Predicting Gas Transport Coefficients of Alternative ... - ACS Publications

Ind. Eng. Chem. Res. 2006, 45, 9211-9223

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CORRELATIONS Predicting Gas Transport Coefficients of Alternative Refrigerant Mixtures J. Moghadasi,† D. Mohammad-Aghaie,*,† and M. M. Papari‡ Department of Chemistry, College of Sciences, Shiraz UniVersity, Shiraz 71454, Iran, and Department of Chemistry, Shiraz UniVersity of Technology, Shiraz 71555-313, Iran

The purpose of this paper is to extract information about intermolecular pair interaction potential energies of mixtures of ozone-friendly refrigerants by the usage of the inversion method and then to predict the dilute gas transport properties of this class of fluids. The components of studied mixtures are nine environmentally acceptable refrigerants consisting of R125, R134a, R152a, R143a, R32, R236ea, R236fa, R227ea, and RC318. Using the inverted pair potential energies, the Chapman-Enskog scheme is employed to calculate transport properties of the aforementioned systems excluding thermal conductivity. The calculation of thermal conductivity using inverted pair potential energies through Schreiber’s scheme is discussed. In the temperature range 200 K < T < 1000 K, the accuracies of the calculated viscosities and the thermal conductivities amount to (2% and at most (15%, respectively. 1. Introduction The depletion of the ozone layer and global warming are still one of the main environmental problems that the world is facing. From the 1930s, chemical product chlorofluorocarbons (CFC) were widely used as refrigerants, blowing agents, cleaning agents, fire extinguishing agents, and spray propellants in refrigeration and air-conditioning, foaming, fire protection, electronics, and medicine industries, because of their favorable physical and chemical properties. In the 1980s, scientists found that CFC substances not only do great damage to the ozone layer of atmosphere but also bring the greenhouse effect to earth and badly affect the telluric environment and human beings’ health.1 Potential alternatives for the chlorofluorocarbons (CFCs) and hydrochlorofluorocarbons (HCFCs) are the hydrofluorocarbons (HFCs), such as HFC-125, HFC-134a, HFC-143a, and HFC32, and their binary and/or ternary mixtures.2 From an engineering point of view, the design of refrigeration or air-conditioning equipment requires the knowledge of transport properties such as viscosity and thermal conductivity of the working fluid in the thermodynamic cycle to determine the size of the heat-transfer equipment and fluid pumps required to meet a specified duty. It should be added that, from a technological and industrial point of view, fluid mixtures are of rather greater importance than pure fluids simply because they are encountered more frequently. It is, therefore, important that it should be possible to present, or predict, the transport properties of multicomponent mixtures of arbitrary compositions from a limited set of information. Therefore, accurate thermal conductivity and viscosity data are required for HFC refrigerant mixtures when designing new refrigeration systems. Compared with research on thermodynamic properties, data for transport properties of HFC refrigerant mixtures are very scarce at present. * Corresponding author. Fax: 98-711-735-4523. E-mail: d_aghaie@ yahoo.com. † Shiraz University. ‡ Shiraz University of Technology.

It is not possible to measure the transport properties of all relevant mixture compositions; hence, prediction methods, based on rigorous theory, become more and more important in the evaluation of transport properties of HFC refrigerant mixtures.3 The results of kinetic and statistical-mechanical theories provide theoretical expressions for various equilibrium and nonequilibrium (transport) properties in terms of the potential energy of interaction between a pair of molecules.4 Thus, the forces between molecules are of interest to scientists in a wide range of disciplines because these interactions control the progress of molecular collisions and determine the bulk properties of matter and, therefore, have received considerably much attention. Consequently, the evaluation of the thermophysical properties of fluids will be straightforward if a pair potential energy is already known. This procedure reduces the need for experimentation to a manageable level. For instance, the most successful and promising approach of this type is the calculation of the transport properties of dilute gases from the known pairwise interaction potential energy using the kinetic theory of gases initiated by Boltzmann5 and developed by Chapman and Enskog.6 This work is concerned with determining effective and isotropic pair potential energies for some alternative refrigerant mixtures from corresponding states correlation of viscosity using the inversion method and then predicting dilute gas transport coefficients of them, encompassing a wide range of temperatures. In preceding works, transport properties of some nonpolar polyatomic gases7,8 and also mixtures of polyatomic gases with noble gases have been reported.9-11 The novelty of the present study is that the method has been extended to predict transport properties of slightly polar ozone-safe refrigerant mixtures. The components of the present mixtures are HFCs including R125 (pentafluoroethane), R134a (1,1,1,2-tetrafluoroethane), R152a (1,1-difluoroethane), R143a (1,1,1-trifluoroethane), R32 (difluoromethane), R236ea (1,1,1,2,3,3-hexafluoropropane), R236fa (1,1,1,3,3,3-hexafluoropropane), R227ea (1,1,1,2,3,3,3heptafluoropropane), and RC318 (octafluoropropane). We employed the method developed by Schreiber et al.12 to predict

10.1021/ie060630v CCC: $33.50 © 2006 American Chemical Society Published on Web 11/18/2006

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Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006

the thermal conductivities of the present mixtures. The Chapman-Enskog version of the kinetic theory of gases6 was used to predict the remaining transport properties including viscosity coefficients, binary diffusion coefficients, and thermal diffusion factors. The predicted viscosities agree with the experiment to within 2% in the temperature range 200-1000 K, which is the industrial interest range. The thermal conductivity is predicted with accuracies of 15% over the same temperature range. Some comparisons have been made for viscosity and thermal conductivity of ternary refrigerant mixtures with literature data. It should be mentioned that, in the limit of zero density, properties of ternary mixtures can be taken as dependent only on binary potential parameters. The close agreement between the predicted transport properties and the experiment demonstrates the predictive power of the inversion scheme.

u(r) would serve as the input information required in calculating collision integrals and, consequently, the transport properties. It is evident that three successive numerical integrations are required to obtain collision integrals. The definition of collision integrals as dimensionless reduced quantities makes calculations of transport properties more convenient. The reduced collision integral is defined as

Ω*(l,s) )

θ ) π - 2b

∫r∞ m

[

r-2 dr 2u(r) b2 1- 2 r mw2

{ ( ) [ ]}

]∫

1 + (-1)l Q(l)(E) ) 2π 1 2(1 + l) Ω(l,s)(T) ) [(s + 1)!(kT)s+2]-1

-1

∞

0

1/2

(1)

(1 - cosl θ)b db (2)

∫0∞ Q(l)(E)

(4)

where range parameter σ denotes the intermolecular separation for which the potential is zero. This quantity is exactly unity for rigid spheres of diameter σ. Numerical differentiation of the aforesaid collision integrals and usage of the recursion relation can generate collision integrals higher than the ones mentioned. That is,4

[

2. Molecular Theory of Transport Properties of Gases The values of transport properties of gases are the macroscopic manifestation of the microscopic motions and interactions of the molecules that comprise the gas. Accurate knowledge of transport properties at low density is essential for the development of accurate theories of transport properties in the dense state. As was stated earlier, statistical mechanics and kinetic theory of gases provide equations showing the relation between macroscopic and microscopic properties of systems at equilibrium and nonequilibrium states, respectively. The transport properties are computed using the method of Sonine’s polynomial expansion of the first-order ChapmanEnskog approximation of Boltzmann’s equation.6 The transport properties appear finally in the Chapman-Enskog theory as a solution of infinite sets of simultaneous algebraic equations and can be expressed formally as ratios of infinite determinants whose elements are the coefficients of the algebraic equations. The coefficients of the equations are complicated functions, which depend on the species, the composition of the mixture, and the integrals related to binary molecular interactions through collision integrals shown by Ω(l,s).6 The superscripts l and s appearing in Ω denote weighting factors that account for the mechanism of transport by molecular collision. For example, for the two transport properties, viscosity and diffusion, the superscripts assume values of l ) 2, s ) 2, and l ) 1, s ) 1, respectively. Collision integrals are related to intermolecular forces according to the following relations,4

Ω(l,s) πσ2

Ω*(l,s+1) ) Ω*(l,s) 1 +

]

1 d ln Ω*(l,s) s + 2 d ln T*

(5)

where reduced temperature T* is usually defined as

T* ) kT/

(6)

where the energy parameter, , represents the depth of the potential energy well. In the higher approximations for transport properties, several recurring ratios of collision integrals are defined for calculation convenience, namely,4

A* )

B* )

Ω*(2,2) Ω*(1,1)

[5Ω*(1,2) - 4Ω*(1,3)] Ω*(1,1) C* )

E* )

F* )

Ω*(1,2) Ω*(1,1) Ω*(2,3) Ω*(2,2) Ω*(3,3) Ω*(1,1)

(7)

(8)

(9)

(10)

(11)

These ratios are weak functions of T*. The magnitude of each of these ratios is approximately unity, and exactly unity for rigid spheres. Collision integrals and their ratios are functions of temperature and the parameters of the selected model for intermolecular forces.

exp(-E/kT)Es+1 dE (3) 3. Applicable Formula for Transport Properties where θ is the scattering angle, Q(l)(E) is the transport collision integral, b is the impact parameter, E is the relative kinetic energy of colliding partners, w is the relative velocity of colliding molecules, rm is the closest approach of two molecules, and kT is the molecular thermal energy. Hence, the potential

The kinetic theory expressions for viscosity, diffusion coefficients, and thermal diffusion factor in terms of collision integrals for pure gases and the mixture of gases are given below:

Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9213

3.1. Viscosity Coefficient. (i) Single compound:4

η)

aij )

5 mkT 1/2 fη 16 π σ2Ω*(2,2)

( )

fη ) 1 +

3 (8E* - 7)2 196

(13)

in which E* is the ratio of collision integrals defined by eq 10. (ii) Mixtures:4

|

H11 H21 l Hν1 x1

ηmix ) -

Hii )

xi2

ν

+

ηi

H12 H22 l Hν2

2xixk

k*i

[(

H11 H21 l H ν1

∑ k)1 η

Hij(i * j) ) -

ηij )

|

ik

‚‚‚ ‚‚‚ • ‚• ‚‚‚ ‚‚‚

H12 H22 l Hν2 x2

H1ν H2ν l Hνν xν

x1 x2 l xν 0

‚‚‚ H1ν ‚‚‚ H2ν • ‚• l ‚‚‚ Hνν

(

mimk

|

5

(mk + mi) 3Aik 2

|

+ *

(

(14)

)]

1/2

)

mk mi

2xixj mimj 5 -1 ηij (m + m )2 3A * i j ik

5 2mimj kT 16 mi + mj π

)

1 2

σij Ωij

*(2,2)

(15)

[(

)]

3 mi + mj kT 8 2mimj π

(16)

(17) (Tij*)

1 + ∆ij kT P σ 2Ω *(1,1)(T *) ij ij ij

1/2

where

xi xi + xj

xij )

It is crucial to note that, although given equations for the viscosity and diffusion are formulated for monoatomic gases, they have also found applicability for polyatomic gases as well. The reason for this is that, in the equations of conservation of mass and momentum for a collision between polyatomic molecules, the center of mass coordinates are more important than the internal coordinates. The Chapman-Enskog version of the kinetic theory of gases6 is applied strictly to molecules that have no internal degrees of freedom. Polyatomic molecules differ not only because they interact through nonspherically symmetric intermolecular pair potentials but also because they possess internal degrees of freedom in the form of rotational and vibrational modes of motion. It is obvious that, because the viscosity and diffusion coefficients are concerned with transporting momentum and mass, respectively, and therefore do not involve in internal degrees of freedom, the ChapmanEnskog theory retains its useful form, but collision integrals must be averaged over all possible relative orientations occurring in collisions. Monchick and Mason13 proposed a simplification of this treatment. The Monchick-Mason collision integrals are given by

〈Ω(2,2)(T)〉 )

∫0∞ ∫02π ∫0∞ γ6{(1 - cos2 φ)b db dψ}

1 4

(18)

where γ ) (m/2kT)1/2w and ψ is the azimuthal angle. For the sake of brevity, we use the notation Ω*(l,s) instead of 〈Ω* (l,s)〉 in the present paper. 3.3. Thermal-Diffusion Factor. The temperature gradients can cause mass fluxes by a process known as thermal diffusion. An expression for the thermal-diffusion factor of a binary mixture is

(

RT ) (6C*ij - 5)

(19)

x1S1 - x2S2

x1 Q1 + x22Q2 + x1x2Q12 2

)

(1 + kT) (22)

where kT is a higher-order correction term for the thermaldiffusion factor. This term is usually negligible compared with experimental uncertainties in RT. The other quantities in eq 22 are4

S1 )

(

m1 2m2 m2 m1 + m2

)

1/2 σ

where P is the pressure and ∆ij is a higher-order correction term of the binary diffusion coefficient, which can be defined as4

aijxij ∆ij ) 1.3(6Cij* - 5)2 1 + bijxij

(20)

exp(-γ2) dγ2 (21)

Aij* is the ratio of the collision integrals defined by eq 7, x is the mole fraction of components, and ηij is the interaction viscosity. In the above equations, subscript i represents the heavier component and subscript j represents the lighter component of the i-j pair. 3.2. Diffusion Coefficient. Diffusion in multicomponent mixtures is entirely described in terms of the binary diffusion coefficients, Dij,4

Dij )

bij ) 10aij[1 + 1.8(mj/mi) + 3(mj/mi)2] - 1

(12)

where m is the mass of a molecule and fη is the higher-order correction factor for the viscosity coefficient that may be defined by4

Ωij*(1,1)(Tij*) x2 8[1 + 1.8(mj/mi)]2 Ωjj*(2,2)(Tij*)

2 11

Ω11*(2,2)(T1*)

σ122Ω12*(1,1)(T*12)

-

4m1m2A12*

+ (m1 + m2)2 15m2(m1 - m2) 2(m1 + m2)2

(

)

2m2 2 Q1 ) m2(m1 + m2) m1 + m2

[(

1/2 σ

(23)

2 11

Ω11*(2,2)(T1*)

σ122Ω12*(1,1)(T12*)

]

8 5 6 * 2 - B m + 3m22 + m1m2A12* (24) 2 5 12 1 5

)

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(

Q12 ) 15

(

)(

m1 - m2 m1 + m2

)

2

)

4m1m2A12* 5 6 * - B12 + 2 5 (m1 + m2)2

2 *(2,2) σ222 Ω22*(2,2) 12 * 8 (m1 + m2) σ11 Ω11 11 - B12 + (25) 5 5 (m m )1/2 σ 2Ω *(1,1) σ 2 Ω *(1,1) 1 2 12 12 12 12

The expressions for S2 and Q2 are obtained from those of S1 and Q1 by interchanging subscripts 1 and 2. The sign convention for RT requires that subscript 1 denotes the heavier component. 3.4. Thermal Conductivity. Because of the fact that the internal degrees of freedom of polyatomic molecules are involved in transporting energy in gases, and on the other hand, in the basic development of the Chapman-Enskog theory, only binary elastic collisions between the molecules are considered and also molecules are taken to be without internal degrees of freedom; thus, this theory cannot be applied to predict thermal conductivity. To calculate the thermal conductivity of the present refrigerant mixtures, we employed the method proposed by Schreiber12 for polyatomic dilute gas mixtures. This method will be introduced in the next section. This scheme has been tested against the available experimental data for some nonpolar mixtures.14 3.4.1. Theory of Thermal Conductivity. The thermal conductivity of a multicomponent polyatomic gas mixture at zero density can be expressed in the form analogous to that for a mixture consisting of monoatomic species,12

|

‚‚ ‚‚ ‚‚ ‚‚

L11 : λ)- L n1 x1

L1n : Lnn xn

x1 : xn 0

|/| | L11 ‚‚ L1n : ‚‚ : Ln1 ‚‚ Lnn

(26)

where xi is the mole fraction of species i and the symbol λ indicates the full formal first-order kinetic theory result obtained by means of expansion in Thijsse basis vectors.15 The resulting expressions for the elements of the determinants, Lij, were first derived by Ross et al.16 and were complicated functions of the effective cross sections and had little value for practical evaluation of thermal conductivity. It was shown for pure polyatomic gases,17 atom-diatom mixtures,18 and atommolecule mixtures19 that accurate and relatively simple expressions can be obtained by means of the Thijsse approximation, which identifies the total energy as the dominant factor in determining thermal conductivity. To simplify, all the quantities that enter the expressions for the elements Lij in eq 26 were replaced by their spherical limits based on the results obtained for an atom-molecule mixture.19 Following the application of Thijsse and spherical approximations to the full results, Schreiber12 derived the relevant determinant elements, Lij, as

Lqq )

xq2 λq

+

∑ µ*q

( )[

25xqxµ R

* 8Aqµ λqµ

C0pq

2

25 4 15 4 * + yµ + yq - 3yµ4Bqµ 4 2 2

* 4yq yµ2Aqµ

Lqq' ) -

( )( )[

+

(

C0pq R

- 2.5

25xqxq'yq2yq'2 R R 55 - 3B*qq' - 4A*qq' * 0 8Aqq'λqq' Cpq C0pq' 4

)]

]

(27)

(28)

where λq is the thermal conductivity of pure molecular species q, λqq′ is the interaction thermal conductivity, C0pq is the ideal-

gas isobaric heat capacity of q, R is the gas constant, and the quantities A* and B* are ratios of effective cross sections given by eqs 7 and 8. In addition, yq is the mass ratio of species q, given by

yq2 )

Mq (Mq + Mq')

(29)

where Mq is the relative molecular weight of species q. The interaction thermal conductivity can be related to the more readily available viscosity, ηqq′,

λqq' )

15 (Mq + Mq') ηqq' R 8 MqMq'

(30)

Evaluation of the thermal conductivity of a multicomponent polyatomic gas mixture thus requires a knowledge of the thermal conductivity and the isobaric heat capacity of each of the pure species. This information is readily available for a large number of fluids as a function of temperature, either in terms of correlations or directly from experimental information. Furthermore, three binary interaction parameters, namely, ηqq′, A*, and B*, are also required as a function of temperature. In the present work, these quantities have been computed from the calculated interaction viscosity and the collision integral ratios obtained from the inversion method outlined in the next section. 4. Determining Pair Potential Energy from the Law of Corresponding States It should be mentioned that, because at low density the viscosity is almost independent of the existence of internal degrees of freedom and, therefore, is nearly unaffected by inelastic collisions, this property is one of the best reliable sources for establishing intermolecular potential energies. In recent years, many attempts have been made to establish the nature of interactions between atoms and molecules. However, no direct way to measure the intermolecular potential is known, and what is generally available are measurements of some macroscopically observable quantities having some functional dependence on the intermolecular forces. Determination of intermolecular potential energy from observable properties such as transport properties, thermodynamic and structural properties of condensed phase such as its crystal structure, elastic constants and enthalpy of sublimation (all experimentally accessible),20 and speed of sound data21-23 has been the subject of extensive research. Historically and traditionally, the earliest estimates of intermolecular potentials were based on macroscopic properties, and they proceeded through parametrized models.24-26 The main disadvantages of this approach were (1) computing limitations restricted the number of adjustable parameters in the model potential energy, to two or three, so that the models used were always too crude, and (2) parameters determined from one property, say, viscosity, were slightly different from another property, e.g., second virial coefficient. The direct inversion scheme, proceeding from data to potential without explicit assumption of a mathematical model, opened a desirable way of tackling the problem. Smith and co-workers27-31 defined an iterative version of this method. Papari and co-workers7-11 have successfully developed this method to infer intermolecular potential energy from the law of corresponding states principle of viscosity and then to predict transport properties of dilute systems using the inverted model

Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9215 Table 1. Reduced Collision Integrals and Their Ratios for Refrigerant Mixtures log T*

Ω*(1,1)

A*

B*

C*

E*

F*

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

2.5802 2.3117 2.0559 1.8207 1.6128 1.4354 1.2887 1.1698 1.0744 0.9979 0.9361 0.8851 0.8422 0.8050 0.7721 0.7422 0.7146 0.6890 0.6653 0.6440 0.6253 0.6097 0.5973 0.5881 0.5817 0.5777

1.0771 1.0938 1.1055 1.1106 1.1099 1.1055 1.0999 1.0951 1.0922 1.0915 1.0929 1.0956 1.0993 1.1033 1.1073 1.1109 1.1136 1.1142 1.1115 1.1046 1.0933 1.0787 1.0626 1.0470 1.0332 1.0221

1.2893 1.2958 1.2844 1.2588 1.2256 1.1914 1.1609 1.1366 1.1189 1.1071 1.0998 1.0958 1.0938 1.0930 1.0928 1.0924 1.0906 1.0859 1.0770 1.0641 1.0486 1.0328 1.0188 1.0077 0.9998 0.9950

0.8470 0.8511 0.8263 0.8231 0.8269 0.8369 0.8515 0.8684 0.8853 0.9007 0.9137 0.9240 0.9318 0.9374 0.9414 0.9441 0.9462 0.9482 0.9509 0.9548 0.9602 0.9668 0.9739 0.9810 0.9873 0.9925

0.9036 0.8909 0.8780 0.8692 0.8672 0.8724 0.8832 0.8973 0.9122 0.9259 0.9374 0.9462 0.9527 0.9570 0.9598 0.9612 0.9614 0.9604 0.9586 0.9570 0.9502 0.9593 0.9639 0.9703 0.9773 0.9840

0.9003 0.8962 0.8947 0.8962 0.9008 0.9086 0.9193 0.9318 0.9451 0.9582 0.9701 0.9805 0.9892 0.9964 1.0023 1.0072 1.0116 1.0161 1.0204 1.0240 1.0260 1.0259 1.0240 1.0208 1.0172 1.0139

potential energy. Since the detail of this procedure has been given elsewhere,7,9 only a brief description will be given here. The necessary requirement to perform the inversion procedure is the determination of G, the inversion function, from an initial intermolecular potential energy, such that the following equations can be applied,27

u ) u * ) GT*

(31)

r ) r* ) (Ω*(2,2))1/2 σ

(32)

for inverse power potential functions, where G is a numerical constant. In this work, G is estimated from the Lennard-Jones (12-6) model potential, following Viehland et al.32 The inversion procedure begins by employing the experimental viscosity collision integral from corresponding states * correlation equations to invert data points (Ω*(2,2) exp , T ) to their corresponding values (u/, r/σ) on the potential energy curve through eqs 31 and 32. The new potential, which is a closer approximation to the true potential energy than the potential of the initial model, is used to calculate collision integrals through eqs 1-3. The above process is repeated until convergence occurs. The convergence condition is judged by the extent to which the calculated collision integrals are in accord with those obtained from the corresponding states correlations and by the degree to which the intermolecular potential energies obtained by the inversion reproduce thermophysical properties consistent with the experimental values. The present results that converged after two iterations are given in the next section. 5. Results and Discussion As already outlined, the inversion procedure is of considerable importance to obtain nonparametric interaction potential energy and transport properties. This scheme relives us of the variation of the selected multiparameter analytic equation parameters for the pair potential function so as to optimize the fit to a wide range of thermophysical data of a material.

In this study, an iterative inversion procedure has been employed to define the intermolecular pair interaction potential energies of refrigerant mixtures from corresponding states correlation for viscosity. Then, using the inverted pair potential energies along with Chapman-Enskog6 version of the kinetic theory of gases and the method proposed by Scheriber,12 transport properties of studied refrigerant mixtures with acceptable accuracies have been predicted. To implement the full inversion procedure, the experimental data should be extended over as wide a temperature range as possible. In this respect, a corresponding states correlation for viscosity collision integral, embracing the temperature range 0.3 e T* e 100, was taken from ref 33 to calculate the reduced viscosity collision integral (Ω*(2,2)). In the case of each refrigerant mixture, a two-iterative inversion procedure, outlined in the previous section, was applied to the calculated reduced viscosity collision integrals to generate isotropic and effective pair potential energies of respective systems. The inversion of viscosity collision integrals, to yield potential energy, requires experimental data over a wide range of temperature. Consequently, to integrate eqs 1-3, over the given range, u(r) should be extrapolated in the long-range region (low temperature). The long-range part of u(r) has the following form:

u(r) )

-C6 r

6

-

C8 r

8

-

C10 ... r10

(33)

The effects of C8 and C10 on the transport properties were so small that we neglected them in our calculations. The value of C6 was estimated from the low-temperature viscosity data using eq 33. Therefore, the effective potential energies obtained from the inversion method were used to perform the integration over the whole range and, in turn, to evaluate the improved kinetic theory collision integrals over the given range. Results for the most commonly needed collision integrals and their ratios, obtained from eqs 7-11, are given in Table 1. Also the calculated collision integrals, Ω*(1,1) and Ω*(2,2), are correlated with the following polynomial equations, which are of third order with respect to ln T*

Ω*(1,1) ) a1 + b1 ln T*+ c1(ln T*)2+ d1(ln T*)3

(34)

Ω*(2,2) ) a2 + b2 ln T*+ c2(ln T*)2 + d2(ln T*)3

(35)

In the case of the ratios of collision integrals, the results are correlated with the fifth-order polynomial equations in terms of T*. The equations have the following form,

y ) a + b ln T*+ c(ln T*)2+ d(ln T*)3+ e(ln T*)4+ f(ln T*)5 (36) where y stands for A*, B*, C*, E*, and F*. Parameters in eqs 34-36 were allowed to vary using the nonlinear least-squares method and are listed in Table 2, along with correlation coefficients (R) and standard errors (ES) for each case. It was found that the results obtained after a particular number of iterations are insensitive to the variation of both the initial approximation of the potential energy and the extrapolations employed, provided of course that the initial potential is a reasonable guess of the true one. Our calculations showed that the results obtained after three iterations were identical to those obtained after two iterations. Therefore, the results of the inversion procedure converged after two iterations.

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Table 2. Least Squares Coefficients, Correlation Coefficients (R), and Standard Errors (SE) for Equations 34-36 a1 b1 c1 d1

parameters in eq 34 1.467 949 1 -0.683 092 58 0.20685684 -0.022 314 45

R ) 0.999 073 57 SE ) 0.026 243 22

a2 b2 c2 d2

parameters in eq 35 1.612 286 6 -0.734 560 9 0.22463863 -0.024 878 562

R ) 0.999 124 55 SE ) 0.028 039 77

a b c d e f

parameters in eq 36, y ) A* 1.104 134 4 -0.002 135 899 8 -0.004 825 715 9 R ) 0.998 412 00 0.021 057 048 SE ) 0.001 544 11 -0.008 087 968 9 0.000 820 147 94

a b c d e f

parameters in eq 36, y ) B* 1.194 359 6 -0.130 281 65 0.019 326 273 R ) 0.999 397 16 0.032 076 64 SE ) 0.003 477 02 -0.014 070 309 0.001 540 887 9

a b c d e f

parameters in eq 36, y ) C* 0.841 573 97 0.055 482 308 0.024 338 154 R ) 0.996 674 61 -0.023 921 18 SE ) 0.005 065 72 -0.006 040 187 2 -0.000 482 327 01

a b c d e f

parameters in eq 36, y ) E* 0.877 201 38 0.039 533 255 0.028 499 695 R ) 0.995 831 07 -0.021 444 204 SE ) 0.003 834 16 0.004 233 458 8 -0.000 234 447 98

a b c d e f

parameters in eq 36, y ) F* 0.910 868 99 0.044 055 969 0.015 225 805 R ) 0.996 674 61 -0.011 561 425 SE ) 0.005 065 72 0.002 485 858 7 -0.000 199 120 53

The crucial benefit of the ratios of collision integrals, obtained from the inversion of the viscosity collision integrals, is that they are expected to be more accurate than those obtained from other corresponding states correlations, because measurements of the viscosity are more practical and accurate than the measurements of other transport properties, say, diffusion and thermal conductivity. Generally, calculations of other properties are considerably eased by the existence of numerical tables of collision integrals and their ratios. A stringent assessment of any potential energy is its ability to reproduce thermophysical properties with acceptable accuracies. In this respect, transport properties of 15 binary refrigerant mixtures have been generated from the present model potential energies. To calculate mixture transport properties, we needed to know binary potential parameters, σ12 and 12. We took the scaling parameters σ and for R134a and R152a from refs 34 and 35, respectively. For the remaining seven refrigerants (R125, R32, R143a, R227ea, R236ea, R236fa, and RC318), we calculated the values of σ from the following equation,36

σ ) 0.809(Vc)1/3

(37)

where Vc is the critical volume. The values of were evaluated using a corresponding states correlation for Ω*(2,2) given in ref

Table 3. Predicted Transport Properties of Equimolar R125-R134A Mixture T (K)

106 η(Pa s)

104 D12 (m2/s)

RT

103 λ (w/(m K))

200 250 273.15 293.15 300 313.15 333.15 353.15 373.15 423.15 473.15 523.15 573.15 623.15 673.15 723.15 773.15 873.15 973.15

8.205 10.281 11.236 12.058 12.336 12.864 13.664 14.443 15.216 17.081 18.868 20.588 22.219 23.793 25.315 26.782 28.194 30.906 33.470

0.0163 0.0253 0.0302 0.0316 0.0362 0.0395 0.0446 0.0498 0.0553 0.0705 0.0868 0.1050 0.1237 0.1443 0.1659 0.1883 0.2123 0.2631 0.3179

-0.0029 0.0008 0.0027 0.0044 0.0050 0.0062 0.0079 0.0096 0.0111 0.0150 0.0183 0.0216 0.0241 0.0264 0.0286 0.0303 0.0319 0.0347 0.0367

7.829 10.950 12.525 13.940 14.435 15.394 16.892 18.415 19.970 23.936 27.995 32.092 36.156 40.169 44.115 47.968 51.685 58.816 65.579

Table 4. Predicted Transport Properties of Equimolar R125-R32 Mixture T (K)

106 η(Pa s)

104 D12 (m2/s)

RT

103 λ(w/(m K))

200 250 273.15 293.15 300 313.15 333.15 353.15 373.15 423.15 473.15 523.15 573.15 623.15 673.15 723.15 773.15 873.15 973.15

8.662 10.757 11.773 12.628 12.922 13.470 14.313 15.133 15.954 17.923 19.809 21.609 23.343 25.014 26.638 28.175 29.683 32.556 35.273

0.0244 0.0378 0.0453 0.0519 0.0543 0.0591 0.0669 0.0749 0.0831 0.1062 0.1307 0.1581 0.1869 0.2177 0.2513 0.2853 0.3214 0.3997 0.4827

-0.0193 -0.0001 0.0089 0.0177 0.0207 0.0263 0.0350 0.0245 0.0513 0.0711 0.0876 0.1033 0.1163 0.1280 0.1391 0.1473 0.1553 0.1693 0.1792

8.694 11.703 13.291 14.711 15.218 16.184 17.714 19.286 20.910 25.092 29.435 33.807 38.148 42.325 46.217 49.638 52.503 55.813 54.815

20 in conjunction with a nonlinear least-squares method. To calculate binary potential parameters, we used the famous combining rule, known as the Lorentz-Berthelot rule, in which the collision diameter is taken to be the arithmetic mean and the well depth is taken to be the geometric mean of those for the pure species:

σ12 ) (σ1 + σ2)/2

(38)

12 ) (12)1/2

(39)

It is crucial to mention that Maitland and Wakeham37 outlined that the experimental values of σ12 agree reasonably with the prediction of eq 38. Expressions 12-17, provided from the Chapman-Enskog version of the kinetic theory, together with the calculated collision integrals obtained from the inverted potential energies were used to compute viscosities of 15 equimolar binary refrigerant mixtures. The viscosity values of the studied binary mixtures in the temperature range 200K < T < 973.15 K are tabulated in Tables 3-17. The calculated interaction viscosities are correlated with the following equation,

ln

()

()

bη cη η T ) aη ln + + 2 + dη η0 T0 T T

(40)

Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9217 Table 5. Predicted Transport Properties of Equimolar R125-R152A Mixture

Table 8. Predicted Transport Properties of Equimolar R134A-R32 Mixture

T (K)

106 η(Pa s)

104 D12 (m2/s)

RT

103 λ (w/(m K))

T (K)

106 η(Pa s)

104 D12 (m2/s)

RT

103 λ (w/(m K))

200 250 273.15 293.15 300 313.15 333.15 353.15 373.15 423.15 473.15 523.15 573.15 623.15 673.15 723.15 773.15 873.15 973.15

7.823 9.716 10.635 11.452 11.707 12.188 12.946 13.738 14.500 16.267 18.066 19.692 21.380 22.882 24.378 25.897 27.255 29.947 32.488

0.0194 0.0298 0.0356 0.0411 0.0429 0.0466 0.0527 0.0592 0.0660 0.0838 0.1040 0.1253 0.1492 0.1735 0.1997 0.2282 0.2568 0.3191 0.3865

-0.0164 -0.0044 0.0012 0.0063 0.0081 0.0119 0.0175 0.0232 0.0285 0.0415 0.0537 0.0644 0.0749 0.0827 0.0906 0.0982 0.1035 0.1141 0.1220

8.190 11.330 12.983 14.505 15.014 15.996 17.580 19.266 20.960 25.238 29.820 34.349 39.131 43.690 48.297 52.975 57.327 65.915 74.079

200 250 273.15 293.15 300 313.15 333.15 353.15 373.15 423.15 473.15 523.15 573.15 623.15 673.15 723.15 773.15 873.15 973.15

8.191 10.111 11.066 11.919 12.196 12.711 13.500 14.325 15.138 17.004 18.909 20.632 22.395 24.025 25.591 27.178 28.659 31.484 34.201

0.0253 0.0386 0.0460 0.0533 0.0557 0.0605 0.0683 0.0767 0.0857 0.1088 0.1353 0.1628 0.1936 0.2258 0.2598 0.2966 0.3346 0.4154 0.5039

-0.0224 -0.0071 -0.0002 0.0057 0.0078 0.0123 0.0193 0.0260 0.0328 0.0484 0.0636 0.0761 0.0882 0.0986 0.1075 0.1162 0.1235 0.1353 0.1456

8.448 11.429 13.031 14.522 15.032 16.017 17.571 19.236 20.934 25.205 29.758 34.184 38.681 42.832 46.618 50.045 52.720 55.619 54.396

Table 6. Predicted Transport Properties of Equimolar R125-R143A Mixture

Table 9. Predicted Transport Properties of Equimolar R134A-R143A Mixture

T (K)

106 η(Pa s)

104 D12 (m2/s)

RT

103 λ (w/(m K))

T (K)

106 η(Pa s)

104 D12 (m2/s)

RT

103 λ (w/(m K))

200 250 273.15 293.15 300 313.15 333.15 353.15 373.15 423.15 473.15 523.15 573.15 623.15 673.15 723.15 773.15 873.15 973.15

8.189 10.075 10.996 11.824 12.088 12.587 13.374 14.197 14.977 16.812 18.668 20.357 22.106 23.659 25.213 26.785 28.188 30.985 33.608

0.0178 0.0272 0.0325 0.0376 0.0393 0.0427 0.0482 0.0542 0.0603 0.0767 0.0951 0.1146 0.1363 0.1585 0.1825 0.2085 0.2345 0.2915 0.3528

-0.0106 -0.0026 0.0009 0.0040 0.0052 0.0074 0.0109 0.0146 0.0178 0.0261 0.0337 0.0405 0.0471 0.0521 0.0572 0.0620 0.0654 0.0723 0.0773

8.176 11.240 12.844 14.328 14.830 15.801 17.355 19.011 20.652 24.797 29.187 33.499 38.023 42.297 46.609 50.961 54.992 62.966 70.511

200 250 273.15 293.15 300 313.15 333.15 353.15 373.15 423.15 473.15 523.15 573.15 623.15 673.15 723.15 773.15 873.15 973.15

7.796 9.537 10.394 11.157 11.421 11.940 12.693 13.468 14.202 16.038 17.813 19.529 21.194 22.795 24.319 25.822 27.263 30.030 32.642

0.0185 0.0281 0.0334 0.0385 0.0403 0.0441 0.0497 0.0557 0.0621 0.0794 0.0984 0.1192 0.1412 0.1656 0.1905 0.2170 0.2455 0.3050 0.3705

-0.0078 -0.0034 -0.0016 3.4E-05 0.0005 0.0016 0.0034 0.0052 0.0070 0.0115 0.0157 0.0197 0.0232 0.0266 0.0294 0.0320 0.0344 0.0384 0.0418

7.951 10.996 12.598 14.076 14.596 15.639 17.229 18.905 20.562 24.899 29.349 33.846 38.343 42.742 47.003 51.190 55.198 62.929 70.480

Table 7. Predicted Transport Properties of Equimolar R134A-R152A Mixture


T (K)

106 η(Pa s)

104 D12 (m2/s)

RT

103 λ (w/(m K))

T (K)

106 η(Pa s)

104 D12 (m2/s)

Rr

103 λ (w/(m K))

200 250 273.15 293.15 300 313.15 333.15 353.15 373.15 423.15 473.15 523.15 573.15 623.15 673.15 723.15 773.15 873.15 973.15

7.275 9.144 10.004 10.752 11.004 11.503 12.230 12.975 13.683 15.450 17.160 18.812 20.415 21.952 23.422 24.871 26.254 28.917 31.431

0.0194 0.0305 0.0363 0.0419 0.0439 0.0480 0.0541 0.0606 0.0676 0.0864 0.1071 0.1298 0.1538 0.1804 0.2075 0.2364 0.2675 0.3324 0.4038

-0.0113 -0.0072 -0.0035 -0.0001 0.0011 0.0033 0.0071 0.0110 0.0149 0.0243 0.0334 0.0416 0.0490 0.0561 0.0619 0.0673 0.0724 0.0804 0.0874

7.825 11.074 12.735 14.246 14.775 15.831 17.456 19.164 20.868 25.350 29.987 34.713 39.472 44.149 48.718 53.227 57.547 65.901 74.067

200 250 273.15 293.15 300 313.15 333.15 353.15 373.15 423.15 473.15 523.15 573.15 623.15 673.15 723.15 773.15 873.15 973.15

7.696 9.420 10.274 11.034 11.297 11.810 12.592 13.338 14.074 15.946 17.706 19.477 21.121 22.740 24.324 25.816 27.284 30.106 32.752

0.0294 0.0444 0.0526 0.0605 0.0634 0.0692 0.0785 0.0878 0.0978 0.1256 0.1553 0.1891 0.2237 0.2618 0.3028 0.3444 0.3892 0.4854 0.5886

-0.0124 -0.0065 -0.0038 -0.0016 -0.0008 0.0007 0.0031 0.0057 0.0083 0.0147 0.0206 0.0265 0.0312 0.0359 0.0401 0.0434 0.0469 0.0523 0.0569

9.033 12.079 13.728 15.260 15.808 16.890 18.618 20.379 22.178 27.304 32.043 37.283 42.394 47.358 52.034 56.144 59.606 63.937 63.595

where η0 ) 1µ Pa s and T0 ) 1 K. Parameters in the above equation were allowed to vary for all systems using the nonlinear least-squares method and are listed in Table 18. The table also contains correlation coefficients and standard errors of the fitting. Figure 1 shows the deviations of the calculated viscosities of four binary refrigerant mixtures from the experimental

values.38-40 Two of these mixtures are R410A (50 wt % R32, 50 wt % R125) and R507 (50 wt % R143a, 50 wt % R125), for which their experimental viscosity values have been measured using an oscillating-disk viscometer over the temperature range 297-403 K. The estimated uncertainty of the reported viscosities is (0.5% at atmospheric pressure.38 The R507 mixture has

9218


Table 11. Predicted Transport Properties of Equimolar R143A-R152A Mixture.

Table 14. Predicted Transport Properties of Equimolar R227ea-R236ea Mixture

T (K)

106 η(Pa s)

104 D12 (m2/s)

RT

103 λ (w/(m K))

T (K)

106 η(Pa s)

104 D12 (m2/s)

RT

103 λ (w/(m K))

200 250 273.15 293.15 300 313.15 333.15 353.15 373.15 423.15 473.15 523.15 573.15 623.15 673.15 723.15 773.15 873.15 973.15

7.385 8.961 9.734 10.418 10.655 11.121 11.860 12.717 13.308 15.070 16.788 18.442 20.120 21.613 23.161 24.674 26.030 28.833 31.348

0.0221 0.0331 0.0391 0.0448 0.0469 0.0510 0.0542 0.0652 0.0725 0.0929 0.1154 0.1400 0.1670 0.1949 0.2255 0.2579 0.2909 0.3639 0.4412

-0.0102 -0.0065 -0.0046 -0.0030 -0.0024 -0.0013 0.0005 0.0004 0.0041 0.0090 0.0137 0.0181 0.0224 0.0259 0.0295 0.0327 0.0353 0.0404 0.0441

8.407 11.518 13.163 14.676 15.211 16.278 18.011 29.098 21.615 26.359 31.305 36.355 41.598 46.550 51.647 56.680 61.342 70.832 79.820

200 250 273.15 293.15 300 313.15 333.15 353.15 373.15 423.15 473.15 523.15 573.15 623.15 673.15 723.15 773.15 873.15 973.15

7.582 9.498 10.384 11.184 11.458 11.974 12.739 13.519 14.320 16.222 18.140 19.958 21.752 23.518 25.160 26.820 28.452 31.442 34.407

0.0103 0.0162 0.0193 0.0223 0.0234 0.0255 0.0288 0.0323 0.0362 0.0464 0.0579 0.0702 0.0837 0.0984 0.1135 0.1298 0.1475 0.1837 0.2244

-0.0022 -0.0028 -0.0024 -0.0020 -0.0019 -0.0016 -0.0009 0.0002 0.0006 0.0026 0.0046 0.0067 0.0085 0.0105 0.0121 0.0136 0.0152 0.0176 0.0198

8.343 10.983 12.349 13.650 14.111 15.056 16.401 17.877 19.440 23.426 27.658


Table 15. Predicted Transport Properties of Equimolar RC318-R236fa Mixture

T (K)

106 η(Pa s)

104 D12 (m2/s)

RT

103 λ (w/(m K))

T (K)

106 η(Pas)

104 D12 (m2/s)

RT

103 λ (w/(m K))

200 250 273.15 293.15 300 313.15 333.15 353.15 373.15 423.15 473.15 523.15 573.15 623.15 673.15 723.15 773.15 873.15 973.15

8.208 9.923 10.777 11.556 11.831 12.369 13.188 13.973 14.740 16.701 18.543 20.403 22.119 23.816 25.476 27.036 28.574 31.533 34.300

0.0275 0.0415 0.0492 0.0566 0.0593 0.0647 0.0735 0.0822 0.0915 0.1175 0.1453 0.1769 0.2093 0.2449 0.2832 0.3223 0.3642 0.4543 0.5510

-0.0212 -0.0112 -0.0066 -0.0026 -0.0011 0.0014 0.0055 0.0101 0.0147 0.0260 0.0366 0.0468 0.0552 0.0635 0.0709 0.0769 0.0829 0.0926 0.1007

8.993 11.937 13.518 15.007 15.544 16.611 18.298 20.023 21.765 26.446 31.228 36.198 40.992 45.626 49.949 53.704 56.834 60.608 59.885

200 250 273.15 293.15 300 313.15 333.15 353.15 373.15 423.15 473.15 523.15 573.15 623.15 673.15 723.15 773.15 873.15 973.15

7.596 9.518 10.384 11.181 11.465 12.004 12.745 13.517 14.325 16.235 18.162 19.991 21.770 23.616 25.218 26.866 28.569 31.532 34.565

0.0095 0.0149 0.0178 0.0206 0.0216 0.0236 0.0266 0.0299 0.0334 0.0429 0.0535 0.0650 0.0774 0.0912 0.1051 0.1202 0.1366 0.1702 0.2080

-0.0054 -0.0073 -0.0063 -0.0055 -0.0051 -0.0045 -0.0026 -0.0008 0.0010 0.0060 0.0112 0.0162 0.0210 0.0258 0.0296 0.0334 0.0372 0.0429 0.0484

6.989 9.883 11.299 12.631 13.115 14.055 15.441 16.926 18.524 22.672 27.214

Table 13. Predicted Transport Properties of Equimolar RC318-R227ea Mixture

Table 16. Predicted Transport Properties of Equimolar RC318-R125 Mixture

T (K)

106 η(Pa s)

104 D12 (m2/s)

RT

103 λ (w/(m K))

T (K)

106 η(Pa s)

104 D12 (m2/s)

RT

103 λ (w/(m K))

200 250 273.15 293.15 300 313.15 333.15 353.15 373.15 423.15 473.15 523.15 573.15 623.15 673.15 723.15 773.15 873.15 973.15

7.751 9.718 10.631 11.468 11.749 12.254 13.031 13.846 14.686 16.584 18.585 20.373 22.239 23.998 25.657 27.376 29.004 32.027 35.025

0.0092 0.0144 0.0172 0.0199 0.0209 0.0227 0.0257 0.0289 0.0324 0.0414 0.0518 0.0625 0.0747 0.0875 0.1010 0.1157 0.1309 0.1633 0.1990

-0.0036 -0.0042 -0.0036 -0.0031 -0.0028 -0.0020 -0.0008 0.0003 0.0015 0.0049 0.0082 0.0113 0.0144 0.0169 0.0194 0.0219 0.0238 0.0274 0.0303

7.510 10.124 11.496 12.812 13.273 14.178 15.509 16.977 18.526 22.303 26.411

200 250 273.15 293.15 300 313.15 333.15 353.15 373.15 423.15 473.15 523.15 573.15 623.15 673.15 723.15 773.15 873.15 973.15

8.150 10.237 11.178 12.044 12.348 12.867 13.650 14.466 15.319 17.219 19.193 20.951 22.739 24.478 26.062 27.689 29.289 32.159 35.044

0.0117 0.0183 0.0219 0.0253 0.0265 0.0288 0.0325 0.0364 0.0407 0.0518 0.0644 0.0776 0.0921 0.1077 0.1238 0.1413 0.1597 0.1981 0.2406

-0.0135 -0.0068 -0.0014 0.0031 0.0047 0.0081 0.0133 0.0185 0.0237 0.0355 0.0471 0.0564 0.0654 0.0732 0.0796 0.0860 0.0915 0.0999 0.1075

7.332 10.402 11.911 13.325 13.830 14.743 16.157 17.653 19.235 23.070 27.205

been proposed as an alternative fluid for refrigeration applications instead of R22. Deviations of the calculated viscosity values of the other two binary refrigerant mixtures, R125R134a and R134a-R32, from literature data are given at three different mole fractions of components. The viscosity measurements of the R125-R134a binary mixture have been carried

out with an oscillating-disk viscometer at temperatures 298.15423.15 K. The experimental viscosities of the binary mixture R134a-R32 embrace the same temperature range, and the uncertainty of the measured viscosity data has been estimated to be within 0.3%.40 Figure 1 shows that the agreement between the predicted and experimental viscosities for the aforementioned

Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9219 Table 17. Predicted Transport Properties of Equimolar R236fa-R134a Mixture T (K)

106 η(Pa s)

104 D12 (m2/s)

RT

103 λ (w/(m K))

200 250 273.15 293.15 300 313.15 333.15 353.15 373.15 423.15 473.15 523.15 573.15 623.15 673.15 723.15 773.15 873.15 973.15

7.571 9.505 10.412 11.177 11.443 11.967 12.758 13.510 14.256 16.168 17.947 19.749 21.425 23.089 24.710 26.219 27.724 31.941 34.780

0.0134 0.0210 0.0252 0.0289 0.0303 0.0331 0.0376 0.0420 0.0468 0.0602 0.0743 0.0905 0.1072 0.1254 0.1452 0.1652 0.1866 0.2330 0.2824

-0.0105 -0.0084 -0.0060 -0.0027 -0.0016 0.0005 0.0037 0.0074 0.0110 0.0201 0.0284 0.0365 0.0434 0.0496 0.0558 0.0605 0.0649 0.0730 0.0788

6.256 8.528 9.664 10.684 11.052 11.793 12.949 14.125 15.385 18.893 22.671

binary mixtures38-40 lies within 2% with average absolute deviation (AAD%) of 1.49% for 53 data points. Figure 2 displays the deviations of predicted viscosities of two ternary refrigerant mixtures, R404A (52 wt % R143a, 44 wt % R125, and 4 wt % R134a) and R407C (52 wt % R134a, 25 wt % R125, and 23 wt % R32) from the ones obtained by Nabizadeh and Mayinger.38 A close look at the deviations reveals a suitable accordance between the predicted viscosities and those given in the literature.38 Furthermore, we have compared the viscosities obtained from the inverted intermolecular potential energies with those calculated through Richenberg’s41 and Davidson’s42 methods. The former method has been recommended by Poling et al.43 as the most consistently accurate method among the other methods cited in the book. Figure 3 illustrates a deviation plot for the calculated viscosities of binary equimolar mixtures R125R134a and R152a-R32 along with those estimated from Richenberg’s41 and Davidson’s42 methods. AADs of the viscosities of the R125-R134a and R152a-R32 systems from the Richenberg’s method were found to be 0.19% and 0.25%, respectively, which shows the desirable harmony of the present work with this method in a broad temperature range. Again in the case of comparison with Davidson viscosities, AADs of 0.16% and 0.24% for the aforementioned mixtures display acceptable accordance between our predicted viscosities and the ones obtained from Davidson’s method.42 Figure 4 shows the deviations of the calculated viscosities of two ternary refrigerant mixtures, R404A (52 wt % R143a, 44 wt % R125, and 4 wt % R134a) and R407C (52 wt % R134a, 25 wt % R125, and 23 wt % R32) from the ones obtained by Davidson’s method.42 Their corresponding AADs were found to be 0.24% and 0.6%, respectively. The binary diffusion coefficients of 15 equimolar refrigerant mixtures were calculated from the inverted potential energies through their corresponding kinetic expressions 18-20. Numerical values of this property in a nearly wide range of temperature are presented in Tables 3-17. Because of the lack of experimental values for mass and binary diffusion coefficients, we could not analyze the accuracies of our results concerning this property. The thermal diffusion factor, RT, is the most sensitive transport coefficient to the details of the intermolecular potential and the most difficult to measure with a high degree of accuracy and describes how a gas mixture can be separated under the influence of a temperature gradient. As the values of this

property increase, separation of the gas mixture becomes easier. This property was calculated for all studied binary refrigerant mixtures using the inverted pair potential energies and relations 22-25. Tables 3-17 represent the values of the thermal diffusion factors of the aforecited mixtures in the temperature range 200 K < T < 973.15 K. In the case of thermal conductivity, in the present study, the predicted interaction viscosities obtained from the inverted pair potential energies were employed to predict interaction thermal conductivities through eq 30. The interaction thermal conductivities λ12 are presented with the following fourth-order polynomial function, which embraces the temperature range 200 K < T < 1000 K.

λ12 ) aλ + bλT + cλT2 + dλT3 + eλT4

(41)

where the above parameters are given in Table 19 for all studied mixtures. Table 20 contains correlation coefficients and standard errors of curve fitting. By employing eqs 26-29, which are based on the Schreiber’s method,12 thermal conductivities of the 15 binary refrigerant mixtures were calculated and listed in Tables 3-17. It should be added that the thermal conductivities of pure species necessary for calculating mixture thermal conductivities were determined from Huber et al.’s method44 for refrigerants (R134a, R152a, and R143a) and from Huber et al.’s method45 for the other six refrigerants (R125, R32, R236ea, R236fa, R227ea, and RC318). In the latter method, with the availability of sufficient dilute-gas thermal conductivity data, fint, instead of being set to 1.32 × 10-3, is fitted to a polynomial in temperature.45 The heat capacities at constant pressure, which are also prerequisite for calculating thermal conductivities, were taken from their respective correlations43,46 for the refrigerants R125, R134a, R152a, R32, and R143a. Because the heat capacities at constant pressure, reported in the above-mentioned references, were valid in the temperature range 50 K < T < 1000 K, our effort was just devoted to predict thermal conductivities embracing the aforecited temperature range for each binary mixture of the above-mentioned refrigerants. In the case of the other four pure refrigerants (R236ea, R236fa, R227ea, and RC318), the heat capacities taken from NIST data47 were used to generate smooth correlations, which are valid in a more limited temperature range and so confine the range of our calculated mixture thermal conductivities. It is noticeable that available literature data on thermal conductivity for this class of fluids is very restricted and does not cover the range of conditions required for engineering applications. Recently, a measurement has been performed on thermal conductivities of some refrigerant mixtures, R404A, R407C, R410A, and R507A.48 The experimental thermal conductivities reported in ref 48 were measured using a modified-steady-state hot wire method. The uncertainty reported by the authors is, in general, less than (1.5% and covers the temperature range 250-400 K. Figure 5 displays the deviations of the thermal conductivities of two binary refrigerant mixtures, R507A (50 wt % R125 and 50 wt % R134a) and R410A (50 wt % R125 and 50 wt %R32), from the experimental data.48 Moreover, Figure 6 depicts a deviation plot for thermal conductivities of two ternary refrigerant mixtures, R407C (52 wt % R134a, 25 wt % R125, and 23 wt % R32) and R404A (52 wt % R143a, 44 wt % R125, and 4 wt % R134a), compared with the experiment.48 In general, the results reveal that Schreiber’s method12 produces thermal conductivities of refrigerant mixtures with accuracies to within 15%. It should be mentioned that the method that we used to

9220


Table 18. Least Squares Coefficients, Correlation Coefficients (R), and Standard Errors (SE) for Equation 40 mixture

aη

bη (K)

cη (K2)

dη

R

SE

R125-R134a R125-R32 R125-R152a R125-R143a R134a-R152a R134a-R32 R134a-R143a R152a-R32 R143a-R152a R143a-R32 RC318-R227ea R227ea-R236ea RC318-R236fa RC318-R125 R236fa-R134a

0.508 352 78 0.496 455 07 0.479 858 73 0.479 161 97 0.522 592 36 0.484 800 48 0.463 399 21 0.442 594 13 0.435 982 53 0.441 974 26 0.558 525 34 0.456 926 26 0.566 947 27 0.533 582 91 0.523 237 58

-230.249 74 -251.673 72 -285.826 64 -285.622 65 -264.881 55 -287.821 73 -329.439 83 -361.601 11 -392.160 23 -361.172 43 -273.909 13 -275.018 31 -280.618 53 -241.194 11 -274.492 47

13 255.922 15 651.941 19 744.434 19 682.762 17 438.654 20 243.064 25 515.635 29 429.362 34 103.434 29 323.595 19 501.683 20 191.432 20 922.944 14 595.766 18 376.747

0.229 768 83 0.327 433 3 0.412 664 82 0.476 133 28 0.084 976 65 0.432 696 17 0.603 199 17 0.779 493 44 0.804 281 79 0.815 195 04 -0.029 965 75 -0.118 951 08 -0.100 828 48 0.095 331 62 0.153 907 11

0.999 995 44 0.999 992 32 0.999 991 03 0.999 991 04 0.999 994 87 0.999 989 38 0.999 991 01 0.999 987 53 0.999 990 75 0.999 987 77 0.999 992 17 0.999 990 03 0.999 989 40 0.999 993 18 0.999 991 21

0.024 781 7 0.032 198 5 0.033 021 5 0.034 999 3 0.024 858 7 0.037 919 8 0.034 610 5 0.041 077 3 0.033 855 1 0.041 985 3 0.035 266 4 0.039 260 2 0.040 405 1 0.031 732 4 0.034 900 6

Figure 1. Deviation plot for the calculated viscosity coefficients of four binary refrigerant mixtures at different temperatures compared with the experiment: R410A (50 wt % R125 and 50 wt % R32)38 (O), R507 (50 wt %R125 and 50 wt % R143a)38 (b), (R125-R134a)39 [X(R125) ) 0.7510 (9), 0.5001 (0), and 0.2508 (4)], and (R134a-R32)40 [X(R134a) ) 0.250 (2), 0.524 (1), and 0.7498 (3)].

Figure 2. Deviation plot for the calculated viscosity coefficients of two ternary refrigerant mixtures, R404A (52 wt % R143a, 44 wt % R125, and 4 wt % R134a) (b) and R407C (52 wt % R134a, 25 wt % R125, and 23 wt % R32) (O) compared with those given in the literature.38

predict the thermal conductivities of the present mixtures has been developed for predicting the thermal conductivity of nonpolar multicomponent molecular mixtures in a dilute-gas limit and has been tested against the available experimental data by Vesovic.14 For the most mixtures he studied, the deviations from the experimental thermal conductivity data are within

Figure 3. Deviation plot for the calculated viscosity coefficients of two binary equimolar refrigerant mixtures compared with those obtained from Richenberg’s method41 [(R125-R134a) (b) and (R152a-R32) (0)] and Davidson’s method42 [(R125-R134a) (O) and (R152a-R32) (4)].

Figure 4. Deviation plot for the calculated viscosities of two ternary refrigerant mixtures, R404A (52 wt % R143a, 44 wt % R125, and 4 wt % R134a) (9) and R407C (52 wt % R134a, 25 wt % R125, and 23 wt % R32) (b) compared with the ones obtained by the Davidson’s method.42

(2.5%. Larger deviations of the calculated thermal conductivities of the present refrigerant mixtures may be attributed to three factors: (1) the polarity of the refrigerant systems, (2) the accuracies of thermal conductivities of pure species, and (3) the mixing rules used to calculate the collision diameter and the well depth of the potential. To evaluate the effect of the thermal conductivities of pure components on the accuracies of the calculated mixture thermal

Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9221 Table 19. Least Squares Coefficients for Equation 41 mixture

aλ (mW/mK)

bλ (mW/mK2)

cλ (mW/mK3)

dλ (mW/mK4)

eλ (mW/mK5)

R125-R134a R125-R32 R125-R152a R125-R143a R134a-R152a R134a-R32 R134a-R143a R152a-R32 R143a-R152a R143a-R32 RC318-R227ea R227ea-R236ea RC318-R236fa RC318-R125 R236fa-R134a

0.167 136 8 -0.383 951 9 0.206 265 3 0.159 983 0 0.026 959 3 0.445 256 9 0.578 256 6 1.137 254 4 1.026 992 1 1.332 1 0.000 101 5 0.000 957 6 0.000 390 9 0.000 437 8 -0.000 991

0.012 667 641 0.020 283 8 0.011 406 418 0.010 587 764 0.012 686 146 0.013 295 275 0.007 577 830 4 0.010 303 758 0.006 588 077 5 0.007 059 006 4 5.600 62E-006 -5.175 74E-006 2.062 15E-006 1.838 65E-006 2.291 01E-005

-3.254 19E-007 -5.359 60E-006 9.727 02E-006 8.793 41E-006 7.821 01E-006 1.664 43E-005 1.768 69E-005 3.137 96E-005 2.456 37E-005 3.777 89E-005 1.200 42E-009 5.856 59E-008 1.980 90E-008 3.388 21E-008 -6.696 12E-008

-5.591 45E-009 -4.831 60E-010 -1.709 68E-008 -1.610 99E-008 -1.493 59E-008 -2.627 60E-008 -2.504 65E-008 -4.167 16E-008 -3.038 02E-008 -4.980 48E-008 7.981 58E-012 -1.160 70E-010 -3.339 01E-011 -7.195 67E-011 1.489 03E-010

3.026 05E-012 4.212 32E-013 7.402 35E-012 7.312 95E-012 6.673 39E-012 1.111 35E-011 1.046 55E-011 1.676 74E-011 1.161 93E-011 2.033 13E-011 -1.497 7E-014 8.368 64E-014 1.966 29E-014 5.305 05E-014 -1.217 06E-01

Table 20. Correlation Coefficients (R) and Standard Errors (SE) for Equation 41 mixture

correlation coeff. (R)

standard error (SE)

R125-R134a R125-R32 R125-R152a R125-R143a R134a-R152a R134a-R32 R134a-R143a R152a-R32 R143a-R152a R143a-R32 R227ea-R236ea RC318-R227ea RC318-R236fa RC318-R125 R236fa-R134a

0.999 993 86 0.999 982 57 0.999 986 21 0.999 984 08 0.999 994 73 0.999 984 07 0.999 984 81 0.999 978 06 0.999 960 53 0.999 981 49 0.994 984 27 0.999 963 82 0.999 979 28 0.999 969 72 0.999 964 98

0.008 410 04 0.000 021 58 0.015 456 36 0.015 260 12 0.010 146 17 0.021 777 95 0.015 753 93 0.030 285 35 0.030 540 30 0.025 909 81 0.004 234 96 0.005 763 16 0.004 493 37 0.006 461 27 0.008 118 18 Figure 6. Deviation plot for the calculated thermal conductivities of two ternary refrigerant mixtures, R407C (52 wt % R134a, 25 wt % R125, and 23 wt % R32) (O) and R404A (52 wt % R143a, 44 wt % R125, and 4 wt % R134a) (9) compared with the experiment.48

Figure 5. Deviation plot for the calculated thermal conductivities of two binary refrigerant mixtures, R507A (50 wt % R125 and 50 wt % R134a) (b) and R410A (50 wt % R125 and 50 wt %R32) (0) from literature data.48

conductivities, typically we selected the R125-R32 binary mixture, calculated the thermal conductivities of R125 and R32 using the two aforementioned methods,44,45 and compared these with the literature data.47 We employed the pure thermal conductivities to Schreiber’s method12 to predict thermal conductivities of the R125_R32 mixture. It was found that the second method45 reduces the errors ∼5%. To assess the effect of mixing rules on the accuracies of the predicted transport properties, we tested this effect on the predicted thermal conductivities of binary and ternary refrigerant mixtures. Five mixing rules other than Lorentz-Berthelot’s were employed to compute the collision diameter and the well depth of the potential. These mixing rules are as follows: Halgren

Figure 7. Deviation plot for the calculated thermal conductivities of binary refrigerant mixture, R507A (50 wt % R125 and 50 wt % R134a) compared with the experiment48 using different mixing rules: Lorentz-Berthelot (b), Halgren HHG49 (4), Tang and Toennies50 (9), Waldman-Hagler51 (3), functional (MADAR-1)52 (O), and fit (MADAR-2)52 (0).

HHG,49 Tang and Toennies,50 Waldman and Hagler,51 functional (MADAR-1), and fit (MADAR-2).52 Figure 7 shows the deviations of thermal conductivities of binary refrigerant mixture R507A (50 wt % R125 and 50 wt % R134a) from the experimental data48 using the six aforementioned mixing rules. Figure 8 compares the effect of mixing rules on the deviations of predicted thermal conductivities of the ternary refrigerant

9222


Literature Cited

Figure 8. Deviation plot for the calculated thermal conductivities of ternary refrigerant mixture, R404A (52 wt % R143a, 44 wt % R125, and 4 wt % R134a) compared with the experiment48 using different mixing rules: Lorentz-Berthelot (b), Halgren HHG49 (4), Tang and Toennies50 (9), Waldman-Hagler51 (3), functional (MADAR-1)52(O), and fit (MADAR2)52(0).

mixture R404A (52 wt % R143a, 44 wt % R125, and 4 wt % R134a) from literature data.48 Figures 7 and 8 show that the effect of mixing rules on the deviations of the calculated thermal conductivities is ∼1%. Results are only slightly in favor of the Halgren HHG mixing rule49 over the other aforecited mixing rules. The analytical form of this mixing rule is as follows:

σij )

ij )

σii3 + σjj3 σii2 + σjj2 4iijj

(ii

1/2

+ jj1/2)2

(42)

(43)

6. Conclusion The acceptable agreement achieved between the calculated and literature values of transport properties indicates that the inversion scheme is, on one hand, a powerful method for generating potential energies from the law of corresponding states of viscosity and, on the other hand, a valuable supplement for obtaining transport properties, especially at either high or low temperatures where direct measurements may be practically impossible. In the case of binary diffusion coefficients and thermal diffusion factors, because of the lack of literature data, we could not make a comparison between the present results and literature. Mixture thermal conductivities, which were calculated on the basis of Schreiber’s method, require only the thermal conductivities and isobaric heat capacities of pure species and a set of interaction parameters, which were calculated in the present work. The accuracies of pure thermal conductivities have a significant effect on the reliability of the predicted mixture thermal conductivities. Also, we found that the mixture transport properties are affected slightly upon changing mixing rules. The Halgren HHG mixing rule reduces the errors at most (1%. Acknowledgment The authors express to Research Committees of Shiraz University and Shiraz University of Technology their sincere thanks for supporting this project and making computer facilities available.

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ReceiVed for reView May 20, 2006 ReVised manuscript receiVed August 23, 2006 Accepted October 2, 2006 IE060630V

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