Literature DB >> 27551607

Absolute Steady-State Thermal Conductivity Measurements by Use of a Transient Hot-Wire System.

H M Roder1, R A Perkins1, A Laesecke1, C A Nieto de Castro2.   

Abstract

A transient hot-wire apparatus was used to measure the thermal conductivity of argon with both steady-state and transient methods. The effects of wire diameter, eccentricity of the wire in the cavity, axial conduction, and natural convection were accounted for in the analysis of the steady-state measurements. Based on measurements on argon, the relative uncertainty at the 95 % level of confidence of the new steady-state measurements is 2 % at low densities. Using the same hot wires, the relative uncertainty of the transient measurements is 1 % at the 95 % level of confidence. This is the first report of thermal conductivity measurements made by two different methods in the same apparatus. The steady-state method is shown to complement normal transient measurements at low densities, particularly for fluids where the thermophysical properties at low densities are not known with high accuracy.

Entities:  

Keywords:  argon; convection; dilute gas; hot wire instrument; steady state; thermal conductivity; transient

Year:  2000        PMID： 27551607      PMCID： PMC4872688          DOI： 10.6028/jres.105.028

Source DB:  PubMed          Journal:  J Res Natl Inst Stand Technol        ISSN： 1044-677X


1. Introduction

The transient hot-wire system has been accepted widely as the most accurate technique for measuring the thermal conductivity of fluids over a wide range of physical states removed from the critical region. However, one of the drawbacks of this system is the need for increasingly larger corrections for the finite wire diameter and the outer boundary in the limit of zero fluid density. This effectively establishes a lower limit in pressure, at approximately 1 MPa for argon (corresponding to 28 kg m−3 in density), where the uncertainty in thermal conductivity increases dramatically. If transient measurements are made on gases below 1 MPa, the thermal conductivity is generally higher than the best theoretical estimates. Below 1 MPa, the linear region in the temperature rise vs the logarithm of time is greatly reduced or no longer exists for transient hot-wire measurements of gases. This curvature in the transient temperature rise is due to extremely large effects of the correction for finite physical properties of the wire at short times, and to the penetration of the transient thermal wave to the outer boundary at longer times, as the thermal diffusivity increases significantly in the limit of zero density. At low densities, the magnitudes of these corrections (comparable to the measured temperature rise itself) make it almost impossible to obtain an accurate mathematical description of the observed transient heat transfer in the hot-wire cells. To overcome these difficulties, researchers often extrapolate measurements along an isotherm from higher densities to obtain the thermal conductivity of the dilute gas. The dilute-gas thermal-conductivity data obtained by such an extrapolation procedure have significantly more uncertainty than the data used in the extrapolation. Near the critical temperature the critical enhancement contributes significantly to the total thermal conductivity even at relatively low densities. This introduces curvature in thermal conductivity isotherms and makes the extrapolation to the dilute gas limit even more uncertain for isotherms near the critical temperature. Furthermore, the 1 MPa restriction makes the transient hot-wire instruments inappropriate for measuring thermal conductivity in the vapor phase at temperatures where the vapor pressure is below 1 MPa. Inconsistencies in dilute-gas thermal conductivities obtained by various researchers using transient hot wires have seriously weakened the credibility of the technique. These problems increase the relative uncertainty at the level of 95 % confidence of the thermal conductivity obtained with the transient hot-wire technique from 0.3 % for measurements at higher densities to about 2 % at the lower densities, which are the focus of the present work. This 2 % relative uncertainty is comparable to the relative uncertainty of measurements obtained with accurate steady-state instruments. The largest uncertainty in steady-state measurements is due to fluid convection and this is known to decrease dramatically in the limit of zero density. Corrections to steady-state measurements actually decrease and become negligible in this dilute-gas region where transient measurements encounter their most serious difficulties. At low densities the transient mode of heat transfer occurs at extremely short real times, where the wire heat-capacity correction is still quite large. This is because the thermal diffusivity of the gas is very large at low densities. This fast approach to steady state at low gas densities is an advantage for steady-state measurements using the same wire geometry. This paper examines the possibility of using the steady-state mode of operation to obtain the thermal conductivity of the dilute gas which is consistent with the higher density data obtained with the transient mode. This would allow any hot-wire instrument to operate in a transient mode at higher densities and in a steady-state mode down to the dilute-gas limit. Measurements at low density have been made on argon gas to test this concept. Argon was selected because the dilute-gas value can be evaluated from the second-order Chapman-Enskog kinetic theory using the known pair-interaction potential. In addition, there are many accurate measurements of the thermal conductivity of argon available in this region using both steady-state and transient techniques. Each series of measurements was made over a wide range of applied powers and included a transient and a steady-state measurement at each power level. Both the transient and the steady-state measurements are compared with the best available predictions from kinetic theory and the other data from the literature. Agreement between measurements using both modes of operation demonstrates the validity of both techniques. Researchers with transient hot wire instruments can potentially select the optimum technique for a given fluid state. The transient hot-wire systems at NIST are completely described in previous papers [1-4]. The apparatus for high-temperature measurements [4] was used in the present work. The transient measurements were of one-second duration, as is typical in our previous measurements. The major change was modification of the data-acquisition system to operate in a steady-state mode, at times up to 40 s. A composite picture of the voltage rises across the Wheatstone bridge, obtained in five transient runs made at the same power level but with different experimental times, is shown in Fig. 1. Time is shown on a logarithmic scale, with the experimental times ranging from 1 s to 40 s. Two linear segments of the voltage rises are clearly visible. The transient thermal conductivity is obtained from the linear portion of finite slope which is proportional to the logarithm of time, while the steady-state thermal conductivity is calculated from the horizontal portion.
Fig. 1

A composite of bridge imbalances from five experiments with different durations for argon at 1 MPa and 300 K. Both the rising transient region and the horizontal steady-state region are shown. Short time curvature is due to the finite wire diameter, while curvature between the transient and steady-state regions is due to penetration of the temperature gradient to the outer cell wall.

In Fig. 2, a typical voltage rise for a measurement in argon gas is shown, at a pressure below 1 kPa (mild vacuum), with an experimental duration of 1 s. This run would normally be evaluated as a transient experiment, the thermal conductivity being obtained from the apparent linear portion between 0.05 s and 0.15 s. There are two reasons to make measurements under these conditions. First, one can show that there is a sufficient section of a horizontal straight line, even at times below 1 s, to obtain a reliable result for steady-state conditions. Secondly, it may be possible to extract values for axial conduction or end effects. In addition, this extreme example shows exactly the origin of the problem with the transient experiment at low densities. The linear region in the temperature rise vs. the logarithm of time is extremely limited, degrading the accuracy of the resulting thermal conductivity data. In other words, a valid constant slope cannot be extracted from such a curve for temperature rise.
Fig. 2

Bridge imbalance at conditions close to vacuum. Here, steady-state heat transfer is reached in less than 1 s and the transient region is almost nonexistent.

2. Transient Mode

Since transient measurements are increasingly unreliable as pressure decreases below 1 MPa, all assumptions made in the application of the theory of the instrument must be carefully examined. The theory for transient hot-wire measurements is well developed [5,6], although proper application of the theory requires significant care and judgment. The hot-wire cells are 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 where q is the power applied divided by length, λ is the thermal conductivity of the fluid, t is the elapsed time, a = λ/ρC is the thermal diffusivity of the fluid, ρ is the mass density of the fluid, C is the isobaric heat capacity of the fluid, r0 is the radius of the hot wire, C = e = 1.781… is the exponential of Euler’s constant, ΔTw is the measured temperature rise of the wire, and δT are corrections to account for deviations from ideal line-source conduction [5,6]. The two most significant corrections account for the finite radius of the wire and for penetration of the fluid temperature gradient to the outer cell wall. The finite wire radius produces the short-time temperature lag relative to the ideal model, as shown in Fig. 1. Penetration of the temperature gradient to the outer wall produces the transition from the linear transient region to the steady-state conduction mode, which is also shown in Fig. 1. It is apparent in Eq. (1) that the thermal conductivity can be found from the slope of the ideal temperature rise as a function of the logarithm of elapsed time. The thermal diffusivity can be found from the intercept of this linear function. The uncertainty of the thermal conductivity at a level of confidence of 95 % is obtained from a linear fit of the ideal temperature rise data to Eq. (1) and is characterized by the parameter STAT. A STAT of 0.003, for example, corresponds to a reproducibility of 0.3 % for the reported thermal conductivity. The principal corrections to the ideal model account for the finite dimensions of the wire δT1, penetration of the expanding thermal wave to the outer boundary δT2, and thermal radiation δT5. The relative magnitude of each correction depends on the fluid properties and the elapsed time in the experiment. In this work we apply the standard corrections [5,6], including the outer-boundary correction and the thermal-radiation correction for a transparent gas. The correction for the finite wire dimensions requires careful examination. The hot wires have finite diameters and their specific heat introduces a temperature lag from the ideal line source model of Eq. (1). The temperature response of an infinitely long wire of finite radius r0 is given [5] by where and where J is the Bessel function of the first kind with order n, Y is the Bessel function of the second kind with order n. In Eqs. (2)–(4) the thermal conductivity of the wire is λw and the thermal diffusivity of the wire is aw. Eqs. (2)–(4) are defined for any point in the wire, and it is the volume-averaged temperature from r = 0 to r = r0 which is required to correct our experimental observed rises in temperature. The resulting correction, δT1f, for the finite wire dimension is Although it is fairly simple to implement this full solution to correct the experimental temperature rise, it is the first-order expansion of this solution which has been recommended because of its simplicity [5,6]. This first-order expansion for a bare wire is [5,6] Before an approximation such as Eq. (6) can be used, the truncation error for the case of our relatively large 12.7 μm platinum hot wire must be evaluated. Figure 3 shows the full solution for the transient temperature rise [Eqs. (2)–(5)], along with the first- [Eq. (6)] and second-order approximations [7], for the case of a 12.7 μm platinum hot wire in argon gas at 300 K and 0.1 MPa. It is apparent in Fig. 3 that the truncation error is quite Eq. (6) cannot be used to correct the data. Both the magnitude of the correction for physical properties of the finite wire and the associated truncation error can be minimized by using thinner hot wires. Table 1 shows how truncation corrections depend on the wire diameter for argon gas at 300 K and 0.1 MPa. Also included in Table 1 are the relative uncertainties in thermal conductivity resulting from a relative uncertainty of 10 % in the thermal diffusivity of the fluid.
Fig. 3

Calculated temperature rise for finite physical properties of the wire for argon gas at 300 K and 0.1 MPa using first-order and second-order approximations as well as the full integral solution.

Table 1

Uncertainties due to corrections for finite wire diameter for argon gas at 300 K and 0.1 MPa. First, differences are examined between the actual heat transfer from a finite diameter wire and the first and second order series approximations of this solution for various wire diameters. Second, the effect of a 10 % error in the thermal diffusivity a used for finite wire diameter correction is calculated for various wire diameters. This correction can introduce significant errors in thermal conductivity measurements with large wires if the thermal diffusivity or wire diameter is not well known.

FluidWire diameter(μm)Ideal slopeSlope after 1st order correction is appliedSlope after 2nd order correction is appliedRelative deviation in λ after 1st order correctionRelative deviation in λ after 2nd order correction
Mathematical approximation errors
Argon  7.04.4456694.4513984.446015−0.12 %−0.01 %
Argon12.54.4456694.5043794.456487−1.32 %−0.24 %
Argon25.04.4456695.5088404.952375−23.91 %  −11.40 %  
FluidWire diameter (μm)Ideal slopeApparent slope with a − 10 % error in thermal diffusivityApparent slope with a +10 % error in thermal diffusivityRelative deviation in λ with a − 10 % error in thermal diffusivityRelative deviation in λ with a +10 % error in thermal diffusivity
Thermal conductivity uncertainty associated with 10 % uncertainty in fluid thermal diffusivity
Argon  7.04.4456694.4445654.446668−0.03 %+0.02 %
Argon12.54.4456694.4410774.449814−0.10 %+0.09 %
Argon25.04.4456694.3983954.487762−1.06 %+0.94 %
In the limit of zero density, the transient thermal wave will penetrate to the outer boundary. The outer-boundary correction, which accounts for penetration of the transient heat pulse [5,6], is given by where g are the roots of J0 (g) = 0 and rb is the radius of the outer boundary. In the limit of infinite time, Eq. (7) approaches the steady-state solution given below. The final corrections, which must be considered because of their increasing significance at low densities, account for compression work δT3 and radial convection δT4. A recent analysis by Assael et al. [8] concludes that δT3 and δT4 must be considered simultaneously and that the previous analysis of Healy et al. [5] is in error. Based on the work of Assael et al. [8] we have set both δT3 and δT4 equal to zero in the present analysis. At low densities, the thermal diffusivity of the fluid increases almost linearly with inverse pressure. Large corrections to the experimental temperature rise are required for both heat-capacity and outer-boundary effects because of this large thermal diffusivity. Any uncertainties in the wire diameter and the fluid properties used in these corrections become increasingly important at low densities. The full heat-capacity correction must be used to correct the measured temperature rises at low densities since this correction is so significant. The measurements at low density must be carefully examined to verify that the thermal wave has not reached the outer boundary since the thermal diffusivity increases so dramatically in this region. Transient results obtained by use of the traditional corrections employed in earlier work [5,6] and the revised corrections as discussed above are shown for argon at 300 K in Fig. 4. The differences are primarily due to setting the compression work δT3 equal to zero [8], use of the full heat-capacity correction of Eqs. (2)–(5), and careful restriction of the regression limits to exclude times where the outer-boundary correction δT2 is significant. The results for all power levels were averaged at each pressure level in Fig. 4. It can be seen that the results at low densities are more linear in terms of density with the revised corrections (hook due to increasing contributions from the outer boundary correction), while the results at the higher densities are not changed appreciably. This linear dependence on density is expected from the kinetic theory of low-density gases.
Fig. 4

Transient results for argon at 300 K with the corrections applied in earlier papers as well as the revised corrections proposed herein. Results were averaged at each pressure.

3. Steady-State Mode

The working equation for the steady-state mode is based on a different solution of Fourier’s law but the geometry is still that of concentric cylinders. The solution can be found in standard texts for the case of constant thermal conductivity (see, for example, Reference [9], page 114). This equation can be solved for the thermal conductivity of the fluid λ; where q is the applied power divided by length, r2 is the internal radius of the outer cylinder, r1 is the external radius of the inner cylinder (hot wire), and ΔT = (T1 − T2) is the measured temperature difference between the hot wire and its surrounding cavity. For the concentric-cylinder geometry described above the total heat flux divided by length, q, remains constant and is not a function of the radial position. Assuming that the thermal conductivity is a linear function of temperature, such that λ = λ0(1 + b), it can be shown that the measured thermal conductivity is given by λ = λ0(1 + b(T1 + T2)/2). Thus, the thermal conductivity that is measured corresponds to the value at the mean temperature of the inner and outer cylinders, where This assumption of a linear temperature dependence for the thermal conductivity is valid for experiments with small temperature rises. The density of the fluid assigned to the measured thermal conductivity is taken from an equation of state [10] using the temperature from Eq. (9) and the experimentally measured pressure. Equation (8) assumes that the dimensions of the wire and cavity are well known and that the wire is perfectly concentric with the outer cylindrical cavity. The diameter of our wire is 13.14 µm and it is known with a relative uncertainty of 0.5 % at a level of confidence of 95 %. This uncertainty contributes a component of relative uncertainty of 0.07 % to the uncertainty of the measured thermal conductivity. Since it is nearly impossible to keep the wire perfectly concentric with the outer cavity, the uncertainty associated with the eccentricity of the wire with respect to the cavity must be assessed. For a wire that is eccentric, the thermal conductivity is given by where b is the distance between the wire’s axis and the axis of the outer cylinder. The eccentricity correction is shown in Fig. 5 for a 12.7 µm diameter wire in a 9 mm diameter cavity. The hot wires in the present cell are concentric with the cavity within 0.5 mm, so it is apparent from Fig. 5 that the relative uncertainty is about 0.2 % due to misalignment of the wire in the cavity. The combined relative uncertainty, due to both the diameter and eccentricity of the wire is 0.3 % in the measured thermal conductivity.
Fig. 5

Error estimate for steady-state results due to the eccentricity of the hot wire. The thermal conductivity measurement is not sensitive to eccentricity for hot wires with small diameters.

While 40 s may seem to be a very short time in comparison to normal steady-state measurements, it still allows the very small wires used in transient hot-wire systems to equilibrate in the gas phase. The time of the steady-state experiments is restricted to 40 s since the temperature of the cell wall T1 is assumed to be the initial cell temperature. Both transient and steady-state measurements of thermal conductivity should be made at several power levels. The thermal conductivity should be valid and free of convection if a plot of the measured values of thermal conductivity as a function of applied power is constant. The present measurements are made over a large range of applied powers, and the powers of the transient measurements overlap those used for the steady-state measurements as much as possible.

4. Data Reduction

Three isotherms were measured for gaseous argon at 300 K, 320 K, and 340 K. There were 13 to 14 different pressure levels covering a density range from 120 kg m−3 down to 2.4 kg m−3. At each pressure level, experimental results were collected at 7 to 11 different applied powers. Transient measurements were made for an experimental time of 1 s, while for the steady-state measurements the total elapsed time used was 40 s. In either case, 250 measurements of the bridge imbalance voltage were obtained. To elucidate the end effects in the experiment, a special series of runs were made using an additional digital voltmeter to measure directly across the long hot wire, the short hot wire, and the bridge. Finally, an abbreviated set of measurements was made for pressures below 1 kPa. In all, we made 883 measurements. The temperatures were measured on the International Practical Temperature Scale of 1968 (IPTS 68) but the effect of converting the temperatures to the International Temperature Scale of 1990 (ITS 90) on the reported thermal conductivity is less than 1 μW m−1 K−1. The steady-state measurements required the development of a new data-analysis procedure. The rises in steady-state voltage as a function of time were always examined to select reliable measurements. Five typical profiles of the bridge imbalance, used to select the appropriate range of power levels, are shown in Fig. 6. Trace b in Fig. 6 is considered reliable since it is nearly horizontal after 20 s. In trace a, the power level is too low, so electronic noise is significant in the imbalance voltages. In traces c, d, and e the power levels are too large, so convection makes a visible contribution.
Fig. 6

Typical voltage-rise profiles for steady-state measurements at T = 340 K. The increase in convection with the increase of applied power is shown in curves a through c. The increase in convection with increase in fluid density is shown in curves d and e.

The next step was to determine the experimental temperature rise ΔT. The experimental voltage rises were averaged over a time interval where they were nearly constant. To find the optimum time to begin the averaging, the last 50 bridge imbalance voltages were averaged to find a reference imbalance voltage. The actual average, Vave, is obtained by averaging the points from the first voltage, which is 0.5 % below this reference imbalance voltage up to the final data point. Solving the bridge equation with Vave yields the change in resistance in the variable arms of the bridge. The resistance change is finally converted into ΔT using the calibration of the wire resistance as a function of temperature. The maximum and minimum values of the voltage rises were also obtained over the range averaged. The difference between them was expressed as a percentage of Vave, and is designated by the parameter TBAND. TBAND is a direct measure of the precision in ΔT at the level of 3 standard deviations. In Fig. 7, the values of TBAND are plotted for all of the steady-state measurements made near 320 K. A final selection of valid measurements was made by rejecting all points with a TBAND larger than 2 %. This is equivalent to rejecting those points that have voltage traces similar to traces c, d, and e in Fig. 6.
Fig. 7

Temperature uncertainty, TBAND, for the 320 K isotherm of argon. Data with TBAND values greater that 2 % were considered invalid due to free-convection contributions.

A correction for radiation was also applied to ΔT. The radiation correction for transparent fluids δT5 was used as given in Ref. [4]. The maximum effect of this correction was 0.13 % at 340 K. Additional corrections have been considered by other authors for steady-state hot-wire systems (see for example Refs. [9,11]). These include corrections for temperature jump, end conduction in the wire, lead-wire conduction, and temperature rise in the outer wall. The temperature-jump correction does not apply because the present pressures are not low enough. The correction for end conduction in the wire and the lead-wire correction were found to be negligible in our experiments because a bridge with a compensating hot wire was used. A special series of measurements was made to determine the size of the end effects by directly measuring the temperature rise of each wire. Temperature gradients in the outer wall were considered negligible for our thick-walled pressure vessel. The primary platinum resistance thermometer (PRT) was mounted on the outside of the pressure vessel. The temperature Tref of the PRT increased by about 30 mK for a series of measurements at a single pressure level. The temperature of the long hot wire, which is inside the cell, increased by a nearly identical amount. Since a measurement series normally consisted of about 20 measurements at time intervals 1 min apart, an average change of 1 mK per measurement was negligible in comparison to the measured ΔT, which was typically from 1 K to 4 K.

4.1 Free Convection

Convection has always been a problem in measuring thermal conductivity; its onset has been associated with the Rayleigh number. One of the major advantages of the transient method is the ability to detect and avoid contributions from convection. The present measurements using the steady-state mode also show the evolution of convection very clearly. Figure 8 shows the deviations between the steady-state measurements near 320 K, calculated before application of the correction for convection, and the thermal conductivity surface for argon [12]. Since the diameter of the hot wires is comparable to the boundary-layer thickness for heat transfer, the standard engineering models for vertical flat plates are not applicable, and so an empirical expression was developed for the thin-wire geometry.
Fig. 8

Correction of steady-state results for free convection along the 320 K isotherm. The baseline is the correlation of Younglove and Hanley [12].

The dimensionless Rayleigh number is commonly used to characterize the onset of free convection. For a concentric-cylinder geometry, the Rayleigh number is given by where gc is the local acceleration of free fall, and is the fluid viscosity. The correction for natural or free convection was obtained from two equations given by Le Neindre and Tufeu [13] for a concentric-cylinder apparatus: and where q is the applied power divided by length of Eq. (1)qmeas is the experimental heat flow determined from the measured voltage and current, qc is the heat transfer by natural convection, K is a numerical apparatus constant, and Ra is the Rayleigh number. Le Neindre and Tufeu use a numerical constant of 720, but also values of d, the thickness of the fluid layer, and l, the length of the internal cylinder. For our system, both d and l are constant and our ratio of d/l is much larger than that of Le Neindre and Tufeu. Since the aspect ratio d/l is constant in our apparatus, it is incorporated into the experimentally determined apparatus constant K for our hot-wire cell. Equations (12) and (13) together give Next, Eq. (8) is applied twice, once for uncorrected conditions and once for corrected conditions. Forming a ratio we can solve for the corrected thermal conductivity or with the use of Eq. (14), The best value for K in Eq. (16) was 1.8435 × 10−6 and was obtained by comparing the experimental points for each isotherm against a parabolic fit of the companion transient measurements. This procedure is justified because the deviations of the companion transient measurements from the thermal conductivity surface of argon are less than 1 % [12]. After applying this correction for free convection, Eq. (16), to all of the steady-state measurements, the resulting deviations are plotted in Fig. 8 for the 320 K isotherm. Figure 8 contains three different regions. For densities from 0 to 40 kg m−3, convection contributes very little to the measured conductivity, i.e., the corrections for convection are less than 1 %. In this region, the Rayleigh numbers range from near 0 to 17 000. For densities between 40 kg m−3 and 80 kg m−3 the corrections given by Eq. (16) gradually increase to about 5 %, while the Rayleigh numbers range up to 65 000. The correction is highly successful, as the band or range of measured values at each pressure level decreases. For steady-state thermal conductivities at densities above about 80 kg m−3, for which some of the Rayleigh numbers range well above 65 000, the corrections for simple natural convection increase to about 12 %. However, the uncertainty bands associated with the uncorrected thermal conductivities no longer decrease. The measured voltage rises suggest that there is flow turbulence in the cells at the larger power levels. This could be along either the short hot wire, the long hot wire, or both. Measurements associated with Rayleigh numbers greater than 70 000 have to be rejected, and are omitted from Fig. 8.

4.2 Results

The transient measurements for all three isotherms are given in Table 2. All of the transient points with a STAT greater than 0.003 were eliminated; thus, 98 points remain at a nominal temperature of 300 K, 105 points remain at 320 K, and 102 points remain at 340 K. Figure 9 shows the deviations between all transient points and the thermal-conductivity surface of argon [12]. Based on Fig. 9 it can be concluded that, over the range of densities shown here, the deviations fit within a band of ±1 %. The problem with the transient measurements at low densities shows up clearly as a systematic deviation. However, as shown later, this systematic deviation must be ascribed to a difference in the dilute gas λ0 values used for the surface [12].
Table 2

Thermal conductivity of argon, transient method

Point no.p(MPa)ρ(kg m−3)Texp(K)λexp(W m−1 K−1)STATRelative dev. (%)q(W m−1)
Nominal temperature 300 K
30017.4440122.916301.6620.020920.0020.880.03941
30027.4441122.832301.8360.020620.003−0.60  0.04485
30037.4440122.736302.0220.020730.001−0.12  0.05063
30047.4441122.636302.2310.020800.0010.190.05676
30057.4442122.525302.4540.020860.0010.440.06324
30067.4438122.409302.6820.020860.0010.380.07007
30077.4440122.297302.9140.020820.0010.140.07725
30087.4441122.169303.1780.021010.0010.980.08479
30107.4436121.893303.7280.020720.001−0.51  0.10090
30226.9077113.988301.3250.020490.0030.010.02960
30236.9078113.916301.4880.020530.0030.170.03433
30246.9079113.836301.6670.020530.0020.150.03941
30256.9076113.744301.8530.020550.0020.240.04484
30266.9076113.652302.0610.020510.0010.000.05062
30276.9077113.560302.2700.020550.0010.140.05676
30286.9078113.460302.4950.020560.0010.130.06324
30296.9078113.348302.7440.020610.0010.300.07007
30306.9078113.237302.9940.020620.0010.300.07726
30436.2650103.038301.5300.020270.0030.280.03433
30446.2650102.962301.7140.020290.0020.350.03942
30456.2651102.886301.9100.020260.0020.160.04485
30466.2651102.806302.1100.020270.0010.150.05063
30476.2651102.714302.3330.020240.001−0.06  0.05676
30486.2652102.622302.5660.020260.001−0.01  0.06325
30496.2652102.527302.8050.020290.0010.120.07008
30506.2652102.423303.0690.020320.0010.210.07727
30625.5849  91.645301.3830.019910.003−0.02  0.02961
30635.5849  91.581301.5570.019900.002−0.11  0.03434
30645.5851  91.521301.7370.019940.0020.030.03942
30655.5849  91.445301.9420.019870.002−0.39  0.04485
30665.5851  91.377302.1500.019970.0020.090.05063
30675.5850  91.293302.3770.019980.0010.090.05676
30685.5848  91.201302.6280.020030.0020.300.06325
30695.5848  91.121302.8590.020010.0010.100.07008
30705.5849  91.034303.1210.020010.0010.040.07727
30824.9512  81.011301.4070.019650.003−0.03  0.02961
30834.9514  80.959301.5890.019660.002−0.04  0.03434
30844.9514  80.899301.7760.019560.002−0.61  0.03942
30854.9516  80.839301.9800.019780.0020.500.04485
30864.9517  80.775302.1980.019760.0010.310.05063
30874.9517  80.707302.4270.019680.001−0.15  0.05677
30884.9518  80.631302.6690.019620.001−0.48  0.06325
30894.9519  80.555302.9260.019700.001−0.16  0.07008
30904.9519  80.475303.1930.019680.001−0.32  0.07727
31024.2568  69.414301.4480.019140.003−1.27  0.02961
31034.2569  69.370301.6240.019280.002−0.61  0.03434
31044.2569  69.318301.8210.019350.002−0.26  0.03942
31054.2570  69.266302.0250.019360.002−0.28  0.04485
31064.2570  69.206302.2580.019380.001−0.24  0.05063
31074.2572  69.150302.4950.019340.001−0.49  0.05677
31084.2569  69.082302.7390.019440.001−0.04  0.06325
31094.2570  69.014303.0090.019420.001−0.23  0.07009
31104.2572  68.946303.2840.019410.001−0.32  0.07728
31233.5548  57.757301.4950.018830.003−1.56  0.02961
31243.5545  57.713301.6780.019050.002−0.42  0.03434
31253.5546  57.669301.8920.019020.002−0.64  0.03942
31263.5546  57.625302.1080.019060.002−0.46  0.04485
31273.5546  57.577302.3350.019070.001−0.49  0.05064
31283.5547  57.529302.5810.019060.001−0.60  0.05677
31293.5547  57.473302.8360.019110.001−0.39  0.06326
31303.5545  57.413303.1120.019120.001−0.39  0.07009
40032.8395  45.944301.6380.018830.003−0.18  0.02960
40042.8397  45.916301.8310.018730.002−0.78  0.03433
40052.8397  45.880302.0470.018810.002−0.38  0.03941
40062.8397  45.844302.2640.018840.002−0.28  0.04484
40072.8397  45.804302.5080.018820.001−0.44  0.05063
40082.8397  45.764302.7650.018790.001−0.70  0.05676
40092.8398  45.720303.0360.018830.001−0.53  0.06324
40102.8397  45.673303.3240.018830.001−0.63  0.07007
40232.1132  34.056301.6810.018500.003−0.60  0.02962
40242.1132  34.032301.8960.018440.003−0.97  0.03435
40252.1131  34.004302.1240.018210.002−2.32  0.03943
40262.1130  33.976302.3460.018460.002−0.95  0.04487
40272.1131  33.948302.5920.018440.001−1.14  0.05065
40282.1132  33.916302.8570.018530.001−0.74  0.05679
40292.1131  33.880303.1410.018470.001−1.15  0.06328
40302.1132  33.848303.4440.018470.001−1.21  0.07011
40451.3696  21.963301.9600.018080.002−1.58  0.03435
40461.3696  21.947302.1950.018070.002−1.73  0.03943
40471.3696  21.927302.4460.018150.002−1.36  0.04487
40481.3697  21.907302.7090.018200.001−1.11  0.05065
40491.3697  21.888302.9950.018170.001−1.39  0.05679
40501.3697  21.864303.2980.018140.001−1.64  0.06327
40840.7010  11.197301.9970.017880.003−1.53  0.03435
40850.7012  11.189302.2560.018030.003−0.75  0.03943
40860.7010  11.177302.5330.018130.003−0.26  0.04487
40870.7010  11.165302.7960.017880.002−1.79  0.05065
40880.7011  11.157303.1090.017930.002−1.56  0.05679
40890.7011  11.145303.4170.017770.002−2.58  0.06327
40900.7011  11.134303.7620.017850.003−2.21  0.07011
41070.3782 6.024302.3520.017630.003−2.49  0.03943
41080.3784 6.020302.6360.017680.002−2.30  0.04487
41090.3784 6.016302.9300.017680.002−2.36  0.05065
41100.3783 6.008303.2540.017650.002−2.62  0.05679
41270.1684 2.681302.2060.017860.003−0.75  0.03435
41280.1686 2.681302.5000.017920.003−0.50  0.03943
41290.1686 2.677302.8170.017940.003−0.51  0.04487
41300.1686 2.673303.1420.018060.0030.100.05065
Nominal temperature 320 K
50018.3272127.678321.4300.021860.002−0.49  0.04215
50028.3273127.630321.5330.021930.002−0.20  0.04501
50038.3273127.590321.6170.021970.002−0.02  0.04796
50048.3273127.542321.7200.021890.002−0.41  0.05100
50058.3275127.494321.8220.021880.001−0.47  0.05414
50068.3273127.442321.9270.021980.001−0.05  0.05737
50078.3275127.394322.0360.021880.001−0.49  0.06069
50088.3277127.346322.1450.021930.001−0.31  0.06411
50098.3275127.286322.2620.021950.001−0.22  0.06762
50108.3275127.234322.3790.021900.001−0.46  0.07123
50217.5805116.005321.5570.021620.002−0.16  0.04501
50227.5807115.925321.7600.021640.002−0.07  0.05100
50237.5808115.837321.9670.021570.001−0.45  0.05737
50247.5808115.745322.1870.021570.001−0.53  0.06411
50257.5808115.645322.4240.021570.001−0.54  0.07123
50267.5809115.542322.6750.021550.001−0.71  0.07871
50277.5812115.438322.9310.021610.001−0.49  0.08658
50287.5811115.322323.2040.021600.001−0.55  0.09482
50297.5812115.202323.4900.021630.001−0.50  0.10343
50307.5812115.078323.7920.021620.001−0.57  0.11241
50416.8721104.995321.5780.021300.002−0.26  0.04501
50426.8720104.915321.7850.021290.001−0.35  0.05100
50436.8720104.836321.9990.021390.0010.070.05737
50446.8721104.744322.2460.021340.001−0.24  0.06411
50456.8721104.652322.4840.021350.001−0.25  0.07122
50466.8721104.560322.7350.021280.001−0.63  0.07871
50476.8721104.460323.0020.021360.001−0.29  0.08658
50486.8721104.356323.2790.021250.001−0.85  0.09481
50496.8722104.248323.5710.021460.0010.060.10343
50506.8721104.128323.8840.021430.001−0.16  0.11241
50616.1871  94.369321.5980.021000.002−0.37  0.04500
50626.1870  94.297321.8090.020990.002−0.50  0.05099
50646.1869  94.133322.2880.021000.001−0.56  0.06410
50656.1869  94.054322.5260.020970.001−0.72  0.07122
50666.1868  93.970322.7730.021040.001−0.43  0.07871
50676.1868  93.882323.0470.021060.001−0.42  0.08657
50686.1867  93.782323.3350.021000.001−0.78  0.09481
50696.1866  93.682323.6290.021030.001−0.70  0.10342
50706.1866  93.578323.9420.021060.001−0.62  0.11241
50825.5060  83.859321.5120.020570.002−1.18  0.04215
50835.5060  83.803321.7050.020730.002−0.43  0.04795
50845.5061  83.735321.9340.020770.001−0.28  0.05414
50855.5061  83.667322.1640.020730.001−0.52  0.06069
50865.5061  83.587322.4300.020680.002−0.82  0.06762
50875.5061  83.515322.6850.020740.001−0.58  0.07492
50885.5061  83.435322.9590.020770.001−0.51  0.08260
50895.5061  83.352323.2500.020640.001−1.21  0.09065
50905.5061  83.268323.5320.020710.001−0.93  0.09908
51034.8431  73.672321.3330.020430.003−0.58  0.03672
51044.8431  73.624321.5180.020420.002−0.64  0.04215
51054.8432  73.572321.7280.020480.002−0.40  0.04796
51064.8431  73.512321.9600.020480.002−0.44  0.05414
51074.8432  73.452322.1980.020480.001−0.52  0.06069
51084.8431  73.388322.4490.020480.001−0.59  0.06762
51094.8431  73.321322.7120.020460.001−0.74  0.07492
51104.8431  73.249323.0000.020490.001−0.66  0.08260
51224.1373  62.842321.1470.020040.0031.140.03166
51234.1373  62.798321.3390.020180.003−0.50  0.03672
51244.1374  62.758321.5410.020230.002−0.31  0.04215
51254.1374  62.706321.7760.020130.002−0.84  0.04796
51264.1375  62.658322.0050.020210.002−0.53  0.05414
51274.1375  62.603322.2660.020120.002−1.02  0.06069
51284.1375  62.547322.5310.020310.001−0.15  0.06762
51294.1375  62.491322.7910.020230.001−0.57  0.07492
51304.1375  62.431323.0810.020200.001−0.80  0.08260
51433.4551  52.320321.3880.019920.003−0.61  0.03672
51453.4549  52.240321.8410.019890.002−0.83  0.04795
51463.4550  52.200322.0770.020040.002−0.12  0.05413
51473.4550  52.156322.3250.019900.001−0.91  0.06069
51483.4546  52.104322.5840.019910.001−0.90  0.06762
51493.4548  52.052322.8890.019940.001−0.86  0.07492
51503.4549  52.000323.1910.019950.001−0.86  0.08259
51632.7632  41.770321.2240.019690.003−0.49  0.03165
51642.7631  41.738321.4290.019610.003−0.93  0.03671
51652.7632  41.706321.6780.019730.002−0.36  0.04214
51662.7633  41.678321.8960.019640.002−0.92  0.04795
51672.7633  41.642322.1590.019660.002−0.84  0.05413
51682.7632  41.602322.4220.019600.002−1.23  0.06068
51692.7633  41.566322.6880.019640.001−1.11  0.06761
51702.7633  41.526322.9850.019660.001−1.08  0.07491
51932.0865  31.455321.2870.019160.003−2.06  0.03165
51942.0865  31.435321.4940.019320.003−1.28  0.03671
51952.0864  31.411321.7260.019290.002−1.50  0.04215
51962.0866  31.387321.9700.019320.002−1.41  0.04795
51972.0865  31.359322.2280.019340.002−1.36  0.05413
51982.0865  31.331322.4990.019330.001−1.48  0.06068
51992.0863  31.299322.7910.019350.001−1.45  0.06761
52002.0863  31.267323.0960.019380.001−1.34  0.07491
52141.3800  20.729321.5810.018970.003−1.91  0.03671
52151.3798  20.713321.8220.019020.002−1.74  0.04214
52161.3798  20.697322.0750.018980.003−2.03  0.04795
52171.3798  20.677322.3460.019020.002−1.85  0.05412
52181.3800  20.661322.6360.018950.001−2.33  0.06068
52191.3800  20.641322.9480.019150.002−1.34  0.06760
52201.3800  20.617323.2670.018950.001−2.45  0.07490
52350.6849  10.251321.9190.018670.003−2.49  0.04214
52360.6849  10.239322.2060.018940.003−1.06  0.04794
52370.6849  10.231322.4940.018700.002−2.48  0.05412
52390.6849  10.211323.1470.018910.002−1.47  0.06760
52400.6849  10.203323.4650.018800.001−2.18  0.07490
52590.3678 5.485322.6250.018640.002−2.27  0.05412
52600.3679 5.481322.9700.018650.002−2.31  0.06067
52770.1732 2.581322.8070.018810.002−1.09  0.05412
52780.1734 2.577323.7220.018850.002−1.09  0.07120
52790.1733 2.565324.7620.018930.002−0.96  0.09062
Nominal temperature 340 K
60018.1063112.607342.8890.022590.001−0.74  0.12914
60028.1064112.786342.4170.022570.001−0.77  0.11441
60038.1065112.954341.9790.022520.001−0.89  0.10058
60048.1066113.118341.5570.022470.001−1.08  0.08763
60058.1067113.267341.1700.022530.001−0.72  0.07556
60068.1068113.403340.8140.022460.001−0.95  0.06439
60187.5842105.344342.7910.022380.001−0.76  0.12412
60197.5843105.453342.4870.022330.001−0.94  0.11439
60207.5845105.566342.1750.022360.001−0.76  0.10507
60217.5846105.664341.8920.022360.001−0.70  0.09614
60227.5845105.762341.6150.022280.001−0.98  0.08761
60237.5848105.859341.3490.022320.001−0.76  0.07948
60247.5848105.949341.1010.022310.001−0.79  0.07173
60366.9060  95.828342.8960.022100.001−0.89  0.12412
60376.9061  95.933342.5670.022080.001−0.89  0.11440
60386.9062  96.027342.2700.021990.002−1.27  0.10508
60396.9065  96.128341.9710.022050.001−0.95  0.09615
60406.9062  96.214341.6830.022120.001−0.55  0.08761
60416.9064  96.304341.4190.022000.001−1.05  0.07948
60426.9065  96.386341.1670.021980.002−1.07  0.07173
60546.2153  86.152342.9830.021870.001−0.77  0.12412
60556.2153  86.241342.6630.021880.001−0.66  0.11440
60566.2154  86.327342.3620.021860.001−0.66  0.10508
60576.2155  86.413342.0610.021730.002−1.20  0.09615
60586.2156  86.499341.7680.021840.001−0.67  0.08762
60596.2155  86.573341.4980.021820.001−0.68  0.07948
60606.2156  86.651341.2260.021850.001−0.49  0.07174
60725.5214  76.433343.1030.021580.001−0.93  0.12412
60735.5215  76.518342.7650.021550.001−1.01  0.11440
60745.5216  76.597342.4430.021580.001−0.82  0.10507
60755.5215  76.675342.1370.021530.001−0.95  0.09615
60765.5216  76.749341.8330.021530.001−0.92  0.08761
60775.5215  76.819341.5520.021510.001−0.96  0.07948
60785.5216  76.886341.2880.021550.001−0.72  0.07173
60904.8340  66.823343.2160.021350.001−0.90  0.12412
60914.8341  66.901342.8640.021340.001−0.85  0.11440
60924.8341  66.971342.5310.021280.001−1.06  0.10508
60934.8340  67.038342.2230.021240.001−1.19  0.09614
60944.8341  67.104341.9150.021320.001−0.76  0.08762
60954.8342  67.170341.6220.021250.001−1.02  0.07948
60964.8341  67.225341.3530.021210.001−1.14  0.07174
61084.1410  57.158343.3450.021060.001−1.12  0.12414
61094.1411  57.221342.9870.021070.001−0.98  0.11441
61104.1409  57.280342.6510.021060.001−0.98  0.10508
61124.1408  57.397342.0050.021000.001−1.12  0.08762
61134.1410  57.455341.7060.020950.001−1.31  0.07949
61144.1411  57.506341.4250.021050.001−0.77  0.07174
61263.4501  47.568343.2970.020850.001−0.97  0.11921
61273.4501  47.619342.9460.020770.001−1.30  0.10969
61283.4499  47.670342.5990.020710.001−1.49  0.10057
61303.4495  47.760341.9550.020800.001−0.94  0.08351
61313.4496  47.806341.6530.020770.001−0.99  0.07556
61323.4496  47.849341.3710.020740.001−1.10  0.06802
61443.4481  47.537343.3060.020810.001−1.18  0.11923
61453.4481  47.592342.9440.020780.001−1.25  0.10970
61463.4483  47.646342.6060.020750.001−1.30  0.10058
61473.4481  47.693342.2730.020720.001−1.37  0.09184
61483.4481  47.740341.9560.020780.001−1.03  0.08351
61493.4481  47.783341.6630.020740.001−1.18  0.07556
61503.4481  47.826341.3730.020740.001−1.08  0.06802
61622.7654  38.056343.4590.020570.001−1.27  0.11920
61632.7655  38.103343.0890.020570.001−1.17  0.10968
61642.7655  38.142342.7380.020490.001−1.52  0.10056
61652.7656  38.189342.3770.020490.001−1.41  0.09184
61662.7655  38.224342.0610.020550.001−1.03  0.08350
61672.7655  38.259341.7460.020500.001−1.25  0.07557
61682.7655  38.294341.4450.020500.001−1.17  0.06802
61912.0713  28.462343.4640.020170.001−2.15  0.11439
61922.0713  28.497343.0730.020250.001−1.65  0.10508
61932.0713  28.528342.7080.020210.001−1.79  0.09615
61942.0714  28.560342.3630.020210.001−1.67  0.08762
61952.0712  28.587342.0280.020240.001−1.43  0.07948
61962.0712  28.614341.7160.020160.001−1.78  0.07174
61972.0713  28.642341.4210.020220.001−1.40  0.06439
61981.3694  18.786343.4810.019950.001−2.15  0.10968
61991.3694  18.809343.0950.019930.001−2.17  0.10055
62001.3694  18.829342.7230.019880.001−2.29  0.09182
62011.3694  18.848342.3680.019890.001−2.19  0.08350
62021.3694  18.868342.0260.019920.001−1.93  0.07555
62031.3695  18.888341.6990.019880.001−2.09  0.06801
62041.3694  18.903341.4050.019760.002−2.63  0.06086
62160.6979 9.559343.5480.019660.001−2.61  0.10506
62170.6979 9.567343.1460.019630.001−2.64  0.09613
62180.6978 9.578342.7560.019580.001−2.84  0.08761
62190.6978 9.590342.3870.019590.001−2.66  0.07947
62200.6979 9.602342.0370.019600.001−2.53  0.07173
62210.6978 9.610341.7030.019520.002−2.89  0.06439
62220.6978 9.618341.3870.019550.002−2.64  0.05744
62340.3486 4.764343.8830.019630.002−2.27  0.10505
62350.3485 4.772343.4400.019630.001−2.18  0.09612
62360.3486 4.776343.0230.019660.002−1.94  0.08760
62370.3485 4.779342.6320.019590.001−2.18  0.07947
62380.3485 4.787342.2590.019650.002−1.80  0.07173
62390.3485 4.791341.8960.019590.002−2.02  0.06438
62400.3486 4.799341.5720.019680.003−1.46  0.05743
62520.1686 2.304343.9870.019860.002−0.86  0.10052
62530.1686 2.304343.5310.019860.002−0.76  0.09180
62540.1686 2.308343.0960.019840.002−0.75  0.08346
62550.1685 2.312342.6900.019820.002−0.73  0.07553
62560.1685 2.312342.3080.019860.002−0.44  0.06800
62570.1685 2.316341.9320.019830.002−0.51  0.06085
62580.1685 2.319341.5890.019840.003−0.36  0.05410
Fig. 9

Transient thermal conductivity results for argon. The baseline is the correlation of Younglove and Hanley [12].

The steady-state measurements are listed in Table 3. As explained before, all steady-state results with TBAND greater than 2 % and all points with Rayleigh numbers greater than 70 000 were not considered. Thus, there are 104 points at a nominal temperature of 300 K, 120 points at 320 K, and 119 points at 340 K. The deviations between all steady-state points and the thermal-conductivity surface of argon [12] are plotted in Fig. 10. It can be concluded from Fig. 10 that over the range of densities shown, the deviations fit within a band of ±2 % at a level of confidence of 95 %. The scales of Figs. 9 and 10 are deliberately the same. Superimposing Figs. 9 and 10 shows that the systematic deviations at low densities are also seen in the steady-state measurements, which is why we ascribe this systematic deviation to a difference in the values for dilute gas thermal conductivity λ0. The agreement between the transient measurements and the steady-state results is quite good; the mean deviation between the two methods is 1 %.
Table 3

Thermal conductivity of argon, steady-state method

Point no.p(MPa)ρ(kg m−3)Texp(K)λexp(W m−1 K−1)TBANDRelative dev. (%)q(W m−1)ΔTRa
Nominal temperature 300 K
30157.4443123.423300.6470.020511.33−0.89  0.014200.66351646
30167.4439123.407300.6670.020821.450.580.015820.72256238
30177.4442123.391300.7130.020711.420.050.017530.79661990
30187.4439123.363300.7570.020301.10−1.97  0.019330.88668872
30346.9070114.295300.6210.020931.462.310.012670.59139372
30356.9071114.283300.6510.020701.441.200.014200.66544239
30366.9069114.263300.6940.020101.61−1.79  0.015820.75550224
30386.9073114.235300.7660.020421.12−0.17  0.019330.89459373
30396.9074114.215300.8120.020341.81−0.58  0.021220.97764788
30516.2656103.318300.8660.020161.21−0.12  0.023191.08858667
30526.2652103.326300.8300.019891.43−1.44  0.021221.01554769
30536.2653103.345300.7880.019941.13−1.20  0.019330.93050217
30546.2652103.357300.7470.019931.22−1.26  0.017540.85045946
30556.2654103.377300.7090.019891.63−1.45  0.015830.77441870
30566.2654103.393300.6720.020191.530.080.014200.69037329
30586.2654103.425300.5920.020791.772.970.011220.53729104
30715.5847  91.824300.8670.020171.181.370.023191.10946858
30725.5847  91.836300.8310.020231.131.690.021221.01843048
30735.5848  91.856300.7830.020251.271.810.019330.93339457
30745.5848  91.872300.7380.020311.072.100.017530.84935936
30755.5848  91.884300.7000.020291.572.010.015820.77132670
30765.5847  91.896300.6630.020321.462.170.014200.69529460
30775.5846  91.908300.6250.020361.502.360.012670.62226388
30944.9524  81.270300.6290.020021.651.960.012670.63921009
30954.9525  81.258300.6700.019971.851.710.014210.71523490
30964.9525  81.246300.7090.019951.321.630.015830.79326058
30974.9526  81.234300.7510.019971.561.710.017540.87428690
30984.9526  81.222300.7970.019901.491.380.019330.96231549
30994.9527  81.210300.8450.019951.121.590.021221.04834349
31004.9528  81.198300.8870.019901.381.360.023191.14237404
31134.2565  69.625300.6100.018911.77−2.28  0.011220.60614468
31154.2560  69.601300.6740.019441.260.460.014200.74217689
31164.2561  69.589300.7270.019381.180.180.015830.82619675
31174.2561  69.581300.7640.019451.130.490.017540.90921640
31184.2562  69.569300.8120.019471.350.580.019330.99723725
31194.2562  69.561300.8460.019511.200.790.021221.08825868
31204.2561  69.545300.9020.019501.370.720.023191.18528147
31343.5546  57.937300.6490.018891.41−0.98  0.012670.68911239
31353.5547  57.929300.6800.019251.440.890.014200.75612333
31363.5546  57.917300.7250.019251.780.890.015830.84113697
31373.5545  57.909300.7700.019240.950.820.017540.92915132
31383.5544  57.897300.8140.019230.900.750.019331.02316635
31393.5544  57.885300.8650.019231.100.710.021221.11918194
31403.5544  57.877300.9100.019241.230.760.023191.21919801
40132.8395  46.096300.7250.018651.54−0.89  0.011220.623  6343
40142.8395  46.092300.7580.018651.52−0.90  0.012670.703  7145
40152.8395  46.084300.8070.018631.22−1.01  0.014200.787  7999
40162.8395  46.076300.8440.018701.48−0.68  0.015820.873  8863
40172.8395  46.068300.8970.018710.89−0.62  0.017530.965  9788
40182.8394  46.060300.9380.018740.91−0.46  0.019331.06010745
40192.8392  46.048300.9870.018750.95−0.45  0.021211.16111757
40202.8392  46.040301.0370.018791.12−0.23  0.023191.26412786
40332.1134  34.180300.7070.018641.920.450.011230.627  3453
40342.1132  34.172300.7530.018411.89−0.82  0.012670.716  3941
40352.1133  34.168300.7950.018631.400.350.014210.793  4360
40362.1132  34.160300.8400.018611.240.230.015830.884  4855
40372.1132  34.152300.8850.018581.480.040.017540.980  5379
40382.1132  34.148300.9320.018510.95−0.34  0.019341.084  5942
40392.1132  34.140300.9900.018430.85−0.78  0.021231.193  6535
40402.1132  34.132301.0500.018330.99−1.32  0.023201.309  7164
40531.3696  22.059300.6910.018301.67−0.04  0.011230.641  1445
40541.3697  22.055300.7390.017891.76−2.34  0.012670.740  1667
40551.3696  22.051300.7890.017911.31−2.24  0.014210.829  1864
40561.3698  22.051300.8220.018261.17−0.29  0.015830.905  2036
40571.3696  22.047300.8690.018300.82−0.09  0.017541.001  2248
40581.3698  22.043300.9220.018260.86−0.30  0.019341.105  2481
40591.3698  22.043300.9750.018250.75−0.38  0.021221.213  2721
40601.3698  22.035301.0300.018251.04−0.39  0.023201.325  2970
40611.3700  22.035301.0910.018260.52−0.38  0.025261.442  3229
40621.3698  22.027301.1470.018220.66−0.60  0.027411.567  3504
40631.3698  22.023301.2110.018210.59−0.70  0.029641.695  3788
40641.3699  22.019301.2750.018220.61−0.62  0.031971.825  4075
40651.3699  22.015301.3530.018100.67−1.32  0.034381.975  4404
40661.3699  22.007301.4170.018110.57−1.29  0.036882.116  4714
40671.3700  22.003301.4940.018120.45−1.27  0.039472.262  5034
40681.3701  21.999301.5690.018110.50−1.31  0.042142.415  5367
40691.3699  21.991301.6410.018180.46−0.94  0.044912.562  5686
40701.3700  21.987301.7240.018170.46−1.01  0.047762.724  6039
40720.7011  11.249300.6610.018341.791.370.011230.641 369
40730.7010  11.245300.7530.017941.36−0.85  0.014210.829 477
40750.7012  11.241300.9570.017991.24−0.65  0.021221.235 708
40760.7010  11.229301.1970.018020.59−0.55  0.029641.722 983
40770.7009  11.221301.3250.018110.47−0.07  0.034381.986  1131
40780.7010  11.217301.4700.018100.53−0.18  0.039472.281  1297
40790.7011  11.213301.6320.018070.39−0.34  0.044912.597  1473
40800.7011  11.205301.7990.018070.34−0.41  0.050702.932  1658
40930.3781 6.056300.7170.018201.871.140.014210.818 135
40940.3780 6.052300.8160.018151.060.850.017541.013 167
40950.3780 6.048300.9190.018100.840.530.021221.229 202
40960.3780 6.044301.0430.018070.720.380.025261.464 241
40970.3780 6.044301.1700.018060.530.260.029641.720 282
40980.3780 6.040301.3070.018020.390.030.034381.998 327
40990.3780 6.040301.4470.018020.32−0.04  0.039472.294 375
41000.3781 6.036301.6140.018010.28−0.13  0.044912.612 426
41110.1684 2.692300.8110.017650.95−1.56  0.017541.041   34
41120.1685 2.692300.9200.017710.79−1.28  0.021221.256   41
41130.1686 2.692301.0250.017900.68−0.24  0.025261.479   48
41140.1686 2.692301.1590.017890.56−0.34  0.029641.737   56
41150.1685 2.689301.2950.017890.42−0.36  0.034382.014   65
41160.1684 2.689301.4470.017870.32−0.49  0.039462.314   74
41170.1685 2.689301.6010.017870.36−0.53  0.044902.633   85
41180.1684 2.685301.7720.017870.22−0.61  0.050702.974   95
41190.1685 2.685301.9580.017870.16−0.66  0.056843.333 106
41200.1686 2.685302.1380.017870.20−0.69  0.063333.713 118
Nominal temperature 320 K
50118.3286128.181320.4140.022191.421.210.015190.66245411
50128.3287128.165320.4480.022121.070.890.016930.73450310
50138.3287128.149320.4830.022121.370.880.018760.80755252
50148.3288128.133320.5240.021981.060.240.020680.88760707
50158.3287128.109320.5660.021841.02−0.40  0.022700.97066368
50317.5812116.492320.4470.021901.341.340.016930.75242454
50327.5812116.480320.4780.021871.061.190.018760.82846742
50337.5813116.460320.5210.021810.790.910.020680.90851240
50347.5815116.444320.5640.021751.320.640.022700.99255906
50357.5815116.428320.6040.021650.920.180.024801.08060833
50367.5816116.412320.6440.021590.85−0.11  0.027011.16965840
50516.8718105.431320.4340.021651.121.570.016930.76935481
50526.8721105.419320.4740.021641.041.530.018760.84739058
50536.8719105.403320.5100.021610.901.390.020680.93042813
50546.8720105.383320.5640.021541.281.030.022691.01746793
50556.8722105.367320.6070.021480.940.770.024801.10650882
50566.8722105.355320.6440.021410.830.410.027011.20055164
50596.8724105.299320.7930.021250.98−0.34  0.034181.49768604
50716.1864  94.785320.3370.021541.902.370.013550.63223485
50726.1863  94.769320.3770.021451.251.960.015190.70826285
50736.1864  94.757320.4220.021441.291.910.016930.78529134
50746.1863  94.745320.4590.021411.241.760.018760.86732135
50756.1864  94.725320.5100.020971.17−0.32  0.020680.97035906
50766.1863  94.709320.5560.021051.140.090.022691.05439002
50776.1863  94.697320.5960.021030.85−0.04  0.024801.14742405
50786.1861  94.677320.6450.021030.88−0.03  0.027011.24145850
50796.1860  94.661320.6890.021021.00−0.09  0.029301.33949437
50806.1860  94.641320.7410.020990.78−0.28  0.031691.44253179
50915.5066  84.234320.2840.021011.961.210.012000.58116950
50925.5067  84.222320.3240.020801.920.240.013550.65919231
50935.5068  84.210320.3600.021051.381.400.015190.72821218
50945.5065  84.194320.4030.021011.091.190.016930.80923571
50955.5066  84.186320.4370.021021.411.220.018750.89325981
50965.5068  84.174320.4830.021021.141.220.020680.98028495
50975.5067  84.158320.5300.020990.971.050.022691.07231151
50985.5070  84.150320.5730.020871.060.510.024801.17234039
50995.5071  84.138320.6270.020731.05−0.18  0.027001.27837084
51005.5070  84.119320.6820.020751.03−0.08  0.029301.37839960
51124.8436  73.944320.2980.020811.501.480.013550.66514844
51134.8438  73.940320.3340.020781.771.370.015190.74316599
51144.8438  73.932320.3690.020771.071.280.016930.82618436
51154.8438  73.916320.4200.020771.081.270.018760.91220342
51164.8439  73.908320.4630.020770.981.260.020681.00222330
51174.8438  73.892320.5060.020801.171.390.022691.09524367
51184.8440  73.884320.5540.020881.131.790.024801.18726408
51194.8441  73.876320.5970.020881.441.780.027001.28728623
51204.8442  73.864320.6500.020941.172.050.029301.38830823
51334.1375  63.018320.3370.020501.401.270.015190.76012223
51344.1375  63.010320.3840.020441.311.010.016930.84713608
51354.1375  62.998320.4370.020391.130.750.018760.93815063
51364.1377  62.994320.4710.020451.201.020.020681.02916509
51374.1377  62.982320.5210.020410.920.830.022691.12818085
51384.1376  62.970320.5720.020411.540.830.024801.22919691
51394.1375  62.954320.6330.020441.120.950.027001.33221327
51404.1373  62.942320.6790.020461.391.010.029301.44023035
51523.4547  52.508320.3000.020251.741.300.013550.692  7658
51533.4547  52.500320.3420.020211.411.110.015190.776  8580
51543.4548  52.496320.3860.020170.990.910.016930.864  9555
51553.4549  52.488320.4340.020090.950.490.018750.96010602
51563.4551  52.484320.4790.020121.110.610.020681.05511644
51573.4549  52.468320.5370.020011.070.070.022691.16112807
51583.4551  52.464320.5840.020041.470.230.024801.26413936
51593.4549  52.452320.6410.020090.750.430.027001.37115094
51603.4552  52.444320.7020.019951.06−0.27  0.029301.49416438
51712.7635  41.893320.3490.019681.43−0.32  0.015190.801  5582
51722.7635  41.889320.3960.019651.00−0.50  0.016930.893  6219
51732.7635  41.881320.4440.019611.07−0.71  0.018750.990  6891
51742.7635  41.873320.4910.019611.04−0.73  0.020671.090  7581
51752.7635  41.866320.5520.019570.80−0.95  0.022691.197  8319
51762.7635  41.862320.5970.019550.78−1.05  0.024801.308  9081
51772.7637  41.854320.6530.019600.80−0.80  0.027001.418  9841
51782.7638  41.846320.7190.019611.14−0.76  0.029301.53610646
51792.7636  41.838320.7750.019641.30−0.64  0.031691.65711471
51802.7637  41.830320.8330.019661.15−0.55  0.034181.78212328
51822.0779  31.419320.3940.019441.20−0.38  0.016930.907  3514
51832.0781  31.415320.4440.019371.26−0.73  0.018761.008  3901
51842.0781  31.411320.4950.019370.92−0.75  0.020681.110  4295
51852.0781  31.403320.5490.019350.81−0.86  0.022691.219  4710
51862.0782  31.399320.6060.019330.86−0.99  0.024801.332  5146
51872.0782  31.395320.6630.019330.55−1.01  0.027001.450  5594
51882.0783  31.391320.7290.019300.64−1.19  0.029301.574  6070
51892.0783  31.383320.7850.019290.58−1.24  0.031691.702  6555
51902.0783  31.375320.8500.019290.64−1.28  0.034171.834  7057
52031.3798  20.813320.3270.019471.391.000.015190.816  1370
52041.3798  20.805320.4340.019341.140.290.018751.013  1699
52051.3796  20.797320.5380.019260.81−0.14  0.022691.230  2059
52071.3798  20.781320.7910.019130.83−0.86  0.031691.727  2880
52081.3798  20.773320.9250.019090.69−1.14  0.036752.006  3338
52091.3800  20.765321.0700.019150.47−0.85  0.042182.293  3809
52101.3798  20.753321.2290.019120.44−1.03  0.047992.610  4324
52220.6847  10.303320.2380.019371.941.670.012000.649 264
52230.6849  10.303320.3240.019331.231.420.015190.823 334
52240.6849  10.299320.4180.019271.261.090.018751.019 413
52250.6848  10.295320.5280.019120.780.300.022691.243 503
52260.6849  10.291320.6560.019001.01−0.39  0.027001.488 601
52270.6847  10.283320.7840.018980.80−0.52  0.031691.748 704
52280.6849  10.283320.9300.018970.42−0.58  0.036752.027 816
52290.6848  10.275321.0740.019000.69−0.49  0.042182.324 932
52300.6849  10.275321.2320.019010.48−0.45  0.047992.641  1058
52440.3681 5.529320.4270.018751.06−1.09  0.018751.048 122
52450.3680 5.525320.5320.018801.25−0.90  0.022691.265 146
52460.3680 5.525320.6450.018921.29−0.26  0.027001.495 173
52470.3680 5.521320.7760.018910.79−0.33  0.031691.755 203
52480.3680 5.517320.9110.018890.81−0.48  0.036752.038 235
52490.3680 5.517321.0670.018880.60−0.56  0.042182.340 269
52500.3680 5.513321.2300.018880.74−0.61  0.047992.662 305
52610.3679 5.505321.4890.018880.67−0.67  0.057413.185 363
52620.3680 5.501321.7740.018800.58−1.21  0.067673.770 429
52630.3680 5.497322.0890.018820.65−1.19  0.078774.384 496
52640.3680 5.493322.4130.018830.60−1.23  0.090715.045 568
52650.1734 2.601320.7740.018720.65−1.06  0.031691.774   45
52660.1733 2.597320.9130.018700.31−1.20  0.036752.060   52
52670.1734 2.597321.0630.018710.28−1.20  0.042182.363   60
52680.1734 2.597321.2270.018720.29−1.17  0.047992.687   68
52690.1733 2.593321.3960.018770.21−0.92  0.054183.024   76
52700.1732 2.589321.5820.018790.13−0.91  0.060733.388   85
52710.1733 2.589321.7690.018790.19−0.95  0.067673.775   95
52720.1733 2.589321.9720.018780.13−1.03  0.074974.184 105
52730.1732 2.585322.1960.018790.11−1.03  0.082654.609 115
527401733 2.585322.4100.018790.17−1.11  0.090715.060 126
Nominal temperature 340 K
60088.1066116.696339.0860.023191.782.520.015240.65231009
60098.1065116.660339.1780.023130.992.270.019910.84039919
60108.1065116.616339.2870.022890.951.200.025191.05650097
60118.1063116.564339.4070.022660.710.190.031101.29361186
60267.5849109.126339.0990.022811.871.820.015240.66627701
60277.5850109.094339.1920.022731.121.450.019910.86135747
60287.5849109.050339.3050.022600.820.880.025191.07944718
60297.5848109.006339.4270.022470.860.280.031101.31854503
60307.5849108.958339.5560.022290.54−0.55  0.037621.57965176
60446.9067  99.287339.1070.022711.832.530.015240.67523177
60456.9067  99.255339.2090.022581.211.950.019910.87630031
60466.9070  99.223339.3170.022551.041.830.025191.09537496
60476.9070  99.179339.4500.022410.751.160.031101.34245834
60486.9070  99.135339.5820.022280.700.570.037631.60854818
60496.9067  99.083339.7260.022100.80−0.28  0.044771.89864562
60626.2157  89.268339.1220.021601.92−1.22  0.015240.71419754
60636.2156  89.232339.2350.021791.31−0.39  0.019910.91525283
60646.2156  89.204339.3400.022200.821.440.025191.12531042
60656.2153  89.164339.4620.022080.890.890.031101.38037987
60666.2155  89.120339.6100.021870.83−0.08  0.037621.66345688
60676.2156  89.080339.7530.021830.90−0.30  0.044771.95653623
60686.2156  89.032339.9140.021730.71−0.77  0.052542.27362158
60805.5216  79.213339.1300.021731.570.590.015240.71515518
60815.5217  79.185339.2410.021710.860.480.019910.92720083
60825.5218  79.157339.3580.021751.110.600.025191.16125109
60835.5219  79.125339.4870.021770.950.670.031101.41830609
60845.5218  79.089339.6180.021750.830.540.037631.69936611
60855.5219  79.049339.7740.021730.990.430.044772.00143020
60865.5219  79.009339.9420.021660.680.090.052542.32849920
60875.5218  78.965340.1060.021641.03−0.04  0.060922.67057107
60885.5219  78.913340.3010.021540.82−0.54  0.069933.04064822
60984.8343  69.270339.1430.021381.490.140.015240.73112075
60994.8342  69.246339.2480.021421.130.270.019910.94715621
61004.8341  69.214339.3760.021430.920.310.025191.18919576
61014.8341  69.186339.5050.021520.860.670.031101.45123842
61024.8341  69.154339.6480.021551.140.780.037621.73928507
61034.8341  69.118339.8100.021561.330.800.044772.05033527
61044.8340  69.082339.9730.021541.680.680.052542.38638917
61164.1414  59.263339.1610.020691.87−2.02  0.015250.760  9130
61174.1415  59.243339.2620.021031.17−0.37  0.019910.97111658
61184.1415  59.219339.3910.021090.97−0.14  0.025201.22014608
61194.1416  59.195339.5210.021130.710.050.031101.49417856
61204.1417  59.167339.6680.021241.230.540.037631.78621312
61214.1415  59.135339.8340.021311.510.810.044782.10525051
61333.4497  49.312339.0560.020651.89−1.00  0.011200.564  4664
61343.4499  49.296339.1580.020691.49−0.82  0.015250.763  6306
61353.4499  49.280339.2730.020690.93−0.87  0.019910.994  8195
61363.4498  49.256339.4050.020700.90−0.85  0.025191.25210305
61373.4499  49.236339.5440.020791.02−0.42  0.031101.53212586
61383.4499  49.212339.7040.020821.09−0.32  0.037631.84315102
61393.4501  49.192339.8590.020911.430.060.044772.17317771
61403.4499  49.160340.0410.020971.570.290.052552.53120635
61523.4481  49.272339.1570.020641.18−1.09  0.015250.765  6316
61533.4481  49.252339.2780.020661.16−1.00  0.019910.995  8196
61543.4481  49.232339.4080.020680.64−0.92  0.025201.25310302
61553.4481  49.212339.5500.020790.76−0.42  0.031101.53212572
61563.4483  49.188339.7060.020810.86−0.40  0.037631.84415100
61573.4483  49.164339.8690.020901.320.020.044782.17517759
61583.4483  49.136340.0420.021022.000.530.052542.52520568
61702.7656  39.461339.1560.020781.300.730.015250.763  4010
61712.7655  39.445339.2830.020441.12−0.97  0.019911.011  5301
61722.7657  39.429339.4240.020440.73−0.98  0.025191.276  6677
61732.7658  39.413339.5660.020450.68−0.96  0.031101.570  8199
61742.7658  39.393339.7270.020480.75−0.86  0.037631.891  9854
61752.7657  39.369339.8890.020591.30−0.37  0.044772.23211599
61762.7658  39.349340.0860.020621.14−0.26  0.052542.60613507
61772.7658  39.325340.2820.020701.700.070.060933.00015503
61782.7659  39.301340.4930.020761.600.340.069933.41917619
61802.0708  29.506339.0620.020221.90−0.88  0.011200.579  1687
61812.0708  29.498339.1650.020201.40−0.97  0.015240.788  2292
61822.0710  29.490339.2930.020151.08−1.26  0.019911.030  2991
61832.0710  29.478339.4230.020180.63−1.16  0.025191.300  3768
61842.0711  29.462339.5800.020170.51−1.24  0.031101.602  4636
61852.0710  29.446339.7410.020180.58−1.24  0.037631.935  5584
61862.0710  29.434339.9220.020160.52−1.38  0.044772.300  6622
61872.0710  29.414340.1170.020180.53−1.29  0.052542.690  7722
61882.0711  29.398340.3250.020260.92−0.97  0.060923.101  8878
61892.0712  29.378340.5360.020361.41−0.53  0.069933.53510089
61902.0712  29.358340.7750.020371.04−0.51  0.079554.00911403
62061.3695  19.475339.1540.020121.49−0.24  0.015250.793 996
62071.3695  19.467339.2770.020111.01−0.32  0.019911.035  1298
62081.3696  19.459339.4180.020070.72−0.55  0.025191.312  1641
62091.3696  19.451339.5660.020070.69−0.63  0.031101.619  2022
62101.3696  19.443339.7420.020050.63−0.75  0.037631.958  2440
62111.3696  19.431339.9260.020030.61−0.89  0.044772.330  2896
62121.3696  19.419340.1300.020010.47−1.07  0.052542.736  3390
62131.3696  19.407340.3480.020000.46−1.16  0.060923.171  3917
62141.3696  19.391340.5750.019990.43−1.26  0.069933.637  4479
62151.3696  19.379340.8350.019970.39−1.39  0.079554.136  5074
62240.6978 9.903339.1550.019631.44−1.67  0.015250.814 262
62250.6980 9.903339.2730.019961.01−0.04  0.019911.045 336
62260.6979 9.899339.4130.019950.71−0.12  0.025191.323 424
62270.6980 9.895339.5670.019930.74−0.24  0.031101.634 523
62280.6979 9.887339.7410.019920.54−0.33  0.037631.977 631
62290.6977 9.883339.9200.019920.55−0.41  0.044772.353 749
62300.6977 9.875340.1290.019870.49−0.69  0.052542.767 878
62310.6977 9.867340.3540.019860.50−0.80  0.060923.209  1015
62320.6977 9.859340.5860.019900.50−0.64  0.069933.675  1158
62330.6977 9.855340.8360.019890.46−0.76  0.079554.182  1314
62420.3486 4.942339.1490.019701.77−0.79  0.015250.811   65
62430.3485 4.942339.2750.019690.99−0.86  0.019911.060   84
62440.3487 4.942339.4060.019690.88−0.89  0.025191.341 107
62450.3486 4.938339.5760.019690.54−0.92  0.031101.655 131
62460.3487 4.938339.7440.019700.64−0.91  0.037632.001 158
62470.3487 4.934339.9330.019700.56−0.98  0.044772.382 188
62480.3487 4.930340.1380.019690.52−1.04  0.052542.795 220
62490.3487 4.926340.3590.019720.43−0.97  0.060923.237 254
62500.3486 4.922340.6010.019740.45−0.92  0.069923.711 290
62510.3487 4.922340.8540.019740.38−0.99  0.079544.221 329
62600.1685 2.389339.1690.019241.78−2.89  0.015250.830   15
62610.1685 2.389339.2920.019431.08−1.94  0.019911.074   20
62620.1685 2.385339.4300.019540.64−1.41  0.025191.352   25
62630.1685 2.385339.5890.019600.56−1.11  0.031101.663   31
62640.1685 2.385339.7600.019640.40−0.94  0.037632.008   37
62650.1684 2.381339.9480.019660.33−0.90  0.044772.387   44
62660.1684 2.381340.1580.019660.25−0.96  0.052542.801   51
62670.1684 2.377340.3810.019670.23−0.94  0.060923.246   59
62680.1685 2.377340.6210.019680.16−0.98  0.069923.724   68
62690.1684 2.377340.8700.019690.15−0.98  0.079554.235   77
Fig. 10

Steady-state thermal conductivity results for argon. The baseline is the correlation of Younglove and Hanley [12].

The steady-state single-wire results are given in Table 4 under several different headings. The single-wire experiments are an attempt to measure the end effects in each wire. At the ends of each wire, heat is flowing from the wire ends to the wire supports. In addition, a part of the applied heat is flowing from the end of the wire through the fluid to the cell ends, a geometry quite different from the center portions of the wire. The lines in Table 4 are in pairs by run and point number. The first line is the regular or normal result calculated from the measured bridge imbalances, while the result on the second line uses the voltages measured directly across the individual hot wires. As a check the added voltmeter was also connected across the bridge. In this case the two lines given in Table 4 are virtually identical. Relative deviations of the data from the thermal conductivity surface of argon [12] are also provided in Table 4. These deviations are shown in Fig. 11 as a function of the applied power q. We see that for the short hot wire the thermal conductivity results are about 20 % above the normal steady-state bridge values, while for the long hot wire the deviation is around 4 % for the higher powers. An inspection of the single-wire voltage profiles indicates that turbulent convection (see trace c of Fig. 6) is first seen in the short-wire cell; however, it carries over into the full-bridge measurement. For the low-temperature system [1], using wire lengths of 0.05 m and 0.10 m, end effects of 8.3 % for the short wire and 4.8 % for the long wire were observed in nitrogen gas. The present wires have lengths of 0.05 m and 0.20 m, with end effects of 16 % and 4 %, respectively. For equivalent wire lengths (short wires), the present end effects are larger by about a factor of 2, quite reasonable given that the present steady-state measurements run considerably longer in time on a different gas.
Table 4

Steady-state results with single wires

Point no.p (MPa)ρ (kg m−3)Texp (K)λexp (W m−1 K−1)TBANDRelative Dev. (%)q (W m−1)Ra
Long hot wire, high pressure
  80012.075329.546339.3230.020381.60−0.140.019912970
  80012.075329.554339.2550.023520.0013.250.019892575
  80022.075229.542339.3820.020341.37−0.360.021953276
  80022.075229.550339.3040.023620.0013.600.021932824
  80032.075429.538339.4400.020331.17−0.430.024093593
  80032.075429.546339.3670.023050.0011.430.024073172
  80042.075429.534339.4910.020311.28−0.520.026333925
  80042.075429.542339.4080.023160.0011.880.026313445
  80052.075229.526339.5570.020310.99−0.560.028674267
  80052.075229.534339.4790.022690.0010.020.028643822
  80062.075429.522339.6120.020250.90−0.840.031114636
  80062.075429.530339.5330.022480.009.160.031084181
  80072.075429.514339.6830.020230.87−0.960.033645010
  80072.075429.522339.6040.022250.008.220.033614560
  80082.075429.510339.7480.020220.68−1.020.036285396
  80082.075429.518339.6660.022160.007.850.036244929
  80092.075529.506339.8190.020240.76−0.930.039015788
  80092.075529.514339.7330.022140.007.720.038985299
  80102.075529.498339.8980.020210.75−1.120.041856207
  80102.075529.506339.8170.021860.006.550.041815743
100012.075729.494339.9670.020230.59−1.060.044786627
100012.075729.502339.8880.021710.005.870.044746180
100022.075929.490340.0380.020210.55−1.170.047817070
100022.075929.498339.9640.021500.004.920.047766651
100032.075929.482340.1200.020220.72−1.110.050957513
100032.075929.490340.0340.021640.005.520.050907030
100042.075929.478340.1970.020220.73−1.140.054177974
100042.075929.482340.1220.021380.004.380.054127547
100052.075929.470340.2830.020260.99−0.970.057508431
100052.075929.474340.2110.021310.004.020.057458023
100062.076029.462340.3600.020281.01−0.880.060938908
100062.076029.470340.2900.021230.013.650.060878516
100072.075929.454340.4550.020331.57−0.640.064469379
100072.075929.462340.3800.021300.013.950.064408959
100082.075929.446340.5390.020361.47−0.540.068099873
100082.075929.450340.4650.021250.013.730.068029463
100092.076129.438340.6300.020371.38−0.510.0718110386
100092.076129.446340.5560.021230.013.580.071749974
100102.076129.430340.7240.020381.37−0.460.0756410907
100102.076129.438340.6390.021330.014.040.0755710432
Long hot wire, low pressure
  80010.16972.405339.2530.019881.310.370.0199120
  80010.16972.405339.1600.024170.0018.060.0198916
  80020.16972.405339.3050.019851.370.230.0219522
  80020.16972.405339.2250.022980.0013.820.0219319
  80030.16972.405339.3590.019851.110.180.0240924
  80030.16972.405339.2790.022670.0012.650.0240621
  80040.16972.405339.4250.019840.990.120.0263326
  80040.16972.405339.3450.022410.0011.590.0263023
  80050.16972.405339.4770.019830.880.090.0286628
  80050.16972.405339.4100.021730.008.850.0286326
  80060.16972.405339.5440.019810.80−0.050.0311031
  80060.16972.405339.4700.021730.008.830.0310728
  80070.16982.405339.6110.019820.67−0.030.0336433
  80070.16982.405339.5400.021530.007.960.0336031
  80080.16962.401339.6820.019800.69−0.150.0362736
  80080.16962.401339.6030.021550.008.010.0362433
  80090.16972.401339.7480.019800.65−0.140.0390139
  80090.16972.401339.6780.021240.006.640.0389736
  80100.16962.401339.8270.019790.48−0.230.0418441
  80100.16962.401339.7570.021110.006.070.0418039
Short hot wire, high pressure
  90012.075629.550339.3220.020481.730.360.019912956
  90012.075629.562339.2200.025770.0020.810.019992365
  90022.075729.550339.3790.020411.36−0.010.021953267
  90022.075729.558339.2670.025670.0020.510.022042615
  90032.075729.542339.4330.020391.55−0.120.024093583
  90032.075729.554339.3180.025210.0019.030.024192919
  90042.075829.538339.4870.020381.15−0.200.026333914
  90042.075829.550339.3710.024750.0017.520.026443245
  90052.075829.534339.5480.020331.12−0.440.028674265
  90052.075829.546339.4200.024780.0017.630.028783524
  90062.075729.526339.6130.020311.10−0.570.031104625
  90062.075729.538339.4740.024760.0017.540.031233822
  90072.075729.522339.6800.020290.94−0.660.033644997
  90072.075729.534339.5420.024320.0016.040.033774200
  90082.075829.518339.7460.020260.89−0.850.036285389
  90082.075829.530339.5830.024710.0017.370.036424453
  90092.075729.510339.8170.020230.78−0.980.039015792
  90092.075729.526339.6440.024610.0017.010.039174802
  90102.075829.502339.8840.020220.78−1.040.041856205
  90102.075829.522339.6970.024660.0017.150.042025135
110012.076529.506339.9500.020190.58−1.220.044786644
110012.076529.526339.7590.024370.0016.180.044965552
110022.076629.502340.0280.020190.83−1.250.047817081
110022.076629.522339.8050.024860.0017.800.048015807
110032.076729.498340.1060.020200.76−1.200.050947527
110032.076729.518339.8680.024900.0017.940.051156167
110042.076629.490340.1840.020210.73−1.210.054177987
110042.076629.510339.9400.024700.0017.240.054396601
110052.076629.482340.2660.020230.94−1.100.057508450
110052.076629.502340.0110.024640.0017.060.057747009
110062.076629.474340.3570.020261.04−0.960.060938921
110062.076629.498340.0910.024640.0017.020.061187415
110072.076629.466340.4400.020311.60−0.770.064469398
110072.076629.490340.1630.024620.0016.940.064727837
110082.076829.458340.5230.020331.45−0.660.068089894
110082.076829.486340.2430.024450.0016.330.068368320
110092.076829.450340.6220.020351.52−0.570.0718110401
110092.076829.478340.3200.024610.0016.870.072118703
110102.076829.442340.7060.020351.39−0.590.0756310930
110102.076829.470340.3920.024550.0016.650.075959171
Voltmeter across the bridge
  70010.16952.401339.2380.019771.19−0.180.0199120
  70010.16952.401339.2380.019770.20−0.180.0199120
  70020.16962.405339.2890.019790.96−0.070.0219522
  70020.16962.405339.2890.019790.15−0.070.0219522
  70030.16972.405339.3460.019780.77−0.140.0240924
  70030.16972.405339.3460.019780.14−0.140.0240924
  70040.16972.405339.4010.019781.07−0.160.0263226
  70040.16972.405339.4010.019780.15−0.160.0263226
  70050.16982.405339.4600.019780.89−0.180.0286628
  70050.16982.405339.4600.019780.13−0.180.0286628
  70060.16972.405339.5310.019760.57−0.280.0311031
  70060.16972.405339.5310.019760.12−0.280.0311031
  70070.16972.401339.5970.019750.54−0.360.0336333
  70070.16972.401339.5970.019750.06−0.360.0336333
  70080.16972.401339.6610.019760.57−0.340.0362736
  70080.16972.401339.6610.019760.11−0.340.0362736
  70090.16972.401339.7330.019770.42−0.270.0390039
  70090.16972.401339.7330.019770.11−0.270.0390039
  70100.16972.401339.8150.019750.41−0.420.0418441
  70100.16972.401339.8150.019750.13−0.420.0418441
Vacuum, short hot wire, very low pressure
120010.00130.020338.8190.0047225.82−318.000.001250
120010.00130.020338.9010.002970.14−564.600.001250
120020.00130.020339.2810.004312.59−358.200.004980
120020.00130.020338.9410.009950.07−98.290.005000
120030.00140.020340.0500.004260.90−364.600.011190
120030.00140.020339.9640.004560.04−334.000.011240
120040.00150.020341.1280.004230.56−369.000.019890
120040.00150.020340.8120.004880.03−306.400.019970
120050.00140.020342.5200.004220.38−371.800.031050
120050.00140.020342.1100.004740.03−319.600.031180
Fig. 11

End effect ratios from “single wire” measurements to determine the effectiveness of end effect compensation by the bridge circuit. The baseline is the correlation of Younglove and Hanley [12].

The very last segment in Table 4 shows a series of runs made at a pressure considerably below ambient with the added voltmeter connected across the short hot wire. The exact pressure was difficult to determine because we did not have an appropriate pressure gage. We estimate that the pressure must be between 2 Pa and 1300 Pa, somewhere between the limit of the forepump and the rather approximate reading of the regular pressure gage. We can be certain that these measurements fall into the Knudsen region where the thermal conductivity is proportional to pressure. The deviations for the two highest power levels in this series fall between the ones for the long and the short hot wires. We might expect the end effects to be smaller under these conditions because the heat flow from the wire ends to the cell ends is reduced. It is shown that for all conditions, except perhaps turbulent convection in either cell, having the short hot wire in the bridge insures sufficient compensation for end effects. This seems to be in agreement with the conclusions of Taxis and Stephan [7]. Finally, the steady-state results, the first line of each pair, are independent of applied power, an excellent verification of the key requirement that the results are free from the influence of natural convection.

4.3 Analysis

Argon was selected as the test fluid because the dilute-gas thermal conductivity λ0 and the first density correction λ1 = (∂λ/∂λρ)T are well known from Chapman-Enskog theory, as is the pair interaction potential for argon. The present results must be compared with the values derived from theory to validate the technique. To make the analysis easier, the large number of points was reduced by averaging as follows. All of the results were shifted from their experimental temperatures to the even temperatures of 300 K, 320 K, and 340 K using the thermal conductivity surface for argon [12]. The mean of this adjustment is −0.35 %, the maximum is −1.25 %. The next step is to average the results for the various power levels at each pressure. This step gives us 13 or 14 points per isotherm. Theory indicates that the thermal conductivity is a nearly linear function of density at low densities. Since values are now available, measured by two different methods in the same apparatus, it is reasonable to combine both transient and steady-state values into one result. Thus the final step is to obtain averaged straight lines for each isotherm from the experimental results. These averaged straight lines become the basis for deviation plots to assess the accuracy of the present steady-state and transient results. The averaged thermal conductivities adjusted to nominal isotherm temperatures are given in Table 5 along with the deviations of these values from the straight lines. From Table 5 it is easy to establish that the mean difference between transient and steady-state measurements is about 1 %, with the steady-state values nearly always higher. While transient and steady-state values agree to within their combined uncertainties, we cannot exclude the possibility that a systematic difference of about 1 % may exist between the two methods. Our straight-line intercepts are the values of the dilute-gas thermal conductivity λ0 and the slopes are values of the first density correction λ1. The coefficients of the lines with their calculated expanded uncertainty at a level of uncertainty of 95 % are given in Table 6.
Table 5

Averaged thermal conductivity of argon, adjusted to even temperature

Averaged transient dataAveraged steady-state data
ρ(kg m−3)λexp(W m−1 K−1)Dev. from line (%)ρ(kg m−3)λexp(W m−1 K−1)Dev. from line (%)
Nominal temperature 300 K
122.48860.020670.09123.395  0.02055−0.60  
113.63610.02043−0.01  114.259  0.020460.06
102.74630.02015−0.05  103.361  0.02007−0.53  
  91.35710.01984−0.19  91.8680.020241.73
  80.76290.01956−0.28  81.2340.019911.42
  69.19790.01922−0.56  69.5810.019340.02
  57.59300.01892−0.63  57.9090.019150.54
  45.82040.01868−0.35  46.0720.01865−0.55  
  33.95580.01831−0.77  34.1560.018470.07
  21.91550.01800−0.85  22.0270.01811−0.25  
  11.16550.01778−0.59  11.2290.018020.74
 6.01620.01752−1.35    6.0440.018021.45
 2.67650.017810.76  2.6890.017770.54
Nominal temperature 320 K
127.4620.02181−0.48  128.149  0.022020.40
115.5740.02146−0.65  116.452  0.021730.50
104.5880.02120−0.51  105.379  0.021480.71
  93.9740.02087−0.75  94.7170.021160.54
  83.5790.02057−0.88  84.1780.020900.64
  73.4720.02035−0.65  73.9080.020801.47
  62.6460.02008−0.57  62.9820.020411.01
  52.1520.01982−0.49  52.4800.020080.77
  41.6540.01955−0.45  41.8620.01959−0.28  
  31.3670.01920−0.86  31.3990.01931−0.29  
  20.6770.01889−1.02  20.7850.019180.49
  10.2270.01867−0.73  10.2910.019081.43
 5.4810.01851−0.92    5.5130.018790.58
 2.5730.018680.41  2.5930.018680.41
Nominal temperature 340 K
115.6290.02243−0.85  116.632  0.023011.57
108.0910.02223−0.82  109.046  0.022620.80
  98.3320.02194−0.92  99.1950.022471.35
  88.4010.02173−0.64  89.1560.021900.05
  78.4340.02144−0.71  79.0770.021710.46
  68.5790.02117−0.71  69.1820.021510.80
  58.6600.02091−0.65  59.2030.021110.23
  48.8000.02065−0.60  49.2280.020810.11
  39.0610.02040−0.52  39.3890.020590.37
  29.2140.02009−0.72  29.4420.02023−0.05  
  19.2830.01977−0.96  19.4310.020050.43
 9.8110.01947−1.17    9.8830.019890.96
 4.8900.01951−0.27    4.9340.019710.74
 2.3650.019711.10  2.3810.019580.45
Table 6

Experimental and theoretical dilute-gas thermal conductivity and first-density coefficientsa for argon. Here, U is the expanded uncertainty (coverage factor k = 2, and thus a 2 standard deviation estimate)

T(K)λ0 (exp)(W m−1 K−1)2U(W m−1 K−1)λ0b(W m−1 K−1)Relative dev. (%)λ0c(W m−1 K−1)Relative dev. (%)λ0d(W m−1 K−1)Relative dev. (%)
Dilute-gas thermal conductivity
3000.017610.000100.0176100.01784−1.300.01772−0.62
3200.018530.000100.01871  −0.920.01883−1.620.01870−0.92
3400.019430.000100.01979−1.850.01966−1.18
T(K)λ1 (exp)(W m−1 K−1)2U(W m−1 K−1)λ1b(W m−1 K−1)Relative dev. (%)λ1e(W m−1 K−1)Relative dev. (%)λ1f(W m−1 K−1)Relative dev. (%)
First-density coefficient
3000.000990.000060.00106  −7.070.00105−5.740.00110−10.77
3200.001060.000060.0010600.001041.790.00110 3.87
3400.001100.000070.001045.710.00110 0.27

λ1 = (∂λ/∂ρ)T W L mol−1 m−1 K−1

Ref. [22]

Ref. [14]

Ref. [15]

Refs. [16,17]

Ref. [18]

The reason that we have used the thermal conductivity surface of Younglove and Hanley [12] in this paper for comparisons, etc., rather than some of the other possible choices, will now become clear. The dilute-gas thermal conductivities of the Younglove and Hanley model [12] are equivalent to the theoretically derived values of Kestin et al. [14]. We conclude from Table 6 that our dilute-gas thermal conductivities are lower than the theoretical values of Kestin et al. [14] as well as those of Aziz [15]. However, they appear to be in better agreement with those of Aziz. Our first density corrections are seen to depend slightly on temperature, unlike the theoretical results, which are constant. They appear to be in better agreement with the values of Rainwater and Friend [16, 17] than with those of Bich and Vogel [18]. Good agreement is found with the previous work using the NIST low temperature instrument [19]. Finally, in Fig. 12 the present results are compared with our earlier ones [1,4,19-22], and with those of other authors [23-28] for a temperature of 300 K, where the baseline is the present least-squares fitted line (coefficients in Table 6). The present results are connected by lines to set them off from the others. All other results were shifted to a temperature of 300 K by using the thermal-conductivity surface of [12]. The largest shift is around 2.3 %, because a few of the original experimental temperatures are as high as 308 K. Figure 12 shows that the present results, including the new steady-state data, are in good agreement with our earlier results [1,4,19-22]. Figure 12 also shows that all of the values assembled here, which include those from transient experiments, steady-state concentric cylinders [24], and steady-state parallel plate systems [23], agree to within 1 %, a truly remarkable result. Comparisons made at the other two temperatures, 320 and 340 K, are quite similar to Fig. 12.
Fig. 12

Deviations of the present results and those of other authors from the line average of the present transient and steady-state results for argon at 300 K.

4.4 Uncertainty

The measurements were deliberately made over a wide range of power levels. For both transient and steady-state results those power levels which were either too small or too large were eliminated. For power levels that are too low, the bridge imbalance becomes comparable to the background noise level and there is significant uncertainty in the measured temperature rises. In general the instruments require a temperature rise of at least 2.5 K to obtain accurate transient results (STAT < 0.003). With power levels that are too high, curvature is found in the relation for ΔT vs ln(t) for the transient measurements, as is typical of convection. The onset of natural convection is also observed in the steady-state measurements as a time-varying steady-state temperature rise. Imposing certain limits on the experimental uncertainty parameter—a maximum STAT of 0.003 for the transient points and a maximum of 2 % in TBAND for the steady-state points—seem to be appropriate restrictions. The uncertainty in transient thermal conductivity data increases at densities below 28 kg m−3 (p = 1 MPa). For valid steady-state measurements of the thermal conductivity, the Rayleigh number must be less than 70 000. With these restrictions it is found that the relative expanded uncertainty of the transient thermal conductivity is 1 % (k = 2, see Fig. 9), the uncertainty of the steady-state thermal conductivity is 2 % (k = 2, see Fig. 10), while the agreement between the two methods is 1 % (see Table 5). The overall agreement between our present results, our earlier transient results and the results of many other authors is a truly remarkable 1 % (see Fig. 12). The steady-state results have a greater uncertainty than the transient ones. This can be attributed to a number of factors. First, the steady-state experiment requires an accurate measurement of the temperature rise ΔT quite similar to the measurement of thermal diffusivity in the transient hot-wire system [20,21]. Second, the steady-state measurement requires accurate determination of the cell geometry: wire radius, cavity radius, eccentricity. Due to the constraints imposed by the onset of convection, the valid temperature rises for steady-state measurements are half those of the transient ones. Table 7 shows the mean temperature rises for both modes of operation. This may serve as a guide for operating hot-wire cells in either the transient or steady-state mode.
Table 7

Survey of the valid temperature rises

T(K)TransientSteady state
No. pointsΔTmean(K)No. pointsΔTmean(K)
300  982.0821041.261
3201052.2151201.473
3401023.3781191.826

5. Summary

It has been demonstrated that the thermal conductivity of argon can be measured with a relative expanded uncertainty (k = 2) of 2 %, using a transient hot-wire system operating in an absolute steady-state mode. The bridge arrangement used in this experiment provides sufficient compensation, eliminating the problem of end effects for both transient and steady-state measurements. The selection of valid steady-state results is based on the shape of the curve of measured voltage rises, on the size of the error in the temperature rise, TBAND, and on the magnitude of the Rayleigh number. Since the fluid gap is quite large in transient hot-wire cells, the steady-state mode is restricted to the low-density gas. The use of the absolute steady-state mode requires no information on the thermophysical properties of the fluid of interest. This is a significant advantage for the measurement of thermal conductivities in the vapor phase of refrigerants where the fluid properties are not known well enough to obtain accurate corrections for transient hot-wire measurements. From a fundamental point of view it can contribute to the determination of accurate pair-interaction potentials for diatomic molecules, especially the long-range weak interactions, a very active research field, by measuring the thermal conductivity of low-temperature vapors.
  1 in total

1.  Second viscosity and thermal-conductivity virial coefficients of gases: Extension to low reduced temperature.

Authors: 
Journal:  Phys Rev A Gen Phys       Date:  1987-10-15
  1 in total
  1 in total

1.  Measurement and Correlation of the Thermal Conductivity of trans-1-Chloro-3,3,3-trifluoropropene (R1233zd(E)).

Authors:  Richard A Perkins; Marcia L Huber; Marc J Assael
Journal:  J Chem Eng Data       Date:  2017-03-29       Impact factor: 2.694
  1 in total

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