Measurements of the Thermal Conductivity of 1,1,1,3,3

Article pubs.acs.org/jced

Measurements of the Thermal Conductivity of 1,1,1,3,3Pentafluoropropane (R245fa) and Correlations for the Viscosity and Thermal Conductivity Surfaces Richard A. Perkins* and Marcia L. Huber Applied Chemicals and Materials Division, National Institute of Standards and Technology, Boulder, Colorado 80305-3337, United States

Marc J. Assael Laboratory of Thermophysical Properties and Environmental Processes, Chemical Engineering Department, Aristotle University, Thessaloniki 54636, Greece S Supporting Information *

ABSTRACT: New experimental data on the thermal conductivity of 1,1,1,3,3pentafluoropropane (R245fa) are reported that cover a wide range of liquid conditions. These new experimental data were made with a transient hot-wire apparatus and cover the liquid phase over a temperature range of 173−344 K and a pressure range of 0.1−71 MPa. The experimental data reported here have an expanded uncertainty (0.95 level of confidence) of less than 1%. The measurements are used with selected literature data to develop correlations for the thermal conductivity. On the basis of this expanded uncertainty and comparisons with experimental data, the thermal conductivity correlation for R245fa is estimated to have a relative expanded uncertainty (0.95 level of confidence) of about 2% at a 95% confidence level for the liquid phase at pressures to 70 MPa and 2% for the vapor phase. In addition, we surveyed literature data and developed a correlation for the viscosity of R245fa. The estimated relative expanded uncertainty (0.95 level of confidence) of this correlation is 3% for the liquid phase at pressures to 40 MPa and 2% for the vapor phase.

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INTRODUCTION

EXPERIMENTAL METHOD Analysis of the sample of R245fa at NIST by gas chromatography (GC) with mass spectrometry (MS) detection, following the protocols of Bruno and Svoronos,8,9 indicated a purity of 99.6 mol %. This analysis revealed the largest impurity as ethane, with lesser quantities of 2,2-dichloro1,1,1-trifluoropropane and 1,1,2-trichloro-1,2,2-trifluoropropane. This two-phase sample was filled into the evacuated measurement cell from the liquid phase. The measurements of thermal conductivity were obtained with a transient hot-wire instrument that has previously been described in detail.10 During an experiment, the hot wires functioned as both electrical heat sources and resistance thermometers to measure the temperature rise. The measurement cell, of volume 25 cm3, consisted of a pair of hot wires of differing lengths operating in a differential arrangement to eliminate errors due to axial conduction. The outer cavity around the hot wires was copper with a diameter of 9 mm. The

R245fa (1,1,1,3,3-pentafluoropropane) is a hydrofluorocarbon that is under consideration for use as a working fluid in organic Rankine cycles with low-temperature heat sources.1−4 It also is of interest for use as a blowing agent5 and in centrifugal chillers.6 It is important to have accurate models for the thermophysical properties of working fluids to efficiently design equipment. Recently, a new equation of state has been developed by Akasaka et al.7 that provides thermodynamic properties for R245fa. It is also necessary to have accurate models for the transport properties thermal conductivity and viscosity. In order to supplement the literature data, we measured the thermal conductivity of R245fa from 173−344 K at pressures up to 71 MPa and developed a wide-ranging correlation for the thermal conductivity. We also developed a wide-ranging viscosity surface using literature data. These correlations, along with the equation of state, can be used to provide thermophysical properties over the entire fluid surface of R245fa including the liquid, gas, and supercritical regions. © XXXX American Chemical Society

Received: April 28, 2016 Accepted: July 7, 2016

A

DOI: 10.1021/acs.jced.6b00350 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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critical density, ρc, and the critical pressure are 427.01 K, 3.875 mol·L−1, and 3.651 MPa. The triple-point temperature is 170 K. The range of state points covered by the present measurements is shown in Figure 1 relative to the vapor-pressure curve

transient hot wires were enclosed by a copper pressure vessel that is capable of operation from 30 to 340 K at pressures to 70 MPa in the liquid, vapor, and supercritical gas phases. Initial cell temperatures, Ti, were determined with a reference platinum resistance thermometer with an expanded uncertainty (0.95 level of confidence) of 0.005 K, and pressures, pe, are determined with a pressure transducer with an expanded uncertainty (0.95 level of confidence) of 7 kPa. The measurements were made with bare platinum hot wires with a diameter of 12.7 μm. All expanded uncertainties in this work are for a coverage factor of k = 2, approximately a 95% confidence interval. The basic theory that describes the operation of the transient hot-wire instrument is given by Healy et al.11 The hot-wire cell was designed to approximate a transient line source as closely as possible, and deviations from this model are treated as corrections to the experimental temperature rise. The ideal temperature rise ΔTid is given by ΔTid =

⎛ 4a ⎞⎤ q ⎡ ⎢ln(t ) + ln⎜ 2 ⎟⎥ = ΔTW + 4πλ ⎢⎣ ⎝ r0 C ⎠⎥⎦

10

∑ δTi i=1

(1) Figure 1. Distribution of experimental thermal conductivity data for R245fa in the T,p plane. Data sets are designated: ◆, Wang et al.;13 ○, Grebenkov et al.;14 +, Yata et al.;15 △, Geller et al.;16 ◊, Dohrn17 (Heinemann et al.18); ▲, present work; ●, critical point; and the dashed line showing the saturation boundary.

where q is the power applied per unit length, λ is the thermal conductivity of the fluid, t is the elapsed time, a = λ/(ρCp) is the thermal diffusivity of the fluid, ρ is the density of the fluid, Cp is the isobaric specific heat capacity of the fluid, r0 is the radius of the hot wire, C = 1.781... is the exponential of Euler’s constant, ΔTw is the measured temperature rise of the wire, and δTi are corrections11 to account for deviations from ideal linesource conduction. During analysis, a line is fit to the linear section, from 0.1 to 1.0 s, of the ΔTid versus ln(t) data, and the thermal conductivity is obtained from the slope of this line. Both thermal conductivity and thermal diffusivity can be determined with the transient hot-wire technique as shown in eq 1, but only the thermal conductivity results are considered here. The experiment temperature, Te, associated with the thermal conductivity is the average temperature at the wire’s surface over the period that was fit to obtain the thermal conductivity.

of R245fa, along with other data in the literature. Our measurements comprise 1138 points in the liquid phase at temperatures from 173 to 344) K at pressures up to 71 MPa, that are reported in the Supporting Information. Table 1 summarizes all experimental thermal conductivity measurements that we are aware of, including the experimental method used, sample purity, and the author’s estimated relative expanded uncertainty U% (0.95 level of confidence).

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THERMAL CONDUCTIVITY CORRELATION We represent the thermal conductivity λ of a pure fluid as a sum of three contributions,

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EXPERIMENTAL RESULTS The results of the transient measurements of the thermal conductivity of R245fa are tabulated in the Supporting Information. Transient experiments were analyzed over the time range from 0.1 to 1.0 s. The reproducibility of the thermal conductivity from relative expanded uncertainty of the slope of the corrected transient-temperature-rise data over this time range is given in the tables of transient data in the Supporting Information. This expanded uncertainty (k = 2) depends on both temperature-rise noise and systematic curvature and is less than 0.2% for the transient measurements in the liquid phase. There are two wide-ranging equations of state available for R245fa.7,12 We selected the most recent one by Akasaka et al.7 to provide density, the critical point, and the triple point. The Akasaka et al.7 equation provides an improvement over the older Lemmon and Span EOS12 especially for vapor pressures, due to the availability of new data and the use of Gaussian bellshaped terms to improve the behavior in the critical region. This equation has an expanded uncertainty of 0.1% for saturated liquid densities, 0.1% for liquid densities below 70 MPa, 0.2% for densities at higher pressures, and 0.3% for vapor densities. It is valid from the triple-point temperature to 440 K with pressures up to 200 MPa. The critical temperature, Tc, the

λ(ρ , T ) = λ 0(T ) + Δλr(ρ , T ) + Δλc(ρ , T )

(2)

where λ0 is the dilute-gas thermal conductivity, which depends only on temperature, Δλr is the residual thermal conductivity, and Δλc is the enhancement of the thermal conductivity in the critical region. The critical enhancement can contribute significantly to the thermal conductivity over a fairly wide region; for example, for water19 the critical enhancement contribution to thermal conductivity exceeds 5% for a region of reduced density approximately from 0.3 to 1.7 for reduced temperatures from approximately 0.95 to 1.1. Both Δλr and Δλc are functions of temperature, T, and density, ρ, with ρ calculated with an equation of state for each given T and p. As mentioned above, in this work we use the recent Helmholtz equation of state of Akasaka et al.7 Dilute-Gas Thermal Conductivity. We represent the dilute-gas thermal conductivity as a polynomial in reduced temperature, 1

λ 0(T )/(W·m−1·K−1) =

∑ Ak (T /Tc)k k=0

(3)

with coefficients Ak, where T is the temperature and Tc is the critical temperature. B



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Table 1. Thermal Conductivity Measurements of R245fa

a

first author

year

methoda

purity (%)

U (%)

no. pts

T range (K)

p range (MPa)

this work Wang13 Grebenkov14 Yata15 Geller16 Dohrn17,b

2016 2006 2004 2000 1999 1999

Thw Thw CCyl Thw SShw Thw

99.6 99.9 99.99 99.9 na na

0.6 2 3.5 1 1 8

1138 148 308 16 50 4

173.2−343.5 244.4−416.2 284.7−436.7 253.4−314.9 234.6−413.6 333.2−382.2

0.1−70.5 0.0−3.0 0.1−10 0.7−14.7 0−2.7 0.1

CCyl coaxial cylinder; na not available; SShw, steady state hot wire; Thw transient hot wire. bAlso published as Heinemann et al.18

Table 2. Coefficients and Parameters for eqs 3 and 4 coefficients Ak/(W·m−1·K−1) for eq 3 A0 = −0.014 364 4

coefficients Bij/(W·m−1·K−1) for eq 4

i

j

1 1 2 2 3 3 4 4 5 5

1 2 1 2 1 2 1 2 1 2 additional parameters

Bij

ρc = 519.435 768 kg·m−3

Tc = 427.01 K

⎛ ⎛ T ⎞⎞⎛ ρ ⎞ Δλr(ρ , T )/(W·m ·K ) = ∑ ⎜⎜Bi ,1 + Bi ,2 ⎜ ⎟⎟⎟⎜⎜ ⎟⎟ ⎝ Tc ⎠⎠⎝ ρc ⎠ i=1 ⎝

(qdξ)ρc

⎞⎤ ⎟⎥ ⎟⎥ 2 ⎥ ⎟ ⎟⎥ ⎠⎦

ρ

(7)

−1

The heat capacity at constant volume, CV(T,ρ) is obtained from the equation of state, and the correlation length ξ is given by

(4)

with coefficients Bi,j, where ρ is the density and ρc is the critical density. This form has been shown to accurately represent other fluorinated refrigerants such as R134a20 and R125.21 Critical Enhancement. Olchowy and Sengers22 developed a complex, theoretically based model for the critical enhancement of thermal conductivity. We use a simplified version of their crossover model,23 Δλc(ρ , T )/(W·m−1·K−1) =

ρCpR 0kBT 6πηξ

(Ω − Ω 0 )

⎡ p ρ ⎤v / γ ⎡ ∂ρ(T , ρ) ξ = ξ0⎢ c 2 ⎥ ⎢ ⎢⎣ Γρc ⎥⎦ ⎢⎣ ∂p

⎤ ⎡ C 2 ⎢⎛⎜ Cp − CV ⎞⎟ arctan(qdξ) + V (qdξ)⎥ ⎜ ⎟ ⎥⎦ Cp π ⎢⎣⎝ Cp ⎠

T

T ∂ρ(TR , ρ) − R ∂p T

⎤v / γ ⎥ ⎥ T⎦

(8)

where the critical amplitudes Γ and ξ0 are system-dependent and are determined by the asymptotic behavior of the equation of state in the critical region. The partial derivative ∂ρ/∂p|T is evaluated with the equation of state at the system temperature T and at a reference temperature, TR. For the reference temperature, we select a value where the critical enhancement is assumed to be negligible: TR = 1.5Tc. The exponents γ = 1.239 and ν = 0.63 are universal constants.23 We have chosen to use values of the critical amplitudes using the generalized method of Perkins et al.25 Γ = 0.060 and ξ0 = 2.04 × 10−10 m. The only parameter left to be determined is the cutoff wavenumber qd (or, alternatively, its inverse, qd−1). This parameter can be obtained by regression of data in the critical region; since we do not have data in this region, we use a value of qd−1 of 0.626 nm that was obtained from the generalized method of Perkins et al.25

(5)

where the heat capacity at constant pressure, Cp(T,ρ), is obtained from the equation of state, kB is Boltzmann’s constant, R0 = 1.02 is a universal constant,24 and the viscosity, η(T,ρ), is obtained from a correlation developed in this work described later in this manuscript. The crossover functions Ω and Ω0 are determined by Ω=

M = 134.047 94 g·mol−1

( )

i

5

−0.120 505 × 10−1 0.937 193 × 10−2 0.652 392 × 10−1 −0.397 844 × 10−1 −0.501 653 × 10−1 0.355 883 × 10−1 0.176 338 × 10−1 −0.141 777 × 10−1 −0.219 652 × 10−2 0.230 154 × 10−2

⎡ ⎛ ⎢ ⎜ 2 −1 Ω 0 = ⎢1 − exp⎜ ⎢ ⎜ π ⎜ (q ξ)−1 + 1 ⎢ 3 ⎝ d ⎣

Residual Thermal Conductivity. We used a polynomial in temperature and density to represent the residual contribution to the thermal conductivity, −1

A1 = 0.038 726 22

(6) C



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THERMAL CONDUCTIVITY CORRELATION RESULTS Since we did not make any measurements in the gas phase, and there have been no additional gas-phase measurements since the work of Wang et al.,13 we selected the dilute-gas correlation proposed by Wang et al.13 and report their coefficients in Table 2. The equation was based on the measurements of Wang et al.13 made with a transient double hot-wire apparatus and has an estimated expanded uncertainty of 2%. Table 1 summarizes all data for the thermal conductivity. We chose to fit only the most accurate data to determine the coefficients for eq 4 and use the other measurements for comparison purposes only. We feel our own extensive measurements in the liquid phase are the most reliable and comprehensive data in the liquid region, and they form the basis of the data set we used to regress coefficients for eq 3. To supplement our own data, we also used the data of Wang et al.13 These measurements cover the saturated liquid region and the vapor phase at pressures up to saturation over the temperature range 243−413 K and were made with transient double hot-wire apparatus with an estimated expanded uncertainty of 2%. We used the fitting program ODRPACK26 to fit the experimental data and present the resulting coefficients in Table 2. Table 3 summarizes comparisons of

Figure 2. Deviations of thermal conductivity data from the present correlation for all data, as a function of temperature. Data sets are designated: ◆, Wang et al.;13 ○, Grebenkov et al.;14 +, Yata et al.;15 △, Geller et al.;16 ◊, Dohrn17 (Heinemann et al.18); and ▲, present work.

Table 3. Evaluation of the R245fa Thermal Conductivity Correlation first author

year publ.

AAD (%)

BIAS (%)

this work Wang13 Grebenkov14 Yata15 Geller16 Dohrn17

2016 2006 2004 2000 1999 1999

0.47 0.69 5.79 3.14 6.75 3.69

0.02 0.01 −5.64 3.14 −0.99 3.69

all experimental data with the correlation. We define the percent deviation as PCTDEV = 100(λexp − λfit)/λfit, where λexp is the experimental value of the thermal conductivity and λfit is the value calculated from the correlation. Thus, the average absolute percent deviation (AAD) is found with the expression AAD = (∑|PCTDEV|)/n, where the summation is over all n points, and the bias percent is found with the expression BIAS = (∑PCTDEV)/n. Figure 2 shows the deviations of experimental data from the correlation as a function of temperature, while Figures 3 and 4 present the deviations as a function of pressure and density, respectively. Our own data and the data of Wang et al.13 are represented to within 2% at a 95% confidence level over the temperature range from near the triple point to 416 K at pressures up to 70 MPa. The uncertainties in the critical region are larger; within 0.5 K of the critical point the expanded uncertainty is estimated to be 10% rising to 30% within 0.1 K of the critical temperature. The equation extrapolates in a physically reasonable manner and can be used over the full range of the equation of state of Akasaka et al.,7 from the triple point to 440 K and pressures up to 200 MPa. At pressures above the range of data, from 70 to 200 MPa, the expanded uncertainty is larger; we estimate ∼10%.

Figure 3. Deviations of thermal conductivity data from the present correlation for all data, as a function of pressure. Data sets are designated: ◆, Wang et al.;13 ○, Grebenkov et al.;14 +, Yata et al.;15 △, Geller et al.;16 ◊, Dohrn17 (Heinemann et al.18); and ▲, present work.

and 530 kg·m−3 illustrates the magnitude of the critical enhancement term.

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VISCOSITY CORRELATION To complement the wide-ranging correlation for the thermal conductivity, we developed a wide-ranging correlation for the viscosity of R245fa. We were unable to locate a wide-ranging correlation in the literature, and it is useful to have correlations available for both transport properties. In addition, the viscosity is used in the critical enhancement model of Olchowy and Sengers,23 used above in the correlation of the thermal conductivity. The goal is to provide a wide-ranging correlation for the viscosity of R245fa that is valid over gas, liquid, and supercritical states and that incorporates densities provided by the equation of state of Akasaka et al.7 Similar to eq 2 for thermal conductivity, we express the viscosity η as the sum of four independent contributions,27 as

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COMPUTER-PROGRAM VERIFICATION Table 4 is provided to assist the user in computer-program verification. The thermal conductivity calculations are based on the tabulated temperatures and densities. The point at 430 K D



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phases. The distribution of the experimental data and the phase boundary in the T,p plane are shown in Figure 5.

Figure 4. Deviations of thermal conductivity data from the present correlation for all data, as a function of density. Data sets are designated: ◆, Wang et al.;13 ○, Grebenkov et al.;14 +, Yata et al.;15 △, Geller et al.;16 ◊, Dohrn17 (Heinemann et al.18); and ▲, present work.

Figure 5. Distribution of experimental viscosity data for R245fa in the T,p plane. Data sets are designated: □, Laesecke and Hafer;33 ∗, Geller et al.;16 ○, Wang et al;32 ▲, Meng et al.;31 ●; critical point; and the dashed line showing the saturation boundary.

Table 4. Sample Points for Computer Verification of the Thermal Conductivity Correlating Equations T (K)

ρ (kg·m−3)

λ (mW·m−1·K−1)

250.0 250.0 430.0 430.0 430.0

0.0 1500.0 0.0 530.0 530.0

8.309 111.40 24.63 66.75a 36.73b

The first viscosity measurements on R245fa were made in 1998 by Laesecke and Hafer33 with a sealed gravitational capillary viscometer, calibrated with toluene. Their measurements were for the saturated liquid over the temperature range 250−283 K and have an estimated expanded uncertainty of 3.4%. The next measurements were made by Geller et al.16 with a capillary tube method and cover the saturated liquid and vapor, as well as single phase vapor at temperatures from 233 to 413 K. The authors estimated expanded uncertainty is 1.2%. Wang et al.32 used an oscillating disc viscometer calibrated with nitrogen to make vapor phase measurements with an estimated expanded uncertainty of 2%. The final set of measurements were made by Meng et al.31 in the liquid phase from 263 to 373 K at pressures up to 40 MPa in a vibrating wire viscometer calibrated with toluene, with an estimated expanded uncertainty of 2.8%. The Dilute-Gas Limit and the Initial-Density Dependence Terms. As was done for the thermal conductivity, we use theory to guide the development of the contributions in eq 9 when possible. The dilute-gas limit viscosity, η0(T) in μPa·s, can be analyzed independently of all other contributions in eq 9. According to the kinetic theory, the viscosity of a pure polyatomic gas may be related to an effective collision cross section, which contains all of the dynamic and statistical information about the binary collision. For practical purposes, this relation is formally identical to that of monatomic gases and can be written as34

a Computed with critical enhancement from eqs 5−8 with a viscosity value of 30.632 μPa·s obtained from the correlation presented in this work. bComputed with critical enhancement set to zero.

η(ρ , Τ) = η0(Τ) + η1(Τ)ρ + Δη(ρ , Τ) + Δηc(ρ , Τ) (9)

where ρ is the molar density, T is the absolute temperature, and the first term, η0(T) = η(0,T), is the contribution to the viscosity in the dilute-gas limit. The linear-in-density term, η1(T) ρ, known as the initial density dependence term, can be separately calculated with the Rainwater−Friend theory.28−30 Unlike thermal conductivity, the critical enhancement term, Δηc(ρ,T) for viscosity is restricted to a very small region very close to the critical point and will be set to zero here. Finally, the term Δη(ρ,T), the residual term, represents the contribution of all other effects to the viscosity of the fluid at elevated densities including many-body collisions, molecularvelocity correlations, and collisional transfer. Table 5 summarizes, to the best of our knowledge, the experimental measurements of the viscosity of R245fa. The data are somewhat limited but do cover a range of liquid and gas Table 5. Viscosity Measurements of R245fa first Author 31

Meng Wang32 Geller16 Laesecke33 a

year

methoda

purity (%)

unc. (%)

no. pts

T range (K)

p range (MPa)

2011 2010 1999 1998

VBW OD Cap Cap

99.9 99.95 nab 97

2.8 2.0 1.2 3.4

107 47 35 65

263−373 300−397 233−413 250−315

1−40 0.1−1.6 0.06−2.83 0.01−0.27

Cap, Capillary; OD, Oscillating Disk; VBW, Vibrating Wire. bna, not available. E



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Table 6. Coefficients and Parameters for eqs 11, 15, and 16 ε/kB = 258.15 K α0 = 0.250 746 b0 = −19.572 881 b3 = 2471.012 5 b6 = −787.260 86 c0 = 0.835 029 35 Tc = 427.01 K

η0(Τ) =

Scaling Parameters σ = 0.588 nm Coefficients αi for eq 11 α1 = −0.603 100 Coefficients34 bi for eq 15 b1 = 219.739.99 b4 = −3375.171 7 b7 = 14.085 455 Coefficients ci for eq 16 c1 = 10.245 205 Additional Parameters ρc = 519.435 768 kg·m−3

0.021357 MT σ 2Sη*(T *)

(10)

where Sη* = S(2000)/(πσ fη) is a reduced effective cross section, M is the molar mass in g·mol−1, σ is the length scaling parameter in nanometers, and fη is a dimensionless higher-order correction factor according to Chapman and Cowling.35 In the above equation, S(2000) is a generalized cross section, often expressed in the functional form

∑ αi(ln T *)i i

T * = kBT/ε

(11) (12)

where, T* is the reduced temperature, ε/kB is an energy scaling parameter in K, and kB is Boltzmann’s constant. Vogel et al.36 have shown that fluids exhibit the same general behavior of the initial density dependence of viscosity, which can also be expressed by means of the second viscosity virial coefficient Bη(T) as Βη (Τ) =

(13)

VISCOSITY CORRELATION RESULTS Table 7 summarizes comparisons of the experimental data with the correlation. We define the percent deviation as PCTDEV = Table 7. Evaluation of the R245fa Viscosity Correlation

(14)

where34 6

Β*η (Τ*) =

(16)

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Βη (Τ) ΝAσ 3

M = 134.047 94 g·mol−1

Coefficients ci are given in Table 6.

The second viscosity virial coefficient can be obtained according to the theory of Rainwater and Friend28,29 as a function of a reduced second viscosity virial coefficient, B*η (T*), as Β*η (Τ*) =

c2 = 0.000 233 562 06

Δη(ρ , T ) = ρr2/3 Tr1/2(c0ρr5 + c1ρr2 /Tr + c 2ρr12 /Tr3)

η1(Τ) η0(Τ)

b2 = −1015.322 6 b5 = 2491.659 7 b8 = −0.346 641 58

body collisions, molecular-velocity correlations, and collisional transfer. Because there is little theoretical guidance concerning this term, its evaluation is based entirely on experimentally obtained data. We used symbolic regression software37 to fit the experimental data to determine the function and coefficients to represent the residual viscosity. This approach has been used successfully to correlate the viscosity surface of hydrogen,38 benzene,39 toluene,40 hexane,41 and heptane.42 We restrict the operators to the set (+,−,*,/) and the operands (constant, Tr, ρr), with Tr = T/Tc and ρr = ρ/ρc. In addition, we use the same form suggested from the hard-sphere model employed by Assael et al.,43 Δη(ρr,Tr) = (ρr2/3Tr1/2)F(ρr,Tr), where the symbolic regression method was used to determine the functional form for F(ρr,Tr). The dilute-gas limit and the initial density dependence terms were calculated for each experimental point (employing eqs 10−15) and subtracted from the experimental viscosity to obtain the residual term. The density values were obtained from the equation of state of Akasaka et al.7 A very simple expression was found:

2

ln Sη*(Τ*) =

α2 = 0.271 008

∑ bi(T *)−0.25i + b7(T *)−2.5 + b8(T *)−5.5 ι= 0

(15)

In the above equations, NA is Avogadro’s constant. The coefficients bi from ref 34 are given in Table 6. Equations 10−15 present an evaluation procedure for the correlation of the dilute-gas limit viscosity, η0(T), and the initial density dependence term, η1(T). If sufficient data are available, one can obtain the parameters ε/kB, σ, and αi from fitting experimental data. We used the lowest density data for each temperature from Wang et al.32 and the low density points from Geller et al.16 to obtain the parameters in Table 6. The Residual Term. As stated earlier, the residual viscosity term Δη(ρ,T) represents the contribution of all other effects to the viscosity of the fluid at elevated densities, including many-

first author

year publ.

AAD (%)

BIAS (%)

Meng31 Wang32 Geller16 Laesecke33 all data

2011 2010 1999 1998

0.45 0.72 3.43 0.63 0.92

−0.11 −0.31 3.24 −0.18 0.26

100(ηexp − ηfit)/ηfit, where ηexp is the experimental value of the viscosity and ηfit is the value calculated from the correlation. The average absolute percent deviation of the fit is 0.92, and its bias is 0.26. Figure 6 shows a comparison of the very lowdensity viscosity data (up to 30 kg·m−3) with the values computed with viscosity correlation model. The agreement is within the experimental expanded uncertainty of the data, and we estimate the expanded uncertainty of the low-density gas phase to be 2%. Figure 7 shows the percentage deviations of all viscosity data from the correlation, as a function of temperature, while Figures 8 and 9 show the same deviations but as a F



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Figure 9. Deviations of the viscosity data from the present correlation for all data, as a function of density. Data sets are designated: □, Laesecke and Hafer;33 ∗, Geller et al.;16 ○, Wang et al.;32 and ▲, Meng et al.31

Figure 6. Deviations of the viscosity data from the present correlation in the dilute gas region. Data sets are designated: ∗, Geller et al.;16 and ○, Wang et al.32

function of the pressure and the density. Comparisons with the data of Meng et al.,31 which extend to 40 MPa in the liquid phase, show that the correlation represents the data to within its level of expanded uncertainty, 2.8%. The saturated liquid data of Laesecke and Hafer33 are consistent with the data of Meng et al.,31 and all points of Laesecke and Hafer33 are represented well within the authors’ estimated expanded uncertainty of 3.4%. Based on comparisons with the available experimental data, we conservatively estimate the expanded uncertainty of the correlation at a 95% confidence level to be 3% for the liquid phase at pressures up to 40 MPa. The data of Geller et al.16 have larger deviations and do not agree with Meng et al.31 and Laesecke and Hafer33 to within their estimated expanded uncertainty levels; this was also reported by Meng et al.31 The data of Wang et al.,32 all in the gas phase, are represented to within 2%, the level of expanded uncertainty of the data. Again, the deviations with the gas-phase data of Geller et al.16 are larger. Although the available experimental data only cover a limited temperature range, 233−413 K, and pressures only to 40 MPa, the correlation extrapolates smoothly and in a physically reasonable manner. The correlation behaves in physically reasonable manner over the full range of the equation of state of Akasaka et al.,7 from the triple point to 440 K and pressures up to 200 MPa; however, uncertainties will be larger especially as the triple point is approached and at high pressures. In the absence of experimental data, it is difficult to estimate the expanded uncertainty in the extrapolation region to high pressures, especially at low temperature (below 250 K) due to the very rapid increase in viscosity in this region; the expanded uncertainty may reach 50% at 200 MPa.

Figure 7. Deviations of the viscosity data from the present correlation for all data, as a function of temperature. Data sets are designated: □, Laesecke and Hafer;33 ∗, Geller et al.;16 ○, Wang et al.;32 and ▲, Meng et al.31

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COMPUTER-PROGRAM VERIFICATION Table 8 is provided to assist the user in computer-program verification. The viscosity calculations are based on the tabulated temperatures and densities.

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CONCLUSIONS We present new experimental data on the liquid-phase thermal conductivity of 1,1,1,3,3-pentafluoropropane (R245fa) that cover a wide range of liquid conditions over a temperature range of 173−344 K and a pressure range of 0.1−71 MPa. The

Figure 8. Deviations of the viscosity data from the present correlation for all data, as a function of pressure. Data sets are designated: □, Laesecke and Hafer;33 ∗, Geller et al.;16 ○, Wang et al.;32 and ▲, Meng et al.31 G



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(8) Bruno, T.; Svoronos, P. D. N. CRC Handbook of Specific Spectroscopic Correlation Charts;. Taylor and Francis CRC Press: Boca Raton, 2005. (9) Bruno, T.; Svoronos, P. D. N. CRC Handbook of Basic Tables for Chemical Analysis, 3rd ed.; Taylor and Francis CRC Press: Boca Raton, 2011. (10) Roder, H. M. A Transient Hot Wire Thermal Conductivity Apparatus for Fluids. J. Res. Natl. Bur. Stand. 1981, 86, 457−493. (11) Healy, J.; DeGroot, J. J.; Kestin, J. The Theory of the Transient Hot-Wire Method for Measuring the Thermal Conductivity. Physica B +C 1976, C82, 392−408. (12) Lemmon, E. W.; Span, R. Short Fundamental Equations of State for 20 Industrial Fluids. J. Chem. Eng. Data 2006, 51, 785−850. (13) Wang, Y.; Wu, J.; Xue, Z.; Liu, Z. Thermal Conductivity of HFC-245fa from (243 to 413) K. J. Chem. Eng. Data 2006, 51, 1424− 1428. (14) Grebenkov, A. J.; Beliayeva, O. V.; Klepatski, P. M.; Saplitsa, V. V.; Timofeyev, B. D.; Tsurbelev, V. P.; Zayats, T. A. Thermophysical Properties of R245fa, ASHRAE Research Project 1256-RP; Joint Institute for Power and Nuclear Research- Sosny: Minsk, Belarus, 2004. (15) Yata, J.; Hori, M.; Niki, M.; Isono, Y.; Yanagitani, Y. Coexistence curve of HFC-134a and thermal conductivity of HFC-245fa. Fluid Phase Equilib. 2000, 174, 221−229. (16) Geller, V. Z.; Bivens, D. B.; Yokozeki, A. In Transport properties and surface tension of hydrofluorocarbons HFC 236fa and HFC 245fa, 20th International Congress of Refrigeration IIR/IIF, Sydney, Australia, 1999. (17) Dohrn, R.; Treckmann, R.; Heinemann, T. Vapor-phase thermal conducitivity of 1,1,1,2,2-pentafluoropropane, 1,1,1,3−3-pentafluoropropane, 1,1,2,2,3-pentafluoropropane and carbon dioxide. Fluid Phase Equilib. 1999, 158−160, 1021−1028. (18) Heinemann, T.; Klaen, W.; Yourd, R.; Dohrn, R. Experimental Determination of the Vapor Phase Thermal Conductivity of Blowing Agents for Polyurethane Rigid Foam. J. Cell. Plast. 2000, 36, 44−56. (19) Huber, M. L.; Perkins, R. A.; Friend, D. G.; Sengers, J. V.; Assael, M. J.; Metaxa, I. N.; Miyagawa, K.; Hellmann, R.; Vogel, E. New International Formulation for the Thermal Conductivity of H2O. J. Phys. Chem. Ref. Data 2012, 41, 033102. (20) Perkins, R. A.; Laesecke, A.; Howley, J.; Ramires, M. L. V.; Gurova, A. N.; Cusco, L. Experimental Thermal Conductivity Values for the IUPAC Round-robin Sample of 1,1,1,2-Tetrafluoroethane (R134a); National Institute of Standards and Technology Internal Report, NISTIR #6605: 2000. (21) Perkins, R. A.; Huber, M. L. Measurement and correlation of the thermal conductivity of pentafluoroethane (R125) from 190 to 512 K at pressures to 70 MPa. J. Chem. Eng. Data 2006, 51, 898−904. (22) Olchowy, G. A.; Sengers, J. V. Crossover from Singular to Regular Behavior of the Transport-Properties of Fluids in the Critical Region. Phys. Rev. Lett. 1988, 61, 15−18. (23) Olchowy, G. A.; Sengers, J. V. A Simplified Representation for the Thermal-Conductivity of Fluids in the Critical Region. Int. J. Thermophys. 1989, 10, 417−426. (24) Krauss, R.; Weiss, V. C.; Edison, T. A.; Sengers, J. V.; Stephan, K. Transport Properties of 1,1-Difluoroethane (R152a). Int. J. Thermophys. 1996, 17, 731−757. (25) Perkins, R. A.; Sengers, J. V.; Abdulagatov, I. M.; Huber, M. L. Simplified Model for the Critical Thermal-Conductivity Enhancement in Molecular Fluids. Int. J. Thermophys. 2013, 34, 191−212. (26) Boggs, P. T.; Byrd, R. H.; Rogers, J. E.; Schnabel, R. B. ODRPACK, Software for Orthogonal Distance Regression, NISTIR 4834; National Institute of Standards and Technology: Gaithersburg, MD, 1992. (27) Dymond, J. H.; Bich, E.; Vogel, E.; Wakeham, W. A.; Vesovic, V.; Assael, M. J. In Transport Properties of Fluids. Their Correlation, Prediction and Estimation; Millat, J., Dymond, J. H., Nieto de Castro, C. A., Eds.; Cambridge University Press, 1996; p 66. (28) Friend, D. G.; Rainwater, J. C. Transport properties of a moderately dense gas. Chem. Phys. Lett. 1984, 107, 590−594.

Table 8. Sample Points for Computer Verification of the Viscosity Correlating Equations T (K)

ρ (kg·m−3)

η (μPa·s)

250.0 250.0 430.0 430.0

0.0 1500.0 0.0 530.0

8.6291 1085.562 14.630 30.632

experimental data have an expanded uncertainty of less than 1%. The measurements are used with selected literature data to develop correlations for the thermal conductivity. Based on the expanded uncertainty of and comparisons with experimental data, the thermal-conductivity correlation for R245fa is estimated to have a relative expanded uncertainty of about 2% at a 95% confidence level for the liquid phase at pressures to 70 MPa and 2% for the vapor phase. In addition, we surveyed literature data and developed a correlation for the viscosity of R245fa. The estimated expanded uncertainty of this correlation is 3% for the liquid phase at pressures to 40 MPa and 2% for the vapor phase. Both correlations may be used over the temperature range from the triple point to 440 K and at pressures to 200 MPa, but the expanded uncertainty in the region where experimental data are unavailable is larger.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.6b00350. Tabulated experimental values (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Dr. Tara Lovestead of NIST for the chemical analysis of the sample. REFERENCES

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Measurements of the Thermal Conductivity of 1,1,1,3,3

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