Nasrallah Laboratoire d’Etudes des Syst�emes Thermiques et Energ�etiques, Ecole Nationale d’Ing�enieurs de Monastir, Avenue Ibn Eljazzar 5019 Monastir, Tunisia Received 16 April 2002; accepted 18 June 2002 Abstract To analyse heat and mass transfer in a metal–hydrogen reactor, the hypothesis that disregards the radiative heat transfer in the reactor, is typically used. In this paper, we take into account the radiative heat transfer and we test the validity of this hypothesis in the case of the LaNi5 and in the case of the magnesium. A theoretical model is conducted for the two-dimensional system where conduction, convection radiation and chemical reaction take place simultaneously. This model is solved by the 5nite volume method. The numerical simulation is used to present the time–space evolutions of the temperature and the hydride density in the reactor and to determinate the sensitivity to some parameters (absorption coe6cient, scattering coe6cient, reactor wall emissivity). ? 2002 International Association for Hydrogen Energy. Published by Elsevier Science Ltd. All rights reserved. 1. Introduction In metal hydrides, hydrogen is stored in the interatomic spaces of the metal. The storage vessel contains powdered metals (often alloys) that absorb hydrogen, and at the same time release heat when the tank is 5lled with hydrogen under pressure. By reducing the pressure and supplying heat, the hydrogen is released. Add to this, hydrogen contains more chemical energy per weight than any hydrocarbon fuel, but it is also the lightest existing substance and therefore prob- lematic to store e9ectively in small containers. Many appli- cations have recently been developed to take advantage of these properties, including rechargeable batteries and heat- ing and cooling systems (heat pumps, heat transformers, refrigerators, thermals compressors, heat storage systems). In general, these alloys are combinations of “A” metals which can absorb H2 independently (rare-earth elements such as La, Ti, Zr, Mg and Ca) with “B”metals which cannot absorb H2 (Fe, Ni, Mn, Co) [1]. The most common exam- ples of hydrogen storing alloys are Fe–Ti hydrides, La-Ni hydrides, Mg-hydrides and Ti–Zr–V series of hydrides [2]. ∗ Corresponding author. Tel.: +216-73-500-511; fax: +216-73- 500-514. E-mail address:
[email protected] (A. Jemni). The work in the scienti5c world is largely aiming at en- hancing the kinetics of current hydrogen storage alloys, as well as looking into new material combinations which may exhibit high hydrogen storage and fast kinetics at low tem- peratures. A lot of work is aiming at controlling the hydride microstructure, to create methods to fully hydrate the ma- terials. Thus, several models describing the heat and mass transfer processes in metal–hydrogen system [3–17], have been done in recent years. The used theoretical models con- sider some simplifying assumptions. Jemni et al. [12–15] proposed a model which is the least restrictive. Using this model, Ben Nasrallah et al. [16] studied, for the LaNi5– H2 system, the validity of the main assumptions considered by di9erent authors. The metal–hydrogen systems can be classed into high-temperature hydrides (Mg-hydrides) and low-temperature hydrides (La–Ni hydrides). For the 5rst class the sorption temperature varies between 150◦C and 400◦C, while for the second class, sorption temperatures varies between 20◦C and 90◦C [2]. Therefore, in the case of the high-temperature hydrides, the e9ect of the radiative heat transfer on the absorption/desorption process can be im- portant. However, all the proposed models were neglected this e9ect. So, the aim of this paper is the study of heat and mass transfer in metal–hydrogen reactor using a model that take into account the radiative heat transfer. We 5rst present 0360-3199/03/$ 30.00 ? 2002 International Association for Hydrogen Energy. Published by Elsevier Science Ltd. All rights reserved. PII: S0360 -3199(02)00141 -6 538 F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 Nomenclature Cp Speci5c heat, J kg−1 K−1 Dp particle diameter, m E activation energy, J mol−1 H conductance between hydride bed and around Kuid, w m−2 K−1 H reactor height, m H=M hydrogen to metal atomic ratio L radiative intensity (w m−2 sr−1) K permeability, m2 M hydrogen mass absorbed or desorbed, kg m−3 s−1 M molecular weight, kg mol−1 P pressure, Pa qr radiative heat Kux (w m−2) R reactor ray, m Rg universal gas constant, j mol−1 K−1 T temperature, K t time, s V gas velocity, m s−1 Greek letters LH reaction heat of formation, J kg−1 Lr thickness of the control volume, m Lt time increment, s Lz thickness of the control volume, m � porosity �r emissivity � thermal conductivity, W m−1 K−1 � dynamic viscosity, kg m−1 s−1 � density, kg m−3 �a absorption coe6cient, m−1 �s scattering coe6cient, m−1 � optical thickness � phase function ; � azimuthal and polar angles � ordinate direction Subscripts A absorption D desorption e e9ective eq equilibrium f cooling or heat Kuid g gas ge gas e9ective s solid Superscripts g gaseous phase s solid phase t top b bottom si side r radial z axial se solid e9ective ss saturated the set of equations which govern heat and mass transfer in the reactor during the sorption phenomena. The resolution of the resulting system of equations was e9ected numeri- cally by the 5nite volume method (MFV). A comparison between the results obtained with and without radiative heat transfer is presented in two cases (LaNi5- andMg-hydrides). Finally, the study of the sensitivity to some parameters (ab- sorption coe6cient, scattering coe6cient, reactor partitions emissivity) is presented. 2. Mathematical model The cylindrical reactor, considered in this paper, ex- change heat through lateral and bases areas at a constant temperature (Fig. 1). The reactor is composed of solid phase (metal-hydride) and a gaseous phase (hydrogen), it is therefore a discontinuous porous media. The equations which govern heat and mass transfer in porous media are generally obtained by changing the scale. We pass from mi- croscopic view, in which the averaging volume ! is small compared to the pores, to the macroscopic view in which the averaging volume is large with regard to the pores. This scale changing permits the conversion from the real discon- tinuous media to a 5ctitious continuous equivalent one. Each macroscopic term is obtained by averaging the microscopic one. We de5ne the average of some microscopic function ’ as ’i = 1 ! ∫ ∫ ∫ !i ’i d!; (1) where ’i is the quantity associated with the i phase. We also de5ne the intrinsic average over a phase i as: ’ii = 1 !i ∫ ∫ ∫ !i ’i d!; (2) where !i is the volume occupied by the phase i in the total averaging volume !. The macroscopic di9erential equations are obtained by taking the average of microscopic equations over the aver- aging volume ! and using closing assumptions. The micro- scopic equations are the mass, the energy and the momentum F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 539 H2 z r Fig. 1. Metal–hydrogen reactor. equations balance. These equations are obtained by using thermodynamic and mechanical laws of continuous media. Several simplifying assumptions are made in order to obtain a closed set of governing equations at macroscopic scale: (1) The viscous dissipation and compression work are neg- ligible, (2) the dispersion term and the tortuosity term are modelled as di9usive Kuxes, (3) the gas phase is ideal from the thermodynamic view point, (4) the medium enclosed by the cylinder is gray, (5) the medium is considered at local thermal equilibrium, it absorbs, emits and anisotropically scatters the radiative energy, (6) thermophysical properties are constant. Considering these assumptions, macroscopic equations governing heat and mass transfer in a metal-hydrogen reac- tor, when the transfers are two-dimensional and depending on the r and z axes, are as follows: Energy equation: (�Cp)e @ @t ( PT ) + Cpg� g gVgr @ @r ( PT ) + Cpg� g gVgz @ @z ( PT ) =�g 1 r @ @r ( r @ @r ( PT ) ) + �g @2 @z2 ( PT )− 1 r @ @r (rqrr) − @ 2 @z2 (qzr ) + mLH 0 + m PT (Cpg − Cps); (3) where �e=j�ge+(1−j)�se; (�Cp)e=jCpg�gg+(1−j)Cps�ss. Momentum equation: The gas velocity within the reactor can be expressed by the Darcy’s law where the gravitational e9ect is neglected: Vgr =− k�g @ @r (Pgg); Vgz =− k�g @ @z (Pgg): (4) Mass balance: For the gas, the mass conservation equa- tion of the hydrogen is j @ @t (�gg) + div(� g gVg) =−m: (5) Assuming that the hydrogen is an ideal gas (�gg=MgP g g =Rg PT ) and considering Darcy’s law, the mass conservation equa- tion of the hydrogen becomes jMg Rg 1 PT @ @t (Pgg) + jMgPgg Rg @ @t ( 1 PT ) − k 'g 1 r @ @r ( r @ @r (Pgg) ) − k 'g @2 @z2 (Pgg) =−m: (6) For the solid, the mass conservation equation of the solid becomes (1− j) @ @t (�ss) = m: (7) Reaction kinetics: The hydrogen mass absorbed or des- orbed, m, per unit time and unit volume is given by [3]: For the absorption case m= Ca exp ( − Ea Rg PT ) Ln ( Pgg Peq ) (�ss − �ss): (8) For the desorption case m= Cd exp ( − Ed Rg PT ) Pgg − Peq Peq �ss: (9) For the LaNi5–Hydrogen system [18]: Cd = 9:57 S −1; Ed = 16:420 kJ mol −1 of H2; Ca = 59:187 S −1 and Ea = 21:170 kJ mol −1 of H2; For the Mg–Hydrogen system [19]: Cd = 5:5× 108 S−1; Ed = 148:510 kJ mol−1 of H2; Ca = 2×105 S−1 and Ea = 100:230 kJ mol−1 of H2: Equilibrium pressure: The basic P–C–T properties of the hydride forming inter-metallic compound are the starting points in analysing the hydride devices. These properties are best presented in the form Van’t Ho9 equation which relate the plateau H2 pressure Peq to absolute temperature T of the hydride, enthalpy change LH , entropy change LS and gas constant Rg: Ln(Peq) = LH Rg PT − LS Rg ; (10) 540 F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 Table 1 The equilibrium pressure polynomial function coe6cients Coe6cients a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 Absorption 0.0075 15.2935 −34.577 39.9926 −26.7998 11.0397 −2.8416 0.446 −0.0391 0.0014 Desorption −1.4654 19.1902 −42.086 49.0869 −33.8194 14.4375 −3.8581 0.6275 −0.0567 0.0021 where LS depends only on the hydrogen-to-metal-atomic ratio (H=M). Jemni et al. [15] found that the evolution of the equilib- rium pressure, for the LaNi5–hydrogen system, is given as a function of temperature and the hydrogen-to-metal-atomic ratio (H=M). The best 5t is obtained with a polynomial func- tion f(H=M) of order 9, whose coe6cients, for the absorp- tion and desorption cases, are given in Table 1. Peq = f(H=M) exp ( LH Rg ( 1 T − 1 Tref )) : (11) Vigeholem et al. [18] demonstrated experimentally that, for the Mg–hydrogen system, the plateau H2 pressure has no slope and proposed the following expression: ln (Pe) =−8941T + 16:23: (12) Radiative Transfer Equation (RTE): In order to deter- mine the divergence of the radiative Kux appearing in the energy equation (3), it is necessary to solve the radiative transfer equation (RTE) that can be written as [20]: d ds (L(�˜; s)) =−(�a + �s)L(�˜; s) + �aL0(T (s)) + �s 4+ ∫ 4+ �(�˜′; �˜)L(�˜′; s) d�′: (13) In the coordinates system (r; �; ) illustrated in Fig. 2, the RTE is written as: 1 r @ @r (r�L(�˜; r˜)) + @ @z (,L(�˜; r˜))− 1 r @ @ (-L(�˜; r˜)) =− (�a + �s)L(�˜; r˜) + �aL0(T (˜r)) + �s 4+ ∫ 4+ �(�˜′; �˜)L(�˜′; r˜) d�′; (14) where �; - and , are the direction cosines and given by: � = sin(�) cos( ); -= sin(�) sin( ) and ,= cos(�); �˜=�e˜r +-e˜ +,e˜z is the unit vector describing the radiation direction, � is the scattering phase function which expressed by a Legendre polynomial series as �(�˜′; �˜) = ∑ CjPj cos(�˜ ′; �˜): Fig. 2. Coordinates system. The expression of the absorption and scattering coe6- cients are given by [21] �s = 1:5 1 dp (1− jr)(1− j)sr ; �a = 1:5 1 dp jr(1− j)sr ; where sr = 1+1:84(1− j)− 3:15(1− j)2 + 7:2(1− j)3 for j¿ 0:3. The knowledge of the radiative intensity within the reactor permits us to calculate the divergence of the radiative heat Kux div(q˜r) = �a ( 4+L0(T )− ∫ 4+ L(�˜; r˜) d� ) : (15) Initial conditions: Initially, the temperature, the pressure and the hydride density in the reactor are assumed to be constant PT (z; r; 0) = T0; (16) Pgg(z; r; 0) = P0; (17) �ss(z; r; 0) = �0: (18) F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 541 Boundaries hydrodynamic conditions: • Taking into account the symmetry about the z-axis, we 5nd that @ @r (Pgg)(z;0; t) = 0: (19) • The wall is impervious and therefore @ @r (Pgg)(z;R; t) = @ @z (Pgg)(H;r; t) = 0: (20) • At the face (z=0) the pressure is assumed to be constant Pgg(0; r; t) = P0: (21) Boundaries thermal conditions: • Taking into account the symmetry about the z-axis, we are able to write @ @r ( PT )(z;0; t) = 0: (22) • The heat Kux continuity through the lateral area (r = R) and the bases areas (z = 0 and H) allows us to write the following equations: − �e @@r ( PT )(z;R; t) + qrr(z; R; t) = h( PT (z; R; t)− Tf ); (23) �e @ @z ( PT )(0; r; t) − qzr (0; r; t) = h( PT (0; r; t)− Tf ); (24) − �e @@z ( PT )(H;r; t) + qzr (H; r; t) = h( PT (H; r; t)− Tf ); (25) where h is the conductance between hydride bed and Kuid around the reactor, considered at the temperature Tf . • We suppose that the side wall (r = R), the bottom wall (z = H) and the top wall (z = 0) are opaque, gray and di9usely emit and reKect radiative energy. L(0; r; �˜) = jbr · L◦( PT (0; r)) + 1− j b r + ∫ ′ �˜ ·˜nb¡0L(0; r; �˜′) ×|�˜′ · n˜b| d�′ for �˜ · n˜b ¿ 0; (26) L(H; r; �˜) = jtr · L◦( PT (H; r)) + 1− j t r + ∫ �˜′ ·˜nt¡0 L(H; r; �˜′) ×|�˜′ · n˜t | d�′ for �˜ · n˜t ¿ 0; (27) L(z; R; �˜) = jsir L◦( PT (z; R)) + 1− jsir + ∫ �˜′ ·˜nsi¡0 L(z; R; �˜′) ×|�˜′ · n˜si | d�′ for �˜ · n˜si ¿ 0; (28) where jtr ; jsir and jbr are, respectively, the emissivity of the top wall, the side wall and the bottom wall of the reactor. The n˜t ; n˜b and n˜si unit vectors are indicated in Fig. 3. The centreline (r = 0) is treated as a 5ctitious, perfectly specular reKecting boundary L(z; 0; �; ,) = L(z; 0;−�; ,): (29) z r is n tn j 2 1 1 2 i i+1 Nr Pi ,j j+1 Nz j_1 i_1 bn Fig. 3. Spatial-discretization (control volume L!). 3. Numerical method 3.1. Mass and energy equations The system of equations, that is presented in the previous sections, is now solved numerically by theMFV based on the notion of control domain as described by Patanker [22]. The advantage of this method is to insure the Kux conservation. The method consists of de5ning a grid of points Pi; j within the calculated domain and then builds around each point a control domain L!(i; j). Fig. 3 shows the mesh used in the numerical resolution. The point Pi; j is located in the center of the control domain. The value of the physical scalar ’ at the Pi; j and at the time t +Lt will be denoted as ’n+1i; j . The equations are integrated on this control domain and over the interval of time [t; t +Lt]. At the boundaries of the reactor, the equations are made discrete by integrating over the half of the control domain and by taking into account the boundary conditions. At the corner we have used the quarter of the control domain. In order to bring the resulting integral equations back to algebraic equations tying together the solution values at the nodes of the grid, we make the following hypotheses: • The Kuxes are constant on the face of the control domain that is perpendicular to them. • The accumulated terms and the source terms can be ap- proximated by their averages on the control domain con- structed around Pi; j . • In order to insure the stability of the numerical scheme, the values of convected quantities at the face of the control domain are assumed to be equal to their values at the grid point situated in the up stream (upwind scheme). 542 F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 Fig. 4. Angular discretization (control angle L�). • The 5rst derivation, which are evaluated on the control domain faces, are approximated by( @’ @r )n+1 i+1=2;j = ’n+1i+1; j − ’n+1i; j Lr : (30) Using an implicit scheme and taking into account these assumptions, the form of the resulting algebraic equations becomes A0’ n+1 i; j = Ae’ n+1 i+1; j + Aw’ n+1 i−1; j + An’ n+1 i; j+1 + As’ n+1 i; j−1 + A1: (31) The resulting system of algebraic equations is solved nu- merically by the iterative line-by-line method scanning. 3.2. Radiative Transfer Equation (RTE) Eq. (14) indicates that intensity depends on spatial posi- tion and angular direction. To discrete this equation a MFV is used. The choice of this method is justi5ed by: (i) this approach has emerged as a popular Kuid Kow solution pro- cedure and has been applied to compute a variety of Kuid Kow and heat transfer processes, (ii) with this method radiative energy is conserved within the control angle, con- trol volume, and globally for any number of control angles and control volumes arranged in any manner. Following the control volume spatial discretization prac- tice (Fig. 3), the angle space is subdivided into L� ∗ M control angles L� (Fig. 4). In order to bring the resulting integral equation back to algebraic equation, we make the following hypotheses: • The radiative Kuxes are constant on the face of the control domain that is perpendicular to them. • The source terms can be approximated by their averages on the control volume L! and on the control angle L�. Taking into account these assumptions, the integration of Eq. (14) over a typical two-dimensional L! and L� gives Ae(i)Dcr(l; m)Li+1=2; j; l;m − Aw(i)Dcr(l; m)Li−1=2; j; l;m +Ab(i)Dcz(l)Li; j+1=2; l;m − Ab(i)Dcz(l)Li; j−1=2; l;m +f(i; l; m)Li; j; l;m+1=2 − f(i; l; m− 1)Li; j; l;m−1=2 =− (�s +�a)L!(i)L�(l)Li; j; l;m+�aL!(i)L�(l)L0i; j + �s 4+ L!(i)(L�(l))2(1 + a)Li; j; l;m + ∑ n;q �s 4+ P(i; l; n)Li; j;n;q; (32) where Ab(i) = 2+(i − 1)(Lr)2; Ae(i) = 2+(i − 1=2)Lz; Aw(i) = 2+(i − 3=2)Lz; Dcr(l; m) = ∫ L�(l) � d� = (sin(mL )− sin((m− 1)L )) × ( L� 2 − 1 4 (sin(2lL�)− sin(2(l− 1)L�)) ) ; Dcz(l) = ∫ L�(l) , d� = L 2 (sin(lL�)2 − (sin(2(l− 1)L�))2); L�(l) = ∫ �(l) �(l−1) ∫ (m) (m−1) sin(�) d� d =−L (cos(lL�))− cos((l− 1)L�); f(i; l; m) =−(Ae(i)− Aw(i)) ( L� 2 − 1 4 (sin(2lL�) −sin(2(l− 1)L�)) ) sin(mL ); f(i; l; m− 1) =(Ae(i)− Aw(i)) ( L� 2 − 1 4 (sin(2lL�) −sin(2(l− 1)L�)) ) sin((m− 1)L ); P(i; l; n) = L!(i)L�(l)L�(n)(1 + a cos((l− n)L�)): F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 543 -1 -1/2 i i+1/2 i+1 j+1 j+1/2 -1/2 j-1 Ω Fig. 5. Typical control domain. z/H 4T r rq σ 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 - Present Work o Dua and Cheng [23] Fig. 6. Comparison of non-dimensional radial heat Kux distribution on the side wall (emitting and absorbing media). � = �aR = 1:0. To relate the intensities at the faces of the control volume to the nodal intensity, the spatial step scheme is used. While adopting the step scheme for the angular discretiza- tion, we obtain: f(i; l; m)Li; j; l;m+1=2 = max(f(i; l; m); 0)Li; j; l;m −max(−f(i; l; m); 0)Li; j; l;m+1; (33) f(i; l; m− 1)Li; j; l;m−1=2 = max(f(i; l; m− 1); 0)Li; j; l;m−1 −max(−f(i; l; m− 1); 0)Li; j; l;m: (34) 4T r r q σ _ z/H 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 - Present Work o Jendoubi [24] Fig. 7. Comparison of non-dimensional radial heat Kux distribution on the side wall (scattering media) for � = �sR = 1:0. 0 1000 2000 3000 20 40 60 80 __ _ _ ___ Te m pe ra tu re (˚ C) 4000 Time (s) Point A Point B Point C Experimental Model Fig. 8. Temperature evolution within the reactor (P0 = 10 bars; Tf = 293 K). In the case of Dcr(l; m)¿ 0 and Dz(l)¿ 0 (Fig. 5), the step scheme lead to Li+1=2; j; l;m = Li; j; l;m; Li−1=2; j; l;m = Li−1; j; l;m; Li; j+1=2; l;m = Li; j; l;m; Li; j−1=2; l;m = Li; j−1; l;m: 544 F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 400 800 0 2 4 6 8 10 ___ Pr es su re (ba rs) 0 1200 1600 2000 Time (s) Experience Model Fig. 9. Pressure evolution within the reservoir. Then, Eq. (32) becomes Ae(i)Dcr(l; m) +Ab(i)Dcz(l) + max(f(i; l; m); 0) +max(−f(i; l; m− 1); 0) +(�a + �s)L!(i)L�(l) − � 4+ LV (i)(L�(l))2(1 + a) Li; j; l;m =Aw(i)Dcr(l; m)Li−1; j; l;m + Ab(i)Dcz(l)Li; j−1; l;m +max(−f(i; l; m); 0)Li; j; l;m+1 +max(f(i; l; m− 1); 0)Li; j; l;m−1 + �aL0i; jL!(i)L�(l) + ∑ n;q � 4+ P(i; l; n)Li; j;n;q (n; q) �= (l; m): (35) The solution process, of Eq. (35), is initiated with the Dcr ¿ 0 and Dcz ¿ 0 conditions by a marching process. This process is repeated for the other directions and a solution is considered to be converged when it satis5ed the following constraint: |Li; j; l;m − L∗i; j; l;m|=Li; j; l;m ¡ 10−6, where L∗i; j; l;m is the value of the intensity in the previous iteration. 4. Results and discussion 4.1. Validity of the RTE numerical solutions To validate our numerical solutions to RTE, we have to consider two problems: • a problem of a 5nite cylindrical enclosure with absorbing and emitting media maintained at a constant temperature (T =100 K). The enclosure is cold (T =0 K) and black. • a problem of a 5nite cylindrical enclosure with scattering and cold media (T = 0 K). The lateral surface of the cylinder is carried to a temperature equals to 100 K and the bases surfaces are maintained to a temperature of zero. These two problems are numerically resolved by the FVM. The comparison (Figs. 6 and 7) between the present numerical results and those presented in References [23,24] shows a good agreement. 4.2. Validity of the numerical heat and mass transfer solutions without radiative heat transfer In order to validate the model without radiative transfer, in the case of the LaNi5–H2 system, we have realized two kinds of comparison with experimental data: • A comparison between the calculated and measured tem- perature [15] at di9erent points, in the absorption case, when the hydrogen inlet pressure is constant. • A comparison between the reactor total mass absorbed when the reactor is connected to a reservoir which initially involves a known quantity of hydrogen. Table 2 Thermophysical properties of materials and data used in computations Absorption Desorption LaNi5 Mg LaNi5 Mg Fluid temperature Tf (K) 293 635 350 670 Initial bed temperature T0 (K) 293 635 350 670 Pressure P0 (bars) 12 23 1 1 Permeability K (m2) 1:6× 10−11 1:6× 10−11 E9ective thermal conductivity �e (w m−1 K−1) 1.32 1 1.32 1 Reactor wall emissivity �r 0.7 0.7 Porosity � 0.5 0.5 F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 545 Fig. 10. Temperature pro5le in the Mg–H2 reactor at selected times (in the absorption case). 546 F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 Fig. 11. Evolution of hydride density in the Mg–H2 reactor at selected times (in the absorption case). F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 547 Fig. 12. Temperature pro5le in the Mg–H2 reactor at selected times (in the desorption case). 548 F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 Fig. 13. Evolution of the hydride density in the Mg–H2 reactor at selected times (in the desorption case). F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 549 Fig. 14. Time–space evolution of the di9erence between the reactor temperature obtained with and without radiative transfer in the absorption case and for the LaNi5–H2 system (Tar : reactor temperature given by the model with radiation, Tsr : reactor temperature given by the model without radiation). 550 F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 Fig. 15. Time–space evolution of the di9erence between the reactor temperature obtained with and without radiative transfer in the desorption case and for the LaNi5–H2 system (Tar : reactor temperature given by the model with radiation, Tsr : reactor temperature given by the model without radiation). F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 551 0 500 1000 1500 2000 2500 3000 3500 0 0.014 0.012 0.008 0.006 0.004 0.002 M as s a bs or be d (kg of H 2/k g o f L aN i5) 0.010 Time (s) Model without radiative transfer Model with radiative transfer 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 M as s d es or be d (kg of H 2/K g o f L aN i5) 500 1000 1500 2000 2500 3000 3500 Time (s) Model without radiative transfer Model with radiative transfer Fig. 16. InKuence of radiative heat transfer on the total mass absorbed and desorbed by the LaNi5. The calculated and the measured temperatures inside the reactor for three points are plotted in Fig. 8. We notice a good agreement between the model and the measured results. Fig. 9 presents the pressure evolutions in the reservoir, during the absorption case, given by the model and the ex- periment. This 5gure shows that there is a good agreement between the experimental and the numerical results. 4.3. Time–space evolution of the temperature and the hydride density The considered reactor has a volume of 235:6 cm3 and it is 5lled with magnesium which absorb hydrogen readily above 575 K at pressures exceeding the equilibrium level and its decomposition requires high temperatures (in excess of 660 K) [18]. The conditions of the simulation and the thermophysical properties are indicated in Table 2. Fig. 10 shows the temperature distribution (in absorption case) after 60, 3600, 7200, 10,800, 14,400 and 18; 000 s. Keeping in mind that the absorption reaction was exother- mic, the temperature in side the reactor increases at 5rst and then decrease as the reaction proceeds, this is because of the reaction velocity decrease. Against the wall, the tem- perature is less than in the center of the reactor due to the external Kuid cooling. After an important period of time, the metal tends towards saturation and the reaction veloc- ity decreases. Consequently, the heat released from the bed becomes too weak, so the problem comes down to the sta- tionary heat conduction and radiation inside an inert porous media. The estimated hydride metal density (in the absorption case) is shown in Fig. 11. We notice that the mass absorbed is higher near the wall where the temperature is low, this is because the reaction velocity decrease with temperature. So, at around 18; 000 s, hydride formation is completed all over the bed of the reactor. Fig. 12 shows the temperature distribution in the reactor (in the desorption case) after 60, 3600, 7200, 10,800, 14,400 and 18; 000 s. It is seen that temperature decrease in the bed since the dehydriding reaction is an endothermic reaction. The center of the bed has the minimum temperature while the wall slightly hotter due to the external heating Kuid. After a substantial period of time, the remaining quantity of hydrogen in the reactor becomes too small. Consequently, the heat needed for the hydrogen dissociation tends to zero, so the temperature in the reactor bed tend to external tem- perature as time proceeds. Fig. 13, shows that the mass desorbed is higher near the wall where the temperature is high. This is because the dis- sociation reaction velocity increase with temperature. When time is long enough, the hydride density inside the reactor tends to a constant. 4.4. E=ect of the radiative heat transfer In order to bring out the e9ect of the radiative heat transfer we simulated the reactor working behaviour with and without taking into account the radiative transfer. The used metal–hydrogen system are the LaNi5–H2 and the Mg–H2. 4.5. Case of LaNi5–hydrogen system Figs. 14 and 15 show the time–space evolution of the di9erence between reactor temperature calculated with and 552 F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 Fig. 17. Time–space evolution of the di9erence between the reactor temperature obtained with and without radiative transfer in the absorption case and for the Mg–H2 system (Tar : reactor temperature given by the model with radiation, Tsr : reactor temperature given by the model without radiation). F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 553 Fig. 18. Time–space evolution of the di9erence between the reactor temperature obtained with and without radiative transfer in the desorption case and for the Mg–H2 system (Tar : reactor temperature given by the model with radiation, Tsr : reactor temperature given by the model without radiation). 554 F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 0 10000 20000 30000 0 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 M as s a bs or be d (kg of H 2/K g o f M g) Time (s) Model with radiative transfer Model without radiative transfer 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 M as s d es or be d (kg of H 2/k g o f M g) Model with radiative transfer Model without radiative transfer 10000 Time (s) 20000 30000 Fig. 19. InKuence of radiative heat transfer on the total mass absorbed and desorbed by the magnesium. 00 4000 8000 12000 16000 20000 σ a σ a σ a .R=533.4 .R=266.7 σ a σ a σ a .R=10. .R=533.4 .R=266.7 .R=10. 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 M as s a bs or be d (kg of H 2/k g o f M g) Time (s) 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 M as s de so rb ed (k g o f H 2/k g o f M g) 4000 8000 12000 16000 20000 Time (s) Fig. 20. Sensitivity to the absorption coe6cient of the media for �sR = 114:3. without radiative heat transfer, respectively, in absorption and desorption case. We notice that this di9erence of tem- peratures does not exceed the 1% of the temperature of the reactor working. The total mass either absorbed and desorbed calculated with and without radiation is plotted in Fig. 16. We notice that the e9ect of radiative transfer on the hydrogen sorption process by the LaNi5, is negligible in regard to the other heat transfer modes. 4.6. Case of Mg–hydrogen system Figs. 17 and 18 show the time–space evolution of the di9erence between the reactor temperature calculated with and without radiative e9ect. We notice, according to these 5gures, that when the transfer of energy by radiation is taken into consideration, the media gotten cold more quickly in the case of absorption and it warms up more quickly in the case of desorption. This observation is explained by the fact F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 555 0 4000 12000 16000 20000 Time (s) σ s.R=228.6 σ s.R=114.3 σs.R=10. σs.R=228.6 σs.R=114.3 σs.R=10. 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 M as s a bs or be d (kg of H 2/k g o f M g) 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 M as s d es or be d (kg of H 2/k g o f M g) 8000 0 4000 12000 16000 20000 Time (s) 8000 Fig. 21. Sensitivity to the scattering coe6cient of the media for �aR = 266:7. 0 4000 8000 12000 16000 20000 Time (s) 0 4000 8000 12000 16000 20000 Time (s) Emissivit Emissivit 1.0 0.7 0.3 0.3 0.7 1.0 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 M as s a bs or be d (kg of H 2/k g o f M g) 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 M as s d es or be d (kg of H 2/k g o f M g) Fig. 22. Sensitivity to the emissivity of the reactor wall. that the radiance accelerates the transfer of heat through the media. Also, these 5gures show that this di9erence of temperatures reaches 40◦C for the absorption and 32◦C for the desorption. It corresponds, respectively, to 11% and 8% of the temperature of reactor working. The evolution of the total mass either absorbed or desorbed, calculated with and without radiative e9ect is represented in Fig. 19. We notice that the e9ect of radiative transfer, reduce the necessary time for the saturation of metal and the one to restore the totality of the quantity of hydrogen. The reduction of time is 48% for the absorption case and 39% for the desorption case. According to these results, and in the considered con- ditions in this study, we can conclude that radiative heat transfer e9ect cannot be neglected in the case of the Mg–H2 reactor. 4.7. Sensitivity to radiative parameters This study shows that, for a good prediction of heat and mass transfer in the case of the Mg–H2 system, the radiative heat transfer must be considered. Consequently, a study of heat andmass transfer sensitivity to radiative parameters (ab- sorption coe6cient, scattering coe6cient, reactor partitions 556 F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537–557 emissivity...) becomes of interest and permit us to determi- nate the parameters that must be known with precision. 4.8. Sensitivity to the absorption coe@cient Fig. 20 shows the time evolution, in the case of Mg– hydrogen system, of the total mass desorbed and absorbed for di9erent values of the absorption coe6cient of the media. These curves show that the kinetics of the absorption and desorption reactions increases with the absorption coe6cient of the media. It is showed that increasing this coe6cient up to 270 gives a substantial improvement rate of hydrogen absorbed or desorbed, whereas an increase above this value yields a little further improvement. Therefore, it is important to determine this radiative parameter with precision. 4.9. Sensitivity to the scattering coe@cient The curves that are presented in Fig. 21 show that the kinetics of the absorption and desorption reactions of the hydrogen by the magnesium are not very sensitive to the scattering coe6cient of the media. 4.10. Sensitivity to the emissivity of the wall reactor We presented in Fig. 22 the evolution of the total mass desorbed and absorbed according to the time, for di9erent values of the emissivity of the reactor wall. We notice that the process of the hydrogen sorption by the magnesium is not very sensitive to this radiative parameter. 5. Conclusions A mathematical model describing the two-dimensional dynamic heat and mass transfer within a metal–hydrogen reactor has been presented and solved by the 5nite volume method. The model takes into account the e9ect of the ra- diative heat transfer. The numerical simulation permitted to present the time–space evolution of the temperature and the hydride density inside the reactor. 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Study of two-dimensional and dynamic heat and mass transfer in a metal--hydrogen reactor Introduction Mathematical model Numerical method Mass and energy equations Radiative Transfer Equation (RTE) Results and discussion Validity of the RTE numerical solutions Validity of the numerical heat and mass transfer solutions without radiative heat transfer Time--space evolution of the temperature and the hydride density Effect of the radiative heat transfer Case of LaNi5--hydrogen system Case of Mg--hydrogen system Sensitivity to radiative parameters Sensitivity to the absorption coefficient Sensitivity to the scattering coefficient Sensitivity to the emissivity of the wall reactor Conclusions References