An implicit curve-fitting method for fast calculation of thermal properties of pure and mixed refrigerants

doi:10.1016/j.ijrefrig.2005.01.014

International Journal of Refrigeration

Volume 28, Issue 6, September 2005, Pages 921-932

https://doi.org/10.1016/j.ijrefrig.2005.01.014 Get rights and content

Abstract

Calculations of refrigerant thermal properties are desired to be very fast and stable in cases of simulation of refrigeration system, etc. The traditional method based on equation of state cannot meet such requirement because of unavoidable iterations in calculation. In this paper, a new calculation method for refrigerant thermal properties is presented. Low order implicit polynomial equations are got by using curve-fitting method at first, and then explicit formulae for calculating refrigerant thermal properties quickly are obtained by getting the analytical solution of these implicit equations. Explicit fast calculation formulae for thermal properties of R22 and R407C, covering the saturated temperature of −60∼80 °C and superheat of 0–65 °C, are presented as examples. The calculation speeds of the formulae of R22 are about 140 times faster than those of REFPROP 6.01 while the formulae of R407C are about 1000 times faster. The total mean relative deviations of the fast calculation formulae for R22 and R407C are less than 0.02%.

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: p_ls=f(T), T_ls=f(p), p_vs=f(T), T_vs=f(p), h_ls=f(T), T_ls=f(h), h_vs=f(T), T_vs=f(h), s_ls=f(T), s_vs=f(T), ρ_ls=f(T), ρ_vs=f(T), C_pls=f(T), C_pvs=f(T). Considering the reversibility requirement, p_s=f(T_s) and T_s=f(p_s) should be derived from the same implicit equation, and so does p_vs=f(T) and T_vs=f(p), h_ls=f(T) and T_ls=f(h), and h_vs=f(T) and T_vs=f(h).

For pure refrigerants, p_ls=f(T) and p_vs=f(T) are same, and so are T_ls=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 h_ls, h_vs, s_ls, s_vs and T_s 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 equations $x = \frac{h - h_{ls}}{h_{vs} - h_{ls}}$ $T = T_{s}$ $s = (1 - x) s_{ls} + x s_{vs}$ $h = (1 - x) h_{ls} + x h_{vs}$ As 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. $h = h_{ls} - C_{pls} (T_{ls} - T)$ $T = T_{ls} - (h - h_{ls}) / C_{pls}$ The method of calculating T_ls h_ls and C_pls 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 T_vs=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

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Citation Excerpt :
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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

Abstract

Introduction

Section snippets

Basic idea of getting explicit formulae based on implicit curve fitting

Kinds of explicit equations for saturated refrigerant thermal properties to be derived

Kinds of explicit equations for superheated refrigerant thermal properties to be derived

Implicit equation form for superheated refrigerant thermal properties

Kinds of explicit equations for two-phase thermal properties

Calculation for two-phase thermal properties of pure refrigerants

Calculation for subcooled refrigerant thermal properties

Fast calculation of thermal properties of R22 and R407C

Conclusions

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Int J Refrigeration

Int J Refrigeration

Int J Refrigeration

Int J Refrigeration

Int J Refrigeration