Ind. Eng. Chem. Res. 1996,34, 2305-2313
2306
Prediction of Metal Hydride Heat Transformer Performance Based on Heat Transfer and Reaction Kinetics M. Ram Gopal and S. Srinivasa Murthy* Refrigeration and Airconditioning Laboratory, Indian Institute of Technology, Madras-600 036, India
The performance of a metal hydride heat transformer is predicted based on heat transfer and reaction kinetics of the paired hydride beds. Results are computed for a heat transformer working with a ZrCrFedLaNis pair. The results show that due to the low effective thermal conductivity of the hydride beds, the performance of the system is controlled initially by heat transfer. However, when the effective thermal conductivity and overall heat transfer coefficient exceed a certain value, the reaction kinetics assume importance. The performance of the system improves significantly with input temperature as heat is supplied in both the half cycles. The performance also improves as the output and heat rejection temperatures decrease. For the ZrCrFeldLaNis system the coefficient of performance lies between 0.27 and 0.30 and the second-law efficiency lies between 0.37 and 0.44.
1. Introduction Metal hydride heat transformers are very promising energy conservation devices because they are compact, environmentally sound, and energy efficient and offer wide operating temperature ranges. Metal hydride heat transformers are considered as the most cost effective means to generate electric power or high-pressure steam from industrial waste heat with temperatures lower than 100 "C for which no conventional process exists which can compete with metal hydride systems (Suda, 1984). In spite of the great potential, very little work has been carried out in this area. Werner (1988) has described the lab model of a two-stage metal hydride heat transformer developed a t IKE, Stuttgart. He has presented the typical operational behavior of the system, which yields a maximum temperature lift of 50 K at a heat output temperature of 130 "C. The design of this system was based on static P-C-T relations. Suda et al. (1991) have developed two-stage hydride heat transformer demonstration units having capacities of 7.72 and 77 kW. The emphasis of their work was the generation of high-pressure steam in the temperature range 120-150 "C using a low-grade heat source at a temperature of 80 "C with ambient air as the cooling medium. Werner and Groll(1991) have built a laboratory model of a two-stage metal hydride heat transformer using three different misch metal-nickel alloys. They have carried out absorption and desorption tests with individual reaction beds to study the dynamic aspects of the hydride reactors. Dantzer and Orgaz (1986) have carried out a thermodynamic analysis of hydride heat transformers and heat pumps. Srinivasa Murthy et al. (1988) have presented a thermodynamic analysis of two-stage ternary metal hydride heat transformers. It has been generally recognized that in order to improve the performance of metal hydride systems it is essential to improve the heat and mass transfer aspects of the hydride beds. Several experimental and theoretical studies on heat and mass transfer characteristics of single hydride beds have been carried out (Suda et al., 1983; Mayer et al., 1987; Da-Wen Sun and Song Jiu-Den, 1989a,b; Choi and Mills, 1990; Ram Gopal and Srinivasa Murthy, 1993). Few experimental studies have been carried out on the dynamic aspects of paired metal hydrides. Tuscher et al. (1984), Anevi et al. (1984), Nagel et al. (1986a,b), and Bjurstrom et al. (1987) have carried out studies on dynamic aspects
of single and paired metal hydride systems. These studies show that sorption kinetics and heat transfer processes in both the reactors are coupled, and together they determine the rate of hydrogen transfer between the two hydrides. From these studies it was concluded that in computer simulations of hydrogen transfer in hydride heat pumps and heat transformers, the conditions in both the reactors must be solved simultaneously. Bjurstrom and Suda (Bjurstrom, 1988) have used a lumped parameter model t o predict the performance of coupled reaction beds. Even though there is good agreement between the predicted and experimental values, the lumped parameter assumption is basically incorrect as the effective thermal conductivity of the hydride bed is extremely low. Groll and Isselhorst (1989) have studied the dynamics of coupled reaction beds. They have used different mathematical models with increasing complexity, the simplest being a onedimensional model with one energy and one mass transfer equation. Their results show that there is very little difference between the computed and measured values of maximum hydrogen transferred. Pons and Dantzer (1992)have simulated the thermal behavior of coupled hydride beds operating in heat pump mode. They assumed infinitely fast reaction kinetics, as a result their model is restricted to relatively long reaction times (about 1h). Recently, the authors have studied the metal hydride refrigerator performance based on heat and mass transfer of coupled beds for the ZrMnFe/ MmNi4.dO.S system (1994a) and the ZrCrFeCuo.$I'io,sZr0.2Cro.sMnl.asystem (1994b). In the above papers, some simplifyingassumptions such as equal absorption/ desorption rates, absence of hysteresis, and plateau slope were made. However, the spatial distribution of temperature and concentration were taken into account. In this paper, the performance of a metal hydride heat transformer is predicted by simultaneously solving the heat transfer and kinetics equations of the paired reactors. This model takes into account the spatial variation of temperature and concentration and the effects of hysteresis and plateau slope. Consequently this model predicts the system performance better than the existing ones. The same model may be used for the design and optimization of the hydride heat transformers, heat pumps, refrigerators, etc. which operate in a dual mode.
0 1995 American Chemical Society 0888-5885/95/2634-23~5~09.00/~
2308 Ind. Eng. Chem. Res., Vol. 34,No. 7, 1995 Fluid
I
REACTOR
I
@
Fluid in
REACTOR @
I
F luld in
Figure 1. Physical model of the metal hydride heat transformer. I
In P
TI
Tm
Th
-
-l/r
Figure 2. Operating cycle of the metal hydride heat transformer.
2. Physical Model Figure 1 shows the physical model of the metal hydride heat transformer system. It consists of one pair of two hydride reactors A and B connected in such a way that hydrogen gas can flow freely between them when the shut-off valve is open. Each reactor consists of a filter tube, a hydride tube, and an outer tube through which the heat transfer fluid flows. Reactor A consists of the low-temperature alloy A (whose hydride form is AH,) while reactor B is filled with the hightemperature alloy B (hydride BH,). One pair of reactors A and B gives an output that is intermittent in nature. To obtain a continuous output two such pairs are operated 180" out of phase. Figure 2 shows the operating cycle on a Clausius-Clapeyron chart. The operating cycle consists of the following four processes: Process 1. At the beginning of the process, hydride AH, is at a uniform temperature Tm and concentration XA,Iand hydride BH, is at a uniform temperature T h and concentration &,I (XBJ Tm). ib soon as the shut-off valve is opened, hydride AH, starts desorbing H2 and hydride BH, starts absorbing it. The energy input is in the form of heat at the intermediate temperature Tmto reactor A through the heat transfer fluid. This heat is used to desorb H2 from the hydride AH,. As bed B absorbs hydrogen gas, it releases heat at high temperature, Th, which is the output of the system. This process is stopped when the required amount of hydrogen gas (usually the maximum transferable amount) is transferred from hydride AH, t o hydride BH,. Process 2. This is a sensible cooling process. During this process reactor A is cooled t o the low temperature Tl while reactor B is cooled to the intermediate temperature T,. The shut-off valve is kept closed during this process. Process 3. During this process the shut-off valve is kept open and hydrogen gas transfer takes place from hydride BH, to hydride AH,. The heat of desorption at reactor B is supplied by the heat transfer fluid flowing through the reactor at the input temperature T, and the thermal mass of the reactor. This is the second
input to the system. The heat of absorption a t reactor A is rejected to the low-temperature sink (the heat transfer fluid and the bed) a t temperature Ti. The process is terminated when the required amount of H2 gas (as in process 1)is transferred from reactor B to reactor A. Process 4. During this process the shut-off valve is kept closed, reactor A is sensibly heated to temperature Tm,and reactor B is sensibly heated to temperature Th, thus completing the cycle. Assumptions. (i) The whole bulk material of the reaction bed is continuous and in the solid phase, Le., heat transfer through the bed is by conduction only. The convection heat transfer between the gas and the hydride particles is neglected. This assumption is made here to simplify the solution procedure, as the inclusion of the convection heat transfer term makes the resulting differential equations highly nonlinear. This is justified because the heat transfer coefficient between hydrogen gas and the solid particles inside the bed is very high due to the fine size of the hydride particles. Moreover, the velocity of hydrogen gas is also small, and hence, the contribution of the convection heat transfer inside the bed may be neglected. Studies have shown that the effect of convection is noticeable in the beginning of the process only when the reaction rate is high (Choi and Mills, 1990). Hence, the error resulting from this assumption may not be significant (Groll and Isselhorst, 1989). (ii) Heat conduction through hydride beds is one dimensional. This assumption is valid if the bed thickness is very small compared to its length. (iii)Initially, the hydride beds of reactors A and B are in equilibrium with the H2 gas in the respective gas spaces of the reactors. (iv) The equilibrium pressures are calculated using static P-C-T relations. A metal hydride heat transformer operates under dynamic conditions. Hence, it would be more appropriate to use dynamic P-C-T relations instead of static relations. However, unlike the static relations, the dynamic P-C-T relations depend upon the heat transfer characteristics of the reactors and the resulting hydrogen flow rates. It is not possible t o establish these relations based on theoretical studies alone. A little experimental data on dynamic aspects of hydride reactors is available. However, almost all these studies deal with the ABS type alloys and are highly specific. Hence, in this study static P-C-T relations are used instead of dynamic relations. It is well-known that the static P-C-T relations predict cycle times that are shorter than the actual cycle times. This is mainly due to the increased hysteresis and plateau slope of the hydrides under dynamic conditions. However, irrespective of whether static or dynamic relations are used, the design procedure and the general performance trends remain unchanged (Groll et al., 1989). (v) Thermal properties of
Ind. Eng. Chem. Res., Vol. 34, No. 7,1995 2307 hydride beds are constant. This assumption is made to simplify the problem formulation even though it is well-known that the effective thermal conductivity varies with hydrogen pressure and concentration. This assumption leads to a slight underestimation of the actual performance of the system (Da-Wen Sun et al., 1989). (vi) The pressure drop through the beds is neglected. This is a simplifying assumption. It is justified because for thin beds, low hydrogen flow rates, and higher pressure levels the pressure drop is not a rate-limiting factor (Choi and Mills, 1990). (vii) The thermodynamic state of hydrogen gas in the combined gas space is uniform throughout, even though it varies with time. (viii) Heat transver between the hydride bed and the surrounding atmosphere is neglected. (ix) Hydrogen gas is considered as a perfect gas. Experimental results show that the error due t o this assumption is less than 1%(Werner and Groll, 1991). (x) It is assumed that at any given instant the average temperature of the reactor material (other than the alloy and heat transfer fluid) is equal to the average temperature of the hydride bed. This is a simplifying assumption. (xi) The difference between the bulk fluid temperature and the initial bed temperature during processes 1and 3 is assumed to be 5 “C.
3. Problem Formulation Initially for processes 1and 3, when the shut-off valve is closed, the hydrogen gas in the respective gas spaces is in equilibrium with the respective hydride beds. Hence the gas pressure in reactors A and B can be estimated by van’t Hoffs equation given by Gambini (1989)
In ‘,A
aA
=
+ bA + f s , A + fhys,A
Where Pg,A and P g , B are the respective gas pressures in reactors A and B. The van’t Hoff equation relates the Gibbs free energy to the fugacity (or pressure in this case) of hydrogen for the metal hydride reaction. The terms fs and fhys take into account the irreversibilities, plateau slope, and hysteresis factors. The plateau slope factor fs is given by
and temperature of Hz gas in the combined gas space immediately after the valve opening is given by
(7) The gas temperature Tg,Iis given by Tg,I
= (nAcg,ATI,A
+ nBCg,BTI,B)
(8)
(ng,ICg,I) nA
and n B are obtained from the ideal gas law given by
(10) Thus, from the initial bed temperatures, from the gas volumes VA and VB, and from the hydride properties, the initial gas pressure and temperature in the combined gas space can be obtained using the above equations. These values are used as the initial conditions of the combined gas space for processes 1 and 3. As stated earlier, the operating cycle consists of two hydrogen transfer processes (process 1 and process 3) and two sensible heat transfer processes (process 2 and process 4). The heat and mass transfer rates during processes 1 and 3, the heat transfer rates during processes 2 and 4, and finally the performance of the system are obtained by applying the energy and mass balance equations to the reactors and the gas space and by solving the resulting equations. The authors have carried out experimental and theoretical studies on single hydride reactors using M d i 4 , & ) . 5alloy (Ram Gopal and Srinivasa Murthy, 1995). A reasonably good agreement was found between the experimental and theoretical results. The present model is based on the single hydride bed model developed by the authors. The governing equations for coupled cylindrical hydride beds are given below. Processes 1 and 3. During these processes, hydrogen transfer takes place between reactor A and reactor B (from reactor A to reactor B during process 1and from reactor B to reactor A during process 3). The governing energy balance and kinetic equations for these processes are
(3) Where x and xm are the bed concentration at a given time and the average concentration of the bed respectively. The hysteresis factor fhys is given by
where the effective density of the bed is given by
= In(Pa/Pd) (4) Pa and P d are the equilibrium pressures during absorption and desorption, respectively. Since the thermodynamic state of the hydrogen gas when the shut-off valve is opened is assumed to be uniform throughout the combined gas space a t any given time instant, the mass and energy balance equations for the Hz gas in the combined gas space immediately after the valve opening are given by
,b is the reaction rate in mol of H h g of hydride-s.
fhys
ng,I
= nA + nB
(5)
(6) ng,ICg,ITg,I = nACg,ATI,A + nBCg,BTI,B Assuming ideal gas behavior for Hz gas, the pressure
@b = @,(I
- 6)
(12)
(13)
p is negative during desorption (for reactor A during process 1 and for reactor B during process 3) and positive during absorption (for reactor B during process 1 and for reactor A during process 3). The expression for axla0 varies depending upon the rate-controlling mechanism. Several expressions have been proposed on the basis of a specific control mechanism. In this model a general kinetic expression suggested by Da-Wen Sun and Song Jiu-Den (1989) is used to find the reaction rate. It is given by
2308 Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995
Processes 2 and 4. These processes are sensible heat transfer processes, hence the governing equations for these processes are The equilibrium pressure P,, is given by the van't Hoff equation given by eqs 1 and 2. Initial Conditions. Initially bed A is at uniform temperature and concentration. Hence,
T=TI at O=O; r i % r % r ,
(15)
X = X , a t O = O ; ri Ir
(16)
%
r,
Boundary Conditions. For 0 L 0, the convective boundary condition at the outer radius of the bed is given by
aT = U(T - Tf) a t r = r, -Ke ar
(17)
Where U is the overall heat transfer coefficient (excluding the heat transfer resistance of the hydride bed). Neglecting the heat transfer between the hydrogen gas and the filter, the boundary condition at the inner radius of the bed is given by
(18)
+
The bed pressure Pb at any time (0 d e ) during processes 1 and 3 is obtained from the mass and energy balance of the combined gas space. When the frictional losses are neglected, the bed pressure Pb is obtained in the following manner. Let n d be the number of moles of H2 desorbed in a small time interval d 0 and na be the number of moles of H2 absorbed in the same time interval. Then from the mass balance, the total number of moles of Hz in the combined gas space a t any time (0 d 0 ) is given by
+
ng,@+d@
- ng,@ + nd - na
(19)
where ng,@is the number of moles of Hz gas in the combined gas space at time 0 and ng,@+d@is the number of moles of H2 gas in the combined gas space at time (0 do). The temperature of H2 gas in the combined gas space after the time interval d 0 is obtained from the energy balance of the combined gas space. In writing the energy balance for the combined gas space, it is assumed that the temperature of the desorbed gas entering into is equal to the average the combined gas space (i.e., Td,@) temperature of the desorbing bed during that time interval d e . The temperature of the gas absorbed in the same interval is taken as the temperature of the gas present in the combined gas space at time 0. Hence,
+
Now using the ideal gas equation, the bed pressure Pb a t time 0 + d 0 is given by
P, =
(ng,@+d&Tg,@+d@)
VT
(21)
The boundary conditions are similar to those of processes 1 and 3. However, the initial conditions are obtained from the final conditions of the previous processes. The analysis has been carried out for 1 kg mass of alloy A (Le., M A = 1 kg). The corresponding mass of alloy B is obtained from the hydrogen capacities of alloys A and B. The performance of the system is designated by the coefficient of performance (COP),the second-law efficiency (vE), and the specific alloy output qo. The specific alloy output qois the total heat output obtained over a cycle for 1 kg of alloys used. This definition is similar to that used by Da-Wen Sun et al. (1992) and Suda (1993). The specific alloy output is the most important performance parameter, as the cost of the system depends mainly on this parameter. The COP gives an indication of the energy efficiency of the system. However, the second-law efficiency VE is more informative than the COP, because VE indicates the efficiency of the system in terms of availability (or exergy), which is far more important than energy. These three quantities, viz., COP, VE, and qo are defined as Qh
COP = -
(23)
Qm
(24)
(25)
where &h is the heat output obtained at temperature Th and Q is the heat i input a t temperature Ti. 0d is the total time taken for one complete cycle, and MA and M B are the masses of the alloys A and B, respectively. In a hydride heat transformer, heat input is supplied during processes 1 and 3 (&&I and &B,& and heat output is obtained during process 1 (QBJ). During process 4, reactor A has to be heated sensibly to the input temperature T, from the heat rejection temperature 21', hence this is an additional heat input (QA,~).Similarly, reactor B has to be sensibly heated to output temperature T h from the input temperature T,. Hence, the heat output is reduced by this amount (QB,~).However, during process 2, reactor A has t o be sensibly cooled to temperature 21' from Tm, and reactor B has to be sensibly cooled to temperature Tmfrom Th. Hence, in practical systems, the performance of the system can be improved by recovering the energy during the sensible cooing (process 2) and supplying this for sensible heating during process 4. Thus the output increases and the input decreases if sensible heat recovery is employed. However, this requires additional heat exchangers for the sensible heat recovery, and the improvement in performance depends upon the effectiveness of the heat exchangers. However, since the emphasis here is on the coupled hydride beds, the sensible heat recovery is not taken into account here. Hence, the heat output &h and input Qm without sensible heat recovery are given by
Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995 2309 1 .o
(26)
I /I
j Th=408 K
Q,
= Q A , ~ -k
+QA,~
QB,~
- Reactor A
, ,/
~
(27)
are the amounts of energy transferred during process 1at reactors A and B, etc. The amount of energy transferred between the hydride bed and the heat transfer fluid (QA,~,&B,~, etc.) is obtained by the following expression:
0.8
& A , ~ , Q B , ~ ,etc.
wO. 6 2
m
P n
Q = f f U A ( T , - Tf) dO
(28)
Where A is the area of the heat transfer surface, T, is the temperature of the bed near the wall ( r = r,), Tfis the bulk fluid temperature, and 0 is the time taken for the particular process. 4. Results and Discussion
z
W
0.2
00
Dantzer and Meunier (1988) have carried out a systematic study t o select the alloy pairs suitable for heat pumps and heat transformers. They have identified seven alloy pairs suitable for heat transformers. Among these, three pairs are vanadium-based alloys and the remaining pairs are based on rare earth metals. Six among the seven pairs suggested require auxiliary refrigeration for heat rejection and/or are too expensive. Only one pair, viz. ZrCrFel.dLaNi5, does not require auxiliary refrigeration for heat rejection a t low temperature, and the high-temperature alloy ZrCrFel.4 is relatively inexpensive due to the presence of Fe. Hence, this alloy is selected for computation even though the COP of this pair is smaller than the other pairs. The following design and operating data were used as input in the analysis: hydride pair (A/B) heat output temperature, T h heat input temperature, T, heat rejection temperature, 2'1 bed thickness (rw- pi) effective thermal conductivity, Ke heat transfer coefficient, UA= UB= U
z
\0.4
ZrCrFel.&aNis 408-418 K 358-368 K 303-313 K 3-6 mm 1.0-15 W/m.K 500-2000 W/m2.K
The operating temperature range and the thermodynamic properties for this pair are taken from Dantzer and Meunier (1988). The kinetic coefficients E and (T for LaNi5 are computed from the Arrhenius plots given by Miyamoto et al. (1983). Unfortunately, the kinetics data for ZrCrFe1.4 are not available in the literature. However, it is well-known that these materials offer fast kinetics similar to LaNis (Ivey and Northwood, 1983). Hence, the kinetic parameters E and (T of ZrCrFel.4 are assumed to be equal to those of LaNi5. The effective thermal conductivity, the bed thickness, the overall heat transfer coefficients, and the three temperatures are varied over a given range to study the effect of these parameters on the system performance. The effective thermal conductivity of powdered hydrides is about 1 W/mK (Suissa et al., 1984). However, in practice, the effective thermal conductivity of the beds can be varied by employing different heat transfer enhancement techniques such as different types of internal fins, meshes, high thermal conductivity powders, metal hydride compacts, etc. These enhancement techniques have yielded effective thermal conductivity values as high as 15 W/m*Kor even more (Suda, 1989, Ron et al., 1991). Hence, the effective thermal conductivity is varied from 1 to 15 W/mK. For computational pur-
A,:,,, , , 100 0 0 300 I
00
,
I
,
200
,
I
I
,
I
I
t
,
/'
0
Time ( s ) Figure 3. Variation of average bed concentrations over a cycle.
/1 !:
1
1:
I:
/":v! , , , , , ,
T1=303 K (rr-ri)=3 mm K,=5 W/mK U=1500 W/mZK
260 0
100
, , , , , , , , , , , , , , , ,~ , ~
, ,
/~
~
200
300
Time (s) Figure 4. Variation of average bed temperatures over a cycle.
poses, the outer radius of the filter (rJ is taken as 2 mm, the void fraction of the beds (EA, E B ) is taken as 0.4 (to take care of the volume expansion of hydride particles which is about 25%),and the specific heat and weight ratio of the heat exchanger material are taken as 500 J k g K and 0.5, respectively (for stainless steel). The system of equations was solved by a fully implicit finite difference method. Processes 1and 3 were terminated when 99% of the available hydrogen is transferred. The sensible heat transfer processes 2 and 4 were terminated when the difference between the hydride bed temperature and the heat transfer fluid temperature is within 0.5 "C. The volume of the gas space of each reactor is assumed to be equal to half the volume of the bed. Relatively short time steps (%0.001s) are taken to ensure accuracy in the bed pressure values. Figures 3 and 4 show the average bed concentrations and temperatures over a cycle. It can be observed that the times taken for the first and second half cycles are not equal. It can also be observed from these figures that the time taken for sensible heat transfer processes 2 and 4 are relatively short when compared with processes 1 and 3. As shown in Figure 4 during
2310 Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995 Th = 408 K T, = 363 K TI = 303 K
15.0
-10.0
t
(d
e a
g 1000-
W
--
Q,
With h steresis Withoul hysteresis
4
0
5.0
h 500:
O j ! \ # < < I
0.0
I
1
'
I
I
5.9
"
'
"
'
~
'
1
~
'
'
"
~
1
"
I
'
15.0
Ke (W/m";) Figure 5. Variation of bed and equilibrium pressures over a cycle.
Figure 7. Effects of effective thermal conductivity and bed thickness on cycle time.
- Reactor A _ _ Reactor B
Th=408 K T,=363 K Ti=303 K
h
M
s
\
- - - -*
6150
/I
W
- - -.
a
,
3
h 0
3
(d
-0
100
200
50
300
Time ( s ) Figure 6. Variation of cumulative energy transfer over a cycle.
processes 1 and 3, as soon as the valve is opened the temperature of the desorbing bed drops below the initial temperature and that of the absorbing bed rises above the initial bed temperature. This is due to the fast kinetics and the poor thermal conductivities of the beds. Figure 5 shows the bed pressure and the average equilibrium pressures of reactors A and B over a cycle. The bed pressure during processes 1 and 3 varies slightly due to the unequal absorption and desorption rates. It is observed that due to unequal reaction rates, the bed pressure is pulled toward the equilibrium plateau pressure of the faster reactor. Figure 6 shows the cumulative energy transfer through reactors A and B over a cycle. The energy transferred through reactor A during processes 1 and 3 is higher than that through reactor B due to the lower heat of formation of hydride B ( H f A= -31.8 kJ/mol and H f , B = -20.7 kJ/mol). The energy transferred during the sensible heat transfer processes 2 and 4 (smaller peaks) is much smaller than that during processes 1 and 3. Figures 7 and 8 show the effects of the effective thermal conductivity and thickness of the beds on cycle time and specific alloy output of the system, respec-
00
u = 1500 W/m2 K Ti, = 408 K With hysteresis T, = 363 K Without hysteresis TI = 303 K I ~ ' ~ ~ ~ " ' ' l ~ " ' 50 10 0 '5 0
'
1 0
Ke (w/m Figure 8. Effects of effective thermal conductivity and bed thickness on specific alloy output.
tively. As shown in Figures 7 and 8, for a given bed thickness and overall heat transfer coefficient, initially the cycle time ( 0 , t ) decreases and the specific alloy output (qo)increases sharply with the effective thermal conductivity. However, beyond a certain optimum value of Ke, the variation in 0,t and qo becomes small. This is because for a given bed thickness, when the effective thermal conductivity is lower than the optimum value, the conduction heat transfer through the hydride beds controls the cycle time and above this optimum value either the overall heat transfer coefficient or the reaction kinetics of the hydrides assumes importance. It can be seen that this optimum value increases with bed thickness. The effect of bed thickness on cycle time and specific alloy output can also be observed from Figures 7 and 8. As the bed thickness increases, the cycle time increases and qodecreases. This is again due to the heat transfer restrictions of the hydride beds on cycle time. For a bed thickness of 3 mm (1cm i.d. tube) an effective thermal conductivity of about 5 W/m-K results in an optimum design and hence gives the best performance
~
'
~
'
1
-
*
110
2 \1604
Ind. Eng. Chem. Res., Vol. 34,No. 7, 1995 2311 O
o
~
160
=) 3 mm
Th = 408 K T, = 363 -K IT, = 303 K
(rx-ri) = 3 mm
K. = 5 W/m K U = 1500 W/mZK Ti = 303 K
/ 120
80
4
a
40
'0
Figure 9. Effect of overall heat transfer coefficient on specific alloy output.
in terms of cycle time and specific alloy output. This also results in the best economic performance as the cost of the system also increases with the amount of heat transfer enhancement required. Since the optimum value of the effective thermal conductivity increases with the bed thickness, it is better t o have thin beds so that the optimum value can be achieved using simple and economic heat transfer enhancement techniques. The effects of hysteresis and plateau slope on system performance can also be observed from Figures 7 and 8. Hysteresis and plateau slope reduce the pressure difference between the reactors, thus reducing the mass transfer rates. Hence, the cycle time increases and the specific alloy output decreases with hysteresis and plateau slope. Hence, neglecting hysteresis and plateau slope leads to an overprediction of the system performance. For given operating temperatures, the effects of the above parameters on COP and 7 ; 1 ~are negligible as the total amount of energy transferred during a cycle remains same. Figure 9 shows the effect of the overall heat transfer coefficient U on the specific alloy output at different values of K,. The overall heat transfer coefficient U as per the definition combines the resistances offered by the reactor wall and the heat transfer fluid flowing outside the bed. As shown in the figure, the specific alloy output of the system improves as U increases. However, the improvement in the specific output decreases gradually as the U value increases indicating that for a given system there exists an optimum overall heat transfer coefficient above which its effect on system performance becomes negligible. The effect of U on cycle time and specific output is more pronounced at higher values of K,. This is due to the fact that the order of magnitude of the internal resistance (hydride bed) and the external resistance (reactor wall heat transfer fluid) become equal at higher values of K, and/or for thinner beds. Thus the wall thickness and fluid velocity are more important for thin beds with higher effective thermal conductivities. Figures 10 and 11 show the effects of the heat source temperature Tm and the heat output temperature Th on the system performance. As shown in Figure 10 for a given heat output temperature, as Tm increases, the cycle time decreases and the specific output increases.
+
Figure 10. Effects of heat source and output temperatures on specific alloy output. 0.30
- COP
0.41
l)e
Y
a 0.27 0 u
F 0 38
Figure 11. Effects of heat source and output temperatures on COP and second-law efficiency.
This is due to the fact that as the source temperature increases, the desorption rate in both the first and second halves of the cycle increases thus reducing the cycle time and increasing the specific output. Similarly, as the output temperature Th increases, the Hz absorption rate by bed B during process 1 decreases, thus increasing the cycle time. As a result, the specific output decreases as Th increases. The worst case occurs when the input temperature is low (358 K) and the source temperature is high (418 K). Figure 11 shows the effects of Th and Tm on the COP and second-law efficiency of the system. "he temperature level of the hydride heat transformer affects the sensible heat loads only. The energy transferred due t o hydrogen absorption and desorption remains the same as the amount of Ha transferred remains constant. Hence the COP increases as T, increases and Th decreases. The secondlaw efficiency 7;1~decreases as the heat source and heat output temperatures increase. This is because, for a given heat output temperature, as the heat source temperature increases, the availability of the input increases, hence 7 ; 1 ~decreases. For a given heat source temperature, as the heat output temperature increases,
2312 Ind. Eng. Chem. Res., Vol. 34,No. 7, 1995 -
0.295
El
0.290
/// I
c, ’20
fi
6ol, ,I, 40
, , , ,
355
,
rt,
!(rw-r,) = 3 mm :K, = 5 W/m K
- U = 1500 W/mZK O m O T ,
-Ti
, , , , , , , -T,; 365
360
4
= 303 K = 308 K ,=, 3f3, 370
Figure 12. Effects of heat source and heat rejection temperatures
o,270
STk
= 408 K
/
o m 0 Ti = 303 K I TI = 308 K 1 *u TI = 313 K -
QBe9p
I
I
I
I
I
,
I
,0.36
Figure 13. Effects of heat source and heat rejection temperatures on COP and second-law efficiency.
5. Conclusions
overall heat transfer coefficient on the specific output is significant when it is very low, the beds are thin, and/ or the effective thermal conductivities are high. The specific output increases significantly with the heat source temperature as it affects the both halves of the cycle. In order to obtain high specific output, the output temperature should be as low as possible, particularly when the source temperature is small. Heat rejection temperatures should be as low as possible to obtain high specific alloy output. The variation in the COP of the system with the above operating and design parameters is not significant, as the total amount of H2 gas transferred and the heats of formation are constant. For the above system, the COP value lies between 0.27 and 0.30with the above parametric values. The second-law efficiency increases as the heat source and heat output temperatures decrease and as the heat sink temperature increases. For the above system, the second-law efficiency lies between 0.37 and 0.44. Since the present model takes the temperature and concentration gradient inside the bed and finite reaction kinetics into account, it predicts the performance better than the existing models. However, a more realistic model can be developed by including the effects of convection heat transfer and dynamic P-C-T relations.
The performance of a metal hydride heat transformer working with the ZrCrFel.fiaNi5 pair is predicted based on the heat transfer and reaction kinetics of the coupled reactors. The effects of important parameters, viz. the effective thermal conductivity and the thickness of the beds, the overall heat transfer coefficient, and the operating temperatures, on system performance are studied. The important system performance factors are the specific alloy output, the COP, and the second-law efficiency. Results show that for a given bed thickness and overall heat transfer coefficient, there exists an optimum effective thermal conductivity beyond which its effect on the alloy output is negligible. This optimum thermal conductivity increases as the bed thickness increases. The thickness of the beds significantly influences the system performance by reducing the specific alloy output. Thus, it is advantageous to employ thin beds so that a high alloy output can be obtained, which reduces the alloy inventory and hence the cost of the system. Apart from this, it is easier and more economical to optimize the beds in terms of the effective thermal conductivities when the beds are thin. The effect of the
Nomenclature a,b = van’t Hoff constants C = specific heat, J k g K COP = coefficient of performance E = activation energy, J/mol fhys = hysteresis factor f s = plateau slope factor Hf= heat of formation, J/mol of HP K = thermal conductivity, W/mK Mmal= molecular weight of alloy, kg/kmol of alloy M = mass of alloy, kg M m = Ratio of mass of alloy A to mass of alloy B n = number of moles of Hz gas Nmol = number of metal atoms per mole of alloy P = pressure,atm Q = energy transferred, J qo = specific alloy output, W k g of alloy r = distance from filter tube center line, m r, = Outer radius of filter tube, m r, = inner radius of hydride tube, m R, = universal gas constant, 8.314 J1mol.K T = temperature, K
on specific alloy output.
the availability of the output increases; however, a t the same time the heat output (&h) decreases and the heat input increases (Qm). The effect of Th on Qh and Qm is more predominant than its effect on the output availdecreases as Th increases. ability, hence, Figures 12 and 13 show the effect of heat rejection temperature Tl on the specific alloy output, the COP, and the second-law efficiency, respectively, a t different heat source temperatures. The heat rejection temperature affects the heat and mass transfer during process 3 and the heat transfer during process 4. Thus, when TI increases, the driving force for mass transfer during process 3 decreases. As a result, the cycle time increases and the specific output decreases as TIincreases. The COP increases slightly as TI increases due to the small variation in the sensible heat load at reactor A during process 4. For the same heat source and output temperatures, as the heat sink temperature Ti increases, the entropy generation decreases. Hence, the second-law efficiency increases as the heat sink temperature increases.
Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995 2313
U = overall heat transfer coefficient, W/m2*K WR = ratio of mass of heat exchanger to mass of alloy X = hydrogen concentration, atoms of Hdatoms of alloy Greek Letters p = reaction rate, mol of H&g of hydrides 6 = void fraction of the bed VE = second law efficiency 0 = time, s O,t = cycle time, s Q = density, kg/m3 B = reaction rate constant, s1 Subscripts a = absorption A = high-temperature hydride, reactor B = low-temperature hydride, reactor b = bed d = desorption e = effective eq = equilibrium F = final f = heat transfer fluid g = gas h = high temperature I = initial 1 = low temperature m = intermediate, metal r = reference T = total
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IE940496R Abstract published in Advance ACS Abstracts, J u n e 15, 1995. @
