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Theoretical analysis of the performance of a regenerative closed Brayton cycle with internal irreversibilities
Enurg>~Conwrs. Mgmr Vol. 38, No. 9, pp. 871-877. 1997 Published by Elsevier Science Ltd Printed in Great Britain
Pergamon
PII: S01%-s904(96)00090-8
0196-8904/97 $17.00 + 0.00
THEORETICAL ANALYSIS OF THE PERFORMANCE OF A REGENERATIVE CLOSED BRAYTON CYCLE WITH INTERNAL IRREVERSIBILITIES LINGEN
CHEN,’ FENGRUI
SUN,’ CHIH WU* and R. L. KIANG3
‘Faculty 306, Naval Academy of Engineering, Wuhan 430033, People’s Republic of China, ‘Mechanical Engineering Department, U.S. Naval Academy, Annapolis, MD 21402, U.S.A. and ‘Naval Surface Weapon Center, Carderock Division, Annapolis, MD 21402, U.S.A.
(Received 21 October
1995)
Abstract-Using the technique of finite-time thermodynamic analysis, the performance of a closed Brayton cycle with regeneration is assessed. Specifically, analytical expressions for the power output and the thermal efficiency, as functions of the pressure ratio and the reservoir temperatures, are derived. The analysis also takes into account all the irreversibilities associated with finite-time heat transfer processes. The paper also shows that the power output can be maximized with judicious selection of parameters such as the heat exchanger surface areas and the heat conductances. Maximum power is obtained when the effectivenesses of the hot- and cold-side heat exchangers are related in a specific manner, and this power output is a strong function of the regenerator effectiveness. Published by Elsevier Science Ltd Brayton
cycle
Irreversible
Finite-time
thermodynamics
INTRODUCTION
The performance of a Brayton cycle using the technique of finite-time thermodynamic analysis has been investigated quite extensively in recent years [l-3]. Various authors would include in their analyses different degrees of realism: endoreversible processes [4-7,9-l 1, 141, irreversible processes [8, 12-141 and finite heat reservoirs (as opposed to infinite reservoirs that can maintain constant temperatures) [9, 13, 141. Both open [6, 71 and closed [4, 5, 8-141 Brayton cycles have been studied. This paper adds regeneration to a closed Brayton cycle and incorporates the combined effects of variable-temperature reservoirs and internal irreversibilities. The irreversible processes incorporated are the non-isentropic compression and expansion and the finite rates of heat transfer in the hot- and cold-side heat exchangers as well as in the regenerator. Despite these complexities, closed form solutions for the power output and the thermodynamic efficiency of the analyzed cycle have been obtained. The availability of these analytical expressions means that performance optimization can be readily realized. An expression for the maximized power has been derived. The resulting equation shows that the maximum power is strongly dependent on the effectiveness of the regenerator. CYCLE
ANALYSIS
A regenerative, irreversible, closed Brayton cycle coupled to variable-temperature heat reservoirs is shown in Fig. 1. In this T-s diagram, the irreversible cycle is 1-2-5-3-4-6-1, and the corresponding endoreversible cycle is l-2s-5-3-4s-6-1. Process 2-5 is an isobaric heat addition process, and 4-6 an isobaric heat rejection process in the regenerator. The working fluid is an ideal gas with a constant thermal capacitance rate CWf(mass flow times the specific heat). The high-temperature (hot-side) heat reservoir is assumed to have a finite thermal capacitance rate G. The inlet and outlet temperatures of the heating fluid are THinand THOU,,respectively. The low-temperature (cold-side) heat reservoir is also assumed to have a finite thermal capacitance 871
872
CHEN et al.:
PERFORMANCE
OF A REGENERATIVE
CLOSED
BRAYTON
CYCLE
rate CL; and the inlet and outlet temperatures of the cooling fluid are TLinand TLaut,respectively. All heat exchangers, including the regenerator, are assumed to be counter-flow heat exchangers with constant heat conductances OH, cL and CR. The heat conductance (a) is defined as the product of the overall heat transfer coefficient (U) and the heat transfer surface area (A) of the heat exchanger. According to the principles of heat transfer and thermodynamics, the rates at which heat is supplied, rejected and regenerated are given by the following expressions: QH =
QH[(THin
-
QH =
CH( TH in -
QH =
Gr(T3
QH =
CHmin(THin
QL =
bJ(T6
QL =
c~(T~out
QL =
Cwr(T4
QL =
CLmin(T6
QR =
G(Ts
-
Tz),
(34
QR =
Gf(Tb
-
T6),
W)
T3) -
(THN~ -
rs)]/ln[(rHi.
-
T~)/(~HwI
-
7’s)],
TH out),
-
(lb)
7’s),
-
UC)
-
Ts)~H,
TL~)
-
-
(T
(14 -
TLJl/ln[(T6
-
TL&/(TI
TLin),
-
TLin)],
(24 (2b)
Td, -
(la)
(24
TLin)cL,
(24
T THin
3
Entropy Fig. 1. T-s diagram of Brayion regenerative cycle.
CHEN et al.:
PERFORMANCE
OF A REGENERATIVE
CLOSED
BRAYTON
CYCLE
873
and QR = CW,(7-4-
(3c)
T2)cR.
where tH and cL are the effectivenesses of the hot-side and cold-side heat exchangers and CRis the effectiveness of the regenerator. The effectivenesses of the heat exchangers are defined as: 6~ = CL =
{I
-
exp[-NH(1
-
C~min/C~max)]}/{l
-
(C,mln/CHmax)eXP[-NH(1
-
- CL.min/CLrnax)]}/{1 - (CL,,,/C~,,,)exp[-NL(l
(1 - exp[-NL(l
CR =
NR/(l
+
CHJCH~~~)]},
- CL~~CL~~~)I},
NR).
(4)
(5) (6)
where CHmlnand CHmaxare the smaller and the larger of the two capacitance rates CH and CWr; and CLminand CLmaxare the smaller and the larger of the two capacitance rates CL and CWf. NH and NL are the numbers of heat transfer units of the hot-side and cold-side heat exchangers. NR is the number of heat transfer units of the regenerator. The heat transfer units are based on the minimum thermal capacitance rates and can be expressed in the following expressions: NH
=
~H/CHmm,
(7)
NL
=
~~ICLrnmt
(8)
NR
=
(9)
a,/&.
The compressor efficiency (Q), the turbine efficiency (qt) and the cycle efficiency (q) are defined as flE= (T1.V - r,)/(K
- TI),
(10)
Q+t = (Tj - T4)/( TJ - z-45)) 1 -
v= For the endoreversible
cycle l-2s-3-4s-1,
(11)
QL/QH.
(12)
the second law of thermodynamics
requires:
T, T, = TJds.
(13)
By denoting the temperature ratio and the pressure ratio of the isentropic compression process as x and 7c, one can write x = Tz,/T, = T,/Tdr = d".
(14)
where m = (k- 1)/k; and k is ratio of the specific heats of the working fluid. Equations (l)-(14) govern the power output and the thermodynamic efficiency of this regenerative closed Brayton cycle. Despite the complexity, a closed form solution can be found. The dimensionless power (P)-and the efficiency (7) are: _ P = ‘I = 1 -
W/(G~TL,,)
(P, - PMP3
=
CLmtnc~(K
-
Y2 +
- P&
Y~)/[CH~,“~H(Y~
-
(15) Yj)],
(16)
where (17) (18) y3 y4 =
=
(~ccwt[l
X3[cRCwT -
6,x,]
YS = PI
Pz =
=
X3[CHmincH(l
{qc(Cw~-
CR) +
x1x2)
P3
=
-
x3(1 -
~RC~F
+
x,(1 X,X&R
-
x1(1
?c[(Cd2
(19)
2cR)&], +
(1
-
(20)
26R)x,}T,
(21)
cR)CLmincL,
X3X4[6R +
-
-
+
(1
-
26R)XS]
2~R)XI]}CHmin6H~,
(22)
qc(Cwf-
(23)
-
CwfexS~Il,
xI~Rx5)}cLmincL,
(24)
874
CHEN et al.:
PERFORMANCE
P4 =
OF A REGENERATIVE
X3X&RCwf
+
Xl = 1 -
x* =
Cwf6l-l
x3
x4
=
X5(1
?I +
+
CLmincL
=
x
Gf
-
1 +
-
X5 = Cwf-
-
CLOSED
BRAYTON
2dX,,
CYCLE
(25)
)I,x-‘,
(26)
-
(27)
CLmin~L~H,
qc,
CLminELy
(29)
CliminEH,
(30)
r = THin/TLin*
(31)
Equations (15) and (16) are the major results of this work. They give, explicitly, the power output and the efficiency of this Brayton cycle as functions of pressure ratios, reservoir temperatures, heat exchanger effectivenesses, compressor and turbine efficiencies, and working fluid thermal capacitance rates.
PERFORMANCE
OPTIMIZATION
Equations (15) and (16) can be used to optimize the performance of a regenerative closed Brayton cycle under a variety of constraints. In one of the simpler scenarios, one may seek the optimal pressure ratio to achieve either maximum power output or maximum efficiency, with all other parameters held constant. Such an exercise can be readily carried out, numerically, using equations (15) and (16). Perhaps the more praCtiCal eXerCiSeS are ones to find the optimal values of OH, gL, gR and Cwf. To find the optimal values of OH, cL and bR, one can approach it in one of two ways. The first is to hold the total heat conductance of the system constant, i.e. bt = flH + crL+ OR= constant, and find the best combination of OH, bL, and CR. This approach has been used in Refs [4, 12-141 for a simple Brayton cycle. The second is to hold the total surface area of all three heat exchangers constant, i.e. A, = AH + AL + AR = constant, and find the most economical combination of AH, AL, and AR. This approach has also been tried for a simple Brayton cycle [5, 8, 10, 111. For the thermal capacitance rate, Cwf, one may optimize C,f/CH for a given CL/C, and pressure ratio. Similar studies for a simple Brayton cycle can be found in Refs [9, 13, 141.These optimizations still need to be done numerically. However, with the explicit solutions of equations (15) and (16), the numerical calculations are greatly simplified. The results should yield optimal values of cH, cL and CRfor either maximized power output or maximized efficiency.
RECONCILIATION
WITH KNOWN
RESULTS
As indicated earlier, different authors have studied different aspects of the Brayton cycle optimization problem. What we have offered in this paper is a unified approach. Our results, thus, take into account the interactions among the controlling parameters. Because of these interactions, some of the conclusions from our study differ from those of previous studies. On the other hand, when known constraints are superimposed on our results, many of the classical results can be recovered. Seven cases are discussed below. 1. Equation (15) shows that the power output of the regenerated cycle is strongly dependent on the effectiveness of the regenerator. Numerical calculations show that the power increases with increasing ERfor a given pressure ratio when that ratio is below a threshold, and the maximum power output is twice the minimum power. Previous studies [15-201, however, indicated that the power output of a regenerated Brayton cycle is independent of the regenerator effectiveness. This significant difference can be attributed to the fact that the internal irreversibility has altered the relationship between the power output and the pressure ratio, such that the recuperator effectiveness comes into play.
CHEN et al.:
PERFORMANCE
OF A REGENERATIVE
CLOSED
BRAYTON
CYCLE
875
2. In the ideal case of infinite heat conductances in the hot and cold heat exchangers, g,, = b,_+m, or equivalently, CH= cL = 1, and in the case of Gf < CH, Cw < CL and CHmin= CLmin= C,r, equation (15) reduces to the classical result [15-201: P = Q(l - x-‘)r - (X - l)/?c.
(32)
The power output is now independent of the regenerator effectiveness, as well as the heat reservoir thermal capacitance rates. 3. When the compression and expansion processes are isentropic, i.e. Q = qt = 1, the power output and the efficiency of the cycle are given by P = (Ps - P6)/(P7- Ps), rj = 1 -
(33) (34)
CLminEL?I/(CHminEH~Z)
where Ps = CH~~~~HZ{C~~ Pg =
[GER
+ CLminEL(l
CLmin6L{[CHmin~H(l p7
Pg=
-
=
CR) +
(CWF)~
-
X4[EHCwrX
-
+
-
X5X7
A’Jb>,
(35)
(36)
Cw,},
(37)
+
(1
-
(38)
26R)&],
(39)
CH m,n~H,
x6 = ERX+ 1 - 2&H
(40)
2cR
(41)
x7
=
tRX-’
+
1 -
-‘z
-
C&(1 - ERX)+
CR)CH~~~~HX
‘12= [Cwf(l -
~Rcrvf]~
LR)]x-’
cw,~Hx,x-‘,
XS = C,r -
VI = (1 -
-
tRX_‘)
-
X4x6]~
-
(1
-
x5x7,
(42)
cR)CLminELX.
(43)
As can be seen, the expressions for the power and the efficiency are different from those of the classical results that assume isentropic processes. Even if we incorporate infinite heat conductances, the classical results are not recovered. The regenerator effectiveness, in either case, strongly affects the power output. 4. When the cycle is coupled to constant temperature heat reservoirs, i.e. CH = CL+ co, one has NH = aH/Cwr, NL = rrL/Cwl,CH= 1 - exp( - NH) and LL = 1 - exp( - NL), and equations (15) and (16) then become P = (Ps - PIO)/PII,
(44)
? = 1-
(45)
tLff3/(cHu4),
where PS = PlO
=
6Hz{fjc(l
~L{x,[x,
PI, = qc[l -
~3 =
-
x1(1
-
t_!4 =
qct[l
+
cR(1
ER)CH~,T -
6,x,]
x,(1
-
-
x,x,) -
6,)x,]
qc[l -
-
x3(1
2cR)(1 -
x3(1
x8 =
6H +
-
-
x3(1
(1 -
-
cL)[cH
CH)] -
-
&[I
EL&H
EL -
+
+
(1
(1
-
-
XIcR(l -
2cH)XI]},
(46) CH)]},
(47)
2&)X,],
(48)
-
eH)(l
-
(49)
6,&L,
CH)CRXI] cL)[cR
+
+ (1
-
X~[CR
+
2cH)x,]
x1(1
-
-
x3(1
ZER)(~
-
-
ER)cL.
CH)],
(50) (51)
As such, the power output is still dependent on the regenerator effectiveness. If 6H = .cL = 1 is imposed in addition, equation (44) is then reduced to equation (32) the classical result, because now Cwf< CH and C,r < CL hold. 5. When cR = 0, the regenerated cycle becomes a simple cycle, and equations (15) and (16) become: P = q=
(PI2
-
P13)lP14,
CLmin~L~S/(CH
min~H?6),
(52) (53)
876
CHEN et al.:
PERFORMANCE
OF A REGENERATIVE
CLOSED BRAYTON CYCLE
where (54) (55) (56)
Equation (52) indicates that the power output of the simple Brayton cycle is dependent on the thermal capacitance rates of the reservoirs even if CH= cL = 1. If, in addition, C,r < CH and Gf < CL, the power then becomes independent of the thermal capacitance rates of the reservoirs. This fact is again obscured in the conventional power cycle analysis. 6. When & = 0 and Q = Q = 1, the cycle becomes an endoreversible closed simple Brayton cycle. Equations (15) and (16) become P = x~[(l - X-‘)r - X +
l]/[Cwf(CHmin~H
+
CLminEL)
-
x9],
?j = 1 - (x-‘7 - l)/(z - x),
(59) (60)
where x9 =
CHminCLmin~H~L.
(61)
From equations (60) and (61), one obtains a relation between the power output and the efficiency of the cycle: P = x~[r - l/(1 -
q)]fl/[Cwf(CHminf%
and if one substitutes the Curzon-Ahlborn
+
CLmincL)
-
x9],
(62)
efficiency, rj = 1 - r-112
(63)
the maximum power of the cycle can be obtained. From the variety of cases discussed in [l-6], the optimal performances of the cycle can all be obtained by sizing the regenerator, the hot and cold heat exchangers, or by choosing proper matching between the working fluid and the reservoirs. CONCLUSIONS By presenting a unified finite-time thermodynamic analysis of a regenerative closed Brayton cycle, this study shows that, in general, the power output is a function of the heat exchanger effectivenesses. Explicit expressions for power and efficiency have been derived that should allow ready optimization of either of these two performance parameters. REFERENCES Advances in Thermodynamics. Volume 4, Finite-time thermodynamics and thermoeconomics. Taylor & Francis, New York, 1990, pp. 1-142. 2. Chen, L., Sun, F. and Chen, W., New developments of finite-time thermodynamics. Nature Journal (in Chinese), 1992, 15, 249-253. 3. Chen, L., Sun, F. and Chen, W., The present state and trend of finite-time thermodynamics for energy systems. Advances in Mechanics (in Chinese), 1992, 22, 479-488. 4. Bejan, A., Theory of heat transfer-irreversible power plant. International Journal of Heat Mass Transfer, 1988, 31,
1.Sieniutycz, S. and Salamon, P., eds,
121 l-1219. 5. Wu, C., Specific power bound of a finite-time closed Brayton cycle. International Journal of Ambienf Energy,
1990,
11, 77-82. 6. Wu, C., Work and power optimization of a finite-time Brayton cycle. International Journal of Ambient Energy, 1990, 11, 129-136. 7. Wu, C., Power optimization of an endoreversible Brayton gas turbine heat engine. Energy Conversion and Management, 1991, 31, 561-565. 8. Wu, C. and Kiang, R. L., Performance of a nonisentropic 1991, 113, 501-503.
Brayton cycle. Trans. ASME J. Engng. Gas Turbine Power,
CHEN et al.:
PERFORMANCE
OF A REGENERATIVE
CLOSED
BRAYTON
CYCLE
871
9. Ibrahim, 0. M., Klein, S. A. and Mitchell, J. W., Optimum heat power cycles for specified boundary. Trans. ASME J. Engng. Gus Turbine Power, 1994, 113, 514-521. 10. Chen, L., Sun, F. and Chen, W., Finite-time thermodynamic analysis for gas turbine closed cycles. Gus Turbine Tech. (in Chinese), 1994, 7, 34-39. 1I. Chen, L., Sun, F. and Chen, W., The maximum power output of gas turbine closed cycles. Ship Engineering (in Chinese), 1993, 00, 40-41. 12. Chen, L., Sun, F. and Chen, W., The optimal power output versus efficiency characteristics of an irreversible gas turbine closed cycle. Journal of Naval Academy of Engineering (in Chinese), 1994, 00, 9-I 3. 13. Chen, L., Sun, F. and Chen, W., Finite-time thermodynamic optimization for irreversible closed Brayton power cycles. Power System Engineering (in Chinese), 1994, 10, 2431. 14. Chen, L., Sun, F. and Chen, W., Thermodynamic optimization of a closed gas turbine cycle powered by solar energy. ACTA Energiae Solaris Sinica (in Chinese), 1994, 15, 274-278. 15. Woods, W. A., Bevan, P. J. and Bevan, D. I., Output and efficiency of the closed cycle gas turbine. Proc. Inst. Mech. Engrg, 1991, 205, 59-66. 16. Frost, T. H., Agnew, B. and Anderson, A., Optimization for Brayton-Joule gas turbine cycle. Proc. Inst. Mech. Engrg., 1992, 206, 283-288. 17. Gandhidasan, P., Thermodynamic analysis of a closed-cycle solar gas turbine plant. Energy Conversion and Management, 1993, 34, 657-661. 18. Tsujikawa, Y. and Nagaoka, M., Determination of cycle configuration of gas turbines and aircraft engines by an optimization procedure. Trans. ASME J. Engrg. Gus Turbine Power, 1991, 113, 100-105. 19. T. Korakianjtis, Marine gas turbines: cycle parameter choices and arrangement effects on performance. Trans. IMarE, 1992, 104, 187-201. 20. Cole, G. H. A., Thermal Power Cycles. Edward Arnold Publ. Inc., London, 1991.
Pergamon
PII: S01%-s904(96)00090-8
0196-8904/97 $17.00 + 0.00
THEORETICAL ANALYSIS OF THE PERFORMANCE OF A REGENERATIVE CLOSED BRAYTON CYCLE WITH INTERNAL IRREVERSIBILITIES LINGEN
CHEN,’ FENGRUI
SUN,’ CHIH WU* and R. L. KIANG3
‘Faculty 306, Naval Academy of Engineering, Wuhan 430033, People’s Republic of China, ‘Mechanical Engineering Department, U.S. Naval Academy, Annapolis, MD 21402, U.S.A. and ‘Naval Surface Weapon Center, Carderock Division, Annapolis, MD 21402, U.S.A.
(Received 21 October
1995)
Abstract-Using the technique of finite-time thermodynamic analysis, the performance of a closed Brayton cycle with regeneration is assessed. Specifically, analytical expressions for the power output and the thermal efficiency, as functions of the pressure ratio and the reservoir temperatures, are derived. The analysis also takes into account all the irreversibilities associated with finite-time heat transfer processes. The paper also shows that the power output can be maximized with judicious selection of parameters such as the heat exchanger surface areas and the heat conductances. Maximum power is obtained when the effectivenesses of the hot- and cold-side heat exchangers are related in a specific manner, and this power output is a strong function of the regenerator effectiveness. Published by Elsevier Science Ltd Brayton
cycle
Irreversible
Finite-time
thermodynamics
INTRODUCTION
The performance of a Brayton cycle using the technique of finite-time thermodynamic analysis has been investigated quite extensively in recent years [l-3]. Various authors would include in their analyses different degrees of realism: endoreversible processes [4-7,9-l 1, 141, irreversible processes [8, 12-141 and finite heat reservoirs (as opposed to infinite reservoirs that can maintain constant temperatures) [9, 13, 141. Both open [6, 71 and closed [4, 5, 8-141 Brayton cycles have been studied. This paper adds regeneration to a closed Brayton cycle and incorporates the combined effects of variable-temperature reservoirs and internal irreversibilities. The irreversible processes incorporated are the non-isentropic compression and expansion and the finite rates of heat transfer in the hot- and cold-side heat exchangers as well as in the regenerator. Despite these complexities, closed form solutions for the power output and the thermodynamic efficiency of the analyzed cycle have been obtained. The availability of these analytical expressions means that performance optimization can be readily realized. An expression for the maximized power has been derived. The resulting equation shows that the maximum power is strongly dependent on the effectiveness of the regenerator. CYCLE
ANALYSIS
A regenerative, irreversible, closed Brayton cycle coupled to variable-temperature heat reservoirs is shown in Fig. 1. In this T-s diagram, the irreversible cycle is 1-2-5-3-4-6-1, and the corresponding endoreversible cycle is l-2s-5-3-4s-6-1. Process 2-5 is an isobaric heat addition process, and 4-6 an isobaric heat rejection process in the regenerator. The working fluid is an ideal gas with a constant thermal capacitance rate CWf(mass flow times the specific heat). The high-temperature (hot-side) heat reservoir is assumed to have a finite thermal capacitance rate G. The inlet and outlet temperatures of the heating fluid are THinand THOU,,respectively. The low-temperature (cold-side) heat reservoir is also assumed to have a finite thermal capacitance 871
872
CHEN et al.:
PERFORMANCE
OF A REGENERATIVE
CLOSED
BRAYTON
CYCLE
rate CL; and the inlet and outlet temperatures of the cooling fluid are TLinand TLaut,respectively. All heat exchangers, including the regenerator, are assumed to be counter-flow heat exchangers with constant heat conductances OH, cL and CR. The heat conductance (a) is defined as the product of the overall heat transfer coefficient (U) and the heat transfer surface area (A) of the heat exchanger. According to the principles of heat transfer and thermodynamics, the rates at which heat is supplied, rejected and regenerated are given by the following expressions: QH =
QH[(THin
-
QH =
CH( TH in -
QH =
Gr(T3
QH =
CHmin(THin
QL =
bJ(T6
QL =
c~(T~out
QL =
Cwr(T4
QL =
CLmin(T6
QR =
G(Ts
-
Tz),
(34
QR =
Gf(Tb
-
T6),
W)
T3) -
(THN~ -
rs)]/ln[(rHi.
-
T~)/(~HwI
-
7’s)],
TH out),
-
(lb)
7’s),
-
UC)
-
Ts)~H,
TL~)
-
-
(T
(14 -
TLJl/ln[(T6
-
TL&/(TI
TLin),
-
TLin)],
(24 (2b)
Td, -
(la)
(24
TLin)cL,
(24
T THin
3
Entropy Fig. 1. T-s diagram of Brayion regenerative cycle.
CHEN et al.:
PERFORMANCE
OF A REGENERATIVE
CLOSED
BRAYTON
CYCLE
873
and QR = CW,(7-4-
(3c)
T2)cR.
where tH and cL are the effectivenesses of the hot-side and cold-side heat exchangers and CRis the effectiveness of the regenerator. The effectivenesses of the heat exchangers are defined as: 6~ = CL =
{I
-
exp[-NH(1
-
C~min/C~max)]}/{l
-
(C,mln/CHmax)eXP[-NH(1
-
- CL.min/CLrnax)]}/{1 - (CL,,,/C~,,,)exp[-NL(l
(1 - exp[-NL(l
CR =
NR/(l
+
CHJCH~~~)]},
- CL~~CL~~~)I},
NR).
(4)
(5) (6)
where CHmlnand CHmaxare the smaller and the larger of the two capacitance rates CH and CWr; and CLminand CLmaxare the smaller and the larger of the two capacitance rates CL and CWf. NH and NL are the numbers of heat transfer units of the hot-side and cold-side heat exchangers. NR is the number of heat transfer units of the regenerator. The heat transfer units are based on the minimum thermal capacitance rates and can be expressed in the following expressions: NH
=
~H/CHmm,
(7)
NL
=
~~ICLrnmt
(8)
NR
=
(9)
a,/&.
The compressor efficiency (Q), the turbine efficiency (qt) and the cycle efficiency (q) are defined as flE= (T1.V - r,)/(K
- TI),
(10)
Q+t = (Tj - T4)/( TJ - z-45)) 1 -
v= For the endoreversible
cycle l-2s-3-4s-1,
(11)
QL/QH.
(12)
the second law of thermodynamics
requires:
T, T, = TJds.
(13)
By denoting the temperature ratio and the pressure ratio of the isentropic compression process as x and 7c, one can write x = Tz,/T, = T,/Tdr = d".
(14)
where m = (k- 1)/k; and k is ratio of the specific heats of the working fluid. Equations (l)-(14) govern the power output and the thermodynamic efficiency of this regenerative closed Brayton cycle. Despite the complexity, a closed form solution can be found. The dimensionless power (P)-and the efficiency (7) are: _ P = ‘I = 1 -
W/(G~TL,,)
(P, - PMP3
=
CLmtnc~(K
-
Y2 +
- P&
Y~)/[CH~,“~H(Y~
-
(15) Yj)],
(16)
where (17) (18) y3 y4 =
=
(~ccwt[l
X3[cRCwT -
6,x,]
YS = PI
Pz =
=
X3[CHmincH(l
{qc(Cw~-
CR) +
x1x2)
P3
=
-
x3(1 -
~RC~F
+
x,(1 X,X&R
-
x1(1
?c[(Cd2
(19)
2cR)&], +
(1
-
(20)
26R)x,}T,
(21)
cR)CLmincL,
X3X4[6R +
-
-
+
(1
-
26R)XS]
2~R)XI]}CHmin6H~,
(22)
qc(Cwf-
(23)
-
CwfexS~Il,
xI~Rx5)}cLmincL,
(24)
874
CHEN et al.:
PERFORMANCE
P4 =
OF A REGENERATIVE
X3X&RCwf
+
Xl = 1 -
x* =
Cwf6l-l
x3
x4
=
X5(1
?I +
+
CLmincL
=
x
Gf
-
1 +
-
X5 = Cwf-
-
CLOSED
BRAYTON
2dX,,
CYCLE
(25)
)I,x-‘,
(26)
-
(27)
CLmin~L~H,
qc,
CLminELy
(29)
CliminEH,
(30)
r = THin/TLin*
(31)
Equations (15) and (16) are the major results of this work. They give, explicitly, the power output and the efficiency of this Brayton cycle as functions of pressure ratios, reservoir temperatures, heat exchanger effectivenesses, compressor and turbine efficiencies, and working fluid thermal capacitance rates.
PERFORMANCE
OPTIMIZATION
Equations (15) and (16) can be used to optimize the performance of a regenerative closed Brayton cycle under a variety of constraints. In one of the simpler scenarios, one may seek the optimal pressure ratio to achieve either maximum power output or maximum efficiency, with all other parameters held constant. Such an exercise can be readily carried out, numerically, using equations (15) and (16). Perhaps the more praCtiCal eXerCiSeS are ones to find the optimal values of OH, gL, gR and Cwf. To find the optimal values of OH, cL and bR, one can approach it in one of two ways. The first is to hold the total heat conductance of the system constant, i.e. bt = flH + crL+ OR= constant, and find the best combination of OH, bL, and CR. This approach has been used in Refs [4, 12-141 for a simple Brayton cycle. The second is to hold the total surface area of all three heat exchangers constant, i.e. A, = AH + AL + AR = constant, and find the most economical combination of AH, AL, and AR. This approach has also been tried for a simple Brayton cycle [5, 8, 10, 111. For the thermal capacitance rate, Cwf, one may optimize C,f/CH for a given CL/C, and pressure ratio. Similar studies for a simple Brayton cycle can be found in Refs [9, 13, 141.These optimizations still need to be done numerically. However, with the explicit solutions of equations (15) and (16), the numerical calculations are greatly simplified. The results should yield optimal values of cH, cL and CRfor either maximized power output or maximized efficiency.
RECONCILIATION
WITH KNOWN
RESULTS
As indicated earlier, different authors have studied different aspects of the Brayton cycle optimization problem. What we have offered in this paper is a unified approach. Our results, thus, take into account the interactions among the controlling parameters. Because of these interactions, some of the conclusions from our study differ from those of previous studies. On the other hand, when known constraints are superimposed on our results, many of the classical results can be recovered. Seven cases are discussed below. 1. Equation (15) shows that the power output of the regenerated cycle is strongly dependent on the effectiveness of the regenerator. Numerical calculations show that the power increases with increasing ERfor a given pressure ratio when that ratio is below a threshold, and the maximum power output is twice the minimum power. Previous studies [15-201, however, indicated that the power output of a regenerated Brayton cycle is independent of the regenerator effectiveness. This significant difference can be attributed to the fact that the internal irreversibility has altered the relationship between the power output and the pressure ratio, such that the recuperator effectiveness comes into play.
CHEN et al.:
PERFORMANCE
OF A REGENERATIVE
CLOSED
BRAYTON
CYCLE
875
2. In the ideal case of infinite heat conductances in the hot and cold heat exchangers, g,, = b,_+m, or equivalently, CH= cL = 1, and in the case of Gf < CH, Cw < CL and CHmin= CLmin= C,r, equation (15) reduces to the classical result [15-201: P = Q(l - x-‘)r - (X - l)/?c.
(32)
The power output is now independent of the regenerator effectiveness, as well as the heat reservoir thermal capacitance rates. 3. When the compression and expansion processes are isentropic, i.e. Q = qt = 1, the power output and the efficiency of the cycle are given by P = (Ps - P6)/(P7- Ps), rj = 1 -
(33) (34)
CLminEL?I/(CHminEH~Z)
where Ps = CH~~~~HZ{C~~ Pg =
[GER
+ CLminEL(l
CLmin6L{[CHmin~H(l p7
Pg=
-
=
CR) +
(CWF)~
-
X4[EHCwrX
-
+
-
X5X7
A’Jb>,
(35)
(36)
Cw,},
(37)
+
(1
-
(38)
26R)&],
(39)
CH m,n~H,
x6 = ERX+ 1 - 2&H
(40)
2cR
(41)
x7
=
tRX-’
+
1 -
-‘z
-
C&(1 - ERX)+
CR)CH~~~~HX
‘12= [Cwf(l -
~Rcrvf]~
LR)]x-’
cw,~Hx,x-‘,
XS = C,r -
VI = (1 -
-
tRX_‘)
-
X4x6]~
-
(1
-
x5x7,
(42)
cR)CLminELX.
(43)
As can be seen, the expressions for the power and the efficiency are different from those of the classical results that assume isentropic processes. Even if we incorporate infinite heat conductances, the classical results are not recovered. The regenerator effectiveness, in either case, strongly affects the power output. 4. When the cycle is coupled to constant temperature heat reservoirs, i.e. CH = CL+ co, one has NH = aH/Cwr, NL = rrL/Cwl,CH= 1 - exp( - NH) and LL = 1 - exp( - NL), and equations (15) and (16) then become P = (Ps - PIO)/PII,
(44)
? = 1-
(45)
tLff3/(cHu4),
where PS = PlO
=
6Hz{fjc(l
~L{x,[x,
PI, = qc[l -
~3 =
-
x1(1
-
t_!4 =
qct[l
+
cR(1
ER)CH~,T -
6,x,]
x,(1
-
-
x,x,) -
6,)x,]
qc[l -
-
x3(1
2cR)(1 -
x3(1
x8 =
6H +
-
-
x3(1
(1 -
-
cL)[cH
CH)] -
-
&[I
EL&H
EL -
+
+
(1
(1
-
-
XIcR(l -
2cH)XI]},
(46) CH)]},
(47)
2&)X,],
(48)
-
eH)(l
-
(49)
6,&L,
CH)CRXI] cL)[cR
+
+ (1
-
X~[CR
+
2cH)x,]
x1(1
-
-
x3(1
ZER)(~
-
-
ER)cL.
CH)],
(50) (51)
As such, the power output is still dependent on the regenerator effectiveness. If 6H = .cL = 1 is imposed in addition, equation (44) is then reduced to equation (32) the classical result, because now Cwf< CH and C,r < CL hold. 5. When cR = 0, the regenerated cycle becomes a simple cycle, and equations (15) and (16) become: P = q=
(PI2
-
P13)lP14,
CLmin~L~S/(CH
min~H?6),
(52) (53)
876
CHEN et al.:
PERFORMANCE
OF A REGENERATIVE
CLOSED BRAYTON CYCLE
where (54) (55) (56)
Equation (52) indicates that the power output of the simple Brayton cycle is dependent on the thermal capacitance rates of the reservoirs even if CH= cL = 1. If, in addition, C,r < CH and Gf < CL, the power then becomes independent of the thermal capacitance rates of the reservoirs. This fact is again obscured in the conventional power cycle analysis. 6. When & = 0 and Q = Q = 1, the cycle becomes an endoreversible closed simple Brayton cycle. Equations (15) and (16) become P = x~[(l - X-‘)r - X +
l]/[Cwf(CHmin~H
+
CLminEL)
-
x9],
?j = 1 - (x-‘7 - l)/(z - x),
(59) (60)
where x9 =
CHminCLmin~H~L.
(61)
From equations (60) and (61), one obtains a relation between the power output and the efficiency of the cycle: P = x~[r - l/(1 -
q)]fl/[Cwf(CHminf%
and if one substitutes the Curzon-Ahlborn
+
CLmincL)
-
x9],
(62)
efficiency, rj = 1 - r-112
(63)
the maximum power of the cycle can be obtained. From the variety of cases discussed in [l-6], the optimal performances of the cycle can all be obtained by sizing the regenerator, the hot and cold heat exchangers, or by choosing proper matching between the working fluid and the reservoirs. CONCLUSIONS By presenting a unified finite-time thermodynamic analysis of a regenerative closed Brayton cycle, this study shows that, in general, the power output is a function of the heat exchanger effectivenesses. Explicit expressions for power and efficiency have been derived that should allow ready optimization of either of these two performance parameters. REFERENCES Advances in Thermodynamics. Volume 4, Finite-time thermodynamics and thermoeconomics. Taylor & Francis, New York, 1990, pp. 1-142. 2. Chen, L., Sun, F. and Chen, W., New developments of finite-time thermodynamics. Nature Journal (in Chinese), 1992, 15, 249-253. 3. Chen, L., Sun, F. and Chen, W., The present state and trend of finite-time thermodynamics for energy systems. Advances in Mechanics (in Chinese), 1992, 22, 479-488. 4. Bejan, A., Theory of heat transfer-irreversible power plant. International Journal of Heat Mass Transfer, 1988, 31,
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CHEN et al.:
PERFORMANCE
OF A REGENERATIVE
CLOSED
BRAYTON
CYCLE
871
9. Ibrahim, 0. M., Klein, S. A. and Mitchell, J. W., Optimum heat power cycles for specified boundary. Trans. ASME J. Engng. Gus Turbine Power, 1994, 113, 514-521. 10. Chen, L., Sun, F. and Chen, W., Finite-time thermodynamic analysis for gas turbine closed cycles. Gus Turbine Tech. (in Chinese), 1994, 7, 34-39. 1I. Chen, L., Sun, F. and Chen, W., The maximum power output of gas turbine closed cycles. Ship Engineering (in Chinese), 1993, 00, 40-41. 12. Chen, L., Sun, F. and Chen, W., The optimal power output versus efficiency characteristics of an irreversible gas turbine closed cycle. Journal of Naval Academy of Engineering (in Chinese), 1994, 00, 9-I 3. 13. Chen, L., Sun, F. and Chen, W., Finite-time thermodynamic optimization for irreversible closed Brayton power cycles. Power System Engineering (in Chinese), 1994, 10, 2431. 14. Chen, L., Sun, F. and Chen, W., Thermodynamic optimization of a closed gas turbine cycle powered by solar energy. ACTA Energiae Solaris Sinica (in Chinese), 1994, 15, 274-278. 15. Woods, W. A., Bevan, P. J. and Bevan, D. I., Output and efficiency of the closed cycle gas turbine. Proc. Inst. Mech. Engrg, 1991, 205, 59-66. 16. Frost, T. H., Agnew, B. and Anderson, A., Optimization for Brayton-Joule gas turbine cycle. Proc. Inst. Mech. Engrg., 1992, 206, 283-288. 17. Gandhidasan, P., Thermodynamic analysis of a closed-cycle solar gas turbine plant. Energy Conversion and Management, 1993, 34, 657-661. 18. Tsujikawa, Y. and Nagaoka, M., Determination of cycle configuration of gas turbines and aircraft engines by an optimization procedure. Trans. ASME J. Engrg. Gus Turbine Power, 1991, 113, 100-105. 19. T. Korakianjtis, Marine gas turbines: cycle parameter choices and arrangement effects on performance. Trans. IMarE, 1992, 104, 187-201. 20. Cole, G. H. A., Thermal Power Cycles. Edward Arnold Publ. Inc., London, 1991.