Identification of key affecting parameters of zeotropic working fluid on subcritical organic Rankine cycle according limiting thermodynamic cycle

Energy Conversion and Management 197 (2019) 111884

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Identification of key affecting parameters of zeotropic working fluid on subcritical organic Rankine cycle according limiting thermodynamic cycle

T

Weicong Xu, Shuai Deng, Li Zhao , Dongpeng Zhao, Ruihua Chen ⁎

Key Laboratory of Efficient Utilization of Low and Medium Grade Energy (Tianjin University), Ministry of Education of China, Tianjin 300350, China

ARTICLE INFO

ABSTRACT

Keywords: Organic Rankine cycle Zeotropic working fluid Limiting thermal efficiency Limiting thermodynamics perfection Thermophysical properties

Attributing to the research on key parameters analysis and working fluid selection, this paper attempted to propose a quantitative method for the performance of zeotropic working fluids used in subcritical organic Rankine cycle. Based on the graphical analysis method in temperature-entropy diagram, the limiting performances, which could be measured using limiting thermal efficiency and limiting thermodynamics perfection, of 4 typical zeotropic working fluids used in simple and regenerative organic Rankine cycle were proposed and calculated. The key affecting parameters and variation mechanism were analyzed as well. The results show that the limiting thermal efficiency of simple organic Rankine cycle increases with the increasing of latent heat of vaporization in both evaporation and condensation process, slope of working fluid saturated liquid line and the decreasing of temperature glide in both evaporation and condensation process, specific heat capacity of liquid working fluid at constant pressure. In addition to the impact of temperature glide in condensation process on limiting thermal efficiency of regenerative organic Rankine cycle, which should be determined according to specific working fluid, the impact of other parameters on limiting thermal efficiency of regenerative organic Rankine cycle is the same as that of simple organic Rankine cycle. The quantitative description of the limiting performance of zeotropic working fluid is of great significance to the analysis and improvement of cycle performance with an in-depth understanding on mechanism and could guide the selection or design of zeotropic working fluid.

1. Introduction

1.2. State of art

1.1. Research background

The research on working fluids has lasted for more than 100 years. In the early stage of research, many scholars used the exhaustive method and trial calculation to select suitable pure and zeotropic working fluids used in ORC and heat pump. The core idea is to calculate the thermal efficiency of each candidate working fluid and then select the working fluid according to the thermal efficiency. Bao et al. [7] reviewed the selection of pure and zeotropic working fluids for ORC from the perspective of effect analysis of working fluids' category and their thermophysical properties on system performance. Abadi et al. [8] listed almost all the proposed zeotropic working fluids used in ORC in existing publications and summarized the advantages and issues. Chen et al. [9] discussed the types of working fluid and the influence of latent heat, density, specific heat on the performance of ORC. The traditional exhaustive method shows simple and convenient advantages in a certain range of candidate working fluids. However, the affecting mechanism of pure and zeotropic working fluids on cycle performance has not been well anatomized using this method. Therefore, in recent years,

Energy shortage and environmental pollution are affecting the development of humanity. Numerous technologies of energy acquisition, transformation, transmission and storage have been put forward to mitigate the energy and environmental crisis [1–4]. In recent years, the proportion of renewable energy and waste heat in energy sources has increased considerably. Whether for traditional energy utilization technologies (such as centralized power generation technology, centralized heating technology, etc.) or new energy utilization technologies, the thermodynamic cycle, which could convert thermal energy into mechanical or electrical energy, is the core. And most of the renewable energy and waste heat are medium and low-temperature heat [5,6]. Therefore, the most commonly used thermodynamic cycles are organic Rankine cycle (ORC) and heat pump. In general, working fluid, which is the carrier of energy transmission and conversion, occupies a dominant position in the research of these two cycles [7].

⁎

Corresponding author. E-mail address: [email protected] (L. Zhao).

https://doi.org/10.1016/j.enconman.2019.111884 Received 18 March 2019; Received in revised form 17 June 2019; Accepted 28 July 2019 Available online 05 August 2019 0196-8904/ © 2019 Elsevier Ltd. All rights reserved.

Energy Conversion and Management 197 (2019) 111884

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Nomenclature

c Carnot cr e exp G_e G_c gen H H1 H2 hse isen L L1 L2 TE Lorenz LTE LTP n net P pre re rH rL rm

Symbols A C c DT FOM f Ja l Q R r S SP s T T TEC VR W wf X Ф ψ αV β α σ η ρ ΔH ΔS Δs

area cycle specific heat capacity (kJ·kg−1·K−1) temperature difference (K) figure of merit objective function Jacob number thermodynamic process line heat transferred (kJ) degree of reaction latent heat of vaporization (kJ·kg−1) entropy (kJ·K−1) size parameter (m) specific entropy (kJ·kg−1·K−1) temperature (K) thermodynamic mean temperature (K) ratio of evaporating temperature and condensation temperature, volumetric flow ratio work (kJ) working fluid other factors flow coefficient loading coefficient volume expansion coefficient (K−1) the slope of working fluid saturated liquid line in T-s diagram (K2·kg kJ−1) the slope of working fluid saturated gas line in T-s diagram (K2·kg kJ−1) zeotropic working fluids selection parameters efficiency density (kg/m3) enthalpy change (kJ·kg−1) entropy change (kJ·kg−1·K−1) specific entropy difference (kJ·kg−1·K−1)

TCave tt

Abbreviations L-ORC LR-ORC LS-ORC ORC R-ORC S-ORC

Subscripts and superscripts b

condensation process Carnot cycle critical point evaporation process expansion process temperature glide in evaporation process temperature glide in condensation process generator high temperature in cycle high temperature of heat source low temperature of heat source heat source isentropic low temperature in cycle low temperature of heat sink high temperature of heat sink thermal efficiency Lorenz cycle limiting thermal efficiency limiting thermodynamics perfection index in Waston equation net output pressure preheating process regenerative process reduced evaporation temperature reduced condensation temperature average of reduced evaporation and condensation temperature average of the turbine outlet temperature and condensation temperatures total to total

limiting organic Rankine cycle limiting regenerative organic Rankine cycle limiting simple organic Rankine cycle organic Rankine cycle regenerative organic Rankine cycle simple organic Rankine cycle

boiling point

many scholars have devoted themselves to the quantitative method with aspects of the relationship between the thermophysical properties of working fluids and the performance of ORC. The core idea is to expose the key affecting parameters of working fluids through the evolution of thermal efficiency expression, and then select the appropriate working fluids according to the key parameters. Using quantitative method, the limitation of working fluid on performance of ORC and key affecting parameters could be directly revealed. The comparison of process of exhaustive method and quantitative method is shown in Fig. 1. At present, the quantitative method is mainly applied in the research of specific thermodynamic process and whole thermodynamic cycle: In terms of actual thermodynamic processes, Zheng et al. [10] have proposed the parameter σ that reflects the temperature difference between zeotropic working fluid and hot fluid. The irreversible loss during constant pressure evaporation or condensation could be calculated using σ, which provides a standard for the selection of zeotropic working fluid in evaporation and condensation processes. Lio et al. [11] have derived the equation of overall efficiency of the expander in ORC

and found that the thermophysical properties of working fluid play a key role. Through experimental research, Xu et al. [12] have proposed a compound parameter αV/ρcp to express the influence of thermophysical properties parameters of pure working fluids on isentropic efficiency of compression process in ORC. In terms of actual whole thermodynamic cycle, many scholars proposed the equation of cycle efficiency that reflects the relationship between thermophysical properties parameters of working fluid and cycle efficiency. Over the past 15 years, more than 10 equations were presented, as listed in Table 1. Most of the quantitative researches focusd on pure working fluid. And it could also be found that no single thermophysical properties parameters that could be confirmed as the exclusive indicator to quantitatively describe the performance of working fluid. The major parameters include latent heat, liquid specific heat, Jacob number, the slope of working fluid saturated liquid and gas line in T-s diagram. However, there is no consistent conclusion has been drawn due to the different cold and heat sources and different setting parameters in different researches. What’s more, there are even obvious inconsistent conclusions. Yamamoto et al. [23] have proposed that the candidate working fluids must have low latent heat. Because low latent 2

Energy Conversion and Management 197 (2019) 111884

W. Xu, et al.

expressed at the present stage, which mainly results from the difference of assumptions and operation conditions in different researches. In order to solve this problem, the calculation of limiting performance for pure working fluid, which idealizes all factors except working fluid, has been put forward in our previous studies [22]. As for zeotropic working fluid, most of the working fluid selection researches still use exhaustive method. Using the Scopus database, the numbers of published papers about research on working fluid selection in ORC in recent 10 years are shown as Fig. 2. Therefore, how to find out key parameters of zeotropic working fluid that affect the performance of ORC and how to quantify describe the effect of key parameters on the ORC efficiency using accurate and general expression are the main issues at this stage.

Exploration on relationship between thermodynamic cycle performance and thermophysical properties of working fluid

Exhastive method

Working fluid 1

Working fluid 2

Quantitative method

η = f (wf, T, X)

Working fluid n Working fluid

System performance calculation model

Tempera -ture

Compon -ent

Key parameters of working fluids (α , β, θ, δ)

1.3. Contribution of this study

η1, η2, ··· ,ηn Working fluid 1

ηmax=η1

Working fluid 2

This paper proposed the limiting thermodynamic cycle, the calculation methods and equations of limiting thermal efficiency (ηLTE) and limiting thermodynamics perfection (ηLTP, is equal to the ratio of ηLTE to the efficiency of Carnot cycle under the same temperatures of heat source and heat sink) of simple organic Rankine cycle (S-ORC) and regenerative organic Rankine cycle (R-ORC) using zeotropic working fluid in subcritical region. And the key thermophysical properties parameters of zeotropic working fluid that affect the ηLTE and ηLTP of SORC and R-ORC are indicated. The ηLTE and ηLTP of S-ORC and R-ORC using four typical zeotropic working fluid are calculated respectively to verify the feasibility of the calculation method. Since the actual cycle must be completed with the actual working fluid, the Carnot cycle as the approaching target of actual cycle for a long time could never be achieved. The limiting cycle provides a new goal that could be achieved in practical cycle by considering the influence of thermophysical properties of working fluids. By analyzing the performance of limiting cycle, not only the ratio of actual cycle and ideal cycle gap caused by thermophysical properties of working fluid could be quantitatively determined, but also a significant reference for the selection of working fluid is provided. The content of each part is arranged as follows: Section 2 presents the methodology of ηLTE and ηLTP; Section 3 presents the parameters of

Working fluid n

Optimal working fluid

Working fluid 1

Fig. 1. Comparison of exhaustive method and quantitative method.

heat would increase the turbine inlet mass flow rate and give the best operating condition. Considering the energy carrying capacity, Tchanche et al. [24] and Al-Sulaiman et al. [25] have suggested that the working fluids with high latent and specific heat are preferable. However, according to the researches from Chen et al. [9] and Xu et al. [22], the working fluids with high latent and low specific heat show better performance. By summarizing the existing researches, it could be found that the research on the selection of pure working fluid began to shift from using exhaustive method to quantitative method. Although the key parameters of pure working fluids that affect the performance of ORC have been confirmed, the consistent and accurate conclusion has not been Table 1 Summary of thermal efficiency equations of ORC. Authors

Year

Zheng et al [10] Lio et al. [11]

2013 2016

Xu et al. [12] Liu et al. [13]

2017 2004

Functions

tt

=f( ,

V/

, R, SP , VR, wf ) 1

TE

1

Mikielewicz et al. [14]

2010 2011

Wang et al. [16]

2012

He et al. [17]

2014

FOM = Ja0.1 TE

( ) Tc Te

2016 2017

Wang et al. [21]

2017

Xu et al. [22]

2018

n +

1 1

TrH TrL

n

1) 1 + (TEC × Ja) 1 1 + Ja 1 1 Carnot

1 + Ja

2016

Ja + 1 TE

Su et al. [20]

Trm TrL

0.8

Ja Carnot + 2 ln 1

exp Carnot

Javanshir et al. [19]

1 1

ln(TEC ) × (TEC

=1

TE

Li et al. [18]

TrH TrL Trm

H(T ) L r (T ) H Ja(T ) Carnot + 1 H

=1 Kuo et al. [15]

nTrm +1 Trm

=

exp

LTE

=1

TE

1

T2

Tc (1 Te

LTE\_S - ORC

1

1

Research object

Zeotropic Pure

Heat transfer process Expansion process

Pure

Whole cycle

Pure

Whole cycle

Pure

Whole cycle

Pure

Whole cycle

Pure

Whole cycle

Pure

Whole cycle

Zeotropic/Pure

Whole cycle

Pure

Whole cycle

Pure

Whole cycle

Pure Pure

cP 1

Working fluid

Compression process Whole cycle

exp\_isen g en

Carnot 2

2 3 c T T1 T3 p 3 T5 T1 T1 r T3 cp T3 T5 T3 cT + 3+ r r T3 T 1 Carnot

cT3 cp,1 T1 T6s + r r T3 1

T3 T1

T TCave Ja ln(TEC ) 1 + T1 TEC 1 TEC Ja + 1 T1 1 (T2 T 1)2 2 sa - b

+ Japre)/ 1 +

= f (TL, TH, , r ) ,

Te + Tc JaPre 2Te LTE\_LR - ORC

3

= f (TL , TH, , r , , c P )

2

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W. Xu, et al. 10

125

100

Number of published articles

Number of published articles

(a) Pure working fluid-exhaustive method

75

50

25

0

2009

2010

2011

2012

10

2013

2014

Year

2015

2016

2017

8

6

4

2

0

2018

Number of published articles

Number of published articles

6

4

2

2010

2011

2012

2013

2014

Year

2010

2011

2012

2013

2015

2014

Year

2015

2016

2017

2018

2017

2018

(d) Zeotropic working fluid-quantitative method

8

2009

2009

10

(c) Zeotropic working fluid-exhaustive method

0

(b) Pure working fluid-quantitative method

2016

2017

8

6

4

2

0

2018

2009

2010

2011

2012

2013

2014

Year

2015

2016

Fig. 2. Activity on working fluid selection research in ORC.

candidate working fluids; Section 4 presents the calculation results and effect analysis; Section 5 summarizes the main conclusions.

the thermophysical properties of working fluid cause the difference between limiting cycle and ideal cycle. Further, the irreversible losses in each thermodynamic process, which may result from heat transfer temperature difference and mechanical loss, are considered in the actual cycle. Fig. 4 shows the relationships between ideal thermodynamic cycle, L-ORC and actual ORC using zeotropic working fluids under heat source with variable temperature. The ideal cycle that composed of variable temperatures heat source and heat sink is defined as the Lorenz cycle [26], which includes two isentropic processes and two polytropic processes. The Lorenz cycle could be considered as the integral of numerous Carnot cycles, which shows the same efficiency as that of an equivalent Carnot cycle at the same thermodynamic mean temperatures of heat source and heat sink [27]. The limiting thermodynamic cycle, which could divide the gap between ideal cycle and actual cycle into two parts, as shown in Fig. 5: (1) irreversible loss caused by thermophysical properties of working fluid;

2. Methodology 2.1. Limiting thermodynamic cycle

TL Entropy (kJ·kg-1·K-1)

Ideal cycle

Temperature (K)

TH

Temperature (K)

Temperature (K)

Limiting thermodynamic cycle is defined as thermodynamic cycle that only consider the thermophysical properties of working fluids, which is used to quantify the role of working fluid in thermodynamic cycle. Under heat source with constant temperature, taking S-ORC as an example, the relationships between ideal thermodynamic cycle, limiting organic Rankine cycle (L-ORC) and actual ORC using pure working fluids are shown in Fig. 3. The ideal cycle is only related to the temperature of heat source and heat sink. In addition to considering the temperature of heat source and heat sink, the L-ORC also considers the thermophysical properties of the actual used working fluid. Therefore,

TH

TL

Entropy (kJ·kg-1·K-1)

Limiting cycle

TH

TL Entropy (kJ·kg-1·K-1)

Actual cycle

thermodynamic process heat source heat sink working fluid Irreversible loss caused by working fluid thermo-physical properties Irreversible loss caused by other factors Fig. 3. Diagram of relationships between ideal cycle, limiting cycle and actual cycle under constant temperature heat source. 4

Energy Conversion and Management 197 (2019) 111884

TH1 TH2 TL1 TL2 Entropy (kJ·kg-1·K-1)

Ideal cycle

TH1 TH2

Temperature (K)

Temperature (K)

Temperature (K)

W. Xu, et al.

TH1 TH2 TL1 TL2

TL1 TL2

Entropy (kJ·kg-1·K-1)

Limiting cycle

Entropy (kJ·kg-1·K-1)

Actual cycle

thermodynamic process heat source heat sink working fluid Irreversible loss caused by working fluid thermo-physical properties Irreversible loss caused by other factors Fig. 4. Diagram of relationships between ideal cycle, limiting cycle and actual cycle under variable temperature heat source.

(2) irreversible loss caused by other factors, such as temperature difference, dissipated heat, friction, etc. This classification method could help researchers understand the role of working fluid in all irreversible losses. The efficiency of limiting thermodynamic cycle, which is defined as ηLTE, represents the maximum efficiency could be achieved only considering the thermophysical properties of working fluid. The thermodynamics perfection of limiting thermodynamic cycle, which is defined as ηLTP, reflects the distance to ideal cycle that results from the thermophysical properties of working fluid. Compared with the actual cycle, the limiting thermodynamic cycle is still an ideal cycle, which ignores irreversible losses due to temperature difference, friction, etc. At the same time, only the irreversible losses due to the thermophysical properties of the working fluid should be considered in limiting thermodynamic cycle. The ηLTE is the maximum efficiency that could be achieved when other factors are ideal in addition to the actual working fluids. Thus, the ηLTE must be greater than the thermal efficiency of the actual cycle. The limiting efficiency is reasonably amplified under some conditions in the calculation of ηLTE. The purpose is to make the calculation simpler and more uniform. In summary, the limiting thermodynamic cycle is used to quantify the effect of working fluids in actual cycle, and ηLTE is between the thermal efficiency of ideal Carnot cycle and actual thermodynamic cycle. Since the role and significance of limiting thermodynamic cycle have been clarified, the following three assumptions should be made to exclude other factors when calculating ηLTE and ηLTP.:

condensation process are perfect. The perfect temperature glide means that the perfect thermal matching could be achieved in both evaporation and condensation processes. Therefore, the temperatures of the working fluid in phase change process are equal to the temperatures of the heat source or heat sink. 3) Both the compression and expansion processes are isentropic processes, which means that the irreversible losses in these two processes are ignored. 2.2. Limiting performance of simple organic Rankine cycle Fig. 6 shows the T-s diagram of S-ORC using different types of zeotropic working fluids under the above assumptions. The gray area represents the gap that results from the thermophysical properties of working fluid. In actual cycle, superheated evaporation may occur during evaporation process. However, the superheated evaporation also results in the temperature difference in phase change evaporation process, which would increase the irreversible loss. As a result, there is no superheated evaporation when the S-ORC using dry or isentropic zeotropic working fluids. For the S-ORC using wet zeotropic working fluid, the superheated evaporation is required to ensure the safety of the expander. Therefore, a suitable superheat temperature is set to ensure that the working fluid at the outlet of the expander remains saturated gas. As shown in Fig. 6, a part of gap between actual cycle and ideal cycle results from the non-phase transitions in evaporation process for S-ORC using all types of zeotropic working fluids. In addition, another part of gap results from non-phase transitions in condensation process and temperature difference in phase change evaporation process induced by superheat for S-ORC using dry and wet zeotropic working fluid respectively.

1) The heat exchange area is infinite. The infinitely large heat exchange area means that there could be no temperature difference in the heat transfer process. 2) The temperature glide of working fluid in both evaporation and

Ideal cycle

Limiting cycle

Actual cycle

Irreversible loss caused by working fluid thermo-physical properties Irreversible loss caused by other factors Fig. 5. Diagram of thermodynamic perfection of limiting and actual thermodynamic cycle. 5

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TL1 TL2

Temperature (K)

Temperature (K)

Temperature (K)

TH1 TH2

(c)

(b)

(a) TH1 TH2 TL1 TL2

TH1 TH2 TL1 TL2

Entropy (kJ·kg-1·K-1)

Entropy (kJ·kg-1·K-1)

Entropy (kJ·kg-1·K-1) Dry working fluids

Isentropic working fluids

thermodynamic process

heat source

Wet working fluids

heat sink

working fluid

the gap

Fig. 6. T-s diagram of S-ORC using zeotropic working fluids: (a) Dry working fluids; (b) Isentropic working fluids; (c) Wet working fluids.

For convenient and unified calculation, only the gap results from the non-phase transitions in evaporation process is considered in limiting SORC (LS-ORC) using different types of zeotropic working fluids, as shown in Fig. 7. Compared with S-ORC, the LS-ORC using dry working fluids ignores the gap that results from the non-phase transitions in condensation process. As for wet working fluids, the LS-ORC ignores a part of gap in evaporation process that results from the superheat. Neglecting these two parts although improves the ηLTE of S-ORC (ηLTE_SORC) and ηLTP of S-ORC (ηLTP_S-ORC), it could simplify and unify the calculation and conform to the principle of the limiting thermodynamic cycle that described in Section 2.1. In addition, the temperature increment of the working fluid during the compression process, which is usually less than 1 °C [28], is also ignored. On these foundations, the LSORC could be represented by cycle C1-2-3-4-1 for all three types of working fluids.

=1

=

(2)

Aa - c - 3 - b - a =

1 sa - b (TH1 + TH2) 2

(3)

1 (TH2 2

TL2 )

TH1

DTe

rc

LTP\_S - ORC

TL2 (4)

DTe

3 2

TL1 DT c TL2 1

4

a

TH1 TH2

c

TL1 DT c TL2

DTe

3 2 4

1 a

b -1

-1

Entropy (kJ·kg ·K ) Dry working fluids thermodynamic process

+

T ln( TL1 ), TL2 L2

re T ln( T H1DT ), DTe H1 e

TL2 )

(5)

>0

<0

LTE\_S - ORC

=

Lornez

(6) LTE\_S - ORC T¯ 1 T¯L H

=

1

LTE\_S - ORC (TL1 TL2)(ln TH2 ln TH1) (TH2 TH1)(ln TL1 ln TL2)

(b) Temperature (K)

Temperature (K)

c

TH1 DTe ) TL2

TL2 )(TH1 DTe sa - b

Using graphical analysis method, the ηLTE_S-ORC could be converted into the ratio of area that is surrounded by thermodynamic process lines in temperature-entropy (T-s) diagram, as shown in formula (1). And the area Aa-1-4-b-a, Aa-c-3-b-a and A1-c-2-1 could be calculated as formulas (2)–(4), where β represents the maximum slope of saturation liquid curve of working fluid in T-s diagram within the range of calculated working conditions [28]. For S-ORC using dry zeotropic working fluid, DTe is equal to the temperature glide in evaporation process (TG_e) and DTc is equal to the temperature glide in condensation process (TG_c) plus the superheat temperature after expansion process, which results from the slope of saturation gas curve of working fluid. When using isentropic zeotropic working fluid, DTe and DTc are equal to TG_e and TG_c respectively. As for wet zeotropic working fluid, DTc is equal to TG_c and DTe is equal to the TG_e plus the superheat temperature in evaporation process, which also results from the slope of saturation gas curve of working fluid. The expression of ηLTE_S-ORC could be obtained by introducing formulas (2)–(4) into formula (1), as shown in formula (5). According to the second law of thermodynamics [29], the maximum

(a) TH1 TH2

=

(TH2

(TH1 + TH2)

(7)

(1)

1 sa - b (TL1 + TL2) 2

cP ln( TL1

Aa - 1 - 4 - b - a Aa - 1 - 2 - 3 - 4 - b - a

Aa - 1 - 4 - b - a =

A1 - c - 2 - 1 =

sa - b =

TL1 + TL2

=1

(c) Temperature (K)

Wnet A1 2 3 4 1 = =1 Q hse Aa - 1 - 2 - 3 - 4 - b - a Aa - 1 - 4 - b - a Aa - c - 3 - b - a A1 - c - 2 - 1

LTE\_S - ORC

LTE\_S - ORC

TH1 TH2 TL1 DTc TL2

b

3 2 4

1 a

Isentropic working fluids heat sink

DTe

b

Entropy (kJ·kg-1·K-1)

Entropy (kJ·kg-1·K-1)

heat source

c

Wet working fluids working fluid

the gap

Fig. 7. T-s diagram of LS-ORC using zeotropic working fluids: (a) Dry working fluids; (b) Isentropic working fluids; (c) Wet working fluids. 6

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entropy change in S-ORC could be calculated using formula (6), where re and rc represent the latent heat in evaporation and condensation process, cP represents the specific heat capacity of liquid working fluids under constant pressure and α represents the maximum slope of saturation gas curve of working fluid in T-s diagram within the range of calculated working conditions. For dry (α > 0) and wet (α < 0) working fluids, Δsa-b could be calculated according the evaporation process and condensation process respectively, as shown in formula (6). For isentropic working fluids, Δsa-b could be calculated using any equations in (6). The expression of ηLTP_S-ORC is shown as formula (7), where TL and TH represent the thermodynamic mean temperature of heat sink and heat source respectively.

and condensation process respectively. And the ηLTP_R-ORC could be also obtained, as shown in formula (16).

Wnet Wnet A1 2 = = Q hse Qeva Qre Aa - 1 - 2 - 3 - d - a A1 2 3 4 1 A A1 - f - 2 - 1 = = 1-f-3-4-1 Ab - e - 2 - 3 - d - b Ab - g - 3 - d - b A e - g - 2 - e =

LTE\_R - ORC

A1 - f - 2 - 1 =

1 (TH2 2

Ab - g - 3 - d - b =

2.3. Limiting performance of regenerative organic Rankine cycle Based on the above three hypotheses, the T-s diagrams of R-ORC using different types of zeotropic working fluids are shown in Fig. 8. Same as that of S-ORC, there is no superheated evaporation when the RORC using dry or isentropic zeotropic working fluids. And the superheat is only used to ensure the safety of the expander when using wet working fluids. The area with blue shadow represents the heat exchanged in the regenerator. Under the limiting conditions, the temperature of exhaust gas after expansion process in regenerator should be higher than the temperature of liquid working fluid after compression process. And the heat released from exhausted gas is equal to the heat absorbed by high pressure liquid working fluid. According to the LS-ORC mentioned in Section 2.2, the T-s diagrams of limiting R-ORC (LR-ORC) using different types of zeotropic working fluids are shown in Fig. 9. The LR-ORC could be represented using cycle C1-2-3-4-1 and two shaded areas Aa-1-e-b-a, Ac-h-4-d-c for all types of working fluids. In the next section, the ηLTE_R-ORC and ηLTP_R-ORC are derived taking isentropic working fluids as an example. The results are also suitable for dry and wet working fluids. Fig. 10 shows a clearer T-s diagram of LR-ORC using isentropic zeotropic working fluids. Line l3-f represents the temperature change of the heat source in S-ORC. The temperature at f point is assumed to be equal to TH2′. In the limiting case, the outlet temperature of cold fluid in the regenerator is equal to the inlet temperature of hot fluid, that is, Te is equal to T4. Line l4-1 represents the temperature change in condensation process in S-ORC. And the temperature at point e and 4 are assumed to be equal to TL1′. The original expression of ηLTE_R-ORC obtained by graphical analysis method is shown as formula (8). And the calculation equations of each part that involved in formula (8) are shown in formulas (9)-(12). The ηLTE_R-ORC could be further derived by introducing formulas (9)–(12) into formula (8), as shown in formula (13). The entropy changes from point a to d and from b to d are expressed as formulas (14) and (15) respectively, where re and rc represents the latent heat in evaporation

Ae - g - 2 - e =

LTE\_R - ORC

(8)

TL2)

(9)

TL2 (10)

d (TH1

TL1 )

+ TH2 ) TH1

(11)

DTe

TL1 (12)

= TL2) (TH2 TL2 )(TH1 (TH2 TL1 )(TH1 DTe

DTe TL2 ) TL1 ) (13)

sa - d =

cP ln

(

rc TL1

TL2

s b - d = sa - d

LTP\_R - ORC

=

) + ln ( ln ( ), <0

TH1 DTe TL2

re DTe

TH1 TH1 DTe

),

>0

TL1 TL2

cP ln

(14)

TL1 TL2

LTE\_R - ORC

(15)

=

Lornez

LTE\_R - ORC T¯ 1 T¯L H

=

1

LTE\_R - ORC (TL1 TL2)(ln TH2 ln TH1) (TH2 TH1)(ln TL1 ln TL2)

(16) 3. Candidate zeotropic working fluids The number of zeotropic working fluids is infinite compared to pure working fluids because the components and the proportions of components of zeotropic working fluids could change independently at the same time. In this paper, four typical zeotropic working fluids are selected as the candidate working fluids, which are composed of different types of pure working fluids, namely R152a/R11, R227ea/R245fa, R1234ze/R141b and R134a/R601. The main thermophysical properties parameters of candidate zeotropic working fluids are listed in Table 2, which are derived from the authoritative database Refprop 9.1 provided by National Institute of Standards and Technology (NIST) [30]. It

(c) Temperature (K)

TH1 TH2 TL1 TL2

TH1 TH2 TL1 TL2

Entropy (kJ·kg-1·K-1)

Entropy (kJ·kg-1·K-1)

Entropy (kJ·kg-1·K-1) Dry working fluids thermodynamic process

1 (TH2 2

DTe

Aa - 1 - e - b - a

(b) Temperature (K)

Temperature (K)

TL1 TL2

TH1

TL2 )

1 sb 2

TLI + TH2

sa - d (TH1 TL1 + TH2 s b d (TH1 + TH2)

(a) TH1 TH2

1 sa - d (TH1 2

A1 - f - 3 - 4 - 1 =

3 4 1

Isentropic working fluids heat source

heat sink

working fluid

Wet working fluids the gap

regenerative process

Fig. 8. T-s diagram of R-ORC using zeotropic working fluids: (a) Dry working fluids; (b) Isentropic working fluids; (c) Wet working fluids. 7

Energy Conversion and Management 197 (2019) 111884

W. Xu, et al.

(b)

DTe

3 2

TL1 DTc TL2 1

4

TH1 TH2

DTe

3 2

TL1 DTc TL2 1

(c) Temperature (K)

TH1 TH2

Temperature (K)

Temperature (K)

(a)

4

TH1 TH2

DTe

TL1 TL2 DTc

Dry working fluids

Isentropic working fluids

thermodynamic process

heat source

heat sink

4 1

Entropy (kJ·kg -1·K-1)

Entropy (kJ·kg -1·K-1)

Entropy (kJ·kg -1·K-1)

3 2

Wet working fluids

working fluid

the gap

regenerative process

Fig. 9. T-s diagram of LR-ORC using zeotropic working fluids: (a) Dry working fluids; (b) Isentropic working fluids; (c) Wet working fluids.

Temperature (K)

4. Results and discussion

TH1 TH2 TL1 TL2

f

g

DTe

In Section 2.2 and 2.3, the detailed calculation methods of ηLTE and ηLTP for both S-ORC and R-ORC have been presented. This part presents the calculation results of ηLTE and ηLTP of for S-ORC and R-ORC using selected zeotropic working fluids. The key affecting parameters and change rules are analyzed according the expressions and T-s diagram firstly and then amended using the calculation results of specific working fluids.

3 2

e

h

4

DTc

1

4.1. Simple organic Rankine cycle It could be found from formulas (5) and (6) that the ηLTE_S-ORC of dry zeotropic working fluid is closely related to TH1, TH2, TL1, TL2, DTe, re, cP, β and the ηLTE_S-ORC of wet zeotropic working fluid is closely related to TH1, TH2, TL1, TL2, DTe, rc, β. For isentropic zeotropic working fluid, two relationships are applicable. As mentioned in Section 2.1, DTe is determined by TG_e and α. And it is noteworthy that DTc is equal to the difference of TL1 and TL2. Therefore, DTc, which is determined by TG_c and α, is also one of the key parameters affecting ηLTE_S-ORC. In summary, the main affecting parameters of ηLTE_S-ORC are shown as formula (17).

c d

a b

Entropy (kJ·kg-1·K-1) LR-ORC thermodynamic process working fluid the gap

heat source heat sink regenerative process

Fig. 10. T-s diagram of LR-ORC using dry zeotropic working fluids.

LTE\_S - ORC

is noteworthy that since this paper also considers the variation of the component proportion, the change ranges of variation of all parameters are presented rather than a specific value. In general, a zeotropic working fluid composed of the same type of pure working fluid should be the same type as pure working fluid. However, the type of zeotropic working fluid composed of different types of pure working fluids varies with the proportion of components. For example, with the increase of proportion R113, the zeotropic working fluids R21/R113 gradually changes from wet working fluid to dry working fluid. Moreover, the temperature glide in phase change process would change with the change of component proportion. The bubble and dew temperature of these candidate zeotropic working fluids under ambient pressure are shown in Fig. 11.

= f (TH1, TH2, TL1, TL2,TG\_e, TG\_c, , , re, rc, c P)

(17)

The calculation results of ηLTE_S-ORC and ηLTP_S-ORC for four candidate zeotropic working fluids are shown in Figs. 12 and 13 respectively. The proportion of component with higher boiling temperature (Tb) in each zeotropic working fluid changes from 0.1 to 0.9 with an interval of 0.1. And the highest temperature in cycle changes from 350 K to critical temperature (Tcr) with an interval of 5 K. The lowest temperature is kept at 300 K in all conditions. In general, the higher the Tb of working fluid, the higher the Tcr. As a result, the Tcr of zeotropic working fluid increases with the increasing of the proportion of component with higher Tb. This is why the top left corner of each figure is blank. The ηLTE_S-ORC increases with the increasing of highest temperature for each zeotropic working fluid, as shown in Fig. 12. This is consistent with the basic knowledge of thermodynamic cycle, which is described as that the higher the temperature difference of the heat source and

Table 2 Thermophysical properties parameters of candidate zeotropic working fluids. Working fluid

Tb (K)

R152a/R11 R227ea/R245fa R1234ze/R141b R21/R113

(249.86, (256.73, (256.33, (281.93,

Tcr (K) 269.67) 288.21) 294.31) 320.64)

(390.71, (374.90, (391.80, (451.48,

r (kJ/kg) 455.21) 427.16) 467.78) 487.21)

(212.16, (131.80, (209.27, (144.35,

8

315.86) 196.09) 234.39) 239.50)

cP (kJ/kg·K)

Type

(0.6464, (0.8165, (0.8176, (0.6202,

Wet/Wet Dry/Dry Isentropic/Isentropic Wet/Dry

1.0089) 0.9205) 0.8813) 13.632)

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Bubble point temperature line Dew point temperature line

290

Bubble point temperature line Dew point temperature line

285

Temperature (K)

Temperature (K)

280 280

270

275 270 265

260

260

(a) R152a/R11

250 0.0 304

0.2

0.4

0.6

0.8

The proportion of R11

(b) R227ea/R245fa

1.0

0.0

0.2

0.4

0.6

0.4

0.6

0.8

The proportion of R245fa

1.0

320

Bubble point temperature line Dew point temperature line

315

296

Temperature (K)

Temperature (K)

310 288 280 272

305 300 295 290

264

285

(b) R1234ze/R141b

256 0.0

0.2

0.4

0.6

0.8

The proportion of R141b

1.0

(d) R21/R113

0.0

0.2

0.8

The proportion of R113

1.0

Fig. 11. Diagram of bubble and dew temperature line of zeotropic working fluids.

heat sink, the higher the efficiency of the thermal power cycle. When the highest temperature keeps constant, the ηLTE_S-ORC increases firstly and then decrease with the increasing of the proportion of component with higher Tb keeps for all four zeotropic working fluids. The highest ηLTE_S-ORC for each zeotropic working fluid occurs at the highest proportion of component with higher Tb. For R152a/R11, R227ea/R245fa, 450 440

(a) R152a/R11

410

20.00

Highset temperature (K)

Highset temperature (K)

430 420

17.50

410 400

15.00

390 12.50

380

R1234ze/R141b and R21/R113, the highest ηLTE_S-ORC are equal to 21.25%, 18.73%, 23.97% and 24.57% respectively. With the increasing of TH2, the ηLTP_S-ORC decreases, as shown in Fig. 13. Under the same TH2, with the increasing of the proportion of component with higher Tb, the ηLTP_S-ORC increases firstly and then decreases for R152a/R11 and R1234ze/R141b. As for R227ea/R245fa LTE(%)

(b) R227ea/R245fa

30.00

400

17.50 27.50

390 380

25.00

15.00

370

22.50

12.50

370

10.00

360

360

20.00

350 0.1

0.2

0.3

0.4

0.5

0.6

The proportion of R11

0.7

0.8

0.9

460

(c) R1234ze/R141b 22.50

440

Highset temperature (K)

350 0.1

20.00

420

17.50

400

0.2

0.3

0.4

0.5

0.6

The proportion of R245fa

0.7

0.8

15.00

380

0.9 17.50

(d) R21/R113

440

Highset temperature (K)

460

7.500

15.00

22.50

420

12.50

20.00

400

10.00

17.50

380

15.00

7.500

12.50 10.00

360

12.50

360

5.000

10.00

0.1

0.2

0.3

0.4

0.5

0.6

The proportion of R141b

0.7

0.8

0.9

0.1

0.2

0.3

0.4

0.5

0.6

The proportion of R113

0.7

0.8

Fig. 12. The change of ηLTE_S-ORC with component proportion and highest temperature in cycle. 9

0.9

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W. Xu, et al. 450 440

(a) R152a/R11

410

Highset temperature (K)

Highset temperature (K)

430 420 85.00

410 400 390 380

90.00

370

LTP (%)

(b) R227ea/R245fa

100.0

400 390

95.00

380

70.00 75.00

90.00

370 80.00

360

360 350 0.1 460

0.2

0.3

0.4

0.5

0.6

The proportion of R11

0.7

0.8

460

(c) R1234ze/R141b

0.2

0.3

0.4

0.5

0.6

The proportion of R245fa

85.00

0.7

0.8

0.9

(d) R21/R113

75.00

80.00

75.00

440

420

Highset temperature (K)

440

Highset temperature (K)

85.00

350 0.1

0.9

80.00

400 85.00

380

75.00

400 85.00 70.00

380 90.00

90.00

360

80.00

420

360 65.00

0.1

0.2

0.3

0.4

0.5

0.6

The proportion of R141b

0.7

0.8

0.9

0.1

0.2

0.3

0.4

0.5

0.6

The proportion of R113

0.7

0.8

0.9

Fig. 13. The change of ηLTP_S-ORC with component proportion and highest temperature in cycle.

and R21/R113, the ηLTP_S-ORC show increase and decrease trend respectively. The highest ηLTP_S-ORC of R152a/R11, R227ea/R245fa, R1234ze/R141b and R21/R113 are 94.96%, 86.72%, 92.73% and 94.08% respectively. Under the same highest temperature in cycle (TH1 is equal to 375 K), the calculation results of ηLTE_S-ORC and ηLTP_S-ORC are shown in Fig. 14. And the detailed parameters of the zeotropic working fluids and key temperatures in the cycle are listed in Table A in Appendix section. For R152a/R11, R1234ze/R141b and R21/R113, with the increasing of the proportion of component with higher Tb, ηLTE_S-ORC decreases firstly and then increases. For R227ea/R245fa, with the increasing of the proportion of R245fa, ηLTE_S-ORC almost remains unchanged first and then increases. When the proportion of component with higher Tb is less than 0.7, R21/R113 shows the best performance among these four zeotropic working fluids. When the proportions of component with higher Tb are equal to 0.8 and 0.9 respectively, R227ea/R245fa and R1234ze/R141b show the best performance. The ηLTP_S-ORC of R152a/R11, R227ea/R245fa and R1234ze/R141b increase with the increasing of the proportion of component with 20

R152a/R11 R227ea/R245fa R1234ze/R141b R21/R113

higher Tb. But for R21/R113, the ηLTP_S-ORC decreases. ηLTP_S-ORC could eliminate the influence of temperature and study the influence of thermophysical properties of the working fluid on limiting performance. For R152/R11, with the increasing of the proportion of component with higher Tb, the re increases first and then decreases, the rc decreases and the β increases. It indicates that β plays a major role in the impact of ηLTP_S-ORC. For R227ea/R245fa and R21/R113, the key thermophysical properties are re and rc. For R1234ze/R141b, re, rc and β all show an upward trend, which lead to the increasing of ηLTP_S-ORC. 4.2. Regenerative organic Rankine cycle From the deduction in Section 2.2, the relationship between ηLTE_Rand key thermophysical properties parameters of working fluid is shown in formula (18). ORC

LTE\_R - ORC

= f (TH1, TH2, TH2, TL1, TL1, TL2,TG\_e, TG\_c, , , re, rc, cP ) (18)

The changes of ηLTE_R-ORC with the component proportion and TH1 120

(a)

(b)

R152a/R11 R227ea/R245fa R1234ze/R141b R21/R113

100

15

η LTP_S-ORC (%)

η LTE_S-ORC (%)

80

10

60

40

5 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Proportion of component with higher Tb

0

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Proportion of component with higher Tb

Fig. 14. The change of ηLTE_S-ORC and ηLTP_S-ORC with component proportion under same highest temperature. 10

0.9

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for the four candidate zeotropic working fluids are shown in Fig. 15. Same as that of S-ORC, the ηLTE_R-ORC increases with the increasing of TH1 for each zeotropic working fluid. The highest ηLTE_R-ORC occurs at the highest proportion of component with higher Tb. The highest ηLTE_RORC are equal to 22.88%, 19.31%, 24.70% and 25.07% for R152a/R11, R227ea/R245fa, R1234ze/R141b and R21/R113 respectively. Fig. 16 shows the changing trend of ηLTP_R-ORC with component proportion and highest temperature in cycle. For all four working fluids, with the increasing of TH2, the ηLTP_R-ORC decreases. Under the same TH2, with the increasing of the proportion of component with higher Tb, the ηLTP_R-ORC increases firstly and then decreases for R152a/ R11 and R1234ze/R141b, which is the same as that of S-ORC. The ηLTP_R-ORC of R227ea/R245fa shows an increase trend in general. As for R21/R113, the ηLTP_S-ORC decreases first and then increases. The highest ηLTP_S-ORC of R152a/R11, R227ea/R245fa, R1234ze/R141b and R21/ R113 are 94.61%, 86.25%, 92.18% and 94.08% respectively. Fig. 17(a) shows the variation of ηLTE_R-ORC in the case of TH1 is equal to 375 K. The detailed data of the zeotropic working fluids and the key temperature in the cycle are listed in Table B in Appendix section. In general, with the increasing of the proportion of component with higher Tb, the ηLTE_R-ORC of R152a/R11 and R1234ze/R141b decrease first and then increase, the ηLTE_R-ORC of R227ea/R245fa increases first and then decreases and finally increases, and the ηLTE_R-ORC of R21/R113 almost remains unchanged. R21/R113 shows the best performance among these four zeotropic working fluids when the proportion of component with higher Tb is less than 0.7. When the proportion is greater than 0.7, R1234ze/R141b shows the highest ηLTE_R-ORC. Fig. 17(b) shows the variation of ηLTP_R-ORC, which is the same as that of ηLTP_S-ORC.

When TH1, TH2, TL1, and TL2 keep constant, with the thermodynamic analysis of real working fluids, the change of every parameter could also lead to the change of other parameters. For dry working fluid, the increase of β results in the increasing of re, DTe (is equal to the TG_e) and the decreasing of cP, which lead to the decreasing of proportion of A1-c2-1 and A1-2-3-4-1. That is the ηLTE_S-ORC increases with the increasing of β, re, TG_e and the decreasing of cP. For wet working fluid, the increasing of β results in the increasing of DTe and the decreasing of cP, which lead to the decreasing of proportion of A1-c-2-1 and A1-2-3-4-1. With the increasing of rc, the proportion of A1-c-2-1 and A1-2-3-4-1 decreases. Therefore, the ηLTE_S-ORC increases with the increasing of β, rc and the decreasing of cP. Under the calculation conditions in this paper, although TH1 and TL2 remain unchanged, TH2 and TL1 would vary with the change of the physical properties of the working fluid. Taking R152a/R11 as an example, with the increasing of the proportion of component with higher Tb, the TH2 decreases firstly and then increases, but TL1 shows the opposite trend. In addition, the β increases with the increasing of the proportion of component with higher Tb. re increases first and then decreases. The interaction of these factors leads to the trend of ηLTE_SORC. And the changing trend of R1234ze/R141b, R21/R113 and R227ea/R245fa could also be analyzed according to data in Table A. For dry working fluid, by combining graphical analysis and calculation results, the increase of TG_e results in the decreasing of TH2, which means that ηLTE_S-ORC decreases with the increasing of TG_e. And TL1 decreases with the decreasing of TG_c and increasing of α. Therefore, ηLTE_S-ORC increases with the decreasing of TG_c and increasing of α. For wet working fluid, the greater the TG_e and α, the greater the degree of superheat required, and the smaller the TH2. That is ηLTE_S-ORC decreases with the increasing of TG_e and α. The greater the TG_c, the smaller the TL1. Therefore, the ηLTE_S-ORC decreases with the increasing of TG_c. ηLTE_R-ORC increases with the increasing of thermodynamic mean temperature of heat source and the decreasing of thermodynamic mean temperature of heat sink, which are determined by TH1, TH2 and TL1, TL2 respectively. In S-ORC, the temperature differences ΔTH (ΔTH = TH2 TH2′) and ΔTL (ΔTL = TL1′ - TL1) increase with the increasing of exchanged heat (Qre) in regenerator. Therefore, Qre increases with the increasing of TL1′ and decreasing of TH2′, which leads to the increasing

4.3. Discussions on impact mechanism The ηLTE_S-ORC is determined by the thermodynamic mean temperature of heat source, heat sink and the proportions of A1-c-2-1 and A12-3-4-1, as shown in Fig. 6. With the increasing of thermodynamic mean temperature of heat source and decreasing of thermodynamic mean temperature of heat sink, the ηLTE_S-ORC increases. Thus, the ηLTE_S-ORC increases with the increasing of TH1, TH2 and the decreasing of TL1, TL2. 450

22.50

(a) R152a/R11

440

20.00

Highset temperature (K)

Highset temperature (K)

430

410

420 410 17.50

400 390

15.00

380

30.00

17.50

400

27.50

390 25.00

380

15.00

370

22.50

12.50

370

360

10.00

360 350 0.1

0.2

0.3

460

0.4

0.5

0.6

0.7

The proportion of R11

0.8

350 0.1

0.9

460

(c) R1234ze/R141b 440

12.50

0.2

0.3

0.4

0.5

0.6

20.00

0.7

The proportion of R245fa

0.8

0.9 17.50

25.00

(d) R21/R113

15.00

440 22.50

420

Highset temperature (K)

Highset temperature (K)

LTE(%)

(b) R227ea/R245fa

22.50 20.00

400 17.50

380

15.00

22.50

420

12.50

20.00

400

10.00

17.50

380

7.500

15.00 12.50

360 0.1

360

10.00

0.2

0.3

0.4

0.5

0.6

The proportion of R141b

12.50 5.000

0.7

0.8

0.9

0.1

0.2

0.3

0.4

0.5

0.6

The proportion of R113

0.7

0.8

Fig. 15. The change of ηLTE_R-ORC with component proportion and highest temperature in cycle. 11

0.9

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W. Xu, et al. 450 440

(a) R152a/R11

410

80.00

Highset temperature (K)

Highset temperature (K)

430 420 410 85.00

400 390 380

90.00

370

LTP (%)

(b) R227ea/R245fa

100.0

400 70.00

390

95.00

75.00

380 80.00

370

90.00

360

360 350 0.1 460

0.2

0.3

0.4

0.5

0.6

The proportion of R11

0.7

0.8

460

(c) R1234ze/R141b

440

0.2

0.3

0.4

0.5

0.6

75.00

420 80.00

400

85.00

380

360 0.1

0.2

0.3

0.4

0.5

0.6

The proportion of R141b

0.7

0.8

0.8

0.9

The proportion of R245fa (d) R21/R113 80.00

75.00

80.00

420

75.00

400 85.00

380

360

90.00

85.00

0.7

440

Highset temperature (K)

Highset temperature (K)

85.00

350 0.1

0.9

0.9

70.00

90.00

0.1

0.2

0.3

0.4

0.5

0.6

The proportion of R113

0.7

65.00

0.8

0.9

Fig. 16. The change of ηLTP_R-ORC with component proportion and highest temperature in cycle.

R152a/R11 R227ea/R245fa R1234ze/R141b R21/R113

20

120

(a)

R152a/R11 R227ea/R245fa R1234ze/R141b R21/R113

100

15

(b)

η LTP_R-ORC (%)

η LTE_R-ORC (%)

80

60

10

40

5 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Proportion of component with higher Tb

0

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Proportion of component with higher Tb

0.9

Fig. 17. The change of ηLTE_R-ORC and ηLTP_R-ORC with component proportion under same highest temperature.

of ηLTE_R-ORC. With the increasing of β, the limiting thermodynamic cycle gradually approaches to the ideal cycle. Therefore, ηLTE_R-ORC would increase with the increasing of β. Due to the inversely proportional to the change of cP and β [31], ηLTE_R-ORC would decrease with the increasing of cP, which could also be obtained according formula (16). When TH1 and TH2 keep constant, the change of TG_e, re and that of β are closely related. The increasing of TG_e results in the increasing of the β and re, which reduces the area of Ae-g-2-e. Thus, the ηLTE_R-ORC increases with the increasing of TG_e. As for re, it could be found from the formula (16) that ηLTE_R-ORC decreases with the increasing of re. However, when TH1 and TH2 remain unchanged, the TG_e and β would also increase with the increasing of re, which improves ηLTE_R-ORC. The increasing of rc would reduce the proportion of A1-f-2-1 in A1-f-3-4-1, which means that the cycle C1-2-3-4-1 is closer to ideal cycle. Thus, ηLTE_R-ORC would increase with the increasing of rc. The main difference between S-ORC and R-ORC lies in condensation process and regenerative process. For wet working fluid, the greater the TG_c, the greater the Qre, the greater the TL1, which results in the change contradiction of ηLTE_R-ORC. And the same contradiction exists for dry

working fluid, both the Qre and TL1 increase with the increasing of TG_c and decreasing of α. Therefore, the change rules of ηLTE_R-ORC with TG_c for wet working fluid and with TG_c and α for dry working fluid should be determined by specific working fluid. For instance, according the calculation results of R152a/R11, the TL1 plays a major role, that is, ηLTE_R-ORC increases with the decreasing of TG_c. According to the formula analysis and calculation results of actual working fluids, the key influence parameters and change rules of ηLTE_SORC and ηLTE_R-ORC are shown in Table 3. Upward arrow indicates positive correlation, downward arrow represents negative correlation. Presenting two arrows indicate that the change rules are related to the specific working fluid. 5. Conclusions This paper presented the calculation methodology of ηLTE_S-ORC, ηLTP_S-ORC, ηLTE_R-ORC and ηLTP_R-ORC to intensify the understanding of the limiting performance of zeotropic working fluids in subcritical ORC. The main conclusions are as follows: 12

Energy Conversion and Management 197 (2019) 111884

W. Xu, et al.

Table 3 Key influence parameters and change rules of ηLTE_S-ORC and ηLTE_R-ORC. TH1

S-ORC R-ORC

↑ ↑

TH2

↑ ↑

TH2′

TL1

– ↓

↓ ↓

TL1′

– ↑

TL2

TG_e

↓ ↓

TG_c

↓ ↓

(1) The limiting thermodynamic cycle of ORC using zeotropic working fluids is proposed to individually evaluate the influence of working fluid on the whole cycle. The ηLTE and ηLTP are presented to quantify the role of working fluid and to quantify the distance to the ideal thermodynamic cycle. The calculation methods of ηLTE_S-ORC, ηLTP_SORC, ηLTE_R-ORC and ηLTP_R-ORC are introduced and the expressions are derived respectively. (2) Through the analysis of the expressions and the calculation of specific zeotropic working fluids, the key influence parameters and change rules of ηLTE_S-ORC and ηLTE_R-ORC are proposed, as shown in Table 3. It is worth noting that some thermophysical properties

re

↓ ↑↓

↑ ↑

rc

↑ ↑

β

↑ ↑

α

cP

>0

<0

↑ ↑

↓ ↑↓

↓ ↓

parameters show different influence rules for ηLTE_S-ORC and ηLTE_RORC, which might guide the selection and design of working fluids for different cycle structures. Acknowledgements This work is supported by National Key Research and Development Plan under Grant No. 2018YFB0905103, National Key Research and Development Plan under Grant No. 2018YFB1501004 and National Nature Science Foundation of China under Grant No.51776138.

Appendix

Table A The detailed data of the zeotropic working fluids and the key temperature of S-ORC. R152a/R11 Proportion of higher temperature component

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

TH1 (K) TH2 (K) TL1 (K) TL2 (K) TG_e (K) TG_c (K) re (kJ/kg) rc (kJ/kg) β (kg·K2/kJ) α (kg·K2/kJ) cP (kJ/kg·K)

375.00 350.62 301.95 300.00 1.4339 1.9469 172.25 267.36 0.17849 −0.84331 1.9181

375.00 349.21 304.66 300.00 3.4311 4.6650 174.17 258.50 0.18924 −0.95167 1.7961

375.00 347.14 308.16 300.00 6.1307 8.1635 177.11 250.43 0.20149 −1.0789 1.6746

375.00 344.67 312.17 300.00 9.5671 12.167 180.30 242.78 0.21562 −1.2550 1.5552

375.00 342.22 316.24 300.00 13.536 16.243 182.78 235.15 0.23213 −1.5408 1.4391

375.00 340.19 319.99 300.00 17.555 19.988 183.75 227.26 0.25171 −2.0339 1.3270

375.00 339.08 322.95 300.00 20.892 22.946 182.60 218.97 0.27495 −2.9277 1.2190

375.00 339.96 324.21 300.00 22.319 24.209 178.54 209.97 0.30213 −4.7424 1.1155

375.00 346.09 321.07 300.00 18.822 21.075 169.39 199.15 0.33571 −9.2161 1.0179

R227ea/R245fa Proportion of higher temperature component TH1 (K) TH2 (K) TL1 (K) TL2 (K) TG_e (K) TG_c (K) re (kJ/kg) rc (kJ/kg) β (kg·K2/kJ) α (kg·K2/kJ) cP (kJ/kg·K)

0.1 375.00 372.93 302.05 300.00 0.54778 1.9454 41.285 118.70 0.25518 3.2990 1.4216

0.2 375.00 370.44 304.51 300.00 1.5483 4.2247 57.366 127.51 0.25344 2.4556 1.3944

0.3 375.00 367.96 306.84 300.00 2.7834 6.3850 71.346 136.55 0.25183 2.1559 1.3826

0.4 375.00 365.89 308.66 300.00 4.0061 8.0905 84.069 145.50 0.25029 1.9924 1.3787

0.5 375.00 364.53 309.80 300.00 4.9643 9.1694 95.587 154.19 0.24880 1.8898 1.3801

0.6 375.00 364.10 310.16 300.00 5.4486 9.5213 105.84 162.52 0.24734 1.8263 1.3856

0.7 375.00 364.76 309.63 300.00 5.3023 9.0353 114.74 170.39 0.24587 1.7937 1.3944

0.8 375.00 366.65 308.02 300.00 4.4066 7.5345 122.24 177.66 0.24437 1.7894 1.4062

0.9 375.00 369.98 305.01 300.00 2.6647 4.7115 128.23 184.08 0.24285 1.8159 1.4205

R1234ze/R141b Proportion of higher temperature component TH1 (K) TH2 (K) TL1 (K) TL2 (K) TG_e (K) TG_c (K) re (kJ/kg) rc (kJ/kg) β (kg·K2/kJ) α (kg·K2/kJ) cP (kJ/kg·K)

0.1 375.00 361.47 309.79 300.00 6.0524 9.7923 108.37 181.62 0.22791 −24.032 1.5162

0.2 375.00 354.49 315.67 300.00 11.766 15.541 134.68 193.38 0.23509 10.989 1.4481

0.3 375.00 349.77 319.24 300.00 15.864 18.957 153.71 202.61 0.24231 5.7209 1.3989

0.4 375.00 347.52 321.05 300.00 18.218 20.684 167.21 210.03 0.24954 4.4786 1.3609

0.5 375.00 347.19 321.41 300.00 18.979 21.001 176.76 215.96 0.25621 4.0355 1.3295

0.6 375.00 348.59 320.40 300.00 18.254 20.003 183.21 220.57 0.26256 3.9293 1.3027

0.7 375.00 351.74 318.01 300.00 16.072 17.668 186.99 223.91 0.26899 4.0580 1.2792

0.8 375.00 356.86 314.11 300.00 12.385 13.846 188.32 225.95 0.27571 4.4550 1.2584

0.9 375.00 364.39 308.34 300.00 7.0801 8.2004 187.20 226.52 0.28288 5.3396 1.2399

(continued on next page)

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Table A (continued) R152a/R11 Proportion of higher temperature component

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

R21/R113 Proportion of higher temperature component TH1 (K) TH2 (K) TL1 (K) TL2 (K) TG_e (K) TG_c (K) re (kJ/kg) rc (kJ/kg) β (kg·K2/kJ) α (kg·K2/kJ) cP (kJ/kg·K)

0.1 375.00 351.47 302.00 300.00 1.9070 1.9983 184.32 221.52 0.32066 −1.7105 1.0865

0.2 375.00 352.92 304.12 300.00 3.8375 4.1171 177.73 214.01 0.32823 −2.2080 1.0724

0.3 375.00 354.50 306.32 300.00 5.7796 6.3158 171.40 206.73 0.33309 −3.0950 1.0584

0.4 375.00 356.35 308.50 300.00 7.6840 8.4992 165.25 199.66 0.33823 −5.1898 1.0445

0.5 375.00 358.76 310.52 300.00 9.4469 10.524 159.18 192.74 0.34370 −16.732 1.0307

0.6 375.00 358.94 312.29 300.00 10.958 12.196 154.39 185.88 0.34958 13.764 1.0155

0.7 375.00 357.56 313.51 300.00 11.905 13.222 150.22 178.96 0.35595 4.8829 0.99949

0.8 375.00 357.83 313.53 300.00 11.617 13.040 145.00 171.68 0.36290 2.9668 0.98438

0.9 375.00 361.67 310.85 300.00 8.7660 10.272 137.52 163.28 0.37054 2.1498 0.97111

Table B The detailed data of the zeotropic working fluids and the key temperature of R-ORC. R152a/R11 Proportion of higher temperature component

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

TH1 (K) TH2 (K) TH2′’ (K) TL1 (K) TL1′ (K) TL2 (K) TG_e (K) TG_c (K) re (kJ/kg) rc (kJ/kg) β (kg·K2/kJ) α (kg·K2/kJ) cP (kJ/kg·K)

375.00 350.43 350.62 301.95 301.95 300.00 15.030 1.9469 172.25 267.36 0.17849 −0.57697 1.9181

375.00 349.63 349.21 304.55 304.66 300.00 16.372 4.6650 174.17 258.50 0.18924 −0.43119 1.7961

375.00 348.35 347.14 307.76 308.16 300.00 18.315 8.1635 177.11 250.43 0.20149 −0.48311 1.6746

375.00 346.69 344.67 311.30 312.17 300.00 20.695 12.167 180.30 242.78 0.21562 −0.57198 1.5552

375.00 345.08 342.22 314.78 316.24 300.00 23.178 16.243 182.78 235.15 0.23213 −0.71305 1.4391

375.00 343.94 340.19 317.81 319.99 300.00 25.379 19.988 183.75 227.26 0.25171 −0.93305 1.3270

375.00 343.30 339.08 320.26 322.95 300.00 26.795 22.946 182.60 218.97 0.27495 −1.1187 1.2190

375.00 344.15 339.96 321.35 324.21 300.00 26.403 24.209 178.54 209.97 0.30213 −1.3928 1.1155

375.00 349.00 346.09 318.97 321.07 300.00 21.375 21.075 169.39 199.15 0.33571 −1.7213 1.0179

R227ea/R245fa Proportion of higher temperature component TH1 (K) TH2 (K) TH2′’ (K) TL1 (K) TL1′ (K) TL2 (K) TG_e (K) TG_c (K) re (kJ/kg) rc (kJ/kg) β (kg·K2/kJ) α (kg·K2/kJ) cP (kJ/kg·K)

0.1 375.00 373.05 372.93 301.91 302.05 300.00 0.5478 1.9454 41.285 118.70 0.25518 1882.7 1.4216

0.2 375.00 370.93 370.44 303.98 304.51 300.00 1.5483 4.2247 57.366 127.51 0.25344 272.93 1.3944

0.3 375.00 368.35 367.96 306.41 306.84 300.00 2.7834 6.3850 71.346 136.55 0.25183 285.80 1.3826

0.4 375.00 366.37 365.89 308.15 308.66 300.00 4.0061 8.0905 84.069 145.50 0.25029 8.3892 1.3787

0.5 375.00 365.06 364.53 309.25 309.80 300.00 4.9643 9.1694 95.587 154.19 0.24880 1302.7 1.3801

0.6 375.00 364.64 364.10 309.62 310.16 300.00 5.4486 9.5213 105.84 162.52 0.24734 297.54 1.3856

0.7 375.00 365.25 364.76 309.13 309.63 300.00 5.3023 9.0353 114.74 170.39 0.24587 583.74 1.3944

0.8 375.00 367.04 366.65 307.62 308.02 300.00 4.4066 7.5345 122.24 177.66 0.24437 2339.1 1.4062

0.9 375.00 370.21 369.98 304.77 305.01 300.00 2.6647 4.7115 128.23 184.08 0.24285 880.68 1.4205

R1234ze/R141b Proportion of higher temperature component TH1 (K) TH2 (K) TH2′’ (K) TL1 (K) TL1′ (K) TL2 (K) TG_e (K) TG_c (K) re (kJ/kg) rc (kJ/kg) β (kg·K2/kJ) α (kg·K2/kJ) cP (kJ/kg·K)

0.1 375.00 362.31 361.47 309.10 309.79 300.00 6.6630 9.7923 108.37 181.62 0.22791 1276.3 1.5162

0.2 375.00 356.78 354.49 313.85 315.67 300.00 11.766 15.541 134.68 193.38 0.23509 2306.8 1.4481

0.3 375.00 353.25 349.77 316.54 319.24 300.00 15.864 18.957 153.71 202.61 0.24231 5259.8 1.3989

0.4 375.00 351.56 347.52 317.94 321.05 300.00 18.218 20.684 167.21 210.03 0.24954 2869.4 1.3609

0.5 375.00 351.14 347.19 318.37 321.41 300.00 18.979 21.001 176.76 215.96 0.25621 2036.7 1.3295

0.6 375.00 352.12 348.59 317.68 320.40 300.00 18.254 20.003 183.21 220.57 0.26256 7649.0 1.3027

0.7 375.00 354.36 351.74 315.98 318.01 300.00 16.072 17.668 186.99 223.91 0.26899 2908.1 1.2792

0.8 375.00 358.56 356.86 312.77 314.11 300.00 12.385 13.846 188.32 225.95 0.27571 2102.0 1.2584

0.9 375.00 365.04 364.39 307.81 308.34 300.00 7.0801 8.2004 187.20 226.52 0.28288 1667.1 1.2399

(continued on next page)

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Table B (continued) R152a/R11 Proportion of higher temperature component

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

R21/R113 Proportion of higher temperature component TH1 (K) TH2 (K) TH2′’ (K) TL1 (K) TL1′ (K) TL2 (K) TG_e (K) TG_c (K) re (kJ/kg) rc (kJ/kg) β (kg·K2/kJ) α (kg·K2/kJ) cP (kJ/kg·K)

0.1 375.00 351.42 351.47 302.00 302.00 300.00 17.503 1.9983 184.32 221.52 0.32066 −1.1452 1.0865

0.2 375.00 353.11 352.92 304.07 304.12 300.00 16.163 4.1171 177.73 214.01 0.32823 −1.2219 1.0724

0.3 375.00 355.10 354.50 306.12 306.32 300.00 14.751 6.3158 171.40 206.73 0.33309 −1.1348 1.0584

0.4 375.00 357.09 356.35 308.14 308.50 300.00 13.156 8.4992 165.25 199.66 0.33823 −1.2251 1.0445

0.5 375.00 359.51 358.76 310.02 310.52 300.00 11.192 10.524 159.18 192.74 0.34370 22,032 1.0307

0.6 375.00 360.07 358.94 311.41 312.29 300.00 10.958 12.196 154.39 185.88 0.34958 2783.7 1.0155

0.7 375.00 359.25 357.56 312.18 313.51 300.00 11.905 13.222 150.22 178.96 0.35595 2301.9 0.99949

0.8 375.00 359.10 357.83 312.53 313.53 300.00 11.617 13.040 145.00 171.68 0.36290 2299.0 0.98438

0.9 375.00 362.19 361.67 310.44 310.85 300.00 8.7660 10.272 137.52 163.28 0.37054 589.47 0.97111

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