# A Fast Calibration Method for Sensors of Atmospheric Detection System
## 2 System Physics Modeling
### Dynamic Heat Transfer Model of System
#### 2.2.2 Dynamic Heat Transfer Model of Tube-Air-ADS  
The structural composition of the Tube-Air-ADS is as shown in Figure 2a. The main components considered in the modeling process are air, glass tube, and foam. In order to simplify the modeling process, the geometry of the system is reduced to a two-dimensional plane rotating around the \(z\) axis, as shown in Figure 2b, and the coordinates of any point in space can be expressed in terms of height and radius. The coordinate sets of air, glass tube, and foam parts can be represented as \(\Omega_{air}(z,r)\), \(\Omega_{tube}(z,r)\), and \(\Omega_{ foam}(z,r)\), respectively. A triangular grid is used to analyze the two-dimensional area of the system, and the coordinates of the sectioning points are calculated by physical field. The results of the grid sectioning are shown in Figure 2c. During the calibration of the detection system, power supply and communication with external equipment are required. The air in the glass tube flows through the holes in the polyethylene cover. The heat transfer of air in the glass tube is studied in the form  
Figure 2: (**a**) Heat transfer model of Tube–Air–ADS structure diagram. (**b**) Two-dimensional structures. (**c**) Grid segmentation. of fluid. The flow of air in the glass tube affects the heat transfer in different areas. The velocity field \(\mathbf{u}\) is updated using the Navier-Stokes equations to describe the air flow in the glass tube [42], and the velocity vector \(\mathbf{u}(z,r,t)\) of the air region \(\Omega_{air}(z,r)\) in the tube at any given time \(t\) (\(t>0\)) can be obtained by iterating the differential equation  
\[\left\{ \begin{array}{l}\rho_{air}\Big{(}\frac{\partial\mathbf{u}}{\partial t}+ \mathbf{u}\cdot\
abla\mathbf{u}\Big{)}=\
abla\cdot\left[-\rho_{air}\mathbf{I} +\mu\big{(}\
abla\mathbf{u}+\
abla\mathbf{u}^{T}\big{)}\right]+\rho_{air} \mathbf{g}+\mathbf{F}\\ \rho_{air}\
abla\cdot\mathbf{u}=0\end{array} \right., \tag{4}\]  
where \(\rho_{air}\) is air density, \(\
abla\cdot\) is the dispersion operator, \(\mathbf{I}\) is unit matrix, \(\mu\) is aerodynamic viscosity, \(\mathbf{g}\) is gravitational acceleration, and \(\mathbf{F}\) is the external force acting on a fluid. Use the special forms of the energy equation mentioned in [43] to update the temperature field \(\mathbf{T}\) in the ADS. The temperature distribution of the air region in the tube at \(t\) moment is related to the solution of the velocity field \(\mathbf{u}(z,r,t)\), while temperature field \(\mathbf{T}(r,z,t)\) changes the velocity field distribution. The temperature field equation for air region \(\Omega_{air}(z,r)\) is  
\[\rho_{air}c_{p\_air}\left(\frac{\partial\mathbf{T}}{\partial t}+\mathbf{u} \cdot\
abla\mathbf{T}\right)-\
abla\cdot\left(k_{air}\
abla\mathbf{T}\right)= Q_{2}+Q_{p}+Q_{vd}, \tag{5}\]  
where \(c_{p\_air}\) is the constant pressure heat capacity of air, \(k_{air}\) is the thermal conductivity of air, \(Q_{2}\) is the heat power transferred from the outside to the air, \(Q_{p}\) is the thermal power generated by the compression or expansion of the air, and \(Q_{vd}\) is viscous dissipation. The heat transfer of the system involves two other solid materials, glass tube wall \(\Omega_{tube}(z,r)\) and insulating foam \(\Omega_{foam}(z,r)\) on the outside of the ADS. The temperature field equation for solid materials is  
\[\rho_{tube}c_{p\_tube}\Big{(}\frac{\partial\mathbf{T}}{\partial t}+\mathbf{u }\cdot\
abla\mathbf{T}\Big{)}-\
abla\cdot\left(k_{tube}\
abla\mathbf{T} \right)=Q_{3}+Q_{led\_tube}, \tag{6}\]  
\[\rho_{foam}c_{p\_foam}\Big{(}\frac{\partial\mathbf{T}}{\partial t}+\mathbf{u }\cdot\
abla\mathbf{T}\Big{)}-\
abla\cdot\left(k_{foam}\
abla\mathbf{T}\right) =Q_{4}+Q_{led\_foam}, \tag{7}\]  
where \(\rho_{tube}\) is tube density, \(c_{p\_tube}\) is the constant pressure heat capacity of tube, \(k_{tube}\) is the thermal conductivity of tube, \(Q_{3}\) is the heat power transferred from the outside to the tube, and \(Q_{led\_tube}\) is the elastic thermal power of tube.