Investigation of Three-Dimensional Heat and Mass Transfer in a Metal Hydride
Reactor
Mahmut D. MAT, Yüksel KAPLAN, Kemal ALDAS
Mechanical Engineering Dept. Nigde University, 51100 Nigde, Turkey
Tel.:+90 388 225 0115, Fax:+90 388 225 0112 e-mail:mdmat@nigde.edu.tr
Abstract
A mathematical model for three-dimensional heat and mass transfer in metalhydrogen reactor is presented. The model considers three-dimensional complex heat, and mass transfer and chemical reaction in the reactor. The main parameter in hydriding processes is found to be the equilibrium pressure, which strongly depends on temperature. Hydride formation enhanced at regions with lower equilibrium pressure.
Hydiriding processes are shown to be two dimensional for the system considered in this study. Effects of heat transfer rate and R/H (radius to height) ratio on hydride formation are investigated. Hydride formation increases significantly with larger heat transfer rate from the boundary walls, however after a certain heat transfer rate, the increase in formation rate is found to be not significant, due to the low thermal conductivity of the metal-hydride systems. The estimated results agree satisfactorily with the experimental data in the literature.

Nomenclature
C p
Specific heat (J kg
-1
K
-1
)
E a
Activation energy (J mole -1 ) h Heat transfer coefficient (Wm
-2
K
-1
)
H Reactor height (m)
R/H Radius/Height ratio
H/M Hydride/Metal ratio
K Permability (m
2
) m Hydrogen mass absorbed (kg m
-3 s
-1
)
M g
Molecular weight (kg mole
-1
)
P Pressure (Pa)
R Universal gas constant (J mole
-1
K
-1
)
T Temperature (K) t Time (s)
V Gas velocity (m s
-1
)
Greek letters

H 0 Reaction heat of formation (J/kg)

Porosity

Thermal conductivity (W m -1 K -1 )

Dynamic viscosity (kg m
-1 s
-1
)

Density (kg m
-3
)
Subscripts e Effective eq Equilibrium f Cooling fluid g Gas s Solid ss Saturated
2

1.Introduction
Recently metal hydrides as a hydrogen storage medium received much research interest due to the superior advantages over the conventional storage methods especially from the safety considerations. The hydrogen can be stored in the hydride beds in solid form at a room temperature. Endothermic reaction, during the hydrogen release, prevents hydrogen leak when the tank is ruptured.
Several parameters strongly affect the heat and mass transfer in the hydride beds.
The main parameters are the equilibrium pressure, effective thermal conductivity, and reaction kinetics. Accurate experimental determination of these parameters play important role in predicting the characteristics of hydriding and dehydriding processes.
The metal hydrides have significant applications areas in automotive industry for hydrogen storage medium, in heat pumps and chemical compressors especially as hydrogen storage medium. There is a large body of numerical and experimental works in the literature on the details of hydriding and dehydriding processes. Mayer et al.
(1987) developed a two-dimensional conduction based mathematical model and applied to several experimental situations. They tested the numerical value of diffusion coefficient with the experimental data. They found that temperature in the bed showed very rapid increase at the beginning of the reaction and did not significantly change in the bed at later times due to low conductivity of the bed.
Sun and Deng (1988) presented a one-dimensional model for heat and mass transfer in hydride beds. This model improved some of the limiting assumptions used in the literature such as hydrogen content dependency of effective thermal conductivity and other thermo physical properties. In a subsequent study, Sun and Deng (1990) extended
3

their model to two-dimensions. The model could be adapted to cylindrical and spherical coordinates. They showed that the model agreed reasonably with the experimental data.
Choi and Mills (1990) investigated the heat and mass transfer in a hydride bed with one-dimensional mathematical model considering the hydrogen flow (convective terms) in the bed. They found that optimum bed thickness should be around 10-20 mm to be used with heat pumps applications depending on the thermal conductivity. Jemni and
Ben Nasrallah (1995a, 1995b, 1997) developed a comprehensive mathematical model considering heat and mass transfer and fluid flow in the hydride bed. Model solved energy equation for both solid and gas phases. They showed that fluid flow did not have a significant effect on the heat and mass transfer characteristics, and thermal equilibrium assumption was valid especially during hydriding processes. However, there was a slight deviation from this assumption in dehydriding process. Jemni and
Ben Nasrallah (1995a) showed that mass transfer was significantly affected from hydrogen inlet pressure and hydrogen inlet temperature and hydrogen absorption was enhanced at high inlet pressure and low inlet temperature. They showed that bed geometry was also an important factor in hydrogen absorption. At small H/R
(Height/Radius) values hydrogen absorption decreased due to resistance hydrogen flow along the z direction. Hydrogen absorbed reached a minimum value when H/R=25.
Beyond this value hydrogen absorption increase with larger H/R ratios.
Jemni et al. (1999) conducted an experimental study to determine the effective thermal conductivity, equilibrium pressure and reaction kinetics in a LaNi
5
-H
2
system.
They used these data in the validation of mathematical model developed at earlier studies (Jemni and Ben Nasrallah, 1995a, 1995b). They showed that energy equation
4

without convective terms and hydrogen and gas mass conservation equation with an appropriate reaction kinetic relation were sufficient to capture the experimental data.
Using a two-dimensional model Nakagawa et al. (2000) investigated the validity of thermal equilibrium in metal-hydride beds. They found that the thermal equilibrium was largely depend on the choice of heat transfer coefficient between gas and solid phases.
Using a heat transfer coefficient relation based on Pe number, Nakagawa et al. (2000) obtained a considerable difference in reacted hydrogen amount with equilibrium and non-equilibrium assumptions. They showed that convection promoted the hydriding reaction at the early times of the process due to enhanced heat removal, however, convection effect reversed at later times, since a gas flow towards the interior prevented the removal of heat.
Although there are a large body of literature on the modeling of hydride reactor, which some of them reviewed above, none of them considered the processes in threedimension. The objective of study extend a simplified mathematical model of Jemni and
Ben Nasrallah (1995a) to three-dimensional geometry and investigate the heat and mass transfer in a metal hydride bed.
2.Mathematical Model
Metal hydride formation is considered in a three-dimensional metal hydride reactor shown in Fig. 1. The system mirrors the experimental configuration of Jemni et al
(1999). The reactor has a 50 mm inner diameter and 100 mm height and filled with a crushed La Ni
5
alloy. Hydrogen is charged from top of the bed with a range of pressure.
5

The bed is cooled from sides and bottom to remove energy released during the hydriding process.
Fluid flow generated due to the pressure gradient in the system is considered not to significantly affect the hydriding characteristics. The findings of Jemni and Ben
Nasrallah (1995a) support this assumption. Jemni and Ben Nasrallah (1995a) found that convective terms do not significantly affect the results especially in the hydriding case.
Therefore, neglecting the fluid flow, equations governing the heat and mass transfer and chemical reaction within the hydrogen storage beds are given as follows.
Energy Equation
Assuming thermal equilibrium between the storage bed and hydrogen a continuum energy equation is solved instead of separate equations for both solid and gaseous phases. Thermal equilibrium assumption is made based on the results of the Jemni and
Ben Nasrallah (1995a). They found that in the hydriding process, the difference between bed temperature and gas temperature in the bed is not significant. The energy equation thus;
(

C ps
) e

T
 t

1 r

 r
( r
 e

T
 r
)

1 r
2

 
(
 e

T
 
)


 z
(
 e

T
 z
)
 m (

H
0 
T ( C pg

C ps
)) (1) where

, C p
,

are effective density, specific heat and thermal conductivity e respectively. (

C ) p e
is calculated as (Jemni and Ben Nasrallah, 1997):
6

(

C p
) e

(
 g
C pg

( 1
 
)
 s
C ps
) (2) where,

is the porosity. The effective thermal conductivity is expressed as;
 e
  g

( 1
 
)
 s
(3) where
 g and
 s are the effective thermal conductivity of the gas and solid phases respectively.
Hydrogen Mass Balance


(
 g
)
 t
  m (5)
Hydrogen gas is assumed to be a perfect gas and obeys a perfect gas law
P g
  g
T g
R / M g
in which R is the universal gas constant and M g
is the molecular weight of the hydrogen.
Mass Balance for Solid Phase
Similar to hydrogen mass balance equation excluding the advection term, mass balance for the solid phase can be expressed as:
( 1
 
)

(

 t s
)
 (6)
7

Reaction Kinetic
The amount of hydrogen absorbed is directly linked to reaction-rate which is expressed as Jemni and Ben Nasrallah (1995a). m
 
C a exp(

E a
RT s
) ln(
P g
P eq
)(
  ss
 s
) (7) where C is material depended constant, a

is the density of the solid phase at ss saturation and P eq
is the equilibrium pressure calculated using the van’t Hoff relation ship (Mayer et al.,1987); ln P eq

A

B
T
(8) where A and B are materials constants deduced from the experimental work of Jemni et al. (1999) as A=52.5 and B=43876.45.
Initial and Boundary Conditions
The hydride bed initially is assumed to have a uniform temperature and pressure and hydrogen is charged at z=0 with a constant pressure. The reaction bed is cooled from the boundary wall by a cooling fluid.
8

The initial and boundary conditions can be expressed as; at t=0
T

T
0 at t

0
T ( r ,

, 0 )

T
0

T ( 0 ,

 r
, z )

0 k

T ( R ,

 r
, z )
 h ( T ( R ,

, z )

T f
) k

T ( r ,

,
 z
H )
 h ( T ( r ,

, H )

T f
)
T ( r , 0 , z )

T ( r , 2

, z )

T ( r , 0 , z )
 


T ( r , 2

,
  z ) where h is the average heat transfer coefficient.
(9)
(10)
(11)
(12)
(13)
(14)
(15)
9

3.Numerical Method
The governing equations are solved numerically with a fully implicit, finite-domain scheme embodied in the PHOENICS computer code (Rosten and Spalding, 1986). The differential equations are integrated over finite control volumes into which the system has been discretized. A representative control volume is shown in Figure 2. In this figure, P represents the cell center, N, S, E, W, H, L are the neighbor cell centers. After integration, the differential equations are transformed into a general algebraic form: a p
 p
  a k
 k
 a o
 p o 
S

(16) where

represents the generic variables solved (T,
 s
,
 g
)
 p
is the unknown nodal value,
 o p
represents
 p
at an earlier time, subscript k designates the neighboring nodes, a represents diffusion coefficients, and S  is a component of the linearized source term. This linearized equation is solved by a guess-and- correct algorithm of
Patankar (1980). PHOENICS allows user to add the source terms that not exist in the program with a code suitable to the logic of the main program.
A 30x60x60 grid system in r,

, z directions respectively are used in all computations after a grid independence test. The results of a typical grid independence test is given in
Table 1. In this table, temperature and H/M values (r=R/2,

=

/2, z=H/2) at t=100sec. calculated using grid system ranging from 10x20x20 to 50x100x100 at r,

and z directions respectively. It is seen that even after 30x60x60 grid system the change in both temperature and H/M values are less than % 0.1. The similar results are obtained at
10

all locations in the solution domain. CPU time for 100 sec. real time takes about 600 sec. in a Pentium II 350 processor.
4. Results
The heat and mass transfer during the hydriding process are presented with temperature and H/M (Hydride/Metal ratio) distributions at r-z direction and several azimuthal directions.
The metal-hydride bed initially is assumed to be a constant temperature T i
=290 K.
The hydriding process is initiated at t>0 charging the bed with hydrogen at a constant temperature and pressure.
Figure 3 shows space-time evolution of temperature field at t=100, 500, 1500, 2500 and 3000 seconds. All this figures are plotted r-z plane keeping
  
is constant.
Temperature profile at t=100 sec. show that starting form the inlet section, temperature increases rapidly in the bed. Cooling effect from the side and top of the bed is evident in this figure exhibiting lower temperatures at these locations. Temperature increase in the bed is attributed to the exothermic hydriding reaction. Heat released due to reaction causes the rapid increase in the temperature of the bed.
At t=500 sec., temperature at the center region is still very high, however, near the cooled surfaces it is significantly reduced. The temperature in the bed at 1500 sec is shown in Fig 3c. It is seen that there is a large temperature gradient between center region and cooled walls. This shows that hydriding process continues in the bed
11

especially central region. The large temperature gradient may also be attributed to low thermal conductivity of the bed.
Temperature distribution at later times is given in Fig.3d and Fig.3e. It is seen that temperature in the bed around 2500 sec. significantly reduced as compared to that of at
100 sec. Again temperature gradient in the bed show that hydriding process is not still completed. It is seen form Fig.3e that at around t=3000 sec reaction almost completed in the bed, expect in the central region. Slight temperature gradient at the center indicates that total thermal equilibrium in such system takes very long times.
Evolution of hydride formation in the bed is presented in Figure 4 with H/M contours at t=100, 500, 1500, 2500 and 3000 sec. Figure 4a shows that at the early times of process, hydride formation initiates all over the bed. However, the hydride formation is higher especially near the inlet and cooled walls. Figure 4b shows that at around t=500 sec. there is a large H/M gradient between center regions and near inlets and cooled wall. This shows that hydride formation is a strong function of temperature in the bed.
The equation governing the reaction kinetics indicates that hydride formation is strongly affected form equilibrium pressure. Equilibrium pressure decreases with the temperature and hydriding process is enhanced at the regions, which has lower equilibrium pressure.
Hydride formed in the bed until around t=2500 sec. is presented in Figure 4d. Since reaction is completed near the inlet and cooled walls continuous reaction at the center results a steep temperature gradient in this regions. Hydriding reaction is almost completed in the bed at around t=3000 sec. (Figure 4e).
Temperature profiles at r-

plane at two locations along the bed axis are presented in
Fig. 5. Temperature profiles are similar in both figures expect the magnitude. It is seen that there is no change in the temperature in

direction. This result indicates that the
12

processes is independent of

and varies only with r and z direction. The similar results are obtained at different times. H/M ratio is also found to be only varies with r-z directions. Therefore, it can be concluded that the hydriding process for conditions considered in this study is two-dimensional. The result may change when fluid flow is considered. However, at high pressure, flow generated in the hydride bed will not be significant to affect the hydriding characteristics.
Figure 6 compares the temperature histories at z=H/2 and r=R/2 in a range of heat transfer rate from the boundary wall. It is seen that reaching the thermodynamic equilibrium takes longer time at lower heat transfer coefficient. Equilibrium time reduces at a higher heat transfer rate from the boundaries. This can be attributed to enhanced heat transfer between the bed and cooling fluid at higher heat transfer rates.
However, after about h=750 W/m
2
K, increasing the heat transfer coefficient does not significantly affect the results. This is a result of low thermal conductivity of the bed material. The similar behavior are observed in temperature histories at other locations, however , not presented due to the brevity
Effect of heat transfer coefficient on the hydride formation is presented in Figure 7.
Since heat transfer rate enhanced at higher heat transfer rates, H/M formation completes faster at higher heat transfer rates. Initially hydride formation is the same for all cases considered since the initial conditions are the same in these simulations.
The effect of R/H (ratio of bed radius to the height) on hydride formation is shown in
Figures 8 and 9. The effect of R/H is studied in range of 0.1

R/H

1. Figure 8 shows change of overall the hydride formation ratios with time at r=R/2 and z=H/2. H/M fraction distribution in whole bed is averaged in this figure. Hydriding reaction increases significantly at low R/H. Low R/H corresponds to thin and tall bed. This result
13

indicates that R/H ratio is a very important factor in reactor design. Since, bed material has very low thermal conductivity, an improved heat transfer rate can be achieved in thin hydride beds. Therefore, to enhance the reaction and reduce the hydriding duration bed radius must be chosen as small as possible.
Figure 9 shows the temperature histories depending on the r/H ratio. It is seen that at
R/H=1 temperature at r=R/2 and z=H/2 does not change significantly until around t=
3000 sec, however, for R/H=0.1 temperature at this location reaches the equilibrium temperature in a short time (around 1000 sec).
The estimated temperature histories at three locations are compared to recent experimental results of Jemni et al. (1999) in Figure 10. It is seen that the results predicted agree satisfactorily with the experimental results. Slight deviation from the experimental results may be attributed to influence of convective terms, which are neglected in this study.
5.Conclusions
A three dimensional numerical analysis of heat and mass transfer in a hydride bed is performed. Mathematical model considers heat and mass transfer and heat release due to the chemical reaction. The results are presented in terms of temporal evolution of temperature field and hydride to metal ratio (H/M) plots at several locations.
It is found that hydride formation in the bed is a strong function of equilibrium pressure and temperature. The hydride formation is higher at regions, which have lower equilibrium pressure. The heat released due to the exothermic reaction must be removed
14

from the bed, since high temperature causes the equilibrium pressure increase; therefore, the rate of hydride formation significantly decreases at higher temperatures.
The hydriding processes under conditions considered in this study is found to be twodimensional.
The predicted results are compared to the experimental data available in the literature. The deviation from the experimental results is attributed to influence of gas flow resulting from pressure gradient. The effects of gas flow on hydriding characteristics will be subject of a further study.
Effect of heat transfer coefficient and R/H (radius to height) ratio of the bed on the hydride formation is investigated. It is found that the hydride formation increases significantly with the cooling rate from the boundary walls. However, after a certain cooling rate, an increases in the cooling rate is not significantly increases the hydride formation.
15

References
Choi H., Mills A. F. 1990. Heat and mass transfer in metal hydride beds for heat pump applications. Int. J. Heat Mass Transfer . 33 :1281-1288.
Jemni A, Ben Nasrallah S.B. 1995a. Study of two-dimensional heat and mass transfer during absorption in a metal-hydrogen reactor. Int. J. Hydrogen Energy . 20 :43-52.
Jemni A, Ben Nasrallah S.B. 1995b. Study of two-dimensional heat and mass transfer during desorption in a metal-hydrogen reactor. Int. J. Hydrogen Energy.
20 :881-891.
Jemni A, Ben Nasrallah S.B. 1997. Heat and mass transfer models in metal-hydrogen reactor. Int. J. Hydrogen Energy . 22 :67-76.
Jemni A, Ben Nasrallah S.B. S. Jilani L. 1999. Experimental and theoretical study of a metal-hydrogen reactor. Int. J. Hydrogen Energy . 24 :631-644,
Mayer U, Groll M, Supper W. 1987. Heat and mass transfer in metal hydride reaction beds: Experimental and theoretical results. J. Less-Common Metals . 131 :235-244.
Nakagawa T., Inomata A., Aoki H., Miura T. 2000. Numerical analysis of heat and mass transfer characteristics in the metal hydride bed. Int. J. of Hydrogen Energy . 25 :339-350.
Patankar S.V. 1980. Numerical Heat and Mass and Fluid Flow, Hemisphere, Washington.
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Rosten H, Spalding D. B. 1986. PHOENICS manual, CHAM, TR/1000, London, UK.
Sun D. W., Deng S. J. 1988. Study of the heat and mass transfer characteristics of metal hydride beds. J. Less-Common Metals . 141 :37-43.
Sun D. W., Deng S. J. 1990. Numerical solution of the two-dimensional non-steady heat and mass transfer problem in metal hydride beds. Int. J. Hydrogen Energy.
15 :807-816.
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