An implicit curve-fitting method for fast calculation of thermal properties of pure and mixed refrigerantsFrigorigènes purs et mélanges de frigorigènes: méthode rapide de calcul des propriétés avec interpolation implicite
Introduction
The EOS (equation of state) method is usually used to predict refrigerant thermal properties in a wide range with high precision [1], [2], [3], [4], [5], [6]. But the calculation speed and stability are limited by unavoidable iterations in calculation. For example, when EOS based software NIST REFPROF 6.01 is used for the three dimensional heat exchanger simulation software [7] to predict the performance of a typical fin-and-tube heat exchanger with R407C as refrigerant, over 10 h are needed for one calculation on a Pentium III PC. But engineers hope the calculation time less than 5 min. In order to successfully simulate the working process of the refrigeration system in a short time, the calculation of refrigerant thermal properties should be very fast and stable.
In order to improve the calculation peed of refrigerant thermal properties, Cleland [8], [9] developed some explicit formulae for several pure refrigerants, including R12, R22, R114, R502, R717 and R134a. Charters [10] and Martin-Dominguez [11] also contributed fast calculation methods for R22.
In recent years, new refrigerants are developed, including some non-azeotropic refrigerant mixtures with large slide temperature difference, such as R407C. The existed fast calculation methods should be further improved in order to be suitable for more kinds of refrigerants and wider effective data range.
The object of this paper is to contribute a new method for fast calculation of refrigerant thermal properties, which can meet the following requirements from simulation of refrigeration appliances: (1) the method is suitable not only for pure refrigerants but also for refrigerant mixtures; (2) the forms of the implicit equations for different kinds of thermal properties and the curve-fitting processes are almost the same; (3) all the calculations for thermal properties are explicit; (4) the related parameters from different explicit formulae are reversible. For example, as far as the two explicit equations h=f(T,p) and T=g(h,p) are concerned, when T1 and p1 are known, h1 can be got from h1=f(T1,p1) and then T2 can be got from T2=g(h1,p1). It is required that T1 and T2 should be equal when the calculating error caused by the computer itself is omitted; (5) the effective data ranges of the fast calculation formulae are as wide as possible.
In order to meet the above requirements, a new method is presented in this paper. An implicit curve-fitting process is done at first to get simplified equations for refrigerant thermal properties, and then explicit formulae are obtained by solving these implicit equations analytically.
The method introduced in this paper has been used to develop software for fast calculation of thermal properties of pure and mixed refrigerants, including R22, R134a, R410A, R407C, R32, R125, etc. When this software is connected to the simulator of heat exchanger [7], making a three dimensional section by section simulation of a fin-and-tube heat exchanger, it usually costs less than 2 min on a Pentium III PC.
Section snippets
Basic idea of getting explicit formulae based on implicit curve fitting
Compared to some complicated EOS of refrigerant, the polynomial form is much simpler for predicting refrigerant thermal properties. If polynomials can be used instead of EOS, the calculation of refrigerant thermal properties can be faster and more stable.
A polynomial can be considered as the remaining low order parts of the expanded form of EOS. More low order parts the polynomial includes, higher accuracy it has. The number of the low order parts should be increased in order to improve the
Kinds of explicit equations for saturated refrigerant thermal properties to be derived
For saturated mixed refrigerants, the following 14 explicit correlations are needed: pls=f(T), Tls=f(p), pvs=f(T), Tvs=f(p), hls=f(T), Tls=f(h), hvs=f(T), Tvs=f(h), sls=f(T), svs=f(T), ρls=f(T), ρvs=f(T), Cpls=f(T), Cpvs=f(T). Considering the reversibility requirement, ps=f(Ts) and Ts=f(ps) should be derived from the same implicit equation, and so does pvs=f(T) and Tvs=f(p), hls=f(T) and Tls=f(h), and hvs=f(T) and Tvs=f(h).
For pure refrigerants, pls=f(T) and pvs=f(T) are same, and so are Tls=f(p
Kinds of explicit equations for superheated refrigerant thermal properties to be derived
For superheated refrigerants, the following 4 explicit correlations: h=f(p,T), T=f(p,h), s=f(p,T) and ρ=f(p,T) need to be developed for a simulation program. Considering the reversibility requirement, h=f(p,T) and T=f(p,h) should be derived from the same implicit equation f(p,T,h)=0.
Implicit equation form for superheated refrigerant thermal properties
As shown in Section 4.1, pressure p is always used as an input parameter for all explicit equations for superheated refrigerants, and temperature T is another important parameter. Let z be one of thermal properties
Kinds of explicit equations for two-phase thermal properties
For two-phase refrigerant, four explicit correlations: h=f(p,x), x=(p,h), s=f(p,h) and T=f(p,h) need to be developed for a simulation program. The correlations h=f(p,T) and T=f(p,h) should be reversible.
Calculation for two-phase thermal properties of pure refrigerants
For pure refrigerants, when p is the saturated parameters hls, hvs, sls, svs and Ts can be got by using the methods explained in Section 3. Then the parameters h, x, s and T can be calculated with the following equationsAs Eqs. (21), (22), (23) are
Calculation for subcooled refrigerant thermal properties
For subcooled refrigerant, h and T are two parameters that are often calculated. For most of refrigeration appliances, the subcooling of refrigerants is limited. So the following explicit formulae can be used for calculating h and T of subcooled refrigerants.The method of calculating Tls hls and Cpls are presented in Section 3.
Fast calculation of thermal properties of R22 and R407C
In this section, we select R22 as a representative of pure refrigerant and R407C as a representative of mixed refrigerant to illustrate how to use the new method. The application ranges of the fast calculation formulae for these two refrigerants are: (1) −60∼80 °C saturated temperature; (2) 0∼65 °C superheat, except −53∼80 °C saturated temperature for Tvs=f(h) of R407C. Two-phase thermal properties of pure refrigerant R22 can be calculated by saturated thermal properties and need no specific fast
Conclusions
A new method of getting explicit fast calculation formulae for refrigerant thermal properties based on implicit curve-fitting method is presented in this paper, which can guarantee the formal uniformity of all the explicit formulae and the calculation reversibility of some thermal property parameters, such as p, T and h, T.
Following conclusions are drawn based on the present work
- (1)
An implicit equation contains one more independent variables than the corresponding explicit equation and includes
References (12)
- et al.
Computer-based refrigerant thermodynamic properties. Part 1. Basic equations
Int J Refrigeration
(1981) - et al.
Computer-based refrigerant thermodynamic properties. Part 2. Program listings
Int J Refrigeration
(1981) - et al.
Computer-based refrigerant thermodynamic properties. Part 3. Use of the program in the computation of standard refrigerant cycles
Int J Refrigeration
(1981) - et al.
Prediction of volumetric and thermodynamic properties of refrigerants: a simplified procedure
Int J Refrigeration
(1994) Calculation of thermodynamic properties of R407C and R410A by the Martin-Hou equation of state—Part I. Theoretical development
Int J Refrigeration
(2002)Computer subroutines for rapid evaluation of refrigerant thermodynamic properties
Int J Refrigeration
(1986)
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2017, Journal of Natural Gas Science and EngineeringCitation Excerpt :This method has been well used for pure refrigerants (e.g. R22, R134a and R32) and mixed refrigerants with fixed compositions (e.g. R410A and R407C) (Ding et al., 2005; Sieres et al., 2012; Zhao et al., 2009). The application range of this method covered the saturated region, the superheated region, the two-phase region and the subcooled region (Ding et al., 2005; Sieres et al., 2012), and was extended to critical pressure (Ding et al., 2007) and supercritical region (Zhao et al., 2009). However, the existing implicit curve-fitting methods for fast calculation of thermodynamic properties of refrigerants mentioned above are only suitable for fixed-composition mixtures.