Performance of a regenerative Brayton heat engine

~

Pergamon

0360-5442(95)00097-6

Energy Vol. 21, No. 2, pp. 71-76. 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0360-5442196 $15.~) + 0.00

PERFORMANCE OF A REGENERATIVE BRAYTON HEAT ENGINE CHIH WU,t

LINGENCHEN,:I:andFENGRUISUN:[:

"tDepartment of Mechanical Engineering, U.S. Naval Academy, Annapolis, MD 21402, U.S.A. and SFaculty 306, Naval Academy of Engineering, Wuhan 430033, People's Republic of China

(Received 30 June 1995)

Abstract--The performance of an endoreversible regenerative Brayton heat engine has been studied. The analysis is focused on minimizing irreversibilities associated with the hot and cold heat exchangers and regenerator.

INTRODUCTION

Efficiency is not the only criterion in the design of a real engine. Maximum power is also important. Curzon and Ahlborn t developed a theoretical model of a Carnot heat engine, which accounted for irreversibilities in the heat-exchange processes between the power cycle and its heat source at TH and heat sink at Tt.. The power was maximized with respect to temperature differences in both the hot and the cold heat exchangers. They t found that the thermal efficiency at maximum power output was T/C-A = I-(TJTH)

t/2 •

(I)

The analysis of heat engines under maximum power output has been applied to the Brayton cycle. Bejan 2 considered an ideal Brayton cycle operating between an infinite heat source and infinite heat sink. Leff3 has shown that the imposition of maximum work adds further constraints on the Braytoncycle temperatures. Ibrahim et al 4 optimized the output power of a closed ideal Brayton cycle for several specified boundary conditions. Wu and Kiang 5 considered the real Brayton cycle efficiency with maximum power output for nonisentropic compression and expansion processes. None of the previous investigations included a regenerator. It is the purpose of this paper to incorporate regeneration into the output/power analysis with finite time thermodynamics. REGENERATIVE BRAYTON CYCLE ANALYSIS

A regenerative Brayton cycle operating between an infinite heat source and an infinite heat sink is shown in Fig. 1. In this T-s diagram, the gas enters the compressor at state 1. At state 2, the cold gas leaving the nonisentropic compressor enters the regenerator, where it is heated to Ts. In an ideal regenerator, the gas will leave the regenerator at the temperature of the hot turbine exhaust (T4), i.e. T5 = T4. The primary heat-addition process takes place between states 5 and 3. The gas enters the gas turbine at state 3 and expands nonisentropically to state 4 or isentropically to state 4s. The hot gas exits the nonisentropic gas turbine at state 4 and enters the regenerator, where it is cooled to state 6 at constant pressure. The cycle is completed by cooling the gas to the initial state. The process l-2s is an isentropic compression and 1-2 takes into account the nonisentropic nature of a real compressor. Process 3-4s is an isentropic expansion and 3--4 a nonisentropic expansion in a real turbine. We consider the nonideal Brayton cycle 1 - 2 - 5 - 3 - 4 - 6 - 1 with its surrounding heat reservoirs. Assuming counter-flow heat exchangers, the governing equations are:

Qn = Uw4n[ ( TH-Ts)-( TH-T O ]/ { ln[ ( TrrTs) / ( TH-T O ] } ,

(2a)

Q . = Cw(T3-Ts),

(2b)

QH = C,,,(TH-Ts)eH,

(2c) 71

72

Chih Wu et al

Heat Soutce

TH

3

,= 2 S / j ~

E

ZS

1 HeatSink

TL

Entropy

S

Fig. I. Temperature-entropydiagram of a regenerativeBrayton cycle.

QL = ULAL[ ( T6-TL)-( TI-TL)/ { In[ ( T6-TL)/ ( T,-TL) ] } ,

(3a)

QL = Cw( T6-TI ),

(3b)

QL = Cw(T6-TL)eL,

(3C)

QR = C , ( T s - T 2 ) ,

(4a)

QR = Cw( Ta-T6) ,

(4b)

QR = C~(T4-T2)eR,

(4c)

tic = (T2~-TI)/(T2-Tj ),

(5)

Th = ( T s - T 4 ) [ ( T a [ T 4 s ) ,

(6)

P = QH-QL,

(7)

rl = 1-QLIQH,

(8)

where QH is heat transferred from the heat source at TH to the cycle, QL heat transferred from the cycle to the heat sink at TL, QR heat transfer from the hot gas leaving the turbine to the cold gas leaving the compressor, UH and UL are, respectively, the overall heat-transfer coefficients in the hot and cold heat exchangers, AH and A L are the corresponding heat-transfer surface areas of the hot and cold heat exchangers, and Cw is the heat capacity rate of the working fluid. Also, ~H, eL and eR are, respectively, the effectiveness of the hot heat exchanger, cold heat exchanger, and regenerator, r/~ and r/t are the corresponding efficiencies of the compressor and turbine, P is the power output of the cycle, and the thermal cycle efficiency. It follows that eH = 1 -- exp(-NTUH),

(9)

eL = 1 -- exp(-NTUL),

(10)

eR = NTUR/(i+NTUR) .

(ll)

where NTUH = Uw4HICw, NTUL = ULAL/Cw,NTUR = A R U R / C w, and NTUH, NTUL and NTUR are the numbers of dimensionless transfer units in the hot, cold and regenerating heat exchangers, respectively.

Regenerative Brayton heat engine

73

Rearranging Eqs. (2-6),

Since 1-2s-3-4s-1 requires that

T3 = E.T. + ( I - E . ) T 5 ,

(12)

T~ = ELYL + (I--EL)T6 ,

(13)

T5 = ERT4 + (1-ER)T2,

(14)

T6 = eRT2 + (I--ER)T4 ,

(J5)

T2., = r/~T2 + (1-r~c)T,,

(16)

T4, = [l - ('Or)-' ]T3 + (rh)-'T4 •

(17)

is an internally reversible Brayton cycle, the second law of thermodynamics

T, T3 = T2~T4~

(18)

and the temperature ratio (X) across the compressor is T 2 f l T , = T31T4~ = X = (p21pl ) ~k-' ~/k = rr ~k-I ~k ,

(19)

where X = (p2]pl) ~k-I)/k, k is the ratio of the specific heats of the working fluid, and "rr = P2/P~ is the pressure ratio across the compressor. Combining Eqs. (7-19) yields T4 = Zi [EHTH + ( 1--ell)( 1--eR)T2]/[ 1 -- ( I--EH(eRZt ] , Tj = Z 2 / [

1 - ( 1--eH)ERZI ] ,

(20) (21)

T 2 = Z3/M 2 ,

(22)

,

(23)

T4 = Z 4 / M , T s = Z s/M,,

(24)

T6 = Z6/M 1 ,

(25)

where Zi = ( l-rh+rhX -I ),

(26)

Z 2 = (l--EL) ( I--ER)Z~EHTH+ [ 1--( I--EH)ERZI ]ELTL + ( 1--EL)[( 1-Er0Zl( I--2ER)+ER]T2 ,

(27)

Z3 = ZT{ ( I--EL)Z,( 1--ER)EHTH + [ l--( 1--EH)ERZI ]ELTL} ,

(28)

Z4 = Z. {['0c-ZT( I--EL)ERIEHTH+ ( l--e.]( 1--eR)ZTeLTL},

(29)

Z5 = Zj [ r/ctR+ZT( 1--e L)( 1--2 ER) ] eRTu + Z7( 1--eR)eLTL ,

(30)

Z6 = rb( 1--eR)ZIeHTH + Z7[ZI( 1-E.)( 1--2ER)+ER]ELTL,

(31)

Z7 = ( X - l + ' o c ) ,

(32)

and Ml = 'Oc[1-(

1 - - E H ) E R Z I ] - Z a ( 1 - - E L ) [ ( i - - E H ) Z I ( I--2EI~.)+ER] .

(33)

Substituting Eqs. (24) and (25) into Eqs. (2) and (3), we obtain the dimensionless power output P* = P/[Cw(TL-T,

er)] = Z 8 / M ~

(34)

74

Chih Wu et al p~ //

0.6

I~'''-----------

05

04

g. ~

o.2

i:5 z

o 1

3

5

?

9

11

'T~

Compressor Compression Ratio

Fig. 2. P* vs ~" for the regenerative Brayton cycle.

and the efficiency

"I/= I-(~L/eH)(ZgM2),

(35)

where T~f is a reference temperature (we set Tref = 0 for convenience), Z8 = { r/d l--Z~M3]--ZTZ~(1--2~R)+~R}TEH - {ZT[M4+ZjMs(1-2ER)]-r/d l-MsERZ,]}¢L,

(36)

Z9 = r/c( 1--eR)ZIEHT --r/c[ I-MsERZt ]+ZT[(ZIMs( 1--2ER)+ER],

(37)

M2 = { r/c[ 1-Z~ ¢R]-ZT( 1--EL)[Z~( 1--2ER)+Ea ] } T-ZT( 1--ER)¢L,

(38)

(39)

T = TH/TL,

M3 = eR'I-EL--eReL ,

(40)

M4 = eHq'ER--IEHER,

(41)

M5 = l - e l l .

(42)

The relationships among the nondimensional power (P*), cycle efficiency (7/) and compression ratio (Tr) are plotted in Fig. 2 (P* vs 7r), Fig. 3 ( r / v s rr), and Fig. 4 (P* vs r/). The plots refer to T=4, EH = 0.9, EL = 0.8, r/c = 0.8, r/t = 0.9, and k = 1.4. The other parameters used in these figures are ER = 0.6, 0.9 and 1, X = 1-1.95 (corresponding to 1r = 1-10.4). p* O.7

O.6

O.S

1.0 ~" o.4

kU o.3

o~ ~2 oJ

0.6 eR -

o

Compressor Compression Ratio

Fig. 3. ~ vs lr for the regenerative Brayton cycle.

Regenerative Brayton heat engine

75

p*

05

04

03

Z

ol

02

03

04

05

06

0.7

Cycle Efficiency

rl

Fig. 4. P* vs r/for the regenerative Brayton cycle.

DISCUSSION (i) It may be seen from Fig. 2 that P* is a strong function of 7r and increases with eR at the same 7r. Specifically, the maximum nondimensional power output (P*) of the Brayton cycle increases from 0.2 to 0,4 (about double) when the regenerator effectiveness (ER) increases from 0.6 to 1. (ii) The broken line in Fig. 2 indicates that the cycle nondimensional power output (P*) is highest when both hot and cold heat exchangers are ideal (eu = 1 and eL = 1). However, these conditions require that both the hot and cold heat exchangers are infinitely large before this theoretical maximum power output can be obtained. (iii) Although a Brayton cycle with perfect regenerator (eR=l) may deliver more output power than a Brayton cycle with a real regenerator (eR < 1 ), this system is not practical because a perfect regenerator also requires an infinitely large regenerator. (iv) P* and r/are reduced to P* = Zlo/M6

(43)

71 = 1-Y,/Yz

(44)

and

for an ideal compressor (rio = 1) and turbine (1"1,= 1), where Zio = [1--M~-I--(I--eL)(eRX+I--2eR)]eHT--[(eH+eL--eHeR)X+Ms(eRX-1+I--2eR)--I]eL,

(45)

M6 = 1--MseRX-I--(1--eL) [eRX+Ms(1--2eR)],

(46)

YI = eL[ (1--eR)X-J eHT--I +M3¢,RX-% I--2eR)+eRX],

(47)

and }I2 = eH{ [ I--ERX-I--(1--EL)( ERX+ I--2eR) ]T--(1--ER)ELX }.

(48)

The plot of the relationships among the nondimensionai output power (P*), cycle efficiency (xl) and compression ratio (FI) are similar to those shown in Figs. 2-4. REFERENCES 1. 2. 3. 4. 5.

F. L. Curzon and B. Ahlborn, Am. J. Phys. 43, 22 (1975). A. Bejan, Int. J. Heat Mass Transfer 31, 1211 (1988). H. S. Left, Am. J. Phys. 55, 602 (1987). O. M. Ibrahim, S. A. Klein, and J. W. Mitchell, Trans. ASME J. Engng Gas Turbine Power 113, 514 (1991). C. Wu and R. L. Kiang, Trans. ASME J. Engng Gas Turbine Power 113, 501 (1991).

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Chih Wu et al

NOMENCLATURE AH = Hot-side heat-exchanger surface area AL = Cold-side heat-exchanger surface area Cw = Heat capacity rate of the working fluid k = Specific heat ratio of the working fluid Mj = Function defined in Eq. (33) M2 = Function defined in Eq. (38) M3 = Function defined in Eq. (40) M4 = Function defined in Eq. (41) M5 = Function defined in Eq. (42) NTUH = Number of dimensionless transfer units in the hot-side heat exchanger NTUL = Number of dimensionless transfer units in the cold-side heat exchanger NTUR = Number of dimensionless transfer units in the regenerator P = Power output of the cycle P* = Dimensionless power output of the cycle P~ = Compressor-inlet pressure P2 = Compressor-outlet pressure QH = Heat added to the cycle QL = Heat removed from the cycle QR = Heat-transfer exchange in the regenerator T~ = Compressor-inlet temperature T2 = Compressor-outlet temperature T2~ = Isentropical compressor outlet temperature T~ = Turbine-inlet temperature /'4 = Turbine-outlet temperature

T4~ = Isentropic turbine-outlet temperature Ts = Regenerator-inlet temperature 7"6= Regenerator-outlet temperature UH = Overall heat-transfer coefficient in the hot-side heat exchanger UL = Overall heat-transfer coefficient in the cold-side heat exchanger UR = Overall heat-transfer coefficient in the regenerator X = Temperature ratio across the compressor Zt = Function defined in Eq. (26) Z2 = Function defined in Eq. (27) Z3 = Function defined in Eq. (28) Za = Function defined in Eq. (29) Z5 = Function defined in Eq. (30) Z6 = Function defined in Eq. (31) Z7 = Function defined in Eq. (32) Z8 = Function defined in Eq. (36) Z9 = Function defined in Eq. (37) Z~o = Function defined in Eq. (45) eu = Hot-side heat-exchanger effectiveness EL = Cold-side heat-exchanger effectiveness ea = Regenerator effectiveness rl = Cycle efficiency rlc-A = Curzon-Ahlborn cycle efficiency Tic = Compressor efficiency r h = Turbine efficiency = Pressure ratio across the compressor T = Source to sink temperature ratio