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DOI:

10.1016/j.energy.2017.11.003

Reference:

EGY 11791

To appear in:

Energy

Received Date: 31 July 2017 Revised Date:

7 October 2017

Accepted Date: 1 November 2017

Please cite this article as: Su W, Zhao L, Deng S, Xu W, Yu Z, A limiting efficiency of subcritical Organic Rankine cycle under the constraint of working fluids, Energy (2017), doi: 10.1016/j.energy.2017.11.003. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

A limiting efficiency of subcritical Organic Rankine cycle

2

under the constraint of working fluids

3 4 5 6 7

Wen Sua, Li Zhaoa,*, Shuai Denga, Weicong Xua, Zhixin Yub a Key Laboratory of Efficient Utilization of Low and Medium Grade Energy (Tianjin University), MOE, Tianjin, 300072, China. Tel: 86-022-27404188, Email: [email protected] b University of Stavanger, 4036 Stavanger, Norway.

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Abstract:

As a theoretical upper bound of cycle efficiency, Carnot efficiency doesn’t

9

contain detailed information on the properties of working fluids. A nature idea

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emerges how to derive the efficiency limit under the constraint of working fluids and

11

how to quantify it by considering the properties of working fluids. Therefore, in this

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contribution, a limiting efficiency is proposed for subcritical Organic Rankine cycle

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(ORC). For the calculation of limiting efficiency, a limiting factor is defined on the

14

basis of the saturated slope of liquid at the reduced temperature 0.9. Furthermore, in

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order to represent the extent to which the practical efficiency approaches to the

16

limiting efficiency, a new expression is proposed for thermodynamic perfectness. 13

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pure fluids and 3 mixtures are employed to demonstrate the effects of working fluids

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on the limiting efficiency and thermodynamic perfectness. For pure working fluids,

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the fluid with a higher critical temperature possesses higher limiting efficiency and

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cycle perfectness. For mixtures, the limiting efficiency generally locates between

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those of pure fluids, while the thermodynamic perfectness varies greatly with the

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composition. Although the proposed limiting efficiency can’t be achieved by practical

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cycles, it can provide guidance for the selection of working fluids and the construction

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of ORC.

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ACCEPTED MANUSCRIPT Key words: Organic Rankine cycle; Working fluids; Limiting efficiency; Thermody

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namic perfectness

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Nomenclature

28

Symbols

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a

Molar Helmholtz energy kJ/kg

30

c

Heat capacity kJ/(kg•K)

31

M

Molar mass

32

ORC

Organic Rankine cycle

33

P

Pressure

34

R

Ideal gas constant

kJ/(kmol•K)

35

s

Specific entropy

kJ/(kg•K)

36

S

Enclosed area in the temperature-entropy diagram

37

T

Temperature

38

v

Molar volume

39

Greeks

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η

41

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γ

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σ

Molecular complexity

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Thermodynamic perfectness

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∆

Difference

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Subscripts

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g/mol

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Pa

K

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m3/mol

%

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Efficiency

Slope of the oblique line (Fig.1)

Limiting factor

%

kJ/kg

1,…8

Thermodynamic state points (Fig.1)

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1s, 2s, 2’

Thermodynamic state points (Fig.1)

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a, b

Points on the axis (Fig.1)

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c

Critical temperature

51

carnot

Performance of Carnot cycle

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eq

Equilibrium

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limit

Performance limit

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L

Liquid

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p

Pressure

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practical

Performance of practical cycle

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r

Reduced

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sup

Superheating temperature

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V

Vapor

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TE D

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1. Introduction

With the increase of global energy demand, Organic Rankine cycle (ORC) has

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been widely employed to utilize the low and medium grade energy, such as solar

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energy, wind energy, geothermal energy and waste heat for the production of

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electricity. In order to convert these energy sources with high efficiency, organic

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substances are employed as working fluids. Even so, in practical engineering, thermal

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efficiency of subcritical ORC is relatively low, around 10% [1, 2]. Compared with the

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Carnot cycle under the same heat source and sink, thermodynamic perfectness of

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ACCEPTED MANUSCRIPT subcritical ORC is usually less than 50%. Being different from the reversible Carnot

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cycle, there always exist different kinds of irreversible losses in the practical cycle,

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due to the heat transfer temperature difference and the flow resistance of working

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fluids. Furthermore, for subcritical ORC, heat exchange between working fluids and

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heat source/sink is an isobaric process and involves the phase change of working

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fluids. While the heat exchange of Carnot cycle is an isothermal process and there is

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no phase change. It has been proven that the efficiency of Carnot cycle is only related

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with the temperatures of heat source/sink via the equation of state of working fluids [3,

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4]. However, for subcritical ORC, the cycle efficiency is a function of working fluids

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and operating conditions, so that a large number of researchers have contributed to the

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selection of working fluids for different applications of ORC [5, 6].

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Thermodynamic properties of working fluids have significant effects on the

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thermal efficiency of subcritical ORC. Numerical literatures have been published to

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reveal the relationship between the properties of working fluids and the cycle

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performance [7, 8]. For instance, Saleh et al. [9] compared cycle performance of 31

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pure working fluids for different types of ORCs, and concluded that the high boiling

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fluids with overhanging saturated vapor line show the highest cycle efficiencies. Yang

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et al. [10] compared cycle performance of 267 working fluids to investigate the

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effects of critical and boiling temperatures on the maximum net output power. The

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optimal combinations of critical and boiling temperatures were proposed for the

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selection of working fluids. In the optimization of cycle parameters for given heat

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sources, Cavazzini et al. [11] noticed that the efficiency of subcritical ORC is highly

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ACCEPTED MANUSCRIPT dependent on the ratio between the critical temperature of working fluid and the inlet

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temperature of heat source. However, an expression derived by Liu et al. [12]

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indicates that the cycle efficiency is a weak function of the critical temperature,

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regardless of the fact that the thermal efficiency for working fluid with the lower

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critical temperature is lower. Furthermore, in order to improve the cycle efficiency

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and reduce the irreversible loss, zeotropic mixtures have been recommended as

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candidates of working fluids. It’s thought that the temperature glide of mixtures can

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alleviate the temperature mismatch in the heat transfer process. Many researches on

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the cycle performance of zeotropic mixtures have been carried out experimentally and

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theoretically [13]. However, it should be noted that the above literatures mainly focus

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on the relationship between the practical efficiency of a specific cycle configuration

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and the properties of working fluids. To the best of authors’ knowledge, few

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researchers have investigated the efficiency limit of subcritical ORC by considering

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the properties of working fluids.

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In addition to selecting the working fluids with appropriate properties, various

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ORC configurations have been proposed and analyzed to enhance the cycle efficiency

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for different applications [14]. For example, Bao et al. [15] proposed an auto-cascade

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ORC for the utilization of solar energy. For the recovery of waste heat, Zhang et al.

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[16] conducted thermo-economic comparison of subcritical ORC based on different

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types of heat exchangers. Li [17] investigated the economic assessment of various

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working fluids for different ORC configurations under the assumption that the net

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power output remained constant. For the geothermal power plants, the

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ACCEPTED MANUSCRIPT exergoeconomic of three ORC configurations, namely simple ORC, ORC with

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internal heat exchanger, regenerative ORC, are compared by Zare [18]. He concluded

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that ORC with internal heat exchanger has the best thermodynamic performance,

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while simple ORC has the best economic performance. Furthermore, some

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researchers introduced the phase separators into the ORC systems to control and

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distribute the working fluid flow, thus achieving the high cycle efficiency with

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proposed system configurations [19, 20]. However, to the best of authors’ knowledge,

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the efficiency of these proposed system configurations is still far from the Carnot

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efficiency, so that the guidance of Carnot efficiency is not strong enough for the

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construction of ORC. A more practical efficiency limit should be proposed to provide

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theoretical guidance for the operating condition and the optimal design of ORC.

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In this study, considering the constraint of working fluids, a limiting efficiency

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is proposed for the subcritical ORC. For the construction of ORC, the limiting

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efficiency gives an upper bound of cycle efficiency. In the process of deriving

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limiting efficiency, a limiting factor is defined from the thermodynamic properties of

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working fluids. Furthermore, according to the proposed efficiency, a new expression

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on thermodynamic perfectness is defined to represent the degree to which the

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practical efficiency approaches to the limiting efficiency. This paper is structured as

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follows: in section 2, the detailed methodology for deriving the limiting efficiency is

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presented. In section 3, the effects of working fluids and cycle temperatures on the

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efficiency limit and the thermodynamic perfectness of ORC are discussed.

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Comparisons between the proposed parameters and the Carnot-related performances

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ACCEPTED MANUSCRIPT 135

are conducted. Finally, conclusions of this work are given in section 4.

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2. Methodology 137

2.1 Derivation of limiting efficiency A temperature-entropy (T-s) diagram for an ideal ORC is shown in Fig.1. The

139

employed working fluids are pure fluid R600 and mixture R600/R134a (50%/50%,

140

mass) respectively, taking as a case study. The figure indicates that the ideal ORC

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consists of two isobaric processes and two isentropic processes. In the isobaric

142

process 3-2, working fluid is heated and vaporized by the heat source. Then, the high

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pressure and temperature vapor conducts expansion to generate work in the isentropic

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process 2-6. Thereafter, the exhausted vapor is condensed into subcooled liquid by the

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heat sink in the isobaric process 6-1. Finally, in the isentropic compression process

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1-3, the liquid at the condensation pressure is pumped into the evaporator. Due to the

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fact that the temperature variation of liquid is not sensitive to the change of pressure

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for the subcritical ORC, the temperature rise of working fluid in the pump can be

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neglected. This means that the temperature at point 3 is equal to the temperature at

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point 1, as shown in Fig.1.

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Fig.1 T-s diagram of subcritical ORC and Carnot cycle for (a) R600, (b) R600/R134a (50%/50%, mass)

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In order to compare the performance of ORC with that of Carnot cycle, Fig.1

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also displays the T-s diagram of Carnot cycle between the highest temperature and the

156

lowest temperature of ORC. It can be observed that the main difference between the

ACCEPTED MANUSCRIPT ORC and the Carnot cycle lies in the thermodynamic processes of liquid working

158

fluid. For the Carnot cycle, the isentropic compression (1-1s) and the isothermal

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expansion (1s-2’) of working fluid in the liquid state can’t be realized in the practical

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engineering, due to the non-sensitive relationship among the pressure, the volume and

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the temperature. As for the ORC, the heat exchange between the heat source and the

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working fluid happens in the isobaric process, considering the thermodynamic

163

properties of liquid fluid. Therefore, compared with the Carnot cycle, there always

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exists an area that can’t be utilized for the ORC in the T-s diagram. In Fig.1, the

165

unexploited area is represented by . For the other areas, they can be

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utilized by selecting appropriate working fluids and adjusting the operating conditions.

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For instance, dry working fluids without superheating can reduce the area S4-2’-2-5-4.

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When working fluids are condensed into the saturated liquid in the condenser, some

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area of can be utilized.

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For the cycle analysis in the T-s diagram, the involved heat and work can be

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denoted by the enclosed areas. For example, the released heat of working fluid in the

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ORC can be represented by . Similarly denotes

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the net output work of ORC. Therefore, the thermal efficiency of ORC can be

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expressed by

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η=

S1−3− 4−5− 2−6−7−8−1 S1−3− 4−5− 2−6−7−8−1 + Sa −1−8−7−6−b −a

(1)

176

According to the surrounded areas in Fig.1, the following inequalities can be

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obtained.

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S1− 3− 4 − 5 − 2 − 6 − 7 −8 −1 < S1− 2' − 2 − 2 s −1

(2)

ACCEPTED MANUSCRIPT 179

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S a −1−8−7 −6 −b − a > S a −1− 2 s −b − a

(3)

Thus, the cycle efficiency satisfies the inequality

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η<

S1− 2' − 2− 2 s −1

(4)

S1− 2' − 2− 2 s −1 + S a −1− 2 s −b − a

Since the area can’t be developed by the ORC, the limiting efficiency of

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ORC is defined from the above inequality as

184

S1−2' −2−2 s −1

Considering the equality

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S1− 2' − 2 − 2 s −1 + S1−1s − 2' −1 = S1−1s − 2 − 2 s −1

we can get

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ηlimit <

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(5)

S1−2' −2− 2 s −1 + Sa −1−2 s −b−a

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ηlimit =

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S1−1s − 2 − 2 s −1 = η carnot S1−1s − 2 − 2 s −1 + S a −1− 2 s −b − a

(6)

(7)

The above inequality indicates that the limiting efficiency of ORC is lower than the

190

Carnot efficiency. According to the presented areas in Fig.1, the Carnot efficiency can

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be simply obtained by:

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T1 T2

(8)

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η carnot = 1 −

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As for the limiting efficiency, Eq. (5) can be transformed into the following equation

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based on the enclosed areas in Fig.1.

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ηlimit = 1 −

Sa −1−2 s −b−a Sa−1s −2−b−a − S1−1s −2' −1

(9)

196

For the calculation of limiting efficiency, the required areas ,

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can be easily determined by the product of the temperature and the

198

entropy. Thus, the key parameter for the limiting efficiency is the area . As

ACCEPTED MANUSCRIPT 199

shown in Fig.1, an oblique straight line is employed to cut out the unexploited area

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from the enclosed area of Carnot cycle. According to the Fig.1, the area of

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triangle can be expressed by

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1 (T2 − T1 ) 2 2β

(10)

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S1−1s − 2' −1 =

Where denotes the slope of the oblique line. Based on Eq. (9) and Eq. (10), the

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limiting efficiency of ORC can be expressed as follows: T1 1 T2 − (T2 − T1 ) 2 2 β ∆ sa − b

(11)

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Where ∆ is the entropy difference in the isobaric process of ORC. Besides the

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temperatures T1, T2, evaporation and condensation pressures are required to calculate

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∆ for a given working fluid. As for the slope , it can be considered as the

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isobaric slope in the liquid phase or the temperature-entropy saturation boundary

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slope of liquid. Due to the fact that the limiting efficiency is the upper bond of cycle

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efficiency, should take a larger value from the isobaric slope and the saturated

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slope.

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2.2 Limiting factor derived from the slope

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For the slope required in Eq. (11), the isobaric slope and the saturated slope of

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liquid are considered. According to the thermodynamic relations, the isobaric slope

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can be derived as

217

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T ∂T = ∂s p c p

(12)

As for the slope of temperature-entropy saturation boundary, it has been theoretically

ACCEPTED MANUSCRIPT 219

derived from the Helmholtz energy for pure and mixed working fluids [21]. The

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corresponding equation is expressed by

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dT aTv dP = − aTv + ds a2 v dT

− a2T eq

−1

(13)

Where a is the Helmholtz energy; = ⁄ represents the partial

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derivatives on the basis of the values m and n. According to the highly accurate

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Helmholtz energy equation of working fluid, the slope of saturated liquid can be

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obtained.

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From Eq. (12) and Eq. (13), it can be seen that the isobaric slope and the

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saturated liquid slope are functions of temperature. A comparison between the isobaric

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slope and the saturated slope of liquid working fluid is presented in Fig.2 for R600

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and R134a. In the figure, the reduced temperature is defined as the ratio of

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temperature to the critical temperature, namely T/Tc. Fig.2 indicates that the positive

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slope of the saturated liquid increases slowly with the increase of reduced temperature

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firstly and then decreases to zero sharply in the vicinity of the critical point. For the

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isobaric slope of liquid fluid, it increases slowly with the increase of pressure in the

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subcritical region. At the low reduced temperature, the isobaric slope almost coincides

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with the saturated slope. However, when the reduced temperature gradually

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approaches to 1, the isobaric slope decreases and is lower than the saturated slope

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obviously. Thus, for the calculation of limiting efficiency of subcritical ORC, the

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saturated liquid slope is employed.

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Fig.2 Comparison between the isobaric slope and the saturated liquid slope for R600

ACCEPTED MANUSCRIPT and R134a

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For the saturated slope of pure working fluids, the authors have obtained the

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liquid slope in the reference [21], based on the Helmholtz energy equations. With the

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increase of reduced temperature, the liquid slope of pure working fluids has a similar

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variation trend, as presented in Fig.3. As for the mixture, the related study indicates

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that the liquid slope of mixture is usually located between the corresponding pure

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fluids' [21]. Since the limiting efficiency is the performance ceiling of subcritical

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ORC, the maximum liquid slope should be employed. However, Fig.3 illustrates that

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the slope of saturated liquid reaches the maximum at different reduced temperatures.

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This phenomenon will increase the computational complexity, when the maximum

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slope is applied for the limiting efficiency. For the sake of simplicity, saturated slope

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at a fixed reduced temperature is adopted for all working fluids in this work. From

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Fig.3, it can be observed that most of working fluids have the maximum slope around

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the reduced temperature 0.9. Therefore, the required slope for the limiting efficiency

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is defined as follows:

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dT ds L ,Tr =0.9

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(14)

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β=

The defined slope represents the temperature-entropy characteristic of

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saturated liquid. It’s a dimensional parameter. In order to get a dimensionless number

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that could represent the properties of working fluids, we define

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γ=

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This definition is similar to the definition of molecular complexity. Molecular

MTc ds R dT L ,Tr =0.9

(15)

ACCEPTED MANUSCRIPT 262

complexity is defined as [22]

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σ=

MTc ds R dT V ,Tr =0.7

(16)

Eq. (16) indicates that molecular complexity is directly related with the saturated

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slope of vapor working fluids at the reduced temperature 0.7. Thus, σ can be

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employed to distinguish whether a working fluid is dry, wet or isentropic [8].

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Furthermore, after the differential transformation of Eq. (16), σ can be directly related

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with the molecular structure of working fluid. However, for the defined parameter γ, it

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is related with the slope of saturated liquid at the reduced temperature 0.9. According

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to Eq. (14) and Eq. (15), the limiting efficiency of Eq. (11) can be transformed into

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the following expression.

T2 −

γR

T1

2 MTc ∆sa − b

(T2 − T1 ) 2

(17)

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ηlimit = 1 −

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Hence, the parameter defined in Eq. (15) is called the limiting factor in this work. Due

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to the fact that the critical temperature is involved in the definition of limiting factor,

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pseudocritical temperature is employed to calculate the limiting factor of mixture.

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Fig.3 Saturated slope of liquid for various working fluids [21]

Based on the calculated limiting factor, the limiting efficiency can be

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determined for a given working fluid. Compared with the Carnot efficiency, the

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limiting efficiency of ORC is derived by considering the properties of working fluid.

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Thus, the limiting efficiency is more reasonable to represent the performance limit of

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ORC. Although the limiting efficiency is derived from the T-s diagram of simple ORC,

ACCEPTED MANUSCRIPT it is applicable to various configurations of subcritical ORC. The reason is that no

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matter what the cycle configuration is, the area in Fig.1 can’t be utilized

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by ORC. Furthermore, compared with the limiting efficiency, the practical efficiency

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is obtained by considering the irreversible loss. The loss is attributed to the

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temperature difference of heat transfer and the flow resistance of working fluids. Thus,

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the practical efficiency of ORC can be improved by taking measures to reduce the

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cycle loss. However, the reduction of cycle loss is always accompanied by the

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increase of cost. For instance, for a given heat amount, the decrease of temperature

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difference will result into the increase of heat exchangers’ area.

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2.3 Thermodynamic perfectness based on limiting efficiency

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Thermodynamic perfectness represents the degree to which the practical cycle

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approaches to the ideal cycle. For a long time, thermodynamic perfectness is defined

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on the basis of Carnot efficiency. The equation can be given by

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η practical ηcarnot

(18)

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ε carnot =

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Where !"# $%&$ ' denotes the efficiency of practical cycle. represents the

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thermodynamic perfectness. However, since the Carnot efficiency is only determined

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by the temperatures of heat source and sink, the corresponding perfectness can’t

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effectively reflect the improvement potential of practical cycle. Thus, in this work, a

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new expression of thermodynamic perfectness is proposed from the limiting

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efficiency. According to the definition of $ #(% , the thermodynamic perfectness for

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the limiting efficiency can be defined as follows:

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ACCEPTED MANUSCRIPT 304

ε limit =

η practical ηlimit

(19)

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3. Results and discussion According to the proposed limiting efficiency and the defined cycle perfectness,

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the effects of working fluids and operating conditions on these two parameters are

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analyzed. In order to compare the cycle efficiency and perfectness among different

309

fluid families, 13 commonly used pure fluids are considered in this work. These

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working fluids consist of 7 kinds of fluid families, namely halohydrocarbons,

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hydrocarbons, olefins, ethers, siloxanes, benzenes and water. Table 1 demonstrates the

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employed working fluids along with their structure, molecular weight, critical

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temperature, limiting factor and fluid behavior. Due to the fact that the proposed

314

limiting efficiency is also suitable for Rankine cycle, water is considered in this study.

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For the limiting factor of pure fluid, the required slope of saturated liquid is

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determined by the Helmholtz energy equation via Eq. (13). Furthermore, according to

317

the definition of molecular complexity in Eq. (16), fluid behavior is predicted. That is,

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σ>0: a dry fluid, σ~0: an isentropic fluid, and σ<0: a wet fluid. Table 1 indicates that

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the employed working fluids consist of dry, isentropic and wet fluids.

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Table 1 Pure working fluids considered in this work

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In order to reveal the effect of composition on the limiting efficiency and the

323

thermodynamic perfectness, three mixtures, namely R134a/R125 (Wet/Wet),

324

R601/R600 (Dry/Dry) and R600/R134a (Dry/Wet), are employed in this study. The

325

reason for selecting these mixtures is that the mixing rules and coefficients, which

ACCEPTED MANUSCRIPT are required to establish the Helmholtz energy equations of mixtures for the

327

calculation of limiting factor, can be obtained from the references for R134a/R125

328

[23], R601/R600 [24] and R600/R134a [24], respectively. Furthermore, the three

329

mixtures are mixed by the same or different fluid families, so that the influence of

330

mixed fluid families on the limiting efficiency can be investigated. Fig.4 presents the

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limiting factor of the mixtures for different mass fractions. It can be seen that the

332

limiting factor of mixture is between the corresponding pure fluids’ and changes

333

monotonically with the increase of mass fraction.

Fig.4 Limiting factor of the considered mixtures for different mass fractions

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3.1 Limiting efficiency

Based on the equation of limiting efficiency, the entropy difference and

338

temperatures are required to calculate the limiting efficiency for a given working fluid.

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Therefore, the following assumptions are made to analyze the limiting efficiency of

340

subcritical ORC, taking as a case study.

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The highest and lowest temperatures of ORC are equal to the temperatures of

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heat source and sink respectively. In this work, the lowest temperature is set to be 298.15 K, while the highest temperature ranges from 310.15 K to Tc.

Pressure drops in the heat exchangers and pipes are ignored.

Superheating temperature of working fluid is set to be 5 K, and the lowest cycle temperature is the temperature of saturated liquid. Since the vapor pressure of pure working fluid is determined by the

ACCEPTED MANUSCRIPT corresponding saturation temperature in the subcritical region, the required entropy

349

difference for the calculation of limiting efficiency can be obtained on the basis of the

350

above assumptions. Fig.5 (a) shows the variation of limiting efficiency with the

351

temperature of heat source for different kinds of pure working fluids. Meanwhile, in

352

order to facilitate the comparison of efficiency, Carnot efficiency is also plotted in the

353

figure. It can be observed that the limiting efficiency and the Carnot efficiency are

354

both proportional to the temperature of heat source under the fixed cold temperature.

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Furthermore, under the same temperatures, the limiting efficiency of any working

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fluid is always lower than the Carnot efficiency. When the temperature difference

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between the heat source and sink is relatively small, the limiting efficiencies of

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different working fluids are almost equal to each other and close to the Carnot

359

efficiency. However, with the increase of temperature difference, the working fluid

360

with a higher Tc has a larger limiting efficiency. This phenomenon can be explained

361

qualitatively by Eq. (17). According to the expression of limiting efficiency, it can be

362

inferred that when the critical temperature becomes infinity, the limiting efficiency is

363

equal to the Carnot efficiency. For the considered working fluids, water, which is

364

employed in Rankine cycle, has the highest limiting efficiency under the same

365

temperature of heat source. Fig.5 (b) presents the curves of limiting efficiency for

366

halohydrocarbons. It can be seen that the distribution of these curves is similar to that

367

illustrated in Fig.5 (a). The rule discussed for different kinds of working fluids is also

368

applicable to the same fluid family. However, compared with the difference of

369

limiting efficiency in Fig.5 (a), the efficiency difference of the same fluid family in

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ACCEPTED MANUSCRIPT 370

Fig.5 (b) is fairly small under the same temperature of heat source.

371

Fig.5 Limiting efficiency for various working fluids: (a) different kinds of working

373

fluids; (b) halohydrocarbons

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In the above analysis of limiting efficiency, superheating temperature of

375

working fluid is set to be constant. In order to investigate the effect of superheating

376

temperature on the limiting efficiency of subcritical ORC, Fig.6 presents the limiting

377

efficiency of R245fa for superheating temperature 0 K, 10 K and 20 K respectively.

378

The reason for selecting this fluid is that R245fa is the most frequently used fluid in

379

ORC experiments [25]. Fig.6 indicates that superheating temperature has no influence

380

on the limiting efficiency at low heat source temperatures. Nevertheless, with the

381

increase of heat source temperature, a higher limiting efficiency can be obtained by a

382

higher superheating temperature. The reason is that when the temperatures of heat

383

source and sink are fixed, a higher superheating temperature means a lower

384

evaporation temperature, thus resulting into a lower evaporation pressure. The

385

decrease of evaporation pressure will lead to the increase of entropy difference in the

386

isobaric process. According to the equation of limiting efficiency, for a given working

387

fluid, the increase of entropy difference contributes to a higher efficiency under the

388

same temperatures of heat source and sink.

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Fig.6 Limiting efficiency of R245fa for different superheating temperatures Unlike the pure working fluids, zeotropic mixtures don’t keep constant

ACCEPTED MANUSCRIPT temperatures in the phase change process. Thus, in the analysis of limiting efficiency

393

for the considered mixtures, the evaporation pressure is determined by the

394

temperature of saturated vapor and the condensation pressure corresponds to the

395

temperature of saturated liquid. Based on the above assumptions of subcritical ORC,

396

the limiting efficiency can be obtained for mixtures at different compositions. The

397

limiting efficiency curves are depicted in Fig.7. It can be seen that the limiting

398

efficiency of any mixture is always lower than the Carnot efficiency, and the change

399

of limiting efficiency with the increase of heat source temperature is similar to that of

400

pure working fluid. For the hydrofluorocarbon mixture R134a/R125, the efficiency

401

curves locate between the curves of R134a and R125. With the increase of R134a

402

mass fraction, the efficiency of corresponding mixture increases from the efficiency

403

of R125 to that of R134a gradually, as shown in Fig.7 (a). The same phenomenon is

404

also observed for hydrocarbon mixture R601/R600 in Fig.7 (b). The reason for this

405

phenomenon is that the limiting factor is proportional to the fraction of mixture for the

406

same constituents, as shown in Fig.4. However, for the mixture R600/R134a, which

407

consists of hydrocarbon and hydrofluorocarbon fluids, there are efficiency curves

408

lower than that of R134a at the R600 mass fraction 10% and 20%, as presented in

409

Fig.7 (c). It can be explained by the fact that R600/R134a has an azeotropic behavior

410

at low compositions of R600. Under the same evaporation temperature, vapor

411

pressure of azeotropic mixture is higher than those of pure working fluids, thus

412

resulting into a lower limiting efficiency of azeotropic mixture.

413

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.

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Fig. 7 Limiting efficiency of the mixtures for different mass fractions:

415

(a) R134a/R125, (b) R601/R600, and (c) R600/R134a

416

3.2 Thermodynamic perfectness For the calculation of thermodynamic perfectness, limiting efficiency and

418

practical efficiency of ORC are both required. Therefore, a case study is conducted to

419

analyze the cycle perfectness in this work. The inlet temperature of heat source for a

420

simple ORC is set to be 453.15 K, as listed in Table 2. Other cycle parameters are also

421

given in Table 2. Furthermore, considering the significant effect of evaporation

422

temperature on the cycle performance for a given heat source, the effect of

423

evaporation temperature on the thermodynamic perfectness is investigated. In this

424

case, the lowest evaporation temperature is assumed to be 30 K higher than the

425

condensation temperature. The assumed minimum temperature difference is possible

426

to drive practical ORC. As for the highest evaporation temperature, it is kept 10 K

427

below the critical point, as also suggested by Delgado-Torres [26]. Meanwhile, it’s

428

assumed that the temperature difference between the inlet temperatures of turbine and

429

the inlet temperature of heat source is larger than 5 K to assure the heat exchange

430

between the working fluid and the heat source. For simplicity, pressure drops in the

431

heat exchangers and pipes are ignored.

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Table 2 Specifications of the subcritical ORC

434

Based on the cycle parameters of subcritical ORC, related efficiency and

435

thermodynamic perfectness can be obtained for various working fluids. Since

ACCEPTED MANUSCRIPT different kinds of working fluids have similar curves of limiting efficiency, only the

437

results of halohydrocarbons are presented in the analysis of the thermodynamic

438

perfectness. The variations of cycle efficiency and perfectness with evaporation

439

temperature are presented in Fig.8. For the given heat source and sink, the Carnot

440

efficiency is a constant, namely 34.21%. However, the limiting efficiency is a

441

function of working fluids and operating conditions, as shown in Fig.8 (a). It can be

442

observed that the fluid with a higher Tc has a higher limiting efficiency.

443

Simultaneously, as the evaporation temperature increases, the limiting efficiency

444

decreases slightly. This is because that the increase of evaporation temperature means

445

the increase of evaporation pressure, thus resulting into a smaller entropy difference

446

between the heat source and sink. According to Eq. (17), the decrease of entropy

447

difference will lead to a decrease of limiting efficiency under the same heat source

448

and sink. Unlike the limiting efficiency, the practical efficiency of ORC increases

449

largely with the increase of evaporation temperature. For the employed working fluids,

450

the highest efficiency 18.83% is obtained by R123 at the evaporation temperature

451

443.15 K. Furthermore, Fig.8 (a) also indicates the efficiency order: !"# $%&$ ' <

452

!'&&% < !$ #(% .

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Fig.8 (b) gives the variation curves of thermodynamic perfectness with the

454

evaporation temperature for different working fluids. It can be seen that both the two

455

types of thermodynamic perfectness increase, as the evaporation temperature

456

increases. For the thermodynamic perfectness $ #(% , a higher $ #(% can be

457

obtained by the fluid with a higher Tc. For the considered working fluids, R123 has

ACCEPTED MANUSCRIPT the highest $ #(% 55.05% at the evaporation temperature 443.15 K. As for the

459

thermodynamic perfectness '&&% , '&&% is always higher than $ #(% for a given

460

working fluid. However, compared with the large difference of $ #(% among

461

working fluids, the difference of '&&% is small, especially at the high evaporation

462

temperature. For the three working fluids, the highest '&&% 75.98% is obtained by

463

R123 at the evaporation temperature 443.15 K.

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Fig.8 Efficiency and thermodynamic perfectness for halohydrocarbons:

466

(a) efficiency; (b) thermodynamic perfectness

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Mixtures are employed to reveal the effect of composition on the thermodynamic

468

perfectness of a practical ORC. Due to the existence of temperature variation in the

469

phase change of mixture, it’s assumed that the pressure of mixture is determined by

470

the bubble temperature. Furthermore, because of the similar laws of mixtures for the

471

thermodynamic perfectness, only the results of R600/R134a are presented here. As

472

shown in Fig.9 (a), for a given fraction, the curves of limiting efficiency and practical

473

efficiency are similar to those of pure working fluid. For different fractions, the

474

corresponding curves of limiting efficiency generally locate between the curves of

475

R134a and R600, as discussed for Fig.7. Due to the azeotropic behavior at low

476

compositions of R600, the curve of limiting efficiency for composition 25% is very

477

close to that of R134a, as shown in Fig.9 (a). As for the practical efficiency of ORC,

478

the figure indicates that the efficiency of mixture R600/R134a is generally lower than

479

that of pure working fluid. This phenomenon has been confirmed in the published

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ACCEPTED MANUSCRIPT literatures [27, 28]. The reason is that the condenser bubble temperature is fixed in the

481

calculation of practical efficiency. Thermodynamic perfectness of R600/R134a is

482

provided in Fig.9 (b). It can be observed that the thermodynamic perfectness increases

483

with the increase of evaporation temperature, and '&&% is always higher than

484

$ #(% for a given composition. Due to the fact that the mixture has a lower practical

485

efficiency, '&&% and $ #(% of mixture are less than those of pure fluids.

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Fig.9 Efficiency and thermodynamic perfectness of the mixture R600/R134a for

488

different mass fractions: (a) efficiency; (b) thermodynamic perfectness

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489

4. Conclusions

In this work, a limiting efficiency for subcritical Organic Rankine cycle (ORC)

491

is derived under the constraint of working fluids. Furthermore, a new expression of

492

thermodynamic perfectness is proposed on the basis of limiting efficiency. Under

493

certain conditions of subcritical ORC, different pure fluids and mixtures are employed

494

to investigate the effect of working fluids on the limiting efficiency and the

495

thermodynamic perfectness. After analyzing these parameters and comparing with the

496

Carnot-related performances, the following conclusions can be drawn.

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(1) For any working fluid, the limiting efficiency is always lower than the

498

Carnot efficiency but higher than the practical efficiency under the same heat source

499

and sink. For pure working fluids, a higher limiting efficiency can be obtained by the

500

fluid with a higher critical temperature. For binary mixtures, the limiting efficiency is

501

generally between the corresponding pure fluids’ under the same temperatures.

ACCEPTED MANUSCRIPT 502

However, when the mixture has an azeotropic behavior, the limiting efficiency may be

503

beyond those of pure fluids. (2) For a practical ORC, the limiting efficiency will decrease slightly with the

505

increase of evaporation temperature. However, for any working fluid, as the

506

evaporation temperature increases, the thermodynamic perfectness will increase.

507

Furthermore, the cycle perfectness defined from the limiting efficiency is always

508

higher than that defined from the Carnot efficiency. The pure fluid with a higher

509

critical temperature generally has higher cycle perfectness, while thermodynamic

510

perfectness varies greatly with the composition for mixture.

511

Acknowledgements

512

This work is sponsored by the National Nature Science Foundation of China

513

(51476110). In addition, the financial support from the China Scholarship Council to

514

the first author is gratefully acknowledged.

515

References

516

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250-kW organic Rankine cycle system. Appl Therm Eng 2015; 80: 339-346.

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[2] Feng Y Q, Hung T C, He Y L, et al. Operation characteristic and performance

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R245fa, R123 and their mixtures. Energ Convers Manage 2017; 144: 153-163.

[3] Tjiang P C, Sutanto S H. The efficiency of the Carnot cycle with arbitrary gas equations of state. Eur J Phys 2006; 27: 719-726. [4] Su W, Zhao L, Deng S. Research on the performance of thermodynamic cycles

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based on the properties of working fluids. Chin Sci Bull, 2017. [5] Hærvig J, Sørensen K, Condra, T J. Guidelines for optimal selection of working

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[6] Su W, Zhao L, Deng S. Simultaneous working fluids design and cycle

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[7] Su W, Zhao L, Deng S. Group contribution methods in thermodynamic cycles:

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[9] Saleh B, Koglbauer G, Wendland M, et al. Working fluids for low-temperature organic Rankine cycles. Energy 2007; 32: 1210-1221. [10] Yang L, Gong M, Guo H, et al. Effects of critical and boiling temperatures on

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system performance and fluid selection indicator for low temperature organic

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mixtures as a working fluid: Advantages and issues. Renew Sust Energ Rev

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ment with various applications part I: Energy and exergy performance

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evaluation. Renew Sust Energ Rev 2016; 53: 477-499.

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[15] Bao J, Zhao L, Zhang W. A novel auto-cascade low-temperature solar Rankine cycle system for power generation. Sol Energy 2011; 85: 2710-2719.

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[16] Zhang C, Liu C, Wang S, et al. Thermo-economic comparison of subcritical

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[20] Yang X, Zheng N, Zhao L, et al. Analysis of a novel combined power and

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ejector-refrigeration cycle. Energ Convers Manage 2016; 108: 266-274.

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ACCEPTED MANUSCRIPT Table Captions: Table 1- Pure working fluids considered in this work

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Table 2- Specifications of the subcritical ORC

ACCEPTED MANUSCRIPT Table 1- Pure working fluids considered in this work

CF3CHCl2 CHF2CF3 CF3CH2F CH3CHF2 CH2FCF2CHF2 CF3CH2CHF2 CH3CH2CH2CH3 CH3CH2CH2CH2CH3 CH2=CFCF3 CHF2OCH2CF3 (CH3)3SiOSi (CH3)3 CH3C6H5 H 2O

Tc (K)

γ

Behavior

152.93 120.02 102.03 66.05 134.05 134.05 58.12 72.15 114.04 150.05 162.38 92.14 18.02

456.83 339.17 374.21 386.41 447.57 427.16 425.13 469.70 367.85 444.88 518.75 591.75 647.10

25.48 22.61 22.10 18.81 29.81 28.94 24.26 30.45 23.69 32.77 52.72 33.18 13.57

Dry Wet Wet Wet Dry Dry Dry Dry Isentropic Isentropic Dry Dry Wet

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R123 R125 R134a R152a R245ca R245fa R600 R601 R1234yf R245mf MM Toluene Water

M (g/mol)

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Molecular structure

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ASHRAE number

ACCEPTED MANUSCRIPT Table 2- Specifications of the subcritical ORC Value

Inlet temperature of heat source (K) Heat sink water inlet temperature (K) Condensation temperature(K) Superheating temperature(K)

453.15 298.15 303.15 5

Subcooling temperature(K) Isentropic pump efficiency [6] Isentropic turbine efficiency [6]

0 65% 85%

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Steady system parameter

ACCEPTED MANUSCRIPT Figure Captions: Fig.1-T-s diagram of subcritical ORC and Carnot cycle for (a) R600, (b) R600/ R134a (50%/50%, mass)

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Fig.2- Comparison between the isobaric slope and the saturated liquid slope for R600 and R134a

Fig.3-Saturated slope of liquid for various working fluids [21]

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Fig.4-Limiting factor of the considered mixtures for different mass fractions

Fig.5-Limiting efficiency for various working fluids: (a) different kinds of

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working fluids; (b) halohydrocarbons

Fig.6-Limiting efficiency of R245fa for different superheating temperatures Fig.7-Limiting efficiency of the mixtures for different mass fractions: (a) R134a/

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R125, (b) R601/R600, and (c) R600/R134a

Fig.8- Efficiency and thermodynamic perfectness for halohydrocarbons: (a) efficiency; (b) thermodynamic perfectness

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Fig.9-Efficiency and thermodynamic perfectness of the mixture R600/R134a for

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different mass fractions: (a) efficiency; (b) thermodynamic perfectness

ACCEPTED MANUSCRIPT Fig.1

450

(a)

2'

1s

2

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400

5

4 Subcritical ORC 350

6 7

8

3

2s

1

300

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Tempreature [K]

Tangent line

Carnot cycle

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1.6

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250

2.0

2.4

2.8

3.2

Specific entropy [kJ/kg•K]

450

(b)

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350

2 5

4 Subcritical ORC

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Tempreature [K]

Tangent line

7

8

3

300

6

2s

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Carnot cycle b

a

250

1.2

1.5

1.8

2.1

2.4

2.7

Specific entropy [kJ/kg•K]

Fig.1 T-s diagram of subcritical ORC and Carnot cycle for (a) R600, (b) R600/R134a (50%/50%, mass)

ACCEPTED MANUSCRIPT Fig.2

2.5

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Slope [kg•K2/kJ]

2.0

1.5

R600 R134a

1.0

0.10Pc 0.25Pc

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0.50Pc

0.5

0.75Pc

Saturated 0.7

0.8

0.9

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1.0

Reudced temperature

Fig.2 Comparison between the isobaric slope and the saturated liquid slope for R600

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ACCEPTED MANUSCRIPT Fig.3

3.5

R123 R236fa R125 R601a R600

R124 R245fa R134a R601 R290

R227ea R142b R152a R600a

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3.0 2.5 2.0 1.5 1.0

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Saturated liquid slope [kg•K2/kJ]

4.0

0.0 0.6

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1.0

Reduced temperature

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Fig.3 Saturated slope of liqud for various working fluids [21]

ACCEPTED MANUSCRIPT Fig.4

30

R134a/R125 R601/R600 R600/R134a

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24

22

0.2

0.4

0.6

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Limiting factor γ

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1.0

Mass fraction of the first component

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Fig.4 Limiting factor of the considered mixtures for different mass fractions

ACCEPTED MANUSCRIPT Fig.5

60

Carnot

(a)

H2O

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Toluene MM

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R601 R245mf R245fa

20

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Efficiency [%]

50

R1234yf

0 300

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600

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Temperature of heat source [K]

40

20

Carnot

R123 R245ca R245fa

R152a R134a

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Efficiency [%]

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Temperature of heat source [K]

Fig.5 Limiting efficiency for various working fluids: (a) different kinds of working fluids; (b) halohydrocarbons

ACCEPTED MANUSCRIPT Fig.6

∆Tsup=0K

30

∆Tsup=10K

25

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Efficiency [%]

Carnot

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Temperature of heat source [K]

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Fig.6 Limiting efficiency of R245fa for different superheating temperatures

ACCEPTED MANUSCRIPT Fig.7

R125 10% 20% 30% 40% 50% 60% 70% 80% 90% R134a

20

Efficiency [%]

15

10

Carnot

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320

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Temperature of heat source [K] 40

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Temperature of heat source [K]

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(c)

Efficiency [%]

25

20

15

Carnot

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Fig.7 Limiting efficiency of the mixtures for different mass fractions: (a) R134a/

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ACCEPTED MANUSCRIPT Fig.8 40

(a)

ηcarnot

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ηpractical ηlimit R134a R245fa R123

10

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400

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Efficiency [%]

30

Evaporation temperature T4 [K]

(b) 70

50 40 30

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Thermodynamic perfectness [%]

80

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εlimit

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R134a R245fa R123

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Evaporation temperature T4 [K]

Fig.8 Efficiency and thermodynamic perfectness for halohydrocarbons: (a) efficiency; (b) thermodynamic perfectness

ACCEPTED MANUSCRIPT Fig.9

40

(a)

ηcarnot

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ηpractical ηlimit R134a 25% 50% 75% R600

0 360

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Efficiency [%]

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Evaporation temperature T4 [K]

(b)

50 40

εlimit εcarnot

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R134a 25% 50% 75% R600

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70

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Evaporation temperature T4 [K]

Fig.9 Efficiency and thermodynamic perfectness of the mixture R600/R134a for different mass fractions: (a) efficiency; (b) thermodynamic perfectness

ACCEPTED MANUSCRIPT Highlights: A limiting efficiency is derived for subcritical Organic Rankine cycle.

A new expression of cycle perfectness is proposed from the limiting efficiency.

The limiting efficiency is lower than the Carnot efficiency for any working fluid.

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