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Optimum performance of a corrugated, collector-driven, irreversible carnot heat engine and absorption refrigerator
Energy Vol. 22, No. 5, pp. 481-485, 1997
Pergamon
PII: S0360-5442(96)00142-9
© 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0360-5442/97 $17.00 + 0.00
OPTIMUM PERFORMANCE OF A CORRUGATED, COLLECTORDRIVEN, IRREVERSIBLE CARNOT HEAT ENGINE AND ABSORPTION REFRIGERATOR SELAHATTiN GOKTUN* and SULEYMAN OZKAYNAK* +i.T.fJ., Maritime Faculty, 81716 Tuzla-istanbul, Turkey and *Turkish Naval Academy, 81704 Tuzlaistanbul, Turkey
(Received 11 April 1996)
Abstract--The technique of energetic optimization is employed to investigate the optimal performance of a corrugated, collector-driven, irreversible Carnot heat engine and an absorption refrigerator. A minimum operating parameter and a relation between the maximum overall efficiencies are obtained for the systems under consideration. © 1997 Elsevier Science Ltd. All rights reserved.
INTRODUCTION
The corrugated sheet collector (CSC) is a simple solar utilization device which transfers the absorbed solar energy to a working fluid by direct contact. From the heat-transfer point of view, the advantage is evident. When a solar collector as a heat source is integrated with a heat engine or a refrigerator, one objective is to determine the operating conditions that will provide optimum system performance. In general, the solar-collector efficiency is decreased by the use of high temperatures for the working fluid. However, for heat engines or refrigerators, the efficiency or coefficient of performance (COP) increases with the supply temperature. This incompatibility in the temperature-efficiency relation indicates that any solar-collector heat engine or refrigerator combination will have an optimum operating temperature. This problem has been extensively studied by many authors [1-8]. The present study is a theoretical investigation on the optimum performance of CSC-driven, irreversible, Carnot heat and absorption refrigeration system by taking into consideration the upper design bound (maximum efficiency and/or maximum COP) of a heat engine and an absorption refrigerator. ANALYSIS
The irreversibilities for a thermodynamic cycle may be considered to be external and internal. External irreversibilities arise from the temperature differences required to transfer heat between the system and the heat source and sink. Internal irreversibilities result primarily from dissipative processes inside the working fluid. The T-S diagram of an irreversible Carnot heat engine and an irreversible Carnot absorption refrigerator are shown in Figs. 1 and 2, respectively. As seen from Fig. 1, the irreversible Carnot heat engine operates between a heat source (CSC) at temperature TH and a heat sink (ambient) at temperature TA. In Fig. 1, AS~ and AS2 are, respectively, the entropy production of the working fluid in two isothermal branches at temperatures TH and TA, and they are defined to be positive. According to the second law of thermodynamics, (QH/TH)
- -
(QA/TA) < O,
(1)
where QH a n d 0A are the heat input to and heat rejected from the heat engine, respectively. In order to describe the effects of internal dissipations in the working fluid on the performance of the heat engine, we introduce the parameter [9]
*Author for correspondence. 481
482
S. G6ktun and S. Ozkaynak
T
TH . . . . . . . . . . . . .
AS~
T~......../
AS2
t
S Fig. 1. T-S diagram of an irreversible Carnot heat engine.
T
(~H
TH HEAT ENGINE QA T^ . . . . .
&S2
..
Q^ REV~GERATOR
AS~
QL
s Fig. 2. T-S diagram of an irreversible Cannot absorption refrigerator.
R = ASI/AS2,
(2)
which characterizes the degree of internal irreversibility in the working fluid. The inequality in Eq. (1) may now be written as ( O H / T . ) - R(QA/TA) = 0,
R < 1,
(3)
where R is called a cycle-irreversibility parameter [9]. From the first law and Eq. (3), the maximum efficiency of the irreversible Camot heat engine is obtained as '}']CI 1 - (TA/RTH) . =
(4)
As seen from Fig. 2, the irreversible Carnot absorption refrigerator may be modelled as a combined
Carnot heat engine
483
cycle, which consists of an irreversible Camot heat engine and an irreversible Camot refrigerator operating between the heat sink (ambient) at temperature TA and the cooled space at temperature TL. Using the second law and Eq. (2) for the refrigerator, (0L/TL)
--
R(OHITH)=
O,
(5)
where QL is the cooling load. From the first law and Eqs. (3) and (5), the maximum COP of the irreversible Carnot absorption refrigerator is seen to be (6)
[3 = TL( RTH - T A ) / T . ( TA - R T L ) .
For steady-state conditions, the thermal efficiency for a CSC is given by [10] rico,, = 0.68
-
7(TH
(7)
TA)/I,
-
where TH and T A are the collector and ambient temperatures, respectively. I is the average value of total solar insolation. The overall system efficiency r/SH for CSC-driven irreversible Carnot heat engine is "I~SH = 'Y~colI'YICI.
(8)
Similarly, the overall system COP [3SA for a CSC-driven, irreversible Carnot absorption refrigerator is [3SA = "0co,,[3 •
(9)
r/sH = [0.68 - 7(TH -- T A ) / I ] [ 1 -- ( T A / R T H ) ] .
(lO)
(0T/SH/0TH) = 0,
(11)
Using Eqs. (4), (7) and (8),
For the extremal condition
the optimum CSC temperature for an irreversible Carnot heat engine becomes Tn(opt) = (Tg/R°5)(a + 1)0.5,
(12)
a = (0.68I)/7TA.
(13)
where
Equation (13) is the operating parameter of a CSC. Substituting Eq. (12) into Eq. (10), the maximum overall efficiency is seen to be 'Y]SH. . . . =
(0.68) {1 + ( l / a ) [ ( 1 / R )
+ 1 - 2 ( ( a + 1)/R)°5]}.
(14)
From Eqs. (6), (7) and (9), we obtain [3SA = [(0.68) -- 7(T H - T A ) / I ] [ ( T L / T H ) ( R T ,
- TA)/(TA -
RTL)].
(15)
For the extremal condition
(c3[3SA/OTH)=
O,
the optimum CSC temperature of the irreversible Camot absorption refrigerator is
(16)
484
S. G0ktun and S. 0zkaynak
(17)
YH(opt) = (TA/R°'5)(Q~ + 1 )0.5.
Substituting Eq. (17) into Eq. (15), the maximum overall COP is found to be given by ~SA . . . . = [ 1 ] S H . m a x ] [ T L R / ( T A
--
RTe)] .
(18)
Using the extremal condition (O'rlSH,max/OR)
(19)
= O,
the minimum value of the operating parameter becomes otm~. = (I/R) - 1.
(20)
It is interesting to note that Eqs. (14) and (18) yield zero for (Xmin. Variations of ~/S.,m~, and with a for different values of R are shown in Figs. 3 and 4, respectively.
~SA.max
RESULTS AND DISCUSSION
Using the energetic optimization technique, the optimal performance of a CSC-driven, irreversible, Camot heat engine and absorption refrigerator system has been investigated. One important consideration in energetically optimizing a solar-driven heat engine or refrigerator is the trade-off between maximum overall system efficiency or COP and minimum operating parameters of the solar equipment. Analyses of such systems have shown that the cycle-irreversibility parameter R has a strong effect on the maximum overall system efficiency and COP for a given value of ami,. As seen in Figs. 3 and 4, the maximum overall system efficiency and COP for the systems under consideration increase with R. This result is due to decreasing loss mechanisms of the systems. When R = 1 (i.e. a reversible heat engine or refrigerator), the maximum overall system efficiency (and the COP) reaches its upper bound. It is seen from Figs. 3 and 4, that the CSC-driven systems under consideration need a minimum value for the total solar insolation to overcome internal irreversibilities for start-up. For example, for T A =303K and R=0.8, Eq. (20) yields aml, =0.25. Using this O/min in Eq. (13), the minimum value for the solar insolation is found to be 780 W / m 2. These results provide a basis for the determination of the optimal operating conditions. TI,,~. R=-I.0
0.2
R=0.9 R=0.8
0.15
R=0.7
R=0.6
0.1 R=0.5
0.05
0
0.5
I
1.5
2
2.5
3 (7.
Fig. 3. The variation of rls.,m~ with (x for different values of R.
Carnot heat engine
485
13, . R=I.0
2.5
2
1.5
1
R--0.9
0.5
R--0~ R=0.7
0
~
0
0.5
1
1.5
r ~
2
2.5 QL
Fig. 4. The variation of/3sA.... with ~ for different values of R. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Howell, J. R. and Bannerot, R. B., Solar Energy, 1977, 19, 149. Gordon, J. M., Solar Energy, 1988, 40, 457. Lee, W. Y., and Kim, S. S., International Journal of Energy Research, 1991, 15, 257. Badescu, V., Energy--The International Journal, 1992, 17, 601. Chen, J., Journal of Applied Physics, 1992, 72, 3778. Eldighidy, S. M., Solar Energy, 1993, 51, 175. Yan, Z. and Chen, J., Journal of Applied Physics, 1994, 76, 8129. GiSktun, S., Renewable Energy, 1996, 7, 67. Wu, C. and Kiang, R. L., Energy--The International Journal, 1992, 17, 1173. Shiang-An, W., Solar Energy, 1979, 23, 333. NOMENCLATURE COP = Coefficient of performance CSC = Corrugated sheet collector 1= Average value of total solar insolation R = Cycle irreversibility parameter S = Specific entropy T = Absolute temperature TA= Ambient temperature T.= Temperature of the CSC TH(opt) ~ Optimum temperature of the CSC TL= Cooled space temperature QA = Heat rejected from the heat engine or refrigerator Heat input to the heat engine
QL ~-"Cooling load
c~= Operating parameter for the CSC Otrn~n= Minimum value of the operating parameter /3 = Maximum COP of the irreversible Carnot absorption refrigerator /3sA = Overall system COP /3SA.... = Maximum value of/3sA A = Difference r/ct = Maximum efficiency of the irreversible Carnot heat engine "qco, = Efficiency of the CSC */sH = Overall system efficiency r/sH.... = Maximum value of "0sn
Pergamon
PII: S0360-5442(96)00142-9
© 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0360-5442/97 $17.00 + 0.00
OPTIMUM PERFORMANCE OF A CORRUGATED, COLLECTORDRIVEN, IRREVERSIBLE CARNOT HEAT ENGINE AND ABSORPTION REFRIGERATOR SELAHATTiN GOKTUN* and SULEYMAN OZKAYNAK* +i.T.fJ., Maritime Faculty, 81716 Tuzla-istanbul, Turkey and *Turkish Naval Academy, 81704 Tuzlaistanbul, Turkey
(Received 11 April 1996)
Abstract--The technique of energetic optimization is employed to investigate the optimal performance of a corrugated, collector-driven, irreversible Carnot heat engine and an absorption refrigerator. A minimum operating parameter and a relation between the maximum overall efficiencies are obtained for the systems under consideration. © 1997 Elsevier Science Ltd. All rights reserved.
INTRODUCTION
The corrugated sheet collector (CSC) is a simple solar utilization device which transfers the absorbed solar energy to a working fluid by direct contact. From the heat-transfer point of view, the advantage is evident. When a solar collector as a heat source is integrated with a heat engine or a refrigerator, one objective is to determine the operating conditions that will provide optimum system performance. In general, the solar-collector efficiency is decreased by the use of high temperatures for the working fluid. However, for heat engines or refrigerators, the efficiency or coefficient of performance (COP) increases with the supply temperature. This incompatibility in the temperature-efficiency relation indicates that any solar-collector heat engine or refrigerator combination will have an optimum operating temperature. This problem has been extensively studied by many authors [1-8]. The present study is a theoretical investigation on the optimum performance of CSC-driven, irreversible, Carnot heat and absorption refrigeration system by taking into consideration the upper design bound (maximum efficiency and/or maximum COP) of a heat engine and an absorption refrigerator. ANALYSIS
The irreversibilities for a thermodynamic cycle may be considered to be external and internal. External irreversibilities arise from the temperature differences required to transfer heat between the system and the heat source and sink. Internal irreversibilities result primarily from dissipative processes inside the working fluid. The T-S diagram of an irreversible Carnot heat engine and an irreversible Carnot absorption refrigerator are shown in Figs. 1 and 2, respectively. As seen from Fig. 1, the irreversible Carnot heat engine operates between a heat source (CSC) at temperature TH and a heat sink (ambient) at temperature TA. In Fig. 1, AS~ and AS2 are, respectively, the entropy production of the working fluid in two isothermal branches at temperatures TH and TA, and they are defined to be positive. According to the second law of thermodynamics, (QH/TH)
- -
(QA/TA) < O,
(1)
where QH a n d 0A are the heat input to and heat rejected from the heat engine, respectively. In order to describe the effects of internal dissipations in the working fluid on the performance of the heat engine, we introduce the parameter [9]
*Author for correspondence. 481
482
S. G6ktun and S. Ozkaynak
T
TH . . . . . . . . . . . . .
AS~
T~......../
AS2
t
S Fig. 1. T-S diagram of an irreversible Carnot heat engine.
T
(~H
TH HEAT ENGINE QA T^ . . . . .
&S2
..
Q^ REV~GERATOR
AS~
QL
s Fig. 2. T-S diagram of an irreversible Cannot absorption refrigerator.
R = ASI/AS2,
(2)
which characterizes the degree of internal irreversibility in the working fluid. The inequality in Eq. (1) may now be written as ( O H / T . ) - R(QA/TA) = 0,
R < 1,
(3)
where R is called a cycle-irreversibility parameter [9]. From the first law and Eq. (3), the maximum efficiency of the irreversible Camot heat engine is obtained as '}']CI 1 - (TA/RTH) . =
(4)
As seen from Fig. 2, the irreversible Carnot absorption refrigerator may be modelled as a combined
Carnot heat engine
483
cycle, which consists of an irreversible Camot heat engine and an irreversible Camot refrigerator operating between the heat sink (ambient) at temperature TA and the cooled space at temperature TL. Using the second law and Eq. (2) for the refrigerator, (0L/TL)
--
R(OHITH)=
O,
(5)
where QL is the cooling load. From the first law and Eqs. (3) and (5), the maximum COP of the irreversible Carnot absorption refrigerator is seen to be (6)
[3 = TL( RTH - T A ) / T . ( TA - R T L ) .
For steady-state conditions, the thermal efficiency for a CSC is given by [10] rico,, = 0.68
-
7(TH
(7)
TA)/I,
-
where TH and T A are the collector and ambient temperatures, respectively. I is the average value of total solar insolation. The overall system efficiency r/SH for CSC-driven irreversible Carnot heat engine is "I~SH = 'Y~colI'YICI.
(8)
Similarly, the overall system COP [3SA for a CSC-driven, irreversible Carnot absorption refrigerator is [3SA = "0co,,[3 •
(9)
r/sH = [0.68 - 7(TH -- T A ) / I ] [ 1 -- ( T A / R T H ) ] .
(lO)
(0T/SH/0TH) = 0,
(11)
Using Eqs. (4), (7) and (8),
For the extremal condition
the optimum CSC temperature for an irreversible Carnot heat engine becomes Tn(opt) = (Tg/R°5)(a + 1)0.5,
(12)
a = (0.68I)/7TA.
(13)
where
Equation (13) is the operating parameter of a CSC. Substituting Eq. (12) into Eq. (10), the maximum overall efficiency is seen to be 'Y]SH. . . . =
(0.68) {1 + ( l / a ) [ ( 1 / R )
+ 1 - 2 ( ( a + 1)/R)°5]}.
(14)
From Eqs. (6), (7) and (9), we obtain [3SA = [(0.68) -- 7(T H - T A ) / I ] [ ( T L / T H ) ( R T ,
- TA)/(TA -
RTL)].
(15)
For the extremal condition
(c3[3SA/OTH)=
O,
the optimum CSC temperature of the irreversible Camot absorption refrigerator is
(16)
484
S. G0ktun and S. 0zkaynak
(17)
YH(opt) = (TA/R°'5)(Q~ + 1 )0.5.
Substituting Eq. (17) into Eq. (15), the maximum overall COP is found to be given by ~SA . . . . = [ 1 ] S H . m a x ] [ T L R / ( T A
--
RTe)] .
(18)
Using the extremal condition (O'rlSH,max/OR)
(19)
= O,
the minimum value of the operating parameter becomes otm~. = (I/R) - 1.
(20)
It is interesting to note that Eqs. (14) and (18) yield zero for (Xmin. Variations of ~/S.,m~, and with a for different values of R are shown in Figs. 3 and 4, respectively.
~SA.max
RESULTS AND DISCUSSION
Using the energetic optimization technique, the optimal performance of a CSC-driven, irreversible, Camot heat engine and absorption refrigerator system has been investigated. One important consideration in energetically optimizing a solar-driven heat engine or refrigerator is the trade-off between maximum overall system efficiency or COP and minimum operating parameters of the solar equipment. Analyses of such systems have shown that the cycle-irreversibility parameter R has a strong effect on the maximum overall system efficiency and COP for a given value of ami,. As seen in Figs. 3 and 4, the maximum overall system efficiency and COP for the systems under consideration increase with R. This result is due to decreasing loss mechanisms of the systems. When R = 1 (i.e. a reversible heat engine or refrigerator), the maximum overall system efficiency (and the COP) reaches its upper bound. It is seen from Figs. 3 and 4, that the CSC-driven systems under consideration need a minimum value for the total solar insolation to overcome internal irreversibilities for start-up. For example, for T A =303K and R=0.8, Eq. (20) yields aml, =0.25. Using this O/min in Eq. (13), the minimum value for the solar insolation is found to be 780 W / m 2. These results provide a basis for the determination of the optimal operating conditions. TI,,~. R=-I.0
0.2
R=0.9 R=0.8
0.15
R=0.7
R=0.6
0.1 R=0.5
0.05
0
0.5
I
1.5
2
2.5
3 (7.
Fig. 3. The variation of rls.,m~ with (x for different values of R.
Carnot heat engine
485
13, . R=I.0
2.5
2
1.5
1
R--0.9
0.5
R--0~ R=0.7
0
~
0
0.5
1
1.5
r ~
2
2.5 QL
Fig. 4. The variation of/3sA.... with ~ for different values of R. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Howell, J. R. and Bannerot, R. B., Solar Energy, 1977, 19, 149. Gordon, J. M., Solar Energy, 1988, 40, 457. Lee, W. Y., and Kim, S. S., International Journal of Energy Research, 1991, 15, 257. Badescu, V., Energy--The International Journal, 1992, 17, 601. Chen, J., Journal of Applied Physics, 1992, 72, 3778. Eldighidy, S. M., Solar Energy, 1993, 51, 175. Yan, Z. and Chen, J., Journal of Applied Physics, 1994, 76, 8129. GiSktun, S., Renewable Energy, 1996, 7, 67. Wu, C. and Kiang, R. L., Energy--The International Journal, 1992, 17, 1173. Shiang-An, W., Solar Energy, 1979, 23, 333. NOMENCLATURE COP = Coefficient of performance CSC = Corrugated sheet collector 1= Average value of total solar insolation R = Cycle irreversibility parameter S = Specific entropy T = Absolute temperature TA= Ambient temperature T.= Temperature of the CSC TH(opt) ~ Optimum temperature of the CSC TL= Cooled space temperature QA = Heat rejected from the heat engine or refrigerator Heat input to the heat engine
QL ~-"Cooling load
c~= Operating parameter for the CSC Otrn~n= Minimum value of the operating parameter /3 = Maximum COP of the irreversible Carnot absorption refrigerator /3sA = Overall system COP /3SA.... = Maximum value of/3sA A = Difference r/ct = Maximum efficiency of the irreversible Carnot heat engine "qco, = Efficiency of the CSC */sH = Overall system efficiency r/sH.... = Maximum value of "0sn