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Study of two-dimensional and dynamic heat and mass transfer in a metal–hydrogen reactor
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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 Systemes Thermiques et Energetiques, Ecole Nationale d’Ingenieurs 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 problematic to store e9ectively in small containers. Many applications have recently been developed to take advantage of these properties, including rechargeable batteries and heating 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 examples 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: +216-73500-514. E-mail address: [email protected] (A. Jemni).
The work in the scienti5c world is largely aiming at enhancing 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 temperatures. A lot of work is aiming at controlling the hydride microstructure, to create methods to fully hydrate the materials. 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 consider 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 important. 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: S 0 3 6 0 - 3 1 9 9 ( 0 2 ) 0 0 1 4 1 - 6
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F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537 – 557
Nomenclature Cp Dp E H H H=M L K M M P qr R Rg T t V
Speci5c heat, J kg−1 K −1 particle diameter, m activation energy, J mol−1 conductance between hydride bed and around Kuid, w m−2 K −1 reactor height, m hydrogen to metal atomic ratio radiative intensity (w m−2 sr −1 ) permeability, m2 hydrogen mass absorbed or desorbed, kg m−3 s−1 molecular weight, kg mol−1 pressure, Pa radiative heat Kux (w m−2 ) reactor ray, m universal gas constant, j mol−1 K −1 temperature, K time, s gas velocity, m s−1
Greek letters LH Lr Lt Lz r
reaction heat of formation, J kg−1 thickness of the control volume, m time increment, s thickness of the control volume, m porosity emissivity thermal conductivity, W m−1 K −1 dynamic viscosity, kg m−1 s−1
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 numerically by the 5nite volume method (MFV). A comparison between the results obtained with and without radiative heat transfer is presented in two cases (LaNi5 - and Mg-hydrides). Finally, the study of the sensitivity to some parameters (absorption coe6cient, scattering coe6cient, reactor partitions emissivity) is presented. 2. Mathematical model The cylindrical reactor, considered in this paper, exchange 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 microscopic view, in which the averaging volume ! is small
a s ;
density, kg m−3 absorption coe6cient, m−1 scattering coe6cient, m−1 optical thickness phase function azimuthal and polar angles ordinate direction
Subscripts A D e eq f g ge s
absorption desorption e9ective equilibrium cooling or heat Kuid gas gas e9ective solid
Superscripts g s t b si r z se ss
gaseous phase solid phase top bottom side radial axial solid e9ective saturated
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 discontinuous 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 1 ’i d!; (1) ’i = ! !i where ’i is the quantity associated with the i phase. We also de5ne the intrinsic average over a phase i as: 1 ’ii = ’i d!; (2) !i !i 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 averaging volume ! and using closing assumptions. The microscopic equations are the mass, the energy and the momentum
F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537 – 557
539
Momentum equation: The gas velocity within the reactor can be expressed by the Darcy’s law where the gravitational e9ect is neglected:
H2
Vgr = −
r
k @ (Pgg ); g @r
Vgz = −
k @ (Pgg ): g @z
(4)
Mass balance: For the gas, the mass conservation equation of the hydrogen is j
Fig. 1. Metal–hydrogen reactor.
For the solid, the mass conservation equation of the solid becomes
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 negligible, (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 reactor, when the transfers are two-dimensional and depending on the r and z axes, are as follows: Energy equation: @ P @ @ (T ) + Cpg gg Vgr (TP ) + Cpg gg Vgz (TP ) @t @r @z 1 @ @2 1 @ @ P r (T ) + g 2 (TP ) − (rqrr ) =g r @r @r @z r @r
(Cp )e
where
@2 z (qr ) + mLH 0 + mTP (Cpg − Cps ); @z 2
(5)
Assuming that the hydrogen is an ideal gas (gg =Mg Pgg =Rg TP ) and considering Darcy’s law, the mass conservation equation of the hydrogen becomes jMg Pgg @ 1 jMg 1 @ g (Pg ) + Rg TP @t Rg @t TP k @2 k 1 @ @ g (Pgg ) = −m: (6) − r (Pg ) − 'g r @r @r 'g @z 2
z
−
@ g (g ) + div(gg Vg ) = −m: @t
(3)
e =jge +(1−j)se ; (Cp )e =jCpg gg +(1−j)Cps ss .
(1 − j)
@ s (s ) = m: @t
(7)
Reaction kinetics: The hydrogen mass absorbed or desorbed, m, per unit time and unit volume is given by [3]: For the absorption case g Pg Ea m = Ca exp − (ss − ss ): Ln (8) Peq Rg TP For the desorption case g Pg − Peq s Ed s : m = Cd exp − Peq Rg TP
(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 ; Ca = 2 × 105 S−1
Ed = 148:510 kJ mol−1 of H2 ; 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 ) =
LS LH − ; Rg Rg TP
(10)
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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 Desorption
0.0075 −1.4654
15.2935 19.1902
−34.577 −42.086
39.9926 49.0869
−26.7998 −33.8194
11.0397 14.4375
−2.8416 −3.8581
0.446 0.6275
−0.0391 −0.0567
0.0014 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 equilibrium 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 function f(H=M ) of order 9, whose coe6cients, for the absorption and desorption cases, are given in Table 1. LH 1 1 Peq = f(H=M ) exp : (11) − Rg T Tref 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 ) = −
8941 + 16:23: T
(12)
Radiative Transfer Equation (RTE): In order to determine 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 ˜ s) + a L0 (T (s)) ˜ s)) = −(a + s )L( ; (L( ; ds s ˜ ; )L( ˜ ˜ ; s) d : + ( 4+ 4+
Fig. 2. Coordinates system.
The expression of the absorption and scattering coe6cients are given by [21] (13)
s = 1:5
1 (1 − jr )(1 − j)sr ; dp
In the coordinates system (r; ; ) illustrated in Fig. 2, the RTE is written as:
a = 1:5
1 jr (1 − j)sr ; dp
1 @ ˜ r)) + @ (,L( ;˜ ˜ r)) − 1 @ (-L( ;˜ ˜ r)) (rL( ;˜ r @r @z r @
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 ˜ r) d : div(˜ qr ) = a 4+L0 (T ) − L( ;˜ (15)
˜ r) + a L0 (T (˜r)) = − (a + s )L( ;˜ s ˜ ; )L( ˜ ˜ ;˜r) d ; + ( 4+ 4+
(14)
where ; - and , are the direction cosines and given by: = sin( ) cos( ); , = cos( );
- = sin( ) sin( )
and
˜ =˜ er +-e˜ +,˜ ez is the unit vector describing the radiation direction, is the scattering phase function which expressed by a Legendre polynomial series as ˜ ; ) ˜ = ˜ ; ): ˜ ( Cj Pj cos(
4+
Initial conditions: Initially, the temperature, the pressure and the hydride density in the reactor are assumed to be constant TP (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 @ (19) (Pgg )(z; 0; t) = 0: @r • The wall is impervious and therefore @ @ (Pgg )(z; R; t) = (Pgg )(H; r; t) = 0: @r @z
(20)
• At the face (z = 0) the pressure is assumed to be constant Pgg (0; r; t) = P0 :
(21)
Boundaries thermal conditions:
• 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: @ P − e (T )(z; R; t) + qrr (z; R; t) = h(TP (z; R; t) − Tf ); (23) @r @ P (24) e (T )(0; r; t) − qrz (0; r; t) = h(TP (0; r; t) − Tf ); @z @ P − e (T )(H; r; t) + qrz (H; r; t) = h(TP (H; r; t) − Tf ); (25) @z 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. 1 − jbr +
˜
˜ ) ·˜nb¡0 L(0; r;
˜ · ˜nb ¿ 0; for (26) t ˜ ) ˜ = jtr · L◦ (TP (H; r)) + 1 − jr L(H; r; L(H; r; ) + ˜ ·˜nt ¡0 ˜ · ˜nb | d ×|
˜ · ˜nt ¿ 0; for (27) si ˜ = jsr i L◦ (TP (z; R)) + 1 − jr ˜ ) L(z; R; ) L(z; R; + ˜ ·˜ns ¡0 ˜ · ˜nt | d ×|
i
˜
×| · ˜nsi | d
˜ · ˜nsi ¿ 0; for
(28)
where jtr ; jsr i and jbr are, respectively, the emissivity of the top wall, the side wall and the bottom wall of the reactor. The ˜nt ; ˜nb and ˜nsi 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; −; ,):
Pi ,j
z
Nz
j+1
ns
i
j
j_1 2
• Taking into account the symmetry about the z-axis, we are able to write @ P (22) (T )(z; 0; t) = 0: @r
˜ = jbr · L◦ (TP (0; r)) + L(0; r; )
nt
(29)
r
1 1
2
i_1
i+1
i
Nr
nb
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 the MFV 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+1 i; 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 approximated by their averages on the control domain constructed 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).
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F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537 – 557
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 + a L!(i)L (l)L0i; j s L!(i)(L (l))2 (1 + a)Li; j; l; m 4+ s + P(i; l; n)Li; j; n; q ; 4+ n;q +
(32)
where Fig. 4. Angular discretization (control angle L ).
Ab (i) = 2+(i − 1)(Lr)2 ; Ae (i) = 2+(i − 1=2)Lz;
• The 5rst derivation, which are evaluated on the control domain faces, are approximated by n+1 n+1 ’n+1 @’ i+1; j − ’i; j = : (30) @r i+1=2;j Lr Using an implicit scheme and taking into account these assumptions, the form of the resulting algebraic equations becomes n+1 n+1 n+1 n+1 A0 ’n+1 i; j = Ae ’i+1; j + Aw ’i−1; j + An ’i; j+1 + As ’i; j−1 + A1 : (31)
The resulting system of algebraic equations is solved numerically by the iterative line-by-line method scanning.
Aw (i) = 2+(i − 3=2)Lz; Dcr (l; m) = d = (sin(mL ) − sin((m − 1)L )) L (l)
Dcz (l) = =
3.2. Radiative Transfer Equation (RTE) Eq. (14) indicates that intensity depends on spatial position 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 procedure 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, control volume, and globally for any number of control angles and control volumes arranged in any manner. Following the control volume spatial discretization practice (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 .
1 L − (sin(2lL ) − sin(2(l − 1)L )) ; 2 4
×
L (l)
L (sin(lL )2 − (sin(2(l − 1)L ))2 ); 2
L (l) =
, d
(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 1 − (sin(2lL ) 2 4
−sin(2(l − 1)L )) =(Ae (i) − Aw (i))
sin(mL ); f(i; l; m − 1) L 1 − (sin(2lL ) 2 4
−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
1.0
j+1 j+1/2
0.8
_ qr
0.6 - Present Work o Jendoubi [24]
r
σ T
-1/2
4
0.4
j-1 -1
-1/2
i
i+1/2
i+1 0.2
Ω
0.0 0.0
Fig. 5. Typical control domain.
0.2
0.4
1.0
Fig. 7. Comparison of non-dimensional radial heat Kux distribution on the side wall (scattering media) for = s R = 1:0.
1.0
0.8
80 Experimental
0.6
Model
Point A
r
qr
60
- Present Work o Dua and Cheng [23]
4
0.4
Temperature (˚C)
σT
0.8
0.6 z/ H
0.2
Point B Point C
__ _ _ __
40
0.0 0.0
0.2
0.4
0.6
0.8
1.0
20
z/ H Fig. 6. Comparison of non-dimensional radial heat Kux distribution on the side wall (emitting and absorbing media). = a R = 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 discretization, 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)
0
1000
− max(−f(i; l; m − 1); 0)Li; j; l; m : (34)
3000
4000
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 ;
f(i; l; m − 1)Li; j; l; m−1=2 = max(f(i; l; m − 1); 0)Li; j; l; m−1
2000 Time (s)
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
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.
10
8 Pressure (bars)
Experience
___ Model
6
4. Results and discussion 4
4.1. Validity of the RTE numerical solutions
2
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.
0 0
400
800 1200 Time (s)
1600
2000
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) Li; j; l; m +max(−f(i; l; m − 1); 0) + )L!(i)L (l) +( a s 2 LV (i)(L (l)) (1 + a) − 4+
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:
= 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 + a L0i; j L!(i)L (l) (35) P(i; l; n)Li; j; n; q (n; q) = (l; m): + 4+ n;q
• A comparison between the calculated and measured temperature [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.
The solution process, of Eq. (35), is initiated with the Dcr ¿ 0 and Dcz ¿ 0 conditions by a marching process. This
Table 2 Thermophysical properties of materials and data used in computations Absorption
Fluid temperature Initial bed temperature Pressure Permeability E9ective thermal conductivity Reactor wall emissivity Porosity
Tf (K) T0 (K) P0 (bars) K (m2 ) e (w m−1 K −1 ) r
Desorption
LaNi5
Mg
LaNi5
Mg
293 293 12
635 635 23
350 350 1
670 670 1
1
1.32
1.32
1:6 × 10−11 0.7 0.5
1:6 × 10−11 0.7 0.5
1
F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537 – 557
Fig. 10. Temperature pro5le in the Mg–H2 reactor at selected times (in the absorption case).
545
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
Fig. 12. Temperature pro5le in the Mg–H2 reactor at selected times (in the desorption case).
547
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 0.014
0.012
0.010 Model without radiative transfer Model with radiative transfer
0.008
0.006
0.004
Mass desorbed (kg of H2/Kg of LaNi5)
0.014
Mass absorbed (kg of H2/kg of LaNi5)
551
0.012 0.010 Model without radiative transfer 0.008
Model with radiative transfer
0.006 0.004 0.002 0.000
0.002 0
500
1000
1500
2000 2500 Time (s)
3000
3500
0
500
1000
1500 2000 Time (s)
2500
3000
3500
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 experiment. 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 exothermic, 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 temperature 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 velocity decreases. Consequently, the heat released from the bed becomes too weak, so the problem comes down to the stationary 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 temperature as time proceeds. Fig. 13, shows that the mass desorbed is higher near the wall where the temperature is high. This is because the dissociation 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).
F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537 – 557 0.09
0.09
0.08
0.08
0.07 0.06 0.05 Model with radiative transfer 0.04
Model without radiative transfer
0.03 0.02 0.01
Mass desorbed (kg of H2/kg of Mg)
Mass absorbed (kg of H2/Kg of Mg)
554
0.07 0.06 0.05 Model with radiative transfer 0.04
Model without radiative transfer
0.03 0.02 0.01
0.00
0.00 0
10000
30000
20000
10000
0
Time (s)
20000
30000
Time (s)
Fig. 19. InKuence of radiative heat transfer on the total mass absorbed and desorbed by the magnesium.
0.09
0.08 Mass desorbed (kg of H2/kg of Mg)
Mass absorbed (kg of H2/kg of Mg)
0.09
0.07 0.06 0.05 0.04
σa .R =533.4 σa.R =266.7 σa.R =10.
0.03 0.02 0.01
0.08 0.07 0.06
σa.R =533.4 σa.R =266.7 σa.R =10.
0.05 0.04 0.03 0.02 0.01 0.00
0.00 0
4000
12000 8000 Time (s)
16000
20000
0
4000
12000 8000 Time (s)
16000
20000
Fig. 20. Sensitivity to the absorption coe6cient of the media for s R = 114:3.
without radiative heat transfer, respectively, in absorption and desorption case. We notice that this di9erence of temperatures 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
0.09
0.09
0.08
0.08 Mass desorbed (kg of H2/kg of Mg)
Mass absorbed (kg of H2/kg of Mg)
F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537 – 557
0.07 0.06 0.05 0.04
σ s.R =228.6 σs.R =114.3 σ s.R =10.
0.03 0.02
555
0.07 0.06
σ s.R =228.6 σ s.R =114.3 σ s.R =10.
0.05 0.04 0.03 0.02 0.01
0.01
0.00
0.00 4000
0
8000 12000 Time (s)
16000
20000
0
4000
8000 12000 Time (s)
16000
20000
0.09
0.09
0.08
0.08 Mass desorbed (kg of H2/kg of Mg)
Mass absorbed (kg of H2/kg of Mg)
Fig. 21. Sensitivity to the scattering coe6cient of the media for a R = 266:7.
0.07 0.06
Emissivit 1.0 0.7 0.3
0.05 0.04 0.03 0.02 0.01 0.00
0.07 0.06 0.05
Emissivit
0.04
0.3 0.7 1.0
0.03 0.02 0.01 0.00
0
4000
8000 12000 Time (s)
16000
20000
0
4000
8000 12000 Time (s)
16000
20000
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 conditions 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 and mass transfer sensitivity to radiative parameters (absorption 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 determinate 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 radiative heat transfer. The numerical simulation permitted to present the time–space evolution of the temperature and the hydride density inside the reactor. These time–space evolutions show that the two dimensional e9ect on the sorption process is important. The results of the numerical simulation show, on the one hand that radiative e9ects on the sorption process are negligible in the case of the LaNi5 –hydrogen system and that they are important for the Mg–H2 system. On the other hand, in the case of the Mg–H2 system, the transfers are sensitive to the value of the absorption coe6cient of the media and are not very sensitive to the scattering coe6cient and the emissivity of the walls reactor. References [1] Anani A. Alloys for hydrogen storage in nickel/hydrogen and nickel/metal hydride batteries. Int J Power Sources 1994;47(3):261–75.
[2] T-Raissi A, Banerjee A, Sheinkopf K. Metal hydride storage requirements for transportation applications. 31st Intersociety Energy Conversion Engineering Conference, Vol. 4, 1996. p. 2280 –5. [3] Mayer U, Groll M, Supper W. Heat and mass transfer in metal hydride reaction beds: experimental and theoretical results. J Less-Common Metals 1987;131:235–44. [4] Suda S, Kobayashi N, Morishita E. Heat transmission analysis of metal hydride beds. J Less-Common Metals 1983;89: 325–32. [5] Da-Wen S, Song-Jiu D. Study on heat and mass transfer characteristics of metal hydride beds. In: Veziroglu TN, editor. Alternate energy sources II. New York: McGraw-Hill, 1988. p. 621–8. [6] Da-Wen S, Song-Jiu D. Study on heat and mass transfer characteristics of metal hydride beds. In: Veziroglu TN, editor. A two-dimensional model. J Less-Common Metals 1989; 155:271–9. [7] Dantzer P, Orgaz E. Hydriding kinetics and the problem of thermal transfer. Z Phys Chem NF 1989;164:1267–72. [8] Nagel M, Komazaki Y, Suda S. Heat transmission in metal hydride beds. J Less-Common Metals 1986;20:45–51. [9] Heung LK. Heat transmission and kinetics of metal hydride reactors. Z Phys Chem NF 1989;164:1415 –20. [10] Dantzer P. Static, dynamic and cycling studies on the hydrogen in the intermetallics. J Less-Common Metals 1987;131: 349–64. [11] El-osery LA, El-osairy MA, Metwally AM, Keshk MM, El-Gammel M. Dynamic simulation of the convective model for metal hydride hydrogen storage beds. Int J Energy Sources 1993;15:523–30. [12] Jemni A, Ben Nasrallah S, Lamloumi J, Percheron Guegan A. Study of heat and mass transfer in a metal hydrogen reactor. Z Phys Chem NF Bd 1994;183:197–203. [13] Jemni A, Ben Nasrallah S. Study of two-dimensional heat and mass transfer during absorption in a metal hydrogen reactor. Int J hydrogen Energy 1995;20(1):43–52. [14] Jemni A, Ben Nasrallah S. Study of two-dimensional heat and mass transfer during desorption in a metal hydrogen reactor. Int J hydrogen Energy 1995;20:43–52. [15] Jemni A, Ben Nasrallah S, Lamloumi J. Experimental and theoretical study of metal–hydrogen reactor. Int J hydrogen Energy 1999;24:631–44. [16] Ben Nasrallah S, Jemni A. Study of heat and mass transfer models in a metal hydrogen reactor. Int J hydrogen Energy 1997;22(1):67–76. [17] Kuznetsov AV, Vafai K. Comparison between the two- and three-phase models for analysis of formation in aluminum-rich castings. Numer Heat Transfer J A 1996;29:859–67. [18] Suda S, Kobayashi N, Yoshida K. Reaction kinetics of metal hydrides and their mixtures. J Less-common Metals 1980;73:119–26. [19] Vgeholm B, Kjeller J, Larsen B, Pedersen AS. Formation and decomposition of magnesium hydride. J Less-Common Metals 1983;89:135–44. [20] Siegel R, Howell JR. Thermal radiation heat transfer, 3rd ed. New York: McGraw-Hill, 1993. [21] Siegel R, Howell JR. Thermal radiation heat transfer, 2nd ed. New York: McGraw-Hill, 1981. [22] Patankar SV. Numerical heat transfer Kuid Kow. New York: Hemi-sphere/McGraw-Hill, 1980.
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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 Systemes Thermiques et Energetiques, Ecole Nationale d’Ingenieurs 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 problematic to store e9ectively in small containers. Many applications have recently been developed to take advantage of these properties, including rechargeable batteries and heating 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 examples 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: +216-73500-514. E-mail address: [email protected] (A. Jemni).
The work in the scienti5c world is largely aiming at enhancing 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 temperatures. A lot of work is aiming at controlling the hydride microstructure, to create methods to fully hydrate the materials. 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 consider 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 important. 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: S 0 3 6 0 - 3 1 9 9 ( 0 2 ) 0 0 1 4 1 - 6
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F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537 – 557
Nomenclature Cp Dp E H H H=M L K M M P qr R Rg T t V
Speci5c heat, J kg−1 K −1 particle diameter, m activation energy, J mol−1 conductance between hydride bed and around Kuid, w m−2 K −1 reactor height, m hydrogen to metal atomic ratio radiative intensity (w m−2 sr −1 ) permeability, m2 hydrogen mass absorbed or desorbed, kg m−3 s−1 molecular weight, kg mol−1 pressure, Pa radiative heat Kux (w m−2 ) reactor ray, m universal gas constant, j mol−1 K −1 temperature, K time, s gas velocity, m s−1
Greek letters LH Lr Lt Lz r
reaction heat of formation, J kg−1 thickness of the control volume, m time increment, s thickness of the control volume, m porosity emissivity thermal conductivity, W m−1 K −1 dynamic viscosity, kg m−1 s−1
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 numerically by the 5nite volume method (MFV). A comparison between the results obtained with and without radiative heat transfer is presented in two cases (LaNi5 - and Mg-hydrides). Finally, the study of the sensitivity to some parameters (absorption coe6cient, scattering coe6cient, reactor partitions emissivity) is presented. 2. Mathematical model The cylindrical reactor, considered in this paper, exchange 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 microscopic view, in which the averaging volume ! is small
a s ;
density, kg m−3 absorption coe6cient, m−1 scattering coe6cient, m−1 optical thickness phase function azimuthal and polar angles ordinate direction
Subscripts A D e eq f g ge s
absorption desorption e9ective equilibrium cooling or heat Kuid gas gas e9ective solid
Superscripts g s t b si r z se ss
gaseous phase solid phase top bottom side radial axial solid e9ective saturated
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 discontinuous 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 1 ’i d!; (1) ’i = ! !i where ’i is the quantity associated with the i phase. We also de5ne the intrinsic average over a phase i as: 1 ’ii = ’i d!; (2) !i !i 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 averaging volume ! and using closing assumptions. The microscopic equations are the mass, the energy and the momentum
F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537 – 557
539
Momentum equation: The gas velocity within the reactor can be expressed by the Darcy’s law where the gravitational e9ect is neglected:
H2
Vgr = −
r
k @ (Pgg ); g @r
Vgz = −
k @ (Pgg ): g @z
(4)
Mass balance: For the gas, the mass conservation equation of the hydrogen is j
Fig. 1. Metal–hydrogen reactor.
For the solid, the mass conservation equation of the solid becomes
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 negligible, (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 reactor, when the transfers are two-dimensional and depending on the r and z axes, are as follows: Energy equation: @ P @ @ (T ) + Cpg gg Vgr (TP ) + Cpg gg Vgz (TP ) @t @r @z 1 @ @2 1 @ @ P r (T ) + g 2 (TP ) − (rqrr ) =g r @r @r @z r @r
(Cp )e
where
@2 z (qr ) + mLH 0 + mTP (Cpg − Cps ); @z 2
(5)
Assuming that the hydrogen is an ideal gas (gg =Mg Pgg =Rg TP ) and considering Darcy’s law, the mass conservation equation of the hydrogen becomes jMg Pgg @ 1 jMg 1 @ g (Pg ) + Rg TP @t Rg @t TP k @2 k 1 @ @ g (Pgg ) = −m: (6) − r (Pg ) − 'g r @r @r 'g @z 2
z
−
@ g (g ) + div(gg Vg ) = −m: @t
(3)
e =jge +(1−j)se ; (Cp )e =jCpg gg +(1−j)Cps ss .
(1 − j)
@ s (s ) = m: @t
(7)
Reaction kinetics: The hydrogen mass absorbed or desorbed, m, per unit time and unit volume is given by [3]: For the absorption case g Pg Ea m = Ca exp − (ss − ss ): Ln (8) Peq Rg TP For the desorption case g Pg − Peq s Ed s : m = Cd exp − Peq Rg TP
(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 ; Ca = 2 × 105 S−1
Ed = 148:510 kJ mol−1 of H2 ; 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 ) =
LS LH − ; Rg Rg TP
(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 Desorption
0.0075 −1.4654
15.2935 19.1902
−34.577 −42.086
39.9926 49.0869
−26.7998 −33.8194
11.0397 14.4375
−2.8416 −3.8581
0.446 0.6275
−0.0391 −0.0567
0.0014 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 equilibrium 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 function f(H=M ) of order 9, whose coe6cients, for the absorption and desorption cases, are given in Table 1. LH 1 1 Peq = f(H=M ) exp : (11) − Rg T Tref 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 ) = −
8941 + 16:23: T
(12)
Radiative Transfer Equation (RTE): In order to determine 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 ˜ s) + a L0 (T (s)) ˜ s)) = −(a + s )L( ; (L( ; ds s ˜ ; )L( ˜ ˜ ; s) d : + ( 4+ 4+
Fig. 2. Coordinates system.
The expression of the absorption and scattering coe6cients are given by [21] (13)
s = 1:5
1 (1 − jr )(1 − j)sr ; dp
In the coordinates system (r; ; ) illustrated in Fig. 2, the RTE is written as:
a = 1:5
1 jr (1 − j)sr ; dp
1 @ ˜ r)) + @ (,L( ;˜ ˜ r)) − 1 @ (-L( ;˜ ˜ r)) (rL( ;˜ r @r @z r @
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 ˜ r) d : div(˜ qr ) = a 4+L0 (T ) − L( ;˜ (15)
˜ r) + a L0 (T (˜r)) = − (a + s )L( ;˜ s ˜ ; )L( ˜ ˜ ;˜r) d ; + ( 4+ 4+
(14)
where ; - and , are the direction cosines and given by: = sin( ) cos( ); , = cos( );
- = sin( ) sin( )
and
˜ =˜ er +-e˜ +,˜ ez is the unit vector describing the radiation direction, is the scattering phase function which expressed by a Legendre polynomial series as ˜ ; ) ˜ = ˜ ; ): ˜ ( Cj Pj cos(
4+
Initial conditions: Initially, the temperature, the pressure and the hydride density in the reactor are assumed to be constant TP (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 @ (19) (Pgg )(z; 0; t) = 0: @r • The wall is impervious and therefore @ @ (Pgg )(z; R; t) = (Pgg )(H; r; t) = 0: @r @z
(20)
• At the face (z = 0) the pressure is assumed to be constant Pgg (0; r; t) = P0 :
(21)
Boundaries thermal conditions:
• 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: @ P − e (T )(z; R; t) + qrr (z; R; t) = h(TP (z; R; t) − Tf ); (23) @r @ P (24) e (T )(0; r; t) − qrz (0; r; t) = h(TP (0; r; t) − Tf ); @z @ P − e (T )(H; r; t) + qrz (H; r; t) = h(TP (H; r; t) − Tf ); (25) @z 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. 1 − jbr +
˜
˜ ) ·˜nb¡0 L(0; r;
˜ · ˜nb ¿ 0; for (26) t ˜ ) ˜ = jtr · L◦ (TP (H; r)) + 1 − jr L(H; r; L(H; r; ) + ˜ ·˜nt ¡0 ˜ · ˜nb | d ×|
˜ · ˜nt ¿ 0; for (27) si ˜ = jsr i L◦ (TP (z; R)) + 1 − jr ˜ ) L(z; R; ) L(z; R; + ˜ ·˜ns ¡0 ˜ · ˜nt | d ×|
i
˜
×| · ˜nsi | d
˜ · ˜nsi ¿ 0; for
(28)
where jtr ; jsr i and jbr are, respectively, the emissivity of the top wall, the side wall and the bottom wall of the reactor. The ˜nt ; ˜nb and ˜nsi 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; −; ,):
Pi ,j
z
Nz
j+1
ns
i
j
j_1 2
• Taking into account the symmetry about the z-axis, we are able to write @ P (22) (T )(z; 0; t) = 0: @r
˜ = jbr · L◦ (TP (0; r)) + L(0; r; )
nt
(29)
r
1 1
2
i_1
i+1
i
Nr
nb
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 the MFV 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+1 i; 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 approximated by their averages on the control domain constructed 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).
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F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537 – 557
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 + a L!(i)L (l)L0i; j s L!(i)(L (l))2 (1 + a)Li; j; l; m 4+ s + P(i; l; n)Li; j; n; q ; 4+ n;q +
(32)
where Fig. 4. Angular discretization (control angle L ).
Ab (i) = 2+(i − 1)(Lr)2 ; Ae (i) = 2+(i − 1=2)Lz;
• The 5rst derivation, which are evaluated on the control domain faces, are approximated by n+1 n+1 ’n+1 @’ i+1; j − ’i; j = : (30) @r i+1=2;j Lr Using an implicit scheme and taking into account these assumptions, the form of the resulting algebraic equations becomes n+1 n+1 n+1 n+1 A0 ’n+1 i; j = Ae ’i+1; j + Aw ’i−1; j + An ’i; j+1 + As ’i; j−1 + A1 : (31)
The resulting system of algebraic equations is solved numerically by the iterative line-by-line method scanning.
Aw (i) = 2+(i − 3=2)Lz; Dcr (l; m) = d = (sin(mL ) − sin((m − 1)L )) L (l)
Dcz (l) = =
3.2. Radiative Transfer Equation (RTE) Eq. (14) indicates that intensity depends on spatial position 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 procedure 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, control volume, and globally for any number of control angles and control volumes arranged in any manner. Following the control volume spatial discretization practice (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 .
1 L − (sin(2lL ) − sin(2(l − 1)L )) ; 2 4
×
L (l)
L (sin(lL )2 − (sin(2(l − 1)L ))2 ); 2
L (l) =
, d
(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 1 − (sin(2lL ) 2 4
−sin(2(l − 1)L )) =(Ae (i) − Aw (i))
sin(mL ); f(i; l; m − 1) L 1 − (sin(2lL ) 2 4
−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
1.0
j+1 j+1/2
0.8
_ qr
0.6 - Present Work o Jendoubi [24]
r
σ T
-1/2
4
0.4
j-1 -1
-1/2
i
i+1/2
i+1 0.2
Ω
0.0 0.0
Fig. 5. Typical control domain.
0.2
0.4
1.0
Fig. 7. Comparison of non-dimensional radial heat Kux distribution on the side wall (scattering media) for = s R = 1:0.
1.0
0.8
80 Experimental
0.6
Model
Point A
r
qr
60
- Present Work o Dua and Cheng [23]
4
0.4
Temperature (˚C)
σT
0.8
0.6 z/ H
0.2
Point B Point C
__ _ _ __
40
0.0 0.0
0.2
0.4
0.6
0.8
1.0
20
z/ H Fig. 6. Comparison of non-dimensional radial heat Kux distribution on the side wall (emitting and absorbing media). = a R = 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 discretization, 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)
0
1000
− max(−f(i; l; m − 1); 0)Li; j; l; m : (34)
3000
4000
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 ;
f(i; l; m − 1)Li; j; l; m−1=2 = max(f(i; l; m − 1); 0)Li; j; l; m−1
2000 Time (s)
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
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.
10
8 Pressure (bars)
Experience
___ Model
6
4. Results and discussion 4
4.1. Validity of the RTE numerical solutions
2
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.
0 0
400
800 1200 Time (s)
1600
2000
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) Li; j; l; m +max(−f(i; l; m − 1); 0) + )L!(i)L (l) +( a s 2 LV (i)(L (l)) (1 + a) − 4+
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:
= 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 + a L0i; j L!(i)L (l) (35) P(i; l; n)Li; j; n; q (n; q) = (l; m): + 4+ n;q
• A comparison between the calculated and measured temperature [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.
The solution process, of Eq. (35), is initiated with the Dcr ¿ 0 and Dcz ¿ 0 conditions by a marching process. This
Table 2 Thermophysical properties of materials and data used in computations Absorption
Fluid temperature Initial bed temperature Pressure Permeability E9ective thermal conductivity Reactor wall emissivity Porosity
Tf (K) T0 (K) P0 (bars) K (m2 ) e (w m−1 K −1 ) r
Desorption
LaNi5
Mg
LaNi5
Mg
293 293 12
635 635 23
350 350 1
670 670 1
1
1.32
1.32
1:6 × 10−11 0.7 0.5
1:6 × 10−11 0.7 0.5
1
F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537 – 557
Fig. 10. Temperature pro5le in the Mg–H2 reactor at selected times (in the absorption case).
545
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
Fig. 12. Temperature pro5le in the Mg–H2 reactor at selected times (in the desorption case).
547
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 0.014
0.012
0.010 Model without radiative transfer Model with radiative transfer
0.008
0.006
0.004
Mass desorbed (kg of H2/Kg of LaNi5)
0.014
Mass absorbed (kg of H2/kg of LaNi5)
551
0.012 0.010 Model without radiative transfer 0.008
Model with radiative transfer
0.006 0.004 0.002 0.000
0.002 0
500
1000
1500
2000 2500 Time (s)
3000
3500
0
500
1000
1500 2000 Time (s)
2500
3000
3500
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 experiment. 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 exothermic, 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 temperature 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 velocity decreases. Consequently, the heat released from the bed becomes too weak, so the problem comes down to the stationary 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 temperature as time proceeds. Fig. 13, shows that the mass desorbed is higher near the wall where the temperature is high. This is because the dissociation 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).
F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537 – 557 0.09
0.09
0.08
0.08
0.07 0.06 0.05 Model with radiative transfer 0.04
Model without radiative transfer
0.03 0.02 0.01
Mass desorbed (kg of H2/kg of Mg)
Mass absorbed (kg of H2/Kg of Mg)
554
0.07 0.06 0.05 Model with radiative transfer 0.04
Model without radiative transfer
0.03 0.02 0.01
0.00
0.00 0
10000
30000
20000
10000
0
Time (s)
20000
30000
Time (s)
Fig. 19. InKuence of radiative heat transfer on the total mass absorbed and desorbed by the magnesium.
0.09
0.08 Mass desorbed (kg of H2/kg of Mg)
Mass absorbed (kg of H2/kg of Mg)
0.09
0.07 0.06 0.05 0.04
σa .R =533.4 σa.R =266.7 σa.R =10.
0.03 0.02 0.01
0.08 0.07 0.06
σa.R =533.4 σa.R =266.7 σa.R =10.
0.05 0.04 0.03 0.02 0.01 0.00
0.00 0
4000
12000 8000 Time (s)
16000
20000
0
4000
12000 8000 Time (s)
16000
20000
Fig. 20. Sensitivity to the absorption coe6cient of the media for s R = 114:3.
without radiative heat transfer, respectively, in absorption and desorption case. We notice that this di9erence of temperatures 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
0.09
0.09
0.08
0.08 Mass desorbed (kg of H2/kg of Mg)
Mass absorbed (kg of H2/kg of Mg)
F. Askri et al. / International Journal of Hydrogen Energy 28 (2003) 537 – 557
0.07 0.06 0.05 0.04
σ s.R =228.6 σs.R =114.3 σ s.R =10.
0.03 0.02
555
0.07 0.06
σ s.R =228.6 σ s.R =114.3 σ s.R =10.
0.05 0.04 0.03 0.02 0.01
0.01
0.00
0.00 4000
0
8000 12000 Time (s)
16000
20000
0
4000
8000 12000 Time (s)
16000
20000
0.09
0.09
0.08
0.08 Mass desorbed (kg of H2/kg of Mg)
Mass absorbed (kg of H2/kg of Mg)
Fig. 21. Sensitivity to the scattering coe6cient of the media for a R = 266:7.
0.07 0.06
Emissivit 1.0 0.7 0.3
0.05 0.04 0.03 0.02 0.01 0.00
0.07 0.06 0.05
Emissivit
0.04
0.3 0.7 1.0
0.03 0.02 0.01 0.00
0
4000
8000 12000 Time (s)
16000
20000
0
4000
8000 12000 Time (s)
16000
20000
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 conditions 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 and mass transfer sensitivity to radiative parameters (absorption 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 determinate 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 radiative heat transfer. The numerical simulation permitted to present the time–space evolution of the temperature and the hydride density inside the reactor. These time–space evolutions show that the two dimensional e9ect on the sorption process is important. The results of the numerical simulation show, on the one hand that radiative e9ects on the sorption process are negligible in the case of the LaNi5 –hydrogen system and that they are important for the Mg–H2 system. On the other hand, in the case of the Mg–H2 system, the transfers are sensitive to the value of the absorption coe6cient of the media and are not very sensitive to the scattering coe6cient and the emissivity of the walls reactor. References [1] Anani A. Alloys for hydrogen storage in nickel/hydrogen and nickel/metal hydride batteries. Int J Power Sources 1994;47(3):261–75.
[2] T-Raissi A, Banerjee A, Sheinkopf K. Metal hydride storage requirements for transportation applications. 31st Intersociety Energy Conversion Engineering Conference, Vol. 4, 1996. p. 2280 –5. [3] Mayer U, Groll M, Supper W. Heat and mass transfer in metal hydride reaction beds: experimental and theoretical results. J Less-Common Metals 1987;131:235–44. [4] Suda S, Kobayashi N, Morishita E. Heat transmission analysis of metal hydride beds. J Less-Common Metals 1983;89: 325–32. [5] Da-Wen S, Song-Jiu D. Study on heat and mass transfer characteristics of metal hydride beds. In: Veziroglu TN, editor. Alternate energy sources II. New York: McGraw-Hill, 1988. p. 621–8. [6] Da-Wen S, Song-Jiu D. Study on heat and mass transfer characteristics of metal hydride beds. In: Veziroglu TN, editor. A two-dimensional model. J Less-Common Metals 1989; 155:271–9. [7] Dantzer P, Orgaz E. Hydriding kinetics and the problem of thermal transfer. Z Phys Chem NF 1989;164:1267–72. [8] Nagel M, Komazaki Y, Suda S. Heat transmission in metal hydride beds. J Less-Common Metals 1986;20:45–51. [9] Heung LK. Heat transmission and kinetics of metal hydride reactors. Z Phys Chem NF 1989;164:1415 –20. [10] Dantzer P. Static, dynamic and cycling studies on the hydrogen in the intermetallics. J Less-Common Metals 1987;131: 349–64. [11] El-osery LA, El-osairy MA, Metwally AM, Keshk MM, El-Gammel M. Dynamic simulation of the convective model for metal hydride hydrogen storage beds. Int J Energy Sources 1993;15:523–30. [12] Jemni A, Ben Nasrallah S, Lamloumi J, Percheron Guegan A. Study of heat and mass transfer in a metal hydrogen reactor. Z Phys Chem NF Bd 1994;183:197–203. [13] Jemni A, Ben Nasrallah S. Study of two-dimensional heat and mass transfer during absorption in a metal hydrogen reactor. Int J hydrogen Energy 1995;20(1):43–52. [14] Jemni A, Ben Nasrallah S. Study of two-dimensional heat and mass transfer during desorption in a metal hydrogen reactor. Int J hydrogen Energy 1995;20:43–52. [15] Jemni A, Ben Nasrallah S, Lamloumi J. Experimental and theoretical study of metal–hydrogen reactor. Int J hydrogen Energy 1999;24:631–44. [16] Ben Nasrallah S, Jemni A. Study of heat and mass transfer models in a metal hydrogen reactor. Int J hydrogen Energy 1997;22(1):67–76. [17] Kuznetsov AV, Vafai K. Comparison between the two- and three-phase models for analysis of formation in aluminum-rich castings. Numer Heat Transfer J A 1996;29:859–67. [18] Suda S, Kobayashi N, Yoshida K. Reaction kinetics of metal hydrides and their mixtures. J Less-common Metals 1980;73:119–26. [19] Vgeholm B, Kjeller J, Larsen B, Pedersen AS. Formation and decomposition of magnesium hydride. J Less-Common Metals 1983;89:135–44. [20] Siegel R, Howell JR. Thermal radiation heat transfer, 3rd ed. New York: McGraw-Hill, 1993. [21] Siegel R, Howell JR. Thermal radiation heat transfer, 2nd ed. New York: McGraw-Hill, 1981. [22] Patankar SV. Numerical heat transfer Kuid Kow. New York: Hemi-sphere/McGraw-Hill, 1980.
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