New equation of state for transport properties: calculation for the thermal conductivity and the viscosity of halogenated hydrocarbon refrigerants

Fluid Phase Equilibria 201 (2002) 309–320

New equation of state for transport properties: calculation for the thermal conductivity and the viscosity of halogenated hydrocarbon refrigerants Mao-Gang He∗ , Zhi-Gang Liu, Jian-Min Yin Division of Thermodynamic and Heat Transfer, School of Energy and Power Engineering, Xi’an Jiaotong University, 710049 Xi’an, China Received 30 November 2001; received in revised form 4 March 2002; accepted 22 March 2002

Abstract A new transport equation of state to estimate the thermal conductivity and the viscosity of the dense fluid for halogenated hydrocarbon refrigerants is presented and the relationships between the reduced residual transport properties and the reduced density were determined. This approach originated from the phenomenological similarity between the reduced residual transport properties and the reduced density in terms of pressure and temperature over the entire thermodynamic surface. The new equation can be used to calculate the thermal conductivity and the viscosity of the dense fluid including the vapor and liquid region with high accuracy, based on the calculation on the transport properties of gases at low pressure. The only input data needed are the critical parameters, molecular weight and acentric factor. The method is based on the concept of the transport equation of state describing the transport properties in terms of pressure and temperature by pressure explicit equations similar to a thermal equation of state. Coherence between the transport properties and equilibrium properties over the entire fluid range was reflected. The absolute average deviation of the thermal conductivity of halogenated hydrocarbon refrigerants is 4.8% with a maximum deviation of 18.0%, and the absolute average deviation of the viscosity of halogenated hydrocarbon refrigerants is 4.4% with a maximum deviation of 15.6%, using the new equation. A new generalized correlation to estimate the thermal conductivity of halogenated hydrocarbon refrigerants at low pressure is also proposed. The range of application of this new formula is for reduced temperatures between 0.6 and 1.2 and for values of the critical compressibility factor between 0.225 and 0.283. The calculation deviation is within ±7%, and the total absolute average deviation is 2.7% compared with the experimental data. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Method of calculation; Transport equation of state; Thermal conductivity; Viscosity; Equation of state; Correlation; Halogenated hydrocarbon refrigerants

∗

Corresponding author. Tel: +86-29-2663863; fax: +86-29-2668789. E-mail address: [email protected] (M.-G. He). 0378-3812/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 2 ) 0 0 0 7 5 - 4

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1. Introduction The transport properties of fluids are important quantities needed in engineering application, especially in numerical analysis of fluid flow and heat transfer. It is necessary to know the transport properties, e.g. the thermal conductivity and the viscosity, of the working fluid in the studied state. However, the database for transport properties is small at present. Although there are many correlation formulas for viscosity and conductivity, the error is often too large to meet the needs of engineering design. The reasons are: (1) it is difficult to measure transport properties accurately, (2) there is no satisfactory theory of transport properties of real dense gases and liquids. Because of the rich supply and the excellent physical and chemical properties, halogenated hydrocarbon substances are widely used in refrigeration equipment. Especially after the chlorofluorocarbon substances, e.g. CCl2 F2 (R12), CHClF2 (R22), CCl3 F (R11), etc. were prohibited to be used and manufactured in the 1990s, many halogenated hydrocarbon substitutes, e.g. CH2 FCF3 (R134a), CHF2 CF3 (R125), CH3 CHF2 (R152a), CH2 F2 (R32), CHCl2 CF3 (R123), etc. are used as refrigerants. Data on transport properties of these new refrigerants are scarce. The aim of this research is to propose a method to estimate the thermal conductivity and the viscosity of halogenated hydrocarbon substances. The molecular kinetic theories describing the transport properties of pure substances as function of temperature are often based on a model for rigid, noninteracting spheres [1]. There are many papers that describe the correlation of the transport properties of a dilute or atmospheric gas as function of temperature. The transport properties of dense gases and liquids are modified using a term dependent on pressure, or are correlated as a function of density based on the value of the dilute gas. In order to describe the thermal conductivity and the viscosity over the entire fluid range, many researchers treat transport properties as a state parameter, dependent on pressure and temperature, or temperature and density, using a called “transport equation of state”, similar to the thermal equation of state f (p, T , ρ) = 0. Transport equations of the state of virial type have already been established for many substances. Those equations have high accuracy, but they contain many adjustable coefficients and need a complete data set covering the entire fluid range to fit. When experimental data over the entire fluid range do not exist, cubic equations of state may successfully predict thermodynamic properties like the PR equation, the RKS equation or the CSD equation. In the same way, when experimental transport property data over the entire fluid range do not exist, it is worthwhile to establish a transport equation of state to predict the thermal conductivity and the viscosity, similar to cubic equations of state. 2. Formulation of the transport equation of state Heckenberger and Stephan found that the shape of transport isotherms in a (p,x) diagram is very similar to the shape of isotherm in a (p,ρ) diagram for the thermophysical properties of water, oxygen and nitrogen [2]. The residual transport property x is defined by: x = x − x0

(1)

where x indicates the thermal conductivity or the viscosity at a certain state, x0 is the thermal conductivity or the viscosity of the dilute gas at the same temperature. The newest experimental data for the halogenated hydrocarbon refrigerant R134a is shown in Figs. 1–3, which show the density, the residual viscosity, and the residual thermal conductivity at a temperature of

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311

Fig. 1. The isothermal density

300 K as a function of pressure. The thermal conductivity and the viscosity of R134a are from reference [3], and the density is calculated using a MBWR equation [4]. The figures show that the profile of the isotherms of the residual thermal conductivity and the residual viscosity of R134a are similar to that of density. This phenomenological similarity indicates that the transport properties can be represented, analogous to a thermal equation of state, by means of pressure-explicit transport equations of state.

Fig. 2. The isothermal residual visicosity.

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Fig. 3. The isothermal residual thermal conductivity.

In ealier studies [5–7] researchers showed that the residual transport properties for nonpolar and polar dense fluids away from the critical region can be described by: (x)r = f (ρr )

(2)

where ρ r is the reduced density (ρr = ρ/ρc ), and (x)r is the reduced residual transport property defined as: λ − λ0 (λ)r = , Γλ (η − η0 ) (η)r = , Γη

2/3

Γλ =

R 5/6 pc 1/6

1/3

Tc M 1/2 NA

2/3

Γη =

M 1/2 pc 1/6

1/3

R 1/6 Tc NA

where λ is the thermal conductivity in W m−1 K−1 and η the viscosity in N s m−2 , the subscript “0” represents the values of the dilute gas at the same temperature. R the universal gas constant, 8314 J kmol−1 K−1 , M the molecular mass in kg kmol−1 , NA the Avogadro’s number, 6.023 × 1026 kmol−1 , and Tc and pc are the critical parameters in K and N m−2 , respectively. Γ λ has the units of m K W−1 and Γ η has the units of m2 N−1 s−1 . For a constant temperature process, p = F (ρ)

(3)

For the residual transport property, Eq. (2) can be written as: ρ = f −1 (x)

(4)

Combining Eqs. (3) and (4) gives p = F [f −1 (x)]

(5)

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313

Table 1 Sources of the thermal conductivity of dense halogenated hydrocarbons Halogenated hydrocarbon

Number of data pointsa

Range of temperature (K)

Range of pressure (Mpa)

Reference

R12 R22 R32 R123 R125 R134a R141b R142b R143a R152a

33 (g), 40 (l) 45 (g), 51 (l) 74 (g), 101 (l) 45 (g), 53 (l) 117 (g), 1271 (l) 162 (g), 135 (l) 17 (g), 30 (l) 9 (g), 40 (l) 26 (g), 24 (l) 40 (g), 58 (l)

253–363 210–348 253–335 263–365 200–354 203–355 194–393 290–370 269–353 223–363

0.10–22.8 0.10–26.6 0.10–30.0 0.10–28.3 0.05–31.0 0.10–68.0 0.10–20.9 0.40–19.0 0.10–31.0 0.10–8.00

[8,9] [8,10–12] [13–17] [10,18–21] [14–17,21–23] [10,18–21,24–27] [15,20,23,24,26] [12,24,28] [27,24] [11,12,19]

a

g: gas, l: liquid.

or in another form p = F [x, T ]

(6)

Obviously, the thermal conductivity and the viscosity can be calculated from the easily measured quantities p and T. At the same time, the above equation must obey the following assumptions: (1) the thermal conductivity and the viscosity are assumed to be state parameters; (2) the residual thermal conductivity and the residual viscosity are strict monotone functions of density. For halogenated hydrocarbon refrigerants, Eq. (4) may be expressed by: ρr = A + B(x)r + C(x)2r

(7)

The coefficients A, B, C were correlated as a function of the acentric factor ω by fitting to the data from the references given in Tables 1 and 2. For the thermal conductivity, we found A = 0.0426756 − 0.001169ω−1 Table 2 Sources of the viscosity of dense halogenated hydrocarbons Halogenated hydrocarbon

Number of data pointsa

Range of temperature (K)

Range of pressure (MPa)

Reference

R22 R32 R123 R125 R134a R141b R142b R143a R152a

54 (l) 17 (l) 37 (l) 8 (l) 52 (l) 9 (l) 9 (l) 6 (l) 8 (l)

250–323 230–323 233–418 251–333 213–423 273–353 273–353 273–323 273–353

0.40–31.0 0.21–3.27 0.02–2.32 0.36–3.96 0.29–2.12 0.03–0.45 0.14–1.38 0.12–0.62 0.26–2.34

[29,30] [31] [29,32] [31] [29,33] [34] [34] [29] [29]

a

l: liquid.

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B = 0.149597 − 0.186854ω C = −0.00174592 + 0.00319773ω and for the viscosity: A = −0.239699 + 0.297845ω−1 B = 0.113585 + 0.686136ω C = 0.000452261 − 0.051216ω When Eq. (7) is used to calculate the thermal conductivity and the viscosity, a thermal equation of state is required to calculate the reduced density. When experimental data are scarce or not available, the cubic equation of state proposed by Peng and Robinson can be used to calculate the reduced density. The PR equation of state is given by [36]: p=

RT a − v − b v(v + b) + b(v − b)

(8)

If Eq. (8) is written in terms of pr , Tr and ρ r , it becomes: pr =

Tr ρr Zc a  ρr2 − 1 − b ρr (1 + b ρr ) + b ρr (1 − b ρr )

(9)

where the two parameters in Eqs. (8) and (9) are: a  = aPc ρc2 = 0.45724 b = bρc =

pc2 α Zc2

0.07780 Zc

and α 0.5 = 1 + k(1 − Tr0.5 ) k = 0.37464 + 1.54226ω − 0.26992ω2 By solving a set of equations including Eqs. (1), (7) and (9), the thermal conductivity and the viscosity at a given pressure and temperature can be obtained. Tables 1 and 2 show the sum of the selected experimental data used to correlate the thermal conductivity and the viscosity of dense halogenated hydrocarbon refrigerants. The deviations between experimental data and calculated results of the thermal conductivity and the viscosity, obtained with the above mentioned transport equation of state, are listed in Tables 3 and 4. According to these tables, the absolute average deviation for all halogenated hydrocarbon substances for the thermal conductivity is 4.8%, the maximum is below 18.0%. The absolute average deviation for the viscosity is 4.4%, the maximum is below 15.6%. A comparison of these results with the results from another transport equation of state proposed by Heckenberger and Stephan [35] is also listed in Tables 3 and 4. The density ρ which is from the PR equations shows relatively large deviations in the liquid region and the critical region. This is a major error source when utilizing this method. Moreover, as Eq. (7)

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315

Table 3 Deviation of the thermal conductivity from the transport equation of state Halogenated hydrocarbon

R12 R22 R32 R123 R125 R134a R141b R142b R143a R152a

Present work

Reference [35]

ρ r (maximum)

Mean (%)

Maximum (%)

Mean (%)

Maximum (%)

2.851 2.811 2.783 2.782 2.785 2.802 2.808 2.801 2.786 2.783

1.8 3.9 4.9 6.3 6.1 4.9 5.8 4.4 6.7 3.6

7.6 9.5 15.9 11.5 18.0 16.0 12.6 7.6 13.5 7.6

7.6 8.5 7.7 14.9 9.1 12.0 15.1 14.3 13.7 12.9

18.2 24.4 34.5 26.0 33.0 37.7 20.2 24.9 25.2 34.5

4.8

12.0

11.6

27.9

Total mean

Table 4 Deviation of the viscosity from the transport equation of state Halogenated hydrocarbon

R22 R32 R123 R125 R134a R141b R142b R143a R152a Total mean

Present work

Reference [35]

ρ r (maximum)

Mean (%)

Maximum (%)

Mean (%)

Maximum (%)

2.726 2.514 2.499 2.462 2.425 2.705 2.658 2.556 2.514

6.3 4.2 2.9 7.4 1.9 4.8 2.9 7.2 2.1

15.0 10.4 4.2 15.6 2.8 5.9 4.7 14.7 5.5

38.6 34.1 43.0 42.4 39.2 38.3 50.1 43.1 49.5

48.5 46.7 47.5 50.9 47.8 41.4 53.9 46.9 50.5

4.4

8.7

42.0

48.2

is a quadratic polynomial, at higher reduced density an unphysical solution occurs easily. The second row of Tables 3 and 4 also show the maximum reduced density of each substance used. Because of the larger in deviation in density in the critical region as a results of the use of the PR equation and the critical enhancement phenomenon of the transport properties, this method is not suitable for the critical region. 3. The transport property η0 at low pressure The transport property x0 of halogenated hydrocarbon refrigerants at low pressure is needed in the above calculation. Nagaoka presented a generalized correlation for the viscosity of gaseous fluorocarbon

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refrigerants at low pressures [37]. This empirical equation is: η0 ηr =  = (0.512Tr − 0.0517)0.82 Zc−0.81 Γη

(10)

where Zc is the critical compressibility factor, and Γη the viscosity parameter defined as Γη =

2/3

M 1/2 pc 1/6

Tc

Nagaoka’s correlation was obtained based on the most reliable experimental data of 16 halogenated hydrocarbon refrigerants, such as R11, R12, R13, R13B1, R14, R21, R22, R23, R113, R114, R115, R142b, R152a, RC318, R500 and R502. The application range is 0.6 ≤ Tr ≤ 1.8 and 0.253 < Zc < 0.282. In a comparison with 116 experimental data, the mean deviation of the equation is <1.62%, and the maximum deviation is 4.97%. When calculating the viscosity of the new refrigerants, such as R134a or R32, by Eq. (10), the deviations are similar to those for R12 or R22. So this equation may be used to calculate the viscosity of the new halogenated hydrocarbon refrigerants. 4. The transport property λ0 at low pressure The popular method to estimate the thermal conductivity of halogenated hydrocarbon refrigerants at low pressure are the Roy–Thodos method [38], the Perelischtien method based on group contributions [39] and the Modified Viswanath method [40]. First of all, these methods were checked by computations. Twelve sets of reliable experimental data for the thermal conductivity of gaseous halogenated hydrocarbon refrigerants have been selected from the literature [41–48] as shown in Table 5. Although, the Roy–Thodos method can be used for all kinds of organic substances, its accuracy for the halogenated hydrocarbon refrigerants is only within 30%. The accuracy of the Perelischtien method based on the group contributions Table 5 Physical constants and sources of the thermal conductivity for halogenated hydrocarbons Halogenated hydrocarbon

M (g mol−1 )

Tc (K)

pc (MPa)

Zc

Γ λ × 103 (m K W−1 )

Number of data points

Range of temperature (K)

Reference

R12 R13 R22 R23 R32 R123 R125 R134a R141b R142b R143a R152a

120.913 104.459 86.468 70.014 52.024 152.931 120.012 102.031 116.950 100.495 84.041 66.051

384.98 301.87 369.30 299.00 351.26 461.56 339.17 374.27 477.31 410.26 345.97 386.41

4.129 3.876 4.990 4.815 5.777 3.500 3.618 4.065 4.250 4.040 3.769 4.512

0.275 0.279 0.273 0.256 0.243 0.283 0.271 0.256 0.225 0.235 0.262 0.271

1.89818 2.03873 2.56444 2.88255 3.67574 1.51533 1.78182 2.05460 1.89835 2.03044 2.81097 2.72301

7 7 7 7 4 7 7 12 4 4 4 6

298–393 283–373 298–393 283–373 285–345 304–365 254–354 252–334 293–353 293–353 293–353 254–354

[41] [41] [41] [41] [42] [43] [44] [43–45] [46] [46] [47] [48]

M.-G. He et al. / Fluid Phase Equilibria 201 (2002) 309–320

317

Fig. 4. Relation between Tr and λr /C for halogenated hydrocarbons.

is within 10% for the halogenated hydrocarbon refrigerants, but it is hard to use and cannot be applied to methane series refrigerants, such as R12, R22, R23, etc. The modified Viswanath method gives a very accurate description of the low pressure thermal conductivity of the 11 halogenated hydrocarbons that were used in the fit of the six constants of this correlation. However, the predictions for other halogenated Table 6 Deviations of Eq. (11) and other methods Halogenated hydrocarbon

R12 R13 R22 R23 R32 R123 R125 R134a R141b R142b R143a R152a Total mean

Present work

Reference [38]

Reference [39]

Maximum

Mean

Maximum

Mean

−5.7 3.5 −4.2 1.4 3.4 3.4 2.7 2.8 6.5 6.3 −1.6 −6.8

4.2 3.3 2.5 0.8 2.9 1.6 1.7 1.0 4.2 3.6 0.7 5.5

−16.9 −25.2 −28.3 −38.7 −33.6 −28.2 −50.7 −50.8 −16.3 −29.2 −52.5 −55.5

15.9 25.1 26.1 36.2 30.9 26.8 49.8 48.6 12.9 28.7 51.3 51.9

2.7

33.7

Maximum

16.088 −2.065 3.304 3.527 −3.640 25.725 −10.522

Reference [40] Mean

12.6 1.1 1.0 2.0 3.1 22.3 8.0 7.2

Maximum

Mean

4.317

4.2

0.951

0.8

1.712 −12.287 −20.075 −17.701 9.279 −1.488 2.498 −8.917

1.2 11.2 19.8 14.5 4.2 0.7 2.5 6.4 6.6

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Fig. 5. Deviation between experimental and estimated thermal conductivity.

hydrocarbons are not so accurate. Therefore, the method to calculate thermal conductivity for halogenated hydrocarbon refrigerants at low pressure has to be reinvestigated. We found that λr , the reduced thermal conductivity, λr = λ/Γλ of function of Tr , the reduced temperature T/Tc , at low pressure can be very well correlated using the following equation: λr = C(−2.10568 + 6.84902Tr ) = A(−2.10568 + 6.84902Tr )e0.00670405M

(11)

This is illustrated by Fig. 4. For methane series substances, C = 0.678e0.00670405M , A = 0.678. For ethane series substances, C = e0.00670405M , A = 1. The application ranges based on the distribution of the original experimental data is 0.6 < Tr < 1.2, 0.225 < Zc < 0.283. Table 6 shows the results of the present correlation and a comparison with deviations from the other three methods. Fig. 5 illustrates the deviations between experimental and estimated thermal conductivity. It is found that Eq. (11) is useful for prediction of the thermal conductivity. Eq. (11) reproduces the experimental data with a mean deviation of 2.7% and a maximum deviation of 7%. 5. Conclusion A new transport equation of state to estimate the residual thermal conductivity and viscosity for halogenated hydrocarbon refrigerants is presented, using the critical parameters (pc , Tc , ρ c ), molecular weight (M) and acentric factor (ω) as input data. A new empirical equation is also presented for the prediction of the thermal conductivity of halogenated hydrocarbon refrigerants at low pressure. Based on the transport properties of gases at low pressure, the thermal conductivity and the viscosity of the dense fluid including

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319

the vapor and liquid region were calculated. Using the new equation to estimate the transport properties of halogenated hydrocarbon refrigerants, the absolute average deviation is 4.8%, the maximum deviation is below 18.0% for the thermal conductivity, and the absolute average deviation is 4.4%, the maximum deviation is below 15.6% for the viscosity. However, this method is not applicable in the critical region. List of symbols A, B, C coefficients M molecular mass NA Avogadro’s number p pressure R universal gas constant T temperature v specific volume x transport property, to indicate the thermal conductivity or the viscosity Z compressibility factor Greek letters η viscosity λ thermal conductivity ρ density ω acentric factor Subscripts 0 low pressure condition r the reduced parameter c the critical parameter References [1] R.C. Reid, J.M. Prausnitz, Bruce E. Poling, The Properties of Gases and Liquids, 4th Edition, McGraw-Hill, New York, 1987, 390 pp. [2] T. Heckenberger, K. Stephan, Int. J. Thermophys. 11 (1990) 1011–1023. [3] R. Krauss, J. Luettmer-Strathmann, J.V. Sengers, K. Stephan, Int. J. Thermophys. 14 (1993) 951–988. [4] R. Tillner-Roth, H.D. Baehr, J. Phys. Chem. Ref. Data 23 (1994) 657–729. [5] L.I. Stiel, G. Thodos, AIChE J. 10 (1964) 26–32. [6] J.A. Jossi, L.I. Stiel, G. Thodos, AIChE J. 8 (1962) 59–65. [7] L.I. Stiel, G. Thodos, AIChE J. 10 (1964) 275–277. [8] T. Makita, Y. Tanaka, Y. Noguchi, H. Kubota, Int. J. Thermophys. 2 (1981) 249–268. [9] M.J. Assael, E. Karagiannidis, W.A. Wakeham, Int. J. Thermophys. 13 (1992) 735–751. [10] M.J. Assael, E. Karagiannidis, Int. J. Thermophys. 14 (1993) 183–197. [11] S.H. Kim, D.S. Kim, M.S. Kim, S.T. Ro, Int. J. Thermophys. 14 (1993) 937–950. [12] O.B. Tsvetkov, Y.A. Laptev, A.G. Asambaev, Int. J. Thermophys. 17 (1996) 597–606. [13] M. Papadaki, W.A. Wakeham, Int. J. Thermophys. 14 (1993) 1215–1220. [14] Y. Tanaka, S. Matsuo, S. Taya, Int. J. Thermophys. 16 (1995) 121–131. [15] M.J. Assael, E. Karagiannidis, Int. J. Thermophys. 16 (1995) 851–865. [16] J. Yata, M. Hori, K. Kobayashi, T Minamiyama, Int. J. Thermophys. 17 (1996) 561–571.

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