Ind. Eng. Chem. Res. 2003, 42, 3163-3178
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CORRELATIONS Model for the Viscosity and Thermal Conductivity of Refrigerants, Including a New Correlation for the Viscosity of R134a Marcia L. Huber,* Arno Laesecke, and Richard A. Perkins Physical and Chemical Properties Division, National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305-3328
We present modifications to an extended corresponding states (ECS) model for thermal conductivity and viscosity originally developed by Ely and Hanley (Ind. Eng. Chem. Fundam. 1981, 20, 323-332). We apply the method to 17 pure refrigerants and present coefficients for the model and comparisons with experimental data. The average absolute viscosity deviation for the 17 pure fluids studied ranges from a low of 0.56% for R236ea to a high of 5.68% for propylene, with an average absolute deviation for all fluids of 3.13% based on a total of 3737 points. The average absolute thermal conductivity deviation for the 17 pure fluids studied ranges from a low of 1.37% for R116 to a high of 6.78% for R115, with an average absolute deviation for all fluids of 3.75% based on a total of 12 156 points. We also present a new correlation for the viscosity of R134a (1,1,1,2-tetrafluoroethane), which is used as a reference fluid for the description of properties of some refrigerants. The new correlation represents the viscosity to within the uncertainty of the best experimental data. 1. Introduction In 1981, Ely and Hanley1 published a model for viscosity in this journal that was based on the principle of extended corresponding states. In 1983, the model was applied to thermal conductivity.2 Since that time, there have been several modifications to the ElyHanley method,3-6 each one improving upon the original model. Most recently, Mathias et al.7 modified the model to include a practical method for describing the critical enhancement of the thermal conductivity. In this paper, we first apply the extended corresponding states (ECS) viscosity model, as developed by Klein et al.,5 to pure fluids used as refrigerants. We present new coefficients for the fluids, but do not change the model from the Klein et al.5 implementation. As part of this process, we develop a new correlation for the viscosity of R134a, which is used as a reference fluid for the calculation of the properties of some refrigerants. We then modify the ECS thermal conductivity model of McLinden et al.6 and apply it to the same 17 pure refrigerants. The modification concerns the treatment of the enhancement in thermal conductivity that occurs in the region of the critical point.
corresponding states principle to the residual contribution only1
η(T,F) ) η*(T) + ∆η(T,F) ) η*(T) + ∆η0(T0,F0) Fη(T,F) (1) where the superscript * denotes a dilute-gas value and the subscript 0 denotes a reference fluid value. The viscosity of the reference fluid is evaluated at a conformal temperature and density, T0 and F0, respectively, given by
T0 ) T/f
(2)
F0 ) Fh
(3)
and
The quantities f and h are called equivalent substance reducing ratios and relate the reference fluid to the fluid of interest using a ratio of critical parameters (denoted by the subscript c) and functions of temperature and density known as shape functions θ and Φ
f)
Tc θ Tc0
(4)
h)
Fc0 φ Fc
(5)
2. Viscosity Model and We represent the viscosity of a pure fluid as a sum of a dilute gas and a residual contribution, then apply a * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: 303.497.5252. Fax: 303.497. 5224. 10.1021/ie0300880
The shape factors can be considered functions of both temperature and density. For small, nonpolar, almost-
This article not subject to U.S. Copyright. Published 2003 by the American Chemical Society Published on Web 05/24/2003
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spherical molecules, they are nearly unity and can be thought of as functions to compensate for deviations from a spherical shape. Different approaches have been used to determine the shape factors; these include an “exact” method in which the thermodynamic surfaces of one fluid are mapped onto another8 and an approximate method that maps the saturation boundary only and produces density-independent shape factors,9 as well as correlations for these quantities.7,10,11 A very recent correlation12 presents the shape factors as functions of the reduced temperature and density, the acentric factor, and the critical compressibility factor. In this work, we generally have available accurate formulations for the thermodynamic properties of the fluids, either in terms of a Helmholtz free energy equation or a PVT equation of state (EOS), and we use a form of the “exact” shape factor method.6 If detailed thermodynamic information on the fluid is not available, then one of the correlation methods for shape factors7,10-12 can be used. However, it is a requirement in this method to first determine the thermodynamic shape factors. The dilute-gas viscosity in eq 1 is found by ChapmanEnskog theory13 as
η*(T) )
5xπmkBT 16πσ2Ω(2,2)
(6)
where η* is the dilute-gas viscosity, m is the molecular mass, kB is the Boltzmann constant, and T is the absolute temperature. We will further assume that a Lennard-Jones 12-6 potential applies and use the Lennard-Jones collision diameter for σ. Neufeld et al.14 gave the following empirical correlation for the calculation of the collision integral Ω(2,2)
Ω(2,2) ) 1.161 45(T*)-0.148 74 + 0.524 87e-0.773 20T* + 2.161 78e-2.437 87T* (7) with the dimensionless temperature T* ) kBT/ and the minimum of the Lennard-Jones pair-potential energy. The range of validity of this empirical correlation is 0.3 < T* < 100. The factor Fη in eq 1 is found using the expression
Fη ) f 1/2h-2/3
( ) M M0
1/2
(8)
where M is the molar mass of the fluid and M0 is the molar mass of the reference fluid. The model as developed to this point is predictive and does not use any information on the viscosity of the fluid (except for the dilute-gas piece that requires the Lennard-Jones and σ parameters). The functions f and h can be found from thermodynamic data. To improve the representation of the viscosity, an empirical correction factor can be used if experimental viscosity data are available. We then evaluate eq 1 at F0,v instead of F0, where5
F0,v(T,F) ) F0(T,F) ψ(Fr)
(9)
and ψ is a polynomial in the reduced density Fr ) F/Fc of the form n
ψ(Fr) )
∑ ckFrk k)0
(10)
where the coefficients ck are constants found by fitting the experimental viscosity data. 2.1. Reference Fluid for Viscosity: Correlation for the Viscosity of R134a. As indicated in eq 1, to evaluate the viscosity of a particular fluid, the value of the residual viscosity of a reference fluid is required. It is not necessary to use the same reference fluid for all refrigerants. When using the model in predictive mode, it is best to select the reference fluid that is most similar in chemical nature to the fluid of interest. The reference fluid should also have a very accurate equation of state and viscosity surface. In accordance with these considerations, R134a is a good candidate for a reference fluid for chlorinated fluorocarbon (CFC and HCFC) and hydrofluorocarbon (HFC) refrigerant fluids. However, to estimate properties of a particular fluid at low reduced temperatures (T/Tc), it is necessary that the equation of state for the reference fluid be valid to the same reduced temperature. R134a has a relatively short saturation boundary, limited by the triple point (from 0.45 < T/Tc < 1.0), and this limits the lower temperature of the estimation. In cases where the model is used with the empirical correction factor ψ, the requirement that the reference fluid be chemically very similar to the fluid of interest is not as important. For this reason, we have chosen to use propane, which has a very long saturation boundary (0.23 < T/Tc
