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Thermoeconomic considerations in the optimum allocation of heat exchanger inventory for a power plant
Energy Conversion and Management 42 (2001) 1169±1179
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Thermoeconomic considerations in the optimum allocation of heat exchanger inventory for a power plant Mohamed A. Antar, Syed M. Zubair * Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, Mail Box 1474, Dhahran 31261, Saudi Arabia Received 18 April 2000; accepted 28 September 2000
Abstract Thermoeconomics is de®ned as the integration of thermodynamics with economics of thermal systems. In this paper, we discuss the thermoeconomics of heat exchanger units in a power plant for cost based optimal design conditions. In this regard, unit cost parameters of hot and cold end heat exchangers in distributing the heat transfer surface area of the power plant are considered for minimum total cost of the heat exchangers. A closed form expression is given in terms of unit costs of the conductances of both heat exchangers, and the results are presented in terms of a unit cost ratio, G, and the hot and cold end heat exchangers costs. The results demonstrate a strong dependence of the total cost function on the absolute temperature ratios as well as the hot to cold end conductance cost ratio. It is also shown that for the case of equal unit costs of the hot and cold end heat exchangers, the total conductance is equally divided between the two heat exchangers. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Thermoeconomics; Power plants; Heat exchangers; Optimization; Finite time thermodynamics
1. Introduction Thermodynamic analysis of power producing systems has gained considerable attention over several decades. Recently, it is combined with heat transfer and thermodynamic analysis in order to provide an additional dimension towards the understanding and improvement of design and performance evaluation of such systems. Among the topics that received signi®cant attention is the optimization of the systems for minimum irreversible losses and, hence, more power and improved thermal eciency. *
Corresponding author. Tel.: +966-3-860-2540; +966-3-860-2949. E-mail addresses: [email protected] (M.A. Antar), [email protected] (S.M. Zubair).
0196-8904/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 0 ) 0 0 1 3 4 - 5
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M.A. Antar, S.M. Zubair / Energy Conversion and Management 42 (2001) 1169±1179
Nomenclature A F G T U Q_ W_
area (m2 ) dimensionless cost ratio given by Eq. (9) unit cost conductance ratio of hot to cold end of power plant temperature (K) overall heat transfer coecient (W/m2 K) heat transfer rate (W) power (W)
Subscripts H high temperature C reversible (Carnot) compartment L low temperature min minimum opt optimum Greek e C c g h s
symbols minimum temperature ratio (TL /TH ) total cost ($) unit conductance cost [$/(W/K)] eciency high side temperature ratio THC =TH Carnot temperature ratio TLC =THC
Several investigators studied thermodynamic optimization of power producing plants for maximum power output or minimum hot and cold end heat exchanger inventories. It is important to note that maximization of power was considered by Novikov [1], El-Wakil [2] and Curzon and Ahlborn [3] in which they considered the simple system shown in Fig. 1. They considered that the two heat exchangers, that is, the hot side and cold side heat exchangers, were the only sources of irreversibility. The same problem was also investigated by Salamon et al. [4,5], Salamon and Nitzan [6], Anderson et al. [7] and De Vos [8]. In 1988, Bejan [9] re-examined this problem and identi®ed three sources of heat transfer irreversibilities: the hot end heat exchanger, the cold end heat exchanger and the direct heat leak to the ambient. He also indicated that if the designer aims at maximizing the power output, then he has to consider the optimum Carnot temperature ratio introduced by Curzon and Ahlborn [3] in addition to the optimum balance between the sizes of the hot and cold end heat exchangers, that is, setting UAH UAL leads to the best thermal design. Bejan [10] used a simple model to prove that a power plant with minimum heat exchanger irreversible losses and ®xed power output has the same eciency as the plant designed for maximum power output. He also showed that the minimization of plant inventory has two degrees of freedom, since two absolute temperature ratios are to be controlled, s TLC =THC and h THC =TH , while earlier investigators considered only a single degree of freedom system because only
M.A. Antar, S.M. Zubair / Energy Conversion and Management 42 (2001) 1169±1179
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Fig. 1. Power plant model with two heat exchangers and internally reversible cycle [9,10].
the hot end temperature, TH , is to be changed for optimum performance. Furthermore, Bejan [11] described several heat engine models, such as the models of Le [12] and Novikov [1], in which the same maximum eciency formula g 1 TL =TH 1=2 can be derived by minimizing the rate of entropy generation. In this paper, he considered the minimum entropy generation rate and criticized the Salamon et al. statement that maximum power and minimum entropy generation are two dierent operating conditions, showing that they are not independent. In another paper, Bejan [13] extended his earlier work [9] and showed that various power plant con®gurations can be maximized by properly dividing the ®xed heat exchangers inventory among the heat transfer components of a power plant. He performed his study on eight examples, including a solar power plant with a total heat exchanger area. The basic assumption applied to all the models is that the overall heat exchanger inventory is ®xed, thus providing guidelines for optimum distribution of the heat transfer areas. It is important to note that in all of the above studies, there is no consideration of the economics of a power plant. This paper aims at adding another dimension to the previous work that did not consider the possibility of dierent unit cost parameters of both the hot and cold end heat exchangers in distributing the total heat transfer area. In this regard, the model considered by Bejan [10] is extended for minimum inventory through a total cost expression of the heat transfer conductances that will be minimized for a given output power.
2. Analysis and discussion Consider the same power producing system that is studied by Bejan [10] where external irreversibilities are considered while the system is considered internally reversible, as shown in Fig. 1. The target in this investigation is to ®nd the minimum total cost of conductance (UA) for a given work output. This can be written in terms of unit cost parameters as
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C cH UAH cL UAL ! ! Q_ H Q_ L C cH cL ; TH THC TLC TL Q_ L Q_ H
cH TH THC
C Q_ H
!
cL
TLC
1 2
! TL
:
3
For an internally reversible system, we have: Q_ L TLC : Q_ H THC
4
Therefore, Eq. (3) can be simpli®ed to give: Q_ H cH TLC cL C : TH 1 THC =TH THC TLC =TH TL =TH
5
Introducing dimensionless absolute temperature ratios as s
TLC ; THC
h
THC ; TH
e
TL TH
and
TLC sh; TH
6
we may write the ®rst law and second law of thermodynamics as: W_ Q_ H g
Q_ L ;
W_ 1 Q_ H
TLC 1 THC
s:
Thus, Q_ H
W_ 1
s
:
Therefore, Eq. (5) can be further simpli®ed to give C cL G s ; W_ =TH 1 s 1 h sh e
7
8
where G is de®ned as the ratio of the unit costs of the conductances of the hot side to the cold side of a power plant, G cH =cL . In terms of dimensionless total cost, we have C= W_ =TH 1 G s F : 9 cL 1 s 1 h sh e Our target now is to minimize the dimensionless cost ratio F with respect to both high side temperature ratio, h and Carnot temperature ratio, s, i.e., minimizing the cost of conductance (UA) for a given work output, the low to high side temperature ratio e and the unit cost ratio G. This gives
M.A. Antar, S.M. Zubair / Energy Conversion and Management 42 (2001) 1169±1179
oF 1 oh 1 s
s2
G h2
1
sh
1173
! e2
0:
10
Simplifying the above equation, we get 2Ge Ge2 2 1 Gh 2 0: h 1 s s2 Solving the above equation results in p p 1 s Ge Gs Ge 1 s h1 and h2 s G 1 s
11
p Ge Gs G 1
p Ge
:
12
It is to be noted that this solution is applicable for all values of G except for G 1, that is, the case in which the unit conductance costs of the hot and cold end heat exchangers are the same. That is the same case as that discussed by Bejan [10]. However, L'HopitalÕs rule is applied to this function to determine the value of h1 when G 1. The solution for this special case is hopt s e=2s, which is the same expression given by Bejan [10]. It is also noticed that the same expression is reached by substituting G 1 in Eq. (11). Notice also that h2 will be discarded, since it does not reduce to the same expression for G 1 and does not give us meaningful results for dierent values of s, e and G. Substituting the value of h1 h in Eq. (9), the results were plotted in Fig. 2 which shows the minimum value of the cost function Fmin versus the ratio of unit costs of conductances of hot to cold side of a power plant, G, for dierent values of s and for e 0:2, 0.3, 0.4, and 0.5, in Figs. 2a± d, respectively. The ®gure indicates that F increases with G for all values of e. Moreover, the ®gure also shows that there is a certain value of Carnot temperature ratio, s at which F is minimum for all values of G (given the speci®ed value of the low to high side absolute temperature ratio, e). This optimum value of s increases with e which indicates that sopt depends on the value of e. This can be con®rmed, mathematically, as follows: Now substituting the value of hopt in Eq. (9) and dierentiating with respect to s to minimize the function F with respect to s gives: oFmin o 1 G s 0: 13 os 1 s 1 h sh e os Simplifying, we get: s
2
e G
2
1
s
This gives the condition p sopt e
2
1 G
p G s
p G e s
p Gs
p Ge
0: e
14
15
Finally, We get the minimum total dimensionless cost with respect to the absolute temperature ratios s and h as:
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Fig. 2. The dimensionless cost function F versus Carnot temperature ratio, s for dierent values of unit cost ratio G (a) e 0:2, (b) e 0:3, (c) e 0:4 and (d) e 0:5.
Fmin;min
C= W_ =TH 1 p cL 1 e
G 1
p e : p h eh e
16
The minimum dimensionless cost Fmin;min is plotted in Fig. 3 as a function of the unit cost ratio G for several values of dimensionless absolute temperature ratio e. The ®gure indicates an increase of the cost with unit cost ratio G. The ®gure also shows that a decrease in the temperature difference between the hot and cold side (i.e. increasing e) results in an increase of the cost which is attributed to using a larger heat transfer surface area. The ratio of the hot end to cold end conductances is calculated and plotted in Fig. 4 which indicates that increasing the hot end unit conductance cost or decreasing the cold end cost would result in an increase in the total cost of the
M.A. Antar, S.M. Zubair / Energy Conversion and Management 42 (2001) 1169±1179
1175
1000
t=TLC/THC= (e)0.5
e = TL/TH = 0.5
100
0.4
Fmin,min
0.3 0.2 0.1 10
1 0.1
1 G = gH/gL
10
Fig. 3. The minimum dimensionless cost versus unit cost ratio G for dierent values of the dimensionless absolute temperature ratio, e TL =TH .
gH(UA)H/gL(UA)L
10
1
0.1 0.1
1 G = gH/gL
10
Fig. 4. The cost ratio of the hot to cold side conductance versus unit cost ratio, G cH =cL .
power plant conductance. This result is also con®rmed in Fig. 3 which shows that increasing G results in an increase in the minimum total dimensionless cost, Fmin;min . It is now desirable to obtain an expression for the ratio between the total conductance costs of the hot side and cold side with respect to the total cost. We note that the absolute temperature
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ratios h and s in the following part will be substituted by their optimum values. This gives the conductance cost of the hot side of a power plant by: W_ 1 1 cH UAH GcL TH 1 s 1 h or in terms of dimensionless cost values, we have cH UAH = W_ =TH G 1 : cL 1 s 1 h
17
Now, from Eqs. (16), (17) and (1), one can write p p G 1 cH UAH G sh e G : Gsh Ge s sh C G 1
18
We note that when we substitute G 1 in the above equation, it reduces to 0.5, which is the value also obtained from Bejan's work [10]. Similarly, for the cold end heat exchanger, one may write cL UAL cL
W_ 1 s TH 1 s sh e
or in terms of dimensionless costs, we have: cL UAL = W_ =TH 1 s : cL 1 s sh e Combining Eqs. (16), (19) and (1), we get: ! cL UAL s s 1 h G s Gsh Ge s C sh e 1 h sh e
19
sh
p G p 3 : 1 G
1 G
20
It is important to mention that when we put G 1, both Eqs. (18) and (20) reduce to a value of 1/ 2 which is the optimum solution given by Bejan [10], i.e. if the unit costs of both the hot and cold side heat exchangers are the same, the optimum ratio reduces to distributing the heat exchanger areas equally on both the cold and hot side of a power plant. Fig. 5 shows the ratios of the costs of the hot and cold side to the total cost. As expected, both ratios are equal to 0.5 when G 1. The ®gure also shows that increasing G increases the hot side conductance cost ratio and decreases the cold side conductance cost ratio. 3. Illustrative example In order to show the applicability of the model presented in this paper, an example is considered where the hot end temperature TH 1000 K, while the cold end temperature TL 300 K e TL =TH 0:3. The cold end unit conductance cost 10 $/(W/K), or (10,000 $/kW/K). Calculate the optimum total conductance cost of the power plant.
M.A. Antar, S.M. Zubair / Energy Conversion and Management 42 (2001) 1169±1179
1177
Fig. 5. The cost ratio of hot or cold side to the total heat exchanger conductance cost.
Solution: On using Eq. (16), one can express the total cost per unit kW power output as p p e e C cL =TH G 10 G p p : 21 p p _ eh e eh e 1 e 1 h 1 e 1 h W This relation is plotted in Fig. 6 as a function of the unit cost ratio G for the given value of e and for optimum h and s. As can be seen from this ®gure, the result is generalized on the same ®gure for dierent values of e to cover a broad range of hot and cold end temperatures e 0:2; 0:3; 0:4 and 0:5, thus providing the total cost per unit power output for dierent cases corresponding to dierent hot and cold end temperatures. We note from this ®gure that there is about a 45% increase in the total cost per unit power produced when the value of the unit cost ratio, G is increased from 1 to 2. It is important to note that the hot and cold side conductances can also be calculated for a speci®ed power output (say 1000 kW). The total cost of conductance C is calculated from Eq. (21). (C $84693.92 for G 0:1). Then, using Eq. (18), we can calculate cL UAL $64,345.95, and similarly, from Eq. (20), we calculate cH UAH $20,347.97. Since, in this example, cL 10,000$/ (kW/K), thus cH 1000 $/(kW/K) (for G 0:1), and hence, the conductances are UAL 6:43 kW/K and UAH 20:35 kW/K. Fig. 7 shows both the hot and cold end conductances versus the unit cost ratio G for this example. As expected, the ®gure shows that UAL UAH when G 1. 4. Conclusion Earlier studies of optimizing the power plant design and performance conditions are extended to consider the unit costs of heat exchanger conductance (or inventory). Optimum values of the absolute temperature ratios are obtained where the high side absolute temperature ratio, h
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Fig. 6. Total cost versus unit cost ratio G for dierent values of the minimum absolute temperature ratio e TL =TH for the illustrative example.
Fig. 7. Hot and cold side conductances versus G for the illustrative example.
depends on the unit cost ratio G G cH =cL , while it is found that the Carnot temperature ratio s is independent of G. Moreover, expressions for the ratio of hot and cold end heat exchanger costs to the total cost are also obtained which show a nonlinear trend. It is also shown that if both the heat exchangers have the same unit cost, i.e. G 1, the heat exchanger costs of the hot or cold end heat exchanger to the total cost is equal to 0.5. In this case, the heat exchangers conductance and,
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1179
hence, the area would be divided equally between the hot and cold end heat exchangers. This latter result represents the special case analyzed by Bejan [10] where the cost of conductance was not considered.
Acknowledgements The support provided by King Fahd University of Petroleum and Minerals (KFUPM) to conduct this investigation is gratefully acknowledged.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Novikov II. The eciency of atomic power stations. J Nucl Ener II 1958;7:125±8. El-Wakil MM. Nuclear power engineering. New York: McGraw-Hill; 1962. p. 162±5. Curzon FL, Ahlborn B. Eciency of a Carnot engine at maximum power output. Am J Phys 1975;43:22±4. Salamon P, Nitzan A, Andresen B, Berry RS. Minimum entropy production and the optimization of heat engines. Phys Rev 1980;A21:2115±29. Salamon P, Band YP, Kafri O. Maximum power from cycling a working ¯uid. J Appl Phys 1982;53:197±202. Salamon P, Nitzan A. Finite time optimization of a Newton's law Carnot cycle. J Chem Phys 1981;74:3546±60. Andresen B, Salamon P, Berry RS. Thermodynamics in ®nite time. Phys Today September, 1984:62±70. De VOS A. Eciency of some heat engines at maximum power conditions. Am J Phys 1985;53:570±3. Bejan A. Theory of heat transfer-irreversible power plants. Int J Heat Mass Transf 1988;31(6):1211±9. Bejan A. Power and refrigeration plants for minimum heat exchanger inventory. ASME J Ener Res Tech 1993;115:148±50. Bejan A. Models of power plants that generate minimum entropy while operating at maximum power. Am J Phys 1996;64:1054±9. Le HS. Thermal eciency at maximum power output: new results for old heat engines. Am J Phys 1987;55:602± 10. Bejan A. Theory of heat transfer-irreversible power plants: II. The optimal allocation of heat exchange equipment. Int J Heat Mass Transf 1995;38(3):433±44.
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Thermoeconomic considerations in the optimum allocation of heat exchanger inventory for a power plant Mohamed A. Antar, Syed M. Zubair * Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, Mail Box 1474, Dhahran 31261, Saudi Arabia Received 18 April 2000; accepted 28 September 2000
Abstract Thermoeconomics is de®ned as the integration of thermodynamics with economics of thermal systems. In this paper, we discuss the thermoeconomics of heat exchanger units in a power plant for cost based optimal design conditions. In this regard, unit cost parameters of hot and cold end heat exchangers in distributing the heat transfer surface area of the power plant are considered for minimum total cost of the heat exchangers. A closed form expression is given in terms of unit costs of the conductances of both heat exchangers, and the results are presented in terms of a unit cost ratio, G, and the hot and cold end heat exchangers costs. The results demonstrate a strong dependence of the total cost function on the absolute temperature ratios as well as the hot to cold end conductance cost ratio. It is also shown that for the case of equal unit costs of the hot and cold end heat exchangers, the total conductance is equally divided between the two heat exchangers. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Thermoeconomics; Power plants; Heat exchangers; Optimization; Finite time thermodynamics
1. Introduction Thermodynamic analysis of power producing systems has gained considerable attention over several decades. Recently, it is combined with heat transfer and thermodynamic analysis in order to provide an additional dimension towards the understanding and improvement of design and performance evaluation of such systems. Among the topics that received signi®cant attention is the optimization of the systems for minimum irreversible losses and, hence, more power and improved thermal eciency. *
Corresponding author. Tel.: +966-3-860-2540; +966-3-860-2949. E-mail addresses: [email protected] (M.A. Antar), [email protected] (S.M. Zubair).
0196-8904/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 0 ) 0 0 1 3 4 - 5
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Nomenclature A F G T U Q_ W_
area (m2 ) dimensionless cost ratio given by Eq. (9) unit cost conductance ratio of hot to cold end of power plant temperature (K) overall heat transfer coecient (W/m2 K) heat transfer rate (W) power (W)
Subscripts H high temperature C reversible (Carnot) compartment L low temperature min minimum opt optimum Greek e C c g h s
symbols minimum temperature ratio (TL /TH ) total cost ($) unit conductance cost [$/(W/K)] eciency high side temperature ratio THC =TH Carnot temperature ratio TLC =THC
Several investigators studied thermodynamic optimization of power producing plants for maximum power output or minimum hot and cold end heat exchanger inventories. It is important to note that maximization of power was considered by Novikov [1], El-Wakil [2] and Curzon and Ahlborn [3] in which they considered the simple system shown in Fig. 1. They considered that the two heat exchangers, that is, the hot side and cold side heat exchangers, were the only sources of irreversibility. The same problem was also investigated by Salamon et al. [4,5], Salamon and Nitzan [6], Anderson et al. [7] and De Vos [8]. In 1988, Bejan [9] re-examined this problem and identi®ed three sources of heat transfer irreversibilities: the hot end heat exchanger, the cold end heat exchanger and the direct heat leak to the ambient. He also indicated that if the designer aims at maximizing the power output, then he has to consider the optimum Carnot temperature ratio introduced by Curzon and Ahlborn [3] in addition to the optimum balance between the sizes of the hot and cold end heat exchangers, that is, setting UAH UAL leads to the best thermal design. Bejan [10] used a simple model to prove that a power plant with minimum heat exchanger irreversible losses and ®xed power output has the same eciency as the plant designed for maximum power output. He also showed that the minimization of plant inventory has two degrees of freedom, since two absolute temperature ratios are to be controlled, s TLC =THC and h THC =TH , while earlier investigators considered only a single degree of freedom system because only
M.A. Antar, S.M. Zubair / Energy Conversion and Management 42 (2001) 1169±1179
1171
Fig. 1. Power plant model with two heat exchangers and internally reversible cycle [9,10].
the hot end temperature, TH , is to be changed for optimum performance. Furthermore, Bejan [11] described several heat engine models, such as the models of Le [12] and Novikov [1], in which the same maximum eciency formula g 1 TL =TH 1=2 can be derived by minimizing the rate of entropy generation. In this paper, he considered the minimum entropy generation rate and criticized the Salamon et al. statement that maximum power and minimum entropy generation are two dierent operating conditions, showing that they are not independent. In another paper, Bejan [13] extended his earlier work [9] and showed that various power plant con®gurations can be maximized by properly dividing the ®xed heat exchangers inventory among the heat transfer components of a power plant. He performed his study on eight examples, including a solar power plant with a total heat exchanger area. The basic assumption applied to all the models is that the overall heat exchanger inventory is ®xed, thus providing guidelines for optimum distribution of the heat transfer areas. It is important to note that in all of the above studies, there is no consideration of the economics of a power plant. This paper aims at adding another dimension to the previous work that did not consider the possibility of dierent unit cost parameters of both the hot and cold end heat exchangers in distributing the total heat transfer area. In this regard, the model considered by Bejan [10] is extended for minimum inventory through a total cost expression of the heat transfer conductances that will be minimized for a given output power.
2. Analysis and discussion Consider the same power producing system that is studied by Bejan [10] where external irreversibilities are considered while the system is considered internally reversible, as shown in Fig. 1. The target in this investigation is to ®nd the minimum total cost of conductance (UA) for a given work output. This can be written in terms of unit cost parameters as
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M.A. Antar, S.M. Zubair / Energy Conversion and Management 42 (2001) 1169±1179
C cH UAH cL UAL ! ! Q_ H Q_ L C cH cL ; TH THC TLC TL Q_ L Q_ H
cH TH THC
C Q_ H
!
cL
TLC
1 2
! TL
:
3
For an internally reversible system, we have: Q_ L TLC : Q_ H THC
4
Therefore, Eq. (3) can be simpli®ed to give: Q_ H cH TLC cL C : TH 1 THC =TH THC TLC =TH TL =TH
5
Introducing dimensionless absolute temperature ratios as s
TLC ; THC
h
THC ; TH
e
TL TH
and
TLC sh; TH
6
we may write the ®rst law and second law of thermodynamics as: W_ Q_ H g
Q_ L ;
W_ 1 Q_ H
TLC 1 THC
s:
Thus, Q_ H
W_ 1
s
:
Therefore, Eq. (5) can be further simpli®ed to give C cL G s ; W_ =TH 1 s 1 h sh e
7
8
where G is de®ned as the ratio of the unit costs of the conductances of the hot side to the cold side of a power plant, G cH =cL . In terms of dimensionless total cost, we have C= W_ =TH 1 G s F : 9 cL 1 s 1 h sh e Our target now is to minimize the dimensionless cost ratio F with respect to both high side temperature ratio, h and Carnot temperature ratio, s, i.e., minimizing the cost of conductance (UA) for a given work output, the low to high side temperature ratio e and the unit cost ratio G. This gives
M.A. Antar, S.M. Zubair / Energy Conversion and Management 42 (2001) 1169±1179
oF 1 oh 1 s
s2
G h2
1
sh
1173
! e2
0:
10
Simplifying the above equation, we get 2Ge Ge2 2 1 Gh 2 0: h 1 s s2 Solving the above equation results in p p 1 s Ge Gs Ge 1 s h1 and h2 s G 1 s
11
p Ge Gs G 1
p Ge
:
12
It is to be noted that this solution is applicable for all values of G except for G 1, that is, the case in which the unit conductance costs of the hot and cold end heat exchangers are the same. That is the same case as that discussed by Bejan [10]. However, L'HopitalÕs rule is applied to this function to determine the value of h1 when G 1. The solution for this special case is hopt s e=2s, which is the same expression given by Bejan [10]. It is also noticed that the same expression is reached by substituting G 1 in Eq. (11). Notice also that h2 will be discarded, since it does not reduce to the same expression for G 1 and does not give us meaningful results for dierent values of s, e and G. Substituting the value of h1 h in Eq. (9), the results were plotted in Fig. 2 which shows the minimum value of the cost function Fmin versus the ratio of unit costs of conductances of hot to cold side of a power plant, G, for dierent values of s and for e 0:2, 0.3, 0.4, and 0.5, in Figs. 2a± d, respectively. The ®gure indicates that F increases with G for all values of e. Moreover, the ®gure also shows that there is a certain value of Carnot temperature ratio, s at which F is minimum for all values of G (given the speci®ed value of the low to high side absolute temperature ratio, e). This optimum value of s increases with e which indicates that sopt depends on the value of e. This can be con®rmed, mathematically, as follows: Now substituting the value of hopt in Eq. (9) and dierentiating with respect to s to minimize the function F with respect to s gives: oFmin o 1 G s 0: 13 os 1 s 1 h sh e os Simplifying, we get: s
2
e G
2
1
s
This gives the condition p sopt e
2
1 G
p G s
p G e s
p Gs
p Ge
0: e
14
15
Finally, We get the minimum total dimensionless cost with respect to the absolute temperature ratios s and h as:
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Fig. 2. The dimensionless cost function F versus Carnot temperature ratio, s for dierent values of unit cost ratio G (a) e 0:2, (b) e 0:3, (c) e 0:4 and (d) e 0:5.
Fmin;min
C= W_ =TH 1 p cL 1 e
G 1
p e : p h eh e
16
The minimum dimensionless cost Fmin;min is plotted in Fig. 3 as a function of the unit cost ratio G for several values of dimensionless absolute temperature ratio e. The ®gure indicates an increase of the cost with unit cost ratio G. The ®gure also shows that a decrease in the temperature difference between the hot and cold side (i.e. increasing e) results in an increase of the cost which is attributed to using a larger heat transfer surface area. The ratio of the hot end to cold end conductances is calculated and plotted in Fig. 4 which indicates that increasing the hot end unit conductance cost or decreasing the cold end cost would result in an increase in the total cost of the
M.A. Antar, S.M. Zubair / Energy Conversion and Management 42 (2001) 1169±1179
1175
1000
t=TLC/THC= (e)0.5
e = TL/TH = 0.5
100
0.4
Fmin,min
0.3 0.2 0.1 10
1 0.1
1 G = gH/gL
10
Fig. 3. The minimum dimensionless cost versus unit cost ratio G for dierent values of the dimensionless absolute temperature ratio, e TL =TH .
gH(UA)H/gL(UA)L
10
1
0.1 0.1
1 G = gH/gL
10
Fig. 4. The cost ratio of the hot to cold side conductance versus unit cost ratio, G cH =cL .
power plant conductance. This result is also con®rmed in Fig. 3 which shows that increasing G results in an increase in the minimum total dimensionless cost, Fmin;min . It is now desirable to obtain an expression for the ratio between the total conductance costs of the hot side and cold side with respect to the total cost. We note that the absolute temperature
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ratios h and s in the following part will be substituted by their optimum values. This gives the conductance cost of the hot side of a power plant by: W_ 1 1 cH UAH GcL TH 1 s 1 h or in terms of dimensionless cost values, we have cH UAH = W_ =TH G 1 : cL 1 s 1 h
17
Now, from Eqs. (16), (17) and (1), one can write p p G 1 cH UAH G sh e G : Gsh Ge s sh C G 1
18
We note that when we substitute G 1 in the above equation, it reduces to 0.5, which is the value also obtained from Bejan's work [10]. Similarly, for the cold end heat exchanger, one may write cL UAL cL
W_ 1 s TH 1 s sh e
or in terms of dimensionless costs, we have: cL UAL = W_ =TH 1 s : cL 1 s sh e Combining Eqs. (16), (19) and (1), we get: ! cL UAL s s 1 h G s Gsh Ge s C sh e 1 h sh e
19
sh
p G p 3 : 1 G
1 G
20
It is important to mention that when we put G 1, both Eqs. (18) and (20) reduce to a value of 1/ 2 which is the optimum solution given by Bejan [10], i.e. if the unit costs of both the hot and cold side heat exchangers are the same, the optimum ratio reduces to distributing the heat exchanger areas equally on both the cold and hot side of a power plant. Fig. 5 shows the ratios of the costs of the hot and cold side to the total cost. As expected, both ratios are equal to 0.5 when G 1. The ®gure also shows that increasing G increases the hot side conductance cost ratio and decreases the cold side conductance cost ratio. 3. Illustrative example In order to show the applicability of the model presented in this paper, an example is considered where the hot end temperature TH 1000 K, while the cold end temperature TL 300 K e TL =TH 0:3. The cold end unit conductance cost 10 $/(W/K), or (10,000 $/kW/K). Calculate the optimum total conductance cost of the power plant.
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Fig. 5. The cost ratio of hot or cold side to the total heat exchanger conductance cost.
Solution: On using Eq. (16), one can express the total cost per unit kW power output as p p e e C cL =TH G 10 G p p : 21 p p _ eh e eh e 1 e 1 h 1 e 1 h W This relation is plotted in Fig. 6 as a function of the unit cost ratio G for the given value of e and for optimum h and s. As can be seen from this ®gure, the result is generalized on the same ®gure for dierent values of e to cover a broad range of hot and cold end temperatures e 0:2; 0:3; 0:4 and 0:5, thus providing the total cost per unit power output for dierent cases corresponding to dierent hot and cold end temperatures. We note from this ®gure that there is about a 45% increase in the total cost per unit power produced when the value of the unit cost ratio, G is increased from 1 to 2. It is important to note that the hot and cold side conductances can also be calculated for a speci®ed power output (say 1000 kW). The total cost of conductance C is calculated from Eq. (21). (C $84693.92 for G 0:1). Then, using Eq. (18), we can calculate cL UAL $64,345.95, and similarly, from Eq. (20), we calculate cH UAH $20,347.97. Since, in this example, cL 10,000$/ (kW/K), thus cH 1000 $/(kW/K) (for G 0:1), and hence, the conductances are UAL 6:43 kW/K and UAH 20:35 kW/K. Fig. 7 shows both the hot and cold end conductances versus the unit cost ratio G for this example. As expected, the ®gure shows that UAL UAH when G 1. 4. Conclusion Earlier studies of optimizing the power plant design and performance conditions are extended to consider the unit costs of heat exchanger conductance (or inventory). Optimum values of the absolute temperature ratios are obtained where the high side absolute temperature ratio, h
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Fig. 6. Total cost versus unit cost ratio G for dierent values of the minimum absolute temperature ratio e TL =TH for the illustrative example.
Fig. 7. Hot and cold side conductances versus G for the illustrative example.
depends on the unit cost ratio G G cH =cL , while it is found that the Carnot temperature ratio s is independent of G. Moreover, expressions for the ratio of hot and cold end heat exchanger costs to the total cost are also obtained which show a nonlinear trend. It is also shown that if both the heat exchangers have the same unit cost, i.e. G 1, the heat exchanger costs of the hot or cold end heat exchanger to the total cost is equal to 0.5. In this case, the heat exchangers conductance and,
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hence, the area would be divided equally between the hot and cold end heat exchangers. This latter result represents the special case analyzed by Bejan [10] where the cost of conductance was not considered.
Acknowledgements The support provided by King Fahd University of Petroleum and Minerals (KFUPM) to conduct this investigation is gratefully acknowledged.
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