International Journal of Refrigeration 28 (2005) 921–932 www.elsevier.com/locate/ijrefrig
An implicit curve-fitting method for fast calculation of thermal properties of pure and mixed refrigerants Guoliang Dinga,*, Zhigang Wua, Jian Liua, Takefumi Inagakib, Kaijian Wangb, Masaharu Fukayab a
b
Institute of Refrigeration and Cryogenics, Shanghai Jiaotong University, 1954 Huashan Road, Shanghai 200030, China Fujitsu General Institute of Air-Conditioning Technology Limited, 1116 Suenaga, Takatsu-Ku, Kawasaki 213-8502, Japan Received 22 July 2004; received in revised form 20 October 2004; accepted 24 January 2005 Available online 14 April 2005
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 K60w80 8C and superheat of 0– 65 8C, 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%. q 2005 Elsevier Ltd and IIR. All rights reserved. Keywords: Refrigerants; Mixture; Calculation; Thermal properties
Frigorige`nes purs et me´langes de frigorige`nes: me´thode rapide de calcul des proprie´te´s avec interpolation implicite Mots cle´s : Frigorige`nes ; Me´lange ; Calcul ; Proprie´te´s thermiques
1. Introduction The EOS (equation of state) method is usually used to predict refrigerant thermal properties in a wide range with
* Corresponding author. Tel.: C86 21 62932110; fax: C86 21 62932601. E-mail address:
[email protected] (G.L. Ding).
0140-7007/$35.00 q 2005 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2005.01.014
high precision [1–6]. 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-andtube 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
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Nomenclature T p r h s CP x
temperature (K) pressure (kPa) density (kg mK3) enthalpy (kJ kgK1) entropy (kJ kgK1 KK1) specific heat at constant pressure (kJ kgK1 KK1) quality (–)
Subscripts s saturated sp superheated tp two-phase ls saturated liquid vs saturated vapor nist REPPROP 6.01 fcf fastcalculation formulae
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 MartinDominguez [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 hZf(T,p) and TZg(h,p) are concerned, when T1 and p1 are known, h1 can be got from h1Zf(T1,p1) and then T2 can be got from T2Zg(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 curvefitting 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-andtube heat exchanger, it usually costs less than 2 min on a Pentium III PC.
2. 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 accuracy of the polynomial. According to the theorem of formula expansion, the number of low order parts in the expanded form of an explicit formula can be increased if the number of independent variables is increased. One simple method of increasing the number of independent variables of an explicit formula is to convert the dependent variable into an independent variable. After the converting process, the polynomial, which includes more low order parts of the expanded form, will become an implicit equation containing one more independent variables. For example, an explicit form yZf(x) contains one independent variable x. If it is changed into g(x,y)Z0, then it has two independent variables x and y. In the condition of the same highest order of independent variables, the polynomial for g(x,y) can get better accuracy than that for f(x) because the polynomial for g(x,y) contains more items than that for f(x). Although the implicit curve-fitting can get better accuracy than explicit curve-fitting under the limitation of the same highest order of variables, explicit formulae still should be got for calculating refrigerant thermal properties in order to avoid iteration in the calculation process. This can be realized in two steps. At first, the source data are regressed with the implicit equation g(x,y)Z0. Then explicit formulae xZg1(y) and yZg2(x) for the actual calculation of x and y, individually, are got by solving equation g(x,y)Z0. In this case, the explicit formulae xZg1(y) and yZg2(x) are the analytical solution of the polynomial equation g(x,y)Z0. But if the highest variable order of the polynomial equation is higher than 4, a general analytical solution form cannot be got [12]. Lower the highest variable order of the polynomial
G. Ding et al. / International Journal of Refrigeration 28 (2005) 921–932
equation has, easier getting explicit formulae from the implicit equation is. Therefore, the highest variable order of the implicit equations in this paper is not larger than 3.
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At the two terminal points (z0, T0) and (zn, Tn), Eq. (1) becomes to the following two equations. f ðu0 ; v0 Þ Z f ðz0 K z0 ; T0 K T0 Þ Z 0
(5)
f ðun ; vn Þ Z f ðzn K z0 ; Tn K T0 Þ Z 0
(6)
3. Implicit curve-fitting and explicit formulae for saturated refrigerant thermal properties
from Eqs. (5) and (6), coefficients a8 and a9 can be obtained:
3.1. Kinds of explicit equations for saturated refrigerant thermal properties to be derived
a8 Z Kðu3n C a1 u2n vn C a2 un vn2 C a3 v3n C a4 u2n
a9 Z 0
For saturated mixed refrigerants, the following 14 explicit correlations are needed: plsZf(T), TlsZf(p), pvsZ f(T), TvsZf(p), hlsZf(T), TlsZf(h), hvsZf(T), TvsZf(h), slsZf(T), svsZf(T), rlsZf(T), rvsZf(T), CplsZf(T), CpvsZ f(T). Considering the reversibility requirement, psZf(Ts) and TsZf(ps) should be derived from the same implicit equation, and so does pvsZf(T) and TvsZf(p), hlsZf(T) and TlsZf(h), and hvsZf(T) and TvsZf(h). For pure refrigerants, plsZf(T) and pvsZf(T) are same, and so are TlsZf(p) and TvsZf(p). Therefore, 12 explicit correlations are needed instead of 14 correlations. 3.2. Implicit equation form for saturated refrigerant thermal properties The following implicit cubic equation containing two variables u and v is applied for saturated refrigerants f ðu; vÞ Z u3 C a1 u2 v C a2 uv2 C a3 v3 C a4 u2 C a5 uv Ca6 v2 C a7 u C a8 v C a9 Z 0
ð1Þ
where a1, a2,.,a9 are coefficients, and u and v indicate two thermal properties of a refrigerant. For example, let uZps and vZTs then we get an implicit equation of ps and Ts, from Eq. (1). Suppose z is one of saturated thermal properties except temperature T (Ts, Tls or Tvs), such as pressure, enthalpy, entropy, etc. Assume total (nC1) groups of data (z0,T0), (z1,T1), and (zn,Tn) are provided for regressing, here (z0,T0) and (zn,Tn) are data of two terminal points of the selected data range. In some cases the regressing accuracy is not satisfied enough if the source data are regressed directly. So the transform of variable z is done before regressing.
z Z f ðzÞ
(2)
Eq. (2) should be simple. For example, z*Zz/100, etc. Now the (nCl) groups of data for regressing are (T0,z0*), (T1,z*1),.(Tn,z*n). Let u Z z K z0
(3)
v Z T K T0
(4)
C a5 un vn C a6 v2n C a7 un Þ=vn
(7)
(8)
So, the implicit Eq. (1) is changed to the following equation: u3 C a1 u2 v C a2 uv2 C a3 v3 C a4 u2 C a5 uv C a6 v2 Ca7 u C a8 v Z 0
ð9Þ
Eq. (9) has only seven coefficients of a1, a2,.,a7 to be determined by regressing. It is simpler than Eq. (1) which has nine coefficients to be regressed. 3.3. Get explicit formula from implicit equation The implicit equation Eq. (9) should be transformed into a standard cubic equation. 8 A Z1 > > > > > < B Z a1 v C a4 Let (10) > C Z a2 v2 C a5 v C a7 > > > > : D Z a3 v3 C a6 v2 C a8 v Eq. (9) becomes Au3 C Bu2 C Cu C D Z 0
Let
8 A Z a3 > > > > > < B Z a2 u C a6 > C Z a1 u2 C a5 u C a8 > > > > : D Z u3 C a4 u2 C a7 u
(11)
(12)
Eq. (9) becomes Av3 C Bv2 C Cv C D Z 0
(13)
Eqs. (11) and (13) are in standard cubic form. Their analytical solutions are given in Appendix. All the explicit equations listed in Section 3.1 can be divided into two types: (1) zZf(T), or (2) TZf(z). Here, T indicates Ts, Tls or Tvs, and z is one of saturated thermal properties except T. When T is known, one suitable root of Eq. (11) in the form of explicit function of A, B, C and D can be got according to Appendix. As A, B, C and D shown by Eqs. (4) and (10), are functions of T, the root of Eq. (11), is an explicit function of T. Eqs. (2) and (3) show that z is a
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function of u, thus the explicit equation zZf(T) can finally be got. For example, suppose p is the variable z to be calculated, and Eq. (2) for p is p*Zp/100. Then uZ(pKp0)/ 100 can be got from Eq. (3). After getting the explicit equation uZu(T) by solving Eq. (11), pZ100u(T)Cp0, which is the explicit from of pZf(T), can be obtained finally. When z is known, Eq. (13) should be solved. A, B, C and D, determined by Eqs. (2), (3) and (12), are functions of z. So the root of Eq. (13) can be expressed as vZv(z), which is an explicit function of z. With the help of Eq. (4), TZ v(z)CT0, which is the explicit form of TZf(z), can be obtained.
4. Implicit curve-fitting and explicit formulae for superheated refrigerant thermal properties 4.1. Kinds of explicit equations for superheated refrigerant thermal properties to be derived For superheated refrigerants, the following 4 explicit correlations: hZf(p,T), TZf(p,h), sZf(p,T) and rZf(p,T) need to be developed for a simulation program. Considering the reversibility requirement, hZf(p,T) and TZ f(p,h) should be derived from the same implicit equation f(p,T,h)Z0.
2 1 1 T T T T2 C a1 vs K C a2 vs K K uvs u u uvs uvs u 3 T T3 1 1 Ca3 vs K C a4 2 K 2 uvs u uvs u 2 T T T T2 Ca5 2vs K 2 C a6 2vs K 2 uvs u uvs u 3 Tvs T 3 1 1 Ca7 2 K 2 C a8 3 K 3 uvs u uvs u 2 T T T T2 Ca9 3vs K 3 C a10 3vs K 3 uvs u uvs u 3 T T3 Ca11 3vs K 3 Z 0 uvs u
ð15Þ
where a1, a2,.,a11, are coefficients; uvs is the corresponding saturated vapor thermal property of u under pressure p. As described in Section 3, both uvs, and Tvs are explicit functions of p. 4.3. Get explicit formula from implicit equation When T is known, let
8 2 3 2 3 2 3 1 Tvs Tvs Tvs 1 Tvs Tvs Tvs 1 Tvs Tvs Tvs > > C a C a C a C a C a C a C a C a C a C a C a A Z K > 1 2 3 4 5 6 7 8 9 10 11 > uvs uvs uvs uvs > u2vs u3vs u2vs u2vs u2vs u3vs u3vs u3vs > < B Z 1:0 C a1 T C a2 T 2 C a3 T 3 > > > C Z a4 C a5 T C a6 T 2 C a7 T 3 > > > : D Z a8 C a9 T C a10 T 2 C a11 T 3 (16) 4.2. Implicit equation form for superheated refrigerant thermal properties then Eq. (15) becomes As shown in Section 4.1, pressure p is always used as an input parameter for all explicit equations for superheated (17) Au3 þ Bu2 þ Cu þ D ¼ 0 refrigerants, and temperature T is another important parameter. Let z be one of thermal properties except T and p. In order to improve regressing accuracy, transform of A is an explicit function of p because uvs and Tvs are variable z to u is done before regressing, explicit functions of p, and B, C and D are explicit functions of T. The solutions of Eq. (17), provided according to Appendix, are therefore explicit functions u Z f ðzÞ (14) of p and T, which can be expressed as uZu(p,T). Being z is a simple function of u, the explicit equation zZ In order to calculate z from u easily, Eq. (14) should be in a f(p,T) can be got finally. For example, suppose h is the simple form. For example, uZz/100. variable need to be calculated, and Eq. (14) for h is uZ In order to guarantee the continuity of thermal properties h/100. After getting the explicit equation uZu(p,T) by between superheated region and saturated region, the solving Eq. (17), hZ100u(p,T), which is the explicit following implicit cubic equation is used to derive the form of hZf(p,T), can be obtained finally. explicit formulae for superheated refrigerant thermal When z is known, u can be got from Eqs. (2) and properties. (3). Let
G. Ding et al. / International Journal of Refrigeration 28 (2005) 921–932
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8 A ¼ a11 þ a7 u þ a3 u2 > > > > > 2 > > < B ¼ a10 þ a6 u þ a2 u 2 C ¼ a9 þ a5 u þ a1 u > > > > > > 1 T T2 T3 1 T T2 T3 1 t T2 T3 > : D ¼ a8 þ a4 u þ u2 K þ a1 vs þ a2 vs þ a3 vs þ a4 2 þ a5 2vs þ a6 2vs þ a7 2vs þ a8 3 þ a9 vs þ a10 3vs þ a11 3vs u3 3 uvs uvs uvs uvs uvs uvs uvs uvs uvs uvs uvs uvs
(18) then Eq. (15) becomes 3
2
AT C BT C CT C D Z 0
(19)
where D is explicit function of p because uvs and Tvs, are explicit functions of p. A, B and C are explicit functions of u. By solving Eq. (19) the method described in Appendix, three roots, which are explicit functions of T, can be obtained. After choosing the right one of them, the explicit formula TZf(p,z), can be got.
5. Implicit curve-fitting and explicit formulae for twophase thermal properties
þ a4 ðpr Þ3 þ a5 ðpr Þ2 þ a6 ðpr Þ þ a7
(24)
where a1,.,a7 are coefficients; pr is a pressure relevant parameter. Usually prZpc/p, where p and pc is the twophase pressure and the corresponding critical pressure, respectively. But sometimes, other forms of pr will also be chosen in order to improve the accuracy. Two variables u and v in Eq. (24) are related to two of the two-phase parameters except p.
C a6 pr C a7
For two-phase refrigerant, four explicit correlations: hZ f(p,x), xZ(p,h), sZf(p,h) and TZf(p,h) need to be developed for a simulation program. The correlations hZ f(p,T) and TZf(p,h) should be reversible. 5.2. 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 equations h K hls hvs K hls
u K v ¼ vð1 K vÞ½a1 ðpr Þ6 þ a2 ðpr Þ5 þ a3 ðpr Þ4
Let A Z a1 p6r C a2 p5r C a4 p4r C a4 p3r C a5 p2r
5.1. Kinds of explicit equations for two-phase thermal properties
xZ
the following implicit equation is used to derive the explicit formulae:
(25)
When u should be calculated, the following explicit equation got from Eq. (24) can be used. u Z v C vð1 K vÞA
(26)
When v should be calculated, the following explicit equation got from Eq. (24) can be used. Av2 K ðA C 1Þv C u Z 0 Solve Eq. (27) and get the explicit formula of v. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A C 1 K ðA C 1Þ2 K 4Au vZ 2A
(27)
(28)
or (20) vZ
T Z Ts
(21)
s Z ð1 K xÞsls C xsvs
(22)
h Z ð1 K xÞhls C xhvs
(23)
As Eqs. (21)–(23) are simple and commonly known, calculation of pure refrigerants in two-phase region need no mention again in this paper. 5.3. Calculation for two-phase thermal properties of mixed refrigerants 5.3.1. Implicit equation and solution method In order to guarantee the continuity of calculating thermal property from two-phase region to saturated region,
A C1 C
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA C 1Þ2 K 4Au 2A
(29)
5.3.2. Getting explicit equation of hZf(p,x) and xZf(p,h) Let uZ(hKhls)(hvsKhls) and vZx, the following explicit equation of hZf(p,x) can be obtained from Eq. (26). h Z hls C ½x C xð1 K xÞAðhvs K hls Þ
(30)
The following two explicit equations of xZf(p,h) can be obtained from Eqs. (28) and (29), respectively. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ls A C 1 K ðA C 1Þ2 K 4A hhKh vs Khls (31) xZ 2A qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ls A C 1 C ðA C 1Þ2 K 4A hhKh vs Khls xZ (32) 2A
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G. Ding et al. / International Journal of Refrigeration 28 (2005) 921–932
Table 1 Main parameters of implicit equations for saturated thermal properties of R22 and R407C Implicit no. R22
R407C
I1a I2a I3 I4 I5 I6 I7 I8 I9 I10
I11 I12 I13 I14 I15 I16 I17 I18 I19 I20
a
Implicit equation
u
v
R22
R407C
R22
R407C
f(pls,Ts)Z0 f(pvs,Ts)Z0 f(hls,Ts)Z0 f(hvs,Ts)Z0 F(sls,Ts)Z0 f(svs,Ts)Z0 f(rls,Ts)Z0 f(rls,Ts)Z0 f(Cpls,Ts)Z0 f(Cpvs,Ts)Z0
(psK37.5049)/100 (psK37.5049)/100 (hlsK133.266)/100 (hvsK360)1/2K4.31202 slsK0.726003 1.95075Ksvs (1463.74Krls)/1000 (0.1rvs)1/3 K0.571122 CplsK1.07099 CpvsK0.563656
(plsK43.6188)/100 (pvsK27.9147)/100 0.821352K100/hls (h1/2vsK19.4068)/4 slsK0.775616 (2.0Ksvs)1/2K0.0788221 (1430.646Krls)1000 r1/2vsK1.17512 (CplsK1.0)1/2K0.520066 (10CpvsK1.0)1/2K2.46368
TsK213.150 TsK213.150 TsK213.150 TsK213.150 TsK213.150 TsK193.150 TsK213.150 TsK213.150 TsK213.150 TsK213.150
TsK213.150 TsK213.150 TsK213.150 TsK213.150 TsK213.150 TsK213.150 TsK213.150 TsK213.150 TsK213.150 TsK213.150
For pure refrigerants, the two equations of f(pls,Ts)Z0 and f(pvs,Ts)Z0 are same.
Similar to the cases for saturated and superheated refrigerant thermal properties, only one of Eqs. (31) and (32) is the suitable equation for two-phase refrigerant thermal properties, which can be selected out by comparing their calculation results with the source data. 5.3.3. Getting explicit equation of sZf(p,h) and TZf(p,h) Let uZ(sKsls)/(svsKsls) and vZ(hKhls)/(hvsKhls), Eq. (33) can be got from Eq. (26). Let uZ(TKTls)/(TvsKTls) and vZ(hKhls)/(hvsKhls) Eq. (34) can be got. h K hls ðh K hls Þðhvs K hÞ s Z sls C C A ðsvs K sls Þ hvs K hls ðhvs K hls Þ2 (33)
h K hls ðh K hls Þðhvs K hÞ T Z Tls C C A ðTvs K Tls Þ hvs K hls ðhvs K hls Þ2 (34)
6. 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 Z hls K Cpls ðTls K TÞ
(35)
T Z Tls K ðh K hls Þ=Cpls
(36)
The method of calculating Tls hls and Cpls are presented in Section 3.
7. 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) K60w80 8C saturated temperature; (2) 0w65 8C superheat, except K53w80 8C saturated temperature for TvsZf(h) of R407C. Two-phase thermal properties of pure refrigerant R22 can be calculated by saturated thermal properties and need no specific fast calculation formulae for them. The data source for regressing come from NIST REFPROP 6.01 [5]. 7.1. Fast calculation formulae for saturated thermal properties of R22 and R407C All the implicit formulae for saturated thermal properties are in the same form as Eq. (9). Table 1 shows the expression of u and v for implicit equations of I1–I20, which are for saturated thermal properties. Table 2 shows the coefficients a1 to a8 for implicit equations of I1–I20. Table 3 shows how to get explicit formulae for saturated thermal properties. 7.2. Fast calculation formulae for superheated thermal properties of R22 and R407C All the implicit equations for superheated thermal properties are in the same form as Eq. (15). Table 4 shows the expression of u and uvs for implicit equations of, I21–I23 for R22, and I24–I26 for R407C, respectively. Table 5 shows the coefficients a1 to a11 of I21– 126. Table 6 shows how to get explicit formulae for superheated thermal properties of R22 and R407C.
G. Ding et al. / International Journal of Refrigeration 28 (2005) 921–932
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Table 2 Coefficients of implicit equations for saturated thermal properties of R22 and R407C Implicit
a1, a5
a2, a6
a3, a7
a4, a8
no. 11
0.98883725495078101!100
0.29540062218367300!101
K0.16473367393392799!100
K0.93613886012967703!103
3
2
5
K0.12911198902310400!104
K0.4445525020821 11100!10 12
0.60140122701540502!10
0.29540062218367300!101
K0.16473367393392799!100
K0.93613886012967703!103
3
2
5
K0.12911198902310400!104
0.17493403788492099!10K3
K0.39109726126599499!10K6
K0.23211221716356198!101
K3
0
K0.23925837271018299!10
0.25644758911717701!10K2
0.69249363065596901!10K4
K0.67772347966420003!101
K0.44455250208211100!10 13
K0.19869418642635200!10
0.98883725495078101!100 K0.23536957237209701!10K1 0.497314155235450903!10
K1
K0.19869418642635200!10
K0.26551003261015698!10
0.60140122701540502!10
14
K0.11842479992554901!10K1
0.17709408794990399!10K3
0.52364848788973495!100
K0.36241240436403602!10K1
K0.68930095590444395!102
0.38883188348537900!101
15
K0.12870694857262799!10K1
0.52312842741896101!10K4
K0.70164869754902995!10K7
K0.51804065800507303!10K1
0.43919457967494900!10K3
0.31377754787782403!10K7
0.90963461037871798!10K1
K0.45679809043052902!10K1
0.15868467464303000!10K1
0.22912897024289700!10K3
0.12945221308136799!10K6
0.56913658960662703!101
K1
K3
1
0.27343167031355399!10K1
0.85972786064023901!10K7
K0.34517192677385400!10K1
16
K0.50701043037114500!10
K0.11877689049049403701!10
K0.63079577831144800!10
17
0.82092546697592507!10K2
K0.61034423081655300!10K4
0.20493040712655002!10K1
K0.29514412227674300!10K4
K0.26501817480917400!100
0.73581858973798399!10K3
18
K0.26051148297331399!10K1
0.35369589897407298!10K3
K0.23263049630462802!10K5
K0.26115310957099500!101
0.52028100415573098!10K1
K0.14768900785032101!10K3
K0.32226595074125299!101
0.32694089293399897!10K1
0.42904264100593498!100
K0.25595963781268601!10K3
K0.11227839635413401!10K5
K0.70405204221522794!102
0.22104493422356100!100
0.10702047839373900!10K3
K0.16190623068005198!102
0.11057305312749300!10K1
0.19359309331881800!102
K0.59823154873992097!10K1
0.30651202346666097!10K4
K0.30256128362725899!104
0.17021909701813399!102
K0.14665195218369100!10K1
K0.67945131522761801!103
0.13546158649061499!101
0.78270683459581802!100
0.35344664036393598!101
K0.27267132962182400!100
K0.90793645688964205!103
K0.58000202291387097!103
K0.32387843681423298!102
0.83700526720381706!105
K0.20669605066390700!104
0.27520544241423200!100
0.19056100652742600!101
K0.12015448977409900!100
K0.47936763819380502!103
K0.32234277678936701!103
K0.12452467499302500!102
0.40687360035874401!105
K0.71292039553989196!103
K0.13277834596419499!100
0.54064058045962698!10K3
K0.20409457657011000!10K6
K0.11701653642864599!102
0.14081522229520399!100
K0.37634676982240098!10K3
K0.34021571441562898!100
0.29216854860840500!10K2
K0.18532590100432900!10K1
0.11286478436621799!10K3
K0.2188977084557600!10K6
0.19121633134604601!101
K0.15619566972722701!10K1
0.31858712632571699!10K4
0.29728484015772499!10K2
K0.11662680826341099!10K4
K0.16208663424498101!10K1
0.87663900761302506!10K4
K0.15773900949847001!10K6
K0.19185769596602599!10K1
0.22616382447427499!10K3
K0.76281725713845203!10K6
K0.87118223820492596!102
0.51914529080541502!10K4
K0.10104895637418499!10K2
0.31455308188919300!10K4
0.21810081597154099!10K7
K0.69735121251174303!100
19 110 111 112 113 114 115 116
K0.37689427869186899!10K2
K0.1752427452542601!10K4
K0.137633220660490000!100
0.27015944494894199!10K2
117
0.11956046520135300!10K2
K0.30842768128926798!10K4
0.50320137164384403!10K7
K0.22878160629037101!101
0.16271222091953200!10K1
K0.27045546741646598!10K4
K0.3006525767836698!100
0.88433617458370903!10K3
118
K0.76412766307885405!10K1
0.92400846126733695!10K2
K0.39433844800312602!10K3
K0.26246630911133501!102
0.2380055363536498!101
0.55592941649771997!10K2
K0.1813321559651499!103
0.62657091664960403!101
119
0.12297118210122600!100
K0.23241427381176500!10K3
0.11565375648951700!10K6
K0.20299908723175701!102
0.6092768539861601!10K1
K0.36718509266973803!10K4
K0.13304329390711400!101
0.18553348019463699!10K2
0.12386126783481399!101
K0.121411898162600!10K2
0.26646389723099199!10K2
K0.19489117187199199!103
0.24696233675443899!101
K0.71487617771057600!10K2
K0.270781760151339399!102
0.15778634049244800!100
120
7.3. Fast calculation formulae of two-phase thermal properties of R407C The implicit equations of I27–I29 for R407C are in the form of Eq. (24). The main parameters and coefficients of these implicit equations are listed in Tables 7 and 8, respectively. The processes of getting explicit expression of htp, x, stp and Ttp are shown in Table 9. 7.4. Accuracy and calculation speed of explicit formulae In order to check the accuracy of the explicit formulae,
140, 3000 and 4000 points, which are distributed uniformly in the application ranges of the developed correlations for saturated, superheated and two-phase thermal properties, are calculated, respectively. The calculation results are compared with those from REFPROP 6.01. Table 10 shows the deviations of the explicit formulae from REFPROP 6.01. It can be found that the total mean relative deviations of the fast calculation formulae for each refrigerant are less than 0.02%. The maximum relative deviations of rspZf(p,T) are less than 2.0% while the maximum relative deviations of other fast calculation formulae are less than 0.6%.
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G. Ding et al. / International Journal of Refrigeration 28 (2005) 921–932
Table 3 Fast calculation formulae for saturated thermal properties of R22 and R407C Explicit form plsZf(T) TlsZf(p) pvsZf(T) TvsZf(p) hlsZf(T) TlsZf(h) hvsZf(T) TvsZf(T) slsZf(T) svsZf(h) rvsZf(T) rvsZf(h) CplsZf(T) CpvsZf(T)
Equations for A, B, C and D; the correct root of cubic Eq. (4)
The explicit formulae
R22
R407C
R22
R407C
Eq. (10), I1; y by Eq. (A4) Eq. (12), I1; y by Eq. (A6) Eq. (10), I2; y by Eq. (A4) Eq. (10), I2; y by Eq. (A4) Eq. (10), I3; y by Eq. (A5) Eq. (12), I3; y by Eq. (A5) Eq. (10), I4; y by Eq. (A5) Eq. (12), I4; y by Eq. (A4) Eq. (10), I5; y by Eq. (A6) Eq. (10), I6; y by Eq. (A6) Eq. (10), I7; y by Eq. (A5) Eq. (10), 18; y by Eq. (A5) Eq. (10), I9; y by Eq. (A5) Eq. (10), I10; y by Eq. (A5)
Eq. (10), I11; y by Eq. (A4) Eq. (10), I11; y by Eq. (A6) Eq. (10), I12; y by Eq. (A4) Eq. (12), I12; y by Eq. (A6) Eq. (10), I13; y by Eq. (A5) Eq. (12), I13; y by Eq. (A5) Eq. (10), I14; y by Eq. (A5) Eq. (12), I14; y by Eq. (A5) Eq. (10), I15; y by Eq. (A5) Eq. (10), I16; y by Eq. (A5) Eq. (10), I17; y by Eq. (A5) Eq. (10), I18; y by Eq. (A5) Eq. (10), I19; y by Eq. (A5) Eq. (10), I20; y by Eq. (A5)
psZ100yC37.5049 TsZyC213.150 psZ100yC37.5049 TsZyC213.150 hlsZ100yC133.266 TlsZyC213.150 hvsZ(yC4.31202)2C360 TvsZyC203.150 slsZyC0.726003 svsZ1.95075Ky rlsZ1463.67K1000y rvsZ10(yC0.571122)3 CplsZyC1.07099 CpvsZyC0.56363
plsZ100yC43.6188 TsZyC213.150 plsZ100yC27.9147 TvsZyC213.150 hlsZ100/(0.821352Ky) TlsZyC213.150 hvsZ(4yC19.4068)2 TvsZyC213.150 slsZyC0.775616 svsZ2.0K(yC0.0788221)2 rlsZ1430.646K1000y rvsZ(yC1.17512)2 CplsZ(yC0.520066)2C1.0 CpvsZ0.1(yC2.46368)2C0.1
Table 4 Main parameters of implicit equations for superheated thermal properties of R22 and R407C Implicit no.
Implicit equation
R22
R407C
121 122 123
124 125 126
f(h,p,T)Z0 f(s,p,T)Z0 f(r,p,T)Z0
u
uvs
R22
R407C
R22
R407C
h/1000 s (0.0001r)1/32
(hK150)/1000 3.5Ks (0.0001r)1/128
hvs/1000 svs (0.0001rvs)1/32
(hvsK150)/1000 3.5Ksvs (0.0001rvs)1/128
Table 5 Coefficients of implicit equations for superheated thermal properties of R22 and R407C Implicit no. 121
122
123
124
125
126
a1, a5, a9
a2, a6, a10
a3, a7, a11
a4, A8, – K0.32267337468324903!100
0.4181952366726502!10K3
0.42458856715409801!10K6
0.18961144369531199!10K9
K0.29151132728135598!10K3
K3.1816658006786500!10K6
K0.15932989240315999!10K9
0.50870618828908200!10K4
0.580391288580298!10K7
0.44494054016817998!10K10
K0.47326605390329603!10K2
0.16553753925425500!10K4
K0.16753290011776399!10K7
0.19023781684775001!10K1
K0.61774773307333704!10K4
0.61275755658211499!10K7
K0.17975059234985299!10K1
0.55925736296820601!10K4
K0.54861470778041500!10K7
–
K0.57060539315396497!10K3
0.12325711729535301!10K5
K0.11349182508341400!10K8
K0.98267513970323805!100
0.98622450403558406!10K3
K0.21504033971899701!10K5
0.20058497572814302!10K8
K0.43215710472342000!10K3
0.95166034294301000!10K6
K0.89677769418126102!10K9
–
K0.12561226675863501!10K2
0.42788077227947001!10K5
K0.10021959568505900!10K8
K0.22361041390789699!100
0.23609694280507501!10K3
K0.74433683485930297!10K6
K0.10101466700110404!10K8
0.97900541837393005!10K5
K0.45470261379367201!10K7
0.28231463896156001!10K9
K0.45441495394325203!10K2
0.11049111202606700!10K4
K0.84844244441961804!10K8
0.15080590731234800!10K1
K0.37847307972458400!10K4
0.30396372386580599!10K7
K0.11567092004007200!10K1
0.301272813133080000!10K4
K0.25097012459648199!10K7
K0.13007255887115000!10K3
K0.46280447606156302!10K6
K0.19313119558760800!10K9
0.25080149718063801!10K3
K0.46280447606156302!10K6
0.3777586045195901!10K9
K0.12106275705373100!10K3
0.22470687264404800!10K6
K0.18499476666820999!10K9
0.34503476352332703!10K1 – K0.23518471294843302!101 0.18943460876188400!101
0.31844738070919588!100
0.14481280048413699!10K1 – K0.25281095714799902!101 0.17004627343363199!101 – K0.9995432316125900!100 0.33265883242424500!100 –
G. Ding et al. / International Journal of Refrigeration 28 (2005) 921–932
929
Table 6 Fast calculation formulae for supherheated thermal properties of R22 and R407C Explicit form
hspZf(p,T) TspZf(p,h) sspZf(p,T) rspZf(p,T)
Equations for A, B, C and D; the correct root of cubic equation
The explicit formulae
R22
R407C
R22
R407C
Eq. Eq. Eq. Eq.
Eq. (10),I24; y by Eq. (11),I21; y by Eq. (10),I25; y by Eq. (10),I26; y by
hspZ1000y TspZy sspZy rspZ10,000y32
hspZ1000yC150 TspZy sspZ3.5Ky rspZ10,000y128
(10), I21; y by Eq. (A6) (11), I21; y by Eq. (A6) (10), I22; y by Eq. (A3) (10),I23; y by Eq. (A6)
Eq. (A6) Eq. (A5) Eq. (A5) Eq. (A6)
Table 7 Main parameters in implicit equations for two-phase thermal properties of R407C Implicit no.
Implicit equation
u
v
pr
127 128 129
f(h,x,p)Z0 f(s,h,p)Z0 f(T,h,p)Z0
(hKhls)/(hvsKhls) (sKsls)/(svsKsls) (TKTls)/(TvsKTls)
x (hKhls)/(hvsKhls) (hKhls)/(hvsKhls)
ln(p/4.634) 4634/p 4634/p
Table 8 Coefficients of implicit equations for twoKphase thermal properties of R407C Implicit no.
a1, a5
a2, a6
a3, a7
a4, –
127
0 0.3522425845539465!10K3 0.573199264237268!10K11 0.595448361006526!10K4 0.254617567195165!10K10 0.752215583211763!10K3
0 K0.102549924206907!10K1 K0.141624313387712!10K8 0.169576348067334!10K3 K0.679929312973732!10K8 K0.940086078992835!10K2
0 0.955106447731523!10K1 0.122686460778061!10K6 0.618634148125263!10K1 0.673476113093236!10K6 K0.183240969306973!10K2
0 – K0.448940726916719!10K5 – K0.316500766354997!10K4 –
128 129
Table 9 Fast calculation formulae for two-phase thermal properties of R407C Explicit form
Equations for u, v and pr
Equations for A
Root of the implicit equation
The explicit formulae
htpZf(p,x) xZf(p,h) stpZf(p,h) TtpZf(p,h)
127 127 128 129
Eq. (25) Eq. (25) Eq. (25) Eq. (25)
u by Eq. (26) v by Eq. (28) u by Eq. (26) u by Eq. (26)
htpZhlsCu(hvsKhls) xZv stpZslsCu(svsKsls) TtpZTlsCu(TvsKTls)
The calculation speed comparison of the explicit formulae and REFPROP 6.01 is shown in Table 11. It can be found that the mean calculation speed of the fast calculation formulae for thermal properties of pure refrigerant R22 and that for refrigerant mixture R407C are about 140 and 1000 times faster than REFPROP 6.01, respectively.
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
8. Conclusions
(1) An implicit equation contains one more independent variables than the corresponding explicit equation and includes more low order parts of the expanded form.
A new method of getting explicit fast calculation
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G. Ding et al. / International Journal of Refrigeration 28 (2005) 921–932
Table 10 Accuracy of fast calculation formulae comparing to source data Explicit form
Deviationa (%) R22
R407C b
plsZf(T) TlsZf(p) pvsZf(T) TvsZf(p) hlsZf(T) TlsZf(T) hvsZf(T) TvsZf(h) slsZf(T) svsZf(T) rlsZf(T) rvsZf(T) CplsZf(T) CpvsZf(T) hspZf(p,T) TspZf(p,h) sspZf(p,T) rspZf(p,T) xtpZf(p,h) htpZf(p,x) stpZf(p,h) TtpZf(p,h) Total meanc a b c
Mean
Max
Meanb
Max
0.0028 0.0004 0.0028 0.0004 0.0001 0.0000 0.0024 0.0004 0.0004 0.0005 0.0002 0.0006 0.0156 0.0134 0.0014 0.0027 0.0024 0.0604 – – – – 0.006%
0.0084 0.0014 0.0084 0.0014 0.0013 0.0002 0.0612 0.0047 0.0015 0.0017 0.0005 0.0012 0.0288 0.0212 0.0075 0.0085 0.0047 0.8627 – – – –
0.0032 0.0004 0.0098 0.0011 0.00165 0.00083 0.0015 0.0015 0.0172 0.0042 0.0026 0.0106 0.0345 0.0065 0.0231 0.0336 0.0172 0.1091 0.0004 0.01872 0.01436 0.00526 0.015%
0.00575 0.00087 0.01963 0.00254 0.07336 0.03107 0.04164 0.0695 0.40412 0.13022 0.00955 0.01407 0.07834 0.00936 0.55589 0.25399 0.07894 1.95663 0.00286 0.10382 0.25327 0.06683
Absolute deviation for xtpZf(p,h), relative deviation (%) for others. P Mean deviationZ1/n n1[(jznistKzfcfj!100)/znist], here z is one kind of refrigerant thermal properties, and n is the calculated points of z. Pm Total meanZ1/m 1[mean deviation], here m is the number of formulae used for one refrigerant.
Therefore, implicit curve-fitting method can get better accuracy than explicit curve-fitting method. (2) In order to get explicit formulae from an implicit equation, the highest order of the variable should not be more than 4. In this paper, cubic implicit equations are used for saturated and superheated refrigerants, and quadric implicit equations are used for two-phase refrigerants. (3) In the cases that the implicit equation has two or more roots, all the roots should be tested in the explicit formulae and only the best one should be selected as the explicit formula for predicting refrigerant thermal properties. (4) Using the method introduced in this paper for pure refrigerant R22 and mixed refrigerant R407C show that the total mean relative deviations of the fast calculation formulae are less than 0.02%, the maximum relative deviations of rspZf(p,T) are less than 2.0% while the maximum relative deviations of other fast calculation formulae are less than 0.6%, and the mean calculation speed of the fast calculation formulae for R22 and R407C are about 140 and 1000 times faster than those of REFPROP 6.01, respectively.
Appendix A. Analytic solution of standard cubic equation Following is a standard cubic equation on form.
Ay3 C By2 C Cy C D Z 0
(A1)
Define
2 DZ4
D A
3
2B BC C 27A 3 K 3A2
2
32
2
5 C4
C A
2
B K 3A 2
3
33 5
(A2)
Eq. (A1) has three roots: (1) three real roots when D%0, or 2) one real root and two complex roots when DO0. All the possible four real roots are listed as Eqs. (A3)–(A6) [12].
G. Ding et al. / International Journal of Refrigeration 28 (2005) 921–932
931
Table 11 Comparison of calculation speed between REFPROP 6.01 and fast calculation formulae Calculation timea (s)
Explicit form
Speed-up times of the calculation time (t1/t2)
REFPROP 6.01 (t1) R22
Fast calculation formulae (t2) R407C
plsZf(T) 11.62 111.56 15.27 100.70 TlsZf(p) pvsZf(T) 11.62 112.38 TvsZf(p) 15.27 120.47 hlsZf(T) 12.17 112.81 – – TlsZf(T) hvsZf(T) 12.03 113.19 TvsZf(h) – – slsZf(T) 12.04 112.80 11.96 113.20 svsZf(T) rlsZf(T) 12.13 111.27 rvsZf(T) 12.16 112.05 12.06 112.72 CplsZf(T) CpvsZf(T) 12.11 113.44 hspZf(p,T) 14.42 73.25 TspZf(p,h) 26.73 143.57 14.38 72.04 sspZf(p,T) rspZf(p,T) 14.63 71.87 htpZf(p,x) – 217.49 xtpZf(p,h) – 330.85 337.75 stpZf(p,h) TtpZf(p,h) 338.68 Total mean speed-up times of the calculation time:
R22
R407C
R22
R407C
0.07 0.08 0.07 0.08 0.07 0.07 0.07 0.08 0.08 0.07 0.07 0.08 0.07 0.07 0.29 0.26 0.38 0.31 – – – –
0.16 0.13 0.12 0.12 0.12 0.12 0.12 0.13 0.12 0.12 0.10 0.12 0.12 0.12 0.17 0.16 0.17 0.18 0.20 0.17 0.20 0.17
165.94 190.82 165.94 190.82 173.81 – 171.89 – 150.55 170.86 173.34 151.96 172.31 173.00 49.70 102.81 37.84 47.20 – – – – 143.05
697.3 805.5 936.5 1003.9 940.1 – 943.2 – 941.5 943.3 927.2 933.8 939.3 945.3 430.9 900.4 423.8 399.3 1318.1 2067.8 1732.0 2052.6 1014.1
a
The time shown in this table is 10,000 times of calculating single-phase refrigerant thermal properties and 3000 times of calculating twophase refrigerant thermal properties on a Pentium III PC.
0 2 y Z 24
B2 3A2
K CA 3
y Z 24
B2 3A2
K CA 3
11
B CC B B CC B B D C 2B33 K BC2 CC B1 B B A 27A 3A CC 5 cosB B arccosBK 2 33=2 CC K B C B3 C 3A B2 KC B CC B @ 24 3A2 A 5 AA @ 3
31=2
0 2
0
0
ðA3)
11
B CC B B CC B B D C 2B33 K BC2 CC B1 2p B B A 27A CC 3A 5 cosB B arccosBK 2 33=2 C CC K B C B3 C 3 3A B2 KC B CC B @ 24 3A2 A 5 AA @ 3
31=2
0
0
ðA4)
11
B CC 31=2 B 2 2 B CC B 3 B C D 2B BC B B C K A 2p C B 3A2 B A C 27A3 K 3A2 CC B1 4 5 y Z2 cosB arccosBK 2 33=2 K CC K B C B3 C 3 3 3A 2 B KC B CC B @ 24 3A2 A 5 AA @ 3
ðA5)
932
G. Ding et al. / International Journal of Refrigeration 28 (2005) 921–932
0 B y ZB @K
D A
0 B CB @K
2B3 C 27A 3
BC K 3A 2
2
D A
3
2B BC C 27A 3 K 3A2
2
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 33 11=3 2 u2 C B2 u D C 2B3 K BC 2 K 2 C u A 27A3 3A2 5 C 4 A 3A 5 C C t4 A 2 3
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 33 11=3 2 u2 C B2 u D C 2B3 K BC 2 K 2 C u A 27A3 3A2 5 C 4 A 3A 5 C K B K t4 A 2 3 3A
References [1] C.Y. Chan, G.G. Haselden, Computer-based refrigerant thermodynamic properties. Part 1. Basic equations, Int J Refrigeration 4 (1981) 7–12. [2] C.Y. Chan, G.G. Haselden, Computer-based refrigerant thermodynamic properties. Part 2. Program listings, Int J Refrigeration 4 (2) (1981) 52–60. [3] C.Y. Chan, G.G. Haselden, Computer-based refrigerant thermodynamic properties. Part 3. Use of the program in the computation of standard refrigerant cycles, Int J Refrigeration 4 (3) (1981) 131–134. [4] S. Devotta, V.R. Pendyala, Prediction of volumetric and thermodynamic properties of refrigerants: a simplified procedure, Int J Refrigeration 17 (1994) 94–100. [5] NIST REFPROP 6.01. National Institute of Standard and Technology, USA. [6] F. de Monte, Calculation of thermodynamic properties of R407C and R410A by the Martin-Hou equation of state—Part
[7]
[8]
[9]
[10] [11]
[12]
ðA6)
I. Theoretical development, Int J Refrigeration 25 (2002) 306– 313. J. Liu, W.J. Wei, G.L. Ding, et al. A general steady state mathematical model for fin-and-tube heat exchanger based on graph theory. Int J Refrigeration, 27 (2004) 965–973. A.C. Cleland, Computer subroutines for rapid evaluation of refrigerant thermodynamic properties, Int J Refrigeration 9 (8) (1986) 346–351. A.C. Cleland, Polynomial curve-fits for refrigerant thermodynamic properties: extension to include R134a, Int J Refrigeration 17 (4) (1994) 245–249. W.W.S. Charters, H.A. Sadafi, Saturation property equation for R22, Int J Refrigeration 10 (2) (1987) 103–104. I.R. Martin-Dominguez, T.W. McDonald, Correlations for some saturated thermodynamic and transport properties of refrigerant R-22, ASHRAE Trans 99 (1993) 344–348. G. Birkhoff, S. MacLane, A survey of modern algebra, 4th ed, Macmillan publishing Co, Inc, New York, 1977.
