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A limiting efficiency of subcritical Organic Rankine cycle under the constraint of working fluids
Accepted Manuscript A limiting efficiency of subcritical Organic Rankine cycle under the constraint of working fluids Wen Su, Li Zhao, Shuai Deng, Weicong Xu, Zhixin Yu PII:
S0360-5442(17)31852-2
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
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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
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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
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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
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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
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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
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Symbols
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a
Molar Helmholtz energy kJ/kg
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c
Heat capacity kJ/(kg•K)
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M
Molar mass
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ORC
Organic Rankine cycle
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P
Pressure
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R
Ideal gas constant
kJ/(kmol•K)
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s
Specific entropy
kJ/(kg•K)
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S
Enclosed area in the temperature-entropy diagram
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T
Temperature
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v
Molar volume
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Greeks
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η
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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
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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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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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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
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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
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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
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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
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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
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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
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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)
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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)
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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
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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
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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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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)
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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
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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
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T ∂T = ∂s p c p
(12)
As for the slope of temperature-entropy saturation boundary, it has been theoretically
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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
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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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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)
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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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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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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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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
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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),
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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
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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
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limiting factor of mixture is between the corresponding pure fluids’ and changes
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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
356
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
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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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Fig.5 (b) is fairly small under the same temperature of heat source.
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Fig.5 Limiting efficiency for various working fluids: (a) different kinds of working
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fluids; (b) halohydrocarbons
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In the above analysis of limiting efficiency, superheating temperature of
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working fluid is set to be constant. In order to investigate the effect of superheating
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temperature on the limiting efficiency of subcritical ORC, Fig.6 presents the limiting
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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
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ORC experiments [25]. Fig.6 indicates that superheating temperature has no influence
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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
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higher superheating temperature. The reason is that when the temperatures of heat
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source and sink are fixed, a higher superheating temperature means a lower
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evaporation temperature, thus resulting into a lower evaporation pressure. The
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decrease of evaporation pressure will lead to the increase of entropy difference in the
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isobaric process. According to the equation of limiting efficiency, for a given working
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fluid, the increase of entropy difference contributes to a higher efficiency under the
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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
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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.
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.
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Fig. 7 Limiting efficiency of the mixtures for different mass fractions:
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(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
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Based on the cycle parameters of subcritical ORC, related efficiency and
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thermodynamic perfectness can be obtained for various working fluids. Since
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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: !"# $%&$ ' <
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!'&&% < !$ #(% .
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Fig.8 (b) gives the variation curves of thermodynamic perfectness with the
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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
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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:
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(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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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
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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
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the first author is gratefully acknowledged.
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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
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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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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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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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ACCEPTED MANUSCRIPT Fig.1
450
(a)
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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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2.5
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Slope [kg•K2/kJ]
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Fig.2 Comparison between the isobaric slope and the saturated liquid slope for R600
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3.5
R123 R236fa R125 R601a R600
R124 R245fa R134a R601 R290
R227ea R142b R152a R600a
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Reduced temperature
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R134a/R125 R601/R600 R600/R134a
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Limiting factor γ
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Mass fraction of the first component
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Fig.4 Limiting factor of the considered mixtures for different mass fractions
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60
Carnot
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R1234yf
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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
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∆Tsup=0K
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Fig.7 Limiting efficiency of the mixtures for different mass fractions: (a) R134a/
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(a)
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Evaporation temperature T4 [K]
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 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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S0360-5442(17)31852-2
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
10
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
12
contribution, a limiting efficiency is proposed for subcritical Organic Rankine cycle
13
(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
15
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
17
pure fluids and 3 mixtures are employed to demonstrate the effects of working fluids
18
on the limiting efficiency and thermodynamic perfectness. For pure working fluids,
19
the fluid with a higher critical temperature possesses higher limiting efficiency and
20
cycle perfectness. For mixtures, the limiting efficiency generally locates between
21
those of pure fluids, while the thermodynamic perfectness varies greatly with the
22
composition. Although the proposed limiting efficiency can’t be achieved by practical
23
cycles, it can provide guidance for the selection of working fluids and the construction
24
of ORC.
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ACCEPTED MANUSCRIPT Key words: Organic Rankine cycle; Working fluids; Limiting efficiency; Thermody
26
namic perfectness
27
Nomenclature
28
Symbols
29
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
40
η
41
42
γ
43
σ
Molecular complexity
44
Thermodynamic perfectness
45
∆
Difference
46
Subscripts
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g/mol
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Pa
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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)
48
1s, 2s, 2’
Thermodynamic state points (Fig.1)
49
a, b
Points on the axis (Fig.1)
50
c
Critical temperature
51
carnot
Performance of Carnot cycle
52
eq
Equilibrium
53
limit
Performance limit
54
L
Liquid
55
p
Pressure
56
practical
Performance of practical cycle
57
r
Reduced
58
sup
Superheating temperature
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V
Vapor
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1. Introduction
With the increase of global energy demand, Organic Rankine cycle (ORC) has
63
been widely employed to utilize the low and medium grade energy, such as solar
64
energy, wind energy, geothermal energy and waste heat for the production of
65
electricity. In order to convert these energy sources with high efficiency, organic
66
substances are employed as working fluids. Even so, in practical engineering, thermal
67
efficiency of subcritical ORC is relatively low, around 10% [1, 2]. Compared with the
68
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
70
cycle, there always exist different kinds of irreversible losses in the practical cycle,
71
due to the heat transfer temperature difference and the flow resistance of working
72
fluids. Furthermore, for subcritical ORC, heat exchange between working fluids and
73
heat source/sink is an isobaric process and involves the phase change of working
74
fluids. While the heat exchange of Carnot cycle is an isothermal process and there is
75
no phase change. It has been proven that the efficiency of Carnot cycle is only related
76
with the temperatures of heat source/sink via the equation of state of working fluids [3,
77
4]. However, for subcritical ORC, the cycle efficiency is a function of working fluids
78
and operating conditions, so that a large number of researchers have contributed to the
79
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
81
thermal efficiency of subcritical ORC. Numerical literatures have been published to
82
reveal the relationship between the properties of working fluids and the cycle
83
performance [7, 8]. For instance, Saleh et al. [9] compared cycle performance of 31
84
pure working fluids for different types of ORCs, and concluded that the high boiling
85
fluids with overhanging saturated vapor line show the highest cycle efficiencies. Yang
86
et al. [10] compared cycle performance of 267 working fluids to investigate the
87
effects of critical and boiling temperatures on the maximum net output power. The
88
optimal combinations of critical and boiling temperatures were proposed for the
89
selection of working fluids. In the optimization of cycle parameters for given heat
90
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
92
temperature of heat source. However, an expression derived by Liu et al. [12]
93
indicates that the cycle efficiency is a weak function of the critical temperature,
94
regardless of the fact that the thermal efficiency for working fluid with the lower
95
critical temperature is lower. Furthermore, in order to improve the cycle efficiency
96
and reduce the irreversible loss, zeotropic mixtures have been recommended as
97
candidates of working fluids. It’s thought that the temperature glide of mixtures can
98
alleviate the temperature mismatch in the heat transfer process. Many researches on
99
the cycle performance of zeotropic mixtures have been carried out experimentally and
100
theoretically [13]. However, it should be noted that the above literatures mainly focus
101
on the relationship between the practical efficiency of a specific cycle configuration
102
and the properties of working fluids. To the best of authors’ knowledge, few
103
researchers have investigated the efficiency limit of subcritical ORC by considering
104
the properties of working fluids.
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In addition to selecting the working fluids with appropriate properties, various
106
ORC configurations have been proposed and analyzed to enhance the cycle efficiency
107
for different applications [14]. For example, Bao et al. [15] proposed an auto-cascade
108
ORC for the utilization of solar energy. For the recovery of waste heat, Zhang et al.
109
[16] conducted thermo-economic comparison of subcritical ORC based on different
110
types of heat exchangers. Li [17] investigated the economic assessment of various
111
working fluids for different ORC configurations under the assumption that the net
112
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
114
internal heat exchanger, regenerative ORC, are compared by Zare [18]. He concluded
115
that ORC with internal heat exchanger has the best thermodynamic performance,
116
while simple ORC has the best economic performance. Furthermore, some
117
researchers introduced the phase separators into the ORC systems to control and
118
distribute the working fluid flow, thus achieving the high cycle efficiency with
119
proposed system configurations [19, 20]. However, to the best of authors’ knowledge,
120
the efficiency of these proposed system configurations is still far from the Carnot
121
efficiency, so that the guidance of Carnot efficiency is not strong enough for the
122
construction of ORC. A more practical efficiency limit should be proposed to provide
123
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
125
is proposed for the subcritical ORC. For the construction of ORC, the limiting
126
efficiency gives an upper bound of cycle efficiency. In the process of deriving
127
limiting efficiency, a limiting factor is defined from the thermodynamic properties of
128
working fluids. Furthermore, according to the proposed efficiency, a new expression
129
on thermodynamic perfectness is defined to represent the degree to which the
130
practical efficiency approaches to the limiting efficiency. This paper is structured as
131
follows: in section 2, the detailed methodology for deriving the limiting efficiency is
132
presented. In section 3, the effects of working fluids and cycle temperatures on the
133
efficiency limit and the thermodynamic perfectness of ORC are discussed.
134
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.
136
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
141
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
143
pressure and temperature vapor conducts expansion to generate work in the isentropic
144
process 2-6. Thereafter, the exhausted vapor is condensed into subcooled liquid by the
145
heat sink in the isobaric process 6-1. Finally, in the isentropic compression process
146
1-3, the liquid at the condensation pressure is pumped into the evaporator. Due to the
147
fact that the temperature variation of liquid is not sensitive to the change of pressure
148
for the subcritical ORC, the temperature rise of working fluid in the pump can be
149
neglected. This means that the temperature at point 3 is equal to the temperature at
150
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)
154
In order to compare the performance of ORC with that of Carnot cycle, Fig.1
155
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
159
expansion (1s-2’) of working fluid in the liquid state can’t be realized in the practical
160
engineering, due to the non-sensitive relationship among the pressure, the volume and
161
the temperature. As for the ORC, the heat exchange between the heat source and the
162
working fluid happens in the isobaric process, considering the thermodynamic
163
properties of liquid fluid. Therefore, compared with the Carnot cycle, there always
164
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
166
utilized by selecting appropriate working fluids and adjusting the operating conditions.
167
For instance, dry working fluids without superheating can reduce the area S4-2’-2-5-4.
168
When working fluids are condensed into the saturated liquid in the condenser, some
169
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
171
denoted by the enclosed areas. For example, the released heat of working fluid in the
172
ORC can be represented by . Similarly denotes
173
the net output work of ORC. Therefore, the thermal efficiency of ORC can be
174
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
177
obtained.
178
S1− 3− 4 − 5 − 2 − 6 − 7 −8 −1 < S1− 2' − 2 − 2 s −1
(2)
ACCEPTED MANUSCRIPT 179
180
S a −1−8−7 −6 −b − a > S a −1− 2 s −b − a
(3)
Thus, the cycle efficiency satisfies the inequality
181
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
183
ORC is defined from the above inequality as
184
S1−2' −2−2 s −1
Considering the equality
186
S1− 2' − 2 − 2 s −1 + S1−1s − 2' −1 = S1−1s − 2 − 2 s −1
we can get
188
ηlimit <
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187
(5)
S1−2' −2− 2 s −1 + Sa −1−2 s −b−a
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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
191
be simply obtained by:
192
T1 T2
(8)
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193
As for the limiting efficiency, Eq. (5) can be transformed into the following equation
194
based on the enclosed areas in Fig.1.
195
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 ,
197
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
200
from the enclosed area of Carnot cycle. According to the Fig.1, the area of
201
triangle can be expressed by
202
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
204 205
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
207
temperatures T1, T2, evaporation and condensation pressures are required to calculate
208
∆ for a given working fluid. As for the slope , it can be considered as the
209
isobaric slope in the liquid phase or the temperature-entropy saturation boundary
210
slope of liquid. Due to the fact that the limiting efficiency is the upper bond of cycle
211
efficiency, should take a larger value from the isobaric slope and the saturated
212
slope.
213
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
215
liquid are considered. According to the thermodynamic relations, the isobaric slope
216
can be derived as
217
218
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
220
corresponding equation is expressed by
221
dT aTv dP = − aTv + ds a2 v dT
− a2T eq
−1
(13)
Where a is the Helmholtz energy; = ⁄ represents the partial
223
derivatives on the basis of the values m and n. According to the highly accurate
224
Helmholtz energy equation of working fluid, the slope of saturated liquid can be
225
obtained.
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From Eq. (12) and Eq. (13), it can be seen that the isobaric slope and the
227
saturated liquid slope are functions of temperature. A comparison between the isobaric
228
slope and the saturated slope of liquid working fluid is presented in Fig.2 for R600
229
and R134a. In the figure, the reduced temperature is defined as the ratio of
230
temperature to the critical temperature, namely T/Tc. Fig.2 indicates that the positive
231
slope of the saturated liquid increases slowly with the increase of reduced temperature
232
firstly and then decreases to zero sharply in the vicinity of the critical point. For the
233
isobaric slope of liquid fluid, it increases slowly with the increase of pressure in the
234
subcritical region. At the low reduced temperature, the isobaric slope almost coincides
235
with the saturated slope. However, when the reduced temperature gradually
236
approaches to 1, the isobaric slope decreases and is lower than the saturated slope
237
obviously. Thus, for the calculation of limiting efficiency of subcritical ORC, the
238
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
241
For the saturated slope of pure working fluids, the authors have obtained the
243
liquid slope in the reference [21], based on the Helmholtz energy equations. With the
244
increase of reduced temperature, the liquid slope of pure working fluids has a similar
245
variation trend, as presented in Fig.3. As for the mixture, the related study indicates
246
that the liquid slope of mixture is usually located between the corresponding pure
247
fluids' [21]. Since the limiting efficiency is the performance ceiling of subcritical
248
ORC, the maximum liquid slope should be employed. However, Fig.3 illustrates that
249
the slope of saturated liquid reaches the maximum at different reduced temperatures.
250
This phenomenon will increase the computational complexity, when the maximum
251
slope is applied for the limiting efficiency. For the sake of simplicity, saturated slope
252
at a fixed reduced temperature is adopted for all working fluids in this work. From
253
Fig.3, it can be observed that most of working fluids have the maximum slope around
254
the reduced temperature 0.9. Therefore, the required slope for the limiting efficiency
255
is defined as follows:
256
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dT ds L ,Tr =0.9
257
(14)
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The defined slope represents the temperature-entropy characteristic of
258
saturated liquid. It’s a dimensional parameter. In order to get a dimensionless number
259
that could represent the properties of working fluids, we define
260
γ=
261
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]
263
σ=
MTc ds R dT V ,Tr =0.7
(16)
Eq. (16) indicates that molecular complexity is directly related with the saturated
265
slope of vapor working fluids at the reduced temperature 0.7. Thus, σ can be
266
employed to distinguish whether a working fluid is dry, wet or isentropic [8].
267
Furthermore, after the differential transformation of Eq. (16), σ can be directly related
268
with the molecular structure of working fluid. However, for the defined parameter γ, it
269
is related with the slope of saturated liquid at the reduced temperature 0.9. According
270
to Eq. (14) and Eq. (15), the limiting efficiency of Eq. (11) can be transformed into
271 272
the following expression.
T2 −
γR
T1
2 MTc ∆sa − b
(T2 − T1 ) 2
(17)
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Hence, the parameter defined in Eq. (15) is called the limiting factor in this work. Due
274
to the fact that the critical temperature is involved in the definition of limiting factor,
275
pseudocritical temperature is employed to calculate the limiting factor of mixture.
277 278
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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
279
determined for a given working fluid. Compared with the Carnot efficiency, the
280
limiting efficiency of ORC is derived by considering the properties of working fluid.
281
Thus, the limiting efficiency is more reasonable to represent the performance limit of
282
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
284
matter what the cycle configuration is, the area in Fig.1 can’t be utilized
285
by ORC. Furthermore, compared with the limiting efficiency, the practical efficiency
286
is obtained by considering the irreversible loss. The loss is attributed to the
287
temperature difference of heat transfer and the flow resistance of working fluids. Thus,
288
the practical efficiency of ORC can be improved by taking measures to reduce the
289
cycle loss. However, the reduction of cycle loss is always accompanied by the
290
increase of cost. For instance, for a given heat amount, the decrease of temperature
291
difference will result into the increase of heat exchangers’ area.
292
2.3 Thermodynamic perfectness based on limiting efficiency
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Thermodynamic perfectness represents the degree to which the practical cycle
294
approaches to the ideal cycle. For a long time, thermodynamic perfectness is defined
295
on the basis of Carnot efficiency. The equation can be given by
296
η practical ηcarnot
(18)
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Where !"# $%&$ ' denotes the efficiency of practical cycle. represents the
298
thermodynamic perfectness. However, since the Carnot efficiency is only determined
299
by the temperatures of heat source and sink, the corresponding perfectness can’t
300
effectively reflect the improvement potential of practical cycle. Thus, in this work, a
301
new expression of thermodynamic perfectness is proposed from the limiting
302
efficiency. According to the definition of $ #(% , the thermodynamic perfectness for
303
the limiting efficiency can be defined as follows:
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ε limit =
η practical ηlimit
(19)
305
3. Results and discussion According to the proposed limiting efficiency and the defined cycle perfectness,
307
the effects of working fluids and operating conditions on these two parameters are
308
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
310
working fluids consist of 7 kinds of fluid families, namely halohydrocarbons,
311
hydrocarbons, olefins, ethers, siloxanes, benzenes and water. Table 1 demonstrates the
312
employed working fluids along with their structure, molecular weight, critical
313
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.
315
For the limiting factor of pure fluid, the required slope of saturated liquid is
316
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,
318
σ>0: a dry fluid, σ~0: an isentropic fluid, and σ<0: a wet fluid. Table 1 indicates that
319
the employed working fluids consist of dry, isentropic and wet fluids.
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Table 1 Pure working fluids considered in this work
322
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
331
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.
339
Therefore, the following assumptions are made to analyze the limiting efficiency of
340
subcritical ORC, taking as a case study.
342 343 344 345 346 347
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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.
355
Furthermore, under the same temperatures, the limiting efficiency of any working
356
fluid is always lower than the Carnot efficiency. When the temperature difference
357
between the heat source and sink is relatively small, the limiting efficiencies of
358
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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372
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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389 390 391
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.
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.
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Fig. 7 Limiting efficiency of the mixtures for different mass fractions:
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(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
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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
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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
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between the working fluid and the heat source. For simplicity, pressure drops in the
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heat exchangers and pipes are ignored.
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Table 2 Specifications of the subcritical ORC
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Based on the cycle parameters of subcritical ORC, related efficiency and
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thermodynamic perfectness can be obtained for various working fluids. Since
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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
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443.15 K. Furthermore, Fig.8 (a) also indicates the efficiency order: !"# $%&$ ' <
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!'&&% < !$ #(% .
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Fig.8 (b) gives the variation curves of thermodynamic perfectness with the
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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
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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:
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(a) efficiency; (b) thermodynamic perfectness
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Mixtures are employed to reveal the effect of composition on the thermodynamic
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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
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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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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
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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.
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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
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the first author is gratefully acknowledged.
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References
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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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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
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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.5-Limiting efficiency for various working fluids: (a) different kinds of
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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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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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450
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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)
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2.5
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Slope [kg•K2/kJ]
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0.75Pc
Saturated 0.7
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Fig.2 Comparison between the isobaric slope and the saturated liquid slope for R600
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R123 R236fa R125 R601a R600
R124 R245fa R134a R601 R290
R227ea R142b R152a R600a
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Reduced temperature
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R134a/R125 R601/R600 R600/R134a
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Mass fraction of the first component
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Carnot
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R1234yf
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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
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∆Tsup=0K
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Fig.7 Limiting efficiency of the mixtures for different mass fractions: (a) R134a/
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(a)
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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 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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