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Prediction of transient heat and mass transfer in a closed metal–hydrogen reactor
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International Journal of Hydrogen Energy 29 (2004) 195 – 208
www.elsevier.com/locate/ijhydene
Prediction of transient heat and mass transfer in a closed 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, Monastir 5019, Tunisia Accepted 12 February 2003
Abstract The metal–hydrogen reactor is usually composed of a porous medium (hydride bed) and an expansion volume (gaseous phase). During the sorption process, the hydrogen 4ow and the heat transfer in the expansion part are badly known and can have some e5ects on the sorption phenomena in the hydride medium. At our knowledge, the hypothesis that neglects those e5ects is typically used. In this paper, a 2D study of heat and mass transfer has been carried out to investigate the transient transport processes of hydrogen in the two domains of a closed cylindrical reactor. A theoretical model is conducted and solved numerically by the control-volume-based 7nite element method (CVFEM). The result on temperature and hydride density distribution are presented and discussed. Moreover, this paper discusses in detail the e5ects of some governing operating conditions, such as dimensions of the expansion volume, height to the radius reactor ratio, and the initial hydrogen to metal atomic ratio, on the evolution of the pressure, 4uid 4ow, temperature and the hydrogen mass desorbed. ? 2003 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. Keywords: LaNi5; Metal-hydrogen reactor; Heat and mass transfers
1. Introduction Depletion of the available fossil energy sources and the very increasing energy demand has made it important to develop a new technique for energy storage. Among those techniques hydrogen storage in the form of metal hydrides is recommended [1,2]. Although hydrogen can be stored as compressed gas, cryogenic liquid, or metal hydride. The latter is seen to be more promising because it o5ers high hydrogen storage capacity as well as long-term stability and safety. Hydriding or dehydriding process is rather complex since simultaneous heat and mass transfer takes place with chemical reaction. The gas motion in a porous medium and changing physical properties also add further complexity to the
∗ Corresponding author. Tel.: +216-73-500-524; fax: +216-73500-514. E-mail address: [email protected] (A. Jemni).
problem. Therefore, there are a large body of experimental and numerical studies in the literature to investigate the details of sorption process and optimisation of metal-hydride systems [3–13]. Mayer et al. [3] developed a mathematical model for transient heat and mass transfer within the metal hydride reaction bed and compared the theoretical results with experimental data. Although their experimental results clearly showed that the temperature pro7le varies not only with r-direction but also with z-direction, their model accounts only for r-direction and the convective e5ect was neglected. Ram Gopal and Srinivasa Murthy [4,5] investigated the e5ects of operation conditions on hydriding and dehydriding characteristics using a one-dimensional mathematical model. They concluded that for a better metal hydride system heat transfer rate must be improved. In a series of papers Jemni and Ben Nasrallah [6–8] examined heat and mass transfer and chemical reaction for both absorption and desorption processes with a comprehensive model. Their numerical simulation results show that
0360-3199/$ 30.00 ? 2003 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. doi:10.1016/S0360-3199(03)00089-2
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F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
Nomenclature Cp E hf H H=M K P r R Rg T t U z
speci7c heat, J kg−1 K −1 activation energy, J mol−1 conductance between hydride bed and heating 4uid, W m−2 K −1 reactor height, m hydrogen to metal atomic ratio permeability, m2 pressure, Pa radial space coordinate reactor radius, m universal gas constant, J mol−1 K −1 temperature, K time, s gas velocity, m s−1 axial space coordinate
Greek symbols JH
reaction heat of formation, J kg−1 porosity
the choice of reactor dimensions, the inlet pressure, and the inlet temperature is very important. Also, the validity of the main assumptions, considered by di5erent authors, has been studied. Shmalkov et al. [9] proposed a theoretical model of heat and mass transfer in metal hydride-hydrogen gas impurities systems. Their model makes it possible to describe the sorption processes taking into account the convective transfer in the metal-hydride modules, for gas mixtures containing impurities that are inactive or low-active to the hydride-forming material. In a subsequent study, Jemni et al. [10] conducted an experimental study to determinate the e5ective thermal conductivity, the equilibrium pressure, and reaction kinetics and validate the theoretical model. A good agreement between measured and theoretical results is obtained. Nakagwa et al. [11] developed a two-dimensional model to evaluate transient heat and mass transfer in the metal hydride bed. They used this model to study the validity of local thermal equilibrium assumption and to determinate the e5ects of the convection term on the heat transfer. Mat and Kaplan [12] examined numerically the metal hydride formation in an Lm-Ni5 storage bed. It is found that hydride formation is important at regions with lower equilibrium pressure. The absorption hydrogen mass increases exponentially at earlier times of hydriding process and slow down after temperature of reaction bed increases due to the heat of reaction. Their theoretical results agree satisfactorily with experimental data in the literature. Aldas et al. [13] studied numerically heat and mass transfer, 4uid 4ow and chemical reaction in a hydride bed with
g ge
dynamic viscosity, kg m−1 K −1 thermal conductivity, W m−1 K −1 gas gas e5ective
Subscripts a d e eq s ss
absorption desorption e5ective equilibrium solid saturated
Superscripts g s
gaseous phase solid phase
z
R
Expansion volume
Hg H
Porous medium
Hp
r Fig. 1. Closed metal–hydrogen reactor.
a general purpose PHOENICS code. As a result, they found that hydride formation takes place faster near the cooled boundary walls. The 4uid 4ow a5ects the temperature distribution in the system, however, it does not signi7cantly improve the amount of hydrogen absorbed. The metal-hydrogen reactor is usually composed of a porous medium (hydride bed) and an expansion volume (gaseous phase). During the sorption process, the hydrogen 4ow and the heat transfer in the expansion part (Fig. 1) are badly known and can have some e5ects on the sorption phenomena in the hydride medium. At our knowledge, the
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
197
Table 1 Speci7c forms of the general conservation equations for the expansion volume Equations
f1
f2
S
Continuity z-momentum
1 u
g g
g g
0 g
0 − @P + 0g g(T − T0 ) @z
r-momentum
v
g
g
g
− @P @r
Energy
T
g cpg
g cpg
g
@P @t
+ U ∇P
Table 2 Speci7c forms of the general conservation equation for the porous medium Equations
f1
f2
S
Continuity for gas Continuity for solid z-momentum
1 1 u
g (1 − )s g
g 0 g
0 0 g
−m m −2
@P @z
r-momentum
v
g
g
g
−2 @P @r
Energy
T
g cpg + (1 − )s cps
g cpg
g + (1 − )s
@P @t
hypothesis that neglects those e5ects is typically used. Besides, there is no available study in the literature that makes it possible to describe the transient heat and mass transfer in the closed metal–hydrogen systems (thermals compressors, hydrogen storage tanks, energy storage reactors). The storage systems can be thermally solicited and therefore the pressure within these systems varies and they can present risks. On the other hand, to optimise the hydride compression systems, a precise prediction of the evolutions of all variables and in particular the pressure is necessary. So the main purpose of this study is to develop a theoretical model that makes it possible to predict the transient heat and mass transfer in the two domains of a closed cylindrical reactor and to discusses in detail the e5ects of various governing operating conditions, such as gaseous part volume, height to the radius ratio of the reactor, and the initial hydrogen to metal atomic ratio, on the evolution of the pressure, 4uid 4ow, temperature and the amount of hydrogen desorbed.
2. Mathematical model The reactor geometry and the coordinate system are depicted in Fig. 1. The used reactor exchanges heat through lateral and bases areas at a constant temperature. This reactor is composed of a porous medium (hydride bed) and a gaseous phase (expansion volume). For the porous medium, the macroscopic di5erential equations are obtained by taking the average of microscopic equations over a representative volume and using closing assumptions. The microscopic equations are the mass, the energy and the momentum equations balance. These equations are obtained by
+ 2 0g g(T − T0 ) − 2 g [ k + −
2 g [ k
+
F √ U ]v k
F √ U ]u k
− um
− vm
+ U ∇P + mJH
using thermodynamic and mechanical laws of continuous medium. In order to establish the governing equations, the following assumptions are considered: • The dispersion and the tortuosity terms are modelled as di5usive 4uxes. • The viscous dissipation is negligible. • The gas phase is ideal from the thermodynamic viewpoint. • The local thermal equilibrium is valid and the radiative heat transfer is negligible. Considering theses assumptions, equations governing heat and mass transfer in the reactor, when the transfers are two-dimensional and depending on the r- and z-axis, are expressed in the following form: @ @( f2 u) 1 @(rf2 v) ( f1 ) + + @t @z r @r 2 @ 1 @ @ + (1) r + S : = @z 2 r @r @r This equation can be written in the following vectored form: @( f1 ) (2) + div( J ) = S ; @t where J is the combined convection-di5usion 4ux: J = J c + Jd ;
Jc = f2 U ;
Jd = − ∇
where U is the 4uid 4ow velocity vector: The meanings of the di5erent terms which are 7gured in Eq. (1), are given for the expansion volume and the porous medium, respectively, in Tables 1 and 2.
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F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
Table 3 The equilibrium pressure polynomial function coeMcients CoeMcients
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
2.1. Reaction kinetics
2.4. Boundaries hydrodynamic conditions
The hydrogen mass absorbed or desorbed, m, per unit time and unit volume is given by [3]: For the hydriding case Ea P (ss − s ): (3) Ln m = Ca exp − Rg T Peq
• Considering the symmetry condition about the z-axis, then: @u (z; 0; t) = 0; (10) @r v(z; 0; t) = 0:
For the dehydriding case Ed P − Peq s : m = Cd exp − Rg T Peq
(4)
For the LaNi5 –hydrogen system [14] Cd = 9:57 S
−1
;
Ca = 59:187 S
Ed = 16:420 kJ=mol −1
of H2 ;
Ea = 21:170 kJ=mol−1 of H2 :
and
• For a closed reactor, the radial velocity and the axial velocity at r = R; z = 0 and z = H are equal to zero: u = v = 0:
−1
2.2. Equilibrium pressure For the LaNi5 –hydrogen system, the evolution of the equilibrium pressure is given as a function of temperature and the hydrogen-to-metal atomic ratio (H=M ). H JH 1 1 ; (5) Peq = f exp − M Rg T Tref
(11)
(12)
2.5. Boundaries thermal conditions • Considering the symmetry condition about the z-axis, then: @T (z; 0; t) = 0: (13) @r • The heat 4ux continuity through the lateral area (r = R) and the bases areas (z = 0 and z = H ) permits to write the following equations:
where f(H=M ) is the equilibrium pressure at the reference temperature Tref . This function f(H=M ) is given by 7tting the experimental data. The best 7t is obtained with a polynomial function of order 9, whose coeMcients, for the hydriding and dehydriding processes, are given in Table 3 [10].
For the porous medium part: @T e (0; r; t) = hf (T (0; r; t) − Tf ); @z @T − e (z; R; t) = hf (T (z; R; t) − Tf ): @r
2.3. Initial conditions
(6)
For the expansion volume part: @T − g (H; r; t) = hf (T (H; r; t) − Tf ); @z @T − g (z; R; t) = hf (T (z; R; t) − Tf ): @r
(7)
3. Numerical method
(8)
The system of equations that is presented in the previous sections is solved numerically by the control-volume-based 7nite element method (CVFEM) [15–18]. The advantages of this method are: (i) It insure the 4ux conservation. (ii) The used control volumes presented more faces (6 faces), that makes it possible to avoid the numerical di5usion. (iii) Six nodes are used for each calculation point, therefore, the
Initially, the temperature, the pressure and the hydride density in the reactor are assumed to be constant: T (z; r; 0) = T0 ; P(z; r; 0) = Peq T0 ; s (z; r; 0) = 0 :
H M
(15)
(16) (17)
; 0
• The gas in the calculation domains is initially motionless: u = v = 0:
(14)
(9)
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
199
z
Fig. 4. v-control volume !v .
r Fig. 2. Spatial-discretisation (↑ axial velocity; → radial velocity; pressure).
An integration of the conservation equation (2), when applied to the control volume !u . (Fig. 2), leads to:
t+Jt
o4 a5 o5
P
+
o3
a4
ω1
a6 a o6 1
a3 o2 a2
o1
+
t+Jt
a1
o6
t
t
o6
a6
t
ai+1
oi t+Jt
+
t+Jt
t
− Fig. 3. u-control volume !u .
5
!u
t
@( f1 ) dv dt + @t i=1
t+Jt
t+Jt
t
oi
ai
J :n2'r ds dt
J :n2'r ds dt
J :n2'r ds dt J :n2'r ds dt
!u
S dv dt = 0;
(18)
where n is a unit outward vector normal to the di5erential length elements ds. stability of the numerical resolution process is improved. (iv) The control volume is treated as the addition of six elements controls volumes which improved the grid 4exibility. The method consists of de7ning a grid of points within the calculated domain and then builds around each point a control domain ! (Fig. 2). The longitudinal cross-section is 7rst divided into three-node triangular elements. Then the centroids of the elements are joined to the midpoints of the corresponding sides. This creates polygonal control volumes around each node in the 7nite element grid (Figs. 3 and 4). The discretization of the longitudinal cross-section is rotated through 2' radians about the axis of symmetry. The result is a discretization of the axisymmetric calculation domain into torus elements of triangular cross-section and torus control volumes of polygonal cross-section. To avoid the oscillatory pressure and velocity 7elds, a staggered grid arrangement is employed, i.e. the axial velocity component and the temperature are stored at the same locations that are di5erent from those of the radial velocity component and from those of the pressure.
4. Results and discussion 4.1. Validity of the theoretical model In order to validate the established model and the numerical method employed, the theoretical results are compared with the experimental data of Jemni et al. [10]. The considered reactor is connected to a reservoir that initially involves a known quantity of hydrogen. The reactor is 7lled with LaNi5 alloy. The initial pressure in the reservoir, the reservoir volume, the inner diameter of the reactor, the inner height of the reactor, the heating 4uid temperature, and the used amount of the alloy are, respectively, 1 bar, 1755 cm3 , 5 cm, 8 cm, 313 K and 422 g. At t = 0, the reservoir is put in contact with the hydride bed. The time-evolutions of the pressure within the reservoir and the total mass desorbed are plotted in Fig. 5. It is seen that numerical results agree satisfactorily with experimental data.
200
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208 3.5E+5 3.5E-4
3.0E+5
3.0E-4
Mass desorb ed (kg)
Pressure (Pa)
2.5E-4
2.5E+5
Experience 2.0E+5
Model
2.0E-4
Experience 1.5E-4
Model 1.0E-4
1.5E+5 5.0E-5
1.0E+5
0.0E+0
0
300
600
900
1200
1500
Time (s)
0
300
600
900
1200
1500
Time (s)
Fig. 5. Calculated and measured evolutions of pressure within the reservoir and the total mass desorbed.
Table 4 Thermophysical properties of materials and data used in computations E5ective thermal conductivity Permeability Porosity Initial bed temperature Heating 4uid temperature
e (W m−1 K −1 ) K(m2 ) (%) T0 (K) Tf (K)
1.32 10−8 50 290 333
4.2. Time-space evolution of the temperature and the hydride density The considered reactor has a volume of 169:56 cm3 and it is heated with a heating 4uid at a constant temperature. The half of the volume of this reactor is 7lled with LaNi5 metal. Initially the hydride bed is at an equilibrium state. The thermo-physical properties of materials and data used in computations are indicated in Table 4. Fig. 6 shows the temperature evolution within the reactor in the desorption case and at the selected times, t = 15, 36, 180, 360, 720 and 1440 s. Initially the hydride bed and the expansion volume were assumed to be at a constant temperature (T = 290 K). The temperature pro7le at 15 s indicates that the temperature near the wall is signi7cantly increased as a result of heat supplied by the heating 4uid. However, the temperature at the centre of the reaction bed (porous medium) remains essentially the same as that of at initially time, showing that heat exchanges with heating 4uid are absorbed by the endothermic desorption reaction near the wall. The temperature pro7le at 180 s shows, on the one hand, that the temperature in porous medium part is clearly increased. On the other hand, for the expansion gaseous
part, there is no important evolution due to the accumulation of the hydrogen mass desorbed. After a long enough time the system comes down to the thermal equilibrium with the heating 4uid. Also, it is noted that the expansion volume part reaches quickly the thermal equilibrium, with the heating 4uid, than the reaction bed. Fig. 7 shows the metal hydride density evolution within the reactor at the selected times mentioned above. At the beginning, the dehydriding process takes place near the walls of the reactor where the desorption equilibrium pressure is important due to the increasing of the temperature. The density pro7les, at 36 and 180 s, indicate that there is an absorption process in the core part of the reaction bed where the temperature is even weak. This phenomenon is explained by the fact that the pressure that reigns in the reactor increases, due to the accumulation of the hydrogen mass desorbed, and becomes higher than the equilibrium pressure at the centre region. Beyond t = 180 s, the desorption reaction takes place in all the reaction bed except near the bottom and lateral surfaces where the absorption reaction takes place. For one elevated enough time, the kinetics reaction decreases considerably and the system o5ers toward an equilibrium state. The distribution of the pressure within the reactor is plotted, at selected times, in Fig. 8. It is seen that the pressure increases during the sorption process, this is due to the increasing of the temperature and essentially to the accumulation of the hydrogen mass desorbed in the reactor. At all times, it is noted that the gradient of the pressure in the reactor is very weak. Fig. 9 shows the velocity distribution at selected times, t =15, 180, 360, and 720 s. Initially the reactor was assumed
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
201
Fig. 6. Temperature pro7le in the LaNi5 reactor at selected times.
to be at rest. At the beginning, the 4uid velocity is important only near the wall, this is due to the natural convection phenomena caused by the heating 4uid. At all times, a very small velocity is observed in the reaction bed compared to the velocity in the expansion gaseous part. This result may be attributed to the small permeability of the hydride bed. Also, the velocity distribution indicates the existence of two cellular 4ow in the expansion volume due to the natural
convection phenomena. When the system reaches the new equilibrium state, the 4uid 4ow disappears. 4.3. Study of the in=uence of the operating parameters Many applications have recently been developed to take advantage of the metal-hydrogen systems properties, including rechargeable batteries and heating and cooling systems
202
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
Fig. 7. Evolution of hydride density in the LaNi5 reactor at selected times.
(hydrogen storage systems, heat pumps, heat transformers, refrigerators, thermals compressors, heat storage systems). These applications require a very precise prediction of the time–space evolutions of all variables and in particular the pressure for the closed systems (Thermals compressors, hydrogen storage systems). Consequently, a study of heat and mass transfer sensitivity to the operating parameters (expansion volume to reactor volume ratio, height to the radius reactor ratio, and initial hydrogen to metal atomic ratio) becomes of interest and permits us to take all
possible precautions in order to have a high safety procedure for the storage systems, and to determine operating parameters permitting to have a housemaid working (eMciency, speed) of compression systems. 4.4. Sensitivity to the reactor geometry The pressure within the reactor and the total hydrogen mass desorbed are plotted for each reactor dimensions as a function of time (Fig. 10). The reactor volume, the amount
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
203
Fig. 8. Pressure pro7le in the LaNi5 reactor at selected times.
of the hydride metal, the physical characteristics, and the boundaries conditions were kept constant during the simulation. It is noticed that, for the di5erent values used of the ratio H=R, the longest time, needed by the reactor to reach the new equilibrium state, is gotten for a ratio equal to 2. So for low values of the ratio H=R, the heat and mass transfer in the reactor are one-dimensional and depend only on z-coordinate. Under these conditions, the resistance to transfers along the z-direction increases with H=R values, hence the necessary time to reach the new equilibrium state increases. For large values of H=R, the transfers are
also one-dimensional and depend only on r-coordinate. When H=R rises, the rPesistance to the transfers according to r-direction decreases. For intermediate values of H=R, the two-dimensional e5ects are important. When H=R increases the resistance to the transfers along the axial direction increases and that along the radial direction decreases. Those competing e5ects explain the existence of a critic value of H=R equal to 2. Also, this result can be explained by the fact that the reactor surface that permits the exchange of the heat with the heating 4uid presents a minimum at H=R equal to 2.
204
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
Fig. 9. Velocity distribution in the reactor LaNi5 at di5erent times. 1.4E+6
1.2E+6
Mass desorbed (kg of H2/kg of metal)
2.5E-4
6
Pressure (Pa)
5 4
1.0E+6
3 2
6. H/R=0.25 5. H/R=0.5 4. H/R=1 3. H/R=6.75 2. H/R=4 1. H/R=2
1 8.0E+5
6.0E+5
6
2.0E-4
5 4 1.5E-4
3 2
6. H/R=0.25 5. H/R=0.5 4. H/R=1 3. H/R=6.75 2. H/R=4 1. H/R=2
1 1.0E-4
5.0E-5
0.0E+0
4.0E+5 0
200
400
600
Time (s)
800
1000
0
200
400
600
800
1000
Time (s)
Fig. 10. In4uence of the height to the radius ratio of the reactor on the total hydrogen mass desorbed and the pressure within the reactor.
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
205
Fig. 11. In4uence of the height to the radius ratio of the reactor on the 4uid velocity pro7le.
Fig. 11 shows the velocity distribution for di5erent values of the height to the radius ratio of the reactor H=R, 0.5, 2, and 6.75. It is seen that, when the H=R ratio increases, the two circulating zone grow in size and the maximum value of the gas velocity increases. 4.5. Sensitivity to the expansion volume to reactor volume ratio Hg =H Fig. 12 presents the mean pressure evolution in the reactor and the total mass desorbed, for di5erent values of the
expansion volume to the reactor volume ratio Hg =H . The physical characteristics, the amount of the hydride metal and the boundaries conditions were kept constant during the simulation. It is noticed that increasing the value of the ratio Hg =H the pressure within the reactor decreases and the total mass desorbed increases. Fig. 13 shows the temperature distribution for di5erent values of the ratio Hg =H , 0.25, 0.5, and 0.75. It is seen that there is an important gradient of temperature in the expansion gaseous part when the ratio Hg =H is lower and an uniform temperature pro7le for the large values of Hg =H .
206
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208 1.4E+6
4.0E-4
Hg/H=0.25
Mass desorbed (kg of H2/kg of metal)
3.5E-4
Hg/H=0.5
1.2E+6
Pressure (Pa)
Hg/H=0.6
1.0E+6
Hg/H=0.75
8.0E+5
6.0E+5
Hg/H=0.75
3.0E-4
Hg/H=0.6 2.5E-4
Hg/H=0.5 2.0E-4
Hg/H=0.25
1.5E-4
1.0E-4
5.0E-5
4.0E+5
0.0E+0
0
200
400
600
Time (s)
800
1000
0
200
400
600
800
1000
Time (s)
Fig. 12. In4uence of the expansion volume to reactor volume ratio Hg =H on the total mass desorbed and the pressure within the reactor.
Fig. 13. In4uence of the expansion volume to reactor volume ratio Hg =H on the temperature pro7le.
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
207
Also, those pro7les indicate that the heat transfer becomes rapid when the ratio Hg =H decreases. The 4uid velocity distributions, for the considered values of Hg =H , are presented in Fig. 14. It is noted that increasing the ratio Hg =H the two circulating zones grow in size and the 4uid velocity increases. 4.6. Sensitivity to the initial hydrogen to metal atomic ratio (H=M )0 In order to evaluate the in4uence of the initial hydrogen to metal atomic ratio (H=M )0 on the pressure evolution within the reactor and the hydrogen mass desorbed, a simulation of the reactor for a series values of the (H=M )0 ratio was done. Fig. 15 shows that increasing the value of this ratio (5:4 ¡ H=M ¡ 6) the reached pressure and the total mass desorbed increase signi7cantly. It is also noted that a decrease of this ratio below the value of 5.4 has no important e5ect on the pressure and the mass desorbed evolutions. This result can be explained by the high equilibrium pressure sensitivity to the hydrogen to metal atomic ratio (H=M ) in the range of (5:4 ¡ H=M ¡ 6) [12]. 5. Conclusions A mathematical model describing hydrogen desorption of a porous lanthanum misch metal bed, in a closed reactor, has been presented and solved numerically by the control-volume-based 7nite element method (CVFEM). The mathematical model includes complex heat and mass transfer, 4uid 4ow and chemical reaction that take place during the sorption processes. The result on temperature, composition distribution and velocity are presented and discussed. The e5ects of various governing operating conditions on the evolution of the pressure and the hydrogen mass desorbed, have been studied. The validity of the theoretical model has been tested by comparison with experimental data of the total mass desorbed and the pressure evolution within the reservoir. A good agreement is obtained. It is found that, at all times, the 4uid velocity in the reaction bed is very small compared to the velocity in the expansion gaseous part there where a two circulating zones take place. These circulating zones grow in size and the 4uid velocity increases when the expansion volume to the reactor volume ratio (Hg =H ) and the height to the radius reactor ratio (H=R) increase. Also, it is found that the expansion volume part reaches quickly the thermal equilibrium, with the heating 4uid, than the reaction bed. The pressure and total hydrogen mass desorbed varied considerably when the initial hydrogen to metal atomic ratio (H=M )0 varies in the range 5:4 ¡ (H=M )0 ¡ 6. The numerical results showed on the one hand that the increase of the expansion part volume a5ect signi7cantly the
Fig. 14. In4uence of the ratio Hg =H on the 4uid velocity.
evolutions of pressure within the reactor and the hydrogen mass desorbed. On the other hand, it is seen that, when the height to the radius reactor ratio is large (H=R2) or too small (H=R2), the necessary time to reach the equilibrium state decreases considerably.
208
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208 1.8E+6
3.0E-4
H/M=5.94
H/M=5.94
Mass desorbed (kg of H2/Kg of metal)
1.6E+6
1.4E+6
H/M=5.82
Pressure (pa)
1.2E+6
1.0E+6
H/M=5.64 8.0E+5
H/M=5.52 H/M=5.40
6.0E+5
4.0E+5
2.5E-4
H/M=5.82 2.0E-4
H/M=5.64
1.5E-4
H/M=5.52 H/M=5.40
1.0E-4
5.0E-5
2.0E+5
0.0E+0
0.0E+0
0
100
200
300
400
Time (s)
0
100
200
300
400
Time (s)
Fig. 15. In4uence of the initial hydrogen to metal atomic ratio H=M on the total mass desorbed and the pressure within the reactor.
References [1] Reilly JJ. Proceedings of the International Symposium Hydride for Energy Storage, Geilo, Norway, 14 –19 August 1977. [2] El-Osery IA, Metwally MM. Proceedings of the International AMSE Conference on Modelling and Simulation, vol. 3.3. Sorrento, Italy, 1986. [3] Mayer U, Groll M, Supper W. Heat and mass transfer in metal hydride reaction beds: experimental and theoretical results. J Less-Common Met 1987;131:235–44. [4] Gopal MR, Murthy SS. Prediction of heat and mass transfer in annular cylindrical metal hydride beds. Int J Hydrogen Energy 1992;17(10):795–805. [5] Gopal MR, Murthy SS. Studies on heat and mass transfer in metal hydride beds. Int J Hydrogen Energy 1995;20(11): 911–7. [6] 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. [7] 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. [8] 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. [9] Shmalkov YUF, Kolosov VI, Solovey VV, Kennedy LA, Zelepouga SA. Mathematical simulation of heat-and-mass transfer processes in metal hydride-hydrogen-gas impurities systems. Int J Hydrogen Energy 1998;23(6):463–8.
[10] Jemni A, Ben Nasrallah S, Lamloumi J. Experimental and theoretical study of metal-hydrogen reactor. Int J Hydrogen Energy 1999;24:631–44. [11] Nakagawa T, Inomata A, Aoki H, Miura T. Numerical analysis of heat and mass transfer characteristics in metal hydride bed. Int J Hydrogen Energy 1999;24(10):1027–32. [12] Mat D, Kaplan Y. Numerical study of hydrogen absorption in an Lm-Ni5 hydride reactor. Int J Hydrogen Energy 2001;26:957–63. [13] Aldas K, Mat D, Kaplan Y. A three-dimensional mathematical model for absorption in a metal hydride bed. Int J Hydrogen Energy 2002;27(10):1063–9. [14] Suda S, Kobayashi N, Yoshida K. Reaction kinetics of metal hydrides and their mixtures. J Less-Common Met 1980;73: 119–26. [15] Balliga BR, Patankar SV. A new 7nite-element formulation for convection-di5usion problems. Numer Heat Transfer 1980;3:393–409. [16] Balliga BR, Patankar SV. A control-volume 7nite element method for two-dimensional 4uid 4ow and heat transfer. Numer Heat Transfer 1983;6:245–61. [17] Schneider GE, Raw MJ. Control-volume 7nite-element method for heat transfer and 4uid 4ow using collocated variables—1. computational procedure. Numer Heat Transfer 1987;11:363–90. [18] Omri A, Ben Nasrallah S. Control volume 7nite element numerical simulation of mixed convection in an air-cooled cavity. Numer Heat Transfer Part A 1999;36: 615–37.
International Journal of Hydrogen Energy 29 (2004) 195 – 208
www.elsevier.com/locate/ijhydene
Prediction of transient heat and mass transfer in a closed 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, Monastir 5019, Tunisia Accepted 12 February 2003
Abstract The metal–hydrogen reactor is usually composed of a porous medium (hydride bed) and an expansion volume (gaseous phase). During the sorption process, the hydrogen 4ow and the heat transfer in the expansion part are badly known and can have some e5ects on the sorption phenomena in the hydride medium. At our knowledge, the hypothesis that neglects those e5ects is typically used. In this paper, a 2D study of heat and mass transfer has been carried out to investigate the transient transport processes of hydrogen in the two domains of a closed cylindrical reactor. A theoretical model is conducted and solved numerically by the control-volume-based 7nite element method (CVFEM). The result on temperature and hydride density distribution are presented and discussed. Moreover, this paper discusses in detail the e5ects of some governing operating conditions, such as dimensions of the expansion volume, height to the radius reactor ratio, and the initial hydrogen to metal atomic ratio, on the evolution of the pressure, 4uid 4ow, temperature and the hydrogen mass desorbed. ? 2003 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. Keywords: LaNi5; Metal-hydrogen reactor; Heat and mass transfers
1. Introduction Depletion of the available fossil energy sources and the very increasing energy demand has made it important to develop a new technique for energy storage. Among those techniques hydrogen storage in the form of metal hydrides is recommended [1,2]. Although hydrogen can be stored as compressed gas, cryogenic liquid, or metal hydride. The latter is seen to be more promising because it o5ers high hydrogen storage capacity as well as long-term stability and safety. Hydriding or dehydriding process is rather complex since simultaneous heat and mass transfer takes place with chemical reaction. The gas motion in a porous medium and changing physical properties also add further complexity to the
∗ Corresponding author. Tel.: +216-73-500-524; fax: +216-73500-514. E-mail address: [email protected] (A. Jemni).
problem. Therefore, there are a large body of experimental and numerical studies in the literature to investigate the details of sorption process and optimisation of metal-hydride systems [3–13]. Mayer et al. [3] developed a mathematical model for transient heat and mass transfer within the metal hydride reaction bed and compared the theoretical results with experimental data. Although their experimental results clearly showed that the temperature pro7le varies not only with r-direction but also with z-direction, their model accounts only for r-direction and the convective e5ect was neglected. Ram Gopal and Srinivasa Murthy [4,5] investigated the e5ects of operation conditions on hydriding and dehydriding characteristics using a one-dimensional mathematical model. They concluded that for a better metal hydride system heat transfer rate must be improved. In a series of papers Jemni and Ben Nasrallah [6–8] examined heat and mass transfer and chemical reaction for both absorption and desorption processes with a comprehensive model. Their numerical simulation results show that
0360-3199/$ 30.00 ? 2003 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. doi:10.1016/S0360-3199(03)00089-2
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F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
Nomenclature Cp E hf H H=M K P r R Rg T t U z
speci7c heat, J kg−1 K −1 activation energy, J mol−1 conductance between hydride bed and heating 4uid, W m−2 K −1 reactor height, m hydrogen to metal atomic ratio permeability, m2 pressure, Pa radial space coordinate reactor radius, m universal gas constant, J mol−1 K −1 temperature, K time, s gas velocity, m s−1 axial space coordinate
Greek symbols JH
reaction heat of formation, J kg−1 porosity
the choice of reactor dimensions, the inlet pressure, and the inlet temperature is very important. Also, the validity of the main assumptions, considered by di5erent authors, has been studied. Shmalkov et al. [9] proposed a theoretical model of heat and mass transfer in metal hydride-hydrogen gas impurities systems. Their model makes it possible to describe the sorption processes taking into account the convective transfer in the metal-hydride modules, for gas mixtures containing impurities that are inactive or low-active to the hydride-forming material. In a subsequent study, Jemni et al. [10] conducted an experimental study to determinate the e5ective thermal conductivity, the equilibrium pressure, and reaction kinetics and validate the theoretical model. A good agreement between measured and theoretical results is obtained. Nakagwa et al. [11] developed a two-dimensional model to evaluate transient heat and mass transfer in the metal hydride bed. They used this model to study the validity of local thermal equilibrium assumption and to determinate the e5ects of the convection term on the heat transfer. Mat and Kaplan [12] examined numerically the metal hydride formation in an Lm-Ni5 storage bed. It is found that hydride formation is important at regions with lower equilibrium pressure. The absorption hydrogen mass increases exponentially at earlier times of hydriding process and slow down after temperature of reaction bed increases due to the heat of reaction. Their theoretical results agree satisfactorily with experimental data in the literature. Aldas et al. [13] studied numerically heat and mass transfer, 4uid 4ow and chemical reaction in a hydride bed with
g ge
dynamic viscosity, kg m−1 K −1 thermal conductivity, W m−1 K −1 gas gas e5ective
Subscripts a d e eq s ss
absorption desorption e5ective equilibrium solid saturated
Superscripts g s
gaseous phase solid phase
z
R
Expansion volume
Hg H
Porous medium
Hp
r Fig. 1. Closed metal–hydrogen reactor.
a general purpose PHOENICS code. As a result, they found that hydride formation takes place faster near the cooled boundary walls. The 4uid 4ow a5ects the temperature distribution in the system, however, it does not signi7cantly improve the amount of hydrogen absorbed. The metal-hydrogen reactor is usually composed of a porous medium (hydride bed) and an expansion volume (gaseous phase). During the sorption process, the hydrogen 4ow and the heat transfer in the expansion part (Fig. 1) are badly known and can have some e5ects on the sorption phenomena in the hydride medium. At our knowledge, the
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
197
Table 1 Speci7c forms of the general conservation equations for the expansion volume Equations
f1
f2
S
Continuity z-momentum
1 u
g g
g g
0 g
0 − @P + 0g g(T − T0 ) @z
r-momentum
v
g
g
g
− @P @r
Energy
T
g cpg
g cpg
g
@P @t
+ U ∇P
Table 2 Speci7c forms of the general conservation equation for the porous medium Equations
f1
f2
S
Continuity for gas Continuity for solid z-momentum
1 1 u
g (1 − )s g
g 0 g
0 0 g
−m m −2
@P @z
r-momentum
v
g
g
g
−2 @P @r
Energy
T
g cpg + (1 − )s cps
g cpg
g + (1 − )s
@P @t
hypothesis that neglects those e5ects is typically used. Besides, there is no available study in the literature that makes it possible to describe the transient heat and mass transfer in the closed metal–hydrogen systems (thermals compressors, hydrogen storage tanks, energy storage reactors). The storage systems can be thermally solicited and therefore the pressure within these systems varies and they can present risks. On the other hand, to optimise the hydride compression systems, a precise prediction of the evolutions of all variables and in particular the pressure is necessary. So the main purpose of this study is to develop a theoretical model that makes it possible to predict the transient heat and mass transfer in the two domains of a closed cylindrical reactor and to discusses in detail the e5ects of various governing operating conditions, such as gaseous part volume, height to the radius ratio of the reactor, and the initial hydrogen to metal atomic ratio, on the evolution of the pressure, 4uid 4ow, temperature and the amount of hydrogen desorbed.
2. Mathematical model The reactor geometry and the coordinate system are depicted in Fig. 1. The used reactor exchanges heat through lateral and bases areas at a constant temperature. This reactor is composed of a porous medium (hydride bed) and a gaseous phase (expansion volume). For the porous medium, the macroscopic di5erential equations are obtained by taking the average of microscopic equations over a representative volume and using closing assumptions. The microscopic equations are the mass, the energy and the momentum equations balance. These equations are obtained by
+ 2 0g g(T − T0 ) − 2 g [ k + −
2 g [ k
+
F √ U ]v k
F √ U ]u k
− um
− vm
+ U ∇P + mJH
using thermodynamic and mechanical laws of continuous medium. In order to establish the governing equations, the following assumptions are considered: • The dispersion and the tortuosity terms are modelled as di5usive 4uxes. • The viscous dissipation is negligible. • The gas phase is ideal from the thermodynamic viewpoint. • The local thermal equilibrium is valid and the radiative heat transfer is negligible. Considering theses assumptions, equations governing heat and mass transfer in the reactor, when the transfers are two-dimensional and depending on the r- and z-axis, are expressed in the following form: @ @( f2 u) 1 @(rf2 v) ( f1 ) + + @t @z r @r 2 @ 1 @ @ + (1) r + S : = @z 2 r @r @r This equation can be written in the following vectored form: @( f1 ) (2) + div( J ) = S ; @t where J is the combined convection-di5usion 4ux: J = J c + Jd ;
Jc = f2 U ;
Jd = − ∇
where U is the 4uid 4ow velocity vector: The meanings of the di5erent terms which are 7gured in Eq. (1), are given for the expansion volume and the porous medium, respectively, in Tables 1 and 2.
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F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
Table 3 The equilibrium pressure polynomial function coeMcients CoeMcients
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
2.1. Reaction kinetics
2.4. Boundaries hydrodynamic conditions
The hydrogen mass absorbed or desorbed, m, per unit time and unit volume is given by [3]: For the hydriding case Ea P (ss − s ): (3) Ln m = Ca exp − Rg T Peq
• Considering the symmetry condition about the z-axis, then: @u (z; 0; t) = 0; (10) @r v(z; 0; t) = 0:
For the dehydriding case Ed P − Peq s : m = Cd exp − Rg T Peq
(4)
For the LaNi5 –hydrogen system [14] Cd = 9:57 S
−1
;
Ca = 59:187 S
Ed = 16:420 kJ=mol −1
of H2 ;
Ea = 21:170 kJ=mol−1 of H2 :
and
• For a closed reactor, the radial velocity and the axial velocity at r = R; z = 0 and z = H are equal to zero: u = v = 0:
−1
2.2. Equilibrium pressure For the LaNi5 –hydrogen system, the evolution of the equilibrium pressure is given as a function of temperature and the hydrogen-to-metal atomic ratio (H=M ). H JH 1 1 ; (5) Peq = f exp − M Rg T Tref
(11)
(12)
2.5. Boundaries thermal conditions • Considering the symmetry condition about the z-axis, then: @T (z; 0; t) = 0: (13) @r • The heat 4ux continuity through the lateral area (r = R) and the bases areas (z = 0 and z = H ) permits to write the following equations:
where f(H=M ) is the equilibrium pressure at the reference temperature Tref . This function f(H=M ) is given by 7tting the experimental data. The best 7t is obtained with a polynomial function of order 9, whose coeMcients, for the hydriding and dehydriding processes, are given in Table 3 [10].
For the porous medium part: @T e (0; r; t) = hf (T (0; r; t) − Tf ); @z @T − e (z; R; t) = hf (T (z; R; t) − Tf ): @r
2.3. Initial conditions
(6)
For the expansion volume part: @T − g (H; r; t) = hf (T (H; r; t) − Tf ); @z @T − g (z; R; t) = hf (T (z; R; t) − Tf ): @r
(7)
3. Numerical method
(8)
The system of equations that is presented in the previous sections is solved numerically by the control-volume-based 7nite element method (CVFEM) [15–18]. The advantages of this method are: (i) It insure the 4ux conservation. (ii) The used control volumes presented more faces (6 faces), that makes it possible to avoid the numerical di5usion. (iii) Six nodes are used for each calculation point, therefore, the
Initially, the temperature, the pressure and the hydride density in the reactor are assumed to be constant: T (z; r; 0) = T0 ; P(z; r; 0) = Peq T0 ; s (z; r; 0) = 0 :
H M
(15)
(16) (17)
; 0
• The gas in the calculation domains is initially motionless: u = v = 0:
(14)
(9)
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
199
z
Fig. 4. v-control volume !v .
r Fig. 2. Spatial-discretisation (↑ axial velocity; → radial velocity; pressure).
An integration of the conservation equation (2), when applied to the control volume !u . (Fig. 2), leads to:
t+Jt
o4 a5 o5
P
+
o3
a4
ω1
a6 a o6 1
a3 o2 a2
o1
+
t+Jt
a1
o6
t
t
o6
a6
t
ai+1
oi t+Jt
+
t+Jt
t
− Fig. 3. u-control volume !u .
5
!u
t
@( f1 ) dv dt + @t i=1
t+Jt
t+Jt
t
oi
ai
J :n2'r ds dt
J :n2'r ds dt
J :n2'r ds dt J :n2'r ds dt
!u
S dv dt = 0;
(18)
where n is a unit outward vector normal to the di5erential length elements ds. stability of the numerical resolution process is improved. (iv) The control volume is treated as the addition of six elements controls volumes which improved the grid 4exibility. The method consists of de7ning a grid of points within the calculated domain and then builds around each point a control domain ! (Fig. 2). The longitudinal cross-section is 7rst divided into three-node triangular elements. Then the centroids of the elements are joined to the midpoints of the corresponding sides. This creates polygonal control volumes around each node in the 7nite element grid (Figs. 3 and 4). The discretization of the longitudinal cross-section is rotated through 2' radians about the axis of symmetry. The result is a discretization of the axisymmetric calculation domain into torus elements of triangular cross-section and torus control volumes of polygonal cross-section. To avoid the oscillatory pressure and velocity 7elds, a staggered grid arrangement is employed, i.e. the axial velocity component and the temperature are stored at the same locations that are di5erent from those of the radial velocity component and from those of the pressure.
4. Results and discussion 4.1. Validity of the theoretical model In order to validate the established model and the numerical method employed, the theoretical results are compared with the experimental data of Jemni et al. [10]. The considered reactor is connected to a reservoir that initially involves a known quantity of hydrogen. The reactor is 7lled with LaNi5 alloy. The initial pressure in the reservoir, the reservoir volume, the inner diameter of the reactor, the inner height of the reactor, the heating 4uid temperature, and the used amount of the alloy are, respectively, 1 bar, 1755 cm3 , 5 cm, 8 cm, 313 K and 422 g. At t = 0, the reservoir is put in contact with the hydride bed. The time-evolutions of the pressure within the reservoir and the total mass desorbed are plotted in Fig. 5. It is seen that numerical results agree satisfactorily with experimental data.
200
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208 3.5E+5 3.5E-4
3.0E+5
3.0E-4
Mass desorb ed (kg)
Pressure (Pa)
2.5E-4
2.5E+5
Experience 2.0E+5
Model
2.0E-4
Experience 1.5E-4
Model 1.0E-4
1.5E+5 5.0E-5
1.0E+5
0.0E+0
0
300
600
900
1200
1500
Time (s)
0
300
600
900
1200
1500
Time (s)
Fig. 5. Calculated and measured evolutions of pressure within the reservoir and the total mass desorbed.
Table 4 Thermophysical properties of materials and data used in computations E5ective thermal conductivity Permeability Porosity Initial bed temperature Heating 4uid temperature
e (W m−1 K −1 ) K(m2 ) (%) T0 (K) Tf (K)
1.32 10−8 50 290 333
4.2. Time-space evolution of the temperature and the hydride density The considered reactor has a volume of 169:56 cm3 and it is heated with a heating 4uid at a constant temperature. The half of the volume of this reactor is 7lled with LaNi5 metal. Initially the hydride bed is at an equilibrium state. The thermo-physical properties of materials and data used in computations are indicated in Table 4. Fig. 6 shows the temperature evolution within the reactor in the desorption case and at the selected times, t = 15, 36, 180, 360, 720 and 1440 s. Initially the hydride bed and the expansion volume were assumed to be at a constant temperature (T = 290 K). The temperature pro7le at 15 s indicates that the temperature near the wall is signi7cantly increased as a result of heat supplied by the heating 4uid. However, the temperature at the centre of the reaction bed (porous medium) remains essentially the same as that of at initially time, showing that heat exchanges with heating 4uid are absorbed by the endothermic desorption reaction near the wall. The temperature pro7le at 180 s shows, on the one hand, that the temperature in porous medium part is clearly increased. On the other hand, for the expansion gaseous
part, there is no important evolution due to the accumulation of the hydrogen mass desorbed. After a long enough time the system comes down to the thermal equilibrium with the heating 4uid. Also, it is noted that the expansion volume part reaches quickly the thermal equilibrium, with the heating 4uid, than the reaction bed. Fig. 7 shows the metal hydride density evolution within the reactor at the selected times mentioned above. At the beginning, the dehydriding process takes place near the walls of the reactor where the desorption equilibrium pressure is important due to the increasing of the temperature. The density pro7les, at 36 and 180 s, indicate that there is an absorption process in the core part of the reaction bed where the temperature is even weak. This phenomenon is explained by the fact that the pressure that reigns in the reactor increases, due to the accumulation of the hydrogen mass desorbed, and becomes higher than the equilibrium pressure at the centre region. Beyond t = 180 s, the desorption reaction takes place in all the reaction bed except near the bottom and lateral surfaces where the absorption reaction takes place. For one elevated enough time, the kinetics reaction decreases considerably and the system o5ers toward an equilibrium state. The distribution of the pressure within the reactor is plotted, at selected times, in Fig. 8. It is seen that the pressure increases during the sorption process, this is due to the increasing of the temperature and essentially to the accumulation of the hydrogen mass desorbed in the reactor. At all times, it is noted that the gradient of the pressure in the reactor is very weak. Fig. 9 shows the velocity distribution at selected times, t =15, 180, 360, and 720 s. Initially the reactor was assumed
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
201
Fig. 6. Temperature pro7le in the LaNi5 reactor at selected times.
to be at rest. At the beginning, the 4uid velocity is important only near the wall, this is due to the natural convection phenomena caused by the heating 4uid. At all times, a very small velocity is observed in the reaction bed compared to the velocity in the expansion gaseous part. This result may be attributed to the small permeability of the hydride bed. Also, the velocity distribution indicates the existence of two cellular 4ow in the expansion volume due to the natural
convection phenomena. When the system reaches the new equilibrium state, the 4uid 4ow disappears. 4.3. Study of the in=uence of the operating parameters Many applications have recently been developed to take advantage of the metal-hydrogen systems properties, including rechargeable batteries and heating and cooling systems
202
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
Fig. 7. Evolution of hydride density in the LaNi5 reactor at selected times.
(hydrogen storage systems, heat pumps, heat transformers, refrigerators, thermals compressors, heat storage systems). These applications require a very precise prediction of the time–space evolutions of all variables and in particular the pressure for the closed systems (Thermals compressors, hydrogen storage systems). Consequently, a study of heat and mass transfer sensitivity to the operating parameters (expansion volume to reactor volume ratio, height to the radius reactor ratio, and initial hydrogen to metal atomic ratio) becomes of interest and permits us to take all
possible precautions in order to have a high safety procedure for the storage systems, and to determine operating parameters permitting to have a housemaid working (eMciency, speed) of compression systems. 4.4. Sensitivity to the reactor geometry The pressure within the reactor and the total hydrogen mass desorbed are plotted for each reactor dimensions as a function of time (Fig. 10). The reactor volume, the amount
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
203
Fig. 8. Pressure pro7le in the LaNi5 reactor at selected times.
of the hydride metal, the physical characteristics, and the boundaries conditions were kept constant during the simulation. It is noticed that, for the di5erent values used of the ratio H=R, the longest time, needed by the reactor to reach the new equilibrium state, is gotten for a ratio equal to 2. So for low values of the ratio H=R, the heat and mass transfer in the reactor are one-dimensional and depend only on z-coordinate. Under these conditions, the resistance to transfers along the z-direction increases with H=R values, hence the necessary time to reach the new equilibrium state increases. For large values of H=R, the transfers are
also one-dimensional and depend only on r-coordinate. When H=R rises, the rPesistance to the transfers according to r-direction decreases. For intermediate values of H=R, the two-dimensional e5ects are important. When H=R increases the resistance to the transfers along the axial direction increases and that along the radial direction decreases. Those competing e5ects explain the existence of a critic value of H=R equal to 2. Also, this result can be explained by the fact that the reactor surface that permits the exchange of the heat with the heating 4uid presents a minimum at H=R equal to 2.
204
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
Fig. 9. Velocity distribution in the reactor LaNi5 at di5erent times. 1.4E+6
1.2E+6
Mass desorbed (kg of H2/kg of metal)
2.5E-4
6
Pressure (Pa)
5 4
1.0E+6
3 2
6. H/R=0.25 5. H/R=0.5 4. H/R=1 3. H/R=6.75 2. H/R=4 1. H/R=2
1 8.0E+5
6.0E+5
6
2.0E-4
5 4 1.5E-4
3 2
6. H/R=0.25 5. H/R=0.5 4. H/R=1 3. H/R=6.75 2. H/R=4 1. H/R=2
1 1.0E-4
5.0E-5
0.0E+0
4.0E+5 0
200
400
600
Time (s)
800
1000
0
200
400
600
800
1000
Time (s)
Fig. 10. In4uence of the height to the radius ratio of the reactor on the total hydrogen mass desorbed and the pressure within the reactor.
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
205
Fig. 11. In4uence of the height to the radius ratio of the reactor on the 4uid velocity pro7le.
Fig. 11 shows the velocity distribution for di5erent values of the height to the radius ratio of the reactor H=R, 0.5, 2, and 6.75. It is seen that, when the H=R ratio increases, the two circulating zone grow in size and the maximum value of the gas velocity increases. 4.5. Sensitivity to the expansion volume to reactor volume ratio Hg =H Fig. 12 presents the mean pressure evolution in the reactor and the total mass desorbed, for di5erent values of the
expansion volume to the reactor volume ratio Hg =H . The physical characteristics, the amount of the hydride metal and the boundaries conditions were kept constant during the simulation. It is noticed that increasing the value of the ratio Hg =H the pressure within the reactor decreases and the total mass desorbed increases. Fig. 13 shows the temperature distribution for di5erent values of the ratio Hg =H , 0.25, 0.5, and 0.75. It is seen that there is an important gradient of temperature in the expansion gaseous part when the ratio Hg =H is lower and an uniform temperature pro7le for the large values of Hg =H .
206
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208 1.4E+6
4.0E-4
Hg/H=0.25
Mass desorbed (kg of H2/kg of metal)
3.5E-4
Hg/H=0.5
1.2E+6
Pressure (Pa)
Hg/H=0.6
1.0E+6
Hg/H=0.75
8.0E+5
6.0E+5
Hg/H=0.75
3.0E-4
Hg/H=0.6 2.5E-4
Hg/H=0.5 2.0E-4
Hg/H=0.25
1.5E-4
1.0E-4
5.0E-5
4.0E+5
0.0E+0
0
200
400
600
Time (s)
800
1000
0
200
400
600
800
1000
Time (s)
Fig. 12. In4uence of the expansion volume to reactor volume ratio Hg =H on the total mass desorbed and the pressure within the reactor.
Fig. 13. In4uence of the expansion volume to reactor volume ratio Hg =H on the temperature pro7le.
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208
207
Also, those pro7les indicate that the heat transfer becomes rapid when the ratio Hg =H decreases. The 4uid velocity distributions, for the considered values of Hg =H , are presented in Fig. 14. It is noted that increasing the ratio Hg =H the two circulating zones grow in size and the 4uid velocity increases. 4.6. Sensitivity to the initial hydrogen to metal atomic ratio (H=M )0 In order to evaluate the in4uence of the initial hydrogen to metal atomic ratio (H=M )0 on the pressure evolution within the reactor and the hydrogen mass desorbed, a simulation of the reactor for a series values of the (H=M )0 ratio was done. Fig. 15 shows that increasing the value of this ratio (5:4 ¡ H=M ¡ 6) the reached pressure and the total mass desorbed increase signi7cantly. It is also noted that a decrease of this ratio below the value of 5.4 has no important e5ect on the pressure and the mass desorbed evolutions. This result can be explained by the high equilibrium pressure sensitivity to the hydrogen to metal atomic ratio (H=M ) in the range of (5:4 ¡ H=M ¡ 6) [12]. 5. Conclusions A mathematical model describing hydrogen desorption of a porous lanthanum misch metal bed, in a closed reactor, has been presented and solved numerically by the control-volume-based 7nite element method (CVFEM). The mathematical model includes complex heat and mass transfer, 4uid 4ow and chemical reaction that take place during the sorption processes. The result on temperature, composition distribution and velocity are presented and discussed. The e5ects of various governing operating conditions on the evolution of the pressure and the hydrogen mass desorbed, have been studied. The validity of the theoretical model has been tested by comparison with experimental data of the total mass desorbed and the pressure evolution within the reservoir. A good agreement is obtained. It is found that, at all times, the 4uid velocity in the reaction bed is very small compared to the velocity in the expansion gaseous part there where a two circulating zones take place. These circulating zones grow in size and the 4uid velocity increases when the expansion volume to the reactor volume ratio (Hg =H ) and the height to the radius reactor ratio (H=R) increase. Also, it is found that the expansion volume part reaches quickly the thermal equilibrium, with the heating 4uid, than the reaction bed. The pressure and total hydrogen mass desorbed varied considerably when the initial hydrogen to metal atomic ratio (H=M )0 varies in the range 5:4 ¡ (H=M )0 ¡ 6. The numerical results showed on the one hand that the increase of the expansion part volume a5ect signi7cantly the
Fig. 14. In4uence of the ratio Hg =H on the 4uid velocity.
evolutions of pressure within the reactor and the hydrogen mass desorbed. On the other hand, it is seen that, when the height to the radius reactor ratio is large (H=R2) or too small (H=R2), the necessary time to reach the equilibrium state decreases considerably.
208
F. Askri et al. / International Journal of Hydrogen Energy 29 (2004) 195 – 208 1.8E+6
3.0E-4
H/M=5.94
H/M=5.94
Mass desorbed (kg of H2/Kg of metal)
1.6E+6
1.4E+6
H/M=5.82
Pressure (pa)
1.2E+6
1.0E+6
H/M=5.64 8.0E+5
H/M=5.52 H/M=5.40
6.0E+5
4.0E+5
2.5E-4
H/M=5.82 2.0E-4
H/M=5.64
1.5E-4
H/M=5.52 H/M=5.40
1.0E-4
5.0E-5
2.0E+5
0.0E+0
0.0E+0
0
100
200
300
400
Time (s)
0
100
200
300
400
Time (s)
Fig. 15. In4uence of the initial hydrogen to metal atomic ratio H=M on the total mass desorbed and the pressure within the reactor.
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