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Maximum power of a•combined-cycle isothermal chemical engine
Pergamon PII:
Applied Thermal Engineering Vol. Il. No. 7. pp. 629-631. 1997 R, 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain. S13594311(96)00082-8 1359-431 l/97 $17.00 + 0.00
MAXIMUM POWER OF A-COMBINED-CYCLE ISOTHERMAL CHEMICAL ENGINE Lingen *Faculty
Chen,*
Fengrui
Sun,*
Chih Wut$
306, Naval Academy of Engineering, Engineering Department. U.S. Naval (Received
and Jianzheng
Gong*
Wuhan 430033, P. R. China and ‘yMechanica1 Academy, Annapolis, MD 21402, USA
10 Nooember
1996)
Abstract-A chemical engine processes mass flow to convert the differences in chemical potentials into work. An isothermal endoreversible chemical engine, in which the sole irreversibility is finite-rate mass transfer, is modeled in this paper. The expression of maximum power from the model using the analogous method of finite-time thermodynamics for a combined-cycle heat engine is derived. An important result shows that the efficiency at maximum power output is half of the maximum efficiency for chemical and heat engines. Q 1997 Elsevier Science Ltd.
INTRODUCTION
The Carnot heat engine proposed in 1824 operates on the reversible and equilibrium principles. As a consequence, this hypothetical engine must operate at an infinitely slow pace. It therefore produces the maximum possible work for a given set of heat source and sink but generates zero power. The Carnot cycle efficiency, long used as the standard against which all real heat engine efficiencies are measured, is unrealistically high. Curzon and Ahlborn [l] started an analysis that accounts for irreversibilities of finite-rate heat transfer to and from a heat engine. Such an engine is able to generate useful power. Because of the external irreversibilities of the engine, its efficiency at maximum power output is less than that of the Carhot heat engine. The Curzon-Ahlborn engine efficiency is now referred to as the finite-time engine efficiency. Since finite-time thermodynamics was advanced in 1975, many researchers have investigated the effect of irreversibilities on thermodynamic processes and cycles. Detailed surveys of finite-time thermodynamics have been given by Sieniutycz [2] and Chen [3,4]. Most of the research work in finite-time thermodynamics has been done on heat engines; only recently have finite-time thermodynamics been extended to chemical reactions and chemical heat engines by Ondrechen [5, 61, DeVos [7-l 11, Chen [I23 and Gordon [13, 141. Heat engines convert temperature differences into work. Similarly, chemical engines convert differences in chemical potentials into work. Chemical potential and mass transfer in chemical heat engines play roles analogous to temperature and entropy in heat engines. There are also engines which exploit both differences in chemical potential and temperature for power production. DeVos [7-l l] has extended the definition of the endoreversible heat engine into a generalized endoreversible engine by generalizing heat reservoirs into heat- and mass-reservoirs, and heat exchangers into heat- and mass-exchangers. If the difference in temperature is zero, a generalized endoreversible engine becomes an isothermal endoreversible chemical engine. The power optimization of a single-cycle isothermal chemical engine was performed by Gordon [13, 141 using the analogous method of finite-time thermodynamics for an endoreversible heat engine. The single cycle isothermal chemical engine analysis is given in Appendix 1. In this paper, we will take another step further beyond refs [7-141 to estimate the maximum power from a combined-cycle isothermal endoreversible chemical engine. The problem is of practical value to many devices, such as chemical reactive devices, mass exchangers, photochemical cells and solid-state devices. IAuthor
to whom correspondence
should
be addressed. 629
Lingen Chen et al
630
COMBINED-CYCLE
ISOTHERMAL
CHEMICAL
ENGINE
A combined-cycle isothermal endoreversible chemical engine is shown in Fig. 1. The cycle is made of topping and bottom single cycles. The configuration of the chemical combined cycle is analogous to the endoreversible combined heat-engine cycle [15-181. The topping cycle receives its mass at p, from the high potential reservoir at pL,,, in time t, and rejects mass at cl2 to the bottom cycle at p3 in time t2. Assuming the two single-chemical cycles have the same total cycle time, then the bottom cycle rejects mass at p4 to the low potential reservoir at p,,,,” in time t,. Therefore
High
chemical
potential
reservoir
lLMAX
N, Chemical
engine
Chemical
engine
Low chemical
potential
reservoir
pMIN
Fig.
1. Combined-cycle
endoreversible
chemical
engine.
Maximum
power of a combined-cycle
chemical
engine
631
(2)
(3) where h,, h, and h, are mass-transfer
coefficients.
Conservation
of mass requires
N, = N> = N, = N . The power
output
(P) and efficiency
(q) of the combined P=
w/T=(u:+
U: and
Combining
W, are the work outputs
equations
chemical
engine
are
Wz)/t,
rl = (u: + K)lbU,,, where
(4)
AAI ,
-
of the topping
(5) (6)
and bottom
cycles, respectively.
K = N(P, - A) >
(7)
W = N(K - ~4) .
(8)
(5)-(X) gives P = [(PI - PA - (~2 -
PJIW 3
(9)
VI= KPI- A4 - (Lb- /4>l/hlax- /4m”> ’ Rearranging
equations
(l)-(3)
yields 3
PI = ~rnal:- Nl(h,t,) PZ ~4
Substituting
equations
(10)
=
~3
~m,n
=
Nl(hh)
+
NlVMJ
(1 l)-( 13) into equation
(9) yields
p = P,.X - PInI”- N[ll(h,&)
+ l/(&t,)
(11) (12)
,
(13)
.
+ ll(Mr
-
~,NlNl~;
(14)
P is a function of two independent variables (t, and N). To find the maximum power output of the combined-cycle chemical engine, taking the derivatives of P with respect to t, and N and setting them equal to zero (aP/dt, = 0 and i3PjaN = 0) gives (t&p, = r/l
+ [(A,) - ’+ (h,) -
%W2,
No,, = Pzr(pL,,, - p,,,,n)1/21 + [(h,) - ’+ (h,) - ‘]h:‘22.
(15) (16)
Substituting equations (15) and (16) into equations (1 I)-(14) yields the optimum chemical potentials of the engine working fluid on the high- and low-potential mass transfer branches ((p&,,, and (JA~)~J, the optimal chemical potential difference between the topping and bottom cycles ((p2 - ,L&,,), and the maximum power output (P,,,,J delivered by the combined-cycle chemical engine in the following equations: @Jo,, = PL,,, - PAP,,,
-
~L,31/21+ KM- ’+ (W - ‘lh:“hKM- ’+ @,I- ‘lW2>
(17)
(~4)opt
-
~nun)l/21 + Kh,) -'+ @J -'lh:'%[(h,) -'+ 6%)m'lh:'2 ,
(18)
=
pm,,
+
Mpmax
(1~1 - &t = bn,,- ~mm>1/21 + Kh,) -'+ (hd-'IV2,
(19)
and Pm,, = h&n,,
- pm,,)/21 + [(h,) - ’+ (hJ - ‘]hY2 .
The efficiency (s+) at the maximum power output equal to 0.5. The maximum efficiency (qWaX)of the to 1. It is noticed that the power optimization of value of p2 and pj, but depends on the difference
(20)
of the combined-cycle chemical engine is again combined-cycle chemical engine is again equal the engine does not depend on each individual of p2 and p, (p2 - &.
Lingen Chen et al.
632
DISCUSSION
1. The combined-cycle engine (with mass-transfer coefficients (h,, h, and h,)) is equivalent to a single-cycle chemical engine (either with mass-transfer coefficients (h, and h,) or with mass-transfer coefficients (h? and h,)) when mass-transfer coefficient h, or h, approaches an infinitely large value. In this case, the maximum power output of the chemical engine is P,,, = WL,,
- ~J2[1
+
(~,lW212,if km
,
(21)
.
(22)
or P,,, = h2(pmar- clm,J2[1 + @z/h,)“‘]*, if h,-too
Equations (21) and (22) have the same power output optimization expression of the single-cycle chemical engine given by equation (A14) in the Appendix. 2. The efficiency at the condition of maximum power output of both the combined-cycle and single-cycle chemical engines is equal to 0.5. However, the maximum power output of the two engines is different. The maximum power output of the combined-cycle is larger than that of the single-cycle if and only if h2 > {(A,) “’ + (h,) _ ‘I2- [(h,) - ’ + (h,) - ‘I”‘} _ 2. The maximum power output of the combined-cycle is only 4/[3 + 2(2)‘:2] = 0.686 of that of the single-cycle when h, = h, = h,. This shows that the mass transfer loss may cause the power decrease of the combined cycle under certain conditions. The key factor is whether the total equivalent mass-transfer coefficient of the combined cycle is larger than that of the single cycle or not. 3. The maximum power output of a single-cycle endoreversible heat engine shown in Fig. 2 is [l] Pmax= tl[(TH)“‘*- ( K)‘:2]/[1 + (U/p)“?]? ,
(23)
where T, and TL are the temperatures of the high- and low-temperature heat reservoirs associated with the heat engine and CI and /? are the heat conductances between the working fluid and the heat reservoirs, respectively. The maximum power output of a reciprocating combined-cycle endoreversible heat engine shown in Fig. 3 is[15, 181 P ,,,dX= j?[( T,)‘l’ - (r,)““]/[ 1 + ((a - ’+ y ‘)b)‘i2]2 , where M, /? and y are the heat conductances between the heat source the two cycles, and between the bottom cycle and the heat sink, The efficiency at the condition of maximum power output of single-cycle endoreversible heat engines is the same; it is known as
(24)
and the topping cycle, between respectively. both the combined-cycle and the Curzon-Ahlborn efficiency
(YICA)[15-181:
qCA=
1 - (TJTH)‘!’ .
(25)
Comparing equation (A14) in the Appendix and equation (20) with equations (23) and (24) shows that the relationships between single- and combined-cycle chemical engines are similar to those between the single- and combined-cycle heat engines. 4. A mass leak directly from the high-chemical-potential source to the low-chemical-potential sink will decrease the efficiency of the combined-cycle chemical engine, because more mass must be transferred to produce the same amount of power as it does for a single-cycle chemical engine [14]. 5. The mass-transfer law affects the power output and efficiency of the combined cycle as it does for a single-cycle chemical engine [14]. If the mass transfer of the system follows the following diffusivity law, N 0~DexpWk
01 t
(26)
where k is the Boltzmann constant and T is the temperature, the efficiency of the combined-cycle chemical engine will increase. 6. An important result is that the efficiency at maximum power output is half of the maximum efficiency for the single- and combined-cycle chemical engines. In fact, this is a nearly universal conclusion for chemical and heat engines. It is shown that the efficiency of an ideal motor powered
Maximum
power
of a combined-cycle
chemical
engine
633
Hot reservoir TEl
Hot Reservoir -f ,
Ql.
2
+,
Cold reservoir
TI_
Fig. 2. Single-cycle
endoreversible
-l-t_ heat engine
Fig. 3. Combined-cycle
endoreversible
heat engine.
by an electrical battery with internal resistance at maximum power output is also 0.5. The battery circuit is shown in Fig. 4, where the voltage of the battery is I&, the electric current is I, the internal and external electrical resistances are R,and R,.The maximum (reversible) efficiency of the plant is 1. In practice, we have
P = Z(v,, - ZR,)
(27)
v,, = Z(R, + R,) ,
(28)
and
where
P is the power output
of the plant.
Lingen
634
Fig. 4. Ideal motor
Substituting
equation
powered
(28) into equation p =
Chen et al.
by electrical
battery
with internal
resistance.
(27) gives
[(KJ21RI[1- (1 + ~)-‘I/(1 + a) 7
(29)
where M.= RJR,. Taking the derivative of P with respect to E and setting it equal to zero (dP/dcc = 0) gives N,,,, = 1 and the efficiency at maximum power output g,,, = l/2. The ratio (qr = Q.&J of the Curzon-Ahlborn efficiency (qCA) at maximum power output to heat the maximum efficiency (qm,, = qCarnot= 1 - r,/T,) for the single and combined endoreversible engine cycles is also nearly 0.5. The ratio is P/r = [l - (zJT”)“*]/(l
- z/r,)
= l/[l + (7JTH)‘i2] 2 0.5.
(30)
If (r,/r,)-, 1, Q = 0.5. The Q versus (TH/TL) relation is shown in Fig. 5; Q rises only slowly above 0.5 for realistic values of (r,/X). 7. The results obtained in this paper provide an upper bound on the power output of combined-cycle chemical engines, such as photovoltaic cells, mass exchangers and electrochemical devices. This analysis does not provide a new technique for synthesizing new high-efficiency solar cells or higher performance fuel cell. Rather the emphasis is on the establishment of fundamental bounds for the performance of a general class of chemical engines. Key sources of irreversibilities are included, but by no means cover all possible sources of irreversibility in real chemical engines. Hence, the results derived in this paper represent bounds rather than accurate representations of the actual performance of chemical engines. The objective of this analysis is similar to that of the finite-time thermodynamic analysis of heat engines. [l-6, 15-181 Depending on the inherent operation mode of the chemical converter, [14] there are two ways to view the physical significance of the optimization procedures presented in this paper. In the first way, chemical engines that can process mass transfer in batch are considered as systems in which a buffer mechanism can be implemented. Mass transfer can be turned on or off during the engine cycle. The optimization with respect to the time spent on the different branches of the cycle is then realistic. For example, the optimal switching time (t,) can be determined by equation (40) in Appendix 1. It is not necessarily mean that the system is amenable to being modified to accommodate the optimal solution given in this paper. However, the optimal solution is established as a bound against which nominally sub-optimal real cycles can be compared.
Maximum
power
of a combined-cycle
chemical
engine
635
II
F-
/
OH 1
Fig. 5. Relationship
2
3
between
efficiency
4
ratio
5
and temperature
6
ratio.
In the second way, the more common chemical converters which operate at constant mass flow rates are considered. Although the times spent on different branches of the cycle cannot be optimized, the distribution areas available for mass transfers are optimized in this anaysis. This is equivalent to the heat-exchanger area distribution optimization for heat transfer in the finite-time thermodynamic heat-engine analysis [ 18- 221. In summary, The procedures and results presented in this paper provide ways and bounds for the optimization performance of chemical engines. The bounds are more reasonable and closer to the real performance of chemical engines than those obtained from the classical reversible limits.
REFERENCES 1. F. L. Curzon and B. Ahlborn.. Amer. J. Phys. 43, 22 (1975). 2. S. Sieniutycz and P. Salamon, Advances in Thermo&namics, Vol. 4. Finite-time Thermodynamics and Thermoeconomics. Taylor and Francis, New York (1990). 3. L. Chen, F. Sun and W. Chen.. Advances in Mechanics 22, 479 (1992). 4. L. Chen, F. Sun and W. Chen.. J. Nature 15, 249 (1992). 5. M. Ondrechen, R. S. Berry and B. Andresen.. J. Chem. Phys. 72, 5118 (1980). 6. M. Ondrechen, B. Andresen and R. S. Berry.. J. Chem. Phys. 73, 5838 (1980). 7. A. DeVos.. Solar Cells 31, 181 (1991). 8. A. DeVos.. J. Phys. Chem. 95, 534 (1991). 9. A. DeVos, P. T. Lansberg, P. Baruch and J. E. Parrott.. J. Appl. Phys. 74, 3631 (1993). 10. A. DeVos, Endoreversible Thermodynamics of Solar Energy Conversion. Oxford University Press, Oxford (1992). 11. A. DeVos.. Solar Energy Mater. Solar Cells 31, 75 (1993). 12. L. Chen, F. Sun and W. Chen.. J. Naval Academy Engng 1, 51 (1993). 13. J. M. Gordon.. J. Appl. Phys. 73, 8 (1993). 14. J. M. Gordon and V. N. Orlov.. J. Appl. Phys. 73, 5303 (1993). 15. M. H. Rubin.. J. Appl. Phys. 53, I (1982). 16. C. Wu.. Energy Convers. Mgmnr 30, 261 (1990). 17. C. Wu, G. Karpouzion and R. L. Kiang.. J. Insf. Energy 65, 41 (1992). 18. L. Chen, F. Sun and W. Chen.. Power Engng 14, 9 (1994). 19. J. B. Woodward.. Trans. ASME J. Energy Res. Technol. 117, 343 (1995) 20. C. Wu, L. Chen and F. Sun.. Energ]> 21, 71 (1996). 21. A. Bejan.. Int. J. Heat Mass Transfer 31, 1211 (1988). 22. A. Bejan.. ht. J. Heat Mass Transfer 38, 433 (1995).
636
Lingen
Chen et al.
APPENDIX SINGLE-CYCLE
ISOTHERMAL
CHEMICAL
ENGINE
A single-cycle isothermal endoreversible chemical engine is shown in Figs Al and A2. The two reservoirs associated with the chemical engine have chemical potentials of pLmaiand pm,“. The rate of mass transfer, which is a function of the chemical potentials of the reservoir and the engine, is denoted by N. In electrochemical and solid-state devices N(p) is referred to as the current-voltage relation, where N is the current, p is the chemical potential of the engine working fluid, and (p 2 pm,“) is the voltage. The condition of maximum chemical potential difference (p = pmar) is the open circuit point, where the mass-transfer rate (current) vanishes. The condition of minimum chemical potential difference (p = pm,“) is the short-circuit point, where the mass-transfer rate (current) is the maximum. An optimization problem for power production exists because it is the product of the mass-transfer rate (current) and exploitable chemical potential (voltage). In the reversible (open-circuit) limit, although full available chemical potential difference can be utilized. it is at the penalty of vanishing rate of mass transfer; therefore, the power delivered by the chemical engine at the open-circuit is zero. In contrast, in the limit of maximum rate of mass transfer (short circuit), the utilizable chemical potential vanishes and hence the power output of the engine is again equal to zero. There is an operating point of maximum power between these two limits. The working fluid of the chemical engine may be a current of electrons for solid-state devices, gas or liquid molecules for mass exchangers in absorption/desorption processes. or chemically reacting species for fuel cells. Assuming the mass transfer is linearly proportional to the difference in chemical potentials, we have NI = Up,,,
>
(Al)
NZ = h&2 - pm&s >
- 0
(A2)
where pI and b are the chemical potentials of the chemical engine fluid on the high- and low-potential mass-transfer branches, t, and t2are the residence times of the two branches, N, and NZ are the mass-transfer rates of the two branches, and h, and h2 are the corresponding proportionality constants for the two mass-transfer branches, respectively. of mass requires Let the total time of the working fluid to complete a cycle be T, then r = t, + t,. Conservation N, = Nl = N
High chemical
potential
(A3)
reservoir
kMAX
Low chemical
potential
reservoir I
Fig. Al.
Single-cycle
endoreversible
chemical
engine.
Maximum
l
(P), efficiency
chemical
engine
637
Constant chemical potential mass transfer process
Fig. A2. The p-N
output
of a combined-cycle
--
kMAX
The power
power
diagram
of single-cycle
(q) and work output
endoreversible
(W) of the chemical
chemical
engine
engine.
are
P = w/s ) v = W/[N(~mnr -
(A4)
~Lmm)l >
(A5)
~=N(P,-P>). Solving
equations
(Al)
(A6)
and (A2) gives PI = ~Lmar- Nl(h,l,)
.
(A7)
w3)
PLY = ~rncn+ N/(&z) Substituting
equations
(A6)+A8)
into equation
(A4) yields
p = %nar - P(m,n- N[ll(h,r,)
+
~lVudlN/~;
(A9)
P is a function of two independent variables (t, and N) for a fixed c value. To find the maximum power output of the chemical engine. taking the derivative of P with respect to I, at constant N and f2 = r-t,. then setting it equal to zero [(~?P/dt,),, = 0] gives (f&t
.
= 7/[1 + (Wz)“‘]
(AlOa)
and (AlOb)
(f&p, = r/U + (WPI Substituting equations (AlO) into equation equal to zero [(CJPjdN), = 0] gives
(A9). taking
Nap, = P,r(p,,, Substituting equations (AlO) and (Al 1) into equations power output of the chemical engine in the following
the derivative -
of P with respect
to N at constant
~nm)l/W + (hlW’l*
(A7t(A9) gives the optimal expressions:
(h/h)“1 , (/&pt = pm + (pmax+ ~(rnmVW + WWI ,
(P&l, = Pmar - (PInax- ~,,,)/2[1
+
5 and setting
it
(Al 1) chemical
potentials
and the maximum
6412) (A13)
and P ma% = h,(pc,a - r~J/2[~
7 ‘1, . + (h,lk)’
(A14)
where Pm.. is the maximum power output of the chemical engine.Substituting equations (Al 1) and (A14) into equation (A5) gives the corresponding engine efficiency qop, = l/2 at the condition of the maximum power output. The maximum efficiency of the engine is qmaX= 1.
Applied Thermal Engineering Vol. Il. No. 7. pp. 629-631. 1997 R, 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain. S13594311(96)00082-8 1359-431 l/97 $17.00 + 0.00
MAXIMUM POWER OF A-COMBINED-CYCLE ISOTHERMAL CHEMICAL ENGINE Lingen *Faculty
Chen,*
Fengrui
Sun,*
Chih Wut$
306, Naval Academy of Engineering, Engineering Department. U.S. Naval (Received
and Jianzheng
Gong*
Wuhan 430033, P. R. China and ‘yMechanica1 Academy, Annapolis, MD 21402, USA
10 Nooember
1996)
Abstract-A chemical engine processes mass flow to convert the differences in chemical potentials into work. An isothermal endoreversible chemical engine, in which the sole irreversibility is finite-rate mass transfer, is modeled in this paper. The expression of maximum power from the model using the analogous method of finite-time thermodynamics for a combined-cycle heat engine is derived. An important result shows that the efficiency at maximum power output is half of the maximum efficiency for chemical and heat engines. Q 1997 Elsevier Science Ltd.
INTRODUCTION
The Carnot heat engine proposed in 1824 operates on the reversible and equilibrium principles. As a consequence, this hypothetical engine must operate at an infinitely slow pace. It therefore produces the maximum possible work for a given set of heat source and sink but generates zero power. The Carnot cycle efficiency, long used as the standard against which all real heat engine efficiencies are measured, is unrealistically high. Curzon and Ahlborn [l] started an analysis that accounts for irreversibilities of finite-rate heat transfer to and from a heat engine. Such an engine is able to generate useful power. Because of the external irreversibilities of the engine, its efficiency at maximum power output is less than that of the Carhot heat engine. The Curzon-Ahlborn engine efficiency is now referred to as the finite-time engine efficiency. Since finite-time thermodynamics was advanced in 1975, many researchers have investigated the effect of irreversibilities on thermodynamic processes and cycles. Detailed surveys of finite-time thermodynamics have been given by Sieniutycz [2] and Chen [3,4]. Most of the research work in finite-time thermodynamics has been done on heat engines; only recently have finite-time thermodynamics been extended to chemical reactions and chemical heat engines by Ondrechen [5, 61, DeVos [7-l 11, Chen [I23 and Gordon [13, 141. Heat engines convert temperature differences into work. Similarly, chemical engines convert differences in chemical potentials into work. Chemical potential and mass transfer in chemical heat engines play roles analogous to temperature and entropy in heat engines. There are also engines which exploit both differences in chemical potential and temperature for power production. DeVos [7-l l] has extended the definition of the endoreversible heat engine into a generalized endoreversible engine by generalizing heat reservoirs into heat- and mass-reservoirs, and heat exchangers into heat- and mass-exchangers. If the difference in temperature is zero, a generalized endoreversible engine becomes an isothermal endoreversible chemical engine. The power optimization of a single-cycle isothermal chemical engine was performed by Gordon [13, 141 using the analogous method of finite-time thermodynamics for an endoreversible heat engine. The single cycle isothermal chemical engine analysis is given in Appendix 1. In this paper, we will take another step further beyond refs [7-141 to estimate the maximum power from a combined-cycle isothermal endoreversible chemical engine. The problem is of practical value to many devices, such as chemical reactive devices, mass exchangers, photochemical cells and solid-state devices. IAuthor
to whom correspondence
should
be addressed. 629
Lingen Chen et al
630
COMBINED-CYCLE
ISOTHERMAL
CHEMICAL
ENGINE
A combined-cycle isothermal endoreversible chemical engine is shown in Fig. 1. The cycle is made of topping and bottom single cycles. The configuration of the chemical combined cycle is analogous to the endoreversible combined heat-engine cycle [15-181. The topping cycle receives its mass at p, from the high potential reservoir at pL,,, in time t, and rejects mass at cl2 to the bottom cycle at p3 in time t2. Assuming the two single-chemical cycles have the same total cycle time, then the bottom cycle rejects mass at p4 to the low potential reservoir at p,,,,” in time t,. Therefore
High
chemical
potential
reservoir
lLMAX
N, Chemical
engine
Chemical
engine
Low chemical
potential
reservoir
pMIN
Fig.
1. Combined-cycle
endoreversible
chemical
engine.
Maximum
power of a combined-cycle
chemical
engine
631
(2)
(3) where h,, h, and h, are mass-transfer
coefficients.
Conservation
of mass requires
N, = N> = N, = N . The power
output
(P) and efficiency
(q) of the combined P=
w/T=(u:+
U: and
Combining
W, are the work outputs
equations
chemical
engine
are
Wz)/t,
rl = (u: + K)lbU,,, where
(4)
AAI ,
-
of the topping
(5) (6)
and bottom
cycles, respectively.
K = N(P, - A) >
(7)
W = N(K - ~4) .
(8)
(5)-(X) gives P = [(PI - PA - (~2 -
PJIW 3
(9)
VI= KPI- A4 - (Lb- /4>l/hlax- /4m”> ’ Rearranging
equations
(l)-(3)
yields 3
PI = ~rnal:- Nl(h,t,) PZ ~4
Substituting
equations
(10)
=
~3
~m,n
=
Nl(hh)
+
NlVMJ
(1 l)-( 13) into equation
(9) yields
p = P,.X - PInI”- N[ll(h,&)
+ l/(&t,)
(11) (12)
,
(13)
.
+ ll(Mr
-
~,NlNl~;
(14)
P is a function of two independent variables (t, and N). To find the maximum power output of the combined-cycle chemical engine, taking the derivatives of P with respect to t, and N and setting them equal to zero (aP/dt, = 0 and i3PjaN = 0) gives (t&p, = r/l
+ [(A,) - ’+ (h,) -
%W2,
No,, = Pzr(pL,,, - p,,,,n)1/21 + [(h,) - ’+ (h,) - ‘]h:‘22.
(15) (16)
Substituting equations (15) and (16) into equations (1 I)-(14) yields the optimum chemical potentials of the engine working fluid on the high- and low-potential mass transfer branches ((p&,,, and (JA~)~J, the optimal chemical potential difference between the topping and bottom cycles ((p2 - ,L&,,), and the maximum power output (P,,,,J delivered by the combined-cycle chemical engine in the following equations: @Jo,, = PL,,, - PAP,,,
-
~L,31/21+ KM- ’+ (W - ‘lh:“hKM- ’+ @,I- ‘lW2>
(17)
(~4)opt
-
~nun)l/21 + Kh,) -'+ @J -'lh:'%[(h,) -'+ 6%)m'lh:'2 ,
(18)
=
pm,,
+
Mpmax
(1~1 - &t = bn,,- ~mm>1/21 + Kh,) -'+ (hd-'IV2,
(19)
and Pm,, = h&n,,
- pm,,)/21 + [(h,) - ’+ (hJ - ‘]hY2 .
The efficiency (s+) at the maximum power output equal to 0.5. The maximum efficiency (qWaX)of the to 1. It is noticed that the power optimization of value of p2 and pj, but depends on the difference
(20)
of the combined-cycle chemical engine is again combined-cycle chemical engine is again equal the engine does not depend on each individual of p2 and p, (p2 - &.
Lingen Chen et al.
632
DISCUSSION
1. The combined-cycle engine (with mass-transfer coefficients (h,, h, and h,)) is equivalent to a single-cycle chemical engine (either with mass-transfer coefficients (h, and h,) or with mass-transfer coefficients (h? and h,)) when mass-transfer coefficient h, or h, approaches an infinitely large value. In this case, the maximum power output of the chemical engine is P,,, = WL,,
- ~J2[1
+
(~,lW212,if km
,
(21)
.
(22)
or P,,, = h2(pmar- clm,J2[1 + @z/h,)“‘]*, if h,-too
Equations (21) and (22) have the same power output optimization expression of the single-cycle chemical engine given by equation (A14) in the Appendix. 2. The efficiency at the condition of maximum power output of both the combined-cycle and single-cycle chemical engines is equal to 0.5. However, the maximum power output of the two engines is different. The maximum power output of the combined-cycle is larger than that of the single-cycle if and only if h2 > {(A,) “’ + (h,) _ ‘I2- [(h,) - ’ + (h,) - ‘I”‘} _ 2. The maximum power output of the combined-cycle is only 4/[3 + 2(2)‘:2] = 0.686 of that of the single-cycle when h, = h, = h,. This shows that the mass transfer loss may cause the power decrease of the combined cycle under certain conditions. The key factor is whether the total equivalent mass-transfer coefficient of the combined cycle is larger than that of the single cycle or not. 3. The maximum power output of a single-cycle endoreversible heat engine shown in Fig. 2 is [l] Pmax= tl[(TH)“‘*- ( K)‘:2]/[1 + (U/p)“?]? ,
(23)
where T, and TL are the temperatures of the high- and low-temperature heat reservoirs associated with the heat engine and CI and /? are the heat conductances between the working fluid and the heat reservoirs, respectively. The maximum power output of a reciprocating combined-cycle endoreversible heat engine shown in Fig. 3 is[15, 181 P ,,,dX= j?[( T,)‘l’ - (r,)““]/[ 1 + ((a - ’+ y ‘)b)‘i2]2 , where M, /? and y are the heat conductances between the heat source the two cycles, and between the bottom cycle and the heat sink, The efficiency at the condition of maximum power output of single-cycle endoreversible heat engines is the same; it is known as
(24)
and the topping cycle, between respectively. both the combined-cycle and the Curzon-Ahlborn efficiency
(YICA)[15-181:
qCA=
1 - (TJTH)‘!’ .
(25)
Comparing equation (A14) in the Appendix and equation (20) with equations (23) and (24) shows that the relationships between single- and combined-cycle chemical engines are similar to those between the single- and combined-cycle heat engines. 4. A mass leak directly from the high-chemical-potential source to the low-chemical-potential sink will decrease the efficiency of the combined-cycle chemical engine, because more mass must be transferred to produce the same amount of power as it does for a single-cycle chemical engine [14]. 5. The mass-transfer law affects the power output and efficiency of the combined cycle as it does for a single-cycle chemical engine [14]. If the mass transfer of the system follows the following diffusivity law, N 0~DexpWk
01 t
(26)
where k is the Boltzmann constant and T is the temperature, the efficiency of the combined-cycle chemical engine will increase. 6. An important result is that the efficiency at maximum power output is half of the maximum efficiency for the single- and combined-cycle chemical engines. In fact, this is a nearly universal conclusion for chemical and heat engines. It is shown that the efficiency of an ideal motor powered
Maximum
power
of a combined-cycle
chemical
engine
633
Hot reservoir TEl
Hot Reservoir -f ,
Ql.
2
+,
Cold reservoir
TI_
Fig. 2. Single-cycle
endoreversible
-l-t_ heat engine
Fig. 3. Combined-cycle
endoreversible
heat engine.
by an electrical battery with internal resistance at maximum power output is also 0.5. The battery circuit is shown in Fig. 4, where the voltage of the battery is I&, the electric current is I, the internal and external electrical resistances are R,and R,.The maximum (reversible) efficiency of the plant is 1. In practice, we have
P = Z(v,, - ZR,)
(27)
v,, = Z(R, + R,) ,
(28)
and
where
P is the power output
of the plant.
Lingen
634
Fig. 4. Ideal motor
Substituting
equation
powered
(28) into equation p =
Chen et al.
by electrical
battery
with internal
resistance.
(27) gives
[(KJ21RI[1- (1 + ~)-‘I/(1 + a) 7
(29)
where M.= RJR,. Taking the derivative of P with respect to E and setting it equal to zero (dP/dcc = 0) gives N,,,, = 1 and the efficiency at maximum power output g,,, = l/2. The ratio (qr = Q.&J of the Curzon-Ahlborn efficiency (qCA) at maximum power output to heat the maximum efficiency (qm,, = qCarnot= 1 - r,/T,) for the single and combined endoreversible engine cycles is also nearly 0.5. The ratio is P/r = [l - (zJT”)“*]/(l
- z/r,)
= l/[l + (7JTH)‘i2] 2 0.5.
(30)
If (r,/r,)-, 1, Q = 0.5. The Q versus (TH/TL) relation is shown in Fig. 5; Q rises only slowly above 0.5 for realistic values of (r,/X). 7. The results obtained in this paper provide an upper bound on the power output of combined-cycle chemical engines, such as photovoltaic cells, mass exchangers and electrochemical devices. This analysis does not provide a new technique for synthesizing new high-efficiency solar cells or higher performance fuel cell. Rather the emphasis is on the establishment of fundamental bounds for the performance of a general class of chemical engines. Key sources of irreversibilities are included, but by no means cover all possible sources of irreversibility in real chemical engines. Hence, the results derived in this paper represent bounds rather than accurate representations of the actual performance of chemical engines. The objective of this analysis is similar to that of the finite-time thermodynamic analysis of heat engines. [l-6, 15-181 Depending on the inherent operation mode of the chemical converter, [14] there are two ways to view the physical significance of the optimization procedures presented in this paper. In the first way, chemical engines that can process mass transfer in batch are considered as systems in which a buffer mechanism can be implemented. Mass transfer can be turned on or off during the engine cycle. The optimization with respect to the time spent on the different branches of the cycle is then realistic. For example, the optimal switching time (t,) can be determined by equation (40) in Appendix 1. It is not necessarily mean that the system is amenable to being modified to accommodate the optimal solution given in this paper. However, the optimal solution is established as a bound against which nominally sub-optimal real cycles can be compared.
Maximum
power
of a combined-cycle
chemical
engine
635
II
F-
/
OH 1
Fig. 5. Relationship
2
3
between
efficiency
4
ratio
5
and temperature
6
ratio.
In the second way, the more common chemical converters which operate at constant mass flow rates are considered. Although the times spent on different branches of the cycle cannot be optimized, the distribution areas available for mass transfers are optimized in this anaysis. This is equivalent to the heat-exchanger area distribution optimization for heat transfer in the finite-time thermodynamic heat-engine analysis [ 18- 221. In summary, The procedures and results presented in this paper provide ways and bounds for the optimization performance of chemical engines. The bounds are more reasonable and closer to the real performance of chemical engines than those obtained from the classical reversible limits.
REFERENCES 1. F. L. Curzon and B. Ahlborn.. Amer. J. Phys. 43, 22 (1975). 2. S. Sieniutycz and P. Salamon, Advances in Thermo&namics, Vol. 4. Finite-time Thermodynamics and Thermoeconomics. Taylor and Francis, New York (1990). 3. L. Chen, F. Sun and W. Chen.. Advances in Mechanics 22, 479 (1992). 4. L. Chen, F. Sun and W. Chen.. J. Nature 15, 249 (1992). 5. M. Ondrechen, R. S. Berry and B. Andresen.. J. Chem. Phys. 72, 5118 (1980). 6. M. Ondrechen, B. Andresen and R. S. Berry.. J. Chem. Phys. 73, 5838 (1980). 7. A. DeVos.. Solar Cells 31, 181 (1991). 8. A. DeVos.. J. Phys. Chem. 95, 534 (1991). 9. A. DeVos, P. T. Lansberg, P. Baruch and J. E. Parrott.. J. Appl. Phys. 74, 3631 (1993). 10. A. DeVos, Endoreversible Thermodynamics of Solar Energy Conversion. Oxford University Press, Oxford (1992). 11. A. DeVos.. Solar Energy Mater. Solar Cells 31, 75 (1993). 12. L. Chen, F. Sun and W. Chen.. J. Naval Academy Engng 1, 51 (1993). 13. J. M. Gordon.. J. Appl. Phys. 73, 8 (1993). 14. J. M. Gordon and V. N. Orlov.. J. Appl. Phys. 73, 5303 (1993). 15. M. H. Rubin.. J. Appl. Phys. 53, I (1982). 16. C. Wu.. Energy Convers. Mgmnr 30, 261 (1990). 17. C. Wu, G. Karpouzion and R. L. Kiang.. J. Insf. Energy 65, 41 (1992). 18. L. Chen, F. Sun and W. Chen.. Power Engng 14, 9 (1994). 19. J. B. Woodward.. Trans. ASME J. Energy Res. Technol. 117, 343 (1995) 20. C. Wu, L. Chen and F. Sun.. Energ]> 21, 71 (1996). 21. A. Bejan.. Int. J. Heat Mass Transfer 31, 1211 (1988). 22. A. Bejan.. ht. J. Heat Mass Transfer 38, 433 (1995).
636
Lingen
Chen et al.
APPENDIX SINGLE-CYCLE
ISOTHERMAL
CHEMICAL
ENGINE
A single-cycle isothermal endoreversible chemical engine is shown in Figs Al and A2. The two reservoirs associated with the chemical engine have chemical potentials of pLmaiand pm,“. The rate of mass transfer, which is a function of the chemical potentials of the reservoir and the engine, is denoted by N. In electrochemical and solid-state devices N(p) is referred to as the current-voltage relation, where N is the current, p is the chemical potential of the engine working fluid, and (p 2 pm,“) is the voltage. The condition of maximum chemical potential difference (p = pmar) is the open circuit point, where the mass-transfer rate (current) vanishes. The condition of minimum chemical potential difference (p = pm,“) is the short-circuit point, where the mass-transfer rate (current) is the maximum. An optimization problem for power production exists because it is the product of the mass-transfer rate (current) and exploitable chemical potential (voltage). In the reversible (open-circuit) limit, although full available chemical potential difference can be utilized. it is at the penalty of vanishing rate of mass transfer; therefore, the power delivered by the chemical engine at the open-circuit is zero. In contrast, in the limit of maximum rate of mass transfer (short circuit), the utilizable chemical potential vanishes and hence the power output of the engine is again equal to zero. There is an operating point of maximum power between these two limits. The working fluid of the chemical engine may be a current of electrons for solid-state devices, gas or liquid molecules for mass exchangers in absorption/desorption processes. or chemically reacting species for fuel cells. Assuming the mass transfer is linearly proportional to the difference in chemical potentials, we have NI = Up,,,
>
(Al)
NZ = h&2 - pm&s >
- 0
(A2)
where pI and b are the chemical potentials of the chemical engine fluid on the high- and low-potential mass-transfer branches, t, and t2are the residence times of the two branches, N, and NZ are the mass-transfer rates of the two branches, and h, and h2 are the corresponding proportionality constants for the two mass-transfer branches, respectively. of mass requires Let the total time of the working fluid to complete a cycle be T, then r = t, + t,. Conservation N, = Nl = N
High chemical
potential
(A3)
reservoir
kMAX
Low chemical
potential
reservoir I
Fig. Al.
Single-cycle
endoreversible
chemical
engine.
Maximum
l
(P), efficiency
chemical
engine
637
Constant chemical potential mass transfer process
Fig. A2. The p-N
output
of a combined-cycle
--
kMAX
The power
power
diagram
of single-cycle
(q) and work output
endoreversible
(W) of the chemical
chemical
engine
engine.
are
P = w/s ) v = W/[N(~mnr -
(A4)
~Lmm)l >
(A5)
~=N(P,-P>). Solving
equations
(Al)
(A6)
and (A2) gives PI = ~Lmar- Nl(h,l,)
.
(A7)
w3)
PLY = ~rncn+ N/(&z) Substituting
equations
(A6)+A8)
into equation
(A4) yields
p = %nar - P(m,n- N[ll(h,r,)
+
~lVudlN/~;
(A9)
P is a function of two independent variables (t, and N) for a fixed c value. To find the maximum power output of the chemical engine. taking the derivative of P with respect to I, at constant N and f2 = r-t,. then setting it equal to zero [(~?P/dt,),, = 0] gives (f&t
.
= 7/[1 + (Wz)“‘]
(AlOa)
and (AlOb)
(f&p, = r/U + (WPI Substituting equations (AlO) into equation equal to zero [(CJPjdN), = 0] gives
(A9). taking
Nap, = P,r(p,,, Substituting equations (AlO) and (Al 1) into equations power output of the chemical engine in the following
the derivative -
of P with respect
to N at constant
~nm)l/W + (hlW’l*
(A7t(A9) gives the optimal expressions:
(h/h)“1 , (/&pt = pm + (pmax+ ~(rnmVW + WWI ,
(P&l, = Pmar - (PInax- ~,,,)/2[1
+
5 and setting
it
(Al 1) chemical
potentials
and the maximum
6412) (A13)
and P ma% = h,(pc,a - r~J/2[~
7 ‘1, . + (h,lk)’
(A14)
where Pm.. is the maximum power output of the chemical engine.Substituting equations (Al 1) and (A14) into equation (A5) gives the corresponding engine efficiency qop, = l/2 at the condition of the maximum power output. The maximum efficiency of the engine is qmaX= 1.