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Thermochemical water splitting cycles: impact of thermal burdens and kinetics
Int. J. Hydrogen Energy, Vol, 5, pp. 507-513 Pergamon Press Ltd. 1980. Printed in Great Britain International Association for Hydrogen Energy
0360-3199/80/0901-0507 $02.00/0
THERMOCHEMICAL WATER SPLITTING CYCLES: IMPACT OF THERMAL B U R D E N S A N D KINETICS B. M. ABRAHAM* Office of Industrial Cooperation, University of Chicago, Chicago, IL 60637, U.S.A. Abstract--Equations are rigorously derived for evaluating the thermal efficiency of a triermochemieal water splitting cycle, from which it is possible to assess the impact of each heat burden or loss separately. The equations of continuity are shown to be coupled, as a consequence, heat flow is found to be the rate determinins process for the operation of a thermochemical water splitting plant. Since heat flow is rate determining, the chemical rate of reaction must be fast relative to heat flow even in the asymptotic approach to completion. The conclusion is drawn that recycling of reactants, which is required if AG/> 0, will probably result in an uneconomical cycle. Unlike a thermomeehanieal engine which can be characterized by a single parameter, the thermal efficiency F/, the thermochemical engine requires two, one, r/, which measures the effective use of heat and a second, z, which measures the effective use of power. The former is defined in the conventional manner, namely the work divided by the heat. The latter is defined as the ratio of the average chemical rate for product in the reaction volume for each stage multiplied by the heat required by the cycle to sprit a mol of water divided by the power of the source :
z =- vR~V~Q/P = v p i Q J P A H , = 1,
v adjusts for the stoichiometry, p~ is the power absorbed at stage i and AH~ is the entbalpy change in the reaction at i. To maintain a balanced plant x must equal unity. It is, also, concluded that hybrid cycles are favoured because of the additional degree of freedom.
NOMENCLATURE thermal burden or penalty = q l / Q ~ low heat rejection temperature when a subscript C heat capacity, kJ/K/mol Gibbs energy, k J/tool G~ AGr(H20 ) = - 237.2 kJ/m Gibbs energy of formation for water highest temperature h enthalpy, k J/tool H running index i stage J rn~ concentration of,,, mol/ce specific rate of production of a at stage i, mol/oe/se¢ M rate of production of product hydrogen, mol/sec number of reactions n power for stage i, kW P~ P power from heat source, kW heat absorbed or rejected by stage i, kJ qi Q total heat required per mol of hydrogen, k J/tool Rr chemical rate for • at stage i r highest useable temperature entropy, k J / K / m o l S temperature, K T velocity, cm/sec V volume of stage i, cc species when superscript A difference thermal conductivity, kW/K/cm thermal efficiency bl c
* Current address: Solid State Science Division, Argonne National Laboratory, Argonne, IL 60439, U.S.A. 507
508 "C
f~
B.M. ABRAHAM through-put efficiency surface across which the divergence is taken INTRODUCTION
A THERMODYNAMIC analysis of water splitting cycles has been published by Abraham and Schreiner [ 1], who showed, with the aid of entropy-temperature (S, T) diagrams, that the thermal efficiency of a cycle is reduced as reaction steps are added, even though there are no irreversibilities in the system. Implicit in their analysis as well as in the analyses of others [2, 3] is the assumption that the system equilibrates at all stages. Unlike a conventional chemical plant in which heat and mass flow streams may be controlled independently, the two streams in a thermochemical water splitting plant (TCP), are coupled. Therefore, any figure of merit based solely on thermodynamic considerations can be very misleading. In the following analyses, mathematical expressions are developed for the impact of a primary heat exchanger on the thermodynamic efficiency; also, a rigorous expression is presented which highlights the effect of various penalties on the efficiency, introduced by adding stages, work, etc. In addition, a second parameter, the through-put efficiency z, is defined. This parameter arises because the heat and mass flow streams are coupled through the equations of continuity. The data necessary to evaluate it should be obtained before efforts are made to assess, economically, any cycle. PRIMARY HEAT EXCHANGER A thermochemical water splitting cycle is a chemical engine, driven by heat, which does work on water to decompose it into the elements and which, in the process, regenerates all reagents. It is useful to relate the thermal efficiency of this engine to the temperature of the heat source, Th. As was shown in reference [1], the minimum heat, which may be termed the ideal value, required to drive a cycle between a reaction temperature, T, and a low temperature, To is: Q,, = - ~ s ( H 2 0 )
×
Td(T,- T~).
(1)
From Equation 1 it is readily seen that the thermal effidency is: rI = ( T , -
T c ) / T ,.
(2)
The heat source for a TCP is usually considered to be a nigh temperature nuclear reactor. For obvious reasons, the reaction vessel which requires the highest temperature cannot be integral with the reactor; a heat exchange fluid is required to transport the heat from the reactor to the reaction vessel. The first correction to the efficiency arises, then from the AT in the primary heat exchanger. Figure 1 displays the indicator diagram for an ideal Carnot engine operating between either Th and T or between T and T and delivering the same amount of work, W. For the former case, the area acdla represents the work; for the latter, the area abfg represents the work. The cross hatched area represents the work lost because the engine had to operate at T, rather than the maximum Th, for an entropy change of AS,. The magnitude of the area representing W is given by: W = Q, - q, = AS, x (T, - To), = Q , x ( T , -- T~)/T,.
(3)
(4)
If the heat had been available at Th, the engine would have performed the work: W'= Qh-q,=S
x (Th - T).
(5)
By substituting the appropriate expressions for W, AS,, and Qh into the standard definition for efficiency, rl =
w/Q,
(6)
THERMOCHEMICAL WATER SPLITTING CYCLES
509
d I i I I I I I I i i ....
I
I
T~
T,
J ¢
I
Xh
TEMPERATURE
FIG. 1. Entropy-temperature indicator diagram for an ideal Carnot engine.
an expression is obtained which reflects the effect of the primary heat exchanger, viz. ~l = (T, - T~)/Th,
(7)
r / = ~/,a x [1 - ( T h - T~)/(T h - L)],
(8)
or
where ,7,~ =
(T~ -
L)ITh.
(9)
From Equation 8, it can be seen that the physical parameters of the heat exchange medium are totally irrelevant to the Carnot efficiency. They become relevant only when the design of the exchanger itself and the power to move the fluid are under consideration. C O R R E C T I O N TERMS The value for the minimum heat required to drive a cycle, given by Equation 1, implies that, as with a mechanical engine, there is but one heat absorbing and one heat rejection step. It was shown in reference [1] that a chemical cycle will require at least two heat absorbing and one heat rejection steps. Nonetheless, the concept of an ideal efficiency and parameters which measure the departure from the ideal are very useful. In order to retain the concept, we proceed in the following manner. The efficiency is defined by Equation 6 where Q is the total heat required for all possible operations. Following Funk and Knoche [4], the heat may be written in terms of Qtd, the heat required for an ideal Carnot engine, plus all heat penalties. Q = Qid + all penalties = Q~ + q~ (reactions in excess of 2) + Q2 (recycle) + qa (operating) + q4 (losses) + etc.
(10)
or, after factoring out Q~, Q = Q,~ x [ l + ~
(11)
510
B.M. ABRAHAM
On substituting Equation 11 into Equations 1, 2, and 6, a general expression for the thermodynamic efficiency is obtained, namely, )7=rha x [ 1 - ( T h - T,)/(Th - Tc)] x 1/ 1 +
b, .
(12)
The first term in brackets is the correction term arising from the primary heat exchanger; the last term arises from the various penalties enumerated in Equation 10. If the individual b~s are not too large, then the last term may be rewritten. To first order,
1/(l+~b,)=l-~,b,1 ~
(13)
and it is seen that each b~ contributes a fractional reduction to the ideal efficiency. P E R F O R M A N C E CRITERIA It is important to keep in mind that a TCP is an energy conversion plant; it is only incidently a mass conversion plant. The primary feedstock to the TCP is heat, presumably from a high temperature reactor (HTGR), which is converted to chemical potential energy in the form of hydrogen. The question to be asked is then, what constraints, ff any, are imposed on a chemical process when the objective is energy conversion rather than mass conversion? Both a TCP and its energy source, the HTGR, must be designed so as to be compatible with respect to power generation, power acceptance, maximum temperature and hydrogen production. In all of the following it will be assumed that the plant operates at ideal efficiency; there is no loss in generality as a result but there is a considerable gain in simplicity. A fixed quantity of energy is required by a chemical process to affect the transformation sought. If an amount of heat Q kJ must be absorbed by the endothermic reactions of a TCP to split 1 mol of water and the rating of the associated HTGR is P kW then the production rate for hydrogen is simply:
KI = P/Q mol/sec.
(14)
In order to make efficient use of the H T G R it will be operated at the rated power level; it follows, therefore, that: / ¢ / = constant.
(15)
If the TCP has n endothermic stages, each stage will receive the requisite amount of power from the HTGR to affect the transformation for that stage so that:
and
P = ~ p~ = constant,
(16)
Q = ~ H~ = constant.
(17)
Regardless of the type of chemical reactor employed at a given stage, whether it be a stirred tank or tubular, it must be operated so as to accept the power. The differential equations describing heat and mass flow in a chemical reactor have been studied extensively in the chemical engineering literature. Following Perlmutter [5], we write two general equations of continuity for production of product m" from reactants mp. V ' m ~ + Fh~ + R~ = 0,
(18)
V2r, Tt + (m~C; + m~C£) L + (ra~C; + m~lr'4C~)AT + R~AH i = 0.
(19)
The conventional chemical engineering problem attempts to determine the operating conditions, concentration, temperature, mass flow, that will produce a steady state condition, i.e. ~h7 = 0, = 0.
(20)
THERMOCHEMICAL WATER SPLITTING CYCLES
511
The TCP problem is somewhat different. At the outset it is assumed that a steady state exists and, in addition, the heat exchange between influent and effluent is perfect. These assumptions are essential as the TCP is continuously operating isothermally, with no mass hold up. Equations 18 and 19 simplify to: V'm~v i + R~ = 0,
(21)
V2xi~ + R~AH i = O,
(22)
m~,t), + R~Vi = O,
(23)
p~ + R~V~AH~ = O,
(24)
and on applying Gauss' theorem
where R~ is now an average rate in the volume V~. Equations 23 and 24 state in a rigorous manner what could be inferred intuitively, namely the product that flows out of a stage i is equal to the production in the stage and the power input is absorbed in the chemical reaction. It follows from the equations that the production rate for hydrogen is equal to the production rate for product at every stage. = vR~V~,
(25)
where v adjusts the stoichiometry; in addition, R~ = I(t/(vV~) = P/vQVi = constant,
(26)
the rate of reaction is independent of the chemical mechanism but depends only on the parameters of the plant, an extraordinary result. Equation 26 is a direct consequence of the fact that the power input is fixed. In contrast to the conventional chemical plant for which the power, the temperature, and the composition are adjustable operating parameters, the chemical cycle of the TCP fixes reactant composition and operating temperature. The one adjustable parameter, the production rate of hydrogen, is determined by the power rating of the heat source. The constraint imposed by Equation 26, namely the production rate of product at every staoe is fixed by the power input not by the chemical kinetics, is a very stringent one indeed. A new parameter, the through-put efficiency, ~, may now be defined: r = vR~ViQ/P = vp~Q/PAH i = 1.
(27)
Unfortunately, z is not readily evaluated. Whereas, the thermal efficiency ff may be readily estimated from tabulations of thermodynamic data, the rate data are very sparse and are affected by reactor geometry and catalysts. Clearly, q alone is not adequate for cycle evaluation; although ~/may be favourable, the cycle may founder because • cannot economically be made equal to unity. IMPACT OF z < 1 The through-put efficiency, z, is a measure of the power utilization. Equations 24 and 27 state the relationships that must apply if a TCP is to operate as designed. Equation 24 is always true as long as 7" = 0. If the condition arises such that •" " m~i["~nj > mjVj['~j.
(28)
i.e. the influx of reactants exceeds the effiux of products, t h e n , becomes less than unity. As a consequence heat will have to be diverted from the stage (dumped) or recycling of reactants will be required or the stage will have to be enlarged. This condition arises either because the design resulted in R~, the average chemical rate, being too slow or AGj i> 0. Obviously, the diversion of heat from a stage in the attempt to bring a plant into balance is inadmissible on two accounts. First, the objective is to take heat, from a capital intensive source utilizing a capital intensive convertor, and to transform it into chemical potential. Any diversion of heat brings the plant close to marginal economy. Second, all other stages become oversized relative to stage j, which in turn diminishes the economy further.
512
B.M. ABRAHAM
Similarly, the attempt to make z = 1 by increasing the volume of the stage to make R~Vi larger and thereby increasing the heat utilization is also uneconomical. An auxiliary separations loop will have to be coupled to stage j to separate the excess reactants. The same argument can be advanced for the case where AG~ >/0; however, in this instance there may be a corrective measure that can be taken. Since the heat of reaction is nearly temperature independent little thermal cost is incurred by raising the reaction temperature of stage j so that AG~ < 0. This is an alternative only for reactions which take place below the maximum useable temperature from the source, T,. Assuming heat exchange between reactants and products is possible then the heat loss will be (C~ - C~)AT. In general ACp ~ 0. As a conclusion from the foregoing arguments, it is important that AG, < 0.
•HeolFeed f~
1/202
~-VVV~/VV,/W-II.
3
/
H2
H20
Feed
FIG. 2. Schematic flow diagram for a four stage thermochemical water splitting plant (TCP). Solid lines material flow; wavy lines heat flow.
The constraints imposed upon a TCP are illustrated diagrammatically in Fig. 2. The cyclic nature of the process is indicated by interconnecting all stages with the only inputs being water and a single heat source; the outputs are H2, 0 2 and waste heat. This diagram may be compared with Fig. 3 which represents a conventional chemical plant. Each stage, it is readily seen, is independent of others insofar as reagents and energy are concerned. Whereas, the TCP has no degrees of freedom; the conventional plant has as many as there are stages. CONCLUSION A thermochemical water spitting cycle is a subtle complex engine. It may be compared to a speed reducing gear box in which each gear represents a stage of the cycle. Though the diameters vary and some rotate clockwise while others rotate anticlockwise, all mesh and are synchronized. A stage for which AGj i> 0 or z < 1 is equivalent to a gear slipping on its shaft or to being of non-commensurate pitch. Eventually the gear box wiU cease to function. The subtlety of the engine now becomes apparent. Not only must the stages be kept few in number to keep thermal efficiency high [1] but they must also be kept few in number to satisfy the kinetic constraints of heat and mass flow.
THERMOCHEMICAL WATER SPLITTING CYCLES
513
pr°duc!
_••S $3 Heo in/ou~/~..
Reocjents in/out
Feed FIG. 3. Schematic diagram for a four stage conventional chemical plant. Solid lines material flow; wavy lines heat flow.
The foregoing analysis has introduced a parameter, in addition to the thermal efficiency, which is essential for the evaluation of a thermochemical cycle. It suggests that the greatest promise will lie with hybrid cycles because of the additional degree of freedom they provide. REFERENCES 1. 2. 3. 4.
B. M. ABRAHAMand F. SCHItEI~mR,Ind. Enong Chem. Fundam. 13, 305 (1974). J. E. FUNKand R. M. RmlqSTROM,Ind. Enon0 Chem., Process Design Develop. 5, 336 (1966). R. H. Wmcror,~, JR. and R. E. ~ N , Science 185, 311 (1974). J. E. Fi.rr~ and K. F. KNOCn~ Paper presented at Twelfth Intersoeiety Energy Conservation and Engineering Conference, Washington, DC, 28 August-2 September 1977. The author wishes to thank Professor Funk for the preprint. 5. D. D. I~RLMUYrER Stability of Chemical Reactors Prentice-Hall, Inc., Englewood Cliffs, NJ (1972), and references therein.
0360-3199/80/0901-0507 $02.00/0
THERMOCHEMICAL WATER SPLITTING CYCLES: IMPACT OF THERMAL B U R D E N S A N D KINETICS B. M. ABRAHAM* Office of Industrial Cooperation, University of Chicago, Chicago, IL 60637, U.S.A. Abstract--Equations are rigorously derived for evaluating the thermal efficiency of a triermochemieal water splitting cycle, from which it is possible to assess the impact of each heat burden or loss separately. The equations of continuity are shown to be coupled, as a consequence, heat flow is found to be the rate determinins process for the operation of a thermochemical water splitting plant. Since heat flow is rate determining, the chemical rate of reaction must be fast relative to heat flow even in the asymptotic approach to completion. The conclusion is drawn that recycling of reactants, which is required if AG/> 0, will probably result in an uneconomical cycle. Unlike a thermomeehanieal engine which can be characterized by a single parameter, the thermal efficiency F/, the thermochemical engine requires two, one, r/, which measures the effective use of heat and a second, z, which measures the effective use of power. The former is defined in the conventional manner, namely the work divided by the heat. The latter is defined as the ratio of the average chemical rate for product in the reaction volume for each stage multiplied by the heat required by the cycle to sprit a mol of water divided by the power of the source :
z =- vR~V~Q/P = v p i Q J P A H , = 1,
v adjusts for the stoichiometry, p~ is the power absorbed at stage i and AH~ is the entbalpy change in the reaction at i. To maintain a balanced plant x must equal unity. It is, also, concluded that hybrid cycles are favoured because of the additional degree of freedom.
NOMENCLATURE thermal burden or penalty = q l / Q ~ low heat rejection temperature when a subscript C heat capacity, kJ/K/mol Gibbs energy, k J/tool G~ AGr(H20 ) = - 237.2 kJ/m Gibbs energy of formation for water highest temperature h enthalpy, k J/tool H running index i stage J rn~ concentration of,,, mol/ce specific rate of production of a at stage i, mol/oe/se¢ M rate of production of product hydrogen, mol/sec number of reactions n power for stage i, kW P~ P power from heat source, kW heat absorbed or rejected by stage i, kJ qi Q total heat required per mol of hydrogen, k J/tool Rr chemical rate for • at stage i r highest useable temperature entropy, k J / K / m o l S temperature, K T velocity, cm/sec V volume of stage i, cc species when superscript A difference thermal conductivity, kW/K/cm thermal efficiency bl c
* Current address: Solid State Science Division, Argonne National Laboratory, Argonne, IL 60439, U.S.A. 507
508 "C
f~
B.M. ABRAHAM through-put efficiency surface across which the divergence is taken INTRODUCTION
A THERMODYNAMIC analysis of water splitting cycles has been published by Abraham and Schreiner [ 1], who showed, with the aid of entropy-temperature (S, T) diagrams, that the thermal efficiency of a cycle is reduced as reaction steps are added, even though there are no irreversibilities in the system. Implicit in their analysis as well as in the analyses of others [2, 3] is the assumption that the system equilibrates at all stages. Unlike a conventional chemical plant in which heat and mass flow streams may be controlled independently, the two streams in a thermochemical water splitting plant (TCP), are coupled. Therefore, any figure of merit based solely on thermodynamic considerations can be very misleading. In the following analyses, mathematical expressions are developed for the impact of a primary heat exchanger on the thermodynamic efficiency; also, a rigorous expression is presented which highlights the effect of various penalties on the efficiency, introduced by adding stages, work, etc. In addition, a second parameter, the through-put efficiency z, is defined. This parameter arises because the heat and mass flow streams are coupled through the equations of continuity. The data necessary to evaluate it should be obtained before efforts are made to assess, economically, any cycle. PRIMARY HEAT EXCHANGER A thermochemical water splitting cycle is a chemical engine, driven by heat, which does work on water to decompose it into the elements and which, in the process, regenerates all reagents. It is useful to relate the thermal efficiency of this engine to the temperature of the heat source, Th. As was shown in reference [1], the minimum heat, which may be termed the ideal value, required to drive a cycle between a reaction temperature, T, and a low temperature, To is: Q,, = - ~ s ( H 2 0 )
×
Td(T,- T~).
(1)
From Equation 1 it is readily seen that the thermal effidency is: rI = ( T , -
T c ) / T ,.
(2)
The heat source for a TCP is usually considered to be a nigh temperature nuclear reactor. For obvious reasons, the reaction vessel which requires the highest temperature cannot be integral with the reactor; a heat exchange fluid is required to transport the heat from the reactor to the reaction vessel. The first correction to the efficiency arises, then from the AT in the primary heat exchanger. Figure 1 displays the indicator diagram for an ideal Carnot engine operating between either Th and T or between T and T and delivering the same amount of work, W. For the former case, the area acdla represents the work; for the latter, the area abfg represents the work. The cross hatched area represents the work lost because the engine had to operate at T, rather than the maximum Th, for an entropy change of AS,. The magnitude of the area representing W is given by: W = Q, - q, = AS, x (T, - To), = Q , x ( T , -- T~)/T,.
(3)
(4)
If the heat had been available at Th, the engine would have performed the work: W'= Qh-q,=S
x (Th - T).
(5)
By substituting the appropriate expressions for W, AS,, and Qh into the standard definition for efficiency, rl =
w/Q,
(6)
THERMOCHEMICAL WATER SPLITTING CYCLES
509
d I i I I I I I I i i ....
I
I
T~
T,
J ¢
I
Xh
TEMPERATURE
FIG. 1. Entropy-temperature indicator diagram for an ideal Carnot engine.
an expression is obtained which reflects the effect of the primary heat exchanger, viz. ~l = (T, - T~)/Th,
(7)
r / = ~/,a x [1 - ( T h - T~)/(T h - L)],
(8)
or
where ,7,~ =
(T~ -
L)ITh.
(9)
From Equation 8, it can be seen that the physical parameters of the heat exchange medium are totally irrelevant to the Carnot efficiency. They become relevant only when the design of the exchanger itself and the power to move the fluid are under consideration. C O R R E C T I O N TERMS The value for the minimum heat required to drive a cycle, given by Equation 1, implies that, as with a mechanical engine, there is but one heat absorbing and one heat rejection step. It was shown in reference [1] that a chemical cycle will require at least two heat absorbing and one heat rejection steps. Nonetheless, the concept of an ideal efficiency and parameters which measure the departure from the ideal are very useful. In order to retain the concept, we proceed in the following manner. The efficiency is defined by Equation 6 where Q is the total heat required for all possible operations. Following Funk and Knoche [4], the heat may be written in terms of Qtd, the heat required for an ideal Carnot engine, plus all heat penalties. Q = Qid + all penalties = Q~ + q~ (reactions in excess of 2) + Q2 (recycle) + qa (operating) + q4 (losses) + etc.
(10)
or, after factoring out Q~, Q = Q,~ x [ l + ~
(11)
510
B.M. ABRAHAM
On substituting Equation 11 into Equations 1, 2, and 6, a general expression for the thermodynamic efficiency is obtained, namely, )7=rha x [ 1 - ( T h - T,)/(Th - Tc)] x 1/ 1 +
b, .
(12)
The first term in brackets is the correction term arising from the primary heat exchanger; the last term arises from the various penalties enumerated in Equation 10. If the individual b~s are not too large, then the last term may be rewritten. To first order,
1/(l+~b,)=l-~,b,1 ~
(13)
and it is seen that each b~ contributes a fractional reduction to the ideal efficiency. P E R F O R M A N C E CRITERIA It is important to keep in mind that a TCP is an energy conversion plant; it is only incidently a mass conversion plant. The primary feedstock to the TCP is heat, presumably from a high temperature reactor (HTGR), which is converted to chemical potential energy in the form of hydrogen. The question to be asked is then, what constraints, ff any, are imposed on a chemical process when the objective is energy conversion rather than mass conversion? Both a TCP and its energy source, the HTGR, must be designed so as to be compatible with respect to power generation, power acceptance, maximum temperature and hydrogen production. In all of the following it will be assumed that the plant operates at ideal efficiency; there is no loss in generality as a result but there is a considerable gain in simplicity. A fixed quantity of energy is required by a chemical process to affect the transformation sought. If an amount of heat Q kJ must be absorbed by the endothermic reactions of a TCP to split 1 mol of water and the rating of the associated HTGR is P kW then the production rate for hydrogen is simply:
KI = P/Q mol/sec.
(14)
In order to make efficient use of the H T G R it will be operated at the rated power level; it follows, therefore, that: / ¢ / = constant.
(15)
If the TCP has n endothermic stages, each stage will receive the requisite amount of power from the HTGR to affect the transformation for that stage so that:
and
P = ~ p~ = constant,
(16)
Q = ~ H~ = constant.
(17)
Regardless of the type of chemical reactor employed at a given stage, whether it be a stirred tank or tubular, it must be operated so as to accept the power. The differential equations describing heat and mass flow in a chemical reactor have been studied extensively in the chemical engineering literature. Following Perlmutter [5], we write two general equations of continuity for production of product m" from reactants mp. V ' m ~ + Fh~ + R~ = 0,
(18)
V2r, Tt + (m~C; + m~C£) L + (ra~C; + m~lr'4C~)AT + R~AH i = 0.
(19)
The conventional chemical engineering problem attempts to determine the operating conditions, concentration, temperature, mass flow, that will produce a steady state condition, i.e. ~h7 = 0, = 0.
(20)
THERMOCHEMICAL WATER SPLITTING CYCLES
511
The TCP problem is somewhat different. At the outset it is assumed that a steady state exists and, in addition, the heat exchange between influent and effluent is perfect. These assumptions are essential as the TCP is continuously operating isothermally, with no mass hold up. Equations 18 and 19 simplify to: V'm~v i + R~ = 0,
(21)
V2xi~ + R~AH i = O,
(22)
m~,t), + R~Vi = O,
(23)
p~ + R~V~AH~ = O,
(24)
and on applying Gauss' theorem
where R~ is now an average rate in the volume V~. Equations 23 and 24 state in a rigorous manner what could be inferred intuitively, namely the product that flows out of a stage i is equal to the production in the stage and the power input is absorbed in the chemical reaction. It follows from the equations that the production rate for hydrogen is equal to the production rate for product at every stage. = vR~V~,
(25)
where v adjusts the stoichiometry; in addition, R~ = I(t/(vV~) = P/vQVi = constant,
(26)
the rate of reaction is independent of the chemical mechanism but depends only on the parameters of the plant, an extraordinary result. Equation 26 is a direct consequence of the fact that the power input is fixed. In contrast to the conventional chemical plant for which the power, the temperature, and the composition are adjustable operating parameters, the chemical cycle of the TCP fixes reactant composition and operating temperature. The one adjustable parameter, the production rate of hydrogen, is determined by the power rating of the heat source. The constraint imposed by Equation 26, namely the production rate of product at every staoe is fixed by the power input not by the chemical kinetics, is a very stringent one indeed. A new parameter, the through-put efficiency, ~, may now be defined: r = vR~ViQ/P = vp~Q/PAH i = 1.
(27)
Unfortunately, z is not readily evaluated. Whereas, the thermal efficiency ff may be readily estimated from tabulations of thermodynamic data, the rate data are very sparse and are affected by reactor geometry and catalysts. Clearly, q alone is not adequate for cycle evaluation; although ~/may be favourable, the cycle may founder because • cannot economically be made equal to unity. IMPACT OF z < 1 The through-put efficiency, z, is a measure of the power utilization. Equations 24 and 27 state the relationships that must apply if a TCP is to operate as designed. Equation 24 is always true as long as 7" = 0. If the condition arises such that •" " m~i["~nj > mjVj['~j.
(28)
i.e. the influx of reactants exceeds the effiux of products, t h e n , becomes less than unity. As a consequence heat will have to be diverted from the stage (dumped) or recycling of reactants will be required or the stage will have to be enlarged. This condition arises either because the design resulted in R~, the average chemical rate, being too slow or AGj i> 0. Obviously, the diversion of heat from a stage in the attempt to bring a plant into balance is inadmissible on two accounts. First, the objective is to take heat, from a capital intensive source utilizing a capital intensive convertor, and to transform it into chemical potential. Any diversion of heat brings the plant close to marginal economy. Second, all other stages become oversized relative to stage j, which in turn diminishes the economy further.
512
B.M. ABRAHAM
Similarly, the attempt to make z = 1 by increasing the volume of the stage to make R~Vi larger and thereby increasing the heat utilization is also uneconomical. An auxiliary separations loop will have to be coupled to stage j to separate the excess reactants. The same argument can be advanced for the case where AG~ >/0; however, in this instance there may be a corrective measure that can be taken. Since the heat of reaction is nearly temperature independent little thermal cost is incurred by raising the reaction temperature of stage j so that AG~ < 0. This is an alternative only for reactions which take place below the maximum useable temperature from the source, T,. Assuming heat exchange between reactants and products is possible then the heat loss will be (C~ - C~)AT. In general ACp ~ 0. As a conclusion from the foregoing arguments, it is important that AG, < 0.
•HeolFeed f~
1/202
~-VVV~/VV,/W-II.
3
/
H2
H20
Feed
FIG. 2. Schematic flow diagram for a four stage thermochemical water splitting plant (TCP). Solid lines material flow; wavy lines heat flow.
The constraints imposed upon a TCP are illustrated diagrammatically in Fig. 2. The cyclic nature of the process is indicated by interconnecting all stages with the only inputs being water and a single heat source; the outputs are H2, 0 2 and waste heat. This diagram may be compared with Fig. 3 which represents a conventional chemical plant. Each stage, it is readily seen, is independent of others insofar as reagents and energy are concerned. Whereas, the TCP has no degrees of freedom; the conventional plant has as many as there are stages. CONCLUSION A thermochemical water spitting cycle is a subtle complex engine. It may be compared to a speed reducing gear box in which each gear represents a stage of the cycle. Though the diameters vary and some rotate clockwise while others rotate anticlockwise, all mesh and are synchronized. A stage for which AGj i> 0 or z < 1 is equivalent to a gear slipping on its shaft or to being of non-commensurate pitch. Eventually the gear box wiU cease to function. The subtlety of the engine now becomes apparent. Not only must the stages be kept few in number to keep thermal efficiency high [1] but they must also be kept few in number to satisfy the kinetic constraints of heat and mass flow.
THERMOCHEMICAL WATER SPLITTING CYCLES
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Feed FIG. 3. Schematic diagram for a four stage conventional chemical plant. Solid lines material flow; wavy lines heat flow.
The foregoing analysis has introduced a parameter, in addition to the thermal efficiency, which is essential for the evaluation of a thermochemical cycle. It suggests that the greatest promise will lie with hybrid cycles because of the additional degree of freedom they provide. REFERENCES 1. 2. 3. 4.
B. M. ABRAHAMand F. SCHItEI~mR,Ind. Enong Chem. Fundam. 13, 305 (1974). J. E. FUNKand R. M. RmlqSTROM,Ind. Enon0 Chem., Process Design Develop. 5, 336 (1966). R. H. Wmcror,~, JR. and R. E. ~ N , Science 185, 311 (1974). J. E. Fi.rr~ and K. F. KNOCn~ Paper presented at Twelfth Intersoeiety Energy Conservation and Engineering Conference, Washington, DC, 28 August-2 September 1977. The author wishes to thank Professor Funk for the preprint. 5. D. D. I~RLMUYrER Stability of Chemical Reactors Prentice-Hall, Inc., Englewood Cliffs, NJ (1972), and references therein.