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An integrated optimization for organic Rankine cycle based on entransy theory and thermodynamics Tailu Li a, Wencheng Fu c, Jialing Zhu b, * a

School of Energy and Safety Engineering, Tianjin Chengjian University, Tianjin 300384, PR China Key Laboratory of Efﬁcient Utilization of Low and Medium Grade Energy, MOE, Tianjin University, Tianjin 300072, PR China c School of Automation, Tianjin University of Technology, Tianjin 300191, PR China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 January 2014 Received in revised form 26 April 2014 Accepted 22 May 2014 Available online 17 June 2014

The organic Rankine cycle has been one of the essential heat-work conversion technologies nowadays. Lots of effectual optimization methods are focused on the promotion of the system efﬁciency, which are mainly relied on engineering experience and numerical simulations rather than theoretical analysis. A theoretical integrated optimization method was established based on the entransy theory and thermodynamics, with the ratio of the net power output to the ratio of the total thermal conductance to the thermal conductance in the condenser as the objective function. The system parameters besides the optimal pinch point temperature difference were obtained. The results show that the mass ﬂow rate of the working ﬂuid is inversely proportional to the evaporating temperature. An optimal evaporating temperature maximizes the net power output, and the maximal net power output corresponds to the maximal entransy loss and the change points of the heat source outlet temperature and the change rates for the entropy generation and the entransy dissipation. Moreover, the net power output and the total thermal conductance are inversely proportional to the pinch point temperature difference, contradicting with each other. Under the speciﬁed condition, the optimal operating parameters are ascertained, with the optimal pinch point temperature difference of 5 K. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Organic Rankine cycle Integrated optimization Thermodynamics Entransy Pinch point

1. Introduction The global energy shortage urges the promotion of the heatwork conversion for low- and medium-temperature heat sources, such as geothermal energy, solar energy, the waste heat and so on. Among the numerous technologies, organic Rankine cycle (ORC) has attracted much attention due to its simple system structure and convenient operating maintenance in the past few decades [1e4]. However, the main problem is that the ORCs driven by relatively low heat sources show lower system efﬁciencies. Researchers have made great efforts to enhance the system performance, focusing on the parameter matching of the ORC with the heat source and the cold source, which can be categorized into two broad types. One is the optimization of the working ﬂuid selection due to that the working ﬂuid is a key factor; and the other is the simulation-based parameter optimization mainly by the ﬁrst and second laws of thermodynamics. Tamamoto et al. [1] investigated the performance of the ORC with R123 and water * Corresponding author. Tel./fax: þ86 22 23085107. E-mail address: [email protected] (J. Zhu). http://dx.doi.org/10.1016/j.energy.2014.05.082 0360-5442/© 2014 Elsevier Ltd. All rights reserved.

theoretically and experimentally, and R123 is preferable. Hung et al. [5] compared the efﬁciencies of ORCs using benzene, ammonia, R11, R12, R134a and R113. Saleh et al. [6] presented a thermodynamic screening of 31 pure working ﬂuids for ORCs, showing that ﬂuids with slightly lower critical temperatures are to be preferred. Aljundi [7] analyzed the inﬂuence of dry ﬂuids on the efﬁciencies of the ORC. Hung [8] investigated benzene, toluene, p-xylene, R113 and R123 in recovering low enthalpy heat sources. Yari [9,10] investigated several dry ﬂuids for the ORC by ﬁrst and second law analyses. Liu et al. [11] investigated the effects of various working ﬂuids on the thermal efﬁciency and on the total heat recovery efﬁciency. Arosio et al. [12] found that PP50 and R134a appear to be the most promising working ﬂuids. Lakew et al. [13] considered are R134a, R123, R227ea, R245fa, R290, and n-pentane. Many researchers have been done a lot of work to optimize the ORC parameters. Hettiarachchi et al. [2] presented an optimum design of an ORC driven by low-temperature geothermal water, with the screening criterion of total heat transfer area to the net power out. Roy et al. [14] conducted a parametric optimization of a waste heat recovery system and considered the power output and efﬁciencies. Rashidi et al. [15] presented a parametric optimization

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T. Li et al. / Energy 72 (2014) 561e573

of the regenerative ORCs, with efﬁciencies and speciﬁc work as the objective functions. Guo et al. [16] showed that optimum evaporation temperature and ﬂuids vary with screening criteria. Chao et al. [17] proposed the optimal evaporation temperature and working ﬂuids. Wang et al. [18] optimized a thermodynamic parameters using genetic algorithm. Cayer et al. [19] and Zhang et al. [20] conducted a parametric investigation for transcritical and subcritical ORC systems. The brief reviews presented above are mainly based on the ﬁrst and second laws of thermodynamics, and these literature did not take the pinch point temperature difference and the heat transfer in the evaporator and condenser in account at the same time. Actually, the two factors have signiﬁcant effect on the ORC performances. The pinch point temperature difference directly determines the net power output, however, the heat transfer in the evaporator and condenser can illustrate the system investment. Most of the previous just studies ﬁxed the value of the pinch point temperature difference based on experience. Moreover, the heat transfer in the evaporator and condenser for the ORC are always simpliﬁed due to that it differs with the working ﬂuid properties and the working condition. Luckily, researchers have introduced and developed a new theory, the entransy theory, to optimize some typical energy utilization systems containing a number of heat transfer processes in heat exchangers, such as evaporative cooling [21], heat exchanger networks in buildings [22,23] and thermal management systems in spacecrafts [24]. Based on the entransy theory and thermodynamics analysis, a theoretical integrated optimization method for the ORC systems is established. The entransy theory is mainly used for the heat transfer processes and the thermodynamic analysis for the expansion process in the turbine and compression process in pump. Furthermore, a novel objective function, Wnet =ððKAÞtotal =ðKAÞc Þ, was deﬁned to optimize the pinch point temperature difference. The main objective of this study was focused on optimizing the ORC system parameters. The cycle parameters, Wnet, Sg, Gdis, Gloss, tgw,out, te, hth, hex, (KA)e, (KA)c, and (KA)total, were calculated, and the optimal operating parameters were also ascertained.

2. Analysis of an ORC system A typical ORC system for power generation can be categorized in three loop circuits according to the working media: the heat source, the working ﬂuid, and the heat sink. The ORC mainly consists of an evaporator, a turbine, a generator, a condenser, a pump, a cooling tower, a cooling pump, and a hot water pump. The heat source transfers heat to the organic ﬂuid, which absorbs heat to generate high-pressure vapor in the evaporator (Figs. 1 and 2, state 1), then the vapor ﬂows into the turbine and its enthalpy is converted into shaft work to drive the generator. The vapor exits the turbine (Figs. 1 and 2, state 2) is led to the condenser where it is liqueﬁed by cooling water. The liquid working ﬂuid at the condenser outlet (Figs. 1 and 2, state 3) is pressurized by the pump and ﬂows into the evaporator (Figs. 1 and 2, state 4). Then a new cycle begins. The Tes diagram of the ORC is shown in Fig. 2. The analysis of an ORC based on thermodynamics and the entransy theory were performed for the working ﬂuids investigated. For simplicity, the following assumptions were made: (1) Geo-plants are operated in a steady state, with a heat source of 105 C. (2) Superheated vapor is considered at the turbine inlet, with a degree of superheat of 5 K, and saturated liquid at the condenser exit. (3) The kinetic and potential energy changes are negligible.

Fig. 1. Schematic diagram of an ORC system.

(4) The thermal loss and the friction loss in the pipes are neglected. There are only two pressures: an evaporating pressure pe, and a condensing pressure pc. (5) The isentropic efﬁciency of the turbine ht, the pump hp, the how water pump hp,hw, and the cooling water pump hp,cw is set to be 0.8, 0.6, 0.75, and 0.75, respectively. (6) Electrical generator efﬁciency is taken as 0.90. (7) Atmospheric condition is taken as 0.101325 MPa and 25 C. (8) The temperature at the condenser outlet t3 is 30 C. In an ORC, there are two different categories of thermodynamic processes as the heat transfer processes in both evaporator and condenser and heat-work conversion processes in both pump and turbine. The temperature differences between the heat source/sink and the working ﬂuid drive the heat transfer in both evaporator and condenser, whereas the absolute temperatures of the working ﬂuid impel the heat-work conversion processes. The heat transfer processes are analyzed by entransy theory and heat-work conversion processes by thermodynamic analyses entransy theory. 2.1. Entransy analysis of the heat exchangers The entransy represents the potential energy of heat in an object corresponding to the analogy of the electrical energy in a capacitor, and it is deﬁned as follows [25]:

● Fig. 2. Tes schematic diagram of an ORC system.

T. Li et al. / Energy 72 (2014) 561e573

and vapor in order. Therefore, the heat transfer process in the evaporator can be divided into three sections, i.e. preheating, evaporating and vapor superheating sections represented by I, II and III, and thus the evaporator can be regarded as the combination of three series-connected counter-ﬂow heat exchangers corresponding to the three sections. The subscript gw and wf stand for geothermal water and the working ﬂuid in the evaporator; e stands for evaporating; and in and out separately stands for the inlet and the outlet. Tgw,1 and Tgw,2 are the temperatures of geothermal water leaving and entering the evaporating section. In Fig. 3, the area between the temperature variation line segments of geothermal water and the working ﬂuid in Section I represents the entransy dissipation rate during the heat transfer in this section, which measures the irreversibility of heat transfer. The entransy dissipation rate during the heat transfer processes is deﬁned as the product of the total heat transfer rate and arithmetic mean temperature difference of the two ﬂuids based on Eq. (4) [27]:

Fig. 3. Teq diagram for the heat transfer in the evaporator.

G¼

1 QT 2

(1)

where Q and T are the heat and the temperature of an object, respectively. The heat transfer associates with the entransy transportation and dissipation. For a counter-ﬂow evaporator, the heat transfer rate, Qe, is expressed as:

Qe ¼ mgw cp;gw Tgw;in Tgw;out ¼ mwf cp;wf ðT1 T4 Þ

(2)

where mgw and mgw are the mass ﬂow rates of geothermal water and the working ﬂuid, respectively. cp,gw is the constant pressure speciﬁc heat of geothermal water. h1 and h4 are the speciﬁc enthalpies of the working ﬂuid at the evaporator inlet and outlet. For the systems operating under steady states, the mass ﬂow rate of the geothermal water, the working ﬂuid and the cooling water are constant. Based on the deﬁnitions of entransy and entransy dissipation [26], the variation of the entransy ﬂux of the speciﬁc ﬂow in the heat exchanger equals to the difference value of the entransy ﬂux at the inlet minus that at the outlet:

Gin Gout ¼

1 1 mcp T 2 mcp T 2 in out 2 2

(3)

where G is the entransy ﬂow rate accompanying with the ﬂuid ﬂow. The entransy dissipation rate during the heat transfer process in a heat exchanger is calculated by:

Gdis ¼ Gh;in Gh;out þ Gc;in Gc;out

(4)

where the subscripts h and c represent the hot ﬂuid and cold ﬂuid in the heat exchanger, respectively.

1 1 2 2 mgw cp;gw Tgw;2 mwf cp;wf Te2 T42 Tgw;out 2 2 Tgw;2 þ Tgw;out Te þ T4 ¼ Qe;1 2 2

fe;1 ¼

2.1.1. Analysis for the evaporator Fig. 3 shows the temperature difference variation between geothermal water and the working ﬂuid in the evaporator versus the heat transfer rate q, Teq diagram, where the upper and lower line segments separately stand for the temperature variations of geothermal water and the working ﬂuid. In the evaporator, accompanying heat absorption and temperature drop, the working ﬂuid experiences such three states as liquid, liquidevapor mixture

1 1 mgw cp;gw mwf cp;wf

!1 ln

(5)

where mgw, cp,gw and mgwcp,gw, respectively, represent the mass ﬂow rate, the constant pressure speciﬁc heat and the heat capacity rate of geothermal water; mwfcp,wf and mwfcp,wf stand for the working ﬂuid in Section I. And Qe,1 is the total heat transfer rate in Section I:

Qe;1 ¼ mgw cp;gw Tgw;2 Tgw;out ¼ mwf cp;wf ðTe T4 Þ

(6)

On the other hand, the area between the e temperature variation line segments of geothermal water and the working ﬂuid in Section I can also be calculated by integrating the area element. Therefore, the entransy dissipation rate is also calculated as:

fe;1

ZQe;1 Tgw Twf dq ¼

(7)

0

where q is the heat ﬂux. The entransy dissipation rate can be expressed as a function of the thermal conductance of the evaporator, i.e. the production of the heat transfer coefﬁcient and the area of the evaporator (KA)e, the heat capacity of the geothermal water and the working ﬂuid in Section I [28]:

1 2 1 1 fe;1 ¼ Qe;1 2 mgw cp;gw mwf cp;wf ðKAÞe;1

e

ðKAÞe;1 ¼

563

ðKAÞe;1

e

!

1 1 mgw cp;gw mwf cp;wf 1 1 mgw cp;gw mwf cp;wf

! !

þ1

(8)

1

where (KA)e,1 is the thermal conductance of Section I of the evaporator, and K and A are the heat transfer coefﬁcient and the heat transfer area. Combining Eqs. (5), (6) and (8) gives the expression of the thermal conductance of Section I in the evaporator:

mwf cp;wf ðTe T4 Þ þ mgw cp;gw Tgw;out Te mgw cp;gw Tgw;out T4

(9)

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T. Li et al. / Energy 72 (2014) 561e573

Similarly, the heat transfer rate Qe,2 in evaporating Section II of the evaporator is calculated as:

Qe;2 ¼ mwf re ¼ mgw cp;gw Tgw;1 Tgw;2

(10)

where re is the latent heat of vaporization of the working ﬂuid. The entransy dissipation rate of the Section II is simpliﬁed as:

fe;2 ¼ Qe;2

Tgw;1 þ Tgw;2 Te 2

ðKAÞe;2

¼

2 Qe;2 1 emgw cp;gw þ 1 e;2 2 mgw cp;gw mðKAÞ e gw cp;gw 1

(11) Through similar derivations, the thermal conductance of Section II in the evaporator is expressed as:

ðKAÞe;2 ¼ mgw cp;gw ln ¼ mgw cp;gw ln

Tgw;1 Te Tgw;2 Te

Fig. 4. Teq diagram for the heat transfer in the condenser.

mwf re þ mgw cp;gw Tgw;2 Te mgw cp;gw Tgw;2 Te

(12)

For Section III in the evaporator, the heat transfer rate Qe,3 is calculated as:

Qe;3 ¼ mgw cp;gw Tgw;in Tgw;1 ¼ mwf cp;wf ðT1 Te Þ

(13)

where T1 is the temperature of superheated vapor of the working ﬂuid at the outlet of the evaporator. The entransy dissipation rate in Section III is expressed as:

fe;3 ¼Qe;3

Tgw;in þ Tgw;1 T1 þ Te 2 2

1 1 2 2 mwf cp;wf Tc2 T32 mgw cp;gw Tcw;1 Tcw;in 2 2 Tc þ T3 Tcw;1 þ Tcw;in ¼ Qe;1 2 2

fc;1 ¼

! 1 2 1 1 ¼ Qe;3 2 mgw cp;gw mwf cp;wf ! 1 1 ðKAÞe;3 mgw cp;gw mwf cp;wf e þ1 ! 1 1 ðKAÞe;3 mgw cp;gw mwf cp;wf e 1

(14)

Combination of Eqs. (13) and (14) gives the expression of thermal conductance of the Section III:

ðKAÞe;3 ¼

1 1 mgw cp;gw mwf cp;wf

!1 ln

ðKAÞe ¼

Qc;1 ¼ mwf cp;wf ðTc T3 Þ ¼ mcw cp;cw Tcw;1 Tcw;in

fc;1

ðKAÞc;1

(16)

i¼1

2.1.2. Analysis for the condenser Fig. 4 shows the Teq diagram of the heat transfer process in the condenser, where the upper and lower line segments separately stand for the temperature variations of the cooling water and the working ﬂuid. The superheated vapor of the working ﬂuid enters

ðKAÞc;1

e

(18)

(15)

1 2 1 1 ¼ Qc;1 2 mwf cp;wf mcw cp;cw

e ðKAÞe;i

(17)

where mcw, cp,cw and mcwcp,cw stand for the cooling water in Section I, and T3 is the temperature of super-cooled ﬂuid of the working ﬂuid at the outlet of the condenser. And Qc,1 is the total heat transfer rate in Section I:

mwf cp;wf ðT1 Te Þ þ mgw cp;gw Tgw;1 T1 mgw cp;gw Tgw;1 Te

As mentioned before, because the evaporator is assumed as the combination of three series-connected counter-ﬂow heat exchangers, the total thermal conductance of the evaporator is calculated as: 3 X

the condenser, condenses and exits as super-cooled ﬂuid. As with the evaporator, the condenser also consists of three sections, i.e. super-cooling and condensing and superheating sections separately represented by I, II and III. Tcw,1 and Tcw,2 are the temperatures of the cooling water entering and leaving the condensing section. Referring to the analysis for condenser, the thermal conductances of Sections I, II and III in evaporator separately have the expressions:

!

1 1 mwf cp;wf mcw cp;cw 1 1 mwf cp;wf mcw cp;cw

! !

þ1

(19)

1

where (KA)c,1is the thermal conductance of Section I of the condenser. Combining Eqs. (17)e(19) gives the expression of the thermal conductance of Section I in the condenser:

T. Li et al. / Energy 72 (2014) 561e573

1 1 mwf cp;wf mcw cp;cw

ðKAÞc;1 ¼

!1 ln

565

mcw cp;cw Tcw;1 Tcw;in þ mwf cp;wf T3 Tcw;1 mwf cp;wf T3 Tcw;1

(20)

Similarly, the heat transfer rate Qc,2 in condensing Section II of the condenser is calculated as:

ðKAÞtotal ¼ ðKAÞe þ ðKAÞc

Qc;2 ¼ mwf rc ¼ mcw cp;cw Tcw;2 Tcw;1

2.2. Thermodynamic and entransy analyses for the system

(21)

where rc is the latent heat of condensation of the working ﬂuid. The entransy dissipation rate of Section II is simpliﬁed as:

fc;2 ¼ Qc;2

Tcw;1 þ Tcw;2 Tc 2

2 Qc;2

ðKAÞc;2 mcw cp;cw

Through similar derivations, the thermal conductance of Section II in the condenser is expressed as:

mcw cp;cw Tc Tcw;1 þ mwf rc mcw cp;cw Tc Tcw;1

(23)

For Section III in the condenser, the heat transfer rate Qc,3 is calculated as:

Qc;3 ¼ mwf cp;wf ðT2 Tc Þ ¼ mcw cp;cw Tcw;out Tcw;2

(24)

where T2 is the temperature of superheated vapor of the working ﬂuid at the inlet of the condenser. The entransy dissipation rate in Section III is expressed as:

T2 þ Tc Tcw;out þ Tcw;2 2 2 ! 1 2 1 1 ¼ Qc;3 2 mwf cp;wf mcw cp;cw ! 1 1 ðKAÞc;3 mwf cp;wf mcw cp;cw e þ1 ! 1 1 ðKAÞc;3 mwf cp;wf mcw cp;cw 1 e

fc;3 ¼Qc;3

1 1 mwf cp;wf mcw cp;cw

!1 ln

(25)

3 X

ðKAÞc;i

(29)

where the h represents the speciﬁc enthalpy. The entropy change of the evaporator is calculated as:

Dse ¼ mwf ðs1 s4 Þ mgw sgw;in sgw;out

Ie ¼ DExgw DExwf þ Wp;gw ¼ Exgw;in Exgw;out Exwf ;1 Exwf ;4 þ Wp;gw ¼ T0 Dse þ Wp;gw

(31)

where Ex and I represent the speciﬁc exergy and the irreversibility; T0 stands for the ambient temperature; Wp,gw is the power consumption of geothermal water pump. The power consumption of the geothermal water pump is calculated as:

. hp;gw rgw

i¼1

Summating the thermal conductance of the condenser and the evaporator yields the expression of the total thermal conductance of heat exchangers in the system:

(32)

where pgw, hp,gw and rcw are the pressure provided by the geothermal water pump, the efﬁciency of the geothermal water pump, and the density for the geothermal water. For the turbine, the isentropic efﬁciency is deﬁned as：

h1 h2;s

(33)

where ht stands for the turbine efﬁciency; h2,s is speciﬁc enthalpy at the turbine outlet experiencing an isentropic process.

mcw cp;cw T2 þ Tcw;2 mwf cp;wf ðT2 Tc Þ mcw cp;cw Tc Tcw;2

(27)

(30)

where s stands for the speciﬁc entropy. Based on Eq. (31), the irreversibility of the evaporator is:

ht ¼ ðh1 h2 Þ

As mentioned before, because the condenser is assumed as the combination of three series-connected counter-ﬂow heat exchangers, the total thermal conductance of the evaporator is calculated as:

ðKAÞc ¼

Qe;i ¼ mgw cp;gw Tgw;in Tgw;out ¼ mwf ðh1 h4 Þ

Wp;gw ¼ mgw pgw

Combination of Eqs. (24) and (25) gives the expression of thermal conductance of Section III:

ðKAÞc;3 ¼

3 X i¼1

(22)

ðKAÞc;2 ¼ mcw cp;cw ln

According to 3.1.1, the heat load of the evaporator can be expressed as:

Qe ¼

1 e þ1 ¼ c;2 2 mcw cp;cw mðKAÞ cp;cw cw 1 e

(28)

(26)

The power outlet of the turbine can be expressed as:

Wt ¼ ht mwf h1 h2;s ¼ mwf ðh1 h2 Þ

(34)

where hm and hg, respectively, stand for the mechanical efﬁciency and the generator efﬁciency; the subscript t represents the turbine. The entropy change of the turbine is calculated as:

Dst ¼ mwf ðs2 s1 Þ Based on Eq. (36), the irreversibility of the turbine is:

(35)

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T. Li et al. / Energy 72 (2014) 561e573

It ¼ Exwf ;1 Exwf ;2 Wt ¼ T0 Dst

(36)

According to 3.1.2, the heat load of the condenser can be expressed as:

Qc ¼

3 X

Qc;i ¼ mwf ðh2 h3 Þ ¼ mcw cp;cw Tcw;out Tcw;in

(37)

i¼1

Wnet ¼ hm hg Wt Wp Wp;gw Wp;cw ¼ hm hg Wp;gw þ Wp þ Wp;cw þ Qe Qc Wp;gw þ Wp þ Wp;cw ¼ hm hg ðQe Qc Þ þ 1 hm hg Wp;gw þ Wp þ Wp;cw

(50)

The heat ﬂow rate released to the environment Q0 is:

Q0 ¼ Cgw Tgw;in Tgw;out þ Ccw Tcw;in T0 Wnet

(51)

The entropy change of the condenser is calculated as:

Dsc ¼ mcw ðs3 s2 Þ mcw scw;in scw;out

(38)

Based on Eq. (39), the irreversibility of the condenser is:

Wp;cw ¼ mcw pcw

hp;cw rcw

(39)

(41)

where h4,s is speciﬁc enthalpy at the pump outlet experiencing an isentropic process. The power consumption of the pump is calculated as:

. Wp ¼ mwf ðpe pc Þ hp rwf

(42)

where pe and pc are the evaporating pressure and the condensing pressure; rwf is the density for the working ﬂuid. The entropy change of the pump is calculated as:

Dsp ¼ mwf ðs4 s3 Þ

(43)

Based on Eq. (31), the irreversibility of the pump is:

Ip ¼ Wp Exwf;4 Exwf ;3 ¼ T0 Dsp

(44)

Net power output:

Wnet ¼ hm hg Wt Wp Wp;cw Wp;gw

(45)

Thermal efﬁciency:

hth ¼ Wnet =Qe

(46)

Exergetic efﬁciency:

hex ¼

Wnet Exgw;in Exgw;out

(47)

System entropy generation:

sg ¼ Dse þ Dsc þ Dsp þ Dst þ Dsp;gw þ Dsp;cw

(48)

During a cycle, the energy conservation equation is expressed as:

Wp;gw þ Wp þ Wp;cw þ Qe ¼ Wt þ Qc From Eq. (46), the following equation can be obtained:

(52)

(40)

where Wp,cw is the power consumption of the cooling water pump; rcw is the density for the cooling water. For the turbine, the isentropic efﬁciency is deﬁned as

hp ¼ h4;s h3 ðh4 h3 Þ

1 1 2 2 2 Cgw Tgw;in þ Ccw Tcw;in Tgw;out T02 Q0 T0 2 2 1 2 2 ¼ Cgw Tgw;in Tgw;out 2Tgw;in T0 þ 2Tgw;out T0 2 2 1 þ Ccw Tcw;in T0 þ T0 Wnet 2

Gloss ¼

Ic ¼ DExwf DExcw þ Wp;cw ¼ Exwf;2 Exwf;3 Excw;out Excw;in þ Wp;cw ¼ T0 Dsc þ Wp;cw

where Cgw and Ccw are the heat capacities of the geothermal water and the cooling water, respectively. The entransy loss [29,30] of an ORC system can be calculated as:

(49)

The entransy dissipation of an ORC system can be calculated as:

Gdis ¼

1 1 2 2 Cgw Tgw;in þ mwf cp;wf ;4e4' Te2 T42 Tgw;out 2 2

1 Qe;2 Te þ mwf cp;wf;g;1'e1 T12 Te2 2 1 1 2 2 Ccw Tcw;in þ mwf cp;wf ;2e2' T22 Tc2 Tcw;out þ 2 2

1 þ Qc;2 Tc þ mwf cp;wf ;3'e3 Tc2 T32 2

1 2 Ccw Tcw;out T02 Ccw Tcw;out T0 T0 þ 2 (53)

2.3. Optimization model of the ORC system For the ORC system in Fig. 1, the optimization objective functions could be various, e.g., the net power output, the total irreversibility, the thermal efﬁciency, the exergetic efﬁciency, the net power output per unit mass ﬂow rate of the heat source, the ratio of total heat transfer area to the net power output, and the electricity production cost. The above-mentioned objective functions are all determined by the unknown parameters, i.e., the thermal conductances of the evaporator and the condenser, the mass ﬂow rate and the temperatures of the working ﬂuid, and can be mathematically expressed as the functions of these parameters. In this paper, the ratio of the net power output to the dimensionless parameter of the total thermal conductance of the evaporator and the condenser to the thermal conductance of the condenser, Wnet/ ((KA)total/(KA)c), is taken as the objective function to demonstrate the optimization modeling and solving processes. The maximization of the objective function is equivalent to the optimal system performance. According to Eq. (28), the total thermal conductance (KA)total is the function of Te, mwf, Tgw,out, and mcw. These four parameters should physically satisfy two constraints separately deﬁned in Eqs. (29), (37) and (50). On the other hand, the ORC system has three degrees of freedom when mgw, Tgw,in, Tcw,in, Tcw,out, Tc are known. Therefore, the constrained optimization problem can mathematically convert into a typical conditional extremum problem which can be solved by the Lagrangian multiplier method. A Lagrange function F is constructed as following:

T. Li et al. / Energy 72 (2014) 561e573

F ¼ ðKAÞtotal þ xðmgw cp;gw ðTgw;in Tgw;out Þ mwf h1 h4

567

þ u mwf h2 h3 mcw cp;cw Tcw;out Tcw;in

! ! mgw pgw mwf pe pc mcw pcw þ j hm hg mwf h1 þ h3 h2 h4 þ 1 hm hg þ þ Wnet hp rwf hp;gw rgw hp;cw rcw " !1 mwf cp;wf Te T4 þ mgw cp;gw Tgw;out Te mwf re þ mgw cp;gw Tgw;2 Te 1 1 ¼ þ mgw cp;gw ln ln mgw cp;gw mwf cp;wf mgw cp;gw Tgw;out T4 mgw cp;gw Tgw;2 Te !1 mwf cp;wf T1 Te þ mgw cp;gw Tgw;1 T1 1 1 ln þ mgw cp;gw mwf cp;wf mgw cp;gw Tgw;1 Te !1 mcw cp;cw Tcw;1 Tcw;in þ mwf cp;wf T3 Tcw;1 mcw cp;cw Tc Tcw;1 þ mwf rc 1 1 þ mcw cp;cw ln ln mwf cp;wf mcw cp;cw mcw cp;cw Tc Tcw;1 mwf cp;wf T3 Tcw;1 !1 mcw cp;cw T2 þ Tcw;2 mwf cp;wf T2 Tc 1 1 þ þ x mgw cp;gw Tgw;in Tgw;out ln mwf cp;wf mcw cp;cw mcw cp;cw Tc Tcw;2 mwf h1 h4 þ u mwf h2 h3 mcw cp;cw Tcw;out Tcw;in þ j hm hg mwf h1 þ h3 h2 h4 mgw pgw mwf ðpe pc Þ mcw pcw þ þ þ 1 hm hg hp rwf hp;gw rgw hp;cw rcw

!

! Wnet (54)

where x, u and j are the Lagrange multipliers. The differential of F with respect to Te offers:

vF ¼ vTe

"

!1

m2gw c2p;gw mwf re mgw cp;gw Tgw;2 Te wf re þ mgw cp;gw Tgw;2 Te !1 # mgw cp;gw mwf cp;wf ðT1 Te Þ 1 1 2 ¼ 0 mgw cp;gw mwf cp;wf mwf cp;wf ðT1 Te Þ þ mgw cp;gw Tgw;1 T1 mgw cp;gw Tgw;1 Te

1 1 mgw cp;gw mwf cp;wf

þ

ln

mwf cp;wf mgw cp;gw þ mgw cp;gw Tgw;out T4 m

(55)

The differential of F with respect to mwf offers:

"

!2 1 1 mgw cp;gw mwf cp;wf mwf cp;wf ðTe T4 Þ þ mgw cp;gw Tgw;out Te mwf cp;wf ðTe T4 Þ þ mgw cp;gw Tgw;out Te mgw cp;gw re 1 þ 2 ln mgw cp;gw Tgw;out T4 mwf re þ mgw cp;gw Tgw;2 Te mwf cp;wf !2 !1 mwf cp;wf ðT1 Te Þ þ mgw cp;gw Tgw;1 T1 1 1 1 1 1 þ ln mgw cp;gw mwf cp;wf mgw cp;gw mwf cp;wf mgw cp;gw Tgw;1 Te m2wf cp;wf !2 cp;wf ðT1 Te Þ 1 1 þ mwf cp;wf mcw cp;cw mwf cp;wf ðT1 Te Þ þ mgw cp;gw Tgw;1 T1 !1 mcw cp;cw Tcw;1 Tcw;in þ mwf cp;wf T3 Tcw;1 1 1 1 þ 2 ln mwf cp;wf mcw cp;cw mwf cp;wf T3 Tcw;1 mwf cp;wf !2 mcw cp;cw Tcw;1 Tcw;in mcw cp;cw rc 1 1 þ mwf cp;wf mcw cp;cw mcw cp;cw Tc Tcw;1 þ mwf rc mcw cp;cw T T m þm c T T

vF ¼ vmwf

1 1 mgw cp;gw mwf cp;wf

wf

cw;1

!1

cw;in

cp;wf ðTe T4 Þ

wf p;wf

3

cw;1

!1 mcw cp;cw T2 þ Tcw;2 mwf cp;wf ðT2 Tc Þ 1 1 1 2 ln mwf cp;wf mcw cp;cw mcw cp;cw Tc Tcw;2 mwf cp;wf

!# cp;wf ðT2 Tc Þ 1 hm hg ðpe pc Þ ¼0 xðh1 h4 Þ þ uðh2 h3 Þ þ j hm hg ðh1 þ h3 h2 h4 Þ þ hp rwf mcw cp;cw T2 þ Tcw;2 mwf cp;wf ðT2 Tc Þ (56)

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T. Li et al. / Energy 72 (2014) 561e573

adopted in Ref. [6] while the fundamental EOS by Tillner-Roth [31] was selected in this paper.

The differential of F with respect to Tgw,out offers:

"

!1 1 1 ¼ vTgw;out mgw cp;gw mwf cp;wf # mgw cp;gw mwf cp;wf ðTe T4 Þ xmgw cp;gw Tgw;out mgw cp;gw Tgw;2 Te Tgw;out T4 vF

(57) The differential of F with respect to mcw offers:

2 vF ¼4 þ vmcw

1 1 mwf cp;wf mcw cp;cw

!1

4. Results and discussion 4.1. Mass ﬂow rate of the working ﬂuid R245fa was selected as the working ﬂuid, and its thermodynamic properties are shown in Table 2. The power output of the turbine is the major factor that inﬂuences the net power output, which can be seen from Eq. (44). The mechanical efﬁciency and the

cp;cw Tcw;1 Tcw;in mcw cp;cw Tcw;1 Tcw;in þ mwf cp;wf T3 Tcw;1

2

mcw cp;cw Tcw;1 Tcw;in þ mwf cp;wf T3 Tcw;1 mcw cp;cw Tc Tcw;1 þ mwf rc c ln p;cw m2cw cp;cw mcw cp;cw Tc Tcw;1 mwf cp;wf T3 Tcw;1 2 1 1 mcw cp;cw T2 þ Tcw;2 mwf cp;wf ðT2 Tc Þ mwf rc cp;cw mwf cp;wf mcw cp;cw þ ln mcw cp;cw Tc Tcw;1 þ mwf r m2cw cp;cw mcw cp;cw Tc Tcw;2 !1 mwf cp;wf ðT2 Tc Þ 1 1 ucp;cw Tcw;out Tcw;in þ þ mwf cp;wf mcw cp;cw mcw cp;cw T þ T ðT Tc Þ mcw m c

1 mwf cp;wf

mcw1cp;cw

ln

2

3 1 hm hg pcw 5¼0 þj hp;cw rcw

cw;2

Eqs. (29), (37), (50) and (55)e(58) compose the optimization equation group which contains seven unknown quantities, i.e. Te, mwf, Tgw,out, mcw, x, u and j. Solving the equation group can directly obtains all these unknown quantities above and design parameters, i.e. (KA)c, (KA)e and mwf, leading to the optimize the ORC system. 3. Validation Numerical solution is validated with the results of Saleh et al. [6] for various working ﬂuids-based ORC without regenerator and for the same operating conditions. The temperature of geothermal heat source used in Ref. [6] was 120 C and the mass ﬂow rate refers to a net power output of 1 MW. The condensing temperature was 30 C, the isentropic turbine efﬁciency was 0.85, and the isentropic pump efﬁciency was 0.65. The temperature difference at pinch point was 10 C. The results of present solutions show very good agreement with the results in Ref. [6] as shown in Table 1. The maximal relative errors of the parameters V1 and hth are 2.66 and 3.48%, respectively. The discrepancies mainly derive from the selection of equation of state (EOS) that the BACKONE EOS was Table 1 Validation of the numerical model with previous published data [6] for various ﬂuids-based ORC. Substance t1, C

t3, C

Pe, MPa Pc, MPa V1, m3/s VFR

hth, % Source

R125 R125 R290 R290 R134a R134a

30.00 30.00 30.00 30.00 30.00 30.00

2.000 2.000 2.000 2.000 2.000 2.000

2.32 2.35 5.91 5.81 7.74 7.48

40.06 40.06 57.14 57.14 67.75 67.75

1.564 1.564 1.079 1.079 0.772 0.772

2.878 2.834 1.063 1.049 0.656 0.639

1.270 1.360 1.667 1.764 2.357 2.483

[6] Present [6] Present [6] Present

wf p;wf

(58)

2

generator efﬁciency, respectively, reach 0.96 and 0.93, whereas the power consumption of the geothermal pump, the cooling pump, and the working ﬂuid pump is relatively small. Therefore, the net power output can be regarded as a linear function of the power output of the turbine. Once the ambient condition and the isentropic efﬁciency of the turbine are ﬁxed, the speciﬁc power output of the turbine is only relating to the evaporating temperature, i.e. the speciﬁc power output of the turbine is proportional to the speciﬁc enthalpy at the turbine inlet. In conclusion, the mass ﬂow rate of the working ﬂuid and the evaporating temperature are two key parameters to an ORC system for the ﬁxed heat source temperatures. Once the parameters for the heat source and heat sink are determined, then the mass ﬂow rate of the working ﬂuid is ascertained. Based on the ﬁrst law of thermodynamics, mwf can be represented by the following equation:

mwf;1 ¼

mgw cp;gw Tgw;in Tgw;out h1 h4

(59)

where h1 and h4 stand for the speciﬁc enthalpy at the turbine inlet and outlet, respectively. However, Eq. (54) is only valid for lower evaporating temperatures. If Te approaches to higher values on the promise of ensuring the pinch point, Eq. (54) will no longer be established, and mwf is determined by the heat provided by the heat source that is used to vaporize the working ﬂuid, which is calculated by:

mwf;2 ¼

mgw cp;gw Tgw;in Te DTpp h1 h40

(60)

where h40 represents the speciﬁc enthalpy of the working ﬂuid reaching the evaporating temperature for x ¼ 0.

T. Li et al. / Energy 72 (2014) 561e573

569

Table 2 Thermodynamic properties of R245fa. Substance

R245fa

Physical data

Environmental data

M (g/mol)

Tb ( C)

Tcri ( C)

Pcri (MPa)

ALT (yr)

ODP

GWP (100yr)

134.05

14.90

154.05

3.640

7.6

0

1030

Fig. 5 demonstrates the variation of the mass ﬂow rate of the working ﬂuid with the evaporating temperature. From Eqs. (54) and (55), both mwf,1 and mwf,2 are inversely proportional to te, but mwf,2 has a higher change rate than mwf,1. Increasing the evaporating temperature consistently promotes the speciﬁc enthalpy change of the working ﬂuid in the evaporator. However, the heat provided by the heat source that is used to vaporize the working ﬂuid is sharply shortened. Moreover, mwf is coupled with the evaporating temperature, and this is due to a one-to-one correspondence between the two parameters, which can also be expressed by the following equation:

mwf

8 mgw cp;gw Tgw;in Tgw;out ; te te;opt > < h1 h4 ¼ > : mgw cp;gw Tgw;in Te DTpp ; te te;opt h1 h40

(61)

4.2. Evaporating temperature Fig. 6 shows the inﬂuence of the evaporating temperature on the mass ﬂow rate of the working ﬂuid, mwf, and the speciﬁc power output and power output of the turbine, wt and Wt. wt is linear with te, and the change rate is 0.33 kW/(kg K). Eq. (34) illustrates that Wt is the production of mwf and wt, thereby illustrating that both wt and mwf should not be too low or too high in order to maximize the net power output. There exists an optimal evaporating temperature, te,opt, maximizing Wt. Combining Eqs. (34), (54) and (55), Wt can also be expressed as below:

8 mgw cp;gw Tgw;in Tgw;out h1 h2 ; te te;opt > < h1 h4 Wt ¼ > : mgw cp;gw Tgw;in Te DTpp h1 h2 ; te te;opt h1 h40 (62)

Fig. 5. The variation of the mass ﬂow rate of the working ﬂuid versus the evaporating temperature.

Sources

Type

[32]

Dry

It can be seen that the parameter, Wt, is a function of h1.

vWt ¼0 vh1

(63)

Combining Eqs. (62) and (63), we can obtain the relation as following:

Te ¼ Tgw;in DTpp

2 Tgw;in Tgw;out ðh2 h4 Þ h1 h40 ðh1 h4 Þ2 ðh2 h4 Þ (64)

An increase in te increases wt but decreases mwf, and the change in mwf can more markedly inﬂuence the power output of the turbine Wt. This is the reason why mwf and Wt show the similar variation trend with te. But the most fundamental reason is that mwf is sharply shortened for te > te,opt. From Eq. (59) and Fig. 6, it can be got that the optimal evaporating temperature te,opt is 87 C. Moreover, te,opt corresponds to the turning point of the change rate of mwf. 4.3. Irreversible loss Fig. 7 illustrates the inﬂuences of the evaporating temperature on the irreversibility of the system components. The irreversible loss caused by the condenser, Ic, ﬁrst increases and then decreases with te, and reaches the maximal value of 1.86 kW for te ¼ te,opt. The reason is that the thermal load of the condenser, Qc, is inversely proportional to te, whereas the temperature of the working ﬂuid at the turbine outlet, t2, is proportional to te. Qc is sharply reduced for te > te,opt due to that the geothermal water outlet temperature, tgw,out, becomes higher than 85 C, thereby resulting in the decrease of Ic. The irreversible loss caused by the evaporator, Ie, is continuing to decline with to te. An inverse relationship is found between the log mean temperature difference of the evaporator, DTe and te. On the other hand, the thermal load of the evaporator, Qe, sharply reduces for te > te,opt due to that the rise of the geothermal water

Fig. 6. The mass ﬂow rate, the power output and the speciﬁc power output with the evaporating temperature.

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T. Li et al. / Energy 72 (2014) 561e573

Fig. 7. The inﬂuence of the evaporating temperature on the irreversibility of the system components.

outlet temperature, thereby resulting in the further decrease of Ie for te > te,opt. Moreover, the change rate of Ie for te > te,opt is slightly lower than that for te < te,opt. For te ¼ te,opt, Ie equals to 3.54 kW. The irreversible loss caused by the pump, Ip, ﬁrst increases and then decreases with te, and there exists an maximal value of 0.14 kW for te ¼ te,opt. As the working ﬂuid at the condenser outlet remains steady, the speciﬁc enthalpy of the working ﬂuid at the condenser outlet, h3, is constant. Furthermore, the isentropic efﬁciency of the pump is also unchanged. Therefore, Ip is only proportional to two parameters mwf and h4. mwf decreases with te, whereas h4 decreases with te and h4. For te < te,opt, h4 has more obvious inﬂuence on Ip, so Ip is consistently increasing. For te > te,opt, mwf has more obvious inﬂuence on Ip, so Ip is consistently decreasing. The irreversible loss caused by the turbine, It, ﬁrst increases and then decreases with te, and there exists an maximal value of 2.50 kW for te ¼ te,opt. As the isentropic efﬁciency of the turbine is invariant, there is a positive correlation between h1, h2 and te. The speciﬁc enthalpy drop of the working ﬂuid, h1 h2, is also proportional to te. Similar with Ip, It depends on h1 h2 and mwf. h1 h2 has more obvious inﬂuence on It for te < te,opt, so It is consistently increasing. For te > te,opt, mwf has more obvious inﬂuence on It, so It is consistently decreasing. The overall irreversible loss of the ORC system, I, continues to decline with te, and the turning point of the change rate of I corresponds to te. For te ¼ te,opt, I equals to 8.03 kW. 4.4. Cycle performance Fig. 8 demonstrates the entropy generation, the entransy dissipation, the entransy loss, net power output, the thermal efﬁciency, the exergetic efﬁciency, and geothermal water outlet temperature with the evaporating temperature. When the evaporating temperature increases, the entropy generation and the entransy dissipation monotonously declines, whereas the entransy loss, the net power output, the thermal efﬁciency, and the exergetic efﬁciency experience going up ﬁrst and then turning down after reaching the maximum value, however, the geothermal water outlet temperature ﬁrst keeps constant and then increases. An increase in the evaporating temperature evidently lowers the log mean temperature difference in the evaporator, DTe, thus reducing the entropy generation of the system. Furthermore, there is no corresponding relation between the Sg and Wnet, so such the minimization of entropy generation will lead to a reduced net power output,

Fig. 8. The entropy generation, entransy dissipation, entransy loss, net power output, thermal efﬁciency, exergetic efﬁciency, and geothermal water outlet temperature with the evaporating temperature.

thereby resulting in a reduced efﬁciency of the cycle. This phenomenon is so called ‘entropy generation paradox’, which is consistent with the arguments presented by Bejan [33], Hesselgreaves [34], Salamon et al. [35], and Haseli [36,37]. The minimum entropy generation corresponds to a much lower net power output. However, the turning point of the change rate for Sg corresponds to Wnet. The entransy dissipation, Gdis, is similar with Sg, and the turning point of the change rate for Gdis corresponds to Wnet. As for Sg and Gdis, the change rates for te < te,opt are evidently lower than those for te > te,opt, which is ascribed to the rise of the geothermal water outlet temperature. From Eq. (45), it can be got that the net power output of the system, Wnet, is proportional to Wt, and Wnet and Wt, maximize at the same evaporating temperature, which is due to that value the power consumptions, Wp, Wp,gw, and Wp,cw, are relatively much smaller compared with Wt and that the mechanical efﬁciency and the generator efﬁciency are high. Wnet comes to its maximal value of 6.27 kW for te ¼ 87 C. Moreover, it can be seen that the entransy loss, Gloss, is linear with Wnet from Eq. (52), and Wnet reaches a peak value of 8298 kJ K at te ¼ 87 C. Three parameters, Wnet, hex, and hth, maximize at different te, and the te corresponding to the maximum of Wnet is the lowest, whereas the te corresponding to the maximum of hth is the highest. For te > te,opt, the promotion of te to a lesser extent can increase the hth and hex, but with the increasing of te, hex, and hth come to their maximum in turn at te ¼ 90 and 94 C, respectively. Fig. 9 analyzes the variations of the entropy generation, the entransy dissipation, the entransy loss, net power output, the thermal efﬁciency, the exergetic efﬁciency, and geothermal water outlet temperature versus the mass ﬂow rate of the working ﬂuid. When the mass ﬂow rate of the working ﬂuid increases, the entropy generation and the entransy dissipation monotonously rise, whereas the entransy loss, the net power output, the thermal efﬁciency, and the exergetic efﬁciency, experience going up ﬁrst and then turning down after reaching the maximum values, however, the geothermal water outlet temperature ﬁrst decreases and then remains unchanged. From the above analysis and Fig. 8, the geothermal water outlet temperature, tgw,out, remains stable for te 87 C, and after that it exceeds 85 C for te > 87 C. This is due to the decrease of the heat absorption of the system for te > 87 C. Furthermore, the change point of te is corresponding to the maximal value of Wnet. According to Figs. 8 and 9, the speciﬁc parameter illustrates the exact opposite variation trend towards te and mwf with any speciﬁed geothermal water inlet temperature, which is because there is an inversely one-

T. Li et al. / Energy 72 (2014) 561e573

Fig. 9. The variations of the entropy generation, entransy dissipation, entransy loss, net power output, thermal efﬁciency, exergetic efﬁciency, and geothermal water outlet temperature versus the mass ﬂow rate of the working ﬂuid.

to-one relationship between te and mwf. Therefore, the variation trends for the seven parameters in Fig. 9 are no more analyzed in detail. 4.5. Thermal conductance Fig. 10 shows the thermal conductance of the evaporator and the condenser, (KA)e and (KA)c, versus the evaporating temperature. When the evaporating temperature te goes up, the thermal conductance of the condenser (KA)c decreases while the thermal conductance of the condenser (KA)e ﬁrst increases until reaching the maximal value at te ¼ 90 C and afterwards consistently drops. For te 87 C, the rising of the te results in the decrease of the log mean temperature difference in the evaporator, DTe, and the increase of the log mean temperature difference in the condenser, DTc. Furthermore, the thermal load of the evaporator Qe remains the same, but the thermal load of the evaporator Qc leads to a slight reduction due to promotion of the power output of the turbine, Wt, which can be seen from Fig. 6. As for te 87 C, the increase of the te results in the rising of the geothermal water outlet temperature, thus reducing the thermal load of the evaporator Qe and the thermal load of the evaporator Qc, which is the leading cause of the reduction of Qe and Qc. It should be specially pointed out that DTc is

Fig. 10. The thermal conductances of the evaporator and condenser versus the evaporating temperature.

571

Fig. 11. The thermal conductances of the evaporator and the condenser and the total thermal conductance with Qe/Qc.

always proportional to te. The variation range of (KA)e is from 0.91 to 6.52 kW/K, whereas the variation range of (KA)c is from 3.50 to 44.75 kW/K. Fig. 11 shows the thermal conductance of the evaporator and the condenser, (KA)e and (KA)c, and the total thermal conductance, (KA)total, with Qe/Qc. The two ratios, (KA)total/(KA)e and (KA)c/(KA)e, are inversely proportional to Qe/Qc, meaning that a higher Qe/Qc leads to reduce (KA)e, (KA)c, and (KA)total. However, the parameter (KA)total/(KA)c presents a direct proportion to Qe/Qc, and it suggests that a higher value of Qe/Qc actually promotes the proportion of (KA)e. From the standpoint of the thermal conductance, a higher evaporating temperature is much more preferable.

4.6. Pinch point temperature difference The pinch point temperature difference, DTpp, is also a key parameter to optimize the ORC system performance, and it is set to be 5 K in Sections 4.1e4.5. The optimal pinch point temperature difference, DTpp,opt, should be ascertained so as to achieve the optimal performance of ORC. From Figs. 6 and 12, it can be easily

Fig. 12. The variation of the net power output versus the evaporating temperature with different pinch point temperature differences.

572

T. Li et al. / Energy 72 (2014) 561e573

output and the total thermal conductance are contradictory and inconsistent. In order to optimize the pinch point temperature difference, a new parameter, fobj, was deﬁned and chosen as the objective function:

fobj ¼ Wnet

. ðKAÞtotal ðKAÞc

(65)

Fig. 14 presents the inﬂuence of the evaporating temperature on the objective function with different pinch point temperature differences. With any speciﬁed DTpp, the objective function fobj ﬁrst goes up and then turns down after reaching the maximum. The maximal value of the objective function fobj,max ﬁrst remains stable and then decreases with DTpp, so fobj,max corresponding to the highest pinch point temperature difference is the optimal pinch point temperature difference DTpp,opt, and DTpp,opt ¼ 5 K. 5. Conclusions

Fig. 13. The variation of the total thermal conductance versus the evaporating temperature with different pinch point temperature differences.

found that the maximal value of the net power output Wnet,max is inversely proportional to DTpp, and the optimal evaporating temperatures corresponding to Wnet,max with different pinch point temperature differences decrease with DTpp. Wnet,max for DTpp ¼ 1e10 K reduces from 6.62 to 5.75 kW, and the gradient is about 0.087 kW/K. From the net power point of view, DTpp should be as low as possible. Fig. 13 is the variation of the total thermal conductance versus the evaporating temperature with different pinch point temperature differences. Similar to Wnet,max in Fig. 12, there is also an inverse relation between the maximal value of the total thermal conductance, (KA)total,max, and DTpp. (KA)total,max for DTpp ¼ 1e10 K reduces from 54.50 to 50.48 kW/K, and the gradient is about 0.4 kW/K2. From the perspective of the total thermal conductance, DTpp should be as high as possible. According to Figs. 12 and 13, the net power output represents the system earnings but neglects the investment, but the total thermal conductance does the reverse. Moreover, the net power

The organic Rankine cycle involves in both heat transfer and thermodynamic processes. We integrate the entransy theory with the thermodynamics to establish a theoretical and mathematical model to convert the ORC optimization into a conditional extremum problem. A global optimization method for the organic Rankine cycle systems is proposed. The main conclusions drawn from the present study can be summarized as follows: (1) For the speciﬁed heat sources, the mass ﬂow rate of the working ﬂuid is inversely proportional to the evaporating temperature. (2) An optimal evaporating temperature maximizes the net power output, which corresponds to the entransy loss and the change point of the heat source outlet temperature. (3) There is a correspondence between the maximal net power output and the change rate of the entropy generation and the entransy dissipation. (4) Both the net power output and the total thermal conductance are inversely proportional to the pinch point temperature difference, and they are contradictory and inconsistent with each other. (5) The optimal pinch point temperature difference is ascertained to be 5K using the ratio of the net power output to the speciﬁc value of the total thermal conductance to the thermal conductance in the condenser within the scope in this paper. Acknowledgement The authors gratefully acknowledge the ﬁnancial support provided by the National High Technology Research and Development Program of China (863 Program) (No. 2012AA053001). References

Fig. 14. The inﬂuence of the evaporating temperature on the objective function with different pinch point temperature differences.

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Nomenclature A: area (m2) Ex: Exergy (kW) G: Entransy (kW K) h: speciﬁc enthalpy (kJ/kg) I: irreversibility rate (kW) K: heat transfer coefﬁcient (W/(m2 C)) M: molar mass (kg/kmol) m: mass ﬂow rate (kg/s) P: pressure (MPa) Q: heat transfer rate (kW) s: speciﬁc entropy (kJ/(kg C)) T: temperature (K) t: temperature ( C) U: intrinsic energy (kJ) W: power (kW) DP: pressure difference (Pa) Greek symbols

h: efﬁciency (%) r: density (kg/m3) F: Entransy dissipation (kW K) Subscripts c: condenser cri: critical cw: cooling water e: evaporator ex: exergetic g: generator gw: geothermal water opt: optimal p: pump pp: pinch point s: isentropic t: turbine th: thermal wf: working ﬂuid 0: environment 1, 2, 3, 4: state points Acronyms ALT: atmosphere life time (yr) GWP: global warming potential ODP: ozone deletion potential ORC: organic Rankine cycle VFR: volumetric ﬂow ratio

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