Prediction and Correlation of the Thermal Conductivity of Pure Fluids

Ind. Eng. Chem. Res. 2002, 41, 989-999

989

Prediction and Correlation of the Thermal Conductivity of Pure Fluids and Mixtures, Including the Critical Region Paul M. Mathias,*,† Vipul S. Parekh,‡ and Edwin J. Miller‡ Aspen Technology, Inc., Ten Canal Park, Cambridge, Massachusetts 02141-2201, and Air Products and Chemicals, Inc., 7201 Hamilton Boulevard, Allentown, Pennsylvania 18195-1501

An engineering correlation is presented for the prediction and correlation of the thermal conductivity of pure fluids and mixtures. The model is valid over the thermodynamic states from the dilute gas to the dense liquid and is applicable to nonpolar as well as polar fluids. A key feature of the model is that it provides a practical method for describing the strong critical enhancements of thermal conductivity in the broad vicinity of the vapor-liquid critical point. Quantitative correlations for the thermal conductivity of pure fluids and mixtures require a separation into background contributions and critical enhancements (Sengers, J. V. Int. J. Thermophys. 1985, 6, 203. Sengers, J. V.; Luettmer-Strathmann, J. In Transport Properties of Fluids: Their Correlation, Prediction and Estimation; Cambridge University Press: New York, 1996; p 113). For the background term, we use the model of Mathias and Parekh (Mathias, P. M.; Parekh, V. S. AIChE Annual Meeting, Chicago, IL, Nov 1996), which uses the principle of extended corresponding states, as previously developed by Ely and Hanley (Ely, J. F.; Hanley, H. J. M. Ind. Eng. Chem. Fundam. 1983, 22, 90). We propose a new phenomenological model to describe the strong critical enhancements. The model has been evaluated and validated through data for a variety of pure fluids and mixtures. The critical-point enhancements have been studied and validated for several pure fluids (methane, ethane, carbon dioxide, and R134a) and the methane-ethane mixture. In the noncritical region, the correlation can be used in the predictive as well as correlative modes, but it is recommended that the correlative model be used if data are available to tune the model parameters. In the critical region, component-specific parameters must be adjusted to quantitatively describe the strong enhancement in thermal conductivity. Introduction A quantitative representation of the transport properties of fluids requires decomposition into a background term and a critical enhancement. This is particularly true for the thermal conductivity, as opposed to the viscosity, as it is well-established that the thermal conductivity of a one-component fluid shows a strong enhancement near the vapor-liquid critical point, whereas the viscosity exhibits only a weak enhancement.1,2 The method of extended corresponding states provides an excellent framework for correlating the background contribution to the thermal conductivity.3,4 Although there are both formal and practical difficulties with developing a model for the background thermal conductivity, the practical approach of Ely and Hanley has resulted in an accurate and predictive representation of the background thermal conductivity. Sengers and co-workers1,5-7 have established the asymptotic variation of the thermal conductivity of pure fluids and mixtures in the vicinity of the vapor-liquid critical point. They have demonstrated that the thermal conductivity of pure fluids is strongly divergent at the critical point. For mixtures, the critical enhancements are significant, but the thermal conductivity remains finite at the critical point, except for mixtures that exhibit azeotropy up to the critical point. We propose a new tractable and phenomenological model, based on * Author to whom correspondence should be addressed. † Aspen Technology, Inc. ‡ Air Products and Chemicals, Inc.

the intrinsic stability of the fluid, to describe the critical enhancement. The model provides a qualitatively correct and accurate description of the thermal conductivity of pure fluids. For mixtures, the model incorrectly predicts a divergent thermal conductivity at the critical point. Although this is strictly incorrect, the region of significant quantitative error is small in practice, and thus the present correlation is expected to be useful as an engineering model. An effective engineering model for thermal conductivity should have the following attributes: it should be valid for all of the pure fluids and mixtures expected to be encountered; it should have reasonable estimation capability for most of the pure fluids and mixtures expected to be encountered; it should have adjustable parameters to improve the correlative capability for pure fluids and mixtures where estimation is expected to be difficult and where reliable data are available; and finally, it should have valid estimation and correlative capability for the critical region where the enhancements in the thermal conductivity are known to be significant. Our proposed model has most of the required attributes of an effective engineering model. The rigorous models of Sengers and co-workers and even simplified versions of the theory8,9 are difficult to apply in engineering calculations. Other engineering models are oversimplified10 or fail to represent the essential feature, the singularity at the critical point.11 In many models, the critical enhancements are simply ignored.12 Ramires et al.13 presented an empirical correlation for propane that could also be extended for other pure fluids, but it allows no clear extension to

10.1021/ie0102854 CCC: $22.00 © 2002 American Chemical Society Published on Web 11/10/2001

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Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002

mixtures. The present model is an attempt to fill a niche with a tractable engineering model that captures the singularity of the thermal conductivity at the vaporliquid critical point. The large thermal conductivities that occur in the vicinity of the critical point offer the potential for highefficiency gains in processes and engines. It is hoped that the present model will be a useful analysis tool in achieving these gains.

λ′x(F,T) ) λ′0(F0,T0)Fλ

where the subscript 0 refers to the reference fluid and

Fλ )

Following ductivity as

fλx1/2hλx-2/3

(7)

(8)

we decompose the thermal conThe equivalent substance reducing ratios fλx and hλx are defined by the mixing rules

(1) fλxhλx )

where λh and ∆λc are the background contribution and the enhancement due to the critical fluctuations, respectively. Background Model The model for the background thermal conductivity follows the proposal of Ely and Hanley.4 λh is divided into two contributions, one arising from the transfer of energy from purely collisional or translational effects, λ′, and the other arising from the transfer of energy via the internal degrees of freedom, λ′′.4 It is further assumed that the latter contribution is independent of density and can be calculated from the modified Eucken correlation for polyatomic gases

η0R

1/2

F0 ) Fhλx; T0 ) T/fλx

λ ) λh + ∆λc

λ′′RMR

( ) M0 Mx

The equivalent temperature (T0) and density (F0) for the reference fluid are defined by the relations

Thermal Conductivity Model Sengers,14

(6)

(

0 ) fint CP,R -

5R 2

)

(2)

where λ′′R is the internal contribution for component R, MR is the molar mass of component R, η0R is the viscosity of the dilute gas in the limit of zero density, 0 is the ideal-gas heat capacity of component R, R is CP,R the gas constant, and fint has a constant value of 1.32. For a mixture, λ′′mix is calculated via the empirical, but phenomenologically reasonable, mixing rule of Li15

λ′′mix(T) )

∑R ∑β xRxβλ′′Rβ

(3)

with

(λ′′Rβ)-1 ) 2[(λ′′R)-1 + (λ′′β)-1]

(4)

It should be noted that the assumption that the internal contribution to the background thermal conductivity is independent of density is not supported by theory, but our experience and that of Ely and Hanley4 indicate that the accuracy and robustness of the overall model is not adversely affected. The translational contribution to the background thermal conductivity is calculated via the extended corresponding states method.4 It is postulated that the translational mixture thermal conductivity is identical to that of a hypothetical pure fluid, denoted by the subscript x

λ′mix(F,T) ) λ′x(F,T)

(5)

The corresponding states principle is then invoked on the hypothetical pure fluid

hλx )

∑R ∑β xRxβfλ

Rβ

hλRβ

(9)

∑R ∑β xRxβhλ

(10)

Rβ

where

fλRβ ) (fλRfλβ)1/2(1 + kλRβ) hλRβ )

(

)

hλR1/3 + hλβ1/3 2

(11)

3

(1 + lλRβ)

(12)

where kλRβ and lλRβ are adjustable binary interaction parameters that are expected to be small relative to unity. In many cases, we expect quantitatively accurate results even when these binary parameters are set to the default value of zero. The equivalent substance reducing ratios for the pure fluids are correlated as ratios of the critical properties and empirical shape factors that are typically close to unity16

fλR )

TcR θ Tc0 λR

(13)

hλR )

vcR φ vc0 λR

(14)

The shape factors are represented as empirical functions of the reduced temperature (TRR) of component R as follows

( (

) )

θλR ) 1 + θλR1 + θλR2 ln(TRR) + θλR3 1 -

1 TR R

(15)

φλR ) 1 + φλR1 + φλR2 ln(TRR) + φλR3 1 -

1 TR R

(16)

The coefficients in eqs 15 and 16 are typically adjusted to obtain a highly accurate description of the thermal conductivity. In the original model of Leach et al.16 and in the application of Ely and Hanley,4 the shape factors are functions of temperature and density, but we have found that dependence on temperature alone is adequate in practice. Through analysis of experimental data, we have also found that reasonable accuracy is obtained, especially for nonpolar fluids, if the coefficients are estimated in terms of the acentric factor (ωR), as we show below.

Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 991 Table 1. Summary of Pure-Component Equation-of-State Parameters compound

Tc (K)

Pc (MPa)

Vc (m3/kmol)

ω

β1

β2

β3

c1

c2

ethane propane butane pentane heptane decane ethylene acetylene benzene toluene nitrogen carbon dioxide methanol ethylbenzene p-xylene n-pentadecane biphenyl cis-decalin hexafluoroethane icosane refrigerant 134a water ethanol ethylene glycol

305.6 370.1 425.3 469.8 540.2 617.6 283.1 309.5 561.7 593.8 126.2 304.2 513.2 617.2 616.2 706.8 789.3 702.3 292.8 767.0 374.2 647.0 516.3 692.8

4.88 4.26 3.79 3.37 2.73 2.11 5.12 6.24 4.86 4.21 3.39 7.38 7.95 3.61 3.51 1.42 3.85 3.24 2.98 1.12 4.07 22.12 6.38 6.62

0.148 0.201 0.256 0.311 0.426 0.602 0.124 0.113 0.257 0.316 0.090 0.094 0.119 0.375 0.379 0.865 0.491 0.480 0.224 1.129 0.199 0.056 0.167 0.198

0.105 0.152 0.201 0.251 0.352 0.486 0.088 0.195 0.210 0.257 0.040 0.225 0.556 0.303 0.303 0.706 0.000 0.294 0.245 0.906 0.336 0.348 0.635 0.638

0.5336 0.6028 0.6727 0.7453 0.8841 1.0604 0.4827 0.6380 0.6452 0.7532 0.4359 0.7080 1.2340 0.8114 0.8597 1.2761 0.6738 0.7628 0.8890 1.6055 0.8672 0.9239 1.3095 1.2366

0.0000 0.0000 -0.0645 0.0000 0.0000 0.0000 0.1188 0.0481 0.2203 0.0000 -0.0729 0.0000 -0.4972 -0.0135 -0.1573 0.1766 1.2827 0.1417 -1.2878 0.0019 -0.1475 -0.3835 -0.1345 0.2896

0.0000 0.0000 0.2735 0.0000 0.0000 0.0000 0.0000 0.0313 0.0000 0.0000 0.3092 0.0000 0.0217 0.2485 0.3636 0.3629 -0.8199 0.1968 2.7257 0.0679 0.1838 0.4375 -0.5533 -0.7206

0.1074 0.1080 0.1657 0.0841 0.0382 -0.1425 0.1558 0.0752 0.1239 0.1840 0.0915 0.0386 -0.1215 0.1180 0.1054 -1.0670 0.4078 0.2367 0.1110 -0.8197 0.0069 -0.0843 -0.0151 -0.2873

-10.6072 0.1010 -36.6620 0.1000 0.0999 -0.1605 -20.4695 -11.7819 -45.2447 -74.8119 -3.3018 -0.0336 0.0000 -77.9220 -92.9789 -8.3045 -147.615 -122.521 0.0100 -493.835 -0.3386 19.2300 0.0000 5.1858

Table 2. Comparison of Thermal Conductivity Predictions (Using Background Model Only) with Data Using the Generalized Shape Factors (eqs 19-24) vapor

liquid

compound

maximum deviation

average deviation

avg abs deviation

maximum deviation

average deviation

avg abs deviation


13.67 11.62 -21.82 -10.37 -20.24 -21.14 44.23 14.55 -28.27 -18.31 12.73 -8.78 -33.22 -19.20 -23.10 -21.83 21.83 -15.14 -12.19 -55.96 -

3.87 1.62 -4.70 -4.06 -13.81 -17.59 11.68 6.17 -7.12 -16.56 5.41 -3.02 -24.57 -17.25 -20.51 -19.64 9.68 -7.23 -0.24 -25.00 -

5.40 1.94 7.00 5.28 13.81 17.59 11.68 6.63 7.12 16.56 5.41 4.10 24.57 17.25 20.51 19.64 9.68 7.99 6.29 25.00 -

12.24 -17.59 27.24 -33.45 -18.83 -19.80 -25.22 -8.11 53.29 -27.88 -3.65 315.04 20.83 19.91 14.17 -40.08 15.63 -20.53 -31.37 -10.96 213.50 309.24 835.71

4.52 -5.13 9.73 -10.35 -9.18 -7.72 -19.68 0.11 16.59 -10.81 0.24 281.22 11.62 11.01 7.33 -37.47 4.40 -14.33 -12.36 -8.93 41.01 280.45 668.98

5.28 5.29 9.73 10.88 9.18 8.18 19.68 3.29 17.10 10.81 1.64 281.22 11.62 11.01 7.80 37.47 7.08 14.33 12.36 8.93 64.24 280.45 668.98

The mixing rule chosen for the molar mass is that proposed by Ely and Hanley4

Mλx-1/2fλx1/2hλx-4/3 )

∑R ∑β xRxβMλ

-1/2

Rβ

fλRβ1/2hλRβ-4/3 (17)

where

1 MλRβ-1 ) (MR-1 + Mβ-1) 2

(18)

Engineering correlations are typically used with temperature and pressure, rather than temperature and density, as the independent variables. We convert from the (T, P) specification to the (T, F) specification by using the models of Mathias and co-workers,17,18 which are modifications of the Peng-Robinson19 equation of state.

The modifications of Mathias and co-workers enable the Peng-Robinson equation of state to provide an accurate description of fluid density and vapor-liquid equilibrium. We have adopted methane as the reference fluid. Ely and Hanley4 have developed a careful correlation for the translational thermal conductivity of methane in terms of its density and temperature. This correlation is used in eq 6 to predict the translational thermal conductivity of the hypothetical pure fluid. As noted by Ely and Hanley,4 the choice of the reference fluid is arbitrary. Ely and Hanley chose methane as the reference fluid mainly because extensive experimental data are available for methane. Also, because the liquid range of methane is limited, Ely and Hanley extended the reference fluid correlations into a pseudo-liquid region.

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Table 3. Summary of Coefficients in the Shape Factor Correlation (eqs 15 and 16) compound

φ1

φ2


0.246 0.224 0.113 -0.076 -0.257 -0.171 0.306 0.172 -0.400 -1.184 0.195 0.182 0.410 -0.487 -0.531 -0.062 -0.461 -0.121 -0.070 -0.101 -0.058 0.459 0.731 0.646

-0.228 -0.084 -0.590 0.066 -0.470 -1.061 -0.464 -0.319 -1.185 -1.544 -0.207 -0.857 0.189 -0.990 -0.780 0.797 -0.859 0.295 0.264 1.469 7.197 -1.010 2.734 -0.199

φ3

θ1

θ2

θ3

0.000 0.016 0.112 0.000 0.000 0.057 0.016 0.000 0.000 0.108 0.128 0.000 0.000 0.342 0.215 0.000 0.000 0.420 0.316 0.000 0.000 0.376 0.324 0.000 0.000 0.141 0.124 0.000 0.000 0.208 -0.055 0.000 0.141 0.529 2.068 -0.939 0.695 0.912 1.090 -0.312 -0.104 0.063 0.029 0.095 0.000 0.156 0.358 0.000 0.000 -0.165 0.013 0.000 -0.193 0.485 1.519 -0.596 0.000 0.470 0.560 0.000 0.000 0.604 0.324 0.000 0.000 0.518 0.556 0.000 0.000 0.407 0.242 0.000 0.000 0.362 0.154 0.000 0.000 1.191 0.126 0.000 -6.552 0.262 -2.236 2.175 0.000 0.022 0.200 0.000 -0.605 -0.259 -0.838 0.136 0.000 -0.172 -0.210 0.126

The work of Ely and Hanley4 on the thermal conductivity of methane provides a sound basis for the reference fluid, and we further note here that Ely and Hanley were careful to eliminate the critical enhancements in their correlation for the reference translational thermal conductivity of methane. An improved correlation is available for the thermal conductivity of methane,20 and other researchers21-23 have used propane as the reference fluid because it has a relatively low freezing point and, hence, an extended fluid region. However, we have retained the original Ely-Hanley approach, which is to use their 1983 correlation for the translational thermal conductivity of methane,4 as our analysis indicates that it provided a sound basis for the correlation.

To develop a predictive model, we have regressed data for several nonpolar species to predict the coefficients in eqs 15 and 16 in terms of the difference in acentric factor between the fluid of interest (ωR) and the reference fluid, methane (ω0), as follows

θλR1 ) (ωR - ω0)0.591

(19)

θλR2 ) (ωR - ω0)0.420

(20)

θλR3 ) 0

(21)

φλR1 ) (ωR - ω0)0.945

(22)

φλR2 ) (ωR - ω0)(-0.184)

(23)

φλR3 ) 0

(24)

Equations 19-24 were developed by simultaneous regression of several sets of data for the thermal conductivity of pure fluids. These pure fluids are the normal alkanes (C1-C10) and the light gases such as oxygen and nitrogen, and the data were taken from the DIPPR database.24 The model described thus far provides a fair predictive and good correlative description of the thermal conductivity of fluids for which the critical enhancements (∆λc) are negligible. In the predictive mode, with the shape factors estimated by eqs 19-24 and the binary interaction parameters in eqs 11 and 12 set equal to zero, the performance of the model is similar to that of the correlation of Ely and Hanley,4 on which the present model is based. In addition, the ability to vary the parameters for the shape factors and the binary interactions permits excellent correlative capability for systems where experimental data are available.

Table 4. Comparison of Thermal Conductivity Predictions (Using Background Model Only) with Data Using the Shape Factor Correlations with Parameters Reported in Table 3 vapor

a

liquid

compound

maximum deviation

average deviation

avg abs deviation

maximum deviation

average deviation

avg abs deviation


7.77 6.65 13.75 18.61 7.73 3.68 19.14 9.62 23.09 20.65 7.83 3.80 12.59 0.37 2.10 3.90 4.01 9.02 4.61 36.14 -

0.12 -0.97 -3.66 2.17 -0.55 -0.18 -2.04 -0.07 -2.61 9.86 -0.02 -0.70 5.03 0.01 -0.07 0.19 0.09 -0.31 0.15 0.70 -

2.66 1.82 4.29 5.26 2.60 1.30 5.00 1.47 2.94 10.51 2.72 1.13 6.33 0.16 1.05 2.12 1.11 1.74 1.19 9.24 -

4.44 3.82 6.70 25.19 9.89 17.22 6.21 5.79 5.22 23.44a 2.79 6.29 0.07 0.97 10.43 4.05 2.73 12.02 6.30 2.0 73.77a 5.78 22.24

1.31 1.48 3.84 -0.84 2.82 -3.73 0.70 -0.64 0.41 -1.60 0.85 3.87 0.00 -0.04 0.27 -0.05 -0.06 -0.98 -0.08 1.35 -18.12 0.73 -2.73

2.34 2.13 4.22 9.37 4.35 6.42 2.58 1.21 1.39 3.79 1.78 3.87 0.03 0.58 3.90 1.93 1.08 3.67 2.41 1.35 22.38 1.97 5.90

DIPPR database includes points with significant critical enhancement.

Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 993

Figure 1. Thermal conductivity of carbon dioxide. Comparison with tabulated values of Vesovic et al. (1990).25

Figure 2. Thermal conductivity vs density for carbon dioxide. Comparison with data of Michels et al. (1962).32

The Ely-Hanley4 model introduces additional correction terms to account for failures of the corresponding-states model. We have not found it necessary to include any such effects in our model. The Ely-Hanley4 model does not describe the strong critical enhancements in the vicinity of the vapor-liquid critical point. An important goal of the present work is to develop an engineering model for this important theoretical and practical effect. Model for Critical Enhancements Sengers and co-workers1,5-7 have established the asymptotic variation of the thermal conductivity of pure fluids and mixtures in the vicinity of the vapor-liquid critical point. These theories have been used to develop correlations for the thermal conductivity of several pure fluids, including carbon dioxide,25 R134a,26 ethane,27 and methane.28 The application of these theories to mixtures is relatively recent, but the experimental data and theo-

retical foundations have been established. The mixtures that have been studied include ethane-carbon dioxide29 and methane-ethane.30 The goal of these studies has been to apply rigorous data analysis and theory to obtain a deep understanding of the variation of the thermal conductivity in the vapor-liquid critical region and to develop highly accurate correlations for specific fluids of interest. Our present aim is to develop a phenomenological model that broadly captures the critical enhancements, with the goal of developing a generalized engineering correlation. We postulate that the critical enhancements to the thermal conductivity of pure fluids can be captured by a thermodynamic quantity that diverges at the critical point. Following an analysis of several possibilities, we chose the bulk modulus δ, a dimensionless quantity related to the inverse of the isothermal compressibility, as this divergent thermodynamic property

δ≡-

V2 ∂P NRT ∂V T,N

( )

(25)

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Figure 3. Thermal conductivity vs density for methane. Comparison with data of Sakonidou et al. (1996).28

Figure 4. Thermal conductivity vs density for ethane. Comparison with data of Mostert et al. (1990).33

δ goes to zero in the limit of intrinsic stability of a pure fluid and, in particular, is zero at the critical point of a pure fluid. To retain easy application to a wide range of pure fluids, we use the Peng-Robinson19 equation to compute δ, recognizing that this simple classical model will not rigorously describe the thermodynamic properties in the close vicinity of the critical point. We propose a simple correlation for the critical enhancement of the thermal conductivity of pure fluids

∆λc ) aδ-b

(26)

Equation 26 provides a reasonable description of the critical enhancement to the thermal conductivity as long as the parameters a and b are optimized for each substance of interest. Next, we propose that eq 26 can be extended to mixtures if the definition of δ is extended to the intrinsic thermodynamic stability of multicomponent mixtures. Guided by the work of Beegle et al.,31 we write δ for a multicomponent mixture as

δmix ≡ -

( )

V2

∂P

∑xR ∂V T,N ,µ NRT R R

(27) β,β*R

where

N)

∑R NR

(28)

Each of the derivatives on the right side of eq 27 will be zero in the limit of intrinsic stability and will have negative values in the metastable and stable regions of the fluid. We choose a mole-fraction average over all of the stability derivatives in order to ensure that eq 27 properly reduces to the pure-component limit, eq 25. Again, the Peng-Robinson18,19 model is chosen as the basis for computing δ in eq 27. To apply eq 26 to mixtures, we need to estimate the parameters a and b. We propose simple mole-fraction averages for these quantities

a)

∑R xRaR

(29)


Figure 5. Thermal conductivity vs density for R134a. Comparison with data of Krauss et al. (1993).26

Figure 6. Thermal conductivity of water-ethanol mixture at 300 K and 0.1 MPa. Comparison to data in Dechema (1988).34

b)

∑R xRbR

(30)

The application of eq 26 to most mixtures is qualitatively incorrect at the critical point as the predicted thermal conductivity will diverge, whereas the true value approaches a high, but finite, enhancement.30 Our analysis, albeit based only on severely limited experience, suggests that the region of unacceptable quantitative error is confined to an extremely small area around the critical point; thus our model will be useful as an engineering correlation. Results for Pure Fluids Table 1 presents the universal pure-component parameters. These parameters are generally readily available and provide an indication of the predictive capability of the thermal conductivity model. The parameters presented in Table 1 are the critical constants (Tc, Pc, Vc), the acentric factor (ω), the parameters for correlating the vapor pressure of pure fluids18 (β1, β2, and β3), and the parameters for the density correction17 (c1 and c2). The predictive results from the background model, which are presented in Table 2, are similar to those available with the Ely-Hanley4 model. The experimen-

tal data used in Table 2 were taken from the 1998 DIPPR database.24 The average errors for nonpolar fluids are typically within 15%, although larger errors are observed for some nonpolar fluids such as toluene and ethylene. It is clear that the predictive model is highly inaccurate for polar fluids such as methanol, ethanol, water, and ethylene glycol. Although predictions from a generalized model are necessary for cases where data are not available, predictive results are rarely equal in accuracy to the experimental data. In cases where the experimental data are available, we recommend that the empirical parameters of eqs 15 and 16 be adjusted to obtain an optimum fit of the data. A set of these optimized constants is presented in Table 3, and Table 4 shows a comparison between experimental data and the correlated background model. The accuracy of the results is clearly improved from the predictive model. In general, we have found that the correlated model will represent vapor and liquid thermal conductivity data of nonpolar fluids to an accuracy of about 2-4% in the noncritical region, which is close to the accuracy of the experimental measurements. The correlative model also provides an effective way to represent the thermal conductivity of polar fluids. No existing methods can predict the thermal conductivity of polar fluids, but Table 4 indicates that the cor-

996


Figure 7. Thermal conductivity of water-ethylene glycol mixture at 313 K and 0.1 MPa. Comparison to data from Rastorguev and Ganier (1967).35

Figure 8. Thermal conductivity of 50/50 methane-ethane mixture. Density ) 6.2036 kmol/m3.

relational model is able to produce a reasonable engineering correlation for fluids such as methanol and ethanol. It should be noted that the critical enhancements are not included in the model and that this causes some large deviations for fluids like nitrogen and water. The accuracy does decrease for highly polar fluids such as water, and we expect that the correlation will have to be adapted to provide an accurate description of the desired domain of application, with an inevitable deterioration in accuracy for conditions outside this region. A key goal of the present paper has been to develop an engineering correlation that captures the strong critical enhancements of the thermal conductivity in the vicinity of the critical point. Figure 1 demonstrates that the model captures the strong enhancements of the saturated vapor and liquid thermal conductivity in the region of the vapor-liquid critical point of carbon dioxide. To focus on the critical region, the rest of the figures in this section highlight the area close to the critical

point. Figures 2-5 present results for the fluids carbon dioxide,32 methane,28 ethane,33 and R134a26 (the reference for each fluid identifies the data sources) and demonstrate that the model quantitatively captures the sharp rise in the thermal conductivity as the critical density and critical temperature are approached. We do note, however, that the constants a and b in eq 26 have been optimized for each fluid and are not universal. The results are sensitive to the values of a and b, and we have not been able to develop a generalized correlation for these constants. We conclude that our model provides a reasonable engineering model for estimating the thermal conductivity of nonpolar pure fluids in the noncritical region. However, even in the noncritical region, the use of the correlative model is recommended to improve the accuracy. The full model provides an effective engineering approach for correlating the thermal conductivity of most pure fluids over the entire fluid region, including the critical region.




Results for Mixtures The mixing rules defined by eqs 9-12 and 17-18 provide a good correlative method for describing the thermal conductivity of mixtures away from the critical region. We have found that the model reasonably predicts the thermal conductivities of nonpolar mixtures even if the binary parameters in eqs 11 and 12 are set to zero. Figures 6 and 7 suggest that the model is able to correlate the mixture thermal conductivity of polar mixtures, such as water-ethanol (data from Dechema34) and water-ethylene glycol (data of Rastorguev and Ganier35) if the binary parameters are adjusted to provide a best fit of the data. As with the pure fluids, a key goal of the present paper has been to develop an engineering model for correlating the critical enhancements in the vicinity of the vapor-liquid critical point. We have applied our

model to the study of the methane-ethane mixture. Figures 8-12 compare calculations from the present model to the data of Sakonidou et al.30 These experiments studied an equimolar methane-ethane mixture, where isochoric measurements of the thermal conductivity were made as a function of temperature. The densities studied range from 6.2 to 10.5 kmol/m3, and the critical density is about 8.5 kmol/m3. The simple equation of state used to describe the mixture (Peng-Robinson18,19 with density correction17 and standard binary interaction parameter equal to 0.007) does not provide a completely accurate description of the vapor-liquid critical point of the equimolar methane-ethane mixture, and hence, the following adjustments were made to the model. The density was multiplied by a factor to ensure that the model critical density (9.00 kmol/m3) coincided with the experimental

998




critical density (8.53 kmol/m3). Thus, the calculations were made at densities 1.06 times the reported values. Also, the plots were shifted in temperature so that the calculated critical temperature (265.5 K) coincided with the experimental critical temperature (263.6 K). Thus 1.9 K was subtracted from the calculated temperature. These corrections are relatively minor but are necessary for the predicted enhancements to be compared with the measured values. Figures 8-12 indicate that the model captures the increasingly large critical enhancements as the critical density and critical temperature are simultaneously approached. Sakonidou et al.30 demonstrated that the experimental thermal conductivity reaches a high, but finite, value at the critical point of the mixture. Our model will incorrectly diverge, but it appears that the region of qualitatively incorrect behavior is confined to a very small area in the vicinity of the critical point,

and this region has not yet been reached in the data presented in Figures 8-12. Conclusions In this paper, we have presented and evaluated an engineering model for the thermal conductivity of pure fluids and mixtures. We have made a special effort to develop a model that describes the strong critical enhancements of the thermal conductivity. The model provides reasonable predictions of the thermal conductivity of nonpolar pure fluids outside the critical region, but we recommend that componentspecific parameters be adjusted to optimize the fit when experimental data are available. The predictive model is not able to provide quantitative predictions of the thermal conductivity of polar pure fluids. The adjustable parameters enable a reasonable quantitative fit of


experimental data for polar fluids in the noncritical region, but problems remain for highly polar fluids such as water. The model is able to describe the strong critical enhancements in the vicinity of pure-fluid critical points, but component-specific parameters must be adjusted to quantitatively capture this effect. The mixture model is able to predict the thermal conductivity of nonpolar mixtures away from the critical region reasonably well, and it can be used with adjustable binary interaction parameters to describe the thermal conductivity of polar liquid mixtures away from the critical region. We have made only limited evaluations of the model for near-critical mixtures. Analysis of the methaneethane mixture suggests that the model captures the critical enhancements in the near-critical region. Very close to the critical point, the model will predict thermal conductivities that are too large, and it will incorrectly diverge at the critical point. However, it appears that the region where the model shows large positive errors is confined to a very small region around the mixture critical point. Literature Cited (1) Sengers, J. V. Transport Properties of Fluids Near Critical Points. Int. J. Thermophys. 1985, 6, 203. (2) Sengers, J. V.; Luettmer-Strathmann, J. The Critical Enhancements. In Transport Properties of Fluids: Their Correlation, Prediction and Estimation; Millat, J., Dymond, J. H., Nieto de Castro, C. A., Eds.; Cambridge University Press: New York, 1996; p 113. (3) Mathias, P. M.; Parekh, V. S. Prediction and Correlation of the Viscosity and Thermal Conductivity of Pure Fluids and Mixtures. Presented at the AIChE Annual Meeting, Chicago, IL, Nov 1996. (4) Ely, J. F.; Hanley, H. J. M. Prediction of Transport Properties. 2. Thermal Conductivity of Pure Fluids and Mixtures. Ind. Eng. Chem. Fundam. 1983, 22, 90. (5) Mostert, R.; Sengers, J. V. Asymptotic Behaviour of the Thermal Conductivity of a Binary Fluid Near the Plait Point. Fluid Phase Equilib. 1992, 75, 235. Also see: Corrigendum. Fluid Phase Equilib. 1993, 85, 347. (6) Luettmer-Strathmann, J.; Sengers, J. V. The Transport Properties of Fluid Mixtures Near the Vapor-Liquid Critical Line. J. Chem. Phys. 1996, 104, 3026. (7) Sakonidou, E. P.; van den Berg, H. R.; ten Seldam, C. A.; Sengers, J. V. The Thermal Conductivity of an Equimolar Methane-Ethane Mixture in the Critical Region. J. Chem. Phys. 1998, 109, 717. (8) Olchowy, G. A.; Sengers, J. V. A Simplified Representation for the Thermal Conductivity of Fluids in the Critical Region. Int. J. Thermophys. 1989, 10, 417. (9) Kiselev, S. B.; Perkins, R. A.; Huber, M. L. Transport Properties of Refrigerants R323, R125, R143a, and R125 + R132 Mixtures in and beyond the Critical Region. Int. J. Refrig. 1999, 22, 509. (10) Asgeirsson, L. S.; Ghajar, A. S. Prediction of Thermal Conductivity for Some Fluids in the Near-Critical Region. Chem. Eng. Commun. 1986, 43, 165. (11) Roder, H. M. The Thermal Conductivity of Oxygen. J. Res. Natl. Bur. Stand. 1982, 87, 279. (12) Shi, L.; Liu, X. J.; Wang, X.; Zhu, M.-S. Prediction Method for Thermal Conductivity of Refrigerant Mixtures. Fluid Phase Equilibr. 2000, 172, 293. (13) Ramires, M. L. V.; Nieto de Castro, C. A.; Perkins, R. A. An Improved Empirical Correlation for the Thermal Conductivity of Propane. Int. J. Thermophys. 2000, 21, 639. (14) Sengers, J. V. Transport Processes Near the Critical Point of Gases and Binary Liquids in the Hydrodynamic Region. Ber. Bunsen-Ges. Phys. Chem. 1972, 76, 234.

(15) Li, C. C. Thermal Conductivity of Liquid Mixtures. AIChE J. 1976, 22, 927. (16) Leach, J. W.; Chappelear, P. S.; Leyland, T. W. Use of Molecular Shape Factors in Vapor-Liquid Equilibrium with the Corresponding States Principle. AIChE J. 1968, 14, 568. (17) Mathias, P. M.; Naheiri, T.; Oh, E. M. A Density Correction for the Peng-Robinson Equation of State. Fluid Phase Equilib. 1989, 47, 77. (18) Mathias, P. M.; Copeman, T. W. Extension of the PengRobinson Equation of State to Complex Mixtures: Evaluation of Various Forms of the Local Composition Concept. Fluid Phase Equilib. 1983, 13, 91. (19) Peng, D. Y.; Robinson, D. B. A New Two Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59. (20) Friend, D. G.; Ely, J. F.; Ingham, H. Thermophysical Properties of Methane. J. Phys. Chem. Ref. Data 1989, 18, 583. (21) Huber, M. L.; Hanley, H. M. J. The Corresponding-States Principle: Dense Fluids. In Transport Properties of Fluids: Their Correlation, Prediction and Estimation; Millat, J., Dymond, J. H., Nieto de Castro, C. A., Eds.; Cambridge University Press: New York, 1996. (22) Baltatu, M. E.; Chong, R. A.; Huber, M. L.; Laesecke, A. Transport Properties of Petroleum Fractions. Presented at the 13th International Symposium on Thermophysical Properties, Boulder, CO, June 1997. (23) NIST Standard Reference Database 4: Thermophysical Properties of Hydrocarbon Mixtures, version 3.01; NIST: Gaithersburg, MD, 1999. (24) American Institute of Chemical Engineers (AIChE) Design Institute for Physical Properties (DIPPR) Project 801, Evaluated Process Design Data; AIChE: New York, 1998. (25) Vesovic, V.; Wakeham, W. A.; Olchowy, G. A.; Sengers, J. V.; Watson, J. T. R.; Millat, J. The Transport Properties of Carbon Dioxide. J. Phys. Chem. Ref. Data 1990, 19, 763. (26) Krauss, R.; Luettmer-Strathmann, J.; Sengers, J. V.; Stephan, K. Transport Properties of 1,1,1,2-Tetrafluoroethane (R134a). Int. J. Thermophys. 1993, 14, 951. (27) Vesovic, V.; Wakeham, W. A.; Luettmer-Strathmann, J.; Sengers, J. V.; Millat, J.; Vogel, E.; Assael, M. J. The Transport Properties of Ethane. II. Thermal Conductivity. Int. J. Thermophys. 1994, 15, 33. (28) Sakonidou, E. P.; van den Berg, H. R.; ten Seldam, C. A.; Sengers, J. V. The Thermal Conductivity of Methane in the Critical Region. J. Chem. Phys. 1996, 105, 10535. (29) Luettmer-Strathmann, J.; Sengers, J. V. Transport Properties of Fluid Mixtures in the Critical Region. Int. J. Thermophys. 1994, 15, 1241. (30) Sakonidou, E. P.; van den Berg, H. R.; ten Seldam, C. A.; Sengers, J. V. Finite Thermal Conductivity at the Vapor-Liquid Critical Line of a Binary Fluid Mixture. Phys. Rev. E 1997, 56, 4943. (31) Beegle, L. B.; Modell, M.; Reid, R. C. Thermodynamic Stability Criterion of Pure Fluids and Mixtures. AIChE J. 1974, 20, 1200. (32) Michels, A.; Sengers, J. V.; van der Gulik, P. S. The Thermal Conductivity of Carbon Dioxide in the Critical Region. II. Measurements and Conclusions. Physica 1962, 28, 1216. (33) Mostert, R.; van den Berg, H. R.; van der Gulik, P. S.; Sengers, J. V. The Thermal Conductivity of Ethane in the Critical Region. J. Chem. Phys. 1990, 92, 5454. (34) Stephan, K.; Heckenberger, T. Thermal Conductivity and Viscosity Data of Fluid Mixtures; Dechema Chemistry Data Series; Dechema: Frankfurt, Germany, 1988; Vol. X, Part 1. (35) Rastorguev, Y. L.; Ganier, Y. A. Thermal Conductivity of Nonelectrolyte Solutions. Russ. J. Phys. Chem. 1967, 41, 717.

Received for review March 30, 2001 Revised manuscript received September 26, 2001 Accepted September 28, 2001 IE0102854

Prediction and Correlation of the Thermal Conductivity of Pure Fluids

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