The Storage of Hydrogen as Metal Hydrides Daniel L. Cummings and Gary J. Powers* Department of Chemical Engineering, Massacbusetts lnstitute of Technology, Cambridge, Massachusetts 02139
Metal hydrides offer a reversible, chemical means -for storing hydrogen and could be used as mobile and stationary fuel sources. The properties and uses of metal hydrides are reviewed. A magnesium hydride bed, used in a hydrogen-powered automobile, is modelled to predict its dynamic response during charging and discharging cycles.
Introduction The recent surge of interest in hydrogen as a generalpurpose fuel has sparked much research into the various means of storing hydrogen. One possible storage system involves the use of reversible reactions between certain metals and hydrogen to form metal hydrides. Such hydrides offer a chemical means for storing hydrogen as a bed of solids a t ambient temperatures and pressures. The purpose of this paper is to model the dynamic behavior of hydride storage beds during charging and discharging of hydrogen. The objective is to determine the bed volume and heat transfer requirements in various dynamic demand situations. An example will be presented involving an automobile equipped with a hydride bed as a fuel supPly. Hydrogen as Fuel A strong case has been made in the last 5 years for an energy regime in which hydrogen would be used as a multipurpose fuel. The use of hydrogen in such an energy economy has several encouraging features. (1) The use of hydrogen avoids dependence on dwindling fossil fuel reserves. (2) Hydrogen is very clean burning. The combustion product is primarily water from which hydrogen is made and hence has the potential for being completely recycled. (3) Hydrogen can be transported more cheaply than electrical energy can be transmitted and unlike electricity, the storage of hydrogen is much more trouble-free. Gregory (1973) has emphasized that hydrogen is not a new energy source, but rather a very efficient means of transporting energy. Serious scientific interest in all aspects of hydrogen fuel was evidenced by the large number of papers devoted to the topic at the 1972 Intersociety Energy Conversion Engineering Conference in San Diego, Calif. Two of the more widely suggested uses for hydrogen are in the areas of (1) energy storage (The use of hydrogen as an energy storage medium for peak-shaving a t electric power plants is currently under investigation a t Brookhaven National Laboratory.) and (2) transportation. The fact that hydrogen fuel can be used in internal combustion engines is now well established. Murray and Schoeppel (1970; 1971a,b) have shown that the efficiency, torque, and power of a hydrogen-fueled internal combustion engine (with minor modification) is comparable to that of a gasoline-fueled engine. In August of 1972, the Urban Vehicle Design Contest was held a t the General Motors Proving Grounds near Milford, Mich. The entry from UCLA, powered by a hydrogen-fueled, six-cylinder Chevrolet engine, took the overall prize in the internal combustion engine class (Fiertag, 1972). Lastly, researchers at Brookhaven National Laboratory found no difficulty in running an unmodified Wankel engine on either hydrogen or gasoline (Wiswall, 1972). 182
Ind. Eng. Chem., Process Des. Develop., Vol. 13,No. 2, 1974
The realization of any scheme for the use of hydrogen hinges on the development of an economical, flexible system for storing hydrogen. There are currently three basic methods for storing hydrogen: (1) compressed gas, (2) liquid, and (3) chemically bound (hydrides). In evaluating alternative techniques for storing hydrogen, the following factors must be considered: (1) cost, (2) weight, (3) volume, (4) ease of operation, (5) maintenance, (6) control, and (7) safety. In the following sections the application of the three main storage techniques to mobile and stationary uses will be evaluated. Hydrogen Storage in an Automobile Table I contains a comparison of the three hydrogen storage methods for use in an automobile. The comparison is based on the energy equivalent to 120 lb of gasoline (about 20 gal of gasoline or 46 lb of hydrogen). The compressed gas scheme, while easy to implement, is extremely heavy and bulky. For example, a storage vessel containing 46 lb of hydrogen at 2000 psia would weigh 2250 lb and occupy 66 ft3. Hydrogen storage as a liquid presents an improvement in weight and volume over the compressed gas idea; however, the difficulty with this concept is the energy leak into the storage vessel. Hydrogen is a liquid at 1-atm pressure a t -253". Very efficient insulated dewars are available but the loss rate is still significant. Hydrogen losses on the order of 5%/day are common with state-ofthe-art cryogenic low volume storage systems. This heat leak causes hydrogen to vaporize from the vessel and hence creates potential safety problems. Furthermore, it does not appear economically feasible in an automobile to recover the hydrogen boil-off in a refrigeration cycle (Jones, 1971). Consequently the hydrogen loss caused by the heat leak would drastically reduce the overall efficiency of an automobile used on an intermittent basis. For example, the average automobile in the United States is in use less than 10% of the time. For vehicles such as trucks, buses, and planes that are in continuous use, the heat leak would not be as serious a problem. The concept of storing and supplying hydrogen by means of a reversible chemical reaction has been of current interest. One useful reaction is the formation of metal hydrides. In its simplest form, this reaction involves the formation of hydrogen-metal bonds by the reaction of the elemental metal with hydrogen gas. The reverse reaction can be accomplished by thermal decomposition of the hydride to yield the metal and pure hydrogen. Thus metal hydrides offer a chemical means for storing hydrogen at high densities without high pressures or low temperatures. For example, the storage of hydrogen as titanium hydride achieves a storage density 1.2 times that possible for liquid storage. The density of liquid hydrogen is very low
'C
Table I. Comparison of Hydrogen Storage Systems"
Storage system Gas at 2000 psi Liquid a t 20°K Magnesium hydride (40% voids) a
Weight of container and fuel, lb
Volume, f t 3
2250 353
66 . O 10.2
692
10.8
Basis: energy equivalent to 120 lb of gasoline (Hoffman,
et al., 1969).
I
\\
-c d 31
however (0.07 g/cm3). Table I1 illustrates a variety of metal hydrides. Of the metal hydrides, only those with a high weight per cent hydrogen would be suitable for an automobile in order that space, weight, and operating range are not seriously affected. Referring to Table 11, it appears that the alkaline hydrides are the most likely candidates. Sodium, magnesium, potassium, and calcium are inexpensive and their hydrides contain 3-8% hydrogen. Figure 1 shows that of the alkaline hydrides MgH2 is preferred because of its relatively high dissociation pressure. A magnesium hydride bed that could contain 46 lb of hydrogen would weigh 700 lb and occupy 11ft3. Stationary Storage In discussing the large-scale, stationary storage of hydrogen, the compressed gas scheme is again unwieldy due to the large number of containers needed to store any significant quantity of gas. The storage of hydrogen liquid can be an attractive alternative when executed on a large scale. A stationary storage facility could combat the heat leak problem by using large, spherical vessels, thereby gaining the advantage of a high volume to surface ratio. Also, at a large facility, it becomes feasible to recover the hydrogen boil-off in a cryogenic refrigeration system. A stationary facility for storing hydrogen as a hydride is not as restricted in weight and volume as a hydride bed in a car. Therefore, the limiting factor in the selection of a hydride will probably be cost. Depending upon the application, vanadium hydride might be useful because it dissociates rapidly at temperatures down to 0"; however, the metal is valued at $4.00/lb. Magnesium metal is relatively inexpensive ($0.38/lb) but the hydride has a moderately high operating temperature (250-300') and a high heat of formation (-15,900 Btu/lb of H2) which may require an auxiliary heat source. Iron-titanium alloy is moderately expensive ($1.92/lb) but the hydride dissociates well at temperatures of 50-100" with a low heat of formation (-4900 Btu/lb of H2). In designing a stationary hydride facility these three factors (cost, storage density, and heat requirements) must. be considered in light of the particular application.
001
1000 I T
I'K)
Figure 1 . Equilibrium dissociationpressure The use of hydride also offers potential safety advantages over liquid storage. A ruptured tank of liquid hydrogen presents an immediate and serious hazard. A ruptured hydride vessel would be much less of a problem. The dissociation pressure of most metal hydrides (Figure 1) is low a t room temperatures so that little hydrogen is present in the bed. Also, kinetics at these temperatures are slow so that no new hydrogen would be released (excepting VH2). If the vessel ruptured while in use, removal of the heat source driving the thermal decomposition should cause the reaction to quench itself. In addition, most hydrides do not react violently with air or water. After considering the various methods for storing hydrogen, it appears that a system using metal hydride could be applied to both mobile and stationary needs. To gain a better understanding of the materials in question the properties of one particular hydride, magnesium hydride, are considered in detail. This discussion of properties will be followed by the development of a mathematical model for hydride beds. Chemical a n d Physical Properties of Magnesium Hydride The overall reaction under consideration is Mg
+ HL = MgH,+ 15,900Btu/lb of
H,
(1)
MgH2 is in that class of hydrides known as alkaline because the mechanism of formation is thought to involve the bonding of 2H- with Mg2+ (Gibb, 1962; Mackay, 1966; Mueller, 1968). The hydride product is a light gray powder. Ellinger and coworkers (1955) have determined the lattice structure to be of the tetragonal rutile type. When formed from pure hydrogen and metal, the hydride is stable in air and decomposes slowly in water. Physical
Table 11. Properties of Certajn Metal Hydrides
Initial and final compositions Li c) LiH Mg * MgH2 Ca CaH2 NaH Na MgsNiHo,,c) Mg2NiH4.2 K o K H VHc, 9: * VH2.o FeTiHo.,c) FeTiHl,o c-) f--)
AH*,Btu/lb of Hz Available wt % H?
- 38,700 - 15,900 - 37,300 -21,500 - 13,800 - 24,300 -8,600 -4,900
12.7 7.7 4.8 4.2 3.5 2.5 2 .o 0.9
Equilibrium temperature at 1 atm of H,, "C
Cost of metal (per lb)
800 290 920 525 250 715 12 0
$8.18 $0.38 $2.20 (CaH2) $0.30 $1 .oo $0.35 $4 .OO $1.92
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 2, 1974
183
T a b l e 111. Physical Properties of Magnesium and Magnesium Hydridea
and coworkers (1961) have reported that, for the reaction of pure magnesium and hydrogen, without the aid of a catalyst or grinding, the maximum yield is 80%. The authors comment that this limit may be caused by the buildup of a hydride shell around the unreacted magnesium core and this shell hinders the diffusion of hydrogen into the core. Indeed, the increase in specific volume from the metal to the hydride supports this theory. Because of this limitation with pure reactants, the reactions studied by b i l l y and Wiswall (1967, 1968) using nickel and copper catalysts are of particular interest. Working a t Brookhaven National Laboratory, b i l l y and Wiswall investigated the following reactions MgJi 2H, = Mg,NiH, 13,800 Btu/lb of H? (3) 2Mg,Cu + 3H, = 3MgHI + MgCu? + 15.500Btu/lbof H? ( 4 )
MgHz Molecular weight Density (70°F), lb/ft3 Melting point, O F Heat capacity (70°F), Btu/lb OF Thermal conductivity (70°F), Btu/hr f t O F Thermal diffusivity (70°F), ft2/hr Powder density (40% voids), lb/ft8 Powder conductivity, Btu/hr f t O F 5
24.312 109 .o 1202
26.336 90.4
0.242
0.319
Decomposes
99 .o
0.8
3.762
0.028
65.4
54.2
6 .O
0.5
+
Taken from the "JANAF Thermochemical Tables,"
MgzNi and MgzCu are the results of alloying magnesium with nickel and copper. By varying the amount of catalytic nickel and copper, the authors discovered that the presence of 510% nickel or copper greatly improved the kinetics of reaction 1. Furthermore, they obtained yields of 9&95% hydride. Table IV contains a summary of the important experimental variables from several studies of MgH2 with various catalysts. Without further experimental work it is not possible to determine the optimum use of catalysts, but it appears that the addition of 5% catalyst would make reaction 1 more suitable for an automobile by improving kinetics and yield. Here, only reaction 1 will be referred to with the understanding that there is a catalyst present.
(1965) and Ellinger, etal. (1955).
property data presented in Table I11 show sharp differences in properties such as thermal conductivity from the metallic magnesium to the salt-like hydride. This is typical of most hydrides. Conductivity of a Hydride Powder Masamune and Smith (1963) have presented a correlation of theoretical and experimental work on the conductivity of beds of spherical particles. The effective conductivity under 1-atm pressure of a bed can be expressed as he =
(1 - PtXl
+
- 6)
-I-( & / k g ) [(l- & ) / k s ]
+ (1 - Pt)6h,
+
(2)
Thermodynamics Thermodynamic data from several sources are reported in Table V. In the automotive example presented later, the equation for equilibrium dissociation pressure developed by Reilly and Wiswall for the reaction with nickel catalyst will be used as the governing relationship
Briefly, t is the void fraction, p and 6 are dimensionless void fractions, and C$ is a dimensionless path length for the interstitial gas. Masamune and Smith report data for the conductivity of glass beads with a diameter of 4.7 x 10-2 cm and a solid conductivity of 0.6 Btu/hr ft OF. They also report data for sintered Cu-Sn alloy particles with a diameter of 4.0 x 10-2 cm and a solid conductivity of 120 Btu/hr ft OF. The size and conductivity of these materials is very similar to that of MgH2 and Mg. One can use the data for glass beads and Cu-Sn particles to evaluate the purely geometrical parameters of eq 2. The result is that the two sets of parameters are nearly identical. Using these geometrical parameters and the appropriate property values for Hz, MgH2, and Mg in eq 2 gives the following values for the effective thermal conductivity: k,, MgH2 = 0.5 Btu/hr f t "F; k,, Mg = 6.0 Btu/hr ft O F .
log ( P )= -4045/T
+ 7.224
where P i s in atmospheres and T i n degrees Kelvin. Kinetics No data for the intrinsic rate of reaction of magnesium and hydrogen are currently available. Data on the pellet composition us. time for the dissociation of MgzNiHr pellets into a vacuum are given in Figure 2. Unfortunately such data intermingle diffusion effects with true kinetic effects. The situation is further complicated by the fact that the hydride becomes highly activated after several cycles of charging and discharging hydrogen. With these factors in mind, a rate constant was derived under the following assumptions: (1) the hydride is activated; (2) the
Catalysts Theoretically the heterogeneous reaction of hydrogen and magnesium can go to completion. However, Dymova Table IV. Synthesis of Magnesium Hydride
Pressure, Ref
Reactants
Henle and Smutny
Mg; 1,3-butadiene; T H F ; H? Mg; Zn; HCl Mg; HP; CHa CzHz; CzHs Mg, allyl iodide; propargyl iodide Mg; Hz Mg; H? Mg; Hz Mg; Hz Mg; Hz Mg; Hz
(1972) Givelel (1968) Lyon (1969)
Faust, et al. (1960) Dymova (1961) Dymova (1961) Dymova (1961) Wiberg (1953) Reilly (1967) Reilly (1968) I a4
Time, hr
Temp, "C
atm
90 20
68 1
6
276-444
1-40
2-6
175 400-450 400-450 380-450 570
5 200-300 200-300 100-200 200 24 24
350 350
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 2, 1974
Wt % catalyst
Yield (% Mg reacted)
Poor 63 %
I? 6 2 6
0.7%
12
1 . 0 % I?
MgI? 6 . 2 % Ni 9.5% c u
99.6% 80 % 80 % 98% 77.5% 98 % 94 %
$4-7, STARTER
i
2 0 1 OC
I
I
HOT E X H A U S T
GASES POWER
EXHAUST GASES
I'
/
Figure 3. Flowsheet of a hydride storage system.
240 'C
in
w
v
HYDRIDE
A
01
0
1
2
3
4
5
6
TIME (min)
Figure 2. Decomposition of MgzNiHr (billy, 1973). decomposition is first order; (3) the equilibrium dissociation pressure must be greater than the ambient hydrogen pressure for the reaction to proceed and the rate is linearly dependent on this pressure difference. The rate expression can be written as
where H = gram moles of Hz as hydride; k(T) = rate constant, hr-1; PD = equilibrium dissociation pressure, atm; PA = ambient pressure, atm; Po = constant = 1 atm. Equation 6 can be solved for k ( T ) with the data from Figure 2. Writing the rate constant as a n Arrhenius expression Then A = 1.9 x 10'0 hr-1 and EA = 18,500 cal/g mol of
Hz. Table V. Thermodynamic Properties of Magnesium Hydride (a) JANAF Thermochemical Tables (1965) AHfo298.16 = - 1 8 . 2 f 2 . 2 kcal/mol So298.36 = 7 . 4 3 f 0.20 cal/mol "K (b) Ellinger, et al. (1955) A H I "= ~ ~ - 1~6 .~0 kcal/mol Activation energy = 53 kcal/mol (c) Stampfer, et al. (1960)
T,
OK 800
AGO,
cal/mol
298
AH", cal/mol
ACP", cal/mol "K
-31.47 -18,350 5.8 + 0.88 =t390 + 8.8 1580 -32.21 -17,830 0.7 f 80 f0.48 f 80 f1.4 -8170 -32.3 -17,790 0 R In f ~ ?= - l 7 , 7 8 5 / T ( " K ) + 3 2 . 2 8
8000 &
600
AS",
cal/mol "K
120
(d) Reilly and Wiswall (1967), copper catalyst A G = ~- 8 . 7~ f 1~ . 0 kcal/mol ~ ~ A H i 0 2 9 8 = - 1 8 . 7 & 1 . 0 kcal/mol A S ~ " Z M= - 3 3 . 4 f 0 . 7 kcal/mol "K Log (P)= -4094/T("K) 7.299 (e) Rielly and Wiswall (1968), nickel catalyst AGr"2os = - 8 . 7 f 1 . 0 kcal/mol AH~"zw = - 1 8 . 5 & 1. O kcal/mol ASt"298 = -33 . O f 0.7 kcal/mol 7.224 Log (P)= -4045/T("K)
,
EXHAUST
GASES
Figure 4. Plate-plate heat exchanger for storing hydride.
The Hydride Storage System. General Description Figure 3 shows a generalized flow sheet for the use of a hydride storage bed. The bed itself is an appropriately designed heat exchanger with the hydride on one side and, during the discharge cycle, a heat source on the other side. One possible source of heat is the exhaust gas from the power plant. The power plant is an electric generator or an automobile engine. It may be necessary to provide additional heat for hydrides with a large heat of formation. This can be done by burning a small amount of hydrogen in an auxiliary firing chamber and passing the exhaust gas directly through the heat exchanger (ie., without gaining any mechanical work). There must also be an initial heat source, a starter, to bring the hydride up to reaction temperature. This starter could be a propane tank or a hydride such as VHz that dissociates a t ambient temperatures. During the recharging cycle, a coolant must be passed through the exchanger to remove the heat of reaction.
Mathematical Description of the Hydride Bed The mathematical description of the hydride bed is quite complex. The physical processes occuring involve unsteady state conduction with convective boundary conditions, reaction in the bed, and a conductivity which depends on location. (The conductivity of the metal is much greater than that of the hydride.) The general conduction equation governing the situation is
+
(7)
+
In order to reduce this problem to manageable size, the following assumptions were made: (1)the hydride bed is a plate-plate heat exchanger, as shown in Figure 4; (2) the temperature distribution in any t-plane parallel with the Ind. Eng. Cham., Process Des. Develop., Vol. 13,No. 2,1974
185
HYDRIDE
REACTION
7
Table VI. Determination of the Rate-Controlline Steo =
Eq 8 q Tg
ZONE
= =
T,,,
REACTED HYDRIDE [ M E TAL)
100-250 Btu/hr f t O F ‘ 0.125 in = 0.0104 f t ksteei 20 Btu/hr f t O F Axxg 0 . 4 in. = 0.033 f t hrg = 6 . 0 Btu/hr f t OF Eq9 Q = (0.5) ( P ) ( V ) ( A H ~ ~ ~=) (4 X . 2H 2 B~t)u f o r P = 60 lb/ft3 V = 0.0001157 f t 3 AH,,, = 15,900 Btujlb of Hz X H ~ = - 0.0765 lb of Hz/lb of MgHz Eq 10 t Q / q = 0.01-0.016 hr h
= = = =
Axsteel
=EXHAUST
GASES
Figure 5. Cross section of a single hydride plate.
wall is uniform (Le., aT/& = 0); (3) there is a center-line symmetry in each plate. Reference to a “plate” of hydride does not mean a block of solid material but rather a rectangular volume of powdered or pelleted material. The cross section through a plate of hydride is shown schematically in Figure 5. There are three regions. The first region is pure metal. All of the hydrogen has been evolved. Heat is being conducted through this region from the exchanger wall to the reaction zone. The center region is pure hydride. Temperatures in this region fall off rapidly from the reaction temperature due to the poor conductivity of the hydride. A simple conduction equation can be written for both of these regions. The reaction zone presents the greatest mathematical difficulty. Reaction is occuring at a rate which is temperature dependent. The material in the region is a varying mixture of metal and hydride with correspondingly varying property values. Also, the boundaries of the region are not immediately well defined. It could be that the reaction zone is a broad band or only a well-defined interface. This region requires a detailed analysis.
The Rate-Controlling Step The metal hydride system presents a classical problem in the design of reactors in which simultaneous heat and mass transfer with reaction occur. However, this particular case is clouded by the change in property values that occurs with reaction. The possible rate-controlling steps are (1) transfer of heat to the reaction zone, (2) diffusion of hydrogen out of the hydride particle, and (3) intrinsic rate of reaction. Consider a plate of MgHz as shown in Figure 5. The system is defined by the following assumptions. (1) The intraplate distance is 1 in. (2) The exchanger wall is lh-in. stainless steel. (3) The metal region is 0.4 in. of pure metal. (4) The hydride region is 0.4 in. of pure hydride. (5) The reaction zone is 0.2 in. of 50% metal and 50% hydride. No assumption is made as to the distribution of the materials in this region. (6) The reaction temperature is 592°F which corresponds to an equilibrium dissociation pressure of 2 atm. ( 7 ) The removal of hydrogen from the bed is such that the ambient pressure is maintained at 1 atm. Recalling that diffusion effects are bound up with existing rate data, only heat transfer and reaction rate need be compared. If the reaction rate is slower than the heat transfer, the reaction zone will be a broad band. If the reaction rate is faster than heat transfer, the reaction zone will be a well-defined interface moving through the bed (i.e.,a square wave). The steady-state rate of heat transfer can be expressed as shown in eq 8. 186
Ind. Eng.
Chern., Process Des. Develop., Vol. 13, No. 2, 1974
(T, - T,,,)/RT = 262 t o 419 Btujhr for 1200’Fo 592’F
In WPo) - - 0.01-0.016 hr -k(T) [(PD - p A ) / p o l = 2atm = 1 atm = 1 atm
-
Eq11t PD PA Pa
Heat transfer time, hr
Reaction rate time, hr
h = 100
(H/Ho) = 0 . 1 (H/Ho)
h = 250
= 0.01
0.016 0.01 0.001 0.002 Melting point of magnesium. b heff as discussed in the automotive example.
4 (Btu/hr) = ( T g- Trm)lR,
(8) where Tg = gas temperature; T,,, = reaction temperature; and RT = thermal resistance = (l/hA) + (Axsteel/ ksteed)
-k (AXMg/kMgA).
The heat needed to decompose the remaining hydride in the reaction zone is
Q(Btu) = ( O . ~ ) ( P () V )(AH,,,Xx H , ) (9) where V = volume of reaction zone and xH2 = weight fraction of hydrogen in the hydride. The time needed to transfer enough heat to decompose all of the remaining hydride in the reaction zone can be written as t
= Ql4
(10)
With regard to kinetics, the time needed to reach a given degree of conversion is
t
= In
(p y ) (11)
(H/H,)/-h(T) -
Equations 8-11 were evaluated over a range of values and the results are presented in Table VI. The rate of reaction is at least one order of magnitude faster than the rate of heat transfer. Hence a sharp reaction front separates the reacted and unreacted regions within the hydride bed. Based on the observations that (1)there is a large change in property values between the metal and its hydride, and (2) that the reaction zone is a well-defined front, the hydride bed can be modeled as a heat conduction problem with phase change. That is, the physical situation in the hydride is analogous to that in which heat is applied to one side of a block of ice. In both cases there is a change in physical properties which follows the movement of a narrow front passing through the system.
The Murray-Landis Method The applicable differential equations for one-dimensiona1 conduction with constant thermal properties (as found in references such as Carslaw and Jaeger, 1959) are
c
a
I
I
I
I
in the hydride region. At the reaction front these equations are coupled by the heat of reaction.
where daldt is the rate of travel of the interface. (In this case AH,,, is taken as negative for the dissociation reaction.) There are no analytical solutions to these equations for a finite slab and the problem must be solved by numerical methods. One such method is that developed by Murray and Landis (1959). A brief outline of the method is given below. The method utilizes a fixed space network as illustrated in Figure 6 for the specific case of a ten-space network. A t some intermediate time the reaction front a will be in the qth segment (Le., at a location qAx 62, where -Ax/2 < 6x < Ax/2). For all points except 0, q, and R the conventional numerical approximations still apply
b
Tq-i
+
dT, dt
-I
aT,., - 2T, (Ax )?
+ T,,,
Tq.1
The reaction front will be traced as it travels through the qth segment and the temperature of this node will be continuously adjusted as the reaction front moves. In the segment q (that containing the reaction front) a discontinuity in the temperature curve occurs at x = a. Two temperatures are calculated for the segment q, one by interpolation from temperatures in the metal region and the other by interpolation from temperatures in the hydride region, In each case the reaction front location, a = qAx 6 x , and the reaction temperature are used to perform the interpolation
+
2 Trx1'(2 bx/Ax)(l
+
2 Tq'
= T r x n ( 2- S x / A x ) ( l - 6xIAx)
+ Gx/Ax) (16 1
Figure 6. Description of variables in Murray-Landis method (Murray, Landis, 1959): (a) situation at time t, q = 3; (b) detail around i = q.
For the starting points q = 0, 1 three adjacent internal network points are not available and the equations must be reduced to alternate slope and curvature approximations. To avoid this starting error, a must be at least 2Ax. A t the wall (i = 0) there is a convective boundary condition
Tlntl=
-
Although 6 x may be positive or negative, both eq 16 and 17 are mathematically defined over the full range of 6x. T, is represented by Tqlwhen 0 < 6x and by Tq2when 0
> 6x.
&
{+g
+ 2T,+1n+
Again, for the terminating points, q = 9, 10 three adjacent internal network points are not available. However, the temperature profile at i = 10 is specified by symmetry
dT,/dx = 0
The temperatures Tqland Tq2are substituted into the uniform difference equations (15) to permit calculation of Tq-land Tq+l. In order to obtain the expression for reaction front motion, the slopes at the reaction front are developed from the first two terms of the difference form of the Taylor series expansion
therefore, Ti+l = Ti-1.Thus eq 15-20 can be applied with an appropriate change of subscripts. For the situation where there is conduction through both the metal layer and the hydride layer with no reaction (i.e., during warm-up), eq 15 applies at all nodes around the interface. At the interface, conduction can be described by requiring that the flux across the interface be continuous. (For a more detailed development, see Carnahan (1969) .) The result is
(19)
Use of Tqland Tq2 in eq 18 and 19 and further substitution into eq 14 results in the reaction front velocity
Therefore, eq 15-22 describe the hydride bed under all conditions where a > 2Ax. For cases where a < 2Ax, the Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 2 , 1974
187
Table VII. Driving Cycle Used to Test the Dynamic Response of the Hydride Beda
Acceleration, mph/sec +2.9 + 2 .o
+o .o -1.1 $2.9 +1.3 + 1 .o + 2 .o
+o .o -2.3 -2.25 a
CumulaSpeed range, Grade, Duration, tive mph % sec time, sec 0-20 20-30 30-30 30-0 0-20 20-30 30-40 40-60 60-60 60-12 12-0
-0.2 -0.1 -2.6
+o .o $5.4 + 1 .o +2.2 +3.6 +O .6 -2.2 -2.1
7 5 11 27.5 7 8 10 10 18 21 5.5
7 12 23 50.5 57.5 65.5 75.5 85.5 103.5 124.5 130
Hass, et al. (1966).
boundary conditions are discussed by Murray and Landis (1959) and will not be taken up here for the reason that if u < 2Ax, the starting error is very significant. To describe the two-dimensional heat transfer situation in the hydride bed, the one-dimensional problem in the x direction will be solved a t a number of locations down the length of the bed. It should be noted that numerical methods are available to solve the problem of a reaction zone that is a broad band (Gloor, 1963). The phase-change analogy is that of a polymer which melts over a range of temperatures. However, one is still faced with the problem of assigning some average property values to a mix of materials as dissimilar as a metal and its hydride. An Automotive Example In order to clarify the above discussion, an example will be presented involving an automobile equipped with a hydride bed. The hydride will be MgH2. The design of the system follows that of the general flowsheet in Figure 3. To evaluate the performance of the hydride storage system, it is necessary to couple three effects: (1) the driving cycle of the vehicle; (2) the power demand on the engine resulting from the driving cycle; and (3) the hydride bed response to the fuel demands of the engine. Acceptable performance of the hydride bed is based on its ability to meet all fuel demands without affecting engine performance.
A Synthetic Driving Cycle Table VI1 contains the driving cycle to be used in this example. It is a modified version of the LA-4 Synthetic Driving Cycle developed by Hass, et al. (1966). The Hydrogen Engine This example considers an Otto cycle engine as the power plant. Tests on hydrogen-fueled internal combustion engines by King and coworkers (1958) have shown that maximum horsepower is developed when the fuel-air ratio is stoichiometric, 2:5. However, in this design, the engine will burn a 2:lO fuel-air mixture for three reasons: (1) combustion knock due to high flame velocity is greatly reduced; (2) NO, emissions (the only pollutant) are greatly reduced; and (3) the higher exhaust flow rate will improve heat transfer in the hydride bed. Overall efficiency remains the same for any Otto cycle, about 25%. The overall stoichiometry for an engine using a 2:l mixture is 2H) 188
+ 2.102 + i.9N2 = 2H2O + 1.102 + 7.9N2
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 2, 1974
On a pound basis 72.5 lb of exhaust are generated per pound of Hz burned. An energy balance around the engine (after that of Judge, 1965) shows that the heat of combustion is spent as (1) 25% power, (2) 20% cooling, (3) 5% engine friction, fan, generator, and (4) 50% exhaust heat. Using this energy balance and the lower heating value of hydrogen (51,000 Btu/lb), the exhaust gas temperature can be calculated
where C, = 0.29 Btu/lb O F is a weighted average heat capacity. Therefore, Tex= 1282°F. The power requirements for the engine are based on findings of Smith, et al. (1969), in developing the "Pittsburgh Driving Cycle." The power requirement is a composite of the power needed to maintain travel a t a constant speed by overcoming grade (gravity), rolling resistance, air resistance, and that needed to accelerate from one speed to another. The rear wheel horsepower requirements for a vehicle moving with acceleration a mph/sec and speed u mph on a G% grade can be closely approximated by the expression
P
= W(c,a
+ c,G + k,)V + k,Su3
(23)
where c1 = 12.15 x 10-5 and c2 = 2.67 x 10-5. S, W, kl, and kz are directly related to the particular vehicle, and kl and 122 were calculated from the data S = 25.0 f t 2 , W = 4212 lb (two passengers), kl = 5.025 x and k~ = 0.432 x 10-5. Due to the efficiency of the engine, 10 hp at the rear wheels implies that the energy equivalent of 40 h p was burned in the engine. Power in the engine can be converted to a fuel demand by Rhp) X
2545(Btu/hr) hp
lb of H2 51,000Btu
- lb of H2 hr
It is assumed that the minimum fuel rate is that for idling and thus if a, G, and u are such that the fuel rate is less than idling, the rate will be that of idling.
The Auxiliary Burner In cases of poor hydride bed response, it may be necessary to bum some of the hydrogen in an auxiliary burner.
Both the fuel rate and fuel-air ratio can be varied. It is assumed that the exit gas will be fed directly into the heat exchanger and hence the temperature cannot exceed 1200°F. An adiabatic flame temperature calculation shows that 1200°F can be reached with a 1:10.7 fuel-air ratio, generating 156 lb of exhaust/lb of Hz burned.
The Hydride Bed The hydride bed is a plate-plate heat exchanger with hydride on one side and the exhaust gases on the other side. Clearly, there is an advantage to keeping the hydride bed as close as possible to the engine in order to keep the inlet gas temperature near its maximum allowable value. If the bed is placed in the trunk and connected to the engine by a bare metal tail pipe, much heat will be lost. Some means of regulating this gas temperature must also be provided. The hydride bed must hold 46 lb of hydrogen at 90% conversion with 5% catalyst. In addition, more hydrogen must be carried if an auxiliary burner is to be used. The dimensions for the bed are given in Table VIII. If the bed can resDond to all fuel demands for an indefinite number of dr&ing cycles without the bed pressure
Table VIII. Bed Weight and Dimensions ~ ~ _ _ _ _ _ _
~~
~~
Feed rate to Approxiauxiliary mate burner, H2capacity, lb of H n / h lb 0 .o 0.1
0.5 1 .o 2 .o
46 . O 46.4 48.0 50 . O 54.0
~ _ _ _ _ _
Magnesium," Catalyst (5% Total Ib by weight) weight, lb 617 622 644 670 724
31 31 32 33 36
Volume, ft3
Height,
Depth,
ft
ft
Length, ft
11.6
3 3 3 3 3
1.25 1.25 1.25 1.25 1.25
3.1 3.2 3.3 3.4 3.6
694 700 724 753 814
11.7
12 .o 12.6 13.6
Based on 90% conversion. * Combined thickness of the hydride plates. Either 14 11-in. plates with two 0.5-in. plates on the ends or 29 0.5-in. plates with two 0.25-in. plates on the ends. 0
Table IX. Heat Exchanger Designsa
Transfer area/volwne, ft2/ft3 Transfer area (auxiliary rate = 0 . O lb of Hz/hr), f t 2 Mass flow rate,* lb/ft2 hr Hydraulic radius [ T H ] , f t NR~ h,c Btu/hr f t 2 O F Transfer area (auxiliary rate = 0.1 lb of Hz/hr), f t 2 Transfer area (auxiliary rate = 0.5 lb of H2/hr), f t 2 Transfer area (auxiliary rate = 1.0 Ib of Hz/hr), f t 2 Transfer area (auxiliary rate = 2.0 lb of H,/hr), f t 2
10-30 (15 0.25-in.
10-30 (30 0.25-in.
10-37 (15 0.1-in.
10-37 (30 0.1-in.
plates)
plates)
plates)
plates)
561
1332.45
1332.45
3260 279 0.00615 22 19
1549 1392 0.002643 48 44
3098 696 0.002643 24 44
561 1630 557 0.00615 44.5 19 1683
3366
1598
3197
1735
3471
1649
3298
1788
3576
1699
3398
1893
3786
1799
3584
a The design number (e.g., 10-30) refers to the figure in Kays and London (1964) from which the data were taken. The length and height of the plates are the same as for the corresponding cases in Table VIII. b At 60 mph. c The flow is laminar and therefore the heat transfer coefficient can be calculated from the standard correlation, Nu = 4.12 = ( h 4 r ~/ )k .
Initialize temperatures
Murray-Landis method,
eouations l15)-(201
c.
I
Compute: moles H2, t e m perature. pressure NC
T
Start driving cycle
00431
condition, equation (211
NO
Figure 7. View of the gas side of two exchanger designs (Kays and London, 1964): (a) exchanger design 10-30; ( b ) exchanger design 10-37.
-
xurr ay La nd iIs met had equations (15)-[20)
e
1I
i
Interface conduction. equation (22) equatlo"
compute: moles H2. temperature, pressure
falling below 1 atm, the design will be considered acceptable. Safety considerations related to air leaking into the hydrogen tank indicate that pressures above 1 atm within the tank are desirable. T h e Heat Exchanger Several heat exchanger configurations as reported by Kays and London (1964) were examined and two sample designs are presented in Table IX and Figure 7. The flow rates given are for a speed of 60 mph, 0% grade, and no acceleration. It can be seen that in all cases the flow is laminar and the heat transfer coefficient can be calculated from the standard correlation given in Table IX. The
Re3ect
1 save Figure 8. Simulation flowsheet. heat transfer area in the table is the area of the gas-side plates. The calculations for the rate-controlling step and the simulation program use a heat transfer area equal to the surface area along the hydride plate ( i e . , the area in the y-z plane). Consequently, an effective heat transfer coefficient was calculated from Ind. Eng. Chem.,
Process Des. Develop., Vol.
13, No. 2, 1974
189
I
0
1
1
l
I
'
l
I
1
!
l
\
0 0
I
I 20
I
I 40
I
I
60
I 80
I
I 100
I
I 120
hr f t OF); (h) design-10-37, 15 plates, auxiliary rate-"= 2.0, kMg = 10.0 (Btu/hr ft OF).
TIME ( S E C O N D S )
Figure 9. Dynamic response of the hydride bed: (a) design 10-30, 15 plates, auxiliary rate = 0.0; (b) design 10-37, 15 plates, auxiliary rate = 0.0; (c) design 10-30, 30 plates, auxiliary rate = 0.0; (d) design 10-30, 30 plates, auxiliary rate = 1.0; (e) design 10-37, 30 plates, auxiliary rate = 0.0. -
c
6 t i
1
1
~
I
I
I
1 1
4
0
W
20 TIME
40
80
60
(SECONDS
1
Figure 12. Effects of varying the starting pressure; (g) design 1037, 15 plates, auxiliary rate = 2.0; (i) design 10-37, 15 plates, auxiliary rate = 2.0.
as the product hA must be equal in both cases. For the two designs in Table IX, the effective heat transfer coefficients were 111 Btu/hr ft OF and 244 Btu/hr ft2 "F for designs 10-30 and 10-37, respectively.
Initial Conditions At the start of each simulation run the hydride bed was assumed to be at a uniform temperature of 70°F with the interface a t u = 2Ax. In all cases the exhaust flow rate for the warm-up was equivalent to the flow rate at 60 mph. The bed was considered to be "warmed-up" when the pressure reached 3 atm. Figure 8 presents a flow sheet of the approach used to find acceptable designs. Discussion of Results The discussion of results will center on the performance of the various heat exchanger designs during the driving cycle. The purpose is to highlight those designs which may merit further study. The two other modes of operation, start-up and recharging, will also be considered. Performance An acceptable hydride bed design must maintain a pressure of 1 atm during the driving cycle. Therefore, the performance data for several simulation runs is presented as pressure us. time curves. Figure 9 presents five such curves. Clearly design e is acceptable as it offers continuous operation above 1 atm. 190
Ind. Eng.
Chem., Process Des. Develop., Vol. 13, No. 2, 1974
Designs c and d are adequate for the first cycle but fail to pass a second cycle. Figure 10 shows the effects of varying the auxiliary fuel rate. Figure 11 shows the effects of varying the thermal conductivity of the metal. When compared with Figure 9, these figures show that decreasing the thickness of the hydride plate is more important than increasing the exhaust flow or the metal conductivity. Figure 12 shows the effects of varying the starting pressure. A higher starting pressure provides some additional heat and fuel a t the start of a run but this can be seen as a transient effect that quickly damps out. Temperature profiles in the bed are very steep near the gas inlet. The front of the bed is rapidly depleted of hydrogen. Only the front 5% of the length of the metal-hydride interface is reacting. These effects are outlined in Figure 13. For the conditions described in this simulation, it appears possible to design a hydride storage bed capable of responding to the dynamic demands of an automobile. The thickness of the hydride plate seems to be the most important design variable. Start-up In the cases just presented, the hydride bed was initially 20% depleted. If the driving cycle were continued until the bed was 80% depleted, most of the depleted hydride (ie., metal) would be at the front of the bed. Starting a bed with this initial condition would require energy to warm up the metal in the front of the bed. Only later would the thermal wave reach the hydride. A rough calculation shows that for an 80% depleted bed
w
Gas
c
I
E
Temperotdre
400
0
5
10
15 TIME
I 0 0 5 01
1
OIST4NCE
I
0.5
03
FROM
GAS
INLET
1 0 75
(0.80) X 617(lb of Mg) X 0.25 (1PF) (700°F - 70°F) = 78,OOOBtu = 1.5 lb of
H,
would be required to warm-up the bed. For the initial conditions given during start-up, this would require about 30 min. If this situation is repeated several times, the overall efficiency and usefulness of the bed declines seriously. Recharging The two major factors in refueling a hydride bed are (1) time to refuel, and (2) temperature rise in the bed due to heat of reaction. In developing a mathematical model of the recharging cycle, one again faces the problem of assigning property values to a material that is part hydride and part metal. In this case there is no well-defined interface. Once hydrogen is pumped into the bed, reaction occurs everywhere. We approached the problem by analyzing extreme cases (Le., the case where the properties are those of the metal and the case where the properties are those of the hydride). The one-dimensional conduction equation is -k
30
35
40
Figure 14. Recharging hydride bed.
150 sec.
ax'
25
(It)
Figure 13. Temperature profile at the front end (gas inlet) of hydride bed: design 10-37, 15 plates, auxiliary rate = 2.0, time =
a2T 8'
20 (MINUTE)
laT =cvat
where Q' is a rate of heat generation. This equation was written in finite difference form and Q' was evaluated at each node by applying rate expression 6. The boundary condition a t the exchanger wall is again described by eq 21 with an additional term for Q'. Four cases were investigated using two plate thicknesses (1 and 0.5 in.) with two sets of property values, those of the metal and those of the hydride. The ambient hydrogen pressure was held constant at 25 atm. In all cases the temperature rise is controlled by thermodynamics. If the temperature of a particle corresponds to an equilibrium dissociation pressure greater than the ambient pressure, the reaction (and heat generation) stops. Therefore the highest temperature in the bed was 790"F, corresponding to a dissociation pressure of 25 atm. Figure 14 shows that in only one case can the hydride bed be refueled as quickly as a gasoline tank. However, this one case is the best possible case and can only be revised downwards. The conclusion to be drawn from the figure is that the refueling will probably take more time than would be acceptable in an automobile. (One way around this problem would be to oversize the bed such
that it holds the required 46 lb of hydrogen when 80% charged. This would decrease the refueling time considerably .) Stationary Beds It should be noted that many of the problems associated with using a hydride bed in an automobile do not carry over to stationary uses. Presumably a stationary bed would be operated a t a steady rate of output and the design need not consider the dynamic demands of automobile driving. Presumably a stationary bed would be operated continuously and there would be no loss of efficiency associated with heating depleted hydride to start the bed again. Presumably a stationary bed would be operated in parallel with another bed so that one can recharge while the other continues operating. Thus several of the problems of mobile use are minimized in a stationary facility. Conclusions (1) Metal hydrides offer a reversible, chemical means for storing hydrogen and there may be applications to both mobile and stationary needs. (2) A specific hydride bed design presented here using a plate-plate heat exchanger can be modeled as a phase change heat transfer problem. (3) For mobile sources the hydride bed appears to be technically feasible but problems with start-up, recharging rate, and system safety need to be investigated further. (4) Most of the discharge and recharging problems are due to heat transfer limitations. A better heat exchanger design is in order. One should consider adding fins on the hydride side of the heat exchanger. (5) Many of the problems in using a mobile hydride bed would not affect a stationary facility operating continuously and a t a steady rate of output. Acknowledgment The authors gratefully acknowledge the support of the Massachusetts Institute of Technology Energy Laboratory and Information Processing Center and the technical assistance of H. Wong and E. Suuberg in the early phases of this work. Nomenclature a = acceleration, mph/sec A = Arrhenius constant, hr-1; area, f t 2 ci,cz = constants in eq 23 C, = heat capacity, Btu/lb O F E A = activation energy, cal/g mol G = percentgrade h = heat transfer coefficient, Btu/hr f t 2 OF H = g mol of hydrogen as hydride, g mol AH,,, = heat of reaction, Btu/lb k = thermal conductivity, Btu/hr ft O F Ind. Eng. Chern., Process
Des. Develop., Vol. 13, No. 2, 1974
191
k(T) = rate constant, hr kl,k2 = constants in eq 23 P = pressure, atm; power, h p PA = ambient pressure, atm PD = equilibrium dissociation pressure, atm Po = constant = 1atm q = node containing reaction interface q = rate of heat transfer, Btu/hr Q' = rate of heat generation, Btu/ft3 R = gaslawconstant RT = thermal resistance, hr "F/Btu S = surface area, f t 2 t = time, hr T = temperature u = velocity V = volume, ft3 W = weight, lb xHZ = weight fraction hydrogen in hydride 6x = distance of reaction interface from center of node, q,
ft Greek Letters a = thermal diffusivity, ft2/hr 8,6 = dimensionless void fraction t = voidfraction 4 = dimensionless path length p = density, lb/ft3 u = depth of reaction interface, f t Subscripts e = effectivevalue g = gas i = ithvalue s = solid 1 = metal 2 = hydride Superscript n = time level n
Literature Cited Carnahan, B., Luther, H. A., Wilkes, J. O., "Applied Numerical Methods," Wiley, New York, N. Y., 1969. Carslaw, H. S.,Jaeger, J. C., "Conduction of Heat in Solids," Clarendon Press, Oxford, 1959.
192
Ind. Eng. Chem., Process Des. Develop., Vol. 13,No. 2, 1974
Dymova, T. N., Sterlyadkina, Z. K., Safronov, V. G., Russ. J. Inorg. Chem., 6,392 (1961). Ellinger, F. H., Holley, C. E., Jr., Mclnteer, B. E., Pavone, D., Potter, R. M., Storitzky, E., Zachariasin, W. H.. J. Amer. Chem. SOC., 77, 2647
(1955). Faust, J. P., etal., J. AppI. Chem., 10,187 (1960). Feirtag, M. D., Tech. Rev., 75 (l), 43 (1972). Gibb, T. R. P., in "Progress in Inorganic Chemistry," F. A. Cotton, Ed., Interscience, New York, N. Y., 1962. Givelel, M., C. R. Acad. Sci., Ser. C, 267 (15),881 (1968). Gloor. W. E., S f E (SOC.Plast. Eng.) Trans., 3 (4)270 (1963). Gregory, D. P., Sci. Amer., 228 (I),13 (1973). Hass, G. C., Sweeney, M. P., Pattison, J. N., Society of Automotive Engineers Paper No. 660546 (1966). Henle, W. K., Smutny, E. J., U. S. Patent No. 3,668,416(1972). Hoffman, K. C., et a/., Society of Automotive Engineers Paper No.
690232 (1969). "JANAF Thermochemical Tables," Clearing House for Scientific and Technical Information, Springfield, Va., 1965. Jones, L. W., Science, 174,367 (1971). Judge, A. W., "Motor Manuals Volume I: Automobile Engines." Robert Bentley, Inc., Cambridge, Mass., 1965. Kays, W. M., London, A. L., "Compact Heat Exchangers," McGraw-Hill, New York, N. Y., 1964. 143 (1958). King, R. O., etal., Trans. €IC, 2 (4), Lyon, R. K., U. S. Patent No. 3,476,165(1969). Mackav. K. M.. "Hvdroaen Comoounds of the Metallic Elements." SDon, London, 1966. Masamune, S., Smith, J. M., Ind. Eng. Chem., Fundam., 2 136 (1963). Mueller, W. M., Blackledge, J. P., Libowitz, G. G., "Metal Hydrides," Academic Press, New York, N. Y., 1968. Murray, R. G., Schoeppel. R. J., Society of Automotive Engineers Paper No. 700608 (1970). Murray, R. G., Schoeppel, R. J., Allen, R. C., U. S. Patent No. 3,572,297
~ . ,
(1971a). Murray, R. G., Schoeppel, R. J., Society of Automotive Engineers Paper No. 719009 (1971b). Murray, W. D., Landis, F., Trans. ASME, Ser. C, 81 (2),106 (1959). Reilly, J. J., Wiswall, R. H.. Inorg. Chem., 6,2220 (1967). Reilly, J. J., Wiswall, R. H., Inorg. Chem., 7,2254 (1968). Reilly, J. J., personal communication, 1973. Smith, R. B., Jr., Meyer, W. A. P., Ayres, R. V., SAE Paper No. 690212
(1969). Stampfer. J. F., Jr.. Holley, C. E., Jr., Suttle, J. F., J. Amer. Chem. SOC.,
82,3504.(1960). Wiberg, E., German Patent No. 862,004(1953). Winsche, W. E., Sheenan, T. V., Hoffman, K. C., Society of Automotive Engineers Paper No. 719006 (1971). Wiswall, R. H., Jr., Reilly, J. M., Paper presented at 7th Intersociety Energy Conversion Conference, San Diego, Calif., 1972.
Receiuedfor review October 15, 1973 Accepted December 19, 1973
