Theoretical descriptions and experimental measurements on the effective thermal conductivity in metal hydride powder beds

Journal of the Less-Common

387

Metals, 160 (1990) 387-395

THEORETICAL DESCRIPTIONS AND EXPERIMENTAL MEASUREMENTS ON THE EFFECTIVE THERMAL CONDUCTIVITY IN METAL HYDRIDE POWDER BEDS DA-WEN SUN* and SONG-JIU DENG Chemical Engineering (China)

Research Institute, South China University of Technology, Guangzhou 510641

(Received November 28, 1988; in final form October 26, 1989)

Summary An experimental apparatus with a computerized data acquisition system has been constructed and used for measuring the effective thermal conductivity K, in hydride beds by means of a steady state method. The effective thermal conductivity in an MlNi,,,Mn,,, (Ml: lanthanum-rich misch metal) hydride bed was measured. The experimental data were compared with theoretical results calculated by using a model (K, model) which was developed by the authors for calculating K, in hydride powder beds, and the comparison shows good agreement. Several factors affecting K, are also discussed.

1. Introduction Since hydriding and dehydriding reactions in metal-hydrogen systems depend on the temperature and are accompanied by a large heat of reaction and since hydriding or dehydriding cycles decrease the particle size of the hydride ultimately to very fine powder, which results in a very low effective thermal conductivity K, in the powdered hydride beds, it is very important to develop reactors with the best possible heat transfer properties. However, from the engineering point of view, it is unrealistic to design reactors with planar configurations in order to increase their heat transfer area, because the strength requirements will make them bulky and heavy. In practice, the technical objective is the optimum design of reactors so that they have the maximum possible volume for containing hydrides, the largest heat transfer area and the lightest weight. Therefore adequate knowledge of the effective thermal conductivity in metal hydride powder beds is essential, since the low K, value of powdered hydrides is a substantial obstacle to any attempt at optimizing reactors. *Present address: Institut fir Kernenergetik waldring 31,700O Stuttgart 80, F.R.G. 0022-5088/90/$3.50

und Energiesysteme,

Universitlt

0 Elsevier Sequoia/Printed

Stuttgart, Pfaffen-

in The Netherlands

388

Several recent studies have been carried out in order to determine experimentally the effective thermal conductivity in various types of hydride bed. The first reported K, experiment was concentrated on the measurement of K, in a T&In,,, hydride bed, and an empirical equation expressed as the sum of two independent terms, functions of pressure and of hydrogen composition, was also derived by correlating the experimental data through a kinetic theory of gas and equilibrium behaviour [ 1, 21. Since a low K, value was observed in these experiments, the measurement of K, in an MmNi,,,Al,,,, (Mm: misch metal) hydride bed with a copper wire matrix to improve the heat transfer characteristics was carried out later, and an improvement in K, of 0.4 W m-l K-l was achieved with the use of a copper wire matrix [3]. Other experiments dealt with the effective thermal conductivities in a TiFe,,s,Mn,,, hydride bed [4], in Mg,Ni and MmNi,Fe hydride beds [5] and in Mg, Mg-lOwt.%Ni and Mg,Ni hydride beds [6]. One study was based on a non-stationary hot-wire method [6] but all the other studies were based on steady state methods. The practical result that K, could be assumed to be equal to l-2 W m- l K- l in engineering applications of hydrides was also reported in ref. 5. Unfortunately, none of these experimental investigations could be used to make theoretical predictions. A theoretical basis will therefore be of great advantage in studying the heat and mass transfer processes in hydride beds, since it can provide a general method for estimating the K, value in such beds. In view of all this, we put forward a theoretical model (K, model), by means of which K, in metal hydride beds could be calculated [7]. In the present study, a brief description of the K, model is given. The effective thermal conductivity in an MlNi,,,Mn,,, hydride bed is calculated and a simpler equation is derived by using the K, model. The MlNi,,,Mn,, hydride (Ml stands for lanthanum-rich misch metal which is very abundant in China) was developed in our university and has proved to be one of the most promising hydrides used in metal hydride energy conversion systems. Because of the lack of experimental data on K, in MlNi,,,Mn,, hydride beds, several experimental measurements were included in the present study. The experimental K, values will be compared with the theoretical results and consistency is shown in the comparisons given. Referring to the K, model, the dependence of K, on a variety of parameters is also discussed in order to understand the heat transfer characteristics in hydride powder beds in more detail. 2. Theoretical description of the effective thermal conductivity In hydriding and dehydriding cycles, metal hydrides decompose into fine powder. A metal hydride bed is actually a bed of finely packed powder. Many investigations have been made in order to develop a method for estimating the effective thermal conductivity in packed beds [8-l 11. However, these studies of packed beds often refer to the beds at low pressure (Q 1 atm) and with packing particles of large diameter. Metal hydride beds are generally composed of fine hydride particles as a stationary phase and hydrogen gas as a continuous phase. They differ from the usual packed beds mainly in two ways. Firstly, the diameter of a hydride particle is less than about 25 ,um. Secondly, the reaction pressure in these

389

hydride beds is above one atmosphere. Generally, the heat transfer in hydride powder packed beds with stationary hydrogen occurs by three mechanisms which are similar to those usually observed in other packed beds [9]: Mechanism 1: through the void fraction by conduction and radiation; Mechanism 2: through a series of consecutive paths amounting effectively to a solid path-length plus gas path-length, Mechanism 3: through the solid phase, the energy flowing from one particle to the next through the area of contact. A metal hydride bed can be considered as an assembly of spherical particles with similar average diameters and with a well-defined contact area, and therefore it can be assumed that in the assembly of the spheres there will be an element which is repeated in identical form in making up the bed. For simplicity, another assumption is adopted, namely that there is no bending of the heat flux lines in the bed. This assumption means that the heat flow in the bed is in one direction. Therefore it can be considered that the three mechanisms operate simultaneously and the sum of their contributions is the total heat flow. From this total heat flow, a fundamental equation for the effective thermal conductivity in the hydride bed can be derived as follows 171: K,={l-1.273(1-~)}(K,*+r,h,,) 1.273(1-s)(l-tan&) . 0.785 -tan

8,

0.215

- i (l-tanB,,)Kt+(l-tanB,)Kf+0.215r,,h,,

II

+ 1.273( 1- E) tan 8°K;

(I)

In this equation, each term on the right-hand side represents the contribution of one of the three mechanisms. The terms h,, and h,, are the heat transfer coefficients for the thermal radiation and can be formulated as follows [8]:

G’b) The assumed thermal conductivity pressure may be written [7] KZ=

of hydrogen

4 1+ N2 - ~val(w?,M~~) Kz in eqn. ( 1) is the assumed thermal conductivity

Kg* which is dependent

on

(3) of the bulk solid which is

390

dependent on the hydrogen-to-metal

atomic ratio. The equation is

K,*=K,(l+/IX)

(4)

Another parameter in eqn. (1) is the contact angle between the solid particles BO.This angle is related to the K, value at zero pressure K,Oby the expression f3,=tan-’

(K:K) 1.273( 1- E)

The theoretical K, value in a hydride bed is obtained by solving the above equations simultaneously. For packed hydride powder beds, the calculation results show that the contributions of the radiant heat transfer are very small. If these contributions are omitted, one may insert eqns. (3)-( 5) into eqn. ( 1) and combine the relative coefficients. Since for a metal hydride bed, the main variables affecting K, are the pressure and hydrogen content as reported by many authors [l-6], eqn. (1) may be simplified to K,=A,+A,X+p

Ad’ A3 +P+A,

A,P+A,XP +A,X+A,P+A,XP

where the coefficients A,-A, have different values for different kinds of metal hydrides. These values can be calculated by inserting the fundamental physical data for each hydride into the specific expressions for A,-A, which can be easily derived by using eqn. ( 1) and eqns. (3)-( 5). These values can also be determined by regression from the experimental data on the effective thermal conductivities VS. the pressure and hydrogen composition if the experiments to measure K, have been done beforehand. 3. Experimental

procedures and results

An experimental apparatus with a computerized temperature acquisition system used in the steady state measurements is shown schematically in Fig. 1. The alloy used in the experiments was MlNi,,,Mn,,, . The alloy was crushed to particles with an average diameter of 0.2 mm and then packed into a reactor. After degassing the system about 10 times, hydrogen of 99.9% purity was introduced into the reactor at a pressure of 30-35 atm and a temperature of 300 “C for about 2-3 hours. About 55 hydriding-dehydriding cycles were performed in order to obtain a sample completely converted to hydride. More than 55 wt.% of the hydride sample decomposed into - 500 mesh powder. The reactor used in the K, measurements consisted of two annular chambers. An MlNi,.,Mq,, hydride sample weighing 267.42 g was packed into the inner chamber with a central tube for water supply. An annular block of polytetrafluoroethylene was used as reference material in the outer chamber in order to reduce the experimental error in measuring the radial heat flow Q, since the low thermal conductivity of the reference material ensured a large temperature gradient. The

391

Y

/

\

Fig. I. A schematic diagram of the experimental apparatus: 1, computerized data acquisition system for temperature; 2, reactors; 3, precision temperature control instrument; 4, vacuum pump; 5, pipes; 6, precision pressure gauge; 7, flow measurement system.

reactor wall was surrounded by an electric heater and the wall temperature was controlled to within * 0.5 “C. The temperature of the cooling water flowing through the central tube was also controlled to within f 0.5 “C. A steady flow of heat through the reference bed to the metal hydride bed was obtained. The radial heat flow Q was estimated by recording the temperature gradient in the reference material of known effective thermal conductivity. The temperature difference AT between two radial points r, and r, in the hydride powder bed was also recorded. Then the effective thermal conductivity K, in the hydride bed could be estimated from the equation K _ Qln(rJ4 r~JcAT

(7)

Figure 2 shows the experimental results (dots) for the effective thermal conductivity in an MlNi,,,M%,, packed powder bed in a hydrogen atmosphere. The results indicate that K, depends strongly on the pressure. The values range from approximately 1.3 W m-’ K-’ at about 3 MPa down to only about 0.3 W m-’ K ’ as the pressure approaches zero. With increasing pressure, a typical S-shaped curve is observed as shown in Fig. 2. Since hydrogen gas has a very long mean free path, the effective thermal conductivity will be affected by the hydrogen pressure over a wide range. In the K, model, the assumed thermal conductivity of hydrogen Kg*is introduced. The typical S-shaped curve shown in Fig. 2 can be explained by the variations in Kg*over a wide pressure range since the way in which hydrogen conducts heat in the voids of the packed hydride powder bed is strongly influenced by the pressure. The range of pressures affecting K, can be divided into three regimes.

392

Fig. 2. Consistency of the theoretical effective thermal conductivity in an MlNi,,,Mu,,, calculated by using the K, model, with the experimental data.

hydride bed,

In the low pressure regime, heat transfer in the hydrogen is by molecular flow. The value of Ki remains almost constant since the mean free path of the gas molecules is considerably greater than the effective size of the voids. Therefore the pressure of the system has no influence on K,.In the middle of the pressure range, heat transfer by hydrogen is in the intermediate flow regime. The assumed thermal conductivity of hydrogen can be expressed by eqn. (3). From the equation, it is clear that Kz is strongly affected by the pressure of the hydrogen. The contribution of Kg*to K, will cause an increase in K, with increasing pressure. In the high pressure regime, hydrogen transfers heat by viscous flow. Since the mean free path of the gas molecules is much less than the effective size of the voids, the process of heat transfer by hydrogen in the voids is similar to that in any large space. Therefore no significant increase in K, is expected with any further increase in the pressure. This phenomenon of an S-shaped graph for the effect of increasing gas pressure can be observed with all packed particle beds. Figure 2 also shows the theoretical result (line) for the effective thermal conductivity in an MlNi,,,Mn 0,5 packed powder bed, calculated using the K, model. Some of the experimental data are also shown in Fig. 2 for comparison. One point stands for each measurement. The consistency of the experimental data with the theoretical result is clear from Fig. 2. Other measured data which are not included in Fig. 2 also show the same consistency. By inserting the physical properties of the MlNi,,,Mn,,, hydride bed into the K, model, the coefficients A,-A, ineqn. (6) can be determined and the expression for K, in the MlNi,,,Mn,, hydride bed as a function of pressure and hydrogen-to-metal atomic ratio is given as

K,=O.l7+0527X+

0.1061P

22.744P+

70.5064XP

0.389 +P+99.26 +307.71X+56.1P+171.47XP

(8)

393

4. Effects of other parameters The effects of other parameters on the effective thermal conductivity in hydride beds can be illustrated with the help of the K, model. Figure 3 demonstrates the effects of the void fraction E and the thermal conductivity of the bulk solid K, on the effective thermal conductivity in hydride beds. It can be seen that K, is less than 2 W m- l K- l for values of K, less than 50 W m- ’ K- l. It is noted that K, < 2 W m- l K- l is the usual range for a powdered hydride bed [5] since only relatively low values of KS have been observed in a number of kinds of alloys. The results in Fig. 3 also indicate that with a decrease in E, K, will increase, as has been shown in many experimental studies. Figure 4 illustrates the relationship between K, and KS at different temperatures. A rise in temperature will cause an increase in K,. The van? Hoff equation predicts that a rise in reaction temperature will result in an increase in reaction pressure. This means that Kz will make a greater contribution to K,. Furthermore, the rise in the reaction temperature also augments the influence of radiative heat transfer on K,. Figure 5 shows the influence of radiative heat transfer on K, as a function of particle size and temperature. The value of the ordinate indicates the ratio of

KS (

V/a )

1, (

v/r

Fig. 3. The relationship between K, and K, with various void fractions (X= 0). Fig. 4. The relationship between K, and K, at different temperatures

400 I

10-7+ 10-5

500 K

10

10-z 1

DpxzoO( m 1 Fig. 5. The contribution of the radiant heat transfer to K, (e = 0.9).

(X- 0).

1

394

radiant heat transfer to overall heat transfer in the bed. It can be seen from Fig. 5 that the radiant heat transfer is extremely small compared with the overall heat transfer especially for a packed bed with very small particles. In hydride beds generally, the particles are very small and the pressure is often above one, atmosphere. Therefore the contribution of the radiative heat transfer can be omitted from the calculation. However, this condition is not valid for packed beds at high temperatures and with large particles.

5. Conclusions The K, model, which can be used to calculate theoretically the effective thermal conductivity in hydride powder beds, has been briefly described. K, was measured experimentally in an MlNi,,,Mn,., hydride bed using a steady state method. The theoretical values of K, for MlNi,,,Mn,,, hydride beds were compared with experimental results and the agreement was found to be quite satisfactory. It is therefore expected that the expression for K, in eqn. (6) will be capable of being used for many different kinds of metal hydrides. Several parameters affecting K, have also been discussed in the present investigation.

References 1 S. Suda, N. Kobayashi and K. Yoshida, ht. J. Hydrogen Energy, 6 (198 1) 52 1. 2 S. Suda, N. Kobayashi, K. Yoshida, Y. Ishido and S. Ono, J. Less-Common Met., 74 (1980) 127. 3 M. Nagel, Y. Komazaki and S. Suda, J. Less-Common Met., 120 (1986) 35. 4 A. Kempf and W. R. B. Martin, Int. J. Hydrogen Energy, I1 (1986) 107. 5 E. Suissa, I. Jacob and Z. Hadari, J. Less-Common Met., 104 (1984) 287. 6 Y. Ishido, M. Kawamura and S. Ono, fnt. J. Hydrogen Energy, 7( 1982) 173. 7 Da-Wen Sun, Song-Jiu Deng and Zu-Xin Li, in T. N. Veziroglu (ed.), Alternative Energy Sources VIII, Vol. 2, Research and Development, Hemisphere, New York, 1989, pp. 6 1 l-62. 8 S. Yagi and D. Kunii, AIChE J., 3 (1957) 373. 9 S. Masamune and J. M. Smith, Ind. Eng. Chem., Fundam., 2 (1963) 136. 10 N. Wakao and D. Vortmeyer, Chem. Eng. Sci., 26 (1971) 1753. 11 W. Schotte, AIChE J., 6 (1960) 63.

Appendix A: Nomenclature D, x h::, K,

K, K, P

average diameter of metal hydride particle (m) emissivity of solid surface heat transfer coefficient for radiation, solid to solid (W mm2 K- ’ ) heat transfer coefficient for radiation, void to void (W mm2 K-l) effective thermal conductivity in hydride powder bed (W m- ’ K- ’ ) thermal conductivity of hydrogen (W m- l K- ’ ) thermal conductivity of bulk solid (W m- ’ K- ’ ) pressure (atm)

395

e ro T

X

amount of heat transferred per unit length per unit time (W m- ’ ) 442 temperature (K) hydrogen-to-metal atomic ratio

Greek symbols a, /3 accommodation

& 6, 4,

coefficient void fraction contact angle between solid particles mean free path length of hydrogen gas at P, (m)