The Numerical Simulation of the Transient Plane Heat Source Method to Measure the Thermophysical Properties of Materials

Abstract

The transient plane heat source method (TPS), also known as the hot disc method, is an experimental method for determining the thermal transport properties of materials. The method’s main element is a sensor made of a nickel metal strip in the shape of a double helix, which is inserted into an insulating polymer film. In this work, we used the finite element method to create a three-dimensional model of the sensors and compared the simulated and experimentally recorded mean temperature rise data. The volume mean temperature rise of the sensor, as determined through simulation, exhibits a high level of resemblance with the corresponding experimental data. Additionally, temperature rise curves of several other materials are also simulated based on the model and the thermal performance parameters are calculated from these data. In the meantime, this paper presents an evaluation and discussion of the current density distribution of the sensor and the temperature distribution during the testing of the sample. This simulation has the potential to be utilized for future geometry and parameter estimate optimization, and provides a theoretical reference for detector design.

1. Introduction

Thermal conductivity represents a significant thermophysical parameter of materials. Over the years, there have been many different methods developed for use in characterization [1,2,3,4,5,6,7,8]. In geological exploration [9,10,11,12], aerospace thermal protection systems [6,13,14,15], and the selection of insulation materials [16,17,18], there are strict restrictions on the thermal conductivity of materials, and so it is necessary to measure the thermal conductivity of materials under various conditions accurately. Traditional methods often suffer from lengthy measurement times and potential inaccuracies [19,20,21]. In this context, the enhancement of testing efficiency while ensuring measurement accuracy has emerged as a central research focus in thermal property-testing technologies. Transient plane heat source (TPS) technology is a highly efficient approach to thermophysical property measurement, having gained considerable application in practice [22,23,24,25]. It was initially developed by Prof. Gustafsson [3,26,27,28] in Sweden as a patented method based on the transient hot wire method and transient tropical methods, also known as the ‘hot disc’ method. The method’s principal advantages include the ability to measure a wide variety of materials’ thermal conductivities, as well as a relatively short testing time, high accuracy, and a straightforward sample preparation process.

Recent advancements in TPS technology have focused on enhancing measurement precision and efficiency through novel approaches and materials. For example, integrating machine learning algorithms with TPS data analysis has shown promise in accurately predicting thermal conductivity in complex materials [29,30]. Furthermore, emerging studies on the impact of nanomaterials in thermal fluids indicate that innovative formulations can significantly boost thermal performance, necessitating the precise measurement capabilities offered by TPS methods [31,32,33]. In the meantime, researchers have worked to improve the TPS technique by combining various kinetic models to enhance measurement precision and efficiency, optimizing sensor designs and increasing heat transfer measurement accuracy [34,35].

This study aims to establish a three-dimensional model of the transient planar source method using finite element modeling (FEM) and to compare the simulated average temperature rise data of PMMA samples with experimental results, thereby validating the feasibility of the model. Additionally, the model will be employed to predict the thermal properties of other materials. The simulation work also seeks to clarify the distribution of current density within the sensor and its relationship with the heat source distribution, providing theoretical support for the optimization of geometric structures and parameters in future thermal performance testing.

2. Mathematical Formulation

A material exhibiting isotropic properties, characterized by a heat diffusion coefficient $a$, a volumetric heat capacity $\rho c$, and a heat source $Q$, is satisfied by the following heat transfer equation [26]:

$$a{\nabla }^{2}T\left(\overrightarrow{r},t\right)+{\displaystyle \frac{Q\left(\overrightarrow{r},t\right)}{\rho c}}={\displaystyle \frac{\partial T\left(\overrightarrow{r},t\right)}{\partial t}}$$

(1)

In the above equation, $\rho$ is the density, $c$ is the material specific heat, and $T\left(\overrightarrow{r},t\right)$ represents the temperature of the point $\overrightarrow{r}$ at the moment $t$. When $Q=0$, $t>0$, the fundamental solution of Equation (1) is as follows [36,37,38]:

$$T=T_0+{\displaystyle \frac{1}{{\left(4\pi at\right)}^{3/2}}}exp(-{\displaystyle \frac{r^2}{4at}})$$

(2)

where $T_0$ represents the initial temperature of the sample. By convolving the above equation with ${\displaystyle \frac{Q\left(\overrightarrow{r},t\right)}{\rho c}}$, the general solution for the heat source $Q\left(\overrightarrow{r},t\right)$ can be obtained. In the case of sensors utilizing the TPS method, the sensor can be approximated as a concentric circle source with a heating power of $P_0=\pi b(m+1)Q_0$, where $Q_0$ is the heat released per unit length, $b$ represents the maximum radius of the rings, and $m$ is the number of concentric rings, with a total heating length of $L={\sum }_{l=1}^{m}2\pi lb/m=(m+1)\pi b$.

By employing the dimensionless integration parameter ${\sigma }^2={\displaystyle \frac{a(t-t^{\prime })}{b}}$, with $t^{\prime }=0$, and defining the characteristic time $\tau ={\displaystyle \frac{\sqrt{at}}{b}}$, as well as the thermal conductivity of the material $\lambda =\rho ac$, it is possible to derive an expression for the average temperature change in the TPS sensor as follows [36]:

$$∆\overline{T}(\tau )={\displaystyle \frac{2\pi }{\pi b(m+1)}}{\displaystyle \frac{P_0}{2{\pi }^{3/2}bm(m+1)\lambda }}{\int }_0^{\tau }{\displaystyle \frac{d\sigma }{{\sigma }^2}}{\displaystyle {\sum }_{k=1}^m}{\displaystyle \frac{kb}{m}}\times {\displaystyle {\sum }_{l=1}^m}lexp\left(-{\displaystyle \frac{k^2+l^2}{4{m^2\sigma }^2}}\right)I_0\left({\displaystyle \frac{kl}{{2m}^2{\sigma }^2}}\right)$$

(3)

$$∆\overline{T}\left(\tau \right)=A+{\displaystyle \frac{P_0}{{\pi }^{\frac{3}{2}}b\lambda }}D\left(\tau \right)$$

(4)

where A represents the initial temperature rise constant, $I_0$ denotes the zero-order Bessel function, and $P_0$ is the input power. $D\left(\tau \right)$ denotes the dimensionless time function, which can be defined by the following equation [3,36,37,38,39,40]:

$$D\left(\tau \right)={\displaystyle \frac{1}{m^2({m+1)}^2}}{\int }_0^{\tau }{\displaystyle \frac{d\sigma }{{\sigma }^2}}{\sum }_{k=1}^m{\displaystyle \frac{kb}{m}}\times {\sum }_{l=1}^{m}lexp\left(-{\displaystyle \frac{k^2+l^2}{4{m^2\sigma }^2}}\right)I_0\left({\displaystyle \frac{kl}{{2m}^2{\sigma }^2}}\right)$$

(5)

The temperature of the TPS detector is determined by measuring the resistance of its nickel strip as a function of temperature change, according to the following expression:

$$R\left(t\right)=R_0[1+\alpha ∆\overline{T}\left(\tau \right)]$$

(6)

where $R_0$ represents the initial resistance of the TPS detector before transient test recording. $R\left(t\right)$ denotes the resistance after a temperature change of $∆\overline{T}\left(\tau \right)$, while $\alpha$ signifies the temperature coefficient of resistance of the detector. Finally, the material’s thermal properties can be calculated using Equation (6) and by fitting the sensor’s temperature change to Equation (4) to determine the material’s thermal properties.

Given the complexity of the dimensionless time function $D\left(\tau \right)$, various assumptions and methods are employed to derive numerical solutions or approximate analytical values [26,39,41,42]. Hot square and hot disc were two insights provided by the original TPS method for use in experiments [26]. In this paper, we chose to use the latter approximation chosen. By applying the TPS method in an experimental setting, the average temperature increase throughout the sensor volume can be determined. Equation (4) is then fitted to the previously recorded experimental data, enabling the calculation of the thermal performance of material around the sensor.

In this paper, we make use of the finite element simulation method to conduct a thorough analysis of a real double helix sensor that has been positioned on a finite-size sample. The simulation takes into account the full dimensions and characteristics of both the probe and the sample, allowing for a detailed and comprehensive performance evaluation. This approach enables us to gain in-depth insights into the temperature and current density distribution across the detector, information that is typically challenging to obtain through experimental methods. By leveraging the validated sensor model, we were able to further simulate and analyze the thermal properties of various other materials, thereby extending the applicability of our findings.

3. Finite Element Simulation Method

In the TPS method, the double helical nickel strip embedded within the Kapton layer functions as both the core component in the actual experimental testing method and the key element in the simulation construction. The position of the sensor in the main body of the testing device is depicted in Figure 1b. The complete test system is illustrated in Figure 1a, which includes a stable adjustable power supply, a constant current output source, a voltage collector, a pressure sensor, a detector body, and a computer. The stable adjustable power supply is responsible for converting alternating current (AC) into direct current (DC) voltage, ensuring a consistent power supply. Meanwhile, the constant current output source can deliver a stable current ranging from 0 to 1 A. The voltage acquisition system is designed to collect the electric potential signals resulting from changes in the sensor’s resistance due to temperature variations. These signals are then used to produce the temperature change data using Ohm’s law and Equation (4). The pressure sensor measures the pressure within the enclosed chamber, providing crucial data for the analysis. The computer processes the collected data and performs fitting calculations to derive meaningful results. The sensor is strategically positioned between two identical samples, forming a sandwich structure, as shown in Figure 1b. In the numerical simulation, the same experimental parameters are applied, allowing for a precise replication of the physical setup. This approach ensures that the finite element method accurately models the sensor’s behavior under test conditions, providing valuable insights into the sensor’s performance and enabling the prediction of its response in various scenarios. By analyzing the actual device, we can validate the simulation results, thereby enhancing the reliability and applicability of the finite element model in studying the thermal properties of different materials.

3.1. Numerical Simulation Settings

The numerical simulation of the TPS test system, utilizing finite element simulation method, necessitates the undertaking of several preparatory steps [43,44,45]. These essential steps include the creation of a geometric model, the selection of appropriate materials, the designation of the desired physical field type, the configuration of the boundary conditions, the definition of the mesh, and the choice of solver type within the standard simulation program [46,47,48]. In the TPS method, a current is passed through a double helix sensor element to generate heat. As shown in Equation (6), both the temperature and resistance of the detector element rise over time. The form of the resulting transient temperature response curve is influenced by the thermal conductivity of the material surrounding the sensor element. For instance, the temperature of a sample with a high thermal conductivity rises at a slower rate.

This paper presents the initial resistance of the sensor element at room temperature, as calculated based on steady-state studies. Additionally, temperature response curves for different samples are obtained based on transient studies. The double helix geometry of the sensor is modeled based on the Archimedes spiral equation, as shown below [49,50]:

$$\left\{\begin{array}{c}x=(a_i+r_b\theta )cos\theta \\ y=(a_i+r_b\theta )sin\theta \\ r_b=(a_f-a_i)/2n\pi \end{array}\right.$$

(7)

where $a_i$ and $a_f$ represent the minimum and maximum radii of the spiral, respectively. Meanwhile, $n$ is employed to indicate the number of turns of the spiral ring. Additionally, $r_b$ represents the spiral growth rate, while $\theta$ denotes the angle of rotation of the spiral. The dimensions of the simulation are essentially identical to those of the hot disc method sensor employed in the experiment. The width of the double helix nickel wire and the spacing between neighboring nickel wires can be controlled with greater ease through the utilization of this equation.

In transient simulations, it is necessary to embed a double helix nickel strip within the Kapton layer, which can enhance the insulation and mechanical strength of the actual detector. Meanwhile, two samples of the same material must also be positioned at the two ends of the detector to form a similar sandwich structure, as illustrated in Figure 2a, which depicts the geometry of the detailed simulation model. Figure 2b illustrates the geometry of the double helix nickel strip. The materials used include PMMA (poly-methyl methacrylate), stainless steel, ceramic, nickel, and Kapton [35,36]. The thermal conductivities in the simulation are 0.19 W·m⁻¹·K⁻¹, 13.56 W·m⁻¹·K⁻¹, 1.45 W·m⁻¹·K⁻¹, 90.7 W·m⁻¹·K⁻¹, and 0.15 W·m⁻¹·K⁻¹, respectively. The specific heats are 1420 J·kg⁻¹·K⁻¹, 8000 J·kg⁻¹·K⁻¹, 2300 J·kg⁻¹·K⁻¹, 445 J·kg⁻¹·K⁻¹, and 1420 J·kg⁻¹·K⁻¹, respectively. The densities are 1190 kg·m⁻³, 460 kg·m⁻³, 657 kg·m⁻³, 8900 kg·m⁻³, and 1090 kg·m⁻³, respectively. The dimensions of the corresponding sensors are shown in

Table 1. The properties of the sensors used in the simulation.

Category	Width (μm)	Kapton Radius (mm)	Thickness (μm)	Total Thickness (μm)
Sensor I	200	6.4	10	50
Sensor II	300	14	10	50

.

This paper aims to use the finite element simulation method to analyze the heat transfer process in TPS method, and to use the model to test other samples. It is mainly completed via two kinds of research simulation: stationary and transient.

3.2. Stationary Study

In the TPS method, the sensor, which consists of a double helix nickel strip embedded in Kapton, serves as the core component. Consequently, a stationary study of the sensor must be conducted first. This study aims to obtain the initial resistance value of the sensor at room temperature and to observe the variation in resistance values at different meshes. In this simulation, the geometry is composed solely of a double helical nickel strip, as illustrated in Figure 2b. Correspondingly, the resistance values calculated from the simulation, along with the parameters modeled using Equation (7), are presented in

Table 2. The modeling parameters and calculated resistance values of the two types of sensors.

Category	${\mathit{a}}_{\mathit{i}}$ (μm)	${\mathit{a}}_{\mathit{f}}$ (μm)	Number of Laps	Spacing (μm)	Resistance (Ω)
Sensor I	200	6400	16	200	11.885
Sensor II	300	14,000	20	300	21.340

.

Then, the AC/DC physical field interface is chosen within the physical field settings of the finite element simulation software (COMSOL 6.1.0.252). The terminals and grounding connections are applied to the two ends of the nickel strip, while the remaining boundaries are configured to be electrically insulated. The ambient temperature for the simulation setup is consistently maintained at room temperature.

The dimensions of the sensor were optimized to make them as close as possible to those of the actual sensor, both in terms of size and resistance values. At this juncture, the resistance value derived from the simulation will be employed to ascertain the power dissipated in the transient simulation, whereby $P=I^{2}R$ and $I$ represents the applied current.

In finite element simulation methods, the size of the mesh affects the accuracy of the computational results [51,52,53,54]. Figure 3 shows the resistance variation curve of the sensor at different meshes, where number 1 to number 9 denote extremely coarse to extremely fine meshes, respectively. The resistance value of the reference is calculated using $R=\rho L/s$, where $\rho$, $L$, and $s$ represent the resistivity, total length of the heating wire, and cross-sectional area of the heating wire, respectively. At this juncture, the resistance values of Sensor I and Sensor II were calculated to be 11.946 Ω and 21.521 Ω, respectively. Among them, the maximum discrepancy between the calculated values of the nine meshes and the theoretical values is less than 1%. Similarly, the maximum error in the calculated values is less than 1%. But the case with the extremely fine mesh is closest to the actual situation. Consequently, the subsequent simulation process of the two types of sensors in this paper is conducted using an extremely fine grid to ensure the accuracy of the calculated results.

3.3. Transient Simulation

The complete model for transient simulation is shown in Figure 2a. As the sensor is capable of functioning as a resistive heating source, the following equation is satisfied [55]:

$$\nabla ·\overrightarrow{J}=q$$

(8)

where $\overrightarrow{J}$ and $q$ are used to denote current density and charge density, respectively. In the finite element simulation software, the current is generated by utilizing the terminal and ground at the boundaries of both ends of the nickel strip. All domains in the model, with the exception of the nickel strips, are electrically insulated ($\overrightarrow{n}·\overrightarrow{J}=0$). In addition to the boundary of the model, there is an adiabatic boundary, i.e., $\overrightarrow{n}·\overrightarrow{q}=0$, where $\overrightarrow{n}$ and $\overrightarrow{q}$ represent the normal vector and the heat flow density vector, respectively. In the simulation, 293 K is set as the initial ambient temperature. Through temperature coupling, Joule heating is used as the heat source for the heat transfer module. In order to transfer heat to the Kapton layer and subsequently to the sample, a pair of thermal contact boundaries is defined and employed. The joint conductance ($h_{\mathrm{i}}$) can be defined as the sum of the constriction conductance ($h_{\mathrm{c}}$) and the thermal conductivity through the interstitial medium, ($h_g$), that is to say, $h_{\mathrm{i}}=h_{\mathrm{c}}+h_g$. The conventional Copper–Mikic–Yovanovich (CMY) [56] model establishes a relationship between $h_{\mathrm{c}}$ and several other variables, including the microscopic surface average slope $m_{asp}$, the mean height of the cusp angle ${\sigma }_{asp}$, the contact pressure $P_r$, and the surface microhardness $H_c$. This relationship is expressed in the following equation [56]:

$${\mathit{h}}_{\mathit{i}}={\mathit{h}}_{\mathit{c}}+{\mathit{h}}_{\mathit{g}}=\mathbf{1.25}{\mathit{\kappa }}_{\mathit{c}}{\displaystyle \frac{{\mathit{m}}_{\mathit{a}\mathit{s}\mathit{p}}}{{\mathit{\sigma }}_{\mathit{a}\mathit{s}\mathit{p}}}}{\left({\displaystyle \frac{{\mathit{P}}_{\mathit{r}}}{{\mathit{H}}_{\mathit{c}}}}\right)}^{\mathbf{0.95}}+{\displaystyle \frac{{\mathit{k}}_{\mathit{g}}}{\mathit{Y}+{\mathit{M}}_{\mathit{g}}}}$$

(9)

where ${\kappa }_{\mathrm{g}}$, $Y,$ and $M_{\mathrm{g}}$ denote the interstitial gas thermal conductivity, mean separation thickness, and gas parameters, respectively.

In Figure 4, we present the temperature distribution along the centerline of the sensor and the two-dimensional temperature distribution within the PMMA sample, which provides a deep understanding of the heat transfer process. As shown in Figure 4a, it is evident that the temperature at the center of the sensor significantly exceeds that of the surrounding area as time progresses. This observed temperature gradient reflects the design characteristic of the heat source being concentrated at the center of the sensor, which leads to the rapid concentration and effective transfer of heat. This phenomenon not only reveals the efficiency of the sensor but also highlights its importance in thermal conductivity measurements. We particularly note that, in the initial stages, the center temperature rises at a significantly faster rate, after which the heat gradually diffuses into the surrounding area, resulting in a typical non-uniform temperature distribution. This dynamic behavior aligns with theoretical predictions, emphasizing the impact of sensor design on the concentration and dissipation of heat flow [57]. In Figure 4b, the two-dimensional temperature distribution of the PMMA sample after 40 s of testing further corroborates this, exhibiting evident symmetry in the internal temperature diffusion, which is indicative of uniform heat propagation. We observe that, at this time point, the temperature peak within the sample closely aligns with the temperature distribution at the center of the sensor, suggesting that the heat transfer path is stable and predictable. This similarity not only supports the effectiveness of our finite element model in replicating the actual heat transfer process but also provides essential theoretical backing for future enhancements in thermal property-testing technologies [58]. These observations deepen our understanding of the mechanisms of heat transfer and offer valuable insights into the practical application of the transient plane source method, particularly as regards future material development and optimization.

Furthermore, the current density through the sensor also can be investigated through finite element simulation. As illustrated in Figure 5a, it can be observed that in the region where the curvature of the detector undergoes the greatest change, the modulus of the current density also exhibits the most significant variation. This phenomenon is also reflected in the current density line graph along the center of the sensor in Figure 5b, where it can be observed that the closer the point of measurement is to the center, where the curvature change is large, the larger the current density change in the nickel strip. In order to observe a clear difference in values, the current through the sensor was set to the maximum current achievable in the actual device.

4. Results and Discussions

In this paper, we analyze the physical modular functionality of the TPS method (Figure 1) and utilize the finite element simulation method to model the critical elements for measuring the thermophysical properties of materials using the TPS method. This includes both transient and stationary studies. Additionally, the Archimedes spiral equation is employed to ensure that the geometry of the design closely resembles that of the actual sensor. Correspondingly, the parameters used in the simulation are detailed in Table 1 and Table 3.

As shown in Figure 6, panels (a) and (b) illustrate the relative resistance and mesh density variation curves for Type I and Type II TPS sensors, respectively. The horizontal axis, labeled with numbers 1 to 9, represents grid sizes ranging from very coarse to very fine meshes. The red curve indicates the variation in mesh density, while the black curve shows the corresponding relative resistance variation in the sensor. The relative change in resistance refers to the absolute value of the difference between the numerical results of the current mesh computation and those of the previous mesh, divided by the results of the previous mesh. This aims to investigate the extent to which the numerical results of the current mesh increase relative to those of the previous mesh as the number of mesh elements increases [59]. It is apparent from the graph that as the mesh density increases, the curve of relative resistance variation approaches zero. However, increasing the mesh density further consumes more computational resources and extends the simulation time. Additionally, for finer mesh resolutions, increasing the mesh density beyond a certain point contributes little to improving the accuracy of the resistance calculation [60,61,62]. In this case, selecting a mesh type with a lower count and a relative change rate approaching zero offers a cost-effective balance between improving computational efficiency and ensuring accuracy. This is also reflected in Figure 3, where the sensor’s resistance converges to a constant value. Consequently, a finer mesh will be employed in subsequent simulations to balance computational accuracy and efficiency. Furthermore, the simulation results exhibit an error of less than 1% when compared to theoretical values.

In the stationary study, the current density distribution in the double helix nickel strip is also investigated in order to gain insight into the causes of heat source distribution, as shown in Figure 5a. From this distribution, it can be seen that the current density increases with increasing geometric curvature, except for a small region linked inside the helix (where the geometric curvature tends to zero). In the central region, due to the significant curvature changes at the edges of the spiral, the current density on both sides of the nickel strip connecting the two spiral lines is higher than at the origin, resulting in a V-shaped pattern, as shown in Figure 5b. The current density increases along the inner curve, with this gradient rise progressively intensifying as it approaches the central ring. The direct manifestation of this is in the form of high temperatures generated at the location of the central area of the sensor, as shown in each of the curves in Figure 4a. Despite this, nickel’s high thermal conductivity and thermal diffusivity result in a uniform temperature distribution across sensor cross-sections over longer timescales, even when the sensor exists significant current density gradients, as shown in Figure 4b. Secondly, the structural design of the double helix curve is conducive to minimizing or completely counteracting the formation of a magnetic field around the nickel strip induced when a DC current flows through it. In this way, any AC impedance that may arise is canceled out. This is especially important for minimizing the rise time of the current pulse through the sensor at the beginning of the transient recording and for collecting accurate data.

In the transient simulation, the TPS method is simulated by placing the detector between two identical media for testing (structure shown in Figure 2a). Since the actual sample size is not infinite in the TPS method, the measurement time needs to be limited to avoid the influence of boundary effects [63,64] on the simulation results. In this simulation, the radius of sensors were set to 6.4 mm and 14 mm, respectively. The test samples were two identical PMMA blocks, each with a height of 20 mm and a radius of 25 mm. The simulation parameters are set to correspond to the actual experiment by selecting the appropriate heating current and measurement time. During the simulation, the temperature rise curves of the sensors were documented and subsequently compared with the recorded experimental data. Here, the calculation of the thermal properties of the material relies on the transient data obtained from the simulation by fitting Equation (4).

Figure 7 presents the experimental and simulation temperature rise curves over time for the two types of sensor under different power levels. The experimental data points represent the average of three tests conducted under the same conditions, with a total uncertainty of less than 7%. The corresponding error bands are also plotted alongside the experimental data. In Figure 7a, the maximum temperature rise error between the experimental and simulation results is less than 0.2 K, while in Figure 7b, it is less than 0.3 K, indicating a high degree of consistency between the experimental and simulation results. Furthermore, it is noteworthy that the increase in power produced similar effects in both simulations and experiments. However, it is important to emphasize that different power levels were applied to the two sensors in order to achieve comparable temperature changes. This adjustment ensures that the temperature variations observed in each sensor are directly comparable, despite the different power levels used. Specifically, larger sensors require more power to achieve the same temperature rise.

Subsequently, additional samples were subjected to further testing using the aforementioned model, the results of which are presented in Figure 8 below. Figure 8a,b illustrate the temperature rise curves over time for the stainless steel and ceramic materials, respectively. Ultimately, Equation (9) was employed for the purpose of fitting, with the objective of calculating the thermal performance parameters of PMMA, stainless steel, and ceramics, respectively. The relative errors were then calculated based on the reference values, and the results are presented in Table 3. The average relative error of both sensors is less than 3%.

Table 3. Calculated thermal conductivity of materials.

Category	Material	Reference (W·m−1·K−1) [36,65]	Calculated (W·m−1·K−1)	Relative Error
Sensor I	PMMA	0.19	0.194	2.11%
	Stainless	13.55	13.81	1.92%
	Ceramic	1.45	1.479	2.00%
Sensor II	PMMA	0.19	0.186	1.96%
	Stainless	13.55	13.27	2.07%
	Ceramic	1.45	1.42	2.03%

Table 3. Calculated thermal conductivity of materials.

Category	Material	Reference (W·m−1·K−1) [36,65]	Calculated (W·m−1·K−1)	Relative Error
Sensor I	PMMA	0.19	0.194	2.11%
	Stainless	13.55	13.81	1.92%
	Ceramic	1.45	1.479	2.00%
Sensor II	PMMA	0.19	0.186	1.96%
	Stainless	13.55	13.27	2.07%
	Ceramic	1.45	1.42	2.03%

5. Conclusions

The aim of this paper is to simulate the working principle of the TPS method by means of the finite element method in order to create a three-dimensional model for test use. The temperature rise profile of the sample is reproduced from this model. Additionally, this paper analyzes the current density distribution in the nickel metal strip within the sensor and its relationship with the heat source distribution through both transient and steady-state studies. The current density tends to increase along the inside of the bar curve. The temperature rise data obtained from the simulation closely match the corresponding experimental data for PMMA. The model is then used again to simulate tests on other materials and the corresponding thermal performance parameters are calculated, and these also match the reference results. This simulation has the potential to be utilized for future geometry and parameter estimate optimization, and provides a theoretical reference for detector design.

Although we performed a detailed modeling of the transient plane source (TPS) method using finite element analysis and achieved good agreement with experimental data, certain limitations remain. Specifically, the model assumes uniform material properties, while in practical applications, materials may exhibit inhomogeneities, defects such as bubbles or impurities, which could influence thermal conductivity and lead to discrepancies between experimental results and simulation outcomes. For future research directions, we propose further exploration in the following areas: First, the scope of the study can be expanded to assess the thermal conductivity of different types of materials, particularly composite materials with complex structures or specialized properties. Second, the development of hybrid approaches that integrate data-driven models with experimental techniques should be pursued to enhance the accuracy and efficiency of thermal conductivity measurements. Third, the impact of environmental factors should be considered to investigate how material properties vary under different conditions, thereby providing a more robust theoretical and experimental foundation for the application of new materials. Finally, we recommend establishing standardized testing procedures to enable the consistent comparison and validation of results across different laboratories and research teams.