## 3 Theory of Dynamic Correction
found in Table 1. The larger the value of \(a\), the larger the value of \(q_{\text{loss}}\)/\(q_{0}\), and \(q_{\text{loss}}\)/\(q_{0}\) increases over time. Of course, air is the best back-filling material of the calorimeter element, and its largest deviation is less than 0.3% (at 30 ms). However, the pressure resistance is poor when air is used as the back-filling material, which limits the application range of the sensor. This is especially the case in high-enthalpy flows, where small particles of metallic or nonmetallic materials exist. Epoxy is commonly used as a filling material for calorimeters to enhance the strength. Unfortunately, its heat flux loss is much higher than that of air, and \(q_{\text{loss}}\)/\(q_{0}\) is more than 10% for a test time of 3 ms. Thus, corrections are needed if high-accuracy heat flux results are expected. According to the energy conservation law, the loaded heat flux on the surface \(q_{0}\) has the following relationship with the absorbed heat flux of the CVD diamond \(q_{\text{b}}\) and the heat flux loss to its surroundings \(q_{\text{loss}}\):  
\[q_{0}=q_{\text{b}}+q_{\text{loss}} \tag{12}\]  
In actuality and for simplicity, \(q_{\text{b}}\) is the heat flux value calculated by Equation (2) from the sensor output temperature signal. Additionally, \(q_{\text{loss}}\) is the heat flux transferred from the calorimeter element to the backing material, namely, the heat flux rate lost from the CVD diamond. \(q_{\text{loss}}\) is calculated from Equation (11) and is brought into Equation (12). We also define the correction coefficient \(\xi\) of the transient calorimeter as:  
\[\xi(a,t)=\frac{2a}{1+a}\sum_{n=0}^{\infty}\left(\frac{1-a}{1+a}\right)^{n}erf \left(\frac{(2n+1)L}{2\sqrt{\alpha_{1}t}}\right) \tag{13}\]  
Then, Equation (12) can be rewritten as  
\[q_{0}=\frac{q_{b}}{1-\xi(a,t)} \tag{14}\]  
where \(\xi\) is a function of the effusivity ratio \(a\) and time \(t\) for a specific calorimeter. The dynamic correction of the calorimeter measurement results can be made using Equation (14) if the thermal diffusion coefficient of the back-filling material can be obtained. The parameters of epoxy in the simulation in Section 2.2 can be obtained from Table 1. The dynamic correction in Equation (14) was first used to correct the simulated results. The calorimetric element radius of \(r_{0}=2.5\) mm is used as an example; the calculation result in Figure 6 was modified using Equation (14), and the results are shown in Figure 9. The error before the correction is 33% at 30 ms, and the overall error after the correction is within 7.8%. This deviation is within 2.3% if the test time is smaller than 12 ms. Although only one-dimensional heat conduction is considered in the modification, it is found that the lateral heat flux loss from the CVD diamond can be ignored. The results are considered excellent for heat flux measurements. \[q_{\text{loss}}=\sqrt{\frac{\rho_{2}c\delta k_{2}}{\pi}}\bigg{[}\frac{T_{b}(t) }{\sqrt{t}}+\int_{0}^{t}\frac{T_{b}(t)-T_{b}(\tau)}{(t-\tau)^{\frac{3}{2}}}d \tau\bigg{]} \tag{10}\]0 \(q_{\text{loss}}\)/\(q_{0}\) versus time based on Equation (11) for three different back-filling materials. It is observed that the measurement error of the transient calorimeter can be reduced to acceptable deviation level after the dynamic correction (Equation (14)) since the thermal diffusivity of the backing material is known. During the manufacturing process of the sensor, the epoxy is attached to the back of the calorimetric element by potting and subsequently solidifies. However, the epoxy is a compound, and the physical parameters are difficult to determine. Meanwhile, the potting quality is difficult to maintain, and the physical parameters of the bottom lining materials are quite different in actual sensors. Therefore, it is necessary to calibrate the physical parameters of the back-filling material before applying the dynamic correction of the NTC calorimeter. Subsequently, the dynamic correction coefficient of each sensor can be obtained.