Recent advances in modeling the vapor-liquid

Fluid Phase Equilibria 432 (2017) 28e44

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Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d

Recent advances in modeling the vapor-liquid equilibrium of mixed working fluids Wen Su, Li Zhao*, Shuai Deng Key Laboratory of Efficient Utilization of Low and Medium Grade Energy, Tianjin University, MOE, Tianjin, 300072, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 July 2016 Received in revised form 7 October 2016 Accepted 18 October 2016 Available online 18 October 2016

Vapor-liquid equilibrium (VLE) properties of mixed working fluids are essential for the study of organic fluids for thermodynamic cycles such as heat pumps or organic Rankine cycles. Thus, the typical and recent models for the VLE calculation of mixed working fluids are reviewed in this paper. The most popular cubic equations of state (EOSs) and two mixing rules, namely the Van der Waals and the excess free energy, are summarized, respectively. VLE data of 13 alternative working fluids at different temperatures are employed to exhibit the capacities and limitations of three predictive models. The calculated VLE results show that the model with Van der Waals mixing rules is only valid for some types of mixture, while the accuracy of the model with excess free energy mixing rules is greatly influenced by the UNIFAC group parameters, which are usually fitted from the VLE data instead of the activity coefficient data. Thus, a generalized model for mixed working fluids should be derived from the molecular theory or the group contribution method, in which the group parameters are fitted from the large VLE data. © 2016 Elsevier B.V. All rights reserved.

Keywords: Vapor-liquid equilibrium Mixed working fluids Cubic equations of state Van der Waals mixing rules Excess free energy mixing rules

1. Introduction Organic working fluids have been widely used in several application domains, such as refrigeration, heat pump and organic Rankine cycle. The physical properties of working fluids directly determine the thermodynamic efficiency of system, the design of system components, the stability and safety of system. Thus, a highefficient, safe and economical operation of thermodynamic cycle is highly dependent on working fluids with appropriate properties. In addition to the thermodynamic properties of working fluids, environmental properties, such as ozone depletion potential (ODP) and global warming potential (GWP), have to be considered in the screening of working fluids with growing concerns regarding the depletion of the ozone layer and the greenhouse effect [1]. Recently, due to high ODP and high GWP, the traditional organic working fluids chlorofluorocarbon (CFC) have been phased out in accordance with the protocol of Montreal and the hydrochlorofluorocarbon (HCFC) will also be phased out in 2040. Therefore, it's urgent to search the high-efficient fluids which are friendly to the environment.

* Corresponding author. Tel./fax: þ86 022 27404188. E-mail address: [email protected] (L. Zhao). http://dx.doi.org/10.1016/j.fluid.2016.10.016 0378-3812/© 2016 Elsevier B.V. All rights reserved.

Although many researchers have proposed suitable pure fluids for different applications, few pure fluids, which are free of all environmental and safety concerns, chemically and thermally stable, and perform efficiently, have been found [2e4]. Thus, mixtures, whose individual drawbacks of the components can be compensated by each other, are recommended as candidates of working fluids. Basically, the mixed working fluids are preferable to pure working fluids on account of energy saving and the flexibility of operation [5e7]. A reliable thermodynamic model is required for evaluating the performance of a given mixture in a thermodynamic cycle. In general, the experimental data of vapor-liquid equilibrium (VLE) for mixed working fluids are used to establish the highly accurate Helmholtz energy equation, which is widely applied to calculate the thermodynamic properties of corresponding mixture [8]. Furthermore, accurate VLE data are essential for designing and modeling thermodynamic processes, which include the evaporation and condensation of working fluids, the constituent separation of mixtures [9]. For the mixed working fluids being in VLE, they may produce different types of phase behavior, because of the different molecular interactions and phase conditions. Although experiments can provide accurate data at specific phase conditions for mixtures, such data are limited and can't be expected to meet the expanding industrial needs for thermodynamic design and

W. Su et al. / Fluid Phase Equilibria 432 (2017) 28e44

development [10]. Besides, given a huge number of mixed working fluids, it's impossible to provide VLE information for every mixture by the time-consuming phase equilibrium experiment with high costs. Therefore, many VLE models have been proposed to provide predictions of VLE for a wide variety of mixtures exhibiting varied phase behaviors, based on the limited experimental data [11e14]. Such calculation models can be categorized into two approaches. The first approach utilizes an equation of state (EOS) to describe the phase behavior for both the liquid and vapor phases. The second approach utilizes an activity coefficient model to describe the liquid phase, while a different model, usually an EOS, is used to describe the vapor phase [10]. Existing studies have demonstrated that the EOS model can be successfully applied to correlate the VLE data of mixed working fluids over broad ranges of temperature and pressure [15e18]. In contrast, the activity coefficient model has been applied successfully at low temperatures where the liquid phase is relatively incompressible. However, this model is deficient in describing the phase behavior at high pressures [10,15]. Thus, for the above reasons, this paper is focused on the EOS model. For the EOS model, cubic EOSs, such as SoaveeRedlicheKwong (SRK) and PengeRobinson (PR), are widely used to predict the VLE of mixed working fluids, due to their inherent simplicity and efficiency [15]. A compositional dependence has to be introduced to account for the molecular interactions in mixtures, in order to predict the phase behavior using cubic EOSs. All extensions of cubic EOSs to mixtures are, at least partially, empirical in nature, because there is no exact statistical mechanical solution relating the properties of dense fluids to their intermolecular potentials, and the detailed information on such intermolecular potentials is unavailable [19]. Therefore, a variety of mixing rules have been proposed to determine the parameters of cubic EOS for mixtures over the years [15,19]. Generally speaking, the proposed mixing rules can be classified into two categories, namely, Van der Waals (VDW) mixing rules and excess free energy (GE) mixing rules. The VDW mixing rules have been used to describe the VLE of mixed working fluids for decades, because of their simplification and relative accuracy for many mixture systems [20e22]. A route for developing excess free energy mixing rules is to combine a cubic EOS with an activity coefficient model, often a local composition model like Wilson, NRTL, UNIQUAC or UNIFAC [12,23e25]. The excess free energy mixing rules have enhanced dramatically the range of cubic EOS's applicability and can give a good prediction for high pressure VLE of mixtures. Furthermore, the group contribution activity coefficient model UNIFAC has been widely used to develop completely predictive models for mixed working fluids [25]. The models, which can be utilized to predict the VLE of mixtures, usually consist of two parts including cubic EOSs and mixing rules. This paper presents a state-of-the-art review of these models and discusses the capabilities and limitations of three predictive models in exhibiting the phase behaviors of mixed working fluids. The methodology of VLE is presented to demonstrate the process of VLE calculations in the next section. In Section 3, commonly-used cubic EOSs are summarized and discussed. Two mixing rules involved the van der Waals and the excess free energy are reviewed respectively in Section 4. In Section 5, the performances of three completely predictive models in predicting the VLE of mixed working fluids are discussed by comparing the calculated results with the experimental data. The knowledge gaps and development directions of VLE models are presented in Section 6. Conclusions of this review on modeling the VLE of mixtures are given in Section 7. 2. Methodology of VLE For the mixture being in VLE, the required thermodynamic

29

condition is that the temperature, pressure and fugacity of each species should be the same in liquid and vapor phases [15]. That is,

TiL ¼ TiV ¼ T

(1)

PiL ¼ PiV ¼ P

(2)

fiL ðT; P; zi Þ ¼ fiV ðT; P; yi Þ

(3)

Where superscripts L and V represent liquid and vapor, respectively. zi, yi sequentially denote the molar fraction of species i in the liquid and vapor phases. The key of VLE calculation is to express the fugacity as a function of temperature, pressure and composition. An EOS is generally used to obtain the fugacity of species i in the vapor phase, and the generalized form is

f V ðT; P; yi Þ ln i yi P

!

1 ¼ RT

# Z∞" vðPV=RTÞ dV ln Z V 1 vNi V T;V;Njsi V

(4) Where V is the total volume, ZV is the vapor compressibility factor computed from the EOS. However, there are two different methods to describe the liquid phase: either the same EOS used for the vapor phase is also used for the liquid phase or the activity coefficient method is employed. For the activity coefficient method, the fugacity of species i in the liquid phase is obtained by

fiL ðT; P; zi Þ ¼ zi gi fiPL ðT; PÞ

(5)

Where fiL ðT; PÞ is the fugacity of pure component i as a liquid at the temperature and pressure of the mixture, and can be calculated from EOSs. At low temperature and pressure, fiL ðT; PÞ is approximately equal to the vapor pressure of pure component. For the activity coefficient gi, many local composition models, such as UNIFAC, NRTL and Wilson, have been proposed to relate gi to composition zi [19]. However, these models are only valid at low pressure and temperature where the liquid phase is relatively incompressible. By contrast, the EOS model can be applied to the liquid phase over a wide range of temperatures and pressures. With the same EOS used for the vapor phase, the fugacity of a species in a liquid mixture is computed from

ln

! # Z∞" fiL ðT; P; zi Þ 1 vðPV=RTÞ dV ln Z L ¼ 1 RT vNi V zi P T;V;Njsi V

(6) Where the liquid compressibility factor ZL is derived from the EOS. This equation differs from Eq. (4) only in that the smallest volume, which is the liquid phase solution to the EOS, is used to calculate the fugacity. Due to the applicability of Eq. (6) over a wide range of temperatures and pressures in calculating the liquid fugacity, the EOS model has been extensively used to predict the VLE of mixed working fluids. Numerous EOSs have been proposed to establish the pressurevolume- temperature relationship of organic fluids. In general, these proposed EOSs can be categorized into the “analytical” and “non-analytic” types [15]. The former means that when T and P are specified, V can be found analytically. This type of EOSs is represented by the cubic EOSs, such as SRK and PR. However, V is calculated numerically in the latter approach. In order to describe the complexity of property behavior, a large number of fitted

30


parameters are introduced into the non-analytic EOSs, such as the Benedict-Webb-Rubin (BWR) EOS and the Wagner formulation [15]. Due to the wide spread use and simple form of cubic EOSs, the review on modeling the VLE of mixed working fluids is limited to cubic EOSs. Fig. 1 shows the flow diagram of the VLE calculation with the cubic EOS. It can be seen that when given the specified value of phase equilibrium temperature T and liquid mole fraction zi, the phase equilibrium pressure P and vapor mole fraction yi are iteratively calculated based on the ratio of fugacity between the liquid and the vapor. The EOS is used to calculate the fugacity of each species for liquid and vapor phases in the process of VLE calculation. The cubic EOS parameters of mixture are obtained from the fluid parameters and molar fraction of species based on the mixing rule, in order to improve the accuracy of prediction on the VLE of mixed working fluids. So far, two types of mixing rules, namely Van der Waals and excess free energy, are frequently used for the VLE prediction of working fluids.

3. Cubic EOSs Many cubic EOSs are available in the literature. A recent comprehensive review on the cubic EOSs is presented by Valderrama [26]. For cubic EOSs, they are of third degree when solved with respect to volume. It means that the volume can be found analytically rather than only numerically at specified values of T and P. To the best of author's knowledge, almost all cubic EOSs express pressure as the sum of two terms, a repulsion pressure PR and an attraction pressure PA.

P ¼ PR þ PA

(7)

The equations of Van der Waals (VDW), SRK and PR are examples. For the repulsion pressure, it's always obtained from the Van der Waals hard sphere equation. That is

PR ¼

RT vb

(8)

Where v is the molar volume, the co-volume parameter b is related to the size of the hard sphere. As for the attraction pressure, it can be generally expressed as [27].

a PA ¼ gðvÞ

(9)

Where g(v) is a function of the molar volume and the co-volume parameter. The energy parameter a, which can be regarded as a measure of the intermolecular attraction force, is a function of temperature. Ever since the appearance of the VDW equation in 1873, many authors have successfully developed different cubic EOSs to establish the relation between P, v and T. The most well-known cubic EOSs together with the expressions often used for estimating their parameters are shown in Table 1. Many of these EOSs employ the repulsive term of VDW, while various expressions for the attractive term. Furthermore, considering that any given twoparameter EOS is intrinsically unable to quantitatively describe the P-v-T properties of complex fluids and their asymmetric mixtures, the third parameter c has been introduced into cubic EOSs, such as RK-PR, volume-translated PengRobinson (VTPR), in order to improve the accuracy of the prediction in the liquid phase. For the parameters involved in the cubic EOSs, two approaches, namely corresponding states principle and data regression, are usually used to get the values [15]. For the corresponding states principle, the energy and co-volume parameters are usually determined by the following two relationships, so that the EOSs can predict the critical temperature and pressure of fluids accurately.

vP ¼0 vv Tc v2 P vv2

Fig. 1. Flow diagram of the VLE calculation.

(10)

! ¼0

(11)

Tc

Applying a specified cubic EOS at the critical point where the first and second derivatives of pressure with respect to volume, one can obtain expressions for a and b in terms of the critical properties. The VDW and RK EOSs are examples. For the data regression, the parameter values are frequently obtained by fitting the data of vapor pressure and saturated liquid density. The acentric factor ⍵, which represents a measure of the acentricity of the molecule, is usually employed to establish the relationship between the energy parameter and the temperature in this approach, as shown in


31

Table 1 The most important cubic EOSs for VLE calculation. EOS

Equation

Parameters of cubic EOSs

P¼

RT vb

RK [29]

P¼

RT vb

SRK [30]

RT a P ¼ vb vðvþbÞ

VdW [28]

a v2

a ¼ 27 64

a T 1=2 vðvþbÞ

a¼

ðRTc Þ2 1 RTc Pc ; b ¼ 8 Pc R2 Tc2:5 0:42748 Pc ; b ¼

a ¼ ac ½1 þ mð1

c 0:08664 RT Pc pffiffiffiffiffi 2 Tr Þ

m ¼ 0:48 þ 1:574u 0:176u2 R2 Tc2 RTc ; b ¼ 0:08664 Pc Pc pffiffiffiffiffi 2 a ¼ ac ½1 þ mð1 Tr Þ ac ¼ 0:42748 PR [27]

RT a P ¼ vb vðvþbÞþbðvbÞ

m ¼ 0:37464 þ 1:54226u 0:26992u2 R2 Tc2 RTc ; b ¼ 0:07780 Pc Pc pffiffiffiffiffi a ¼ ac ½1 þ mð1 Tr Þ2 ac ¼ 0:45724 PRSV [19]

RT a P ¼ vb vðvþbÞþbðvbÞ

m ¼ m0 þ m1 ð1 þ Tr0:5 Þð0:7 Tr Þ m0 ¼ 0:378893 þ 1:4897153u 0:1713184u2 þ 0:0196554u3 ac ¼ 0:45724 CSD [31]

R2 Tc2 RTc ; b ¼ 0:07780 Pc Pc

P¼

RT 1 þ y þ y2 y3 a v vðv þ bÞ 1 y3

a ¼ 4a

ðRTc Þ2 ; 4a ¼ a0 expða1 Tr þ a2 Tr2 Þ Pc

y¼

b 4v

b ¼ 4b

RTc ; 4 ¼ b0 þ b1 Tr þ b2 Tr2 Pc b

PT [32]

RT a P ¼ vb vðvbÞþcðvþbÞ

RK-PR [33]

RT P ¼ vb

a ðvþcbÞþ

2

c a ¼ Ua ðRTPcc Þ ½1 þ Fð1 Tr Þ2 ; b ¼ Ub RT Pc ; k 2 3 ðRTc Þ RTc ; b ¼ Ub ; a ¼ Ua 2 þ Tr Pc Pc

vþ1c b 1þc

1=2

c c ¼ Uc RT Pc

c ¼ d1 þ d2 ðd3 Zc Þd4 þ d5 ðd3 Zc Þd6 VTPR [34]

RT a P ¼ vþcb ðvþcÞðvþcþbÞþbðvþcbÞ

a ¼ 0:45724

R2 Tc2:5 RT aðTÞ; b ¼ 0:0778 c Pc Pc

aðTÞ ¼ TrNðM1Þ expðLð1 TrNM ÞÞ; c ¼ vpR vexp at Tr ¼ 0:7 Note: some variables used in the parameter expression of cubic EOSs are not specified. Their values are related with the fluids or involved with complex calculation. Please consult the corresponding reference for these variables.

Table 1. Meanwhile, owing to the fact that two-parameter EOSs are widely used in engineering calculation of VLE for mixed working fluids, mixing rules for energy and co-volume parameters are reviewed in the next section.

4. Mixing rules The concept of mixing rule is that the mixture can be assumed as a one fluid. For fixed compositions, the mixture properties and their variations with T and P are the same as the pure fluid with appropriate parameter values. To describe all pure components as well as mixtures, the mixture parameters must vary with composition so that if the composition is actually for a pure component, the model describes that substance [15]. In general, there are two kinds of widely used mixing rules, namely Van der Waals and excess free energy, for energy and co-volume parameters of cubic EOSs in the VLE calculation of mixed working fluids. Thus, comprehensive reviews are presented for these mixing rules.

4.1. Van der Waals The first successful method of generalizing a pure fluid EOS to mixtures is the one-fluid model proposed by van der Waals. The underlying assumption of this model is that the same EOS used for pure fluids can be used for mixtures if a satisfactory way is found of obtaining the mixture EOS parameters. One of the exact mixing rules is derived from statistical mechanics by virial EOS. For mixtures, the only composition dependence of the second virial coefficients Bm is given by

Bm ¼

N X N X

xi xj Bij

(12)

i¼1 j¼1

Since Eq. (12) is exact, it can be considered low-density boundary conditions that should be satisfied for mixtures by other EOSs when expanded into the virial form [15]. Thus, in order to satisfy the quadratic mixing rule for the second virial coefficient, the quadratic dependence on mole fractions of mixture parameters is commonly used in the VDW mixing rules [15].

am ¼

N X N X

xi xj aij

(13)

xi xj bij

(14)

i¼1 j¼1

bm ¼

N X N X i¼1 j¼1

The following combining rules, namely, the geometric mean rule for the cross energy and the arithmetic mean rule for the cross co-volume parameter, are frequently used to obtain the cross coefficients aij and bij from the corresponding pure component parameters:

aij ¼

bij ¼

pffiffiffiffiffiffiffiffi ai aj 1 kij bi þ bj 1 lij 2

(15)

(16)

Where kij denotes the binary interaction parameter for aij, lij is the

32


binary interaction parameter for bij. Generally, for a specified mixture, these interaction parameters can be accurately obtained by fitting the measured phase equilibrium and volumetric data [22]. Furthermore, lij is usually set to zero in the VLE calculation for most mixtures, in order to simplify the binary interaction parameters fitting process. As a result of that, the co-volume parameter of mixtures can be derived by

bm ¼

N X

x i bi

(17)

i¼1

However, when the mixture is in high pressure phase equilibrium, the accuracy of VLE calculation can be improved, if lij is determined by the experimental data [35]. For the binary interaction parameter kij, it's generally thought kij is symmetric, i.e., kij equals kji. Furthermore, the value of kij has a decisive effect on the phase behavior of mixtures. When fitting VLE data with the VDW mixing rules, it is found that kij is approximately zero for mixtures, which have small differences between the molecular structures of species, such as Hydrocarbon (HC) mixtures, Hydrofluoro carbon (HFC) mixtures, whereas for some other mixtures such as HC with HFC, it is not only nonzero but also changes with the compositions [29,30,35]. On the other hand, a theoretical value of kij can be determined, owing to the fact that the geometric mean rule employed for the cross-energy parameter can be derived from the theory of Mie potential energy and London dispersion forces [36]. However, the interaction parameters calculated via this theoretical approach are often much higher than those actually from a cubic EOS-based data fitting of the experimental data, because there is no direct relation between the attractive part of the intermolecular potential and the energy parameter of the cubic EOS. Thus, such theoretical approach is usually used by adding adjustable parameters. For example, a semi-empirical correlation of kij, which is derived from the molecular theory, was proposed for non-polar systems [36]. So far, according to the geometric mean rule presented by Eq. (15), a variety of expressions have been proposed to get the interaction parameter from molecular theories and VLE experiments. Table 2 summarizes the proposed equations for kij. It can be seen that the factors associated with kij involve the temperature, the critical parameters, the composition, the acentric factor, the ionization potential and the infinite dilution activity coefficients. Some of these methods are applicable for various kinds of mixtures, such as the linear relationship between kij and temperature [37], the Margules-type and Van Laar-type expressions of kij relating with composition [19,38]. However, more expressions of kij are aimed at some types of mixtures. For instance, kij was proposed as a function only of critical temperature and compressibility factor for binary light hydrocarbon systems [39]. With regard to HFC and HC mixtures, Chen et al. [40,41] supposed that each pure component in a mixture has a mixing factor and kij can be calculated out by a simple operation of these two mixing factors. Thereafter, the correlation was modified with the introduction of acentric factors and the critical properties [42]. In 2016, Zhang et al. [43] introduced the contribution of mixing factor to kij for mixtures containing HCs, HFCs, PFCs, HFOs, CO2, RE170 and R13I1. Empirical correlations were developed in terms of the absolute difference between the acentric factors of the components and the critical compressibility factor ratio, based on the fitted kij values of mixtures [44,45]. In addition, since 2004, a group contribution model, called PPR78, has been proposed to determine kij from the mere knowledge of chemical structures of molecules within the mixture by Jaubert and Mutelet [11,46e48], in order to realize the accurate prediction of interaction parameters for different mixtures. In the

model PPR78, the PR EOS is used and kij is a function of the temperature, the pure components' critical temperatures, critical pressures and acentric factors. So far, the PPR78 model is able to represent the phase behavior of any fluid containing alkanes, alkenes, aromatic compounds, cycloalkanes, permanent gases, mercaptans and water. For the binary interaction parameter lij, it has the same symmetry characteristic as kij. However, lij has little effect on the VLE calculation results of mixtures, compared with the effect of kij. Thus, lij is generally set to zero for mixtures and less attention is paid to the research of lij. Yet, for non-ideal and high pressure phase equilibrium, additional adjustable parameter lij can achieve a good agreement between the calculation of VLE model and the experimental data [15]. On the other hand, on the basis of Mie potential energy and London dispersion theory, the cross co-volume can be related to the hard sphere diameter [36]. Thus, many equations, which are different from Eq. (16), have been proposed for cross covolume of mixtures, according to the hard sphere mixing rules. A summary of cross co-volume bij is presented in Table 3. It can be seen that most of the cross co-volume expressions are derived from hard sphere mixing rules, such as the geometric mean of co-volume [51], the method of Coutinho et al. [36] and the approach proposed by Lee and Sandler [52]. However, real molecules are not hard spheres. Therefore, some authors corrected these expressions with temperature, composition and shape factor. For example, Weber and Brunner [37] thought that the interaction parameter lij is a linear function of temperature. Wang and Zhong [53] proposed a composition dependent cross parameter for VLE calculations of hydrogen containing systems at different temperatures. Han et al. [54] introduced molecular shape factors to correct bij and derived a theoretical formula, which could reflect the actual situations of mixture systems.

4.2. Excess free energy As mentioned above, the liquid behavior of highly non-ideal system can be described by activity coefficient models. Meanwhile, considering the poor capacities of VDW mixing rules in the VLE prediction of these systems, the excess free energy mixing rules, which combine the strengths of the cubic EOSs and the activity coefficient models, have been widely investigated, since the publication of the Huron-Vidal (HV) mixing rule [12,29]. Many different models, called EOS/GE, have been available in a very large number of literatures. In these models, the energy parameter of cubic EOS is often derived by incorporating the excess free energy into the EOS. Thus the cubic EOS with excess free energy mixing rules can be applied to the VLE calculation of polar and non-polar mixtures at both low and high pressures. The starting point for deriving most EOS/GE models is the equality of the excess Gibbs energy from an EOS and from an explicit activity coefficient model at a suitable reference pressure [55].

GE RT

EOS

¼

P

GE RT

Model (18) P

Where a suitable reference pressure P can be the infinite pressure or the zero pressure. The excess Gibbs energy from a cubic EOS is calculated by

Table 2 Binary interaction parameter kij for cross-energy parameter. Binary interaction parameter kij ¼

k0ij

þ

Comments

[37]

TT

kij is set to be an exponential function of temperature, in order to model the phase behavior of asymmetric alkane þ alkane binary systems.Tc1 is the critical temperature of the more volatile componen.

[18]

The function is recommended in the literature for methane(i) with higher n-alkanes (j). kij increases monotonically and tends asymptotically to a maximum of 0.12 for an infinite carbon number in the second compound.

[49]

For binary light hydrocarbon systems. kij was proposed as a function only of critical temperature and compressibility factor. Thus, cross-energy parameter can be obtained without any adjustable parameter.

[39]

The Margules-type expression provides a good correlation of VLE data of highly non-ideal systems by adding an additional composition dependence and parameters to kij. kij is dependent on the composition. Their relationship is expressed as Van Laar-type

[19]

CN

20j

kij ¼ 0:12 1 e 0

1Zcij pffiffiffiffiffiffiffiffiffi

2 TCi TCj A TCi þTCj

kij ¼ 1 @

Zcij ¼

Zci þZcj 2

kij ¼ xi Kij þ xj Kji kij ¼ Kij Kji =ðxi Kij þ xj Kji Þ kij ¼ ki kj ki ¼ 0:30ui þ 0:031n0:1 F =ui kij ¼

0:5 uj Pc;j

!

0:5 ui Pc;i

ki

Tc;j

kij ¼ ðki þ kj Þ=2

kij is assumed to be the difference between the mixing factors of the pure components. ki is the mixing factor of the component with a higher vapor pressure and kj is the mixing factor of the component with a lower vapor pressure. The mixing factor of pure HFC and HC can be determined by the acentric factor and the number of F. The interaction parameter kij was modified with the introduction of the acentric factors and the critical properties of the components. In order to avoid the mistake of kij ¼ -kji, all involved components are arranged in a certain sequence.

!

Tc;i

kj

kii ¼ ki

kij ¼ wj ki þ wi kj 0:45 kij ¼ 0:6858 þ 0:002713 ui uj 0:6842ðZci =Zcj Þ kij ¼ 0:16339 þ 0:30504ðui uj Þ 0:14921ðZci =Zcj Þ ! qffiffiffiffiffiffiffiffiffiffiffiffi kij ¼ 0:171 Ii Ij ln IIij 0 kij ¼ 1 A@ kij ¼ 1 12

bj bi

1n3 pffiffiffiffiffiffi 3 bi bj A

[42]

The binary interaction parameters of HFC mixtures are calculated by the average of mixing factors.

[41]

Constant wi is the contribution coefficient of component j mixing with component i. kij can be determined by summing the contribution coefficient of each pure component times its contribution. On the basis of the fitted kij values for refrigerant mixtures, the empirical function is correlated, in terms of the absolute difference between the acentric factors of the components and the critical compressibility factor ratio. Based on the interaction parameters of HFC mixtures, the new correlation for kij can be developed.

[43] [45] [44]

The correlation relates kij to the ionization potentials of the component species according to the low temperature VLE data for binary systems containing hydrogen, helium, and neon.

[50]

The theoretical combining rule is derived from interaction potential function and London-Mie theory. The adjustable parameter A, which is determined by experimental data, is introduced to improve the prediction accuracy.

[36]

The equation is derived based on the HuroneVidal mixing rules. Adjustable parameters q1q2,q3 are obtained by fitting experimental data of each mixture. kij is calculated by the composition, the pure component EOS parameters and infinite dilution activity coefficients

[17]

bij

qffiffiffi ai aj

12

bi bj

qffiffiffi aj ai

b RT þ 12 pjffiffiffiffiffiffi ai aj

qi q

q

Tri2 Prj3

kij ¼ Kij xi þ Kji xj vap

Kij ¼ 1

[38] [40]


kij is a linear function of temperature. k0ij , k1ij are determined by the experimental data.

c1

kij ¼ k0ij e

References

T k1ij 1000

∞ ai ½bj =bi ðln g∞ ji þ ln fj ðT; Pi

[19]

Þ li Þ=εi

2ðai aj Þ1=2 0pffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi12 3,2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 aj ðTÞ ai ðTÞaj ðTÞ ai ðTÞ 6 A 7 5 kij ¼ 4Eij ðTÞ @ 5 42 bi bj bi bj 2

Ng X Ng 1X 298:15 ðaik ajk Þðail ajl ÞAkl Eij ðTÞ ¼ 2 T

Group contribution model PPR78, is used to determine kij from the mere knowledge of chemical structures of molecules within the mixture. So far, the PPR78 model is able to represent the fluid phase behavior of any fluid containing alkanes, alkenes, aromatic compounds, cycloalkanes, permanent gases, mercaptans and water.

[11]

Bkl Akl 1

k¼1 l¼1

33

34


Table 3 Cross co-volume parameter bij. Cross co-volume parameter qffiffiffiffiffiffiffiffi bij ¼ bi bj 1=3

bij ¼ ðbii

2=3

þ bij Þ=2Þ3=2

ðb þb Þ T bij ¼ i 2 j 1 l0ij l1ij 1000 1=3

bij ¼ ½1 lij þ ðlij lji Þxi ððbii

4j 4i 4ij

4ij bij ¼ bm ¼

13 1=3

bi

ðbi þbj Þ ð1 2

C1 Þ þ C2 @

1=3

þbj 2

A

! qffiffiffiffiffiffiffiffi qj qi bi bj q ij

PN

* i¼1 ½xi xj ð1 mij Þbij þ mij bij bij ¼ ðbi þ bj Þ=2;

GE ðT; p; xi Þ ¼ RT

Zm ðT; p; xi Þ

X

* b*ij ¼ f ðbi ; bj Þ bij is any one of the non-classical mixing rules for co-volume and mij can be considered a [51] binary mixing parameter used to proportion the overall mixing rule. Furthermore, five forms of parameter mij have been proposed

! xi Zi ðT; pÞ

i

ln Zm ðT; p; xi Þ 0

B @

X

! xi ln Zi ðT; pÞ

i vm ðT;p;x Z iÞ ∞

[51]

The composition dependent cross parameter for the co-volume is recommended for VLE [53] calculations of hydrogen containing systems at different temperatures. The theoretical combining rule is derived from interaction potential function and London- [36] Mie theory. The adjustable parameters C1, C2, which are determined by experimental data, are introduced to improve the prediction accuracy. The molecular shape factors (q,4) are used to correct the geometric mean of co-volume [54] parameters.

1=3

þ bij Þ=2Þ3

0 bij ¼

References

The cross parameter is a geometric mean of co-volumes of species.

Owing to the fact that the co-volume is about proportional to the cubic of hard sphere [36] diameter s, i.e.bfs3, the equation can be derived from the hard spherediameter additive. The expression is obtained based on the relation between the hard sphere diameter and co- [52] volume. l0ij , l1ij are determined by the experimental data. [37]

1=3

þ bij Þ3 =8

2=3

bij ¼ ððbii

Comments

X Zm 1 dv xi vm i

vi ðT;p;x Z iÞ ∞

1 Zi 1 C dvA vi (19)

Meanwhile, there exists a rigorous thermodynamic relation between the excess Gibbs energy GE and the activity coefficient gi. Their relationship is expressed by

GE ¼ RT

X

xi ln gi

(20)

i

Thus, an explicit activity coefficient model, such as UNIFAC, Wilson or NRTL can be used to get the value of GE at a suitable reference pressure. Generally, a linear mixing rule, given by Eq. (17), is employed to get the co-volume parameter of cubic EOSs in EOS/ GE models. When submitting Eq. (17) into Eq. (19), the energy parameter can be derived in terms of the excess free energy GE. Meanwhile, the constants involved in the expression of energy parameter are always related with the used EOS [15]. Many GE mixing rules have been proposed since the HV mixing rule. Meanwhile, a large number of literatures have been published to summarize the existing GE mixing rules and discuss the capacities and limitations of these rules [56]. Thus, only the most successful mixing rules for mixed working fluids are presented in this paper. They are listed in Table 4. In general, these mixing rules can be categorized as those following the infinite or zero reference pressure assumption and those not having a specific reference pressure. As the first successful combination of an EOS and excess free energy, the basic assumption of HV mixing rule is the use of the infinite pressure as the reference pressure [57]. For non-ideal mixtures, HV shows good correlation capabilities but is not satisfactory for extrapolation over a range of temperatures. Furthermore, it was observed that HV is superior to the VDW for correlating the VLE data of highly non-ideal mixtures. However, it's possible to reduce the HV to the VDW by choosing specific algebraic forms for GE. On the other hand, one can't use the local composition model parameters obtained at low pressure, and the interaction

parameters of the GE expression must be re-estimated using an EOS with the HV mixing rule, because the excess Gibbs energy is very pressure dependent. The excess Gibbs energy at infinite pressure can be very different from the value at the experimental pressure. Consequently, a fundamental shortcoming of the HV mixing rule is the use of the pressure-dependent excess Gibbs energy in the EOS rather than the excess Helmholtz energy, which is much less pressure dependent. The Wong-Sandler (WS) mixing rule is also based on the infinite reference pressure but with significant modifications compared to that in HV [58]. It's assumed that the following equation is approximately true:

AE ðT; P ¼ ∞; xÞzAE ðT; low

P; xÞzGE ðT; low

P; xÞ

(21)

Where AE is the excess Helmholtz free energy. The first of these equalities is based on the relative insensitivity of the excess Helmholtz energy with pressure. The second of these equalities results from the relation:

GE ¼ AE þ PvE

(22) E

At low pressure, the term Pv is neglected so that the second equality of Eq. (21) can be derived. As GE is highly dependent on pressure, the WS mixing rule is derived by equating the excess Helmholtz energy AE to that of an EOS at infinite pressure. According to the equalities given by Eq. (21), the WS mixing rule uses VLE information only at low pressure, and it can also be used to predict the VLE of mixtures at high pressure. The low-pressure activity coefficient model parameters reported in data banks, such as the Dortmund Data Bank (DDB) [59], could be used directly with good accuracy in the WS mixing rule. Thus, it is no need to refit any experimental data for the activity coefficient model parameters. The second advantage of WS over HV is that the proper composition dependence of the second virial coefficient is assured by introducing a second virial coefficient binary interaction parameter kij. The value of kij is usually obtained by fitting experimental VLE data. Meanwhile, various approaches have also been proposed to predict the extra interaction parameter introduced in the WS mixing rule. For instance, Orbey and Sandler [19] proposed a slightly reformulated version of the WS mixing rule, which can go smoothly to the VDW mixing rule. So that, the long term accumulated binary interaction parameters for VDW mixing rule can be used in this modified mixing rule. Thereafter, Valderrama and


35

Table 4 GE mixing rules for cubic EOSs. Authors

Mixing Equations rules

Huron et al. [57]

HV

am ¼ bm bm ¼

Wong and Sandler [58]

WS

Orbey and Sandler [19]

MWS1

Han et al. [61]

MWS2

Michelsen [63]

MHV1

Michelsen [64]

MHV2

xi bi

aj a a Bij ¼ b ¼ bi i þ bj 1 kij Þ=2 RT ij RT RT P X x B þ x2 B2 þ ða 1Þe xi ln bi GE a¼ þ xi ai kij ¼ 1 1 x1 x2 ðB1 þ B2 Þ RTC WS i X

PSRK

Ahlers and Gmehling [66]

VTPR

The equation of second virial coefficient introduces the binary interaction parameter in a manner similar to that of the van der Waals mixing rule. Thus, the interaction parameters used in van der Waals can be employed in this rule.

kij is simplified as a function of composition-dependent and excess Gibbs energy-dependent. The VLE calculation shows that the correlated accuracy is as good as or better than that of the original WS mixing rule.

ai GE þ bi C WS XX

X a GE a xi i ¼ xi xj b 1 þ WS RT ij RTb C RT i

pffiffiffiffiffiffiffiffi

According to the fitted value of kij for mixtures containing n-alkanols and ai aj a 1 Bij ¼ b ð1 kij Þ ¼ ðbi þ bj Þ carbon dioxide, the interaction parameter is correlated as a function of the RT RT ij 2 reduced temperature TR and the polar parameter m. 0:1730 11:39 36:76 2 þ 20:51 þ j þ 74:08 j kij ¼ 0:4680 TR TR TR X m2 T a GE TR ¼ j ¼ 2:83 am ¼ bm xi i þ WS Tc Tc vc bi C XX

X a GE a xi xj b xi i bm ¼ 1 þ WS RT ij RTbi C RT " # X am a 1 GE X b ¼ xi i þ MHV1 þ xi ln bi bm RT bi RT C RT i X xi bi bm ¼

q1 ða

X

xi ai Þ þ q2 ða2

X i

am a a ¼ i bm RT i bi RT

bm ¼

xi a2i Þ ¼

X

GE X b þ xi ln bi RT i

xi bi

SRK þ MHV1 þ UNIFAC

X

xi ai GE bi 0:53087 1 0 3=4 3=4 4=3 N X N X bii þ bij A ¼ xi xj bij bij ¼ @ 2 i¼1 j¼1

am ¼ bm

bm

cm ¼ HVOS

The value of GE can be calculated by the existing activity coefficient model parameters. In addition, the theoretically correct quadratic composition dependence of the second virial coefficient is satisfied in this mixing rule. The binary interaction parameter kij in the cross second virial coefficient is usually obtained from fitting experimental VLE data.

xi

i

Holderbaum and Gmehling [65]

The mixing rule is derived by supposing that the excess volume be zero at infinite pressure. It violates the imposed by statistical thermodynamics quadratic composition dependency of the second virial coefficient.

pffiffiffiffiffiffiffiffi

ai aj a 1 ð1 kij Þ Bij ¼ b ¼ ðbi þ bj Þ RT ij 2 RT X E a G xi i þ WS am ¼ bm bi C XX

X a GE a xi xj b xi i bm ¼ 1 þ WS RT ij RTbi C RT

a¼


xi ai GE þ HV bi C

aj a a Bij ¼ b ¼ bi i þ bj 1 kij Þ=2 RT ij RT RT X a GE am ¼ bm xi i þ WS bi C XX

X a GE a xi xj b xi i 1 þ WS bm ¼ RT ij RTbi C RT

bm

MWS3

X

X

am ¼ bm

Valderrama and Zavaleta [60]

Comments

X

An assumption of the MHV1 model is that the ratio of the zero-pressure liquid molar volume to the co-volume parameter b is the same for the mixture and for each of its pure components. The mixing rule is derived from the linear approximation of the q function. Quadratic extrapolation parameters were used to ensure continuity of the function q and its derivatives. In this mixing rule, the relation between the excess Gibbs free energy from an EOS and from an activity coefficient model takes the quadratic form.

The method combines the SRK cubic EOS with the group contribution model UNIFAC via MHV1 mixing rules. In order to predict thermodynamic properties over a large temperature and pressure range, the UNIFAC parameters are refitted to vaporeliquid equilibrium data around atmospheric pressure VTPR EOS is combined with the excess Gibbs energy. However, Only the residual part of the excess Gibbs energy, which is calculated by the modified UNIFAC, is used in the mixing rule for the parameter a. The exponent 3/4 in the nonlinear combination rule for the cross-parameter bij, and the linear mixing rule for constant c. This model leads to a significantly better description in the case of asymmetric systems

xi ci

X am GE 1 X b a ¼ þ x ln xi i þ bi bm RT C HVOS RT C HVOS i i bi RT i X xi bi bm ¼

This model is developed at the limit of infinite pressure, and it is assumed that there is a universal linear algebraic function that relates the liquid molar volumes to their close packed hard-core volumes (continued on next page)

36


Table 4 (continued ) Authors

Mixing Equations rules

Coniglio et al. GCVM [69]

Boukouvalas et al. [70]

LCVM


CHV

Comments

X X am GE b a ¼ C1 þ C2 xi ln xi i þ bi bm RT RT bi RT i i X xi bi bm ¼

Depending on the choice of values for C1 and C2, the generalized expression can yield different mixing rules/methods

X E am l 1l G 1l X b a þ xi ln xi i þ ¼ þ bi bm RT bi RT C HV C MHV1 RT C MHV1 i i X xi bi bm ¼

It is a hybrid model and lacks a firm theoretical basis because it combines GE of the HV model, which is evaluated at infinite pressure, and GE of MHV1, which is evaluated at zero pressure. Experimental value l¼0.36 is obtained from the fitting of experimental data.

X am GE 1d X b a ¼ x ln xi i þ þ bi bm RT C CHV RT C CHV i i bi RT i X xi bi bm ¼

The mixing rule has an infinite reference pressure and performs equally well in comparison to the other models for VLE correlation-prediction of mixtures of molecules with large size differences.

Zavaleta [60] correlated kij as a function of the reduced temperature and the polar parameter for the modified mixing rule, based on the experimental data of mixtures containing n-alkanols and carbon dioxide. Han et al. [61] proposed a completely predictable model for kij based on the equality of GE calculated from a cubic EOS at zero pressure and from a solution model. Only compositions and activity coefficient parameters are required in the calculation of kij. The three modified WS mixing rules are also presented in Table 4. The problem with the HV model resulting from the pressure dependence of the excess Gibbs energy led a number of investigators to propose EOS mixing rules based on the idea of combining activity coefficient models and EOSs at low or zero pressures, as originally suggested by Mollerup [62]. The exact mixing rule obtained at zero pressure is

X GE X bm am ai þ ¼q xi ln xi q RT bi bm RT bi RT i i

(23)

The function q requires solving for the liquid density at zero pressure from the cubic EOS for each species in the mixture. However, there may not be a liquid density solution for one or more of the pure components at the temperature of the mixture and zero pressure. Therefore, mixing rules are developed by taking some sort of approximate extrapolation technique. Most of them are based on a linear or a quadratic approximation of the q function. The most well-known approximate zero reference pressure mixing rules, namely MHV1 and MHV2 were proposed by Michelsen and co-workers [63,64]. The activity coefficient models, such as UNIFAC and UNIQUAC can be directly used in these mixing rules. Owing to the fact that the linear mixing rule is used for the co-volume parameter, the proposed GE mixing rules at low pressure do not satisfy the theoretically justified mixing rule for the second virial coefficient. Furthermore, Gmehling and co-workers [65] derived a group contribution EOS for VLE of mixtures, namely predictive Soave-RedlicheKwong (PSRK), based on the MHV1 mixing rule. In PSRK, the modified group contribution model UNIFAC is employed to calculate the activity coefficients. This model gives very good results for thermodynamic properties, especially for VLE at highpressures. However, the large deviations between the calculations and the experimental data occur for asymmetric systems. Thus, some modifications of PSRK model were proposed later, such as the VTPR. In VTPR, the logarithmic term of co-volume in the mixing rule for the energy parameter and the combinatorial part in the UNIFAC model are skipped simultaneously. A nonlinear mixing rule for the co-volume parameter, instead of the linear mixing rule, is employed [66,67]. With these two modifications, better results are obtained for VLE of asymmetric systems. On the other hand, in

order to avoid calculations that a liquid phase solution to the EOS may not exist for one or more of the pure components at the temperature of interest, Orbey and Sandler [68] proposed HuronVidal-Orbey-Sandler (HVOS) mixing rule based on the equality given by Eq. (21). It's assumed that the ratio between the liquid molar volumes and hard-core volumes is unity for pure fluids and their mixtures at infinite pressure. Thus, this model is in agreement with the spirit of the Van der Waals hard core concept. Meanwhile, from the HVOS expression presented in Table 4, it can be seen that HVOS mixing rule is algebraically similar to the commonly used zero-pressure models, such as MHV1. Considering that the mixing rules at infinite or zero pressure have a weak performance on calculating the VLE of sizeasymmetric mixtures, a new type of mixing rules is suggested. Coniglioet al [69] proposed a generalized expression for this type. It can be written by

X X am GE bm a ¼ C1 þ C2 þ xi ln xi i bm RT RT b b i i RT i i

(24)

For the co-volume parameter, such models use the linear mixing rule. The first successful mixing rule of this type is called LCVM, which was proposed by Tassios and co-workers [70]. As indicated in Table 4. LCVM combines the HV with MHV1 mixing rules linearly and can't be derived from Eq. (18), owing to the unknown reference pressure. Thus, it is uncertain whether the UNIFAC parameters estimated from low pressure data can be used to calculate GE. Parameters of this mixing rule must be refitted based on the VLE data. Although this model lacks a firm theoretical basis, the authors have shown that the LCVM performs very well for VLE predictions involving a wide variety of systems and is particularly suitable for systems consisting of molecules of very different sizes. The reason for this success was analyzed comprehensively by Georgios and Philippos [12]. Subsequently, it was found that in the proposed mixing rules, the size difference of the species appears twice, first in the excess free energy term and again in the logarithmic term of co-volume. Thus, a new parameter was introduced into the HVOS mixing rule by Orbey and Sandler [71], in order to reduce the effect of this double counting of the size difference. The new mixing rule is called CHV, as listed in Table 4. Even though the CHV and LCVM models look quite similar, there are two differences between them. First, unlike the LCVM model, the CHV model has a well-defined reference pressure. Second, in the CHV model, the value of adjustable parameter only influences the logarithmic term of covolume as it is not a multiplier of the excess energy term. In contrast, in the LCVM model, the value of adjustable parameter affects both terms.


5. Discussion

37

Table 6 Experimental data sets of the considered mixtures.

Recently, a large amount of experimental studies have been conducted to measure VLE data for mixed working fluids [9,72e79]. In these related literatures, four kinds of mixtures, namely HCs þ HCs binaries, HCs þ HFCs binaries, HFCs þ HFCs binaries and trifluoroiodomethane (R13I1) þ HCs/HFCs binaries are mainly considered. The reason is that these mixtures are considered to be alternative working fluids for thermodynamic cycles, and VLE data for these mixtures can provide useful information for calculation of thermodynamics properties. Furthermore, these related literatures also show that various VLE models, which combine cubic EOSs, like PR and SRK, and mixing rules, like VDW and MHV1, have been used to correlate the experimental data of mixtures. For the cubic EOSs, PR has been widely used to describe the phase equilibrium of working fluids. Furthermore, when the excess free energy mixing rules are used in the VLE calculation, the activity coefficient model NRTL is always employed to calculate GE. In order to predict the VLE accurately, the parameters of NRTL are fitted based on the measurements of VLE data. However, the fitted interaction parameters are only valid for the corresponding mixtures. Therefore, predictive models are needed to calculate the great number of alternative mixtures, considering that VLE experiments are very time-consuming and expensive. In fact, when the group contribution method UNIFAC is used to predict the activity coefficient model, any excess free energy mixing rule can be predictable. Meanwhile, some predictive VDW mixing rules have also been proposed for mixed working fluids, as stated in Table 2. In order to compare the performance of VLE models with different mixing rules, three predictive models, namely PR þ VDW, PR þ MWS2þUNIFAC and PR þ MHV1þUNIFAC, are applied to predict the VLE behavior of alternative mixed working fluids, as listed in Table 5. These models are also called VDW, MWS2 and MHV1 methods respectively. The VDW is selected on account of the completely predictive ability for HFC and HC mixtures. The MWS2 and MHV1 have different reference pressures. The reason for selecting MWS2 and MHV1 is that the UNIFAC parameters fitted by VLE data can be directly used in these mixing rules. The group interaction parameters required for the VLE prediction of mixed working fluids can be obtained from Refs. [80] and [81]. Due to the importance of

Mixed working fluids

Year

N

T/K

Data sources

R290 þ R600 R290 þ R600a R152a þ R245fa R134a þ R245fa R125 þ R245fa R134a þ R290 R134a þ R600a R290 þ R245fa R600 þ R245fa R600a þ R245fa R13I1þR290 R13I1þR600a R13I1þR152a

2005 2005 2015 2001 2000 2011 1999 2000 2016 2001 2014 2013 2015

16 34 36 31 21 36 32 32 51 40 32 36 36

270.00e310.00 260.00e320.00 323.15e353.15 293.15e313.15 298.18, 313.22 255.00e285.00 303.15e323.15 298.18,313.22 303.15e373.15 293.15e313.15 258.15e283.15 263.15e293.15 258.15e283.15

[82] [82] [74] [83] [84] [85] [86] [84] [72] [83] [78] [79] [75]

fugacity in VLE calculation, the fugacity expressions based on the three models are also given in Table 5. In order to exhibit the capacities and limitations of these predictive models in predicting the VLE of alternative mixed working fluids, the published VLE data sets of 13 working fluids at different temperatures are used. As indicated in Table 6, these mixed working fluids can be categorized into HCs þ HCs binaries, HFCs þ HFCs binaries, HCs þ HFCs binaries and R13I1þ HCs/HFCs binaries. For some working fluids, VLE data were measured in recent years, and their data haven't been utilized to get the parameters of the predictive models. Thus, performances of different models for the four types of mixtures are discussed and compared. Table 7 presents the molecular structures, critical parameters and acentric factors of pure working fluids involved in these mixtures. The molecular structures and thermodynamic properties of pure fluids are essential to predict the VLE of mixed working fluids for the predictive models. The calculated results from three predictive models are presented in Figs. 2e14 for 13 mixed working fluids respectively. In Figs. 2e14, the points (,) are the experimental data of liquid phase, the points (B) are the experimental data of vapor phase, the red dot lines are the results of the VDW model, cyan dash lines are the results from the MWS2 model, and the black dash-dot lines are from the MHV1 model. In general, the results from the VDW, MWS2, and MHV1 models are all quite similar, especially for the MWS2 and MHV1. Due to the fact that the same values of UNIFAC

Table 5 Three predictive models for the VLE calculation of mixed working fluids. Models PR þ VDW

Fugacity expressions

Remarks

0 P 1 0 2 xj aij pffiffiffi 1 fi ðT; P; xi Þ bi am b C Z þ ð1 þ 2ÞBA B j pffiffiffi ðZ 1Þ lnðZ BÞ pffiffiffi i Aln@ ¼ ln @ xi P bm am bm 2 2bm RT Z þ ð1 2ÞB

kij¼kikjki ¼ 0:30ui þ 0:031n0:1 F =ui bm ¼

B ¼ bm P=RT PR þ MWS2þUNIFAC

0 pffiffiffi 1 f ðT; P; xi Þ C am C2 C Z þ ð1 þ 2ÞBA pffiffiffi 1 ln@ ln i ¼ 1 ðZ 1Þ lnðZ BÞ pffiffiffi xi P bm 2 2bm RT am bm Z þ ð1 2ÞB C1 ¼

2 X

a Q a ln g xj b 1 i WSi 2 1D j RT ij ð1 DÞ bi RT C

B ¼ bm P=RT

D¼

X j

PR þ MHV1þUNIFAC

xj

ai GE þ bi RT C WS RT

Q¼

XX i

j

C2 ¼ RTDC1 þ RTbm

ai ln gi þ bi RT C WS

bm b þ i 1 bi b

B ¼ bm P=RT

a xi xj b RT ij

0 pffiffiffi 1 f ðT; P; xi Þ b 1 ai A Z þ ð1 þ 2ÞBA pffiffiffi þ MHV1 ln@ ¼ i ðZ 1Þ lnðZ BÞ pffiffiffi ln i xi P bm 2 2 bi RT C Z þ ð1 2ÞB A ¼ ln gi þ ln

C WS ¼ 0:62323

CMHV1¼0.53

P

xi bi

38


Table 7 Molecular structures, critical parameters and acentric factors of pure working fluids. ASHRAE number

Refrigerant name

Molecular structure

Tc/K

Pc/MPa

u

R290 R600a R600 R125 R134a R152a R245fa R13I1

Propane Isobutane Butane Pentafluoroethane Norflurane Difluoroethane Pentafluoropropane Trifluoroiodomethane

CH3CH2CH3 CH(CH3)3 CH3CH2CH2CH3 CHF2CF3 CF3CH2F CHF2CH3 CF3CH2CHF2 CF3I

369.89 407.81 425.13 339.17 374.21 386.41 427.16 396.44

4.25 3.63 3.80 3.62 4.06 4.52 3.65 3.95

0.15 0.18 0.20 0.31 0.33 0.28 0.38 0.18

1.0

VDW MWS2 MHV1

0.8

2.4

300K

2.0

353.15K

1.6

P/MPa

0.6

P/MPa

VDW MWS2 MHV1

0.4

0.8

270K

0.2

1.2

323.15K 0.4

0.0 0.0

0.2

0.4

0.6

0.8

0.0

1.0

0.2

Molar fraction of R290

0.4

0.6

0.8

1.0

Molar fraction of R152a

Fig. 2. VLE prediction results for the R290 þ R600 system. Experimental data: liquid phase (,); vapor phase (B). The model: the red dot lines for VDW; the cyan dash lines for MWS2; the black dash-dot lines for MHV1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. VLE prediction results for the R152a þ R245fa system. Experimental data: liquid phase (,); vapor phase (B). The model: the red dot lines for VDW; the cyan dash lines for MWS2; the black dash-dot lines for MHV1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

interaction parameters are used for MWS2 and MHV1, the results of these two models overlap on the scale of the figures in some cases. From these figures, it can be seen that some mixed working fluids

have obviously azeotropic behaviors, which mean that the first derivative of the pressure with respect to the composition equals to zero at the azeotropic point for the binary system. The phase

1.6

1.0

320K

VDW MWS2 MHV1

VDW MWS2 MHV1

313.15K

0.8

300K

P/MPa

P/MPa

1.2

0.8

0.6

0.4

280K

293.15K 0.4

0.2

260K

0.0

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Molar fraction of R290 Fig. 3. VLE prediction results for the R290 þ R600a system. Experimental data: liquid phase (,); vapor phase (B). The model: the red dot lines for VDW; the cyan dash lines for MWS2; the black dash-dot lines for MHV1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

0.0

0.2

0.4

0.6

0.8

1.0

Molar fraction of R134a Fig. 5. VLE prediction results for the R134a þ R245fa system. Experimental data: liquid phase (,); vapor phase (B). The model: the red dot lines for VDW; the cyan dash lines for MWS2; the black dash-dot lines for MHV1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)


39

1.5

VDW MWS2 MHV1

2.0

323.2K

1.2

P/MPa

P/MPa

313.22K

0.8

0.9

303.2K

0.6

298.18K

0.4

0.0 0.0

0.2

0.4

0.6

0.8

0.3

1.0

0.0

0.2


behavior at the azeotropic point satisfies the following thermodynamic criterion:

dP dx

0.4

0.6

0.8

1.0

Molar fraction of R134a

Fig. 6. VLE prediction results for the R125 þ R245fa system. Experimental data: liquid phase (,); vapor phase (B). The model: the red dot lines for VDW; the cyan dash lines for MWS2; the black dash-dot lines for MHV1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

VDW MWS2 MHV1

1.2

1.6

Fig. 8. VLE prediction results for the R134a þ R600a system. Experimental data: liquid phase (,); vapor phase (B). The model: the red dot lines for VDW; the cyan dash lines for MWS2; the black dash-dot lines for MHV1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

N pexp pcal 1 X i i AARD P ¼ 100 N i¼1 pexp i

(26)

N 1 X exp yi ycal i N i¼1

(27)

AAD y ¼

¼0

(25)

T

Furthermore, Table 8 represents the deviations between the data calculated by these three models and the experimental data. Two statistical parameters, namely the average absolute relative deviation of pressure (AARD P) and the average absolute deviation of vapor composition (AAD y) are used for every mixture. The parameters are defined as follows:

Where N is the number of experimental data, the subscripts exp and cal mean experimental and calculated values respectively. For the VLE predictions of HCs þ HCs binaries, the results for R290 þ R600, R290 þ R600a at different temperatures are shown in Figs. 2 and 3 respectively. The two mixtures are the most symmetric fluids in the studied mixtures. As can be seen, two excess Gibbs 1.4

VDW MWS2 MHV1

0.8

VDW MWS2 MHV1

1.2

313.22K

1.0

285K 0.8

P/MPa

P/MPa

0.6

275K

0.6

0.4

265K

0.4

298.18K 255K

0.2

0.2 0.0

0.0

0.2

0.4

0.6

0.8

1.0

Molar fraction of R134a Fig. 7. VLE prediction results for the R134a þ R290 system. Experimental data: liquid phase (,); vapor phase (B). The model: the red dot lines for VDW; the cyan dash lines for MWS2; the black dash-dot lines for MHV1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

0.0

0.2

0.4

0.6

0.8

1.0

Molar fraction of R290 Fig. 9. VLE prediction results for the R290 þ R245fa system. Experimental data: liquid phase (,); vapor phase (B). The model: the red dot lines for VDW; the cyan dash lines for MWS2; the black dash-dot lines for MHV1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

40


0.7

VDW MWS2 MHV1

2.0

373.15K

VDW MWS2 MHV1

283.15K 0.6

1.6

273.15K

1.2

P/MPa

P/MPa

0.5

343.15K 0.8

0.4

263.15K 0.3

0.4

0.2

303.15K 0.0 0.0

0.2

0.4

0.6

0.8

0.1

1.0

0.0

0.2


0.4

0.6

0.8

1.0

Molar fraction of R13I1

Fig. 10. VLE prediction results for the R600 þ R245fa system. Experimental data: liquid phase (,); vapor phase (B). The model: the red dot lines for VDW; the cyan dash lines for MWS2; the black dash-dot lines for MHV1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 12. VLE prediction results for the R13I1þR290 system. Experimental data: liquid phase (,); vapor phase (B). The model: the red dot lines for VDW; the cyan dash lines for MWS2; the black dash-dot lines for MHV1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

energy models, MWS2 and MHV1 give similar results for the HCs þ HCs binaries at different temperatures. Compared with the excess Gibbs energy models, the VDW model gives less accurate results, and slightly underestimates the phase equilibrium pressure for the VLE system R290 þ R600. However, for the mixture R290 þ R600a, VDW presents a bit better correlation between the calculated results and the experimental data. The reason for these phenomena is that the binary interaction parameter of the used VDW is assumed to be the difference between the mixing factors of the pure components. However, this assumption has no foundations of theory and the correlation is developed only based on the experimental data of HCs/HFCs mixtures. Thus, the accuracy of VLE predictions for mixed working fluids depends on the number of experimental data used in the fitting process of VDW parameters.

The predictions for the VLE of R152a þ R245fa, R134a þ R245fa and R125 þ R245fa at different temperatures are shown in Figs. 4, 5 and 6 respectively. The best results are obtained by the VDW model, which gives good agreements with the experimental data for all the HFCs þ HFCs binaries. The MWS2, MHV1 models give good results only for the mixtures R134a þ R245fa, R125 þ R245fa. Although the three mixtures are homogeneous zeotropes, the results from the excess free energy models deviate far from the experimental data of R152a þ R245fa, and with the increase of temperature, the deviations increase obviously. This is caused by the fitted UNIFAC parameters. The early experimental data of R134a þ R245fa and R125 þ R245fa were used to fit the interaction parameters of UNIFAC groups in Ref. [80]. Thus, it's reasonable that the accurate VLE predictions are obtained for R134a þ R245fa and

0.5

VDW MWS2 MHV1

0.6

0.4

0.5

293.15K

283.15K

0.4

0.3

P/MPa

P/MPa

VDW MWS2 MHV1

313.15K

293.15K

0.3

273.15K 0.2

263.15K

0.2

0.1 0.1 0.0

0.2

0.4

0.6

0.8

1.0

Molar fraction of R600a Fig. 11. VLE prediction results for the R600a þ R245fa system. Experimental data: liquid phase (,); vapor phase (B). The model: the red dot lines for VDW; the cyan dash lines for MWS2; the black dash-dot lines for MHV1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

0.0

0.2

0.4

0.6

0.8

1.0

Molar fraction of R13I1 Fig. 13. VLE prediction results for the R13I1þR600a system. Experimental data: liquid phase (,); vapor phase (B). The model: the red dot lines for VDW; the cyan dash lines for MWS2; the black dash-dot lines for MHV1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)


VDW MWS2 MHV1

0.4

283.15K

P/MPa

0.3

273.15K

0.2

263.15K

0.1 0.0

0.2

0.4

0.6

0.8

1.0

Molar fraction of R13I1 Fig. 14. VLE prediction results for the R13I1þR152asystem. Experimental data: liquid phase (,); vapor phase (B). The model: the red dot lines for VDW; the cyan dash lines for MWS2; the black dash-dot lines for MHV1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

R125 þ R245fa. However, the experimental data of R152a þ R245fa were measured in 2015 and not used in Ref. [80]. So, for the fitted parameters of groups, the phase behavior of the new mixtures may not be described well by the excess Gibbs energy mixing rules. For the mixture R152a þ R245fa, MWS2, MHV1 overestimates the phase pressure greatly. The prediction results for more asymmetric binary systems are presented in Figs. 7e11 for R134a þ R290, R134a þ R600a, R290 þ R245fa, R600 þ R245fa and R600a þ R245fa respectively. These mixtures belong to HCs þ HFCs binaries. For the considered mixed working fluids, the VDW model correlates the data well. In some cases, the accuracy of the MHV1 can be comparable with the VDW model. However, for most mixtures, the MHV1 underestimates the pressure. As for the MWS2, it is the least accurate. Perhaps, the reason is that when fitting the UNIFAC group interaction parameters, VLE data with a specified GE mixing rule, instead of the activity coefficient data, are used. Therefore, for some mixing rules, such as MWS2, MHV1, the fitted parameters may be not applicable. In addition, most of the HCs þ HFCs binaries have obviously azeotropic behaviors, such as R134a þ R290, R134a þ R600a. As can be seen, all the models can predict the

Table 8 VLE calculated results from the three predictive models: PR þ MWS2þUNIFAC, PR þ MHV1þUNIFAC. Mixed working fluids

R290 þ R600 R290 þ R600a R152a þ R245fa R134a þ R245fa R125 þ R245fa R134a þ R290 R134a þ R600a R290 þ R245fa R600 þ R245fa R600a þ R245fa R13I1þR290 R13I1þR600a R13I1þR152a

AARD P

PR

þ VDW,

AAD y

VDW

MWS2

MHV1

VDW

MWS2

MHV1

3.39 0.72 0.56 0.74 0.83 0.91 0.78 0.93 1.76 0.71 2.52 4.17 3.67

1.67 2.52 14.32 0.77 0.80 6.67 2.70 8.83 10.95 11.47 0.40 0.77 1.18

1.77 2.29 16.96 0.71 0.83 2.54 1.17 7.25 8.62 8.78 0.47 0.29 0.53

0.0099 0.0091 0.0033 0.0022 0.0029 0.0077 0.0054 0.0025 0.0056 0.0048 0.0077 0.0129 0.0434

0.0092 0.0084 0.0323 0.0033 0.0032 0.0167 0.0139 0.0304 0.0308 0.0470 0.0022 0.0029 0.0024

0.0089 0.0083 0.0365 0.0031 0.0028 0.0062 0.0044 0.0245 0.0242 0.0348 0.0029 0.0043 0.0037

41

azeotropic point accurately, that's why the UNIFAC parameters obtained from ref [80] are still widely used to get the azeotropic point of mixed working fluids, even though large deviations exist between the calculated results and the VLE data. The results for the correlation of the R13I1 þ HCs/HFCs binaries at different temperatures are given in Figs. 12e14. For the three mixtures R13I1þR290, R13I1þR600a and R13I1þR152a, the MHV1 model gives the most accurate results. When presenting the phase behaviors of R13I1þR290 and R13I1þR600a, another EOS/GE model, MWS2 has the same capacity as MHV1. However, for the binary R13I1þR152a, MWS2 slightly underestimates the phase pressure. The reason for the differences between the results of these two models is that MHV1 was used to correlate the R13I1 þ HCs/HFCs binaries in Ref. [81], in order to obtain the UNIFAC group interaction parameters. Thus, best results are obtained with the MHV1 model. Although the same parameters are used for the activity coefficient in the MWS2, the good agreement with the experimental data for the mixtures can't be assured by the MWS2 model. For the VDW model, large deviations between the predicted results and experimental data exist in these three mixtures. The reason is that the VDW model was proposed only based on the experimental data of HCs/HFCs. Thus, it's not appropriate to use VDW model in the VLE prediction of R13I1 þ HCs/HFCs binaries.

6. Knowledge gaps and development directions Many cubic EOSs and mixing rules have been widely combined to predict the phase behavior of mixed working fluids. The accuracy of a model depends on the used cubic EOS, the chosen mixing rule, and the mixture studied. So far, the most accurate descriptions on the VLE of mixture are always obtained by the models, whose interaction parameters are fitted based on the mixture's experimental data. However, considering the huge number of alternative mixed working fluids, predictive models are needed urgently to calculate the thermodynamic properties of working fluids. In general, two-parameter cubic EOSs are accurate enough to describe the VLE behavior of working fluids. Thus, the key for VLE prediction of mixed working fluids lies in the mixing rules for parameters of EOSs. For the Van der Waals mixing rules, a variety of expressions have been proposed to obtain the binary interaction parameter kij. But, most of these correlations are only valid for some types of mixture, and the empirical equations from the VLE data can't guarantee the VLE accuracy of mixed working fluids. Although the cross co-volume parameter can be correlated with the hard sphere diameter theoretically, the interaction parameter for cross covolume has little effect on the VLE calculation in comparison with that of the cross energy parameter. Thus, a generalized VDW model for mixed working fluids should be derived from the molecular theory or group contribution methods. The group contribution values can be obtained from the large VLE data fitting. As for the excess free energy mixing rules, the energy parameter is expressed by the excess free energy GE, the co-volume parameter is generally obtained using a linear mixing rule. When the group contribution method UNIFAC is used for the activity coefficient, any model with GE mixing rules can be predictable. However, the UNIFAC group interaction parameters fitted from VLE data by a GE mixing rule may not be suitable for other EOS/GE models. The reason for this is that the interaction parameters aren't derived from the activity coefficient data. Therefore, a predictive model, which consists of a cubic EOS, a GE mixing rule and corresponding UNIFAC group parameters, should be developed in the future, given the unavailability of activity coefficient data for most of the mixed working fluids.

42


7. Conclusions Recent advances in modeling the vapor-liquid equilibrium of mixed working fluids have been reviewed. Based on the methodology of VLE calculation for mixed working fluids, the most frequently-used cubic EOSs has to combine with the mixing rules, so that the phase behavior of mixtures can be described by the EOSs with parameters obtained from the mixing rules. The most successful cubic EOSs in expression of the VLE of mixtures have been presented in this review. Thereafter, two types of mixing rules, namely Van der Waals and excess free energy are summarized respectively. For the Van der Waals mixing rules, extensive researches focus on the cross energy parameter of the EOS, which has a significant effect on the VLE prediction. Although quite a few semi-empirical correlations have been proposed to obtain the cross co-volume parameter, a linear mixing rule is always used in the VLE calculation of mixed working fluids for simplicity. In the excess free energy mixing rules, the energy parameter is expressed by the excess free energy. When the group contribution activity coefficient method UNIFAC is used to calculate the excess free energy, the formed models can predict the phase behavior of mixed working fluids without experimental data. So, experimental data of 13 mixed working fluids at different temperatures are used to evaluate the performances of three predictive models VDW, MWS2, MHV1 in this review. The results show that no predictive models can give good agreements with the experimental data for all the mixed working fluids. Therefore, it's thought that a generalized model should be derived from the molecular theory or the group contribution method, in which the group parameters are fitted from the large VLE data. Acknowledgements This work is sponsored by the National Nature Science Foundation of China (51476110). Nomenclature

Symbols a A AARD P AAD y b B CFC DDB EOS f G GWP HC HFC HCFC k l N ODP P PR SRK T v VDW

Energy parameter Excess Helmholtz free energy Average absolute relative deviation of pressure Average absolute deviation of vapor composition Co-volume parameter Second virial coefficients Chlorofluorocarbon Dortmund Data Bank Equation of state Fugacity Free energy Global warming potential Hydrocarbon Hydrofluorocarbon Hydrochlorofluorocarbon Binary interaction parameter for energy parameter Binary interaction parameter for co-volume parameter Number of experimental data Ozone depletion potential Pressure PengeRobinson SoaveeRedlicheKwong Temperature Molar volume Van der Waals

VLE x y z Z

Vapor-liquid equilibrium Molar fraction Molar fraction of vapor phase Molar fraction of liquid phase Compressibility factor

Greeks

g

Activity coefficient

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