## 3 Measurement System for Time Constant Based on High Temperature Furnace
### Analysis of Heat Transfer Environment in High Temperature Furnace  
In practical engineering, there are usually two or three heat transfer modes at the same time, which is called composite heat transfer. In the measurement system for time constant of the thermocouple based on high temperature furnace, the thermocouple is quickly inserted and stays in the uniform temperature zone at a distance of 120 mm from the furnace mouth, and the thermocouple is heated. There are three heat transfer modes: radiative heat transfer from furnace wall to the thermocouple, convective heat transfer from the gas to the thermocouple, and heat conduction from the surface of the thermocouple junction to the inside. For bare thermocouple, the thermal resistance of heat conduction is very small relative to that of radiative and convection heat transfer in the heat transfer environment of high temperature furnace. The thermal resistance of heat conduction can be neglected, and it is considered that the temperature distribution inside the thermocouple junction is approximately uniform at any time. Therefore, the heat transfer modes between the thermocouple and the environment in the high temperature furnace are mainly radiative heat transfer and convective heat transfer. Because the heat transfer surface area of the thermocouple junction is much smaller than the inner wall area of the high temperature furnace, the quantity of radiation heat transfer can be expressed as  
\[\Phi_{r}=\varepsilon A\sigma\Big{[}(T_{\omega}/100)^{4}-(T/100)^{4}\Big{]}, \tag{4}\]  
where \(\varepsilon\) is the emissivity of the thermocouple, \(A\) is the heat transfer area, \(\sigma\) is the Steffan-Boltzmann constant, \(T_{\omega}\) is the temperature of the furnace wall, and \(T\) is the temperature of the thermocouple. In order to analyze and calculate complex heat transfer problems, Equation (4) can be generally converted into a function of the surface heat transfer coefficient as  
\[\Phi_{r}=h_{r}A(T_{\omega}-T)\, \tag{5}\]  
\[h_{r}=\frac{\varepsilon\sigma\Big{[}(T_{\omega}/100)^{4}-(T/100)^{4}\Big{]}}{ T_{\omega}-T}, \tag{6}\]  
\(h_{r}\) can be defined as surface heat transfer coefficient of radiative heat transfer. Assuming that \(\varepsilon=0.4\), when the temperature of the furnace wall \(T_{w}\) is constant, the surface heat transfer coefficients \(h_{r}\) are obtained at different temperatures of the thermocouple, as shown in Figure 6. It can be seen from the figure that the surface heat transfer coefficient is related to the temperatures of the furnace wall and the thermocouple. When the temperature of the furnace wall \(T_{w}\) is constant, \(h_{r}\) increases with the increase of the temperature of the thermocouple. When the temperature of the thermocouple is constant, \(h_{r}\) increases with the increase of the temperature of the furnace wall \(T_{w}\). For convection heat transfer, according to [PERSON]'s law of cooling, the quantity of convection heat transfer can be expressed as  
\[\Phi_{c}=h_{c}A\big{(}T_{g}-T\big{)}, \tag{7}\]  
where \(h_{c}\) is the convective heat transfer coefficient, \(A\) is the convective heat transfer area which is approximately equal to the radiative heat transfer area, and \(T_{g}\) is the temperature of the gas around the thermocouple. In the high temperature furnace, the total quantity of heat transfer of the thermocouple can be expressed as the sum of the quantity of radiative heat transfer and convection heat transfer.