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Optimal choice of the performance parameters of an absorption heat transformer
Heat Recovery Systems & CHP
Pergamon
OPTIMAL OF
089164332(94)E0028-I
CHOICE AN
OF THE
ABSORPTION
PERFORMANCE HEAT
Vol. 15, No. 3, pp. 249-256, 1995 ElsevierScienceLtd Printed in Great Britain 0890-4332/95 $9.50+ .00
PARAMETERS
TRANSFORMER
JINCAN CHEN Department of Physics, Xiamen University, Xiamen 361005, The People's Republic of China (Received with revisions
18 March 1994)
Abstract--The optimal performance of an absorption heat transformer, i.e. a type II absorption heat pump, is investigated by using the cyclic model with continuous flow. The effect of thermal resistances between the heat transformer and the heat reservoirs is considered in the model. A general expression related to the rate of heat-pumping, the coefficientof performanceand the overall heat transfer area of the heat transformer is derived. The expression is used to optimise the main performance parameters of the heat transformer. The maximum rate of heat-pumping and the corresponding coefficientof performance are calculated. For a given overall heat transfer area of the heat transformer, the optimal relation of the heat transfer areas of the heat exchangers is obtained. The problems concerning the optimal choices of other performance parameters are discussed. The results obtained here can not only enrich the theory of finite time thermodynamics, but also provide some new theoretical bases for the optimal design and operation of real absorption heat transformers.
1. I N T R O D U C T I O N Absorption heat pumps, which can be driven by "low-grade" heat energy rather than "high-grade" work, are a class of currently developing heat p u m p equipment. They have a large potential for saving energy and decreasing environmental pollution, so many countries, such as these in Europe, America, and Japan, have paid great attention to the application of absorption heat pumps [1-8]. Absorption heat pumps have many categories, but they may, in general, be classified as two types according to the cases Th > Tp or Th < Tp, where Th and Tp are the temperatures of the heat source driving the heat pumps and the heating-space, respectively. The first type of absorption heat pumps operate in the case of Th > Tp, while the second type of absorption heat pumps operate in the case of Th < T 0. The aim of applying the second type of absorption heat p u m p is to provide thermal energy to the heating-space at temperature Tp by absorbing heat from the heat source at temperature T h lower than Tp and releasing some heat to the heat sink at temperature Te lower than Th, SO the second type of absorption heat pumps are also referred to as temperature boosters, or heat transformers [3-8]. An absorption cycle normally transfers heat between three temperature levels [8]. According to the theory of classical thermodynamics, the cyclic model of an absorption heat transformer is a reversible three-heat-source cycle, which consists of three reversible adiabatic and three reversible isothermal processes [9]. In order to achieve the coefficient of performance of the reversible cycle, the isothermal parts of the cycle must be carried out infinitely slowly, so that the working fluid comes into thermal equilibrium with the external heat reservoirs. However, under these conditions the rate of heat-pumping of the heat transformer would be zero because it takes an infinite time to provide a finite amount of heat to the heating-space. No practical engineer wants to design or build a heat transformer that runs infinitely slowly without producing any rate of heat-pumping. Therefore, it is necessary to develop the new theory of absorption heat transformers further. It is known that a real irreversible absorption heat transformer is not as efficient as the classical three-heat-source cycle, because there are always thermal resistances between the heat transformer and the external heat reservoirs and other irreversibilities in a real heat transformer. In order to expound the effect of the irreversibilities on the performance of thermodynamic cycles, finite time thermodynamics has been successfully developed [10--18]. In this paper, the theory of finite time thermodynamics is used to analyse the performance of an absorption heat transformer affected by thermal resistances. 249
250
JINCANCHEN 2. T H E C Y C L I C M O D E L
For a class of absorption heat transformers operating among three heat reservoirs (the heat source at temperature Th, heating-space at temperature Tp, and heat sink at temperature Te (Tp > Th > Te)) it may be assumed that the working fluid of the cycle flows continuously and carries out a reversible three-heat-source cycle, whereas there are thermal resistances between the working fluid and the three heat reservoirs and the temperatures of the working fluid are different from the temperatures of the heat reservoirs, so that heat transfer is carried out under a finite temperature difference, as shown schematically in Fig. 1. Such a cyclic model is referred to as an endoreversible cycle, which is often adopted in the investigation of finite time thermodynamics [10-20]. According to the first law of thermodynamics, we have Oh
-
-
Qe - 0 r = 0,
(1)
where Qh is the heat absorbed from the heat source at temperature Th by the working fluid per cycle and Q, and Qp are, respectively, the heats released to the heat sink at temperature Te and pumped to the heating-space at temperature Tp by the working fluid per cycle. The different parts of the working fluid exchange heat with three heat reservoirs during the full cycle time t, so that the rate of heat-applying, the rate of heat-pumping and the rate of heat-releasing of the cycle are, respectively, qh = Q h / t , H = Q p / t , and q~ = Q e / t . Because an endoreversible cycle is an irreversible cycle, its performance is dependent on the heat transfer law. We often assume that the heat transfer between the working fluid and the heat reservoirs obeys a linear law [1 l, 12], i.e. qh = k h A h ( T h -- T1),
(2)
II = kpAp(Tz -
Tp)
(3)
q~ = k c A e ( T 3 -
To),
(4)
and
where kh, kp and k~ are, respectively, the heat transfer coefficients between the working fluid and the three heat reservoirs at temperatures Th, Tp and T~; Ah, Ap and Ae are, respectively, the heat transfer areas between the working fluid and the three heat reservoirs at temperatures Th, Tp and T~; and TI, /'2 and /'3 are, respectively, the temperatures of the working fluid carrying out three isothermal processes. The heat transformer exchanges heat only with the three heat reservoirs at
.
//" •
c\,b~'\a
\- \,,,
6
r'-
~
T2
i-'-
T,
0 0.00
Fig. I. Schematic diagram of an absorption heat transformer.
I
i
0.20
0.40
i~t 0.60
0.80
Fig. 2. The H/IIm,xversus ~ curves of an absorption heat transformer. Plots are presented for Th= 80°C, Tp= 120°C and T~= 25°C. The curves a, b, and c correspond to the cases of B = -0.5, 0, and 0.5, respectively.
Optimal choice of performance parameters
251
temperatures Th, Tp and Te, so the overall heat transfer area of the heat exchangers of the heat transformer is A = Ah + Ap + A~.
(5)
The cyclic model mentioned above is a useful model. The heat transfer areas of the heat exchangers, which are important performance parameters of real heat transformers, are included in the model besides the irreversibility of heat conduction. Thus, some significant results concerning the heat transfer areas of the heat exchangers can be obtained by using the model to optimise the performance of absorption heat transformers. 3. THE COEFFICIENT OF PERFORMANCE AND THE RATE OF HEAT-PUMPING According to the characteristics of the above model and the second law of thermodynamics, we have Oh TI
Qp /'2
Qe=0" T3
(6)
Using equations (1) and (6), we can obtain the coefficient of performance of an absorption heat transformer: = Q_.2 = TI -
Qh
T3
T2
T,
(7)
T2- T3
and using equations (1)-(5) and (7), we can obtain the rate of heat-pumping: kpA H=
1
1
--+bl T: - rp
!P( Th - T, )
(8) 1-~v
k b2 -
-
1 -
'
-
~P T3 - T~
where bl = kp/kh and b 2 = kp/k e. For the sake of convenience, let x = T3/TI, y = T3/T2 and z = T3. Then, equations (7)and (8) may be written as 1--X
~v = _ _ 1-y
(9)
and II =
kpA 1-y__x +b2 x _ - y _ l ' 1 --XXTh--Z 1 --XZ -- T e
~ +Y b l z -yTp
(10)
respectively. 4. A GENERAL EXPRESSION Our problem now is to determine the optimal coefficient of performance of an absorption heat transformer under a given rate of heat-pumping and a given overall heat transfer area of the heat exchangers. For this purpose, we introduce the Lagrangian Ae= ~P + 2 1 1 = 1 - x + 2
kpA
1 -y
y
1 -y
z-yTp+bl -
-
-
x
l --xxTh--z -
-
-
(11) +b2
x -y
l-xz-
1
T,
From the Euler-Lagrange equations aAe ~gx
--=0,
dLa
d~
----0,
--=0,
dy
dz
(12)
we can find that, under the given rate of heat-pumping and the given overall heat transfer area
252
JINCANCFIEN
of the heat exchangers, when the coefficient of performance of the heat transformer is optimal, x, y, and z must satisfy the following relations: xT h - z = ~ll(z
(13)
- yTp)
and z - Te = ~ 2 ( z
(14)
-yTp).
Solving equations (9), (13) and (14), we obtain x =
v, r0 + (l - B ) ( 1 - v , ) r .
B T T h + (1 - B ) r p
y =
(15)
'
To + B C e - I)Th
(16)
B ~ T h + (1 - B)Tp
and z =
(17) (l
+
'
where B = (1 - x/~2)/(l + x/~t ). From equations (15)-(17), we find that the relations between the temperatures of the working fluid in the three isothermal processes of the cycle and the optimal coefficient of performance are given by T~=
1 l+x~
l T~ - 1 + ~
~
~ThTe4- ~ T p T e 4 - ~ / - ~ 2 ( l - ~)ThTp ~PTe + (l _ B ) ( l _ ~ ) T p , ~ThTe+w/~TpT¢+~(l - ~)Th Tp L + B(q' - 1)T~
(18) (I9)
and T3 =
1 ~ThT~-t-x//~ITpT~4- %/~2(1 - ~ ) T h T p 1 4- x/bT B ~ T h 4- (1 - B)Tp
(20)
respectively. Substituting equations (18)-(20) into equation (8), we obtain H = KA
T ( ~ P , - ~P)Th(Tp - T¢)
(21)
~'Te + B2(~ ' - l ) ~ ' T h + (1 - B)~(I - ~ ) r ~ '
where X = khkp/(~-~h + ~/kT~): and ~gr = (1 - T~/Th)Tp/(Tp - Te) < 1 is the coefficient of performance of a reversible absorption heat transformer [8, 9]. Equation (21) determines the optimal coefficient of performance of an endoreversible absorption heat transformer under the given rate of heat-pumping and the given overall heat transfer area of the heat exchangers. It is a general expression which may be used to discuss the optimal performance of an endoreversible absorption heat transformer. 5. THE O P T I M A L R E L A T I O N OF HEAT T R A N S F E R AREAS In order to make an endoreversible absorption heat transformer obtain the optimal performance under the given rate of heat-pumping and the given overall heat transfer area of the heat exchangers, the temperatures of the working fluid in the three isothermal processes of the cycle must satisfy equations (l 8)-(20) and the heat transfer areas between the working fluid and the three heat reservoirs at temperatures Th, Tp and Te must still satisfy a certain optimum relation. Using equations (21) and (1)-(4), we find that the relations between the heat transfer areas of the heat exchangers and the overall heat transfer area A are given by Ah = A
x//~
kUT~+ (1 - B)(I - ~)Tp
l + ~
~Te + B 2 ( ~ -- 1)~Th + (l -- B)2(1 - ~ ) T p '
l 1+ ~
~Te + B ( ~ - 1)~Th ~T~ 4- B 2 ( ~ - 1)~Th + (1 -- B)2(I - ~ ) T p
Ap = A - -
(22)
(23)
Optimal choice of performance parameters
253
and A, = A
x/~2
B(1 - ~ ) ~ T h + (1 - B)(1 - ke)Tp
(24)
1 + x/r~t ~Te + B2( ~t - l)~'Th + (1 -- B)2(1 -- ~ ) T p ' respectively. From equations (22)-(24), we obtain a simple and useful optimum relation of the heat transfer areas:
+,/ZAo.
(25)
Equation (25) is an important result, which can play an instructive role for engineers to design the heat exchangers of real absorption heat transformers. When kh = kp = k,, equation (25) may be simply written as .4 h = Ap + A e
(26)
Ah:Ap:A, = [Te + Tp(1 - ~ ) / ~ ] : T¢: Tp(1 - ~ ) / ~ ,
(27)
and another simple relation:
can be deduced from equations (22)-(24). Equation (26) clearly shows that when the heat transfer coefficients between the working fluid and the heat reservoirs are the same, the heat transfer area between the working fluid and the heat source at temperature Tn, should be equal to the sum of the heat transfer areas between the working fluid and the heat reservoirs at temperatures Tp and T, and both An and (A 0 + A,) should be equal to A/2. Equation (27) further shows that when T, = Tp(1 - ~)/~', both Ap and A, should be equal to A/4. In general, Te # To0 - ~)/qt, so that one of the two heat transfer areas is larger than A/4 and the other is smaller than A/4. 6. T H E M A X I M U M R A T E OF H E A T - P U M P I N G AND T H E C O R R E S P O N D I N G C O E F F I C I E N T OF P E R F O R M A N C E It is clearly seen from equation (21) that when ~v = 0 or ~u = ~/'r, H = 0, this implies that when ~u is equal to some value, H has a maximum. Using equation (21) and the extreme condition 0II 0--~ = O,
(28)
we can find that when the optimal coefficient of performance
Tp- T. + D(x/Th T, - T.) the rate of heat-pumping attains the maximum
n°a = r,A(,/%_ j Z ) 2
r. Tp - Te + 2 D ( x / ~ h T~ - T,) - D2(x/~h -- x/-Tee)z'
(30)
where K ' = khkd(x/~h + X/~) ~ and O = (x/~h/x/~p)(x/~ - x/~p)/Cx/~h + x / ~ ) . In such a case, the temperatures o f the working fluid in the three isothermal processes of the cycle are determined by 1 TI =
-
l + ~
[To - (1 - S ) T . ] ( 1 - ~ )
+
(1 --
B)T'
= Tim ,
r2= T. (To-x/b2TO(I-~/To/TOTh+(v/~,To+x/~TOT'=_T2m, 1 + x/~
(31) (32)
Br~rp(l-x//~/Th)+(Te-BTh)T"
and I
T~ = 1 + ~
(r~ - v @ r p ) ( l -x/Te/Th)Th +(x/b~Te+~/b~Th)T'=- T3m, Brh(1 - x//-~/Th) + (! -- B)T'
(33)
254
JINCANCHEN
respectively, and the rate of heat-supplying of the heat source at temperature Th, per unit heat transfer area, is given by x~t ( T e - TO)Th(I -- Tx/-T~/Th)+ (Th-- T¢)T' =_(qh/Ah)m, l +xf~[T-(l-B)Tp](l~)+(1-B) T" where T ' = T p - Te+ D[(ThT~) '/2- T¢]. When kh = kp = k e, equations (29)-(34) may be simply written as q~ =
(34)
A,
Tm= (1 -- X/-~T~) TO~T~'TP l-Im,x = -~ A ( ~ h - -
Zh(
~ e e ) 2 ToG --- T¢'
(35)
(36)
//~ee~,
(37)
T~m= I+X/Toj
(38)
~m=s~l +q re/
(39)
T~m= T
1 +X/T,]
and (qh/Ah)m = T
1 - V Th/I
(40)
respectively.
Obviously, limax and I//m are two important performance parameters of absorption heat transformers. Like the maximum power output Pm,x and the corresponding efficiency r/m obtained by Curzon and Ahlborn, concerning a Carnot heat engine [10], they are conducive to the further understanding of the cyclic performance of real absorption heat transformers. 7. THE O P T I M A L C H O I C E OF SEVERAL P E R F O R M A N C E P A R A M E T E R S (1). Equation (21) clearly shows that if, and only if, FI = 0, T = T r. Thus none of the coefficients of performance in real absorption heat transformers can attain T r. This result indicates that the bound Tr of classical thermodynamics does not have a very large instructive significance for real absorption heat transformers and it is important to establish the bounds of finite time thermodynamics. Using equations (21) and (30), we can obtain the 1-Ill-Ira,x-- T curves of an endoreversible absorption heat transformer, as shown in Fig. 2. Figure 2 clearly shows that when FI < H . . . . we can obtain two different T for a given H, where one is smaller than Tm and the other is larger than ~m" When ~u < 7Jm, H decreases as T decreases, so the coefficients of performance smaller than ~m are not the optimal values of the coefficient of performance. It is thus clear that, although the coefficient of performance of an absorption heat transformer affected by thermal resistances is always smaller than ~u, the heat transformer does not operate in the optimal working states if its coefficient of performance is smaller than T m. In other words, the optimal coefficient of performance of an endoreversible absorption heat transformer should be situated between T m and q'r, i.e.
~r > ~tt >t tPm.
(41)
It is shown once again that Tm is an important performance parameter of absorption heat transformers, because it not only determines the optimal coefficient of performance at the maximum rate of heat-pumping, but also provides a lower bound for the optimal coefficient of performance of endoreversible absorption heat transformers.
Optimal choice of performance parameters
255
W h e n ~u > ~/m, H is a monotonically decreasing function o f ~, so that consideration must be simultaneously given to both the coefficient o f performance and the rate o f heat-pumping. In the case o f an unlimited energy source driving heat transformers, it is reasonable for heat transformers to operate in the state o f the m a x i m u m rate o f heat-pumping. However, for an energetically limited source, one should, in general, place particular emphasis o n the coefficient o f performance o f heat transformers because it is beneficial to the rational use o f limited source energy. (2). Using equations (41), (2), and (18), we can determine that the optimal region o f the rate o f heat-supplying o f the heat source at temperature Th, per unit heat transfer area, is 0 < qh/Ah <~ (qh/Ah)m.
(42)
It can be further proven that when ( q h / A h ) > (qh/Ah)rn, both H and ~ decrease with an increase in the value o f (qh/Ah), SO that the operating states o f (qh/Ah) > (qh/Ah)m are not the optimal states o f absorption heat transformers. Therefore, it is naturally not allowable to choose these operating states o f (qh/Ah) > (qh/Ah)m in the design and operation o f absorption heat transformers. (3). Using equations (41) and (18)-(20), we can determine that the optimal regions o f the temperatures o f the working fluid in the three isothermal processes o f the cycle are
Th > T1/> T~m,
(43)
T2m ~ Z 2 > Tp,
(44)
T3m ~ T 3 ~> Ze,
(45)
and
respectively. 8. C O N C L U S I O N It has been shown that the effect o f thermal resistance on the performance o f an absorption heat transformer can be investigated by using an endoreversible cycle model with continuous flow. Some new t h e r m o d y n a m i c bounds concerning important performance parameters o f absorption heat transformers are determined. These b o u n d s are more realistic and useful than those o f classical thermodynamics. They will play an instructive role in the optimal design and operation o f absorption heat transformers. Acknowledgement--The author thanks Dr Alexis De Vos for helpful comments and discussions.
REFERENCES 1. R. W. Lazzarin, G. A. Longo and F. Piccininni, An open cycle absorption heat pump. Heat Recovery Systems & ClIP 12(5), 391-396 (1992). 2. W. Kern, A two-stage absorption heat pump for heating and cooling. Newslett. lEA Heat Pump Center 7(3), 22-23 (1989). 3. F. Ziegler and P. Riesch, Absorption cycle. A review with regard to energetic efficiency.Heat Recovery Systems & CHP 13(2), 147-159 (1993). 4. I. W. Eames and S. Aphornratana, Research on heat-operated heat pumps and refrigerators. J. Institute of Energy 66, 29-39 (1993). 5. J. M. George and S. S. Murthy, Experiments on a vapour absorption heat transformer. Int. J. Refrig. 16, 107-119 (1993). 6. M. Ikeuchi, T. Yumikawa, E. Ozaki and G. Yamanaka, Design and performance of a high-temperature-boost absorption heat pump. ASHRAE Trans. 91(2), 2081-2094 (1985). 7. S. Iyoki and T. Uemura, Performance characteristics of the water-lithium bromide-zinc chloride-calcium bromide absorption refrigerating machine, absorption heat pump and absorption heat transformer. Int. J. Refrig. 13, 191-196 (1990). 8. K. Abrahamsson and ~ Jernqvist, Carnot comparsion of multi-temperature level absorption heat cycles. Int. J. Refrig. 16, 240-246 (1993). 9. F. F. Huang, Engineering Thermodynamics. Macmillan, New York (1976). 10. F. L. Curzon and B. Ahlborn, Efficiencyofa Carnot engine at maximum power output. Am. J. Phys. 43, 22-24 (1975). I I. P. Salamon and A. Nitzan, Finite time optimizations of a Newton's law Carnot cycle. J. Chem. Phys. 74, 3546-3560 (1981). 12. M. J. Ondrechen, M. H. Rubin and Y. B. Band, The generalized Carnot cycle: a working fluid operating in finite time between finite heat sources and sinks. J. Chem. Phys. 78, 4721-4727 (1983). HRS 15/3--B
256 13. 14. 15. 16. 17. 18. 19. 20.
JINCAN CHEN B. Andresen, Finite Time Thermodynamics. University of Copenhagen, Copenhagen (1983). A. De Vos, Reflections on the power delivered by endoreversible engines. J. Phys. D: Appl. Phys. 20, 232-236 (1987). J. Chen and Z. Yan, Unified description of endoreversible cycles. Phys. Ret,. A39, 4140-4147 (1989). J. Chen and Z. Yan, Equivalent combined systems of three heat-source heat pumps. J. Chem. Phys. 90, 4951-4955 (1989). C. Wu, Power optimization of an endoreversible Brayton gas heat engine. Energy Convers. & Mgmt 31,561 565 (1991). A. De Vos, Endoreversible thermodynamics and chemical reactions. J. Phys. Chem. 95, 4534-4540 (1991). J. Chen and Z. Yan, Optimal performance of endoreversible cycles for another linear heat transfer law. J. Phys. D: Appl. Phys. 26, 1581-1586 (1993). M. J. Gordon and V. N. Orlov, Performance characteristics of endoreversible chemical engines. J. Appl. Phys. 74, 5303 5309 (1993).
Pergamon
OPTIMAL OF
089164332(94)E0028-I
CHOICE AN
OF THE
ABSORPTION
PERFORMANCE HEAT
Vol. 15, No. 3, pp. 249-256, 1995 ElsevierScienceLtd Printed in Great Britain 0890-4332/95 $9.50+ .00
PARAMETERS
TRANSFORMER
JINCAN CHEN Department of Physics, Xiamen University, Xiamen 361005, The People's Republic of China (Received with revisions
18 March 1994)
Abstract--The optimal performance of an absorption heat transformer, i.e. a type II absorption heat pump, is investigated by using the cyclic model with continuous flow. The effect of thermal resistances between the heat transformer and the heat reservoirs is considered in the model. A general expression related to the rate of heat-pumping, the coefficientof performanceand the overall heat transfer area of the heat transformer is derived. The expression is used to optimise the main performance parameters of the heat transformer. The maximum rate of heat-pumping and the corresponding coefficientof performance are calculated. For a given overall heat transfer area of the heat transformer, the optimal relation of the heat transfer areas of the heat exchangers is obtained. The problems concerning the optimal choices of other performance parameters are discussed. The results obtained here can not only enrich the theory of finite time thermodynamics, but also provide some new theoretical bases for the optimal design and operation of real absorption heat transformers.
1. I N T R O D U C T I O N Absorption heat pumps, which can be driven by "low-grade" heat energy rather than "high-grade" work, are a class of currently developing heat p u m p equipment. They have a large potential for saving energy and decreasing environmental pollution, so many countries, such as these in Europe, America, and Japan, have paid great attention to the application of absorption heat pumps [1-8]. Absorption heat pumps have many categories, but they may, in general, be classified as two types according to the cases Th > Tp or Th < Tp, where Th and Tp are the temperatures of the heat source driving the heat pumps and the heating-space, respectively. The first type of absorption heat pumps operate in the case of Th > Tp, while the second type of absorption heat pumps operate in the case of Th < T 0. The aim of applying the second type of absorption heat p u m p is to provide thermal energy to the heating-space at temperature Tp by absorbing heat from the heat source at temperature T h lower than Tp and releasing some heat to the heat sink at temperature Te lower than Th, SO the second type of absorption heat pumps are also referred to as temperature boosters, or heat transformers [3-8]. An absorption cycle normally transfers heat between three temperature levels [8]. According to the theory of classical thermodynamics, the cyclic model of an absorption heat transformer is a reversible three-heat-source cycle, which consists of three reversible adiabatic and three reversible isothermal processes [9]. In order to achieve the coefficient of performance of the reversible cycle, the isothermal parts of the cycle must be carried out infinitely slowly, so that the working fluid comes into thermal equilibrium with the external heat reservoirs. However, under these conditions the rate of heat-pumping of the heat transformer would be zero because it takes an infinite time to provide a finite amount of heat to the heating-space. No practical engineer wants to design or build a heat transformer that runs infinitely slowly without producing any rate of heat-pumping. Therefore, it is necessary to develop the new theory of absorption heat transformers further. It is known that a real irreversible absorption heat transformer is not as efficient as the classical three-heat-source cycle, because there are always thermal resistances between the heat transformer and the external heat reservoirs and other irreversibilities in a real heat transformer. In order to expound the effect of the irreversibilities on the performance of thermodynamic cycles, finite time thermodynamics has been successfully developed [10--18]. In this paper, the theory of finite time thermodynamics is used to analyse the performance of an absorption heat transformer affected by thermal resistances. 249
250
JINCANCHEN 2. T H E C Y C L I C M O D E L
For a class of absorption heat transformers operating among three heat reservoirs (the heat source at temperature Th, heating-space at temperature Tp, and heat sink at temperature Te (Tp > Th > Te)) it may be assumed that the working fluid of the cycle flows continuously and carries out a reversible three-heat-source cycle, whereas there are thermal resistances between the working fluid and the three heat reservoirs and the temperatures of the working fluid are different from the temperatures of the heat reservoirs, so that heat transfer is carried out under a finite temperature difference, as shown schematically in Fig. 1. Such a cyclic model is referred to as an endoreversible cycle, which is often adopted in the investigation of finite time thermodynamics [10-20]. According to the first law of thermodynamics, we have Oh
-
-
Qe - 0 r = 0,
(1)
where Qh is the heat absorbed from the heat source at temperature Th by the working fluid per cycle and Q, and Qp are, respectively, the heats released to the heat sink at temperature Te and pumped to the heating-space at temperature Tp by the working fluid per cycle. The different parts of the working fluid exchange heat with three heat reservoirs during the full cycle time t, so that the rate of heat-applying, the rate of heat-pumping and the rate of heat-releasing of the cycle are, respectively, qh = Q h / t , H = Q p / t , and q~ = Q e / t . Because an endoreversible cycle is an irreversible cycle, its performance is dependent on the heat transfer law. We often assume that the heat transfer between the working fluid and the heat reservoirs obeys a linear law [1 l, 12], i.e. qh = k h A h ( T h -- T1),
(2)
II = kpAp(Tz -
Tp)
(3)
q~ = k c A e ( T 3 -
To),
(4)
and
where kh, kp and k~ are, respectively, the heat transfer coefficients between the working fluid and the three heat reservoirs at temperatures Th, Tp and T~; Ah, Ap and Ae are, respectively, the heat transfer areas between the working fluid and the three heat reservoirs at temperatures Th, Tp and T~; and TI, /'2 and /'3 are, respectively, the temperatures of the working fluid carrying out three isothermal processes. The heat transformer exchanges heat only with the three heat reservoirs at
.
//" •
c\,b~'\a
\- \,,,
6
r'-
~
T2
i-'-
T,
0 0.00
Fig. I. Schematic diagram of an absorption heat transformer.
I
i
0.20
0.40
i~t 0.60
0.80
Fig. 2. The H/IIm,xversus ~ curves of an absorption heat transformer. Plots are presented for Th= 80°C, Tp= 120°C and T~= 25°C. The curves a, b, and c correspond to the cases of B = -0.5, 0, and 0.5, respectively.
Optimal choice of performance parameters
251
temperatures Th, Tp and Te, so the overall heat transfer area of the heat exchangers of the heat transformer is A = Ah + Ap + A~.
(5)
The cyclic model mentioned above is a useful model. The heat transfer areas of the heat exchangers, which are important performance parameters of real heat transformers, are included in the model besides the irreversibility of heat conduction. Thus, some significant results concerning the heat transfer areas of the heat exchangers can be obtained by using the model to optimise the performance of absorption heat transformers. 3. THE COEFFICIENT OF PERFORMANCE AND THE RATE OF HEAT-PUMPING According to the characteristics of the above model and the second law of thermodynamics, we have Oh TI
Qp /'2
Qe=0" T3
(6)
Using equations (1) and (6), we can obtain the coefficient of performance of an absorption heat transformer: = Q_.2 = TI -
Qh
T3
T2
T,
(7)
T2- T3
and using equations (1)-(5) and (7), we can obtain the rate of heat-pumping: kpA H=
1
1
--+bl T: - rp
!P( Th - T, )
(8) 1-~v
k b2 -
-
1 -
'
-
~P T3 - T~
where bl = kp/kh and b 2 = kp/k e. For the sake of convenience, let x = T3/TI, y = T3/T2 and z = T3. Then, equations (7)and (8) may be written as 1--X
~v = _ _ 1-y
(9)
and II =
kpA 1-y__x +b2 x _ - y _ l ' 1 --XXTh--Z 1 --XZ -- T e
~ +Y b l z -yTp
(10)
respectively. 4. A GENERAL EXPRESSION Our problem now is to determine the optimal coefficient of performance of an absorption heat transformer under a given rate of heat-pumping and a given overall heat transfer area of the heat exchangers. For this purpose, we introduce the Lagrangian Ae= ~P + 2 1 1 = 1 - x + 2
kpA
1 -y
y
1 -y
z-yTp+bl -
-
-
x
l --xxTh--z -
-
-
(11) +b2
x -y
l-xz-
1
T,
From the Euler-Lagrange equations aAe ~gx
--=0,
dLa
d~
----0,
--=0,
dy
dz
(12)
we can find that, under the given rate of heat-pumping and the given overall heat transfer area
252
JINCANCFIEN
of the heat exchangers, when the coefficient of performance of the heat transformer is optimal, x, y, and z must satisfy the following relations: xT h - z = ~ll(z
(13)
- yTp)
and z - Te = ~ 2 ( z
(14)
-yTp).
Solving equations (9), (13) and (14), we obtain x =
v, r0 + (l - B ) ( 1 - v , ) r .
B T T h + (1 - B ) r p
y =
(15)
'
To + B C e - I)Th
(16)
B ~ T h + (1 - B)Tp
and z =
(17) (l
+
'
where B = (1 - x/~2)/(l + x/~t ). From equations (15)-(17), we find that the relations between the temperatures of the working fluid in the three isothermal processes of the cycle and the optimal coefficient of performance are given by T~=
1 l+x~
l T~ - 1 + ~
~
~ThTe4- ~ T p T e 4 - ~ / - ~ 2 ( l - ~)ThTp ~PTe + (l _ B ) ( l _ ~ ) T p , ~ThTe+w/~TpT¢+~(l - ~)Th Tp L + B(q' - 1)T~
(18) (I9)
and T3 =
1 ~ThT~-t-x//~ITpT~4- %/~2(1 - ~ ) T h T p 1 4- x/bT B ~ T h 4- (1 - B)Tp
(20)
respectively. Substituting equations (18)-(20) into equation (8), we obtain H = KA
T ( ~ P , - ~P)Th(Tp - T¢)
(21)
~'Te + B2(~ ' - l ) ~ ' T h + (1 - B)~(I - ~ ) r ~ '
where X = khkp/(~-~h + ~/kT~): and ~gr = (1 - T~/Th)Tp/(Tp - Te) < 1 is the coefficient of performance of a reversible absorption heat transformer [8, 9]. Equation (21) determines the optimal coefficient of performance of an endoreversible absorption heat transformer under the given rate of heat-pumping and the given overall heat transfer area of the heat exchangers. It is a general expression which may be used to discuss the optimal performance of an endoreversible absorption heat transformer. 5. THE O P T I M A L R E L A T I O N OF HEAT T R A N S F E R AREAS In order to make an endoreversible absorption heat transformer obtain the optimal performance under the given rate of heat-pumping and the given overall heat transfer area of the heat exchangers, the temperatures of the working fluid in the three isothermal processes of the cycle must satisfy equations (l 8)-(20) and the heat transfer areas between the working fluid and the three heat reservoirs at temperatures Th, Tp and Te must still satisfy a certain optimum relation. Using equations (21) and (1)-(4), we find that the relations between the heat transfer areas of the heat exchangers and the overall heat transfer area A are given by Ah = A
x//~
kUT~+ (1 - B)(I - ~)Tp
l + ~
~Te + B 2 ( ~ -- 1)~Th + (l -- B)2(1 - ~ ) T p '
l 1+ ~
~Te + B ( ~ - 1)~Th ~T~ 4- B 2 ( ~ - 1)~Th + (1 -- B)2(I - ~ ) T p
Ap = A - -
(22)
(23)
Optimal choice of performance parameters
253
and A, = A
x/~2
B(1 - ~ ) ~ T h + (1 - B)(1 - ke)Tp
(24)
1 + x/r~t ~Te + B2( ~t - l)~'Th + (1 -- B)2(1 -- ~ ) T p ' respectively. From equations (22)-(24), we obtain a simple and useful optimum relation of the heat transfer areas:
+,/ZAo.
(25)
Equation (25) is an important result, which can play an instructive role for engineers to design the heat exchangers of real absorption heat transformers. When kh = kp = k,, equation (25) may be simply written as .4 h = Ap + A e
(26)
Ah:Ap:A, = [Te + Tp(1 - ~ ) / ~ ] : T¢: Tp(1 - ~ ) / ~ ,
(27)
and another simple relation:
can be deduced from equations (22)-(24). Equation (26) clearly shows that when the heat transfer coefficients between the working fluid and the heat reservoirs are the same, the heat transfer area between the working fluid and the heat source at temperature Tn, should be equal to the sum of the heat transfer areas between the working fluid and the heat reservoirs at temperatures Tp and T, and both An and (A 0 + A,) should be equal to A/2. Equation (27) further shows that when T, = Tp(1 - ~)/~', both Ap and A, should be equal to A/4. In general, Te # To0 - ~)/qt, so that one of the two heat transfer areas is larger than A/4 and the other is smaller than A/4. 6. T H E M A X I M U M R A T E OF H E A T - P U M P I N G AND T H E C O R R E S P O N D I N G C O E F F I C I E N T OF P E R F O R M A N C E It is clearly seen from equation (21) that when ~v = 0 or ~u = ~/'r, H = 0, this implies that when ~u is equal to some value, H has a maximum. Using equation (21) and the extreme condition 0II 0--~ = O,
(28)
we can find that when the optimal coefficient of performance
Tp- T. + D(x/Th T, - T.) the rate of heat-pumping attains the maximum
n°a = r,A(,/%_ j Z ) 2
r. Tp - Te + 2 D ( x / ~ h T~ - T,) - D2(x/~h -- x/-Tee)z'
(30)
where K ' = khkd(x/~h + X/~) ~ and O = (x/~h/x/~p)(x/~ - x/~p)/Cx/~h + x / ~ ) . In such a case, the temperatures o f the working fluid in the three isothermal processes of the cycle are determined by 1 TI =
-
l + ~
[To - (1 - S ) T . ] ( 1 - ~ )
+
(1 --
B)T'
= Tim ,
r2= T. (To-x/b2TO(I-~/To/TOTh+(v/~,To+x/~TOT'=_T2m, 1 + x/~
(31) (32)
Br~rp(l-x//~/Th)+(Te-BTh)T"
and I
T~ = 1 + ~
(r~ - v @ r p ) ( l -x/Te/Th)Th +(x/b~Te+~/b~Th)T'=- T3m, Brh(1 - x//-~/Th) + (! -- B)T'
(33)
254
JINCANCHEN
respectively, and the rate of heat-supplying of the heat source at temperature Th, per unit heat transfer area, is given by x~t ( T e - TO)Th(I -- Tx/-T~/Th)+ (Th-- T¢)T' =_(qh/Ah)m, l +xf~[T-(l-B)Tp](l~)+(1-B) T" where T ' = T p - Te+ D[(ThT~) '/2- T¢]. When kh = kp = k e, equations (29)-(34) may be simply written as q~ =
(34)
A,
Tm= (1 -- X/-~T~) TO~T~'TP l-Im,x = -~ A ( ~ h - -
Zh(
~ e e ) 2 ToG --- T¢'
(35)
(36)
//~ee~,
(37)
T~m= I+X/Toj
(38)
~m=s~l +q re/
(39)
T~m= T
1 +X/T,]
and (qh/Ah)m = T
1 - V Th/I
(40)
respectively.
Obviously, limax and I//m are two important performance parameters of absorption heat transformers. Like the maximum power output Pm,x and the corresponding efficiency r/m obtained by Curzon and Ahlborn, concerning a Carnot heat engine [10], they are conducive to the further understanding of the cyclic performance of real absorption heat transformers. 7. THE O P T I M A L C H O I C E OF SEVERAL P E R F O R M A N C E P A R A M E T E R S (1). Equation (21) clearly shows that if, and only if, FI = 0, T = T r. Thus none of the coefficients of performance in real absorption heat transformers can attain T r. This result indicates that the bound Tr of classical thermodynamics does not have a very large instructive significance for real absorption heat transformers and it is important to establish the bounds of finite time thermodynamics. Using equations (21) and (30), we can obtain the 1-Ill-Ira,x-- T curves of an endoreversible absorption heat transformer, as shown in Fig. 2. Figure 2 clearly shows that when FI < H . . . . we can obtain two different T for a given H, where one is smaller than Tm and the other is larger than ~m" When ~u < 7Jm, H decreases as T decreases, so the coefficients of performance smaller than ~m are not the optimal values of the coefficient of performance. It is thus clear that, although the coefficient of performance of an absorption heat transformer affected by thermal resistances is always smaller than ~u, the heat transformer does not operate in the optimal working states if its coefficient of performance is smaller than T m. In other words, the optimal coefficient of performance of an endoreversible absorption heat transformer should be situated between T m and q'r, i.e.
~r > ~tt >t tPm.
(41)
It is shown once again that Tm is an important performance parameter of absorption heat transformers, because it not only determines the optimal coefficient of performance at the maximum rate of heat-pumping, but also provides a lower bound for the optimal coefficient of performance of endoreversible absorption heat transformers.
Optimal choice of performance parameters
255
W h e n ~u > ~/m, H is a monotonically decreasing function o f ~, so that consideration must be simultaneously given to both the coefficient o f performance and the rate o f heat-pumping. In the case o f an unlimited energy source driving heat transformers, it is reasonable for heat transformers to operate in the state o f the m a x i m u m rate o f heat-pumping. However, for an energetically limited source, one should, in general, place particular emphasis o n the coefficient o f performance o f heat transformers because it is beneficial to the rational use o f limited source energy. (2). Using equations (41), (2), and (18), we can determine that the optimal region o f the rate o f heat-supplying o f the heat source at temperature Th, per unit heat transfer area, is 0 < qh/Ah <~ (qh/Ah)m.
(42)
It can be further proven that when ( q h / A h ) > (qh/Ah)rn, both H and ~ decrease with an increase in the value o f (qh/Ah), SO that the operating states o f (qh/Ah) > (qh/Ah)m are not the optimal states o f absorption heat transformers. Therefore, it is naturally not allowable to choose these operating states o f (qh/Ah) > (qh/Ah)m in the design and operation o f absorption heat transformers. (3). Using equations (41) and (18)-(20), we can determine that the optimal regions o f the temperatures o f the working fluid in the three isothermal processes o f the cycle are
Th > T1/> T~m,
(43)
T2m ~ Z 2 > Tp,
(44)
T3m ~ T 3 ~> Ze,
(45)
and
respectively. 8. C O N C L U S I O N It has been shown that the effect o f thermal resistance on the performance o f an absorption heat transformer can be investigated by using an endoreversible cycle model with continuous flow. Some new t h e r m o d y n a m i c bounds concerning important performance parameters o f absorption heat transformers are determined. These b o u n d s are more realistic and useful than those o f classical thermodynamics. They will play an instructive role in the optimal design and operation o f absorption heat transformers. Acknowledgement--The author thanks Dr Alexis De Vos for helpful comments and discussions.
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