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Entropy and entransy analyses and optimizations of the Rankine cycle
Energy Conversion and Management 68 (2013) 82–88
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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Entropy and entransy analyses and optimizations of the Rankine cycle WenHua Wang, XueTao Cheng, XinGang Liang ⇑ Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Article history: Received 27 October 2012 Received in revised form 7 December 2012 Accepted 24 December 2012
Keywords: Entropy generation Entransy loss The Rankine cycle Heat-work conversion Phase change
a b s t r a c t The Rankine cycle has been widely used in industry, and the optimization of the Rankine cycle is of great significance to energy utilization improvement. In this paper, both the concepts of entropy generation and entransy loss are applied to analyzing the thermodynamic processes of the Rankine cycle that receives heat from the exhaust-heat boiler with a hot stream and releases heat to a cold stream. In order to get the maximum output power, the mass flow rate of the working medium is optimized. Theoretical analysis and numerical results show that both the maximum entransy loss rate and the minimum entropy generation rate correspond to the maximum output power when the inlet temperatures and the heat capacity flow rates of the streams are prescribed. When the inlet temperatures or the heat capacity flow rates of the streams is not prescribed, smaller entropy generation rate does not lead to larger output power but larger entransy loss rate still leads to larger power output for this kind of cases. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The rapid increase in global energy consumption has been boosting the development of the energy conservation technologies [1–4]. The optimization design of the Rankine cycle is of great significance to the improvement of energy utilization because the cycle is widely used in the thermal power plants [5], the solar power and geothermal power plants [6–8], the cogeneration technology [9–11], the comprehensive polygeneration energy system [12– 14], etc. Many researchers have paid their attention on this topic, such as Gu et al. [3], Yaghoubi et al. [4], and Ahmadi et al. [12–14]. The Rankine cycle is a typical heat-work conversion process with phase change and entropy generation is most widely used in its optimizations. The concept of entropy was proposed by Clausius when he analyzed the Carnot cycle [15]. As entropy generation represents the ability loss of doing work, the minimum entropy generation is thought to correspond to the optimal performance of the corresponding thermodynamic cycle [16]. For instance, Yaghoubi et al. [4] optimized the mass flow rate of a solar power plant consisting of the Rankine cycle based on the entropy generation minimization method with the fixed ambient conditions. However, there are heat-work conversion optimization cases in which the minimum entropy generation dese not correspond the best performance. In the investigation of the refrigeration systems, Klein and Reindl [17] found that minimizing the entropy generation rate does not always result in the same design as maximizing the system performance unless the refrigeration capacity is fixed.
⇑ Corresponding author. Tel./fax: +86 10 62788702. E-mail address: [email protected] (X.G. Liang). 0196-8904/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.enconman.2012.12.020
Cheng et al. [18] revealed that the entropy generation could not be related to the net work change in the air standard cycle when the heat absorbed by the working fluid is from the combustion reaction of the gas fuel. Therefore, for the optimization of heatwork conversion processes, the applicability and preconditions of the entropy theory need further discussions. In recent years, the concept of entransy [19] was developed for the optimization design of thermal system, and was applied to heat transfer optimization [19–32]. In heat-work conversion optimization, Cheng et al. [33] defined the concept of entransy loss and extended the entransy theory to heat-work conversion optimizations. In the analyses of the air standard cycle [18], the Stirling engine cycle [34] and the one-stream heat exchanger networks [35], the results show that the increase in entransy loss rate always leads to the increase in output power. However, the applicability of the entransy theory to optimizing thermodynamic cycles stills needs further investigations because there are not many reports. In the present report, the Rankine cycle with water as the working fluid that uses waste heat from a hot stream to produce power is analyzed and optimized. The concept of entransy loss is applied to the analyses of the Rankine cycle for the first time. The cycle system is optimized with the concepts of entropy generation, entransy loss and entransy dissipation. The relationships between the concepts of entropy generation, entransy loss, entransy dissipation and the output power for the Rankine cycle are derived and demonstrated. There are optimum values for the mass flow rate of the working fluid and the thermal conductance that correspond to the maximum output work.
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Nomenclature C c G_ loss G_ dis _ m NTU Q_ r S_ f S_ g DS_ s T T0 UA _ W
Subscripts c cold side eh from hot stream to environment ec from cold stream to environment g gas state h hot side h1 superheating section h2 evaporating section h3 preheating section hin2 out of the superheating section hin3 out of the evaporating section in into the system l liquid state min the minimum value max the maximum value out out of the system
heat capacity flow rate, W/K specific heat capacity, J/(kg K) entransy loss rate, W K entransy dissipation rate, W K mass flow rate, kg the number of transfer units heat transfer rate, W the latent heat, J/kg entropy flow rate, W/K entropy generation rate, W/K entropy change rate, W/K specific entropy, J/(kg K) temperature, K environment temperature, K thermal conductance, W/K output power, W
Greek symbols e effectiveness
The applicability and the relevant preconditions of the entropy generation, entransy loss and entransy dissipation for the Rankine cycle are discussed. 2. Entransy, entransy dissipation and entransy loss Entransy is proposed to describe heat transfer ability [19]. It represents the potential energy of heat in a body corresponding to the electrical energy in a capacitor based on the analogy between electrical and thermal systems, and it is defined as
Evh ¼
1 Q T; 2 vh
ð1Þ
where Qvh = McvT is the thermal energy or the heat stored in an object with constant volume which may be referred to as the thermal charge, in which M is the mass and cv is the specific heat capacity. The temperature T represents the thermal potential. In heat transfer processes, it was found that the entransy always decreases in spontaneous heat transfer [19]. The decreased part is called entransy dissipation [19], which has been used in heat transfer optimizations [19–32]. The balance equation of entransy dissipation can be derived from the energy conservation of heat transfer. For instance, multiplying the energy equation of heat conduction by temperature leads to the entransy balance equation for heat conduction [19],
T qc v
@T _ ðq_ rTÞ; ¼ r ðqTÞ @t
ð2Þ
where q is the density, q_ is the heat flux vector, t is the time, the left term is the entransy change of the infinitesimal element, the first term on the right-hand side is the entransy flow into the element, and the second term on the right is the local rate of entransy dissipation Gdis. Based on the Fourier’s law, the entransy dissipation per unit time and per unit volume is k(rT)2 [19], in which k is the thermal conductivity and rT is the temperature gradient. The extremum entransy dissipation was proved appropriate as the criterion for the optimal heat transfer performance. Guo et al. [19] derived the extremum entransy dissipation principles based on the entransy balance equation for heat convection. The extremum value of entransy dissipation was found to relate to minimum heat transfer temperature difference for fixed heat flow, or maximum heat transfer rate for fixed transfer temperature difference at steady
state. The principles are summarized into the minimum entransydissipation-based thermal resistance principle. These principles were used to optimize heat conduction [19–22], heat convection [23–25], thermal radiation [26,27], heat exchangers [28,29] and thermal networks [30–32]. For heat-work conversion, the concept of entransy dissipation was found to be not applicable [29]. Cheng and Liang [33] further defined the concept of entransy loss,
Gloss ¼ GH GL ;
ð3Þ
where GH is the entransy flows into the system, GL is the entransy flows out of the system, and applied this concept to heat-work conversion optimizations. In thermodynamic processes, it can be found that some of the entransy flow from the high temperature heat sources is dissipated in heating or cooling the working fluid, some is used to do work, and the rest flows into the low temperature heat sources [33]. For an endoreversible thermodynamic cycle, the entransy loss is the difference between entransy flows into and out of the system, which is also the summation of the entransy dissipation due to heat transfer and the entransy variation (work entransy) resulting from heat-work conversion [33]. Some researches have shown that the concept of entransy loss is appropriate for heatwork conversion optimization [18,33–36]. 3. Analyses of entropy generation and entransy loss For a common endoreversible cycle, assume that Ch is the heat capacity flow rate, and Th,in is the inlet temperature of the hot stream; Cc is the heat capacity flow rate, and Tc,in is the inlet temperature of the cold stream; Q_ h is the heat transfer rate between the hot stream and the working fluid, and Q_ c is the heat transfer rate between the cold stream and the working fluid; T0 is the environment temperature. According to the energy conservation, the _ is output power W
_ ¼ Q_ h Q_ c : W
ð4Þ
The endoreversible cycle and the environment can be treated as one system. According to the first law of thermodynamics, the heat flow rate released to the environment Q_ 0 is
_ Q_ 0 ¼ C h ðT h;in T 0 Þ þ C c ðT c;in T 0 Þ W ¼ C h ðT h;in T 0 Þ þ C c ðT c;in T 0 Þ Q_ h þ Q_ c ;
ð5Þ
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where the first term on the right-hand side is the net heat rate from the hot stream, and the second term is the net heat rate from the cold stream. For the optimization based on the concept of entropy generation, Bejan [37] pointed out that the entropy generation which is associated with dumping the used streams into the environment must be taken into account when he discussed the power plant optimization. The entropy balance equation for the whole system can be expressed as [15]
_ S_ g þ S_ f ¼ DS;
ð6Þ
where DS_ is the entropy change rate with time, S_ f is the entropy flow rate into the system, and S_ g is the entropy generation rate in the system. When the system is steady, DS_ equals to zero and the entropy generation rate of the whole system is
T0 T0 Q_ 0 þ C c ln þ ; S_ g ¼ S_ f ¼ C h ln T h;in T c;in T 0
Fig. 1. The device flow diagram of the Rankine cycle.
ð7Þ
Based on Eqs. (4) and (5), Eq. (7) can be changed as
T0 T0 C h ðT h;in T 0 Þ þ C c ðT c;in T 0 Þ 1 _ þ C c ln þ W: S_ g ¼ C h ln T0 T0 T h;in T c;in ð8Þ As the environment temperature T0 is fixed. Eq. (8) shows that the minimum entropy generation corresponds to the maximum output work if Ch, Cc, Th,in and Tc,in are prescribed. The concept of entransy loss is defined as the sum of the entransy variation due to heat transfer and that due to output work, and it is also equal to the difference between entransy into the system and that out of the system [33],
1 G_ loss ¼ C h T 2h; 2
in
1 T 20 þ C c T 2c; 2
in
T 20 Q_ 0 T 0 ;
ð9Þ
where the first term on the right-hand side is the net entransy flow rate taken in from the hot stream, and the second term is the net entransy flow rate taken in from the cold stream. The last term is the entransy loss rate due to the heat release from the cycle into the environment. With Eqs. (4) and (5), Eq. (9) can be changed into
1 G_ loss ¼ C h ðT h; 2
in
1 _ T 0 Þ2 þ C c ðT c;in T 0 Þ2 þ T 0 W: 2
ð10Þ
It can be seen that the maximum entransy loss also corresponds to the maximum output power for prescribed Ch, Cc, Th,in and Tc,in. Eqs. (8) and (10) shows that both the concepts of entropy generation and entransy loss can be used to optimize common endoreversible cycles with prescribed Ch, Cc, Th,in and Tc,in. In the next section, we apply the concepts of entropy generation, entransy loss and entransy dissipation to the optimization analyses of the Rankine cycle. 4. Analyses and optimization of the Rankine cycle The Rankine cycle is an ideal steam power cycle, which was proposed by Rankine in 1859 [38]. It is used as a thermodynamic standard for rating the performance of steam power plants. The Rankine cycle is also a typical endoreversible cycle with phase change process. We discuss the waste heat generating system with water as the working fluid, in which the preheating section, evaporating section and superheating section are heated by one hot stream in an exhaust-heat boiler [39]. Fig. 1 is the cycle flow diagram. Water is heated at constant pressure in the exhaust-heat boiler. The subcooled water is converted to the saturated water in the preheating section, then the saturated water is changed to
Fig. 2. The T S diagram of the Rankine cycle.
the saturated vapor in the evaporating section, and the saturated vapor is transformed to the superheated vapor in the superheating section. The superheated vapor expands in the turbine to do work and generate electricity. The waste steam at low pressure flows into the condenser, which is a counter flow heat exchanger as well, being cooled down to saturated water. The pressure of the saturated water is increased by a pump and then the water is sent back to the exhaust-heat boiler. Then, one cycle finishes. Fig. 2 is the T–S diagram of the cycle. The working medium undergoes four processes: heating at constant pressure 4–5–6–1; reversible adiabatic expansion 1–2; cooling at constant pressure _ is 2–3; and reversible adiabatic compression 3–4. Assume that m the mass flow rate of the working fluid, Th,out and Tc,out are the outlet temperatures of the hot and cold streams in the exchangers, respectively. Thermal conductances of the superheating section, evaporating section and preheating section in the exhaust-heat boiler, which is defined as the product of heat transfer coefficient and heat transfer area, are UAh1, UAh2 and UAh3, respectively. Thermal conductance of the low temperature exchanger is UAc. _ is to be optimized so as The mass flow rate of the working fluid m _ maximum for prescribed Th,in, Tc,in, Ch to make the output power W _ and m. _ and Cc. It is necessary to set up the relationship between W Assume that cg is the specific heat capacity of water vapor, cl is the specific heat capacity of water, and rh is the latent heat of vaporization at high pressure. The heat transfer rates of the three sections in the exhaust-heat boiler are
_ g ðT 1 T 6 Þ ¼ C h;min1 eh1 ðT h;in T 6 Þ ¼ C h ðT h;in T h;in2 Þ; Q_ h1 ¼ mc _ _ h ¼ C h; min2 eh2 ðT h;in2 T 6 Þ ¼ C h ðT h; in2 T h;in3 Þ; Q h2 ¼ mr
ð11Þ
ð12Þ _ l ðT 5 T 4 Þ ¼ C h;min3 eh3 ðT h;in3 T 4 Þ ¼ C h ðT h; in3 T h;out Þ: ð13Þ Q_ h3 ¼ mc
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where Th,in2 is the outlet temperature of the hot stream at the superheating section and the inlet temperature of the evaporating section; Th,in3 is the outlet temperature of the hot stream at the evaporating section and the inlet temperature of the preheating _ g ; C h Þ; C h; max 1 ¼ maxðmc _ g ; C h Þ; C h; min 3 section; C h; min 1 ¼ minðmc _ l ; C h Þ. As the heat capacity _ l ; C h Þ and C h; max 3 ¼ maxðmc ¼ minðmc flow rate of the working fluid during phase change process is much larger than Ch, Ch,min2 = Ch. The effectivenesses of the superheating section eh1, evaporating section eh2 and preheating section eh3 in the exhaust-heat boiler are [40]
eh1 ¼
1 exp ðNTU 1 Þð1 C hmin 1 =C h; max 1 Þ ; ð14Þ 1 C hmin 1 =C h; max 1 exp ðNTU 1 Þð1 C hmin 1 =C h; max 1 Þ
eh2 ¼ 1 expðNTU 2 Þ; eh3
ð15Þ 1 exp ðNTU 3 Þð1 C h: min 3 =C h; max 3 Þ ; ð16Þ ¼ 1 C hmin 3 =C h; max 3 exp ðNTU 3 Þð1 C hmin 3 =C h; max 3 Þ
_ ¼ Q_ h1 þ Q_ h2 þ Q_ h3 Q_ c : W The optimization of the Rankine cycle is expressed as
(
_ max W; _ 2 ð0; m _ max Þ; s:t: m
ð17Þ
NTU 2 ¼ UAh2 =C h; min 2 ;
ð18Þ
NTU 3 ¼ UAh3 =C h; min 3 :
ð19Þ
As the temperature of the working fluid remains the same during the phase change process, then we have
T2 ¼ T3; T5 ¼ T6:
ð20Þ
Similarly, for the condenser, the heat transfer rate is
Q_ c ¼ C c; min ec ðT 2 T
c;in Þ
_ 2 ðs2 s3 Þ: ¼ C c ðT c;out T c; in Þ ¼ mT
ð21Þ
During the condensing process, the temperature of the working fluid remains the same and the heat capacity of the working fluid is much larger than Cc, then we have
C c; min ¼ C h ;
ð22Þ
ec ¼ 1 expðNTU c Þ;
ð23Þ
where ec is the effectiveness of the condenser and NTUc is the number of transfer units of condenser,
NTU c ¼ UAc =C c; min :
ð24Þ
As process 1–2 and process 3–4 are both adiabatic process, there are
s2 ¼ s1 ;
ð25Þ
s3 ¼ s4 :
ð26Þ
The latent heat of vaporization rh in Eq. (12) can be determined by looking up the thermodynamic chart of water and vapor [41] according to saturated temperature T6. State 4 is sub-cooled water at high pressure, and its temperature could be also determined by looking up the thermodynamic chart of water and vapor [41] according to its entropy s4 and the saturated temperature T6 at high pressure. We can also determine the entropy s1 and s3 according to their corresponding temperatures. Eqs. (11,12,13) and (21) include eight equations, and eight individual system variables can be solved by these equations. There are seven individual system variables, including T1, T2, T6, Th,in2, Th,in3, Th,out and Tc,out. One of the thermal conductances among UAh1, UAh2 and UAh3 can be treated as the eighth individual variable if the thermal conductance of the condenser is prescribed. We can obtain the temperatures of all the cycle point from T1 to T6, the heat transfer rates, Q_ h1 ; Q_ h2 ; Q_ h3 and Q_ c , and the outlet temperature of the streams, Th,out and Tc,out. At the same time, the thermal conductance of the unknown section could be designed. _ is the difference between the absorbed The output power W heat flow and the released heat flow. It is
ð28Þ
_ max is the maximum mass flow rate of the working fluid in where m the calculation. By considering the entropy generation associated with dumping the used streams into the environment, we get the total entropy generation rate of the Rankine cycle shown in Fig. 1,
S_ g ¼
T h;out T c;out T T Q_ Q_ _ l ln 5 þ mc _ g ln 1 þ h2 c þ C c ln þ mc T h;in T c; in T4 T6 T6 T2 ! _ _ Q eh Q ec T0 T0 þ þ C c ln þ þ C h ln ; T h;out T0 T c;out T0
!
C h ln
where NTU1, NTU2 and NTU3 are the numbers of transfer units of the corresponding section in the exhaust-heat boiler,
NTU 1 ¼ UAh1 =C h; min 1 ;
ð27Þ
ð29Þ where the terms in the first bracket on the right-hand side are the entropy generation rates due to heat transfer between the working fluid and the heat sources, the terms in the second bracket are the entropy generation rates due to dumping the used streams into the environment. Q_ eh and Q_ ec are the heat transfer rates from the outlet streams to the environment, which can be expressed as
Q_ eh ¼ C h ðT h;out T 0 Þ; Q_ ec ¼ C c ðT c;out T 0 Þ:
ð30Þ ð31Þ
Based on Eqs. (29)–(31), the total entropy generation rate of the Rankine cycle can be calculated. The total entransy loss rate can be expressed as Eq. (10) if the whole system including the Rankine cycle and the environment is considered. For this heat-work conversion problem, let’s discuss whether the entransy dissipation can be used for its optimization. Considering the entransy dissipation due to discharging the used streams into the environment as well, the total entransy dissipation rate can be expressed as 1 1 1 _ l T 24 T 25 Q_ h2 T 6 þ mc _ g T 26 T 21 G_ dis ¼ C h T 2h; in T 2h;out þ mc 2 2 2 1 2 2 þ C c T c;in T c;out þ Q_ c T 2 2 1 1 þ C h T 2h;out T 20 þ C c T 2c;out T 20 Q_ eh T 0 Q_ ec T 0 Þ ; 2 2
ð32Þ where the terms in the first square bracket on the right-hand side is the entransy dissipation rates due to heat transfer between the hot stream and the working fluid, and the terms in the second square bracket are the entransy dissipation rates due to heat transfer between the cold stream and the working fluid. The terms in the last square bracket on the right hand side are the entransy dissipation rates due to discharging the used streams into the environment. 5. Numerical examples and discussions Some numerical examples are discussed in this section. In the first one, Th,in = 773.16 K, Tc,in = 283.16 K, Ch = 1000 W/K, Cc = 10,000 W/K, UAc = 5000 W/K, the specific heat capacity of water cl is 4200 J/(kg K), the specific heat capacity of water vapor cg is 2060 J/(kg K), and the environment temperature T0 is 300 K. For this case, the thermal conductances of the superheating section and evaporating section in the exhaust-heat boiler are prescribed, UAh1 = 5000 W/K and UAh2 = 5000 W/K. The calculated variations
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Fig. 3. The variations of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate with the mass flow rate for UAh1 = 5000 W/K and UAh2 = 5000 W/K.
Fig. 4. The variation of the output power with the thermal conductance of the preheating section.
of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate with the mass flow rate are shown in Fig. 3 where the subscript max denotes the maximum value of the corresponding parameter in the calculation range. Both the maximum entransy loss rate and the minimum entropy generation rate correspond to the maximum output power. The extremum of the entransy dissipation rate does not correspond to the maximum work output because it does not consider the influence of work output on the change of entransy. The variation of the output power with the thermal conductance of the designed preheating section is shown in Fig. 4. It can be found that larger conductance of the preheating section does not always correspond to larger output power. There is an optimum value for the thermal conductance of the preheating section which is 1493.36 W/K. The thermal conductance UAh3 and the mass flow rate _ are dependent variables. When UAh3 is determined, the mass flow m _ is determined as well under the prescribed condition. As the rate m maximum output work corresponds to the certain value of mass flow rate, then there must be an optimum value of the thermal conductance UAh3 corresponding to the maximum output work. In the second example, the thermal conductance of the superheating section and preheating section in the exhaust-heat boiler are prescribed (UAh1 = 5000 W/K and UAh3 = 3000 W/K), the other parameters are the same as those in the first example. The variations of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate with the mass flow rate are shown in Fig. 5. The results are similar to those in Fig. 3.
Fig. 5. The variations of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate with the mass flow rate for UAh1 = 5000 W/K and UAh3 = 3000 W/K.
Fig. 6. The variation of the output power with the thermal conductance of the evaporating section.
Fig. 7. The variations of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate with the mass flow rate for UAh2 = 5000 W/K and UAh3 = 4000 W/K.
The variation of the output power with the thermal conductance of the designed evaporating section is shown in Fig. 6. There is also an optimum value for the thermal conductance of the evaporating section, which is 3884.49 W/K. In the third example, the thermal conductance of evaporating section and preheating section in the exhaust-heat boiler is prescribed, UAh2 = 5000 W/K and UAh3 = 4000 W/K. The variations of the output power, the entropy generation rate, the entransy loss
W.H. Wang et al. / Energy Conversion and Management 68 (2013) 82–88
Fig. 8. The variation of the output power with the thermal conductance of the designed superheating section.
rate and the entransy dissipation rate with the mass flow rate in Fig. 7 are also similar to the above two cases. Fig. 8 shows the variation of the output power with the thermal conductance of the designed superheating section. The optimum value for the thermal conductance of the superheating section is 709.67 W/K. For the above examples where the heat capacity flow rates and the inlet temperatures of the streams are prescribed, both the minimum entropy generation principle and the entransy loss are applicable to the optimization of the Rankine cycle, while the concept of entransy dissipation is not. We further discuss some cases in which the heat capacity flow rates or the inlet temperatures of the streams are not prescribed. First, we calculate the dependence of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate on the inlet temperature of the hot stream Th,in with _ ¼ 0:1114 kg=s. The results are in prescribed mass flow rate m Fig. 9. The output power and the entransy loss rate both increase monotonically with increasing inlet temperature of the hot stream, while the entropy generation rate and the entransy dissipation rate first decrease and then increase. The extremum entropy generation rate and entransy dissipation rate do not correspond to the maximum output power. The variations of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate with the heat capacity rate of the hot stream Ch with prescribed mass flow _ ¼ 0:1114 kg=s shown in Fig. 10 are similar to those in Fig. 9. rate m Besides the entropy generation rate, the entropy generation number is also used as an important optimization criterion for heat-work conversion processes. The entropy generation number is a dimensionless parameter [37], which is defined as
NS ¼ S_ g =C min ;
ð33Þ
where Cmin is the minimum heat capacity rate of all the fluids. When Th,in changes, the heat capacity rates of the fluids do not change, so Cmin remains constant. When Ch changes, according to the prescribed parameters, we get
_ g ¼ 289:64 W=K: C min ¼ mc
ð34Þ
As Cmin is a constant under these two conditions, the variation tendencies of entropy generation number and entropy generation rate are consistent. For the other definition of entropy generation number by Bejan [42], there is
NS ¼ S_ g =S_ g; min :
ð35Þ
As S_ g; min is fixed and is independent of Th,in and Ch, the entropy generation number has the same variation tendency as that of the entropy generation rate.
87
Fig. 9. The variations of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate with the inlet temperature of the hot stream.
Fig. 10. The variations of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate with the heat capacity flow rate of the hot stream.
The absorbed heat from the hot stream comes from the preheating section, the evaporating section and the superheating section. With the increase in the inlet temperature Th,in or the heat capacity flow rate Ch of the hot stream, the evaporating temperature of the working medium T6 increases while the condensing temperature T2 decreases. Due to the limitation of the saturated vapor line, the latent heat of evaporation rh decreases, which makes the absorbed heat Qh2 in the evaporating section decrease. As the thermal conductances of the evaporating section and the superheating section are fixed, the mean temperature difference between the working medium and the hot stream decreases with increasing Th,in, which makes the absorbed heat from the superheating section Qh1 decrease as well. The absorbed heat from the preheating section Qh3 increases due to the increase in T6 and the decrease in T2. For the specific example in this paper, the absorbed heat in the evaporating section and the superheating section takes larger proportion in the total absorbed heat, and then the total absorbed heat Qh decreases with increasing Th,in. On the other hand, with decreasing condensation temperature T2, the released heat decreases. However, as T2 approaches to the temperature of the cold stream, the decreasing rate of the released heat reduces, and the curve tends to be flat. That is the reason why the output work, which is the difference between the absorbed heat and the released heat, increases rapidly at first, and then the increasing rate reduces gradually, which makes the variation of output power in Figs. 9 and 10 appear similar.
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The entropy generation exactly corresponds to the exergy destruction in essence. When the inlet temperature or the heat capacity flow rate of the hot stream increases, the exergy flow into the system increases monotonically, and so does the output power. It is obvious in Figs. 9 and 10 that the output power increases rapidly at first with increasing input heat from the hot stream, and its increasing rate is higher than that of the exergy flow. Then, the increasing rate of the output power reduces gradually with increasing input heat, and is finally lower than the increase in exergy flow. The exergy destruction rate, which is the difference between the exergy flow into the system and the output power, as well as the entropy generation rate, decreases first and then increases with Ch and Th,in. Therefore, the minimum entropy generation rate does not correspond to the maximum output power. In addition, the increase in the inlet temperature or heat capacity flow rate of the hot stream will loss meanings after a specific value because the thermal properties of water and vapor restricts the evaporating temperature. For entransy loss, when the inlet temperature or heat capacity flow rate of hot stream increases, the output power increases as well. The change tendency of the entransy loss rate is consistent with that of the output power. 6. Conclusions The concept of entransy loss is applied to analyzing the Rankine cycle, as well as the concepts of entropy generation and entransy dissipation. Their application preconditions are obtained. Both the minimum entropy generation rate and the maximum entransy loss rate correspond to the maximum cycle output power for fixed heat capacity flow rates and inlet temperatures, but the maximum entransy dissipation rate does not. For the mass flow rate of the working fluid and the thermal conductance, there are optimal values that lead to the maximum output work. The output power increases with increasing inlet temperature or heat capacity flow rate of the hot stream if the inlet temperature or the heat capacity flow rate of the hot stream is not prescribed. Larger entransy loss rate still relates to larger output power, while smaller entropy generation rate does not. The concept of entransy loss is suitable for the output power optimization of the Rankine cycle for all the cases discussed in this paper. Acknowledgments The present work is supported by the Natural Science Foundation of China (Grant No. 51136001) and the Tsinghua University Initiative Scientific Research Program. References [1] Myat A, Thu K, Kim YD. A second law analysis and entropy generation minimization of an absorption chiller. Appl Therm Eng 2011;31:2405–13. [2] Mistry KH, Lienhard JH, Zubair SM. Effect of entropy generation on the performance of humidification–dehumidification desalination cycles. Int J Therm Sci 2010;49:1837–47. [3] Gu W, Weng YW. Second law analysis of evaporator for organic Rankine cycle system. In: Jacksonville FL, editor. Proceedings of the 2nd international conference on energy sustainability law analysis of evaporator for organic Rankine cycle system; 2009. [4] Yaghoubi M, Azizian K, Kenary A. Simulation of Shiraz solar power plant for optimal assessment. Renew Energy 2003;28:1985–98. [5] Zhang B, Ni W D, Li Z. Analysis of conventional power plant, IGCC and coal gasification SOFC hybrid with CO2 mitigation. Coal Convers 2005;28:1–7. [6] Wagar WR, Zamfirescu C, Dincer I. Thermodynamic performance assessment of an ammonia–water. Rankine Cycle Power Heat Prod 2010;51:2501–9. [7] Ronan KM, William JS. Optimal concentration and temperatures of solar thermal power plants. Energy Convers Manage 2012;60:226–32. [8] Roberto G. A novel design approach for small scale low enthalpy binary geothermal power plants. Energy Convers Manage 2012;64:274–363.
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Beijing: High Eduction Press; 2007 [in Chinese]. [16] Zeng DL, Ao Y, Zhang XM, et al. Engineering thermodynamics. Beijing: High Eduction Press; 2002 [in Chinese]. [17] Klein SA, Reindl DT. The relationship of optimum heat exchanger allocation and minimum entropy generation rate for refrigeration cycles. J Energy Res 1998;120:172–8. [18] Cheng XT, Wang WH, Liang XG. Entransy analysis of open thermodynamic systems. Chin Sci Bull 2012;57:2934–40. [19] Guo ZY, Zhu HY, Liang XG. Entransy—a physical quantity describing heat transfer ability. Int J Heat Mass Transfer 2007;50:2545–56. [20] Chen LG, Wei SH, Sun FR. Constructal entransy dissipation minimization for ‘volume-point’ heat conduction. J Phys D: Appl Phys 2008;41:195506. [21] Xie ZH, Chen LG, Sun FR. Constructal optimization for geometry of cavity by taking entransy dissipation minimization as objective. Sci China Ser E: Tech Sci 2009;52:3413–504. [22] Cheng XT, Xu XH, Liang XG. Homogenization of temperature field and temperature gradient field. Sci China Ser E: Tech Sci 2009;52:2937–42. [23] Chen Q, Ren JX. Generalized thermal resistance for convective heat transfer and its relation to entransy dissipation. Chin Sci Bull 2008;53:3753–61. [24] Wu J, Cheng XG, Meng JA, et al. Potential capacity dissipation extreme and entropy generation minimization in laminar convective heat transfer. J Eng Thermophys 2006;27:100–2. [25] Liu W, Liu ZC, Jia H, et al. Entransy expression of the second law of thermodynamics and its application to optimization in heat transfer process. Int J Heat Mass Transfer 2011;54:3049–59. [26] Cheng XT, Liang XG. Entransy flux of thermal radiation and its application to enclosures with opaque surfaces. Int J Heat Mass Transfer 2011;54:269–78. [27] Cheng XT, Xu XH, Liang XG. Radiative entransy flux in enclosures with nonisothermal or non-grey, opaque, diffuse surfaces and its application. Sci China: Tech Sci 2011;54:2446–56. 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Contents lists available at SciVerse ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Entropy and entransy analyses and optimizations of the Rankine cycle WenHua Wang, XueTao Cheng, XinGang Liang ⇑ Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Article history: Received 27 October 2012 Received in revised form 7 December 2012 Accepted 24 December 2012
Keywords: Entropy generation Entransy loss The Rankine cycle Heat-work conversion Phase change
a b s t r a c t The Rankine cycle has been widely used in industry, and the optimization of the Rankine cycle is of great significance to energy utilization improvement. In this paper, both the concepts of entropy generation and entransy loss are applied to analyzing the thermodynamic processes of the Rankine cycle that receives heat from the exhaust-heat boiler with a hot stream and releases heat to a cold stream. In order to get the maximum output power, the mass flow rate of the working medium is optimized. Theoretical analysis and numerical results show that both the maximum entransy loss rate and the minimum entropy generation rate correspond to the maximum output power when the inlet temperatures and the heat capacity flow rates of the streams are prescribed. When the inlet temperatures or the heat capacity flow rates of the streams is not prescribed, smaller entropy generation rate does not lead to larger output power but larger entransy loss rate still leads to larger power output for this kind of cases. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The rapid increase in global energy consumption has been boosting the development of the energy conservation technologies [1–4]. The optimization design of the Rankine cycle is of great significance to the improvement of energy utilization because the cycle is widely used in the thermal power plants [5], the solar power and geothermal power plants [6–8], the cogeneration technology [9–11], the comprehensive polygeneration energy system [12– 14], etc. Many researchers have paid their attention on this topic, such as Gu et al. [3], Yaghoubi et al. [4], and Ahmadi et al. [12–14]. The Rankine cycle is a typical heat-work conversion process with phase change and entropy generation is most widely used in its optimizations. The concept of entropy was proposed by Clausius when he analyzed the Carnot cycle [15]. As entropy generation represents the ability loss of doing work, the minimum entropy generation is thought to correspond to the optimal performance of the corresponding thermodynamic cycle [16]. For instance, Yaghoubi et al. [4] optimized the mass flow rate of a solar power plant consisting of the Rankine cycle based on the entropy generation minimization method with the fixed ambient conditions. However, there are heat-work conversion optimization cases in which the minimum entropy generation dese not correspond the best performance. In the investigation of the refrigeration systems, Klein and Reindl [17] found that minimizing the entropy generation rate does not always result in the same design as maximizing the system performance unless the refrigeration capacity is fixed.
⇑ Corresponding author. Tel./fax: +86 10 62788702. E-mail address: [email protected] (X.G. Liang). 0196-8904/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.enconman.2012.12.020
Cheng et al. [18] revealed that the entropy generation could not be related to the net work change in the air standard cycle when the heat absorbed by the working fluid is from the combustion reaction of the gas fuel. Therefore, for the optimization of heatwork conversion processes, the applicability and preconditions of the entropy theory need further discussions. In recent years, the concept of entransy [19] was developed for the optimization design of thermal system, and was applied to heat transfer optimization [19–32]. In heat-work conversion optimization, Cheng et al. [33] defined the concept of entransy loss and extended the entransy theory to heat-work conversion optimizations. In the analyses of the air standard cycle [18], the Stirling engine cycle [34] and the one-stream heat exchanger networks [35], the results show that the increase in entransy loss rate always leads to the increase in output power. However, the applicability of the entransy theory to optimizing thermodynamic cycles stills needs further investigations because there are not many reports. In the present report, the Rankine cycle with water as the working fluid that uses waste heat from a hot stream to produce power is analyzed and optimized. The concept of entransy loss is applied to the analyses of the Rankine cycle for the first time. The cycle system is optimized with the concepts of entropy generation, entransy loss and entransy dissipation. The relationships between the concepts of entropy generation, entransy loss, entransy dissipation and the output power for the Rankine cycle are derived and demonstrated. There are optimum values for the mass flow rate of the working fluid and the thermal conductance that correspond to the maximum output work.
W.H. Wang et al. / Energy Conversion and Management 68 (2013) 82–88
83
Nomenclature C c G_ loss G_ dis _ m NTU Q_ r S_ f S_ g DS_ s T T0 UA _ W
Subscripts c cold side eh from hot stream to environment ec from cold stream to environment g gas state h hot side h1 superheating section h2 evaporating section h3 preheating section hin2 out of the superheating section hin3 out of the evaporating section in into the system l liquid state min the minimum value max the maximum value out out of the system
heat capacity flow rate, W/K specific heat capacity, J/(kg K) entransy loss rate, W K entransy dissipation rate, W K mass flow rate, kg the number of transfer units heat transfer rate, W the latent heat, J/kg entropy flow rate, W/K entropy generation rate, W/K entropy change rate, W/K specific entropy, J/(kg K) temperature, K environment temperature, K thermal conductance, W/K output power, W
Greek symbols e effectiveness
The applicability and the relevant preconditions of the entropy generation, entransy loss and entransy dissipation for the Rankine cycle are discussed. 2. Entransy, entransy dissipation and entransy loss Entransy is proposed to describe heat transfer ability [19]. It represents the potential energy of heat in a body corresponding to the electrical energy in a capacitor based on the analogy between electrical and thermal systems, and it is defined as
Evh ¼
1 Q T; 2 vh
ð1Þ
where Qvh = McvT is the thermal energy or the heat stored in an object with constant volume which may be referred to as the thermal charge, in which M is the mass and cv is the specific heat capacity. The temperature T represents the thermal potential. In heat transfer processes, it was found that the entransy always decreases in spontaneous heat transfer [19]. The decreased part is called entransy dissipation [19], which has been used in heat transfer optimizations [19–32]. The balance equation of entransy dissipation can be derived from the energy conservation of heat transfer. For instance, multiplying the energy equation of heat conduction by temperature leads to the entransy balance equation for heat conduction [19],
T qc v
@T _ ðq_ rTÞ; ¼ r ðqTÞ @t
ð2Þ
where q is the density, q_ is the heat flux vector, t is the time, the left term is the entransy change of the infinitesimal element, the first term on the right-hand side is the entransy flow into the element, and the second term on the right is the local rate of entransy dissipation Gdis. Based on the Fourier’s law, the entransy dissipation per unit time and per unit volume is k(rT)2 [19], in which k is the thermal conductivity and rT is the temperature gradient. The extremum entransy dissipation was proved appropriate as the criterion for the optimal heat transfer performance. Guo et al. [19] derived the extremum entransy dissipation principles based on the entransy balance equation for heat convection. The extremum value of entransy dissipation was found to relate to minimum heat transfer temperature difference for fixed heat flow, or maximum heat transfer rate for fixed transfer temperature difference at steady
state. The principles are summarized into the minimum entransydissipation-based thermal resistance principle. These principles were used to optimize heat conduction [19–22], heat convection [23–25], thermal radiation [26,27], heat exchangers [28,29] and thermal networks [30–32]. For heat-work conversion, the concept of entransy dissipation was found to be not applicable [29]. Cheng and Liang [33] further defined the concept of entransy loss,
Gloss ¼ GH GL ;
ð3Þ
where GH is the entransy flows into the system, GL is the entransy flows out of the system, and applied this concept to heat-work conversion optimizations. In thermodynamic processes, it can be found that some of the entransy flow from the high temperature heat sources is dissipated in heating or cooling the working fluid, some is used to do work, and the rest flows into the low temperature heat sources [33]. For an endoreversible thermodynamic cycle, the entransy loss is the difference between entransy flows into and out of the system, which is also the summation of the entransy dissipation due to heat transfer and the entransy variation (work entransy) resulting from heat-work conversion [33]. Some researches have shown that the concept of entransy loss is appropriate for heatwork conversion optimization [18,33–36]. 3. Analyses of entropy generation and entransy loss For a common endoreversible cycle, assume that Ch is the heat capacity flow rate, and Th,in is the inlet temperature of the hot stream; Cc is the heat capacity flow rate, and Tc,in is the inlet temperature of the cold stream; Q_ h is the heat transfer rate between the hot stream and the working fluid, and Q_ c is the heat transfer rate between the cold stream and the working fluid; T0 is the environment temperature. According to the energy conservation, the _ is output power W
_ ¼ Q_ h Q_ c : W
ð4Þ
The endoreversible cycle and the environment can be treated as one system. According to the first law of thermodynamics, the heat flow rate released to the environment Q_ 0 is
_ Q_ 0 ¼ C h ðT h;in T 0 Þ þ C c ðT c;in T 0 Þ W ¼ C h ðT h;in T 0 Þ þ C c ðT c;in T 0 Þ Q_ h þ Q_ c ;
ð5Þ
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where the first term on the right-hand side is the net heat rate from the hot stream, and the second term is the net heat rate from the cold stream. For the optimization based on the concept of entropy generation, Bejan [37] pointed out that the entropy generation which is associated with dumping the used streams into the environment must be taken into account when he discussed the power plant optimization. The entropy balance equation for the whole system can be expressed as [15]
_ S_ g þ S_ f ¼ DS;
ð6Þ
where DS_ is the entropy change rate with time, S_ f is the entropy flow rate into the system, and S_ g is the entropy generation rate in the system. When the system is steady, DS_ equals to zero and the entropy generation rate of the whole system is
T0 T0 Q_ 0 þ C c ln þ ; S_ g ¼ S_ f ¼ C h ln T h;in T c;in T 0
Fig. 1. The device flow diagram of the Rankine cycle.
ð7Þ
Based on Eqs. (4) and (5), Eq. (7) can be changed as
T0 T0 C h ðT h;in T 0 Þ þ C c ðT c;in T 0 Þ 1 _ þ C c ln þ W: S_ g ¼ C h ln T0 T0 T h;in T c;in ð8Þ As the environment temperature T0 is fixed. Eq. (8) shows that the minimum entropy generation corresponds to the maximum output work if Ch, Cc, Th,in and Tc,in are prescribed. The concept of entransy loss is defined as the sum of the entransy variation due to heat transfer and that due to output work, and it is also equal to the difference between entransy into the system and that out of the system [33],
1 G_ loss ¼ C h T 2h; 2
in
1 T 20 þ C c T 2c; 2
in
T 20 Q_ 0 T 0 ;
ð9Þ
where the first term on the right-hand side is the net entransy flow rate taken in from the hot stream, and the second term is the net entransy flow rate taken in from the cold stream. The last term is the entransy loss rate due to the heat release from the cycle into the environment. With Eqs. (4) and (5), Eq. (9) can be changed into
1 G_ loss ¼ C h ðT h; 2
in
1 _ T 0 Þ2 þ C c ðT c;in T 0 Þ2 þ T 0 W: 2
ð10Þ
It can be seen that the maximum entransy loss also corresponds to the maximum output power for prescribed Ch, Cc, Th,in and Tc,in. Eqs. (8) and (10) shows that both the concepts of entropy generation and entransy loss can be used to optimize common endoreversible cycles with prescribed Ch, Cc, Th,in and Tc,in. In the next section, we apply the concepts of entropy generation, entransy loss and entransy dissipation to the optimization analyses of the Rankine cycle. 4. Analyses and optimization of the Rankine cycle The Rankine cycle is an ideal steam power cycle, which was proposed by Rankine in 1859 [38]. It is used as a thermodynamic standard for rating the performance of steam power plants. The Rankine cycle is also a typical endoreversible cycle with phase change process. We discuss the waste heat generating system with water as the working fluid, in which the preheating section, evaporating section and superheating section are heated by one hot stream in an exhaust-heat boiler [39]. Fig. 1 is the cycle flow diagram. Water is heated at constant pressure in the exhaust-heat boiler. The subcooled water is converted to the saturated water in the preheating section, then the saturated water is changed to
Fig. 2. The T S diagram of the Rankine cycle.
the saturated vapor in the evaporating section, and the saturated vapor is transformed to the superheated vapor in the superheating section. The superheated vapor expands in the turbine to do work and generate electricity. The waste steam at low pressure flows into the condenser, which is a counter flow heat exchanger as well, being cooled down to saturated water. The pressure of the saturated water is increased by a pump and then the water is sent back to the exhaust-heat boiler. Then, one cycle finishes. Fig. 2 is the T–S diagram of the cycle. The working medium undergoes four processes: heating at constant pressure 4–5–6–1; reversible adiabatic expansion 1–2; cooling at constant pressure _ is 2–3; and reversible adiabatic compression 3–4. Assume that m the mass flow rate of the working fluid, Th,out and Tc,out are the outlet temperatures of the hot and cold streams in the exchangers, respectively. Thermal conductances of the superheating section, evaporating section and preheating section in the exhaust-heat boiler, which is defined as the product of heat transfer coefficient and heat transfer area, are UAh1, UAh2 and UAh3, respectively. Thermal conductance of the low temperature exchanger is UAc. _ is to be optimized so as The mass flow rate of the working fluid m _ maximum for prescribed Th,in, Tc,in, Ch to make the output power W _ and m. _ and Cc. It is necessary to set up the relationship between W Assume that cg is the specific heat capacity of water vapor, cl is the specific heat capacity of water, and rh is the latent heat of vaporization at high pressure. The heat transfer rates of the three sections in the exhaust-heat boiler are
_ g ðT 1 T 6 Þ ¼ C h;min1 eh1 ðT h;in T 6 Þ ¼ C h ðT h;in T h;in2 Þ; Q_ h1 ¼ mc _ _ h ¼ C h; min2 eh2 ðT h;in2 T 6 Þ ¼ C h ðT h; in2 T h;in3 Þ; Q h2 ¼ mr
ð11Þ
ð12Þ _ l ðT 5 T 4 Þ ¼ C h;min3 eh3 ðT h;in3 T 4 Þ ¼ C h ðT h; in3 T h;out Þ: ð13Þ Q_ h3 ¼ mc
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where Th,in2 is the outlet temperature of the hot stream at the superheating section and the inlet temperature of the evaporating section; Th,in3 is the outlet temperature of the hot stream at the evaporating section and the inlet temperature of the preheating _ g ; C h Þ; C h; max 1 ¼ maxðmc _ g ; C h Þ; C h; min 3 section; C h; min 1 ¼ minðmc _ l ; C h Þ. As the heat capacity _ l ; C h Þ and C h; max 3 ¼ maxðmc ¼ minðmc flow rate of the working fluid during phase change process is much larger than Ch, Ch,min2 = Ch. The effectivenesses of the superheating section eh1, evaporating section eh2 and preheating section eh3 in the exhaust-heat boiler are [40]
eh1 ¼
1 exp ðNTU 1 Þð1 C hmin 1 =C h; max 1 Þ ; ð14Þ 1 C hmin 1 =C h; max 1 exp ðNTU 1 Þð1 C hmin 1 =C h; max 1 Þ
eh2 ¼ 1 expðNTU 2 Þ; eh3
ð15Þ 1 exp ðNTU 3 Þð1 C h: min 3 =C h; max 3 Þ ; ð16Þ ¼ 1 C hmin 3 =C h; max 3 exp ðNTU 3 Þð1 C hmin 3 =C h; max 3 Þ
_ ¼ Q_ h1 þ Q_ h2 þ Q_ h3 Q_ c : W The optimization of the Rankine cycle is expressed as
(
_ max W; _ 2 ð0; m _ max Þ; s:t: m
ð17Þ
NTU 2 ¼ UAh2 =C h; min 2 ;
ð18Þ
NTU 3 ¼ UAh3 =C h; min 3 :
ð19Þ
As the temperature of the working fluid remains the same during the phase change process, then we have
T2 ¼ T3; T5 ¼ T6:
ð20Þ
Similarly, for the condenser, the heat transfer rate is
Q_ c ¼ C c; min ec ðT 2 T
c;in Þ
_ 2 ðs2 s3 Þ: ¼ C c ðT c;out T c; in Þ ¼ mT
ð21Þ
During the condensing process, the temperature of the working fluid remains the same and the heat capacity of the working fluid is much larger than Cc, then we have
C c; min ¼ C h ;
ð22Þ
ec ¼ 1 expðNTU c Þ;
ð23Þ
where ec is the effectiveness of the condenser and NTUc is the number of transfer units of condenser,
NTU c ¼ UAc =C c; min :
ð24Þ
As process 1–2 and process 3–4 are both adiabatic process, there are
s2 ¼ s1 ;
ð25Þ
s3 ¼ s4 :
ð26Þ
The latent heat of vaporization rh in Eq. (12) can be determined by looking up the thermodynamic chart of water and vapor [41] according to saturated temperature T6. State 4 is sub-cooled water at high pressure, and its temperature could be also determined by looking up the thermodynamic chart of water and vapor [41] according to its entropy s4 and the saturated temperature T6 at high pressure. We can also determine the entropy s1 and s3 according to their corresponding temperatures. Eqs. (11,12,13) and (21) include eight equations, and eight individual system variables can be solved by these equations. There are seven individual system variables, including T1, T2, T6, Th,in2, Th,in3, Th,out and Tc,out. One of the thermal conductances among UAh1, UAh2 and UAh3 can be treated as the eighth individual variable if the thermal conductance of the condenser is prescribed. We can obtain the temperatures of all the cycle point from T1 to T6, the heat transfer rates, Q_ h1 ; Q_ h2 ; Q_ h3 and Q_ c , and the outlet temperature of the streams, Th,out and Tc,out. At the same time, the thermal conductance of the unknown section could be designed. _ is the difference between the absorbed The output power W heat flow and the released heat flow. It is
ð28Þ
_ max is the maximum mass flow rate of the working fluid in where m the calculation. By considering the entropy generation associated with dumping the used streams into the environment, we get the total entropy generation rate of the Rankine cycle shown in Fig. 1,
S_ g ¼
T h;out T c;out T T Q_ Q_ _ l ln 5 þ mc _ g ln 1 þ h2 c þ C c ln þ mc T h;in T c; in T4 T6 T6 T2 ! _ _ Q eh Q ec T0 T0 þ þ C c ln þ þ C h ln ; T h;out T0 T c;out T0
!
C h ln
where NTU1, NTU2 and NTU3 are the numbers of transfer units of the corresponding section in the exhaust-heat boiler,
NTU 1 ¼ UAh1 =C h; min 1 ;
ð27Þ
ð29Þ where the terms in the first bracket on the right-hand side are the entropy generation rates due to heat transfer between the working fluid and the heat sources, the terms in the second bracket are the entropy generation rates due to dumping the used streams into the environment. Q_ eh and Q_ ec are the heat transfer rates from the outlet streams to the environment, which can be expressed as
Q_ eh ¼ C h ðT h;out T 0 Þ; Q_ ec ¼ C c ðT c;out T 0 Þ:
ð30Þ ð31Þ
Based on Eqs. (29)–(31), the total entropy generation rate of the Rankine cycle can be calculated. The total entransy loss rate can be expressed as Eq. (10) if the whole system including the Rankine cycle and the environment is considered. For this heat-work conversion problem, let’s discuss whether the entransy dissipation can be used for its optimization. Considering the entransy dissipation due to discharging the used streams into the environment as well, the total entransy dissipation rate can be expressed as 1 1 1 _ l T 24 T 25 Q_ h2 T 6 þ mc _ g T 26 T 21 G_ dis ¼ C h T 2h; in T 2h;out þ mc 2 2 2 1 2 2 þ C c T c;in T c;out þ Q_ c T 2 2 1 1 þ C h T 2h;out T 20 þ C c T 2c;out T 20 Q_ eh T 0 Q_ ec T 0 Þ ; 2 2
ð32Þ where the terms in the first square bracket on the right-hand side is the entransy dissipation rates due to heat transfer between the hot stream and the working fluid, and the terms in the second square bracket are the entransy dissipation rates due to heat transfer between the cold stream and the working fluid. The terms in the last square bracket on the right hand side are the entransy dissipation rates due to discharging the used streams into the environment. 5. Numerical examples and discussions Some numerical examples are discussed in this section. In the first one, Th,in = 773.16 K, Tc,in = 283.16 K, Ch = 1000 W/K, Cc = 10,000 W/K, UAc = 5000 W/K, the specific heat capacity of water cl is 4200 J/(kg K), the specific heat capacity of water vapor cg is 2060 J/(kg K), and the environment temperature T0 is 300 K. For this case, the thermal conductances of the superheating section and evaporating section in the exhaust-heat boiler are prescribed, UAh1 = 5000 W/K and UAh2 = 5000 W/K. The calculated variations
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Fig. 3. The variations of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate with the mass flow rate for UAh1 = 5000 W/K and UAh2 = 5000 W/K.
Fig. 4. The variation of the output power with the thermal conductance of the preheating section.
of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate with the mass flow rate are shown in Fig. 3 where the subscript max denotes the maximum value of the corresponding parameter in the calculation range. Both the maximum entransy loss rate and the minimum entropy generation rate correspond to the maximum output power. The extremum of the entransy dissipation rate does not correspond to the maximum work output because it does not consider the influence of work output on the change of entransy. The variation of the output power with the thermal conductance of the designed preheating section is shown in Fig. 4. It can be found that larger conductance of the preheating section does not always correspond to larger output power. There is an optimum value for the thermal conductance of the preheating section which is 1493.36 W/K. The thermal conductance UAh3 and the mass flow rate _ are dependent variables. When UAh3 is determined, the mass flow m _ is determined as well under the prescribed condition. As the rate m maximum output work corresponds to the certain value of mass flow rate, then there must be an optimum value of the thermal conductance UAh3 corresponding to the maximum output work. In the second example, the thermal conductance of the superheating section and preheating section in the exhaust-heat boiler are prescribed (UAh1 = 5000 W/K and UAh3 = 3000 W/K), the other parameters are the same as those in the first example. The variations of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate with the mass flow rate are shown in Fig. 5. The results are similar to those in Fig. 3.
Fig. 5. The variations of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate with the mass flow rate for UAh1 = 5000 W/K and UAh3 = 3000 W/K.
Fig. 6. The variation of the output power with the thermal conductance of the evaporating section.
Fig. 7. The variations of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate with the mass flow rate for UAh2 = 5000 W/K and UAh3 = 4000 W/K.
The variation of the output power with the thermal conductance of the designed evaporating section is shown in Fig. 6. There is also an optimum value for the thermal conductance of the evaporating section, which is 3884.49 W/K. In the third example, the thermal conductance of evaporating section and preheating section in the exhaust-heat boiler is prescribed, UAh2 = 5000 W/K and UAh3 = 4000 W/K. The variations of the output power, the entropy generation rate, the entransy loss
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Fig. 8. The variation of the output power with the thermal conductance of the designed superheating section.
rate and the entransy dissipation rate with the mass flow rate in Fig. 7 are also similar to the above two cases. Fig. 8 shows the variation of the output power with the thermal conductance of the designed superheating section. The optimum value for the thermal conductance of the superheating section is 709.67 W/K. For the above examples where the heat capacity flow rates and the inlet temperatures of the streams are prescribed, both the minimum entropy generation principle and the entransy loss are applicable to the optimization of the Rankine cycle, while the concept of entransy dissipation is not. We further discuss some cases in which the heat capacity flow rates or the inlet temperatures of the streams are not prescribed. First, we calculate the dependence of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate on the inlet temperature of the hot stream Th,in with _ ¼ 0:1114 kg=s. The results are in prescribed mass flow rate m Fig. 9. The output power and the entransy loss rate both increase monotonically with increasing inlet temperature of the hot stream, while the entropy generation rate and the entransy dissipation rate first decrease and then increase. The extremum entropy generation rate and entransy dissipation rate do not correspond to the maximum output power. The variations of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate with the heat capacity rate of the hot stream Ch with prescribed mass flow _ ¼ 0:1114 kg=s shown in Fig. 10 are similar to those in Fig. 9. rate m Besides the entropy generation rate, the entropy generation number is also used as an important optimization criterion for heat-work conversion processes. The entropy generation number is a dimensionless parameter [37], which is defined as
NS ¼ S_ g =C min ;
ð33Þ
where Cmin is the minimum heat capacity rate of all the fluids. When Th,in changes, the heat capacity rates of the fluids do not change, so Cmin remains constant. When Ch changes, according to the prescribed parameters, we get
_ g ¼ 289:64 W=K: C min ¼ mc
ð34Þ
As Cmin is a constant under these two conditions, the variation tendencies of entropy generation number and entropy generation rate are consistent. For the other definition of entropy generation number by Bejan [42], there is
NS ¼ S_ g =S_ g; min :
ð35Þ
As S_ g; min is fixed and is independent of Th,in and Ch, the entropy generation number has the same variation tendency as that of the entropy generation rate.
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Fig. 9. The variations of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate with the inlet temperature of the hot stream.
Fig. 10. The variations of the output power, the entropy generation rate, the entransy loss rate and the entransy dissipation rate with the heat capacity flow rate of the hot stream.
The absorbed heat from the hot stream comes from the preheating section, the evaporating section and the superheating section. With the increase in the inlet temperature Th,in or the heat capacity flow rate Ch of the hot stream, the evaporating temperature of the working medium T6 increases while the condensing temperature T2 decreases. Due to the limitation of the saturated vapor line, the latent heat of evaporation rh decreases, which makes the absorbed heat Qh2 in the evaporating section decrease. As the thermal conductances of the evaporating section and the superheating section are fixed, the mean temperature difference between the working medium and the hot stream decreases with increasing Th,in, which makes the absorbed heat from the superheating section Qh1 decrease as well. The absorbed heat from the preheating section Qh3 increases due to the increase in T6 and the decrease in T2. For the specific example in this paper, the absorbed heat in the evaporating section and the superheating section takes larger proportion in the total absorbed heat, and then the total absorbed heat Qh decreases with increasing Th,in. On the other hand, with decreasing condensation temperature T2, the released heat decreases. However, as T2 approaches to the temperature of the cold stream, the decreasing rate of the released heat reduces, and the curve tends to be flat. That is the reason why the output work, which is the difference between the absorbed heat and the released heat, increases rapidly at first, and then the increasing rate reduces gradually, which makes the variation of output power in Figs. 9 and 10 appear similar.
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The entropy generation exactly corresponds to the exergy destruction in essence. When the inlet temperature or the heat capacity flow rate of the hot stream increases, the exergy flow into the system increases monotonically, and so does the output power. It is obvious in Figs. 9 and 10 that the output power increases rapidly at first with increasing input heat from the hot stream, and its increasing rate is higher than that of the exergy flow. Then, the increasing rate of the output power reduces gradually with increasing input heat, and is finally lower than the increase in exergy flow. The exergy destruction rate, which is the difference between the exergy flow into the system and the output power, as well as the entropy generation rate, decreases first and then increases with Ch and Th,in. Therefore, the minimum entropy generation rate does not correspond to the maximum output power. In addition, the increase in the inlet temperature or heat capacity flow rate of the hot stream will loss meanings after a specific value because the thermal properties of water and vapor restricts the evaporating temperature. For entransy loss, when the inlet temperature or heat capacity flow rate of hot stream increases, the output power increases as well. The change tendency of the entransy loss rate is consistent with that of the output power. 6. Conclusions The concept of entransy loss is applied to analyzing the Rankine cycle, as well as the concepts of entropy generation and entransy dissipation. Their application preconditions are obtained. Both the minimum entropy generation rate and the maximum entransy loss rate correspond to the maximum cycle output power for fixed heat capacity flow rates and inlet temperatures, but the maximum entransy dissipation rate does not. For the mass flow rate of the working fluid and the thermal conductance, there are optimal values that lead to the maximum output work. The output power increases with increasing inlet temperature or heat capacity flow rate of the hot stream if the inlet temperature or the heat capacity flow rate of the hot stream is not prescribed. Larger entransy loss rate still relates to larger output power, while smaller entropy generation rate does not. The concept of entransy loss is suitable for the output power optimization of the Rankine cycle for all the cases discussed in this paper. Acknowledgments The present work is supported by the Natural Science Foundation of China (Grant No. 51136001) and the Tsinghua University Initiative Scientific Research Program. References [1] Myat A, Thu K, Kim YD. A second law analysis and entropy generation minimization of an absorption chiller. Appl Therm Eng 2011;31:2405–13. [2] Mistry KH, Lienhard JH, Zubair SM. Effect of entropy generation on the performance of humidification–dehumidification desalination cycles. Int J Therm Sci 2010;49:1837–47. [3] Gu W, Weng YW. Second law analysis of evaporator for organic Rankine cycle system. In: Jacksonville FL, editor. Proceedings of the 2nd international conference on energy sustainability law analysis of evaporator for organic Rankine cycle system; 2009. [4] Yaghoubi M, Azizian K, Kenary A. Simulation of Shiraz solar power plant for optimal assessment. Renew Energy 2003;28:1985–98. [5] Zhang B, Ni W D, Li Z. Analysis of conventional power plant, IGCC and coal gasification SOFC hybrid with CO2 mitigation. Coal Convers 2005;28:1–7. [6] Wagar WR, Zamfirescu C, Dincer I. Thermodynamic performance assessment of an ammonia–water. Rankine Cycle Power Heat Prod 2010;51:2501–9. [7] Ronan KM, William JS. Optimal concentration and temperatures of solar thermal power plants. Energy Convers Manage 2012;60:226–32. [8] Roberto G. A novel design approach for small scale low enthalpy binary geothermal power plants. Energy Convers Manage 2012;64:274–363.
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