## 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]:  
\[a\
abla^{2}T(\overrightarrow{r},t)+\frac{Q(\overrightarrow{r},t)}{\rho c}= \frac{\partial T(\overrightarrow{r},t)}{\partial t} \tag{1}\]  
In the above equation, \(\rho\) is the density, \(c\) is the material specific heat, and \(T(\overrightarrow{r},t)\) 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]:  
\[T=T_{0}+\frac{1}{\left(4\pi att\right)^{3/2}}exp(-\frac{r^{2}}{4 at}) \tag{2}\]  
where \(T_{0}\) represents the initial temperature of the sample. By convolving the above equation with \(\frac{Q(\overrightarrow{r},t)}{\rho c}\), the general solution for the heat source \(Q(\overrightarrow{r},t)\) 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 \(\sigma^{2}=\frac{a(t-t^{\prime})}{b}\), with \(t^{\prime}=0\), and defining the characteristic time \(\tau=\frac{\sqrt{at}}{b}\), 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]:  
\[\Delta\overline{T}(\tau)=\frac{2\pi}{\pi b(m+1)}\frac{P_{0}}{2\pi^{3/2}bm(m+1) \lambda}\int_{0}^{\tau}\frac{d\sigma}{\sigma^{2}}{\sum}_{k=1}^{m}\frac{kb}{m} \times\sum_{l=1}^{m}lexp\left(-\frac{k^{2}+l^{2}}{4m^{2}\sigma^{2}}\right)I_{0 }\left(\frac{kl}{2m^{2}\sigma^{2}}\right) \tag{3}\]  
\[\Delta\overline{T}(\tau)=A+\frac{P_{0}}{\pi^{2}\frac{b}{b}\lambda}D(\tau) \tag{4}\]where \(A\) represents the initial temperature rise constant, \(I_{0}\) denotes the zero-order Bessel function, and \(P_{0}\) is the input power. \(D(\tau)\) denotes the dimensionless time function, which can be defined by the following equation [3; 36; 37; 38; 39; 40]:  
\[D(\tau)=\frac{1}{m^{2}(m+1)^{2}}\!\int_{0}^{\tau}\frac{d\sigma}{\sigma^{2}}\! \sum\
olimits_{k=1}^{m}\!\frac{kb}{m}\times\sum\
olimits_{l=1}^{m}\!exp\! \left(-\frac{k^{2}+l^{2}}{4m^{2}\sigma^{2}}\right)I_{0}\!\left(\frac{kl}{2m^{2 }\sigma^{2}}\right) \tag{5}\]  
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:  
\[R(t)=R_{0}\!\left[1+\alpha\Delta\overline{T}(\tau)\right] \tag{6}\]  
where \(R_{0}\) represents the initial resistance of the TPS detector before transient test recording. \(R(t)\) denotes the resistance after a temperature change of \(\Delta\overline{T}(\tau)\), 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 \(D(\tau)\), 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.