Fast Calculation Method for Supercritical CO2
Thermodynamic Properties
Zhao Dan，Ding Guoliang，Wu Zhigang
Institute of Refrigeration and Cryogenics，Shanghai Jiao Tong University，Shanghai（200240）E-mail：danzhao@sjtu.edu，glding@sjtu.edu，wzg@sjtu.edu
Abstract
Calculations of refrigerant thermal properties are desired to be very fast and stable in cas of simulation of refrigeration system, etc. The implicit curve-fitting method is a quite good method for the fast and stable calculation of refrigerant thermodynamic properties, and it has been ud for saturated, two-pha, superheated and subcooled refrigerants. In this paper, new fast calculation method for supercritical CO2 is prented. The results shows that the calculation speeds of the fast calculation formulae for supercritical CO2 are more than 100 times faster than tho of REFPROP 7.1 while the total mean relative deviation of the fast calculation formulae from REFPROP 7.1 is less than 1.8 %.
Keywords:CO2；Fast calculation；Thermodynamic properties；Supercritical
1. Introduction
Since simulation is applied in designing refrigeration products more widely than ever before, more attentions should be paid to develop the methods for the calculation of refrigerant thermodynamic properties which need very strict requirements on stability and speed.[1] The EOS (equation of state) method is usually ud to predict refrigerant thermal properties in a wide range with high precision.[2-8] But the calculation speed and stability are limited by unavoidable iterations in calculation. The explicit polynomial regression method is another simple yet fast calculation method for refrigerant thermodynamic properties, and the stability of this method is also better than EOS while the accuracy is still satisfied [9, 10]. Cleland [11, 12] prented a method to speed up the calculation of refrigerant thermodynamic properties and gave the correlations for R12, R22, R114, R502, R717 (NH3) and R134a for the saturation temperature of -60°C ~ 60°C.But this method cannot guarantee the calculation reversibility, and so divergence might happen in simulation unless extremely high regressing accuracy is applied.
In order to ensure calculation reversibility of refrigerant thermodynamic properties besides calculation rapidness and stability, Ding et al. [1, 17] developed the implicit curve-fitting method. The implicit curve-fitting method is suitable for both pure refrigerants and refrigerant mixtures and it has b
een ud to predict the thermodynamic properties for saturated liquid, saturated vapour, two pha, superheated and subcooled refrigerants. This method can be divided into two steps: (1) construct the implicit curve-fitting model which depends on refrigerant pha; (2) get correlations for calculating refrigerant thermodynamic properties, namely, explicit fast calculation formulae, by solving the standard cubic equation which is transformed from the implicit curve-fitting model. Two analytical solutions from the same implicit curve-fitting model can ensure reversibility. However, the implicit curve-fitting method for the supercritical refrigerants is not reported. And all the published fast calculation methods can not be ud in the supercritical region.
Natural refrigerant CO2 is often ud under the working condition that the pressure is higher than the critical pressure. For example, supercritical CO2 often works in the heat pump and automobile air conditioning. Therefore a new implicit curve-fitting method to predict supercritical CO2 thermodynamic properties is necessary.
The purpo of this paper is to prent an implicit curve-fitting method to calculate the thermodynamic properties for supercritical CO2. The calculation results of the fast calculation formulae of CO2 will be compared with the data produced by REFPROP 7.1 in order to show the curve-fitting accuracy, and the calculation speed of the fast calculation formulae of CO2 will be comp
茶杯蛋糕ared with that of REFPROP 7.1.策划人生
Specialized Rearch Fund for the Doctoral Program of Higher Education (Grant No. 20050248019) National Natural Science Foundation of China (Grant No. 50576053)
2. Curve-fitting model for the fast calculation of supercritical CO2
thermodynamic properties
For the implicit curve-fitting method, the explicit equations should be the analytical solutions of an implicit polynomial equation, namely, implicit curve-fitting model. So it is very important to construct the curve-fitting model for the fast calculation of supercritical CO 2 thermodynamic properties. The curve-fitting model should be bad on the curve figure, and Fig.1 shows the pressure-enthalpy chart.
Fig 1 pressure-volume chart
The curves in the chart are the isothermal lines with uniform distribution. We can obrve that the difference between enthalpy values on the adjacent isothermal lines under the same pressure is almost the same. So we can get
001
k
三候是什么意思A A kC ≅ (1) Where k is the proportionality coefficient, C is a constant. Martin-Hou equation can be written as following form
23231234567823
()()a a T a T a T a a T a T a T RT一和就造句子
p v b v b v b ++++++−=+−−− (2)
For the isothermal lines with temperature T0, equation (2) can be written as following form
2323
01203040560708023000()()RT a a T a T a T a a T a T a T p v b生存与毁灭
v b v b ++++++−=+
−−− (3) Since constant b is difficult to regress by the least square method, so we neglect it. Under the same
pressure, equation (3) minus equation (2) is
23230000123422222222000002
3
2
3
000567822222222
000011
()()()()(11
(
)()()()0T T T T T T T T R a a a a v v v v v v v v v v T T T T T T a a a a v v v v v v v v −+−+−+−+−+−+−+−+−= (4)
The equation just reflects the relationship mentioned above. Suppo temperature T 0 is a constant and
its value is higher than critical temperature. For the isothermal lines with temperature T 0, v 0 only depend on pressure P , and v 0 is a function of pressure P, that is v 0=f(P). The function can be obtained by using the implicit curve-fitting method for sutured CO 2 thermodynamic properties [13]. Since specific
P 0
volume v 0 can be calculate when pressure P is known, the equation (4) is the implicit correlation about temperature T and specific volume v.
We obrve that not only specific volume have the relationship under same pressure, other thermal properties also does. For example, the enthalpy also has the relationship as shown in fig 2. Therefore, it is reasonable that the equation (4) can also be ud as the implicit correlation about temperature T and other thermal properties when the specific volume v is replaced by other thermal properties.
Fig 2 pressure-enthalpy chart
In order to improve the precision of the equation, some low order items can be added to the equation (4) and the implicit curve-fitting model for the supercritical CO 2 can be obtained as follows.
2323000123000023230004567222222220000232300089101133333333000011
()()()()11
()()()()
11
()()()(0
T T T T T T a a a u u u u u u u u T T T T T T a a a a u u u u u u u u T T T T T T a a a a u u u u u u u u
−+−++−+−+−+−+−+−+−+−+−+−= (5)
where, a 1, a 2, …a 11 are the coefficients to be regresd; u is a simple function of z. For example, u =
z/100, etc. It can be written as u=f(z), and z is one of supercritical thermal properties except temperature T and pressure P , such as enthalpy, entropy, density, etc. After the transform of the thermal properties, the precision of the model can be improved. T 0 is a constant and its value is hig
her than critical temperature. u 0 is a simple function of z 0, that is u 0=f(z 0).For the isothermal lines with temperature T0, z 0 only depend on pressure P, so z 0 is a function of pressure P, that is z 0=f(P). The function can be obtained by using the implicit curve-fitting method for saturated refrigerants thermodynamic properties [13].
Considering the usual application conditions of supercritical CO2, the expected application range of the fast calculation formulae is: temperature in 304.15K ～393.15 K; pressure in 7.3773MPa ～12.0 MPa (the critical temperature and critical pressure for CO2 are 304.15K and 7.3773 MPa ,respectively.) The data source for regressing comes from NIST REFPROP 7.1 [2].
For the implicit curve-fitting model equation (5), we t T 0=318.15K. The coefficients a 1, a 2, …a 11 can be regresd by using the data source from NIST REFPROP 7.1 [2]. Table 1 lists the coefficients for implicit curve-fitting model.
P
Table 1 Coefficients of implicit equations for thermodynamic properties of supercritical CO2 Implicit form
f(p,T,s)=0 f(p,T,h)=0 f(p,T,ρ) =0 f(p,T,λ)=0 f(p,T,µ)=0 u s h 1/7
5-ρ/1000 100-100λ (106µ)1/6 a 1
-3.8341×10-3 -4.9998×10-4 -8.7673×10-7 -6.3423×10-6 5.2334×10-3 a 2
1.9232×10-5 1.9449×10-6 1.7061×10-7 1.5205×10-7 -3.2281×10-5 a 3
-2.5091×10-8 -6.3650×10-10 -5.1530×10-10 -3.7780×10-10 5.0088×10-8 a 4
-1.5015×100 -2.3208×100 -4.5585×100 -9.5058×101 -1.7590×100 a 5
1.2500×10-2留学费用一览表
2.3283×10-3 7.0312×10-4 -1.5100×10-3 -1.6951×10-2 a 6
-6.7218×10-5 -7.9691×10-6 -4.8476×10-6 -1.0032×10-5 1.0686×10-4 a 7
9.1176×10-8 5.5420×10-10 8.2396×10-9 3.8006×10-8 -1.6892×10-7 a 8
7.5499×10-1 1.7956×100 6.9332×100 3.0122×103 1.0341×100 a 9
-1.1733×10-2 -2.4629×10-3 -3.4146×10-3 1.8826×10-1 1.3926×10-2 a 10
6.4670×10-5
7.1942×10-6 1.9655×10-5 -3.9069×10-4 -
8.9374×10-5 a 11 -8.8767×10-8 3.0685×10-9 -2.7975×10-8 -2.0877×10-7 1.4361×10-7
3. Get explicit fast calculation formula from implicit curve-fitting model for supercritical CO2 thermodynamic properties
When T is known, let
23200000123456222
00000003230000789101123333000002312323456723
89101111
(1
)
1.0T T T T T A a a a a a a u u u u u u u T T T T a a a a a u u u u u B a T a T a T C a a T a T a T D a a T a T a T =−+++++++++++=+++=+++=+++ (6)
then Eq. (5) becomes
023=+++D Cu Bu Au (7)
A is an explicit function of p becau u 0 is explicit functions of p , and B, C and D are explicit functions
of T . The solutions of Eq. (7), provided according to Appendix, are therefore explicit functions of p and T , which can be expresd as u=u(p, T). Being z is a simple function of u , the explicit equation z=f(p, T) can be gotten finally. For example, suppo h is the variable need to be calculated, and u = h/100. After getting the explicit equation u=u(p,T) by solving Eq. (7), h =100 u(p,T), which is the explicit form of h = f(p, T), can be obtained finally. When z is known,. Let
21173210622
951232
0008412342
00000
2323300000056789101122233330000000
11
(1
A a a u a u
B a a u a u
C a a u a u T T T
D a a u u a a a a u u u u u T T T T T T a a a a a a a u
u u u u u u u =++=++=++=++−+++++++++++ (8)
then Eq. (5) becomes
023=+++D CT BT AT (9)
where, D is explicit function of p becau u 0 are explicit functions of p. A, B and C are explicit functions of u . By solving Eq. (9) with 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 T =f(p, z) can be gotten.
With the above method, the fast explicit calculation formulae can be obtained and table2 lists the explicit calculation formulae for thermodynamic properties of supercritical CO 2
Table 2 Explicit calculation formulae for thermodynamic properties of supercritical CO2 form form A, B, C and D of cubic equation (4) （y is the root of
equation 5） s=f(p,T) (6) (A5)
s=y T=f(p,s) f(p,T,s)=0 s (8) (A4) T=y
h=f(p,T) (6) (A5) h=y 7 T=f(p,h) f(p,T,h)=0 h 1/7
(8) (A5) T=y ρ= f(p,T) (6) (A5) ρ=5000-1000y T=f(p,ρ) f(p,T,ρ) =0 5-ρ/1000(8) (A4) T=y λ=f(p,T) f(p,T,λ)=0 100-100λ(6) (A5) λ=1-y/100 µ=f(p,T) f(p,T,µ)=
(106µ)1/6
(8) (A5)
µ=106y 6
4. Accuracy and calculation speed of explicit fast calculation formula for supercritical CO2 thermodynamic properties
蜂蜜烤鸡腿In order to verify the calculation accuracy and speed of the fast calculation formulae, the calculation results and time of the fast calculation formulae are compared with tho of REFPROP 7.1. 10, 000 points, which are distributed uniformly in the application ranges of the developed correlations for supercritical thermal properties, are calculated, respectively. Table 3 shows the deviations of the explicit formulae from REFPROP 7.01. It can be found that the total mean relative deviations of the fast calculation formulae are less than 1.8 %, and deviations for about 91.12% data points which are calculated by the fast calculation formulae are less than 1% and deviations for about 91.12% data points are less than 2%. In addition, the data points with higher deviations concentrate in a small region near critical point. Table 3 also shows the calculation speed comparison of the explicit formulae and REFPROP 7.01. It can be found that the mean calculation speed of the fast calculation formulae for thermal properties of supercritical CO 2 is about 130 times faster than REFPROP 7.01.
In the condition that the temperature is calculated when other thermal properties are given, the speed advantage of the fast calculation formulae are more obvious.
Table 3 Comparison of calculation accuracy and speed between REFPROP 7 and fast calculation
formulae
Relative Deviation Calculation time c 7.3773 MPa ～12.0MPa
304.15 ～393.15 K 304.15 ～318.15 K 318.15 ～393.15 K Formula mean a (%) max b
(%) mean a (%) max b (%) mean a (%) max b (%) t NIST (s ) t FCF
(s ) NIST FCF t t s=f(p,T) 0.09 12.96 0.36 12.96 0.03 0.22 4.46 0.066 67.58 T=f(p,s) 0.03 0.46 0.04 0.46 0.03 0.16 21.46
0.058 370.00
h=f(p,T) 0.32 6.55 1.09 7.00 0.17 1.06 4.52 0.078 57.94 T=f(p,h) 0.12 1.03 0.24 1.03 0.10 0.37 18.5 0.066 280.47 ρ=f(p,T) 0.37 27.22 1.09 28.62 0.23 0.86 4.51 0.062 72.70 T=f(p,ρ) 0.07 0.43 0.11 0.43 0.06 0.21 1.96 0.054 36.31 λ=f(p,T) 1.80 58.96 6.69 58.96 0.86 6.66 5.47 0.066 82.86 µ=f(p,T)
1.32 24.76 6.52 24.76 0.31 3.34 5.47 0.066 8
2.79
Total mean d 0.52 16.55 2.02 16.78 0.22 1.61 8.29 0.065 131.3
a FCF
NIST NIST 1
mean (||/)
N
i Y
Y Y E N =−=
∑ where Y FCP is the data calculated by the fast calculation formulae, Y NIST is
gotten from REFPROP 7, N is the number of the test points, and N=10000.
b max FCF NIST NIST max(||/)E Y Y Y =− where Y FCP is the data calculated by the fast calculation formulae, Y NIST is gotten from REFPROP 7.
c The calculation is done on a PC with the CPU AMD Turion(tm) 64 X2. Each point of the function for supercritical thermodynamic property has been calculate
d 10000 times. d []1
(1/)Mean deviation m Total mean m =∑, where m is the number of formulae.
5. Conclusions
The new fast calculation method for calculating thermodynamic properties of supercritical CO 2 can guarantee the formal uniformity of all the fast calculation formulae and the calculation reversibility of the thermal property parameters required in refrigeration system simulation. Being there are no iterations, the fast calculation formulae can guarantee the high calculation speed and absolute stability. Ca study shows that the calculation speeds of the fast calculation formulae for supercritical CO 2 are more than 100 times faster than tho of REFPROP 7.1 while the total mean relative deviation of the fast calculation formulae from REFPROP 7.1 is only 1.8 % and deviations for
about 91.12% data points which are calculated by the fast calculation formulae are less than 1% and deviations for about 91.12% data points are less than 2%. In addition, the data points with higher deviations concentrate in a small region near critical point. The results show that the stability, speed and preci of the fast calculation formulae for supercritical CO 2 satisfy the requirements of simulation of the refrigeration system.
References
[1] Ding GL, Wu ZG , Wang KJ, Masaharu Fukaya. Extension of the applicable range of the implicit curve-fitting
婚礼妈妈method for refrigerant thermodynamic properties to critical pressure. Int J Refrig.2007, 30(3): 418-432. [2] NIST REFPROP 7.1. National Institute of Standard and Technology. USA.
[3] C.Y . Chan, G .G . Halden, Computer-bad refrigerant thermodynamic properties. Part 1. Basic equations,
Int J Refrigeration, 4(1) (1981) 7-12.
