Short Hot Wire Technique for Measuring the Thermal Conductivity and Thermal Diffusivity of Various Materials

H. Q. Xie, H. Gu, X. Zhang, C and M. Fujii

Institute for Materials Chemistry and Engineering, Kyushu University, Kasuga 816-8580, Japan C Corresponding author’s Tel & Fax: 092-583-7638, E-mail: xzhang@cm.kyushu-u.ac.jp

Transient short hot wire (SHW) technique is developed for simultaneously determination of the thermal conductivity and thermal diffusivity of various materials such as liquids, gases, or powders. A metal wire with (or without) insulation coating serves both as a heating unit and as an electrical resistance thermometer and the wire is calibrated using water and toluene as the standard samples. This SHW method includes correlating the experimental data with numerically simulated values based on a two-dimensional heat conduction model. When a linear relation between temperature rise and logarithmic heating time is valid, the thermal conductivity and thermal diffusivity are obtained by comparing the slope and the intercept of the measured and calculated. On the other hand, when the relation between temperature rise and logarithmic heating time is nonlinear, the thermal conductivity and thermal diffusivity are extracted from a curve fitting method by using downhill simplex method. This technique is applied here using air as a testing sample. The effect of natural convection is investigated and the accuracy of this measurement is estimated to be 2% for thermal conductivity and 7% for thermal diffusivity. KEY WORDS: short hot wire; thermal conductivity; thermal diffusivity.

1. INTRODUCTION Many different techniques, which are classified into two categories consisting of steady state methods like parallel plate method and transient methods like laser flash method, have been developed to determine thermal conductivity and thermal diffusivity. Of the techniques developed for fluids, the transient hot wire (THW) method has been regarded by many as a technique producing generally excellent results and it has been used widely for measurements of the thermal conductivities, and in some cases, the thermal diffusivities of fluids with a high degree of accuracy [1-5]. Modified hot wire apparatuses also have been reported for different measurement purposes [6-8]. The most attractive advantage of this method for application to fluids is its capacity of experimentally eliminating convective error and the data obtained are more reliable than those obtained using the steady state method [2]. In practical application, it requires the attainment of the very high precision of the measurement of the absolute temperature rise in order to determine the thermal diffusivity [2]. Due to the advances of data acquisition device, it is not very difficult to measure the temperature rise accurately. However, this method is rather difficult to be applied for

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electrically conducting or highly corrosive fluids, like molten carbonates, because the rather long wire must be insulated with anticorrosive and electrically insulating materials [9]. Also there are some technical difficulties to keep a homogeneous initial temperature of the samples in a rather long vessel. These disadvantages could be much reduced if the long hot wire is replaced by a much shorter wire. From this consideration, Fujii and Zhang [9-11] developed a new method which used a metal wire as short as about 10 mm to measure the thermal conductivity and thermal diffusivity using a relatively smaller sample cell. The metal wire serves both as a heating unit and as an electrical resistance thermometer. For corrosive and/or electrical conductive samples, a thin layer of insulation coating on the surface of the hot wire is needed. Substantially different from the conventional THW method, the rather short hot wire can not be considered as an infinite heat source and the analytical solution [12] of temperature evolution of THW is not applicable in this short hot wire (SHW) measurement. The average temperature rise of SHW is affected by the themophysical properties as well as the dimensions of the hot wire, insulation coating, and the sample. Therefore, a two-dimensional heat conduction model is needed to investigate the temperature evolution of the SHW when heating power is applied. This temperature history is related not only to the properties of the wire and the heating power, but also to the thermophysical properties of the surrounding samples. Therefore, the thermal conductivity and the thermal diffusivity of the sample can be evaluated from this temperature history.

In the present paper, two-dimensional heat-conduction equations for SHW are numerically analyzed by a finite difference method with an alternating direction implicit (ADI) method. The effects of the thermophysical properties as well as the dimensions of the hot wire, insulation coating, and the sample are investigated. A SHW apparatus and the measurement procedure are described. A curve fitting method by matching experimental data and numerical values is proposed to extract thermal conductivity and thermal diffusivity. 2. PRINCIPLE OF MEASUREMENTS 2.1. Physical Model and Numerical Analysis

The physical processing of the short hot wire is similar to that of normal hot wire technique [2]. According to the actually used experimental setup described in Section 3.1, the hot wire probe is immersed in an isotropic medium and initially kept at equilibrium with the medium. The probe is subjected at time t=0 to a step heating with a constant current generating a power, , per unit volume. The temporal evolution of the temperature of the wire depends on the thermophysical properties of probe and the tested sample. Figure 1 shows the physical model and the coordinate system. In this two-dimensional heat conduction system which is symmetric with respect to the z-axis, the governing equations are non-dimensionalized as follows.

vq

2

Fig. 1. Physical model of SHW method.

In hot wire,

1

12

2

2

2

1

11

d

chhh

d

h

RR

ZRRRRFo+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

=∂∂ θθθθ (1a)

In insulation layer,

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

=∂∂

2

2

2

2

2

11ZRRRRFo

iii

d

i θθθθ (1b)

In sample,

2

2

2

2 1ZRRRFo

ffff

∂∂

+∂∂

+∂∂

=∂∂ θθθθ

(1c)

where θ, R, and Z are the dimensionless temperature and radial and longitudinal coordinates, respectively. Fo is the Fourier number. These are defined as

f

hvrqTT

λ

θ 20−

= , hrrR = ,

hrzZ = , 2

h

f

rt

Foα

= (1d)

The parameters Rc1, Rd1, and those appear in the boundary conditions are the thermal conductivity and thermal diffusivity ratios of corresponding layer and are defined as

h

fdR

αα

=1 , h

fcR

λλ

=1 , i

fdR

αα

=2 , h

icR

λλ

=2 , i

fcR

λλ

=3 (1e)

3

100 101 102 1030

0.5

1

1.5

2

Fo

θ v Numerical simulated

0.6

0.9

1.2

1.5

1.8

Fo

θ v

Linear fitting

10024 480

Fig. 2. Dependence of non-dimensional temperature increase on Fourier number ((a) Calculated results; (b) Calculated and linear fitting results in the range of 24 – 480).

where λ and α are the thermal conductivity and the thermal diffusivity, respectively. Here in all these equations the subscripts h, i, and f denote short hot wire, insulation layer, and the tested sample, respectively.

These equations are numerically solved by a finite difference method with an ADI method, under relevant initial and boundary conditions including the continuity conditions for temperature and heat flux at the interfaces between the hot wire and the insulation coating and between the coating and the sample. 2.2. Linear Relation between the Temperature Rise and the Logarithmic Time

In the experiments, the volume-averaged temperature of SHW is measured. To correlate the calculated value to the experimental results, let us analyze the non-dimensional volume-averaged hot wire temperature rise Vθ as a function of the non-dimensional heating time . Here Fo Vθ is expressed as

∫ ∫= LV RdRdZ

L 010

2 θθ (2)

In Eqs. (1) and (2), the evolution of vθ is related to the dimensionless parameters presented in equation (1e). For a short hot wire with known thermophysical properties, vθ depends highly on the thermophysical properties of the tested sample. To elucidate this dependence, let us investigate the histories of a short hot wire vθ when testing different samples with relative high and low

thermal conductivity, respectively. Figure 2 shows the dependence of Vθ on . In this calculation, the hot wire was taken as a platinum wire with an aspect ratio L (l/2r

Foh) of 200 and the

insulation coating was taken as alumina layer with a dimensionless thickness of 0.04, respectively. The thermal conductivities of the hot wire (platinum), the coating (alumina), and the sample (distilled water), are taken as 71.4 w m-1 K-1, 36.0 w m-1 K-1, and 0.610 w m-1 K-1, respectively. The thermal diffusivities of the hot wire, the coating, and the sample, are taken as 25.2 mm2/s, 11.9

4

mm2/s, and 0.147 mm2/s, respectively. The non-dimensional parameters of equation (1) are calculated from these assumed parameters. In the Fourier number range of 24 to 480, which corresponds to the real experimental time from 0.1 s to 2 s, Vθ is linearly dependent on

(Fig. 2(b)). The result enables us to correlate the

)ln(Fo

Vθ and to a linear equation, equation (3), with the coefficients A and B determined by the least square method.

)ln(Fo

BFoAv += lnθ (3)

The measured temperature rise of a wire could also be approximated by a linear equation with coefficients a and b in the above time range as

btaT += ln (4)

Using the definition of θ and , Equation (3) can be dimensionalized as Fo

⎟⎟⎠

⎞⎜⎜⎝

⎛++= B

rA

rqtA

rqT

h

hvhvv 2

22

lnln αλλ

(5)

Comparing the corresponding coefficients of equations (4) and (5), the thermal conductivity and thermal diffusivity of a sample are expressed as

aA

lVI

aArq hv π

λ == (6a)

⎟⎠⎞

⎜⎝⎛ −=

AB

abrh exp2α (6b)

where V and I are the voltage and current supplied to the wire. Equations (6a) and (6b) are similar to those obtained for the conventional transient hot wire (THW) method. It is worth noting that the A and B are related to the dimensions of wire and coating and the thermophysical properties of wire, coating, and the sample, whereas they are constants in conventional THW with a large aspect ratio L. The detailed effects of aspect ratio L, insulation coating thickness δ, and the thermophysical properties of wire, coating, and the sample on the wire temperature increase have been investigated previously [9]. 2.3. Nonlinear Relation between ΘV and Foln

The linear relation between the wire’s temperature increases and the logarithmic time is valid when the heat capacity of the hot wire and coating and the heat loss from the wire to the terminals are negligible. It requires that the thermophysical properties of the test sample are in a proper range. In measurement of such a sample, the effect of heat capacity in the wire can be negligible when the Fourier number large enough. Furthermore, the effect of heat loss from the tip is not large enough to

5

change the exponential temperature rise. Fortunately, many of the liquids, like water, toluene, glycol etc., belong to these classes of materials. This is the key point that the SHW method is capable of measuring the thermal conductivity and thermal diffusivity of liquid simultaneously. However, some other materials with low thermal conductivity or thermal diffusivity, this linear relationship does not exist. It is impossible to determine their thermal conductivity and thermal diffusivity through the reported procedure. Figure 3 shows the dependence of the calculated dimensionless temperature increase vθ on the logarithmic Fourier number . In the calculations, the wire and the coating were taken the same as those for Fig. 2. For Fig.3, the thermal conductivity and the thermal diffusivity of the supposed sample are 0.05 Wm

Foln

-1K-1, and 0.25×10-4 m2s-1. Markedly different from Fig. 2(b), for this sample with low thermal conductivity, the calculated dimensionless temperature rise increases nonlinearly with the logarithmic Fourier number (Fig.3(b)). The previously used procedure to determine the thermal conductivity and thermal diffusivity by the slope and the intercept of the experimental line and of the calculated line could not be used. Therefore, to these kinds of samples, a curve fitting method by matching the experimental data and numerical simulated values should be used for determining the thermal conductivities and thermal diffusivities. This least square fitting is performed using a downhill simplex method and the thermal conductivity and the thermal diffusivity are set as free fitting parameters to minimize the square variance between the measured and the calculated temperature rise. As a measure of the reliability of the fitting, the square variance, S, of the fitting is calculated as

(7) (∑=

−=N

i

fi

mi TTS

1

2)

where and are the measured and fitted temperature rise, respectively. N is the experimental data number. A smaller variance represents higher quality of the fitting.

miT f

iT

100 101 102 1030

0.5

1

1.5

2

Fo

θ v

0.6

0.8

1

1.2

1.4

1.6

Fo

θ v

Numerical simulated

Linear fitting

10040 800

Fig. 3. Dependence of nondimensional temperature increase on Fourier number

((a) Calculated results; (b) Calculated and linear fitting results in the range of 40 – 800).

6

3. MEASUREMENT SYSTEMS AND PROCEDURES 3.1. Apparatus Setup

Fig. 4 describes the SHW cell used in these measurements. A short platinum (Pt) wire is welded at both ends to Pt lead wires of 1.5 mm in diameter. The length and the diameter of the wire are determined from calibration using distilled water and toluene as the standard samples. After calibration, the naked probe was coated with a thin alumina layer for insulation by using sputtering apparatus. The detailed setup of sputtering device and the coating process have been described in Ref. [13]. The thickness of the insulation coating is determined from calibration once again by using the same standard samples. 3.2. Electrical Systems

A block diagram of the measurement system is shown in Fig. 5. The system consists of a dc powder supply and voltage and current measuring and control systems, that is, digital multimeters (Keithley 2002), a personal computer (PC), and PI/O controllers. The heating and measuring are controlled by the PC through a GPIB board. The voltage changes of the probe, form which the wire’s temperature rise can be calculated, are collected and transferred to the PC. The recorded data were processed by a self-developed program.

(6)

(5)

(4) (3)

(2)

(1) (6)

(5)

(4) (3)

(2)

(1)

Fig.4. Schematic diagram of SHW cell ((1) hot wire; (2) thermocouple; (3) voltage leads;

(4) current leads; (5) Pt holder; (6) vessel).

7

Digitalmultimeter

Digitalmultimeter

PI/O controlswitch

Rdummy

R probe

R std

P.C.

D.C.Power supply

Digitalmultimeter

Digitalmultimeter

PI/O controlswitch

Rdummy

R probe

R std

P.C.

D.C.Power supply

Digitalmultimeter

Digitalmultimeter

PI/O controlswitch

Rdummy

R probe

R std

P.C.

D.C.Power supply

Digitalmultimeter

Digitalmultimeter

PI/O controlswitch

Rdummy

R probe

R std

P.C.

D.C.Power supply

Digitalmultimeter

Digitalmultimeter

PI/O controlswitch

Rdummy

R probe

R std

P.C.

D.C.Power supply

Digitalmultimeter

Digitalmultimeter

PI/O controlswitch

Rdummy

R probe

R std

P.C.

D.C.Power supply

Digitalmultimeter

Digitalmultimeter

PI/O controlswitch

Rdummy

R probe

R std

P.C.

D.C.Power supply

Digitalmultimeter

Digitalmultimeter

PI/O controlswitch

Rdummy

R probe

R std

P.C.

D.C.Power supply

Digitalmultimeter

Digitalmultimeter

PI/O controlswitch

Rdummy

R probe

R std

P.C.

D.C.Power supply

Digitalmultimeter

Digitalmultimeter

PI/O controlswitch

Rdummy

R probe

R std

Digitalmultimeter

Digitalmultimeter

PI/O controlswitch

Rdummy

R probe

R std

P.C.

D.C.Power supply

P.C.P.C.

D.C.Power supplyD.C.Power supply

Fig. 5. Schematic diagram of the measurement system.

3.3. Measurement Procedures 3.3.1 Calibration of hot wire.

The measurements of the radius and the length of the platinum wire may introduce some error, especially for a short wire. Therefore, and l are calibrated by using distilled water and toluene as standard substances. From equation (6), the following expressions are obtained.

hr l

hr

aAVIl

πλ= (8a)

⎟⎠⎞

⎜⎝⎛ −=

ab

ABrh expα (8b)

Fig. 6 shows an example of measured temperature increase using water as a standard sample. The reproducibility is in 0.5%. Using least square method, the measured data are fitted to a linear expression shown in equation 4, with the slope, a, and the intercept, b. To obtain the values of the length and radius of the hot wire, the following procedure is proposed as: (1) firstly, assume l and

, these values are used for numerical simulation; (2) the numerically simulated non-dimensional temperature rises as a function of logarithmic Fourier number are plotted like Fig. 2 and the slope A and intercept B are determined by the first order least square method in the period corresponding to the actual measurement time; (3) l and are calculated from Eqs. 8(a) and 8(b); (4) if the difference between the values of l and in step (3) and those in step (1) is larger than a preset value, the procedure should be repeated from step (1) to step (3) using the newly obtained values of l and . The iteration procedure is carried out till the difference between the values of l and in step (3) and those in step (1) is within an expected error. Usually a few time iteration is sufficient to obtain the converged values of l and .

hr

hr

ir

ir hr

hr

8

1

1.5

2

2.5

t / s

∆T

/ o C

First runSecond runThird run Linear fitting

0.1 1 2

Fig. 6. Measured temperature increase as a function of heating time.

0 0.1 0.2 0.3 0.4 0.5 0.60

0.5

1

1.5

t / s

∆T /

o C

Measured Best fitting 0.95λ, α 1.05λ, α

0 0.1 0.2 0.3 0.4 0.5 0.60

0.5

1

1.5

t / s

∆T

/ o C

Measured Best fitting

λ, 1.15α λ, 0.85α

(a) With λ variation. (b) With α variation. Fig. 7. Experimental and calculated temperature rises.

3.3.2 Determination of thermal conductivity and thermal diffusivity

To most of liquid materials like water, glycol, and oil, the linear relation between the wire’s temperature increases and the logarithmic time is obtained in experimental range (Fig. 2). The procedure for determination of thermal conductivity λ and thermal diffusivity α has been described before and this method has been successfully used to measure the thermal conductivity and thermal diffusivity of various fluids, polymer melts, and even molten carbonates [4-6].

For samples with temperature evolution shown in Fig.3, the previously used procedure to determine the thermal conductivity and thermal diffusivity by the slope and the intercept of the experimental line and of the calculated line could not be used. Therefore, a curve fitting method by matching the experimental data and numerical simulated values should be used for determining the thermal conductivities and thermal diffusivities. Section 4.1 presents an example of these measurements for determining the thermal conductivity and diffusivity of gases, like air.

9

4. SAMPLE MEASUREMENTS AND DISCUSSION 4.1. Measurements of the Thermal Conductivity and Thermal Diffusivity of Air Here we use air as testing sample to illustrate the capability of SHW in determining the thermal conductivity and thermal diffusivity of gases. Fig.7 depicts the experimental Pt wire’s temperature rise values with the corresponding best-fitting curve for air at 15 oC and under atmospheric pressure. The best estimates are: 0242.0=λ W m-1 K-1 and m410205.0 −×=α 2 s-1, with a minimum variance. In Section 4.2, we discussed the effect of natural convection. Under the present experimental conditions, no natural convection took place. Experiment and fitting are clearly in excellent agreement at the time range from initial time to 1 s. Fig. 7(a) illustrates that the calculated values are very different from the best fitted or the experiment data if we used a thermal conductivity of 5% different from the best fitted thermal conductivity value in the calculations. The calculation results are not so sensitive to thermal diffusivity value as to thermal conductivity value. However, as shown in Fig. 7(b), the difference between the calculated values and the best fitted or experimental data is clearly seen if a thermal diffusivity value of 15% different from the best fitted thermal diffusivity was used in the calculation. The consistence of the experiments and the numerical simulation undoubtedly demonstrated that the genuine values of the conductivity and diffusivity are the same to the estimates. We are currently using this method to measure the thermal conductivity and diffusivity of packed carbon nanofibers (PCNFs). The preliminary results show that SHW measurement combined with the curve fitting method is capable of determining the thermal conductivity and thermal diffusivity of packed powder like PCNFs. 4.2. Convection Consideration

Comparing to conventional hot wire method, in the measurement of gas with low viscosity, the effect of natural convection becomes of particular important because the onset time of this effect is further reduced as the aspect ratio decreases. The transient natural convection for a long vertical wire with large aspect ratio has been studied by Goldstein and Briggs [14], Pantaloni et al. [15], and Ro et al. [16]. Regarding to the critical Fourier number for SHW, Zhang et al. [17] experimentally obtained an expression as

Foc

332*

9.2 LRalFoc

−= (9a)

RalNuRal =* (9b)

PrPr 2

3

νβ TlgGrRal ∆

== (9c)

λhlNu = (9d)

where is modified Rayleigh number, L the aspect ratio, Rayleigh number, Nusselt number,

*Ral Ral NuPr Prantel number, g the gravitational acceleration, β volumetric expansion

coefficient, T∆ the temperature rise, the length of hot wire, υ the dynamic viscosity, λ the thermal conductivity, and the heat transfer coefficient. From the definition of Fourier number,

lh

10

the onset time of natural convection is expressed as ct

αFocr

t hc

2

= (10)

For the measurements discussed in Section 4.1, the critical time is estimated to be 2.8 s. Because the temperature rise is only about 1.1 oC (Fig. 7), the discrepancy of temperature rise caused by natural convection is not obvious till 3 s after heating. To clarify this discrepancy, we applied a larger heating current to the wire. As shown in Fig. 8, the largest temperature increase is about 4.85 oC and the critical time is estimated to be 1 s from Eq. 10. Actually the onset time of natural convection is experimentally determined as about 1.2 s (range C marked in Fig. 8). Therefore, the effect of natural convection is negligible in a measurement mentioned in Section 4.1. 4.3. Estimation of Accuracy To the measurement with linear relation of the temperature rise to the logarithmic time, the relative errors of the thermal conductivity and thermal diffusivity have been evaluated elsewhere [13]. To the measurement with nonlinear relation between the temperature rise and the logarithmic time, the thermal conductivity and thermal diffusivity are evaluated by best-fitting the experimental data with calculated wire’s temperature rises with assumed thermal conductivity and thermal diffusivity. The errors of this method come from measurement as well as calculation and fitting. The relative errors of the thermal conductivity and thermal diffusivity are estimated as

( )21

2222

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛= λδδδδ

λδλ f

ll

II

VV (11a)

( )21

22

2⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛= αδ

δαδα f

rr

i

i (11b)

where λδf and αδf are fitting errors for λ and α , respectively. and are determined l ir

Fig. 8. Dependence of temperature increase on heating time.

11

from calibration using water and toluene with known thermophysical properties and their relative errors are estimated from Eqs. (8a) and (8b) as

212222

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

aa

AA

II

VV

ll δδδδδ (12a)

2122

21

21

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛=

ab

AB

rr

i

i δδδ

(12b)

Substituting Eqs. (12a) and (12b) into Eqs. (11a) and (11b), we have

( )21

22222

22⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛= λδδδδδ

λδλ f

aa

AA

II

VV (13a)

( )21

222

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛= αδδδ

αδα f

ab

AB (13b)

The best-fitting errors of 1.0 and 5.0 % are for δfλ and δfα, respectively. From Eqs. 13a and 13b, the total errors of the measurement are estimated to be 2 for thermal conductivity and 7% for thermal diffusivity, respectively. 5. CONCLUSIONS

A transient short hot wire (SHW) method for simultaneous studies of the thermal conductivity and thermal diffusivity was described. Two-dimensional heat-conduction equations together with appropriate initial and boundary conditions for SHW were numerically analyzed by a finite difference method with an alternating direction implicit (ADI) method. There are two kinds of temperature evolution tendencies corresponding to the different testing samples. For the measurements with proportional relation between temperature rise and logarithmic heating time interval, the thermal conductivity and thermal diffusivity are obtained from the slope and the intercept of the measured temperature rises and those of calculated non-dimensional temperature rise. For the measurements with nonlinear relation between temperature rise and logarithmic heating time interval, the thermal conductivity and thermal diffusivity are extracted from a curve fitting method by matching the experimental data and numerical values. In SHW apparatus, a metal wire serves both as a heating unit and as an electrical resistance thermometer and the wire was calibrated using water and toluene with known thermophysical properties. This method was successfully applied here using air as a testing sample. The effect of natural convection is negligible in the experimental heating time range and the accuracy of this measurement is estimated to 2% for thermal conductivity and 7% for thermal diffusivity.

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