Second-law-based analysis of vapor-compression refrigeration cycles: Analytical equations for COP and new insights into features of refrigerants

Energy Conversion and Management 138 (2017) 426–434

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Second-law-based analysis of vapor-compression refrigeration cycles: Analytical equations for COP and new insights into features of refrigerants Weiwu Ma, Song Fang, Bo Su, Xinpei Xue, Min Li ⇑ School of Energy Science and Engineering, Central South University, Changsha 410083, China

a r t i c l e

i n f o

Article history: Received 12 November 2016 Received in revised form 7 February 2017 Accepted 8 February 2017

Keywords: Vapor-compression refrigeration cycle Heat pump Second-law analysis Entropy generation analysis Coefficient of performance

a b s t r a c t This article reports a second-law-based analysis of the vapor-compression refrigeration cycle, which leads to a set of explicit theoretical formulas for the coefficient of performance (COP). These analytical expressions provide a fast and accurate approach to computer simulations of the vapor-compression cycle without recourse to thermodynamic diagrams or equations of state. The second-law-based analysis yields specific expressions for the entropy generations of irreversible processes, enabling us to evaluate the thermodynamic features of the refrigerant and to elucidate the thermodynamic mechanisms behind the effects of the cycle processes, including superheat, subcooling, and throttling processes. In particular, these processes can interact, therefore this paper presents a global entropy generation analysis for evaluating the impact of the interacted processes on COP. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction The vapor-compression (VC) refrigeration cycle is a classic topic in the field of engineering thermodynamics. It is also the most used refrigeration and heat pump cycle. In particular, the VC chillers and heat pumps have been widely used in various large-scale applications. Two good examples of these are the combined heating, cooling & power (CCHP) systems [1,2] and ground-coupled heat pump (GCHP) systems [3–6]. One of central issues in these large-scale systems is how to efficiently calculate the coefficient of performance (COP) of the VC units, which is directly related to design, analysis, and simulation of VC units and the whole systems. A CCHP system is a complicated distributed energy system that involve a diverse spectrum of equipment and subsets, including power generators (combustion gas turbines, internal combustion engines, etc.), waste heat recovery units (absorption chillers, heat exchangers, etc.), and VC chillers and heat pumps [7,8]. CCHP systems are generally designed for one or a group of buildings. So, in this case, the high-frequency perturbations in the systems may be attenuated and delayed by some subsets, such as building envelopes, outdoor pipelines, and thermal energy storage equipment. Therefore, the characteristic time of a typical CCHP system is large compared with that of VC chillers and heat pumps. Similar ⇑ Corresponding author at: Energy Building, School of Energy Science and Engineering, Central South University, Changsha, Hunan 410083, China. E-mail address: [email protected] (M. Li). http://dx.doi.org/10.1016/j.enconman.2017.02.017 0196-8904/Ó 2017 Elsevier Ltd. All rights reserved.

situations also occur in GCHP systems. A GCHP system is a heat pump system that use the ground as the heat reservoir by ground heat exchangers [3,5]; the time constant of the GCHP systems are dominated mainly by the ground heat exchangers, which is on the order-of-magnitude from several hours to decades [9,10]. To sum up, it is generally acceptable to ignore the transient processes within VC units when designing or analyzing these and other similar large-scale systems. And static thermodynamic models should be sufficiently accurate for these engineering problems. Static models for the VC cycles can be divided into empirical and theoretical groups. Empirical models may contain several empirical constants [11], which are determined by fitting experimental data to the models for a particular chiller or heat pump. Empirical models are widely used because of their simplicity [12–15]; but they provide little insight into the underlying thermodynamic processes, making them difficult to generalize. On the other hand, COP can be calculated by enthalpy balance equations in conjunction with a complete and accurate property diagrams or equations of state [16–18]. This approach is very general but can be computationally intensive and complicated if highly accurate equations of state are involved [19]. Although one can develop by curve-fitting simplified empirical property relations for common refrigerants [20,21], the difficulty will be insuperable if complete experimental data is unavailable for new refrigerants. In this context, accurate analytical equations for COP will be very desirable; they can provide an easy-use basis for analysis, simulation, and optimum design of vapor-compression refrigeration cycles.

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Nomenclature COP T h

g

c r s q w b

coefficient of performance temperature (K) enthalpy (J/kg) isentropic efficiency of compressor specific heat (J/(kg K)) heat of evaporation (J/kg) specific entropy (J/K s) heat flow (J/s) input power (W) mean volumetric expansivity

Subscripts i state point (1, 10 , 2, 2s, 200 . . .) R refrigerating cycle

It is well known that the COP of Carnot refrigerator, ignoring all irreversible losses, is simple and gives only the theoretical limit of the performance. Thus, a large number of researchers have studied various endoreversible Carnot refrigerators using finite-time thermodynamics [22], which can incorporate irreversible losses associated with finite temperature-difference heat transfer and heat leakage from the heat sink to the heat source [22–25]. Although this approach can yield analytical expressions for COP, it fails to deal with all internal irreversible processes associated with the feature of the working fluid. In order to deal with the internal loss, Shelton and Grossmann developed a simple analytical equation for calculating COP of VC refrigeration cycle based on the idea-gas assumption and several fundamental thermodynamic relations [26]. Moreover, Alefeld derived several analytical expressions for COP of refrigerators and heat pumps by explicitly using the second law of thermodynamics [27]. The use of the second law enable all possible irreversible losses to be considered. But, Alefeld’s model ignored superheating in the evaporator and sub-cooling in the condenser, which may exert significant influence on the performance of the VC cycle. To the best knowledge of the authors, a theoretical COP equation is still unavailable for the VC cycles involving superheat in the evaporator and/or subcooling in the condenser. The purpose of this work is to develop theoretical COP equations for vapor-compression refrigeration cycles based on a second-law-based analysis. Our second-law analysis not only addresses superheating and sub-cooling processes but also provides more rigorous mathematical derivation than that reported by Alefeld. These analytical expressions are accurate, precise, and applicable in a general sense because few assumptions are used in the analysis. The theoretical expressions demonstrate that in conjunction with the condensing and the evaporating temperatures, COP depends upon some properties of the refrigerant easy to determine: specific heat capacities of the liquid and vapor states and the latent heat of vaporization at the evaporating temperature. The second-law analysis leads to global entropy-generation analysis that results in quantitative estimation of irreversible losses associated with the thermodynamic processes and eventually guides us to elucidating the thermodynamic mechanisms behind the effects of these processes, including superheat, subcooling, and throttling processes.

H C p m in out l h l; 1 gen; i 1  10 2  2s

heating cycle Carnot cycle pressure mean input output low-temperature heat-carrier fluids high-temperature heat-carrier fluids 1–10 process generation of ith process process from 1 to 10 process from 2 to 2s

Fig. 1. Pressure-Enthalpy diagram for a vapor-compression refrigeration cycle.

indicated, but we should be aware of that they differ from the evaporating temperature T 1 and the condensing temperature T 3 due to the irreversibility of the heat transfer processes. Conventionally, the coefficient of performance (COP) of the vapor-compression refrigeration cycle COPR can be calculated by [28]

COP R ¼ gi

h10  h4 h2s  h10

where gi is the isentropic efficiency of the compressor, and hi denotes the enthalpies of the state points i (i = 1, 2, . . .), as labeled in Fig. 1. As discussed above, Alefeld derived several analytical expressions for the efficiencies of refrigerators and heat pumps by using the first and the second laws of thermodynamics. According to Alefeld [27], COP can be calculated by




Fig. 1 shows a typical VC refrigeration cycle in a pressureenthalpy diagram. In Fig. 1, the temperatures of the heat-carrier fluids flowing through the evaporator and the condenser are not

gi gc 1  cTr 1



COP R ¼ 1 þ 21g

2. Model development

ð1Þ

c

cT 1 r




1

gc

1

gc




2  c T T 3 T1  1 þ pr 1 TT2s3 T T3 1

ð2Þ

where gc ¼ T 1 =ðT 3  T 1 Þ is the COP of the Carnot refrigerator operating between T 1 and T 3 ; r is the heat of evaporation of the working fluid at the evaporating temperature. The corresponding efficiency of heating, COPH , is

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COP H ¼ 1 þ COP R

ð3Þ

Eq. (2) reveals that COPR depends on three refrigerant-related nondimensional variables: cT 1 =r; cp T 1 =r, and ðT 2s  T 3 Þ=ðT 3  T 1 Þ. For some refrigerants, the irreversibility of the desuperheating process 2s  2 is insignificant. Ignoring this part yields




COP R ¼

gi gc 1  cTr 1 1 þ 21g

c

cT 1 r




1

gc

1



gc

1



ð4Þ

The analytical expression developed by Shelton and Grossmann is [26]

COP R ¼ gi gc

r  cðT 3  T 1 Þ r

ð5Þ

Eqs. (2) and (5) are analytical theoretical equations for COP. They are easy and convenient to use compared to Eq. (1) because Eq. (1) requires a high precision pressure-enthalpy chart or an equation of state for computing enthalpies of the refrigerant. 2.1. New analytical expressions for COP This section presents new analytical formulas for COP that extend Eq. (2) by involving subcooling in the condenser and superheating in the evaporator. The theoretical derivation is similar to Alefeld’s work but more mathematically rigorous. The following second-law analysis frequently uses the entropic (or thermodynamic) average temperature, which is defined as

1 1 ¼ Tm q

Z

2

1

cp dT T

ð6aÞ

  q41 q110 q110 q110 q q 0 þ  þ  41 þ 11 T1 T1 T1 T 110 T l;1 T l;2 !   Z 1 Z 10 qin q110 q110 cp cp    dT þ dT ¼ T1 T1 T 110 T T 1 4

h2  h1 s2  s1

ð6bÞ

Applying the first and the second laws of thermodynamics to the reverse Rankine cycle yields

qout ¼ qin þ win

ð7aÞ

n X q q sgen;i ¼ out  in T Tl h i¼1

ð7bÞ

where T l and T h are entropic average temperatures of the lowtemperature and the high-temperature heat-carrier fluids. Eq. (7a) is substituted for qout in Eq. (7b), giving

  n X Th  Tl qin þ T h sgen;i win ¼ Tl i¼1

ð8Þ

The following is to determine the entropy generations sgen;i to develop an analytical expression for COP. First, the thermal process in the evaporator is divided into two stages: the first stage is the evaporating process from the state 4 to state 1 (Fig. 1), and the second stage is the superheating process 1–10 . Accordingly, the entropy generation due to heat transfer in the evaporator sgen;e consists of two parts:



sgen;e ¼ q41

1 1  T 1 T l;1





þ q110

1

T 110

1  T l;2



ð9Þ

where q41 and q110 denote the heat exchanged during 4–1 and 1–10 processes, respectively; T l;1 and T l;2 denote the corresponding entropic average temperatures of the heat-carrier fluid during these processes; T 110 is the average temperature of the refrigerant during process 1–10 ; T 1 is the evaporating temperature. Eq. (9) can be rewritten as the following equivalent form

ð11Þ

The derivation of Eq. (11) uses Eq. (6a). According to the additivity of definite integral, the last two terms of Eq. (11) are equal to

qin ¼ Tl

Z

10

4

cp dT ¼ T

Z

1 4

cp dT þ T

Z

10

1

cp dT T

ð12Þ

Eq. (11) can reduce to Eq. (13) by substituting Eq. (12) for the last two terms:

sgen;e ¼

    qin qin q 0 q 0   11  11 T1 Tl T1 T 110 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl ffl} Sgen;1

ð13Þ

Sgen;2

where the minus sign before sgen;2 implies that superheating in the evaporator results in an decrease in irreversible loss. This is because the superheat can decrease the temperature difference between the refrigerant and the heat-carrier fluid. Next, the entropy generation during the compression process 10 -2 is determined by

sgen;3 ¼ s2  s10

ð14Þ

Then, similar to the evaporating process, the condensing process is considered as a four-stage process, including 2-2s, 2s-200 , 200 -3, and 3–30 . The corresponding entropy generation becomes



   1 1 1 1 þ q2s200   T h;1 T 22s T h;2 T 2s200     1 1 1 1 þ q200 3 þ q330   T h;3 T 3 T h;4 T 330

sgen;c ¼ q22s

or

Tm ¼

ð10Þ

sgen;e ¼

ð15Þ

where q22s ; q2s200 ; q200 3 , and q330 denote the heat transferred during 2-2s, 2s-200 , 200 -3, and 3–30 processes, respectively; T h;1 ; T h;2 ; T h;3 , and T h;4 are the entropic average temperatures of the heat-carrier fluid during these processes. T 22s ; T 2s200 , and T 330 denote the average temperatures of the working fluid during 2–2s; 2s–200 , and 3–30 processes, respectively; T 3 is the condensing temperature. Eq. (15) also has the following equivalent form  sgen;c ¼

       00 00 qout qout q q q q q 0 q 0  þ 22s  22s þ 2s2  2s2  33  33 Th T3 T3 T 22s T3 T 2s200 T 330 T3 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflffl ffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} |fflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflffl ffl} Sgen;4

Sgen;5

Sgen;6

Sgen;7

ð16Þ

Like sgen;2 , minus sign occurs before sgen;7 . Finally, the entropy generation of the throttling process can be written as

sgen;8 ¼ s4  s30 ¼ s4  s4f þ s4f  s3 þ s3  s30

ð17Þ

We substitute sgen;1 and sgen;4 into Eq. (8) and get

win ¼

  X T3  T1 qin þ T 3 sgen;i T1

ð18Þ

Sum of sgen;3 and sgen;5 is

sgen;3 þ sgen;5 ¼ s2  s10 þ

q22s 1  gi  ðs2  s2s Þ ¼ win T3 T3

ð19Þ

And substituting Eq. (19) into Eq. (18) yields

win ¼

  qin T 3  T 1 T3 X þ sgen;i gi T1 gi

ð20Þ

Furthermore, sgen;2 ; sgen;6 , and sgen;8  sgen;7 can be written as follows by using the two-term series expansion lnð1 þ xÞ  x  x2 =2:

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h10  h1 T 0  T1 T 0  ðs10  s1 Þ ¼ cp 1  ln 1 T1 T1 T1  2 cp;1 T 10  T 1  2 T1

ð21Þ

sgen;2 ¼

sgen;6 ¼

h2s  h200  T3

Z

ð22Þ

where the definitions

 2 cp dT cp T 2s  T 3  T 2 T3

2s 00

2

ð23Þ

  h3  h30 sgen;8  sgen;7 ¼s3  s30 þ s4f  s3 þ s4  s4f  s3  s30  T3 T 1 h4  h4f h3  h30 ¼c ln þ þ T T1 T3 " 3 # 2 1 T3  T1 ðT 3  T 30 ÞðT 3  T 1 Þ c  2 T1T3 T1

ð24Þ

qin

gi gc

( 1þ

ð26Þ

" #)  2  2 T 3  T 1 cp T 1 T 2s  T 3 T 3  T 30 cp;1 T 3 T 10  T 1 cT 3  2c  þ 2qin T3 T3  T1 T3  T1 T1 T3  T1 T1

According to the definition of COP of the refrigeration cycle, we get

1þ

T 3 T 1 2qin

 cp T 1 T3




T 2s T 3 T 3 T 1

2

gi gc T T

 2c T33 T310 

cp;1 T 3 T1




T 10 T 1 T 3 T 1

2

If we ignore the superheating process, Eq. (31) can be simplified as

8 91 cp T 1 T 1 2 cT 1 cT 3 = < s  2 1 þ r T3 r h ir COP R ¼ gi gc 1 þ cðT T Þ : ; 2 g 1  30 1

ð32Þ

r

ð25Þ

ð27Þ

COP R ¼

ð31Þ

T 1 T T r ¼ TT130T ; 1 ¼ T33 T310 ; s ¼ br  1 are used. cp 1

c

Here, c is the average specific heat of the liquid refrigerant within the temperature range from T 1 to T 3 ; cp and cp;1 are the average specific heats of the refrigerant vapor within the temperature ranges, respectively, from T 3 to T 2s and from T 1 to T 10 . In most applications, cp;1 can be well approximated by the specific heat of the vapor at state 1. Substituting Eqs. (22), (23) and (26) for sgen;2 ; sgen;6 , and sgen;8  sgen;7 in Eq. (20) leads to win ¼

8 91 c T cp T 1 T 1 < ðr þ sÞ2  21 cTr 1  r2 p;1r 3 þ cTr 3 = r T3 h i COP R ¼ gi gc 1 þ c ðT 0 T Þ cðT T Þ : ; 2gc 1  30r 1 þ p;1 1r 1

 þ cTT 13 ð28Þ

Since we have

qin  r  cðT 30  T 1 Þ þ cp;1 ðT 10  T 1 Þ

ð29Þ

T 2s  T 3 T 10  T 1 br ¼ þ 1 T3  T1 T 3  T 1 cp

ð30Þ

where b is the mean volumetric expansivity of the working fluid. Substituting Eqs. (29) and (30) into Eq. (28) yields

3. Results and discussion To validate Eq. (31), we compare Eqs. (31) with (1) and (2) using eight working fluids (R600a, R134a, R124, R600, R22, R290, R152a and R717). Eq. (1) is used together with NIST standard reference fluid thermodynamic and transport properties [19]. Thus Eq. (1) should be thought as a benchmark against which the accuracy of Eq. (2) and the new expressions is measured. Figs. 2–5 illustrate how COPs vary with the evaporating temperature (Fig. 2), with the condensing temperature (Fig. 3), with the subcooling degree (Fig. 4), and with the superheating degree (Fig. 5). The calculated COPs by Eqs. (1), (2) and (31) are shown in these figures for comparison. Eq. (2) is compared only in Figs. 2 and 3 because Eq. (2) ignores the superheat and the subcooling. The range of the evaporating temperature is so determined that the lower limits of the evaporating temperature are higher than the normal boiling points of the fluids. The condensing temperatures are chosen according to the evaporating temperatures, ensuring that the temperature differences between the condensing and the evaporating temperatures are about 25–50 °C. Whereas this choice is somewhat arbitrarily, it is enough for the purpose of validation. As shown in Figs. 2–5, Eq. (31) matches Eq. (1) very well, and it has a high degree of accuracy compared to Eqs. (2) and (5). The advantage of Eqs. (2) and (5) is the simple form, especially Eq. (5). In fact, except for dry fluids R717 and R22, the maximum relative errors of Eq. (31) for other refrigerants are within 3%. The maximum relative errors of R717 and R22 are approximately 5%. The varying ranges of the abscissas are large enough compared with the practice; so, it is rational to conclude that the accuracy of Eq. (31) is acceptable for engineering applications.

Fig. 2. Comparisons of Eqs. (31), (1), (2), and (5) (COP vs. evaporating temperature).

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Fig. 3. Comparisons of Eqs. (31), (1), (2), and (5) (COP vs. condensing temperature).

Fig. 4. Comparisons of Eqs. (31), (5), and (1) (COP vs. subcooling).

Fig. 6 compares Eq. (32) with Eq. (1). Eq. (32) is the COP equation ignoring the entropy generation due to the superheat in the evaporator. As an example, 10  C superheat is used. As shown in Fig. 6, the discrepancies between Eqs. (32) and (1) are small, and the relative errors are approximately 4%. This implies that superheat within 10  C has insignificant impact on the calculation COP for the eight common refrigerants.

Eq. (31) differs from Eqs. (1) in two aspects, which are responsible for the discrepancy shown in Figs. 2–5. The first source of the errors is a result of the power series lnð1 þ xÞ  x  x2 =2 used in the derivation of Eqs. (21)–(24). To improve the accuracy, ones can use the three-term series (i.e.,x  x2 =2 þ x3 =3) for lnð1 þ xÞ. But the final equations will be more complicated than Eq. (31). Considering the accuracy illustrated in Figs. 2–5, the three-term

W. Ma et al. / Energy Conversion and Management 138 (2017) 426–434

431

Fig. 5. Comparisons of Eqs. (31), (5), and (1) (COP vs. superheating).

series seems to be unnecessary. The second source is due to the average values for c; cp ; cp;1 and b used in the analytical equations. For refrigeration circles with large superheating or subcooling, it is difficult to determine the accurate average specific heats cp and cp;1 . We used the arithmetic averages for cp and cp;1 in the validation study; the accuracy is satisfactory, validating that the theoretical expressions are robust to the heat capacities. Explicit expressions of COP are a powerful tool for the simulation and simulation-based optimum design of refrigeration and heat pump systems. In general, COP of the refrigeration cycles is calculated by the enthalpy balance equation, in conjunction with a property diagram or a state equation. Consequently, COP is the result of the division of two enthalpy differences (Eq. (1)). The enthalpies are relatively large so that they must be computed with higher accuracy than required for COP [27]. As a result, the enthalpies must be determined from highly accurate equations of state (e.g., state equations consisting of a huge number of substance specific parameters). These equations require great computational resources and intensive programming work. By contrast, the analytical expressions for COP are explicit functions of several dimensionless variables in the same order-ofmagnitude of COP; thus the calculated COP is relatively insensitive to the uncertainties of the input parameters on the one hand, the analytical solutions can eliminate some iterative computation and thus simplify computation and programming on the other. These features could be very desirable when modeling largescale energy systems involving chillers and heat pumps, such as distributed energy systems [1,2], and ground-coupled heat pump systems [3,4].

None of the derived analytical solutions is an explicit function of the pressures of the working fluids. The effect of operating pressures is implicitly included in the dimensionless temperatures T 3 =T 1 or ðT 2s  T 3 Þ=ðT 3  T 1 Þ. The ratios of temperatures determine the ratios of pressures for a given fluid. If we approximate the fluid as an ideal gas during the isentropic expansion process, we can also derive analytical expressions comprising the pressure ratios by substituting ðP 3 =P 1 Þk1=k for T 2s =T 10 . 4. Entropy generation analysis The methodology developed in this paper not only provides the analytical expressions for COP but also throw new light on the impact of working fluids on the thermodynamic performance of the vapor-compression refrigeration cycle. As revealed by Eq. (20), the performance of the cycle depends on quantities qin ; sgen;2 ; sgen;6 ; sgen;7 and sgen;8 , which are correlated directly with the substance properties r; c; cp ; cp;1 and b. In conjunction with the operating temperatures, these parameters form a group of dimensionless variables characterizing the working fluid. Among these dimensionless variables, r; cp;1 T 3 =r and cp;1 ðT 10  T 1 Þ=r are associated with the entropy generation of superheat sgen;2 . cp T 1 =r; r and s are related with the entropy generation during desuperheating process sgen;6 . Variables 1; cT 1 =r and cðT 30  T 1 Þ=r are linked with the entropy generation during subcooling sgen;7 . cT 1 =r is relating to the entropy generation of the throttling process sgen;8 . More specifically, sgen;2 is proportional to cp;1 ; sgen;6 increases with the heat capacity and the volumetric

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Fig. 6. Comparisons of Eqs. (32), (5), and (1) (COP vs. evaporating temperature).

expansivity of the vapor (cp and b); sgen;7 and sgen;8 are proportional to the heat capacity of the saturated liquid c. The performance of the cycle depends directly upon the total entropy generation. The contribution of each entropy generation to the overall entropy generation reflects the impact of the individual process on COP, which may be negatively or positively. As shown in Eqs. (16) and (17), sgen;6 and sgen;8 contribute positively to the total entropy generation, whereas sgen;7 contribute negatively to the overall entropy generation. These equations explain why the desuperheating and the throttling processes lead to decreases in COP and why an increase in subcooling leads to an increase in COP. It is well known that the impact of the superheat in the evaporator depends on the nature of the refrigerant. Increasing superheat may result in a decrease (e.g., R717 and R22) or an increase (e.g., R600a and R134a) in the performance. But, no satisfactory analysis is available in the literature to explain the underlying thermodynamic mechanism. The superheat in the evaporator exerts a global influence on the vapor-compression cycle by interacting with the desuperheating in the condenser. Thus, it is necessary to perform a global entropy generation analysis to evaluate the contribution of the superheat to the overall entropy generation. As shown in Eqs. (22), (23) and (30), the superheat degree is associated with the entropy generations of both the superheating and the desuperheating processes (i.e., sgen;2 and sgen;6 ). An increase in the superheat contributes to a decrease in the loss in the evaporator (there is a minus sign before sgen;2 in Eq. (13)), while it can increase the loss in the condenser because sgen;6 in the condenser increases with the superheat in the evaporator. Therefore, we need to calculate sgen;6  sgen;2 to identify the overall effect of the superheat (Fig. 7). The overall effect depends on the increasing rates of

sgen;6 and sgen;2 with the superheat degree. More precisely, it depends on the combined dimensionless entropy generation T 1 ðsgen;6  sgen;2 Þ=qin . As shown in Fig. 8, when T 1 ðsgen;6  sgen;2 Þ=qin decreases with the superheat, COP increases; on the contrary, when T 1 ðsgen;6  sgen;2 Þ=qin increases with superheat, COP decreases with superheat. The results are derived from the second law of thermodynamics; thus, it seems that the results provide a reliable explanation for the impact of the superheat on COP of the VC refrigeration cycle.

Fig. 7. Variations of sgen;6  sgen;2 with superheating temperature.

W. Ma et al. / Energy Conversion and Management 138 (2017) 426–434

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based optimum design of refrigeration and heat pump systems without recourse to high-accurate equations of state. Furthermore, a global entropy generation analysis is also performed; and the key findings from this analysis are listed below: 1. It appears that the effect of superheat on COP can be quantitatively estimated by dimensionless entropy generation T 1 ðsgen;6  sgen;2 Þ=qin ; 2. Dimensionless variable cT 1 =r can be used as a key parameter characterizing working fluids. COP generally decreases with increasing cT 1 =r, implying that cT 1 =r can be a property indicator of comparing refrigerants; 3. The impact of operating pressures on COP is implicitly included in the dimensionless temperature T 3 =T 1 or ðT 2s  T 3 Þ=ðT 3  T 1 Þ.

Fig. 8. COP of different fluids under different superheating temperature.

Finally, and perhaps most importantly, all the findings are drawn from a second-law-based analysis; so the methodology used here should have very general applicability to other thermodynamic problems. Acknowledgement

Analytical expressions show that c=r is a key parameter characterizing the working fluid. In particular, Eq. (4) indicates that COP is a function of dimensionless variables gc ; gi and cT 1 =r. Among them, cT 1 =r is the only one characterizing the influence of the refrigerant. In fact, Shelton and Grossmann have pointed out that cT 1 =r allows quick comparison of refrigerants and straightforward estimation of cycle performance [26]. This view is further supported by the findings of this work. What is more, our analysis showed that cT=r is also a key quantity characterizing working fluids for organic Rankine power cycles [29]. It is important to remember that c=r can be easily determined by a throttling experiment whereby the ratio of vapor to liquid after throttling is measured. Therefore, it could be argued that a throttling experiment measuring c=r is probably enough for evaluating the thermodynamic performance of new working fluids. Compared to this workload, constructing experimentally validated equations of state or property diagrams is highly intensive for a new working substance.

5. Conclusions To date, the vapor-compression refrigeration cycle is proclaimed as the most developed thermodynamic cycle. Yet, while building a wide knowledge of impacts of the cycle processes on COP that are definitely related with the refrigerant, the underlying mechanisms of these impacts still need a satisfactory explanation from the viewpoint of thermodynamics. It is common that some thermodynamic processes in a system have a global influence, i.e., the processes are correlated with others. The superheat in the evaporator is a case in point, which is correlated with the desuperheating process in the condenser. This kind of global influence retard a thorough understanding of the impact of individual processes on the cycle performance. Moreover, a fast approach to the calculation of COP is central to many large-scale applications of VC chillers and heat pumps, as used in CCHP and GCHP systems. It is highly desirable for us to have explicit analytical formulas of COP derived from the laws of thermodynamics because such theoretical expressions not only provide fast and simple way of determining COP but also lend direct and fresh insights into the impacts of individual processes on the performance of the VC cycle. This article reports development of a set of analytical expressions for the COP of VC refrigeration cycles, which is derived from a second-law analysis. The analytical expressions provide an easy-use theoretical approach to simulation and simulation-

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