## Novel Transient Calorimeter (NTC)
### Influence of Non-Ideal Heat Conduction  
Equation (1) is applied with the assumption that the heat loss from the CVD element can be assumed to be negligible. However, the calorimeter element must be fixed and sealed during sensor manufacturing. Thus, the measurement is affected because the packaging causes heat loss of the calorimeter element, which leads to inaccuracy. Numerical simulations were conducted to understand the heat exchange between the CVD diamond and its surroundings due to the straightforward operation and detailed information. The governing equation is the axisymmetric unsteady heat conduction equation:  
\[\frac{\partial T(r,z,t)}{\partial t}=\frac{k_{i}}{\rho_{i}c_{i}}\bigg{(}\frac{ \partial^{2}T}{\partial r^{2}}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac {\partial^{2}T}{\partial z^{2}}\bigg{)}(i=1,\ 2,\ 3,\ 4) \tag{4}\]  
where \(r\) and \(z\) are the radial and axial directions of the calculation model, respectively; the subscripts 1, 2, 3, and 4 represent the CVD diamond, air/epoxy insulation, fiber-reinforced plastics (FRP sensor base), and stainless steel (test model), respectively, as shown in Figure 4. Equation (4) is solved using the finite difference method for spatial discretization and a fourth order Runge-Kutta method for time integration [36]. Since the thickness of the platinum film on the back of the CVD diamond is only about 0.2 \(\upmu\)m, and the diameter of the platinum film lead line is only 0.1 mm, the heat dissipation by the platinum film and lead line on is ignored in this calculation. Due to the axial symmetry of the computational model, only half of the geometry is considered, as shown in Figure 4. In the present calculation, the CVD diamond thickness is fixed at \(L\) = 0.2 mm, which is used in the author's home-made calorimeters; the radius is \(r_{0}\) = 2.5, 5 and 10 mm, respectively; the radius of the back-filling material is \(r_{1}\) = \(r_{0}\) - 0.5;  
Figure 4: Schematic drawing of the simulated model (not to scale). Figure 3: Output voltage of the NTC and Type-E thermocouple sensors. the outer diameter of the FRP support is \(r_{2}=r_{0}+0.2\), and the outer diameter of the stainless steel model is \(r_{3}=r_{2}+5\). The initial temperature of the calculation model is set to \(T_{\infty}=293\) K for the calculation. A constant uniform heat flux (\(q_{0}=1.0\) MW/m\({}^{2}\)) is loaded on the top surface. Thus, the boundary conditions at the top surface are as follows:  
\[\left(\frac{\partial T}{\partial z}\right)_{z=0}=\frac{q_{0}}{k_{i}}\;(i=1,\; 3,\;4)\;t>0 \tag{5}\]  
Other boundary conditions are shown in Figure 4. In addition, the temperature and heat flux satisfy the continuity condition at the interface between the two different materials. It also needs to be noted that the back-filling material is air or epoxy insulation, both of which have very low heat conductivity and reduce the heat loss from the CVD to its surroundings. These materials are commonly used in calorimeters, where epoxy can be used to enhance the strength of the sensor. The model material here is made of stainless steel, which is commonly used in shock tunnel experiments; and its physical parameters are shown in Table 1.  
The equations were solved by our developed C++ program in this study and structured grids of \(250\times 300\) are applied. The zones inner the calorimeter and the calorimeter/back-filling material interface are incorporated with clustered points to provide good spatial resolution. The temperature distribution in the calculation model at different times were output during the calculation process to understand the heat transfer inside the senor. Temperature distributions were generated by the software Tecplot 360 as shown in Figure 5.  
Figure 5 shows the simulated temperature distribution inside the calculation model for \(r_{0}=2.5\) mm at the moment of \(t=12\) ms; Figure 5a shows the results for the air as back-filling material, and Figure 5b depicts the result of using epoxy. As expected, a more complicated heat conduction process occurs around the CVD diamond.