## 3 Theory of Dynamic Correction  
According to the numerical calculation results, the heat flux loss on the back of the calorimeter increases over time. Thus, a dynamic correction is necessary to obtain accurate measurement results. Only the heat flux loss from the bottom of the CVD diamond was considered in the present corrections, and the heat exchange in the lateral direction was ignored. However, it will become evident in the following discussion that the lateral heat exchange does not have a significant influence on the corrections. The heat dissipation effect of the back-filling material is simplified as a one-dimensional heat conduction problem of two different materials, as shown in Figure 7, where \(\rho_{1}\), \(c_{1}\), and \(k_{1}\) are the density, the specific heat, and the thermal conductivity of the calorimeter element (CVD diamond in the present study). \(\rho_{2}\), \(c_{2}\), and \(k_{2}\) are the density, the specific heat, and the thermal conductivity of the backing material (air or epoxy). According to reference [14], the calculation formula of the heat penetration depth for the lining material is as follows:  
\[x=4\sqrt{\alpha t} \tag{6}\]  
Figure 6: \(q_{\text{b}}/q_{0}\) versus time for different radii of the CVD diamond. The effective test time of transient facilities is typically in the order of milliseconds. The heat penetration depth of epoxy would only be 0.4 mm at a longer test time of 100 ms. Thus, the backing material can be regarded as a one-dimensional semi-infinite body, where the surface temperature is monitored using the platinum thin-film resistance, which is labeled \(T_{b}\) in Figure 7.  
The heat conduction equation for a two-layer semi-infinite system as shown in Figure 7 is defined as  
\[\frac{\partial^{2}T_{1}}{\partial x^{2}}=\frac{1}{\alpha_{1}}\frac{\partial T _{1}}{\partial t},\ \frac{\partial^{2}T_{2}}{\partial x^{2}}=\frac{1}{\alpha_{2}}\frac{\partial T _{2}}{\partial t} \tag{7}\]  
where subscript 1 is the calorimeter element and subscript 2 is the substrate. Their boundary conditions are as follows  
\[ \begin{array}{ll}x=0&-k_{1}\frac{\partial T_{1}}{\partial x}=q_{0}\\ x=l&-k_{1}\frac{\partial T_{1}}{\partial x}=k_{2}\frac{\partial T_{2}}{ \partial x},T_{1}=T_{2}\\ x=\infty&T_{2}=0\end{array}  \tag{8}\]  
Thus, by taking the Laplace transforms, the temperature at the back of the calorimeter element or the surface of the substrate is obtained by solving Equations (7) and (8) [14]:  
\[\overline{T_{L}}=\frac{2\eta_{0}\sqrt{\alpha_{1}}}{k_{1}P^{\frac{3}{2}}\bigg{[} (1+a)e^{L\sqrt{\frac{p}{\alpha_{1}}}}-(1-a)e^{-L\sqrt{\frac{p}{\alpha_{1}}}} \bigg{]}} \tag{9}\]  
where \(a=\sqrt{\frac{\rho_{2}c\delta k_{2}}{\rho_{1}c_{1}k_{1}}}\) is the ratio of the back-filling material to calorimetric element material; \(P\) is the Laplace operator. Subsequently, the heat flux loss to the substrate is obtained by \(\overline{T_{L}}\) and the one-dimensional semi-infinite heat conduction theory:  
\[q_{\text{loss}}=\sqrt{\frac{\rho_{2}c\delta k_{2}}{\pi}}\bigg{[}\frac{T_{b}(t) }{\sqrt{t}}+\int_{0}^{t}\frac{T_{b}(t)-T_{b}(\tau)}{(t-\tau)^{\frac{3}{2}}}d \tau\bigg{]} \tag{10}\]  
By combining Equations (9) and (10), the following expression is obtained using the Taylor expansion and the inverse Laplace transform:  
\[\frac{q_{\text{loss}}}{q_{0}}=\frac{2a}{1+a}\sum_{n=0}^{\infty}\bigg{(}\frac{1 -a}{1+a}\bigg{)}^{n}erfc\bigg{(}\frac{(2n+1)L}{2\sqrt{\alpha_{1}t}}\bigg{)} \tag{11}\]  
where \(\alpha_{1}\) is the thermal diffusion coefficient of the calorimeter element. Detailed processes of the above derivation can be found in [14; 36]. Figure 8 shows the relationship between \(q_{\text{loss}}/q_{0}\) and time for three commonly used back-filling materials, where the calorimeter element is CVD diamond; its thermal diffusion coefficient can be  
Figure 7: The simplified one-dimensional heat transfer model between the calorimeter element and the back-filling material.