Measurement of the thermal properties of materials by the hot plate method considering the convection coefficient around the device

Measurement methods based on the hot plate did not consider the exchange coefficients around the device. These measurements were based on analytical solutions of the unidirectional (1D) heat conduction equation. This paper describes an experimental method to simultaneously estimate the thermal conductivity and thermal diffusivity of materials, taking into account the exchange coefficient around the device. This method is based on the measurement of the temperature at the center of a heating element inserted between two samples, with the unheated surface of the samples being kept constant. As a first step, a two-dimensional (2D) heat transfer model was solved using the finite volume method (FVM) and implemented in MATLAB®. To validate the computational code developed with the 2D model, a comparative study was performed with the full 3D model simulated in COMSOL Multiphysics. This 2D model was then used to perform a sensitivity analysis on the different thermophysical parameters and the convection coefficient. The Levenberg–Marquardt algorithm was used to calculate the estimates. An experimental study was performed on several material samples, leading to an estimation of their thermal properties. In addition, a comparative study based on the asymmetric hot plate (AHP) method was performed, and the results obtained showed a relative error of less than 4%. Therefore, it can be concluded that the proposed model is validated to provide reliable thermophysical properties of materials.


Abstract
Measurement methods based on the hot plate did not consider the exchange coefficients around the device. These measurements were based on analytical solutions of the unidirectional (1D) heat conduction equation. This paper describes an experimental method to simultaneously estimate the thermal conductivity and thermal diffusivity of materials, taking into account the exchange coefficient around the device. This method is based on the measurement of the temperature at the center of a heating element inserted between two samples, with the unheated surface of the samples being kept constant. As a first step, a two-dimensional (2D) heat transfer model was solved using the finite volume method (FVM) and implemented in MATLAB ® . To validate the computational code developed with the 2D model, a comparative study was performed with the full 3D model simulated in COMSOL Multiphysics. This 2D model was then used to perform a sensitivity analysis on the different thermophysical parameters and the convection coefficient. The Levenberg-Marquardt algorithm was used to calculate the estimates. An experimental study was performed on several material samples, leading to an estimation of their thermal properties. In addition, a comparative study based on the asymmetric hot plate (AHP) method was performed, and the results obtained showed a relative error of less than 4%. Therefore, it can be concluded that the proposed model is validated to provide reliable thermophysical properties of materials. Step-Wise Transient

Introduction
The building sector plays an important role in the global energy balance. According to the International Energy Agency (IEA.), the building sector will account for 132 ExaJoule (EJ) of energy consumption in 2021, or 30% of the world's final energy consumption [1]. Reducing energy consumption helps mitigate negative environmental impacts [2]. To reduce this energy consumption, it is imperative to improve the intrinsic properties of the building envelope, which is the main component of the passive systems. The energy efficiency of buildings has become a priority objective of energy policies worldwide. The design of energy-efficient buildings requires the judicious choice of building materials. This choice requires a perfect knowledge of their thermal properties, particularly the thermal conductivity and thermal diffusivity [3]. These thermal properties can be obtained using several thermal characterization methods. These methods can be divided into two categories : transient and permanent. The guarded hot plate [4] and the thermal flux meter method [5] are the most widely used devices in the steady-state category. By definition, steady-state methods determine the thermal conductivity only. Several transient methods are available, such as the hot disc [6], the dynamic plane source (DPS) [7], the step-wise transient method (SWT) [8], the flash method [9], the three-layer method [10], the transient hot wire (THW) [11], the hot plate method [12], the hot strip method [13]. Each of the above-mentioned methods works for a specific range of materials and assumptions to be verified in advance with a certain precision. For example, the transient hot wire (THW) technique introduced by [11] was developed to measure the thermal conductivity of solids. The uncertainty analysis of this method shows that the uncertainty of the determined solid thermal conductivity is better than 1%, while the uncertainty of the product measurement (density × specific heat capacity) is 5%. The hot disc method [6] estimated thermal diffusivity and thermal conductivity. The method has a low accuracy in the estimation of the thermal diffusivity and its sensitivity becomes very low for materials with high thermal conductivity [14]. The extended dynamic plane source (EDPS) is a modification of the DPS method for the thermal characterization of low-conductive materials [15]. This method was developed to simultaneously determine the thermal conductivity and diffusivity of materials. It is based on the assumption of one-dimensional heat flow in a finite sample. The precision of this method is estimated to 5% [16]. This method is improved to reduce the impact of lateral losses by making measurements under a vacuum. But this approach is only applicable for materials with conductivity between (0.1-2) W m −1 K −1 [17]. This excludes insulating materials and materials with high conductivity from the scope of application. The 'step-wise' method [8,18] is an experimental method for measuring the diffusivity and thermal conductivity of a solid material. This method assumes that heat transfer is unidirectional (negligible lateral heat losses). It is also possible to perform measurements under vacuum to reduce the impact of lateral losses, but this approach is not valid for insulating materials ; their thermal conductivities would not be the same under normal pressure because they are porous. However, these simple methods are based on assumptions that are not always easy to verify, such as : negligible inertia of the heating element, semi-infinite medium, and unidirectional transfer.
Among the transient methods, the three-layer, flash, and hot plate methods have recently attracted attention because of their ability to measure both thermal properties simultaneously in a single experiment.
The hot plate method at a temperature measurement is initially a method for measuring effusivity [19]. In the classical version of this method, a thin, flat heating element is inserted between two samples of the material to be characterized. A thermocouple placed at the center of the heating element was used to measure the temperature. The advantage of this hot plate method is that it is suitable for both large and very small material thicknesses [20,21] used the hot-plate method to characterize insulating and super-insulating materials of low thickness. The thermal conductivity was estimated with good precision from a simple steady-state measurement and the heat capacity was estimated by exploiting the thermogram of the transient part. It has been shown that the precision of the heat capacity depends on the samples. It was then shown that a device with two samples of different thicknesses improved the precision of heat capacity estimation. This method is based on the assumption of unidirectional transfer. It requires beforehand checking the validity condition of the unidirectional model. To improve the estimates of the thermo-physical parameters, Jannot and al. [22,23] proposed hot-plate methods with two temperature measurements on both sides of the sample. But somehow, the exchange coefficient on the lateral sides and the longitudinal heat transfer in the heating element make the heat flow three-dimensional, and thus the assumption of a 1D flow in the center of the sample may not be verified beyond a certain time.
Some authors have used methods to straighten the flux lines using a passive guard. To respect the assumption of the 1D heat flux, either measurement are made at the center of the sample neglecting lateral losses [24], which does not reflect the reality in the long term, guard rings are used to straighten the flow lines, which requires the use of large materials [25]. It was recognized that it was highly desirable to surround the heating element and the entire sample with temperature-controlled 'guards' designed to block the flow of heat from the lateral directions and to ensure that the flow of heat through the sample is essentially unidirectional. Even with great care, this guarded hot plate approach remains imperfect and requires theoretical and experimental corrections. In particular for measurements on insulators with low thermal conductivity [26,27] proposed a new approach for straightening flow lines using a passive steady-state guard. It is shown that by skillfully choosing the ambient temperature between the temperatures of the two faces of the sample, a relationship between them can be established to make the flow direction unidirectional at the measurement point.
The aforementioned methods do not consider the exchange coefficients around the device. Other methods that consider the exchange coefficient have also been developed. For example, the 'Flash' method, developed by Parker [28] is a commonly used method for measuring the thermal diffusivity of a material. Modeling of heat transfer in the sample has allowed several authors to propose methods for estimating the thermal diffusivity from the experimental thermogram [29,30]. This method is not adapted to the thermal characterization of insulating materials [31]. Insulating materials are porous materials, so the light flux sent on the front side of the sample will not be absorbed in-depth, but only on the surface. Moreover, to obtain a sufficiently hightemperature increase on the unheated side, it is necessary to produce a very high-temperature increase on the heated side. This method is based on the assumption of a constant and identical convective heat transfer coefficient on all the faces.
The three-layer method [32] is a method for measuring the thermal conductivity, thermal diffusivity, and exchange coefficient of low-density insulation materials. The method is based on a three-layer system, in which the inlet and outlet temperatures are measured after applying a short heat flow to one layer. This method allows the estimation of the thermal conductivity of materials with very low conductivity ( ) < -k 0.15 W m K , 1 1 with a precision better than 5%. The precision of thermal diffusivity depends on the density of the material. This method does not apply to materials with high conductivity [33].
The analytical methods developed, which are based on the assumption of unidirectional heat transfer, do not consider lateral losses around the device. Methods aimed at simultaneously estimating the thermophysical properties considering lateral losses have also been studied. However, most of these methods do not apply to certain materials.
The objective of this study was to develop a 3D numerical model to simultaneously determine the thermal conductivity and thermal diffusivity of materials taking into account the convection coefficient around the device.
In this work, a numerical hot plate method that considers lateral losses is proposed for the simultaneous estimation of the thermophysical properties of materials in the transient regime. At first, a numerical code was developed. This allowed the temperature evolution at the center of the material to be determined. A sensitivity study of the numerical model to the thermophysical parameters and convection coefficient was conducted.
Afterward, COMSOL Multiphysics was also used to perform a numerical experiment to test the ability of the numerical model to estimate the thermal conductivity, thermal diffusivity, and convection coefficient. The Levenberg-Marquardt algorithm was used to calculate estimates. Finally, to confirm the quality of the estimation, an experimental study on a range of materials was performed. A comparison of the results with those of the asymmetric hot plate method (AHP) was made.

Mathematical model
The 3D heat transfer model is based on the assumption that the horizontal surfaces at the ends of the device are maintained at the uniform and constant temperature T . 0 The vertical sides were in contact with the ambient air at T amb with a uniform surface heat transfer coefficient by natural convection h as shown in figure 1.
We have a symmetrical model, so 1/4 of the material has been considered. If the temperature at any point of the heating element is T (x, y, z, t), and λ is its thermal conductivity and ρc its heat capacity by volume, then the heat transfer equation is written : The initial and boundary conditions can be written : The system exchanges by convection on the lateral surfaces x = L ; y = L.
Where λ is the thermal conductivities of the sample, ρc is the thermal capacities. We consider e and e S as the thickness of the sample and the heating element respectively. The temperature on the back side of the sample is T 0 and h is the convection coefficient of convective heat transfer, which is assumed to be uniform along the walls. The heat flux density produced by the heating element is AE . 0 The temperature field T (x, y, z, t) is obtained by solving equations (1)-(6) using the finite volume method (FVM). The process starts by integrating equation (1) over element C that enables recovering its integral balance form, as : Then, the volume integral is transformed into a surface integral using the divergence theorem. Then by replacing the surface integral with a summation over the faces of the control volume, the equation (7) becomes: To better perform the discretization, the surface integral at each face of the element must be evaluated.
To obtain the temporal discretization, we used the implicit Euler scheme which is unconditionally stable.
represent the surfaces of the neighboring node faces, T C and T C 0 represent the temperature of the central node C at time t and time ( ) -D t t respectively. Further, we need a profile that approximates the variation in T between its neighbors. The temperature gradients along the six faces of the cell can, in turn, be expressed in terms of the six adjacent nodal temperatures as follows: By substituting equation (11) into equation (10), the finite volume form for an interior node can be organized as follows: Equation (12) can be rearranged as follows : We pose: Equation (13) can be rewritten as : The imperfect contact between the heating element interface and the sample causes a contact resistance effect, which is considered a reducing agent of the flux of heat. This contact thermal resistance Rc is non-negligible. 70 W m . 2 The dimensions of the material are : L = 5 cm ; e = 3 cm. In order to verify that the mesh size does not significantly affect the results, a sensitivity study was performed. For the 3D model on Comsol, we chose 120 × 120 × 72 nodes, representing the number of nodes along the x-axis, y-axis, and z-axis respectively. The number of meshes chosen for our 2D model is presented in table 1, as well as the thermal properties of the material. Figure 2 shows the evolution of the temperature at the different points (M , 1 M 2 ) between the computational code developed under MATLAB and the simulation of the 3D model under COMSOL. The results show that the curves are perfectly matched, which guarantees that the algorithm and the boundary conditions are correctly implemented in the program. Thus, we can conclude that the developed code is valid.

Description of the experimental approach for the measurement
The experimental measurements were performed using a MINCO HK 5489 heating element consisting of a flat resistor inserted between two insulating polymer films with a heated area of 100 ± 1 mm × 100 ± 1 mm and a thickness of 2.2 × 10 −4 ± 0.01 mm. The heating element was connected to a power generator (AIM-TTI ® , PL330, 32V-3A, PSU). A K-type thermocouple (wire diameter 0.03 mm) connected to an acquisition unit (USB, TC-08, UK) was fixed at its center to record the evolution of the temperature versus time. The two samples were placed on either side of a heating element. The samples and heating element had the same dimensions (10 × 10 cm). The side of the sample on the side of the heating element is called the 'front side' as opposed to the 'back side' which is the unheated side of the system. On the back side of the samples, aluminum blocks are placed to keep the temperature constant ( ) T . 0 The whole system is placed in an environment with an ambient temperature ( ) T amb and a uniform heat transfer coefficient h via natural convection. A clamping device was used to exert a slight clamping pressure on the heating element/sample/aluminum block assembly. Matched thermocouples are used to measure the temperatures of the back side of the samples T , 0 the temperature of the front panel T c as well as the ambient temperature T amb around the device. The features described above are shown in figure 3, which represents the version of the experimental device used for the measurements described in this paper.
The temperature in the aluminum blocks is assumed to be uniform. The assumption of uniformity is validated with the Biot number = l Bi hL which is less than 0.1, thereby defining the criterion for thermal accommodation. Considering that h = 10 --W m K , 2 1 the thermal conductivity of the block l = --200 W m K b 1 1 and the dimensions of sample 10 ×10 cm, we obtain a Biot number lower than 0.1 leading to Bi = 0.005 so that the temperature of the aluminum blocks can be considered as uniform [21]. The temperature ( ) T e t , 0 increased after a certain time (corresponding to a long time). The relative difference between the minimum and maximum temperature ( ) T e, t 0 is less than 1%, so the temperature of the unheated side ( ) T e, t 0 can be considered constant. Therefore, the temperature of the aluminum blocks can be assumed to be homogeneous.
The 3D heat flux at the center of the heating element created by lateral convective losses can be amplified by the heat fluxes that occur in the heating element in the (Ox) and (Oy) directions. The properties of the heating element are not perfectly known, nor are the properties of the heat flux produced by the heating element (uniform, etc). The fin effect in the heating element is not considered in this model. However, the thermal conductivity of the heating element is very different from the thermal conductivity of the material [27].

Reduced sensitivity analysis
Using equation (18), the reduced sensitivities of the heated face temperature T c to the parameters that remain unknown in the model for the thermal characterization of the sample, namely λ, a, Rc, and h, were numerically calculated. The reduced sensitivity of the temperature T to a parameter X is defined as [34]:  Table 1. Thermal properties of the material. The reduced sensitivities of thermal diffusivity, thermal conductivity, contact resistance, and convection coefficient were calculated to highlight the sensitive parameters that can theoretically be estimated and the insensitive parameters that do not influence on the observed quantity and the accuracy of their estimation. The theoretical results are plotted in figure 4. The calculations were carried out considering the following parameters of the heating element and the table 2.
According to figure 4, several observations can be made from this reduced sensitivity study : • For a given sample thickness, figure 4 shows that the exchange coefficient became sensitive only after 500 s for the wood-based material and 200 s for the polystyrene material. This finding may also be explained by the fact that lateral losses take time to be felt at the measurement point in high-conductivity materials. This sensitivity seems to be de-correlated from the other parameters, its separate estimation will be possible.
• It was observed that the reduced sensitivity of thermal conductivity was very large. It is noted that it is not correlated with the sensitivities to the convection coefficient h and thermal diffusivity ; this will allow an accurate estimation of λ over the time interval of the experiment.
• It is noted that the sensitivity of the thermal diffusivity a is sufficiently large to allow a good estimation over the entire duration of the experiment.
• It should be noted that the contact resistance Rc is sensitive at the beginning of the experiment and then remains constant, and an estimation of this parameter is possible. The sensitivity of the contact resistance Rc is very low, which would lead to a huge uncertainty in the numerical calculation of its values.
This sensitivity analysis shows that the estimation of the thermophysical parameters is always possible for all the chosen materials.

Heating element calibration
According to the proposed model, the electrical resistance of the heating element Rel and its surface S must be determined to calculate the heat-flux density AE . 0 To avoid any uncertainty due to the measurement of Rel and S with measuring tools, it is best to estimate their ratio .

Justification for choosing a 2D model
A 1D model is generally chosen by [35,36] to model the heat transfers at the center of the heating element of the device presented in figure 3. The 1D modelization allows to obtain an analytical model simple to use. But the range of validity of these models is small since the effect of lateral convection is not taking into account. To show the significant effect of lateral exchanges in the measurement of thermal conductivity and thermal diffusivity, 1/4 of the top of the device (figure 1) was simulated using COMSOL software [37]. To simplify the analysis, the   Standard deviation = 1.506% thermal contact resistances between the heating element and the sample were neglected. To neglect the contact resistances, the thicknesses were chosen so that the ratio e/λ > 0.01 [20]. The simulations carried out for the two materials used were considered a 'numerical experiment'. A white noise following a normal distribution of mean zero and standard deviation s = K 0.01 bruit is added to the simulated temperature curve ( ) T t c thus providing an experimental character for the simulation. A mesh sensitivity study was performed to ensure that the type of mesh does not significantly affect the results. Calculations were performed using the materials listed in table 4. The calculations were performed considering the following parameters : L = 10 cm, e = 3 cm, The time t , max corresponding to a relative difference of 0.1% between the two simulated temperatures, was determined. The results are shown in figure 7.  deviate from each other for laterite (vermiculite) samples. These results show that the 1D model is not adequate to model the heat transfers at the center of the device at long time. 3D modeling is therefore necessary. Figure 8 represents the temperature as a function of time of our 2D and 3D models for = -h 10 Wm K .

1
Results show that the 2D model can represents the 3D model. Therefore, our 2D model can be used to perform the estimations. 2D model has the advantage of being less time consuming. However, for a long time, we notice  Table 4. Thermal properties of some representative cases of materials. an increasing gap between the models for relatively well-conducting samples (i.e., laterite), this may have an effect on the estimated thermophysical parameters.

Estimation of the time of the experiment
To estimate simultaneously a, λ and h, the time of the experiment must be long enough for the model to be sensitive to h but not too long for h to remain constant. Indeed, h varies according to the temperature. The time of the experiment is the time needed to simultaneously estimate a, λ and h with a minimum of error. To determine the error of the estimates, a numerical experiment under COMSOL Multiphysics is performed by simulating the 3D model presented above ( figure 1). The temperature at the center of the heating element (T exp ) is then collected. By minimizing the difference between T exp and T mod (the temperature at the center of the heating element given by our 2D model) by minimizing the quadratic deviations, the parameters a, λ and h are determined. The algorithm used to converge to the minimum quadratic criterion is that of [38,39], which combines the gradient method with Newton's method. The estimation is done by minimizing a sum of squared temperature residuals (D), which is given by the relation below : The estimated values of a, λ and h were compared with the values initially used in the simulation using COMSOL Multiphysics (see figure 9) to obtain the estimation error. The results obtained by calculating the estimation errors for different time intervals are given in figure 9 for laterite and vermiculite. The results indicate that beyond 1000 it is possible to estimate all 3 parameters with relative error less than 3%. We note a small increase in the error of estimating h in the long term for laterite. This shows that our method is limited in measurement time.

Estimation parameters from experience
To validate this theoretical study, experimental measurements were performed on four different sample pairs available in our laboratory. The materials are of industrial type, they are homogeneous and isotropic.
• E1 : wood-based material, two samples with a thickness of 1.8 cm.
• E2 : PVC-based material, two samples with a thickness of 2 cm.
• E3 : material based on polystyrene, two samples with a thickness of 4.9 cm • E4 : material based on polystyrene, two samples of a thickness of 2.5 cm The uncertainty on the surface of the heating element is about 2%. We must add the uncertainty on the thickness of the sample estimated at 1% and on the heat flux produced in the heating element, estimated at 0.6%. The sum of these uncertainties leads to an overall uncertainty of 3.6% to which must be added the estimation error due to the measurement of noise on T. Three measurements were performed for each pair of samples to estimate measurement errors. The time interval between the two measurements was at least 1 h. The measurements were performed after the disassembly and reassembly of the entire measuring block. The values of the parameters λ, a, Rc, and h were estimated by minimizing the sum of the squared errors between the experimental curve T exp (t) and the theoretical curve ( ) T t .
mod Figure 10 represents an example of an experimental curve and the theoretical curve with the estimated parameters ; the residuals defined as the difference between the points between the experimental points and the model are also represented on the same figure. Figure 10 shows that the estimated curve and the experimental curve are perfectly superposed. The residuals are not signed, they are flat and centered on 0, and they reflect the absence of model bias. The estimated parameters, the standard deviations, and the standard deviation of the residuals are also presented in tables 5-8.
Tables 5-8 present a review of the experimental results. To verify our numerical model, the asymmetric hot plate method (AHP) developed by Jannot and al. [24] was also used to estimate the thermal conductivity and effusivity of the different materials. Thermal diffusivity can be obtained from the thermal effusivity. Table 9 lists the relative errors between the measured thermal properties of the different materials using the two methods.
• The estimated values of thermal conductivity λ for the four materials are in good agreement with those measured by the AHP method. The standard deviation of the measurements is compared to the mean value, leading to a standard deviation of less than 5% for thermal conductivity. The reproducibility of the method was very good for the estimation of λ.
• The thermal diffusivity a measured by this method is very close to the value estimated by the AHP method. The relative error is less than 4% for the 4 materials considered. Moreover, the small standard deviations of the residuals show that the parameters are correctly estimated.   • For the estimation of the convection coefficient h, the standard deviation of the measurements is very low for the insulating materials (Polystyrene, PVC). This shows that this parameter is well estimated for these materials.
• On the other hand, the standard deviation on the contact resistance is very high, this confirms the reduced sensitivity study which shows that the sensitivity of the contact resistance is very low. This leads to a large uncertainty on the estimation.

Conclusion
This paper presents a simple experimental method for the simultaneous estimation of the thermal conductivity and thermal diffusivity of materials, considering the exchange coefficient around the device. A threedimensional (3D) heat-transfer model was proposed. This model was solved numerically using the finite volume method (FVM). A numerical calculation code allowing the evolution of the temperature along the entire sample was developed. This numerical calculation code was validated by simulation using COMSOL Multiphysics of the 3D model. A sensitivity study showed that our model is sensitive to the parameters a, λ and h at long time. The effect of lateral convection must therefore be taken into account. We have shown that a 2D model is sufficient to take it into account. The analysis of the estimation errors allowed the estimation of the time of the experiment for different types of materials. We then used our 2D model to estimate the parameters of four materials based on wood, PVC, and Polystyrene. The estimates were in good agreement with other measurement methods (relative error 4%). Therefore, it can be concluded that the proposed model has been validated to provide reliable thermo-physical properties of the materials.

Data availability statement
No new data were created or analysed in this study.