Measurement of the Thermophysical Properties of Anisotropic Insulation Materials with Consideration of the Eﬀect of Thermal Contact Resistance

materials

Article Measurement of the Thermophysical Properties of Anisotropic Insulation Materials with Consideration of the Eﬀect of Thermal Contact Resistance

Dongxu Han, Kai Yue * , Liang Cheng, Xuri Yang and Xinxin Zhang

School of Energy and Environmental Engineering, University of Science and Technology Beijing, Beijing 100083, China; [email protected] (D.H.); clﬁ[email protected] (L.C.); [email protected] (X.Y.); [email protected] (X.Z.) * Correspondence: [email protected]

Received: 13 January 2020; Accepted: 13 March 2020; Published: 17 March 2020

Abstract: A novel method involving the effect of thermal contact resistance (TCR) was proposed using a plane heat source smaller than the measured samples for improving measurement accuracy of the simultaneous determination of in-plane and cross-plane thermal conductivities and the volumetric heat capacity of anisotropic materials. The heat transfer during the measurement process was mathematically modeled in a 3D Cartesian coordinate system. The temperature distribution inside the sample was analytically derived by applying Laplace transform and the variables separation method. A multiparameter estimation algorithm was developed on the basis of the sensitivity analysis of the parameters to simultaneously estimate the measured parameters. The correctness of the algorithm was verified by performing simulation experiments. The thermophysical parameters of insulating materials were experimentally measured using the proposed method at different temperatures and pressures. Fiber glass and ceramic insulation materials were tested at room temperature. The measured results showed that the relative error was 1.6% less than the standard value and proved the accuracy of the proposed method. The TCRs measured at different pressures were compared with those obtained using the steady-state method, and the maximum deviation was 8.5%. The thermal conductivity obtained with the contact thermal resistance was smaller than that without the thermal resistance. The measurement results for the anisotropic silica aerogels at different temperatures and pressures revealed that the thermal conductivity and thermal contact conductance increased as temperature and pressure increased.

Keywords: anisotropic material; cross-plane thermal conductivity; in-plane thermal conductivity; small-plane heat source; thermal contact resistance

1. Introduction Anisotropic insulating materials are vastly applied in the construction industry and other related fields owing to the excellent thermal insulating properties. Accurate thermophysical parameters are imperatively needed for the effective practical application of these materials. However, the thermal contact resistance (TCR) in thermal property measurement systems is unavoidable in many cases and will affect the measurement accuracy of the thermophysical parameters of anisotropic insulation materials. In a thermal property measurement system, TCR usually exists at interfaces between a sensor or a heating source element and the measured sample or between two samples because of the incomplete contact between two parts. The TCR can be characterized by the ratio of the temperature difference to the average heat flux density. Studies have shown that TCR is a nonlinear problem affected by multiple

Materials 2020, 13, 1353; doi:10.3390/ma13061353 www.mdpi.com/journal/materials Materials 2020, 13, 1353 2 of 12 factors such as the roughness and topography of contact surfaces [1–3], external pressure loading [4–6], properties of intermediate media, and time-dependent temperature condition. Some theoretical models have been proposed by statistically analyzing the morphological characteristics of contact surfaces [7–9], temperature distributions [10], and deformation characteristics at interfaces. In practice, the accurate value of TCR is difficult to obtain through only theoretical analysis, and experimental methods are widely used to measure TCR [11–14]. According to the definition of TCR, the steady-state method was developed for TCR measurement by determining the temperatures on the contact interface and applied heat flux [15–17]. For transient methods, TCR is known by measuring the changes in correlated parameters under an applied thermal disturbance [18–20]. For instance, the intrinsic relationship between TCR and contact resistance is employed to determine TCR by measuring contact resistance [21]. The laser flash method has been proposed to measure the specific heat and TCR of a two-layer material [22]. Some studies have attempted to improve the accuracy of measuring the thermophysical parameters by establishing a modified measurement model involving the TCR for certain specific methods. The degree of influence of TCR on the accuracy of measuring the thermophysical parameters of materials depends on the method itself. These methods are divided into steady-state and transient methods. The TCR of ethylene tetrafluoroethylene sheets, Nafion membranes, and gas diffusion layers is measured using the hot-plate method with a more detailed consideration of TCR [23].The deviation of the measured results of ethylene tetrafluoroethylene sheets reaches 66.7% in comparison with the results measured using the traditional transient plane source method. Xu et al. [24] applied the effective medium theory to study the influence of thermal resistance at the interface between an inclusion and a substrate on the thermal conductivity of concrete. They showed that interfacial thermal resistance plays an important role in predicting the effective thermal conductivity of cement materials with coarse inclusions. Cheng et al. [25] provided a new thermal probe method by considering the thermal capacity of a thermal probe and TCR between the probe and the sample to improve the accuracy of measuring the thermal conductivity and volumetric heat capacity. Several liquid and solid samples have been measured, and the results show that TCR greatly affects the thermal conductivity of solid samples. In the measurement of the thermal diffusivity of stainless steel 304, oxygen-free copper, and aluminum nitride ceramic using the laser photothermal method, the change in TCR with temperature is measured and an empirical formula for the TCR-temperature relationship is established [26]. In our previous work, we proposed a small-plane heat source method for the simultaneous measurement of the in-plane and cross-plane thermal conductivities of anisotropic materials [27]. Given that TCR is at a contact interface between a heat source element and a sample, this work was performed to investigate a novel method in which the effect of TCR is considered to enhance the accuracy of the simultaneous measurement of thermal conductivities and volumetric heat capacity. The TCR between the heat source element and the sample in the measurement system was also determined at different pressures and temperatures.

2. Methods and Veriﬁcation

2.1. Physical Mathematical Model The measurement model with consideration of the eﬀect of the TCR between the heating element and measured sample was established on the basis of our previously proposed small-plane heat source method, and the schematic is shown in Figure1. As shown in Figure1a, a plane source element with a side length of 2a was located between two samples with the same side length of 2b, i.e., samples I and II, and used as a heat source. The temperature increase at the measuring points was determined using a thermocouple with ﬁne wires (d = 0.05 mm). Figure1b shows the computational domain in the model and three spatial dimensions. Materials 2020, 13, 1353 3 of 12 Materials 2020, 13, x FOR PEER REVIEW 3 of 14

(a) (b)

FigureFigure 1. 1. SchematicSchematic diagram: diagram: (a (a) )experimental experimental system; system; ( (bb)) the the modeled modeled sample sample and and location location of of thermal thermal contactcontact resistance resistance (TCR). (TCR).

AA three-dimensional mathematicalmathematical model model was was constructed constructed for describingfor describing the heat the transferheat transfer during duringthe measurement the measurement process process when thewhen heat the capacity heat capa andcity thickness and thickness of the of heat the sourceheat source element element were wereneglected. neglected. The governingThe governing equation equation and theand boundary the boundary condition condition were were expressed expressed as follows. as follows.

∂T(x, y, z,∂tT)(x, y, z∂, t2)T(x, y, z∂,2tT)(x, y, z∂, t2)T(x, y∂, z2T,t()x, y,∂z,2Tt)(x,∂y2,Tz(,xt,)y , z, t) ρc ρc = λ = λx + λ + λy λ λz (1) (1) ∂t ∂tx ∂x2 ∂x2y ∂y2 ∂y2z ∂z2 ∂z2 wherewhere ρ and c are thethe densitydensity andand specificspecific heat heat capacity capacity of of the the sample; sample;T (Tx(,xy, ,yz,, zt,) t is) is the the temperature temperature of λ λ λ ofthe the sample sample at diatff differenterent times; times; and andx, λy,x, and λy, andz are λ thez are thermal the thermal conductivities conductivities of the measuredof the measured sample samplein the x ,iny ,the and x,z ydirections,, and z directions, respectively. respectively. GivenGiven thatthat thethe increase increase in thein temperaturethe temperature of the of sample the sample during measurementduring measurement did not remarkably did not remarkablyincrease the increase thermal conductivity,the thermal conductivity, the specific heat the capacity specific and heat density capacity of theand sample density were of the considered sample wereconstant considered with respect constant to temperature. with respect The to temperature. dimensions of The the dimensions sample were of supposed the sample to bewere large supposed enough toto be satisfy large the enough requirements to satisfy for the the requirements assumption offor semi-definite the assumption medium. of semi-definite Therefore, medium. the upper Therefore, boundary theplane upperz = boundaryb was set asplane the z isothermal = b was set condition as the isothermal at room temperaturecondition at roomT0. Given temperature the symmetry T0. Given of thethe symmetry measurement of the model, measurement the left boundary model, the plane left boundaryx = 0 and plane front boundaryx = 0 and front plane boundary y = 0 were plane set asy =adiabatic 0 were set boundary as adiabatic conditions. boundary The temperaturesconditions. The of thetemperatures right boundary of the plane rightx =boundaryb and back plane boundary x = b andplane backy = boundaryb were not plane affected y = b by were the not heating affected element by the during heating the element experiment during under the experiment the control under of the theheating control power. of the Then, heating the power. right boundary Then, the plane rightx boundary= b and back plane boundary x = b and plane backy boundary= b were alsoplane set y as= bthe were adiabatic also set boundary as the adiabatic conditions. boundary The boundaryconditions. condition The boundary of the condition lower boundary of the lower plane boundaryz = 0 was planeexpressed z = 0 aswas follows: expressed as follows:

∂ ∂T(x, y, 0, t) q = hc(Th T) z = 0 =T (λxz, y,0,t) (2) q = h (T − T )− = = −λ − ∂z (2) c h z 0 z ∂z where q is half of the heat flux density; hc is the thermal contact conductance (TCC) between the whereheating q elementis half of and the the heat sample flux density; which is h thec is reciprocalthe thermal of contact the TCR; conductance and T is the (TCC) temperature between at the the heatingmeasuring element point. and the sample which is the reciprocal of the TCR; and T is the temperature at the measuring point. The initial condition was set as room temperature T0. The sample size should be sufficiently largerThe than initial the penetrationcondition was depth set ofas heatroom flux temperature to ensure that T0. theThe heat sample flux hassizeno should thermal be esufficientlyffect on the largerz = b condition than the penetration plane of the depth semidefinite of heat medium.flux to ensure that the heat flux has no thermal effect on the z = b condition plane of the semidefinite medium.

2.2. Analytical Solution

First, the governing equation and boundary conditions were transformed with the Laplace transform L(ΔT (x, y, z,t)) = θ (x, y, z, p) and could be written as follows:

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2.2. Analytical Solution First, the governing equation and boundary conditions were transformed with the Laplace transform L(∆T(x, y, z, t)) = θ(x, y, z, p) and could be written as follows:

∂2θ(x, y, z, p) ∂2θ(x, y, z, p) ∂2θ(x, y, z, p) ρcpθ(x, y, z, p) = λx + λy + λz (3) ∂x2 ∂y2 ∂z2

q ∂θ(x, y, 0, p) = hc(θ θ) z = = λz (4) p h − | 0 − ∂z where p is the Laplace parameter of t; and θ is the Laplace transform of the temperature increase. θ(x, y, z, p) could be transformed into X(x, p)Y(y, p)Z(z, p) by applying the variable separation method. Then, the temperature increase was analytically obtained as follows:

X∞ X∞ θ(x, y, z, p) = Bmn cos(αmx) cos(δn y) exp( Amnz) (5) − m = 1 n = 1

The Amn and Bmn were calculated in terms of the boundary conditions as follows: s 2 2 ρcp + λxαm + λyδn Amn = (6) λz p 4φ0 λzρcp h sin(αmα) sin(δnα) Bmn = − (7) p 2 ph λzρcpAmnb αm δn The inverse Laplace transform was applied to obtain the analytical solution of the temperature increase in the time domain which is derived as:

P∞ P∞ 4φ0 sin(αma) sin(δna) 1 ∆T(x.y.z.t) = cos(αmx) cos(δn y) q b2 √h αm δn 2 2 m = 1 n = 1 axαm+ayδn q  q q ρcp q   ρcp z λ   exp( z a α2 + a α2 )er f c( z a α2 + a α2 √t)   λz x m y n √ x m y n  1  − q2 t − −   ρcp  √λzρcp  q q z q   ρcp 2 2 λz 2 2   exp(z axα + ayα )er f c( + axα + ayα √t)  λz m n 2 √t m n + P∞ 2φ0 sin(αma) ( ) 2 α cos αmx = b √h m m 1 q  q q ρcp q   ρcp z λ   exp( z a α2 )er f c( z a α2 √t)   λz x m √ x m  1  − q2 t − −  (8) 2  ρcp  √λzρcp √axα  q q z q  m  ρcp 2 λz 2   exp(z axα )er f c( + axα √t)  λz m 2 √t m + P∞ 2φ0 sin(δna) ( ) 2 δ cos δn y = b √h n m 1 q  q q ρcp q   ρcp z λ   exp( z a α2 )er f c( z a α2 √t)   λz y n √ y n  1  − q2 t − −  q  ρcp  λ ρc a δ2  q q z q  √ z p y n  ρcp 2 λz 2   exp(z ayα )er f c( + ayα √t)  λz n 2 √t n 2 q 2 q q 2a φ z ρcp ρcp ρcp + 0 4t ( ) z er f c(z ) 2 π exp 4tλ λ 4tλ b √λzhρcp − z − z z where αx and αy are the thermal diﬀusion coeﬃcient in the x-direction and y-direction, respectively; mπ nπ αm = b (m = 1, 2, 3 ...); and δn = b (n = 1, 2, 3 ...). The thermal conductivities and volumetric heat capacity of the sample and the TCC could be accurately estimated using Equation (8) and appropriate parameter estimation methods. The analytical Materials 2020, 13, 1353 5 of 12

solution for isotropic materials could be obtained by setting λx = λy = λz into Equation (8). Given the reaction time of the thermocouple, the temperature measured at time t should be the one obtained at time t t . Substituting t = t t into Equation (8), the analytical solution with the reaction time for − d − d the temperature of anisotropic materials could be achieved. Herein, the reaction time should satisfy 0 t 0.005t, and t is the total test time [28]. ≤ d ≤ 2.3.Materials Comparison 2020, 13, x of FOR Analytical PEER REVIEW and Numerical Solutions 6 of 14

FLUENTThe numerical 16.0. The simulationnumerical simulation of the heat results transfer of of the this temperature measurement increase model at was the performed measuring using point FLUENT(0,0,0.005) 16.0. for Theisotropic numerical and simulationanisotropic resultsmaterial ofs the were temperature compared increase with those at the obtained measuring from point the (0,0,0.005)analytical for solution. isotropic Table and 1 anisotropic shows the materials thermal–ph wereysical compared properties with used those for obtained the calculation. from the analytical solution. Table1 shows the thermal–physical properties used for the calculation. Table 1. Thermal–physical properties used in numerical simulations. Table 1. Thermal–physical properties used in numerical simulations. Parameters Isotropic Material Anisotropic Materials Parameters Isotropic Material Anisotropic Materials λz (W·m−1·K−1) 0.4 0.2 1 1 λx = λy λ(W·mz (W m−1·K− −K1)− ) 0.40.4 0.2 0.4 · · 1 1 λx = λy (W m K ) 0.4 0.4 q (W·m−2) · − · − 80 80 q (W m 2) 80 80 b (m) · − 0.05 0.05 b (m) 0.05 0.05 ρcp (J·m−3·k−1)3 1 1599750 1500000 ρcp (J m k ) 1,599,750 1,500,000 · − · − a (m)a (m) 0.02 0.02 0.02 0.02 t (s) t (s) 200 200 200 200 2 1 hc (W−2 m−1 K ) 150 150 hc (W·m ·K· −) · − 150 150 Coordinates (x,y,z) (m) (0,0,0.005) (0,0,0.005) Coordinates (x,y,z) (m) (0,0,0.005) (0,0,0.005)

FigureFigure2 2shows shows the the comparative comparative results results between between the the numerical numerical and and analytical analytical solutions solutions for for isotropicisotropic (see(see FigureFigure2 a)2a) and and anisotropic anisotropic (see (see Figure Figure2b) 2b) materials materials at at the the measuring measuring point point (0,0,0.005). (0,0,0.005). TheThe maximum maximum deviation deviation in thein temperaturethe temperature increase increa forse the for two the materials two materials was less thanwas 1.6%,less than indicating 1.6%, theindicating correctness the correctness of the analytical of the solution analytical in thissolution study. in this study.

(a) (b)

FigureFigure 2. ComparisonComparison between between the numerical the numerical and analytical and analytical solutions: solutions: (a) isotropic (a) and isotropic (b) anisotropic and (b) anisotropicmaterials. materials.

2.4.2.4. SimultaneousSimultaneous EstimationEstimation ofof ThermophysicalThermophysical ParametersParameters Firstly,Firstly, thethe sensitivitysensitivity analysisanalysis waswas carriedcarried outout andand thethe sensitivitysensitivity coecoefficientsfficients werewere calculatedcalculated asas followsfollows to to relate relate the the magnitude magnitude of of the the thermal thermal parameters parameters to to the the change change in in the the temperature temperature increase increase at theat the measuring measuring point. point. The sensitivityThe sensitivity coeffi cientcoeffici forent the for cross-plane the cross-plane thermal conductivitythermal conductivity was obtained was + ∂T(ti,α) + by X = αjXj(ti, α) = αj , where X represents the sensitivity coefficient of an estimated ij + ∂αj ∂T(t ,ijα) + obtained by X = α X (t ,α) = α i , where X represents the sensitivity coefficient of ij j j i j ∂α ij j α λ λ ρ α an estimated parameter j (i.e., xy , or z , or C p , or hc ) to the temperature T(ti , ) . These coefficients were also used in the inverse problem solving to identify the parameters. Using the values listed in Table 1, the results of sensitivity coefficients calculation for the estimated thermophysical parameters and TCC are shown in Figure 3.

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parameter αj (i.e., λxy, or λz, or ρCp, or hc) to the temperature T(ti, α). These coeﬃcients were also used in the inverse problem solving to identify the parameters. Using the values listed in Table1, the results of sensitivity coeﬃcients calculation for the estimated thermophysical parameters and TCC are shown Materialsin Figure 20203., 13, x FOR PEER REVIEW 7 of 14

0.4

0.2

0.0

-0.2

T(K) -0.4 + X (ρcp) X+(λ ) -0.6 xy X+(h ) -0.8 c X+(λ ) -1.0 z 0 40 80 120 160 200

t(s) FigureFigure 3. 3. ResultsResults of of the the parameter parameter sensitivity sensitivity analysis. analysis.

AsAs can can be be seen seen in in Figure Figure 33,, the curves of the se sensitivitynsitivity coecoefficientsfficients of of four four estimated estimated parameters parameters were all nonlinear and uncorrelated to each other. It It was was indicated indicated that that the the sensitivity sensitivity coefficients coefficients of of thethe estimated estimated parameters parameters were were linea linearlyrly independent independent of one one another, another, and there was no no set set of of correlated correlated parameters. According According to to the the basic basic principle principle of of parameter parameter estimation estimation [29], [29], the the thermal thermal conductivities conductivities λxy and λλz, volumetric heat capacity, and TCC could be estimated simultaneously. xy and z , volumetric heat capacity, and TCC could be estimated simultaneously. Then, an improved Gauss–Newton algorithm was developed to simultaneously obtain four Then, an improved Gauss–Newton algorithm was developed to simultaneously obtain four estimated parameters [30] by the least squares analysis in which the sum of the square errors between estimated parameters [30] by the least squares analysis in which the sum of the square errors between the calculated and experimental results of the temperature increase of the measuring point in the sample the calculated and experimental results of the temperature increase of the measuring point in the was minimized. A series of simulation experiments were performed to verify the correctness of the sample was minimized. A series of simulation experiments were performed to verify the correctness method for estimating multiple parameters and the stability of the estimation results. The temperature of the method for estimating multiple parameters and the stability of the estimation results. The signal obtained by the analytical solution was added with different random noises to be used as the temperature signal obtained by the analytical solution was added with different random noises to be experimental measurement data. The signal can be expressed as ∆T = ∆T [1 + (εr and ε/2)], where used as the experimental measurement data. The signal 0can be expressed− as ε is the value of random error,∆T0 is the temperature increase obtained by the analytical solution at the Δ = Δ + ε −ε ε Δ measuringT T0[1 point,( rand and rand/is2)] used , where to generate is the random value matrices.of random In error, the simulationT0 is the experiments, temperature the increasein-plane andobtained cross-plane by the thermal analytical conductivities solution at ofthe the measuring anisotropic point, materials and wererand setis used to 0.4 to W generatem 1 K 1 · − · − randomand 0.2 Wmatrices.m 1 K In1, the and simulation the volumetric experiments, heat capacity the in-plane was set and to 1,500,000cross-plane J m thermal3 K 1. conductivities The values of · − · − · − · − ofother the parametersanisotropic werematerials the same were as set those to 0.4 listed W·m in− Table1·K−1 1and for 0.2 the W·m isotropic−1·K−1, materials. and the volumetric Figure4 shows heat capacitythe comparison was set ofto the1,500,000 increase J·m in−3 temperature·K−1. The values between of other the simulatedparameters and were estimated the same results as those as well listed as inthe Table residual 1 for errors the isotropic for isotropic materials. and anisotropic Figure 4 shows materials. the comparison of the increase in temperature betweenIt was the seen simulated that the estimatedand estimate andd simulatedresults as results well ofas thethe increase residual in errors temperature for isotropic agreed well,and anisotropicand the residual materials. errors were small. All the estimated errors of the estimated parameters of the isotropic and anisotropic materials were less than 2.8% at different random errors. This result verified the feasibility and effectivity of the developed method for the simultaneous estimation of the in-plane and cross-plane thermal conductivities, volumetric heat capacity, and TCC based on the proposed small-plane heat source method with TCR.

Materials 2020, 13, 1353 7 of 12 Materials 2020, 13, x FOR PEER REVIEW 8 of 14

(a) (b)

Figure 4. Comparison of the increase in temperature be betweentween the simulated and estimated results: ( a)) isotropic material, εε= =0.03; 0.03; (b ()b isotropic) isotropic material, material,ε =ε0.05;= 0.05; (c) ( anisotropicc) anisotropic material, material,ε = ε0.03;= 0.03; and and (d) anisotropic(d) anisotropic material, material,ε = 0.05.ε = 0.05.

3. ExperimentalIt was seen that Measurement the estimated and simulated results of the increase in temperature agreed well, and the residual errors were small. All the estimated errors of the estimated parameters of the 3.1. Experimental System isotropic and anisotropic materials were less than 2.8% at different random errors. This result verified the feasibilityFigure5 shows and effectivity the measurement of the developed system establishedmethod for accordingthe simultaneous to the proposed estimation method. of the in-plane As can beand seen, cross-plane the system thermal mainly conductivities, included a power volumetric supply, heat a computer-basedcapacity, and TCC data based acquisition on the system,proposed a pressuresmall-plane loading heat source device method (see Figure with5 a),TCR. a Cr20Ni80 alloy heating element with a size of 40 mm × 40 mm 0.1 mm, and a resistance of 10.2 Ω (see Figure5b), and the tested samples consisting of × three3. Experimental squares of Measurement the same size but different thicknesses (see Figure5c, i.e., samples A, B, C, and D). The constant-temperature muffle furnace shown in Figure5d is a box-type resistance furnace with an3.1. accuracy Experimental of 1SystemC (SRJX-8-13, Tianjin, China) and was used for the experimental measurements ± ◦ performedFigure at5 shows different the high measurement temperatures. system The establis powerhed supply according (PA36-2A, to the KENWOOD, proposed method. Akaho, As Japan) can providedbe seen, the a constant system currentmainly output included for heating.a power Thesupp multi-channelly, a computer-based data acquisition data acquisition instrument system, (MX-100, a YOKOGA,pressure loading Tokyo, device Japan) (see was Figure used to 5a), record a Cr20Ni80 the temperature alloy heating rise at element the measuring with a pointsize of of 40 the mm sample × 40 whichmm × 0.1 was mm, detected and a resistance by a K-type of 10.2 thermocouple Ω (see Figure with 5b), a diameter and the tested of 0.05 samples mm (OMEGA, consisting Norwalk, of three CT,squares USA). of the same size but different thicknesses (see Figure 5c, i.e., samples A, B, C, and D). The constant-temperature muffle furnace shown in Figure 5d is a box-type resistance furnace with an accuracy of ±1 °C (SRJX-8-13, Tianjin, China) and was used for the experimental measurements performed at different high temperatures. The power supply (PA36-2A, KENWOOD, Akaho, Japan) provided a constant current output for heating. The multi-channel data acquisition instrument (MX- 100, YOKOGA,Tokyo, Japan) was used to record the temperature rise at the measuring point of the sample which was detected by a K-type thermocouple with a diameter of 0.05 mm (OMEGA,

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Materials 2020, 13, 1353 8 of 12 Norwalk, CT, USA).

FigureFigure 5.5. SchematicSchematic ofof thethe measurementmeasurement system.system. (a)) experimentalexperimental system;system; ((b) heatheat sourcesource plane;plane; ((cc)) samples;samples; ((dd)) highhigh temperaturetemperature furnace.furnace. 3.2. Experimental Verification 3.2. Experimental Verification First, the thermophysical parameters of two kinds of standard reference materials (SRMs), First, the thermophysical parameters of two kinds of standard reference materials (SRMs), including a glass fiber insulating material (Sample A) and ceramic material (Sample B) produced by the including a glass fiber insulating material (Sample A) and ceramic material (Sample B) produced by Institute of Standardization of National Building Materials (China), were measured using the proposed the Institute of Standardization of National Building Materials (China), were measured using the method with consideration of the effect of TCR to validate its feasibility and accuracy. The standard proposed method with consideration of the effect of TCR to validate its feasibility and accuracy. The value of cross-plane thermal conductivity of sample A was 0.0331 0.0002 W m 1 K 1, which was standard value of cross-plane thermal conductivity of sample A was± 0.0331 ± 0.0002· − W·m· − −1·K−1, which obtained by the guarded hot-plate method at 28 C. The standard value of the cross-plane thermal was obtained by the guarded hot-plate method◦ at 28 °C. The standard value of the cross-plane conductivity of Sample B was 0.052 0.0002 W m 1 K 1 as obtained by the heat flow meter. Table2 thermal conductivity of Sample B was± 0.052 ± 0.0002· − · W·m− −1·K−1 as obtained by the heat flow meter. shows the experimental results of the Samples A and B. Table 2 shows the experimental results of the Samples A and B.

Table 2. Results of verification experiments. Table 2. Results of verification experiments. λ SD λ SD h SD ρC SD Sample z λ xy± P λ ± SD±1 1 ±1SD1 hSD±±2 1 ρCSD3± ±1 Sample (Wz m− K− ) (Wxym− K− ) (W m− K− ) (J m− PK− ) · · · · · ·−2 −1 · · A(W·m 0.0334−1·K−0.00011) 0.0452(W·m−1·K0.0004−1) 173.0(W·m1.665·K ) 1.595(J·m0.0011−3·K−1) ± ± ± ± A B 0.0334 0.0528 ± 0.00010.0012 0.0452 0.156 ±0.00473 0.0004 122.9 173.0 ±1.967 1.665 3.033 1.595 0.0484± 0.0011 ± ± ± ± B 0.0528 ± 0.0012 0.156 ± 0.00473 122.9 ± 1.967 3.033 ± 0.0484 As can be seen, the relative measurement errors of Samples A and B were 0.91% and 1.53% in comparisonAs can be withseen, thethe standardrelative measurement values, respectively. errors of Thus, Samples it was A and concluded B were that0.91% the and established 1.53% in measurementcomparison with system the could standard be used values, to measure respective thely. thermal Thus, properties it was concluded with good that accuracy. the established measurementThen, the system TCR of could the insulation be used tileto measure material the (Sample thermal C, properties ceramic insulation with good tiles) accuracy. were measured usingThen, the traditional the TCR of steady-state the insulation method tile material [17] and (Sample the proposed C, ceramic method insulation at diff tiles)erent were pressures measured and temperatures.using the traditional The measured steady-state TCCs method are shown [17] in an Tabled the3 .proposed method at different pressures and temperatures.The TCCs The obtained measured by the TCCs proposed are shown method in increasedTable 3. as the temperature and pressure increased. This variationThe TCCs in obtained the TCC wasby the rational proposed and in method agreement increased with that as measuredthe temperature with the and steady-state pressure method.increased. The This values variation of the in TCCsthe TCC measured was rational with theand proposedin agreement method with were that smallermeasured than with those the obtainedsteady-state by themethod. steady-state The values method, of the and TCCs the measured deviation with was the within proposed 8.5%. method This phenomenon were smaller might than bethose due obtained to the di ffibyculty the steady-state in completely method, satisfying and the the requirement deviation was of 1Dwithin heat 8.5%. transfer This in phenomenon the practical might be due to the difficulty in completely satisfying the requirement of 1D heat transfer in the

Materials 2020, 13, 1353 9 of 12 measurement system for the steady-state method, and heat was loss through the thermocouples and through radiation and convection.

Table 3. Experimental results of thermal contact conductivity using the proposed methods and the steady state.

Thermal Contact Conductivity (W m 2 K 1) SD · − · − ± T(◦C) P (kPa) Proposed Method Steady-State Method Deviation (%) 2 94.1 0.35 102.1 0.45 8.5 ± ± 200 5 98.5 0.42 105.8 0.36 7.4 ± ± 8 102.4 0.25 110.7 0.21 8.1 ± ± 2 97.3 0.46 104.4 0.71 7.3 ± ± 300 5 101.9 0.42 108.8 0.43 6.8 ± ± 8 104.8 0.38 112.7 0.58 7.5 ± ± 2 99.2 0.57 105.5 0.65 6.4 ± ± 400 5 105.7 0.55 112.9 0.54 6.8 ± ± 8 108.9 0.43 116.6 0.32 7.1 ± ±

3.3. Experimental Measurement at High Temperatures The thermophysical parameters and the TCCs of an anisotropic high-temperature-resistant material (Sample D, silica aerogel material) with a large difference in its in-plane and cross-plane thermal conductivities were experimentally measured at room and high temperatures by using the proposed method. For comparison, the same thermophysical parameters of the same sample were measured using the small-plane heat source method without the effect of TCR which was reported in our previous work [27]. The corresponding changes with the increase in temperature are shown in Figure6. It was indicated that an increase in temperature led to an increase in thermal conductivities, volumetric heat capacity, and TCC. The cross-plane and in-plane thermal conductivities and volumetric heat capacity increased by 20.2%, 37.9%, and 12.3% when the temperature increased to 800 ◦C compared with those at 28 ◦C, respectively. The TCC of the material also increased by 21.7%. This result indicated that the thermophysical parameters of the material and the TCR of the measurement system might greatly change with the temperature. The change in the in-plane thermal conductivity was larger than that in the cross-plane thermal conductivity. The thermal conductivities measured by the method with the effect of the TCR were less than those obtained by the method without the TCR. The maximal deviation was 8.21% when the TCC between the sample and the heating element was 125.1 W m 1 K 1, · − · − because the increase in the temperature of the sample calculated by the mathematical model of the measurement system would be larger than that measured by the experimental measurement if the TCR was not considered. Hence, leading the measured value of the thermal conductivities estimated using the experimental data was larger than the desired value. In addition, the in-plane thermal conductivity showed higher difference between the TCR consideration and without the TCR consideration as the temperature increased. A possible reason might be that the change with temperature in the in-plane thermal conductivity was larger than that in the cross-plane thermal conductivity, and, correspondingly, the thermal resistance in the in-plane direction was relatively smaller. Subsequently, the effect of the TCR on the heat transfer in the in-plane direction was larger than that in the cross-plane direction in response to increasing temperature. The uncertainty in experimental results was mainly affected by the temperature measurement, the determination of the power supply, the location of the measuring point, and the time delay. In this study, the uncertainties caused by temperature measurement, time counting, and location of the measuring point were 0.43%, 0.2%, and 0.3%, respectively. The uncertainty of output power from the power supply was 0.2%. The maximum difference in temperature of the results from parameter estimation between the experimental measurement and theoretical calculation was less than 0.06 ◦C, Materials 2020, 13, x FOR PEER REVIEW 10 of 14

practical measurement system for the steady-state method, and heat was loss through the thermocouples and through radiation and convection.

Table 3. Experimental results of thermal contact conductivity using the proposed methods and the steady state.

Thermal Contact Conductivity (W·m−2·K−1) ± SD T (°C) P (kPa) Proposed Method Steady-State Method Deviation (%) 2 94.1 ± 0.35 102.1 ± 0.45 8.5 200 5 98.5 ± 0.42 105.8 ± 0.36 7.4 8 102.4 ± 0.25 110.7 ± 0.21 8.1 2 97.3 ± 0.46 104.4 ± 0.71 7.3 300 5 101.9 ± 0.42 108.8 ± 0.43 6.8 8 104.8 ± 0.38 112.7 ± 0.58 7.5 2 99.2 ± 0.57 105.5 ± 0.65 6.4 400 5 105.7 ± 0.55 112.9 ± 0.54 6.8 8 108.9 ± 0.43 116.6 ± 0.32 7.1

3.3. Experimental Measurement at High Temperatures

Materials 2020, 13The, 1353 thermophysical parameters and the TCCs of an anisotropic high-temperature-resistant 10 of 12 material (Sample D, silica aerogel material) with a large difference in its in-plane and cross-plane thermal conductivities were experimentally measured at room and high temperatures by using the proposed method. For comparison, the same thermophysical parameters of the same sample were and the average residual error was less than 0.04 ◦C. The relative uncertainty was 2.0% based on the measured using the small-plane heat source method without the effect of TCR which was reported absolute averagein our previous value work of the [27]. temperature The corresponding increase. change Thus,s with the the increase overall in uncertaintytemperature are of shown the measurement in system wasFigure 2.3%. 6.

0.195 0.032 Without consideration of TCR Without consideration of TCR With consideration of TCR 0.180 With consideration of TCR 0.030

) 0.165 -1 ) K -1

0.028 -1 K -1 0.150 (Wm xy

(Wm 0.026 z λ λ 0.135 0.024 0.120

0.022 0 200 400 600 800 0 200 400 600 800 Temperature(℃) Temperature(℃) Materials 2020, 13, x FOR PEER REVIEW 11 of 14 (a) (b) 130 2.70 Without consideration of TCR h 2.65 125 c With consideration of TCR 2.60 120

) 2.55 ) 1 -1 - K K 2 115 -3 - 2.50 (Jm

p 2.45 (Wm c c

110 ρ h 2.40

105 2.35

2.30 100 0 200 400 600 800 0 200 400 600 800 Temperature(℃) Temperature(℃) (c) (d) Figure 6. FigureResults 6. Results of experimental of experimental measurement measurement at at different diﬀerent temperatures: temperatures: (a) cross-plane (a) cross-plane thermal thermal conductivity;conductivity; (b) in-plane (b) in-plane thermal thermal conductivity; conductivity; ( cc)) TCC; TCC; and and (d) (volumetricd) volumetric heat capacity. heat capacity.

4. ConclusionsIt was indicated that an increase in temperature led to an increase in thermal conductivities, volumetric heat capacity, and TCC. The cross-plane and in-plane thermal conductivities and In thisvolumetric study, theheat small-plane capacity increased heat by source 20.2%, method 37.9%, and was 12.3% improved when the bytemperature considering increased the etoff ect of TCR on the thermophysical800 °C compared parameterwith those at measurement.28 °C, respectively. The The TCC heat of transfer the material process also increased in the measurementby 21.7%. was This result indicated that the thermophysical parameters of the material and the TCR of the theoreticallymeasurement modeled system and might analytically greatly change solved. with A the multiparameter temperature. The change estimation in the in-plane method thermal was developed on the basisconductivity of sensitivity was larger analysis than that to in simultaneously the cross-plane thermal measure conductivity. the desired The thermal thermophysical conductivities parameters and TCR.measured The thermophysical by the method parameterswith the effect of of ditheff erentTCR were samples less than and those TCR obtained in the by system the method were subjected to simulationwithout and the experimentalTCR. The maximal measurements deviation was 8.21% at di ffwhenerent the temperatures TCC between the and sample pressures and the to validate heating element was 125.1 W·m−1·K−1, because the increase in the temperature of the sample calculated the proposedby the method. mathematical The model comparison of the measurement of thermal syst conductivityem would be larger between than that the measured by valuethe and the standard valueexperimental showed measurement that the relativeif the TCR error was not was consid lessered. than Hence, 2%, proving leading the the measured correctness value of the the proposed method. Thethermal measured conductivities result estimated with the using effect the of experiment TCR wasalsmaller data was than larger that than without the desired this value. effect, In indicating that TCRaddition, should the be in-plane considered. thermal conductivity Uncertainty showed analysis higher difference showed between that the the uncertaintyTCR consideration of the whole and without the TCR consideration as the temperature increased. A possible reason might be that the system waschange 2.3% with which temperature could in meet the in-plane the requirement thermal cond ofuctivity an experimental was larger than measurement.that in the cross-plane The proposed method couldthermal be conductivity, used to simultaneously and, correspondingly, determine the th theermal in-plane resistance and in cross-plane the in-plane thermaldirection conductivities,was volumetricrelatively heat capacity, smaller. Subsequently, and TCC with the effect high of measurement the TCR on the accuracyheat transfer for in various the in-plane anisotropic direction materials. was larger than that in the cross-plane direction in response to increasing temperature. Author Contributions:The uncertaintyConceptualization, in experimental results K.Y. and was X.Z.;mainly methodology, affected by the K.Y.; temperature software, measurement, L.C.; validation, X.Y.; formal analysis,the determination L.C.; investigation, of the power D.H.; supply, resources, the location L.C.; data of the curation, measuring D.H.; point, writing—original and the time delay. draft In preparation, K.Y.; D.H. andthis study, X.Y.; writing—review the uncertainties caused and editing, by temperatur K.Y.; D.H.e measurement, and X.Y.; visualization, time counting, D.H.; and location supervision, of the K.Y.; project administration,measuring K.Y.; point funding were 0.43%, acquisition, 0.2%, and K.Y. 0.3%, All respectively. authors have The readuncertainty and agreed of output to power the published from the version of the manuscript.power supply was 0.2%. The maximum difference in temperature of the results from parameter estimation between the experimental measurement and theoretical calculation was less than 0.06 °C, Funding: The authors would like to appreciate the financial support received from the National Natural Science and the average residual error was less than 0.04 °C. The relative uncertainty was 2.0% based on the Foundation of China (Grant No.51890894) and the Scientific and Technological Innovation Foundation of Shunde Graduate School,absoluteUSTB average (Grant value of No. the BK19AE012). temperature increase. Thus, the overall uncertainty of the measurement system was 2.3%.

4. Conclusion In this study, the small-plane heat source method was improved by considering the effect of TCR on the thermophysical parameter measurement. The heat transfer process in the measurement was theoretically modeled and analytically solved. A multiparameter estimation method was developed on the basis of sensitivity analysis to simultaneously measure the desired thermophysical parameters and TCR. The thermophysical parameters of different samples and TCR in the system were subjected

Materials 2020, 13, 1353 11 of 12

Acknowledgments: Thanks to Yan Zhang for her help in the establishment of the mathematical model, Fuyong Su for his guidance in the methodology, and Ruifeng Dou for providing the experimental equipment. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Nomenclature

α thermal diffusivity (m2 s 1) · − a half-length of heat source (m) b half-length of sample (m) λ thermal conductivity (W m 1 K 1) · − · − c special heat capacity (J kg 1 K 1) · − · − q half heat flux density in the element (W m 2) · − T0 initial temperature of the system (K) θ Laplace transform of temperature 3 1 ρcp volumetric heat capacity (J m K ) · − · − p Laplace parameter of t X+ sensitivity coefficient hc thermal contact conductance D the deep of heat transfer (m) ρ density (kg m 3) · − x the direction of X y the direction of Y z the direction of Z

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