## 3 Mathematical Model and Numerical Simulation
### Numerical Simulation  
A finite-element model was established using COMSOL 4.4 software to estimate the cooling capacity of the thermal control system, and the thermal transfer process was mainly analyzed to simulate the temperature rise and the thermal distribution in the laser cabinet. The heat dissipation of the mid-infrared OPO laser is a multi-physics field coupled process with heat conduction, heat convection and gas turbulent flow. The material properties and size of the finite-element model are shown in Table 2. The whole laser cabinet structure including fins was made by aluminum material. The aluminum 6063-T83 with thermal conductivity 201 W/(m-K) was defined as the cabinet material in the software. When the thermal control system works, the cold side of the TEMs absorbs the heat produced by the pump laser module, and releases heat at the hot side simultaneously. This heat quantity was taken as the heat source in the model and delivered through thermal conduction to the laser cabinet. A 500 W boundary heat source was defined to simulate the quantity of heat produced by the TEMs. There were 14 cooling fins at the bottom of the laser cabinet. The shape of each fin was 300 \(\times\) 50 \(\times\) 2 mm 3, and the fin spacing was 20 mm. Seven fans were defined in the software, and the flow rate was set at 0.011 m\({}^{3}\)/s. The thermal conductivity of the heat pipe was set in segmented mode to make a closer equivalent simulation of its physical features. The thermal conductivity of the evaporation section was set at \(2\times 10^{4}\) W/(m\(\cdot\)K), while the thermal conductivity of the condensation section was set at 500 W/(m\(\cdot\)K). The finite-element mesh model is shown in Figure 8 and the size unit of this model is in millimeters, the results for the temperature analysis of the model at ambient environment temperatures of 25 degrees and 45 degrees are shown in Figure 9, and the temperature units are degrees. It can be deduced that the highest temperature rise occurred at the left edge of the laser cabinet and with almost the same value at the different ambient temperatures. It is assumed that the two groups of TEMs were close to the left edge of the cabinet with the highest temperature at the hot side, since the simulated heat source was 500 W and the temperature rise was almost 20 K, thus the thermal resistance (\(R_{3}\)) for the TEMs was 0.04 K/W. The other two groups of TEMs in the middle of the cabinet almost had a 15 K temperature rise at the hot side and the thermal resistance (\(R_{3}\)) for them was 0.03 K/W. The cooling capacity of the system is discussed in two situations--at 25 \({}^{\circ}\)C (room temperature) and 45 \({}^{\circ}\)C (high temperature). From the temperature rising simulation results in Figure 9, we approximately calculated the TEC performance parameters, \(R_{m}\), \(K_{m}\) and \(S_{m}\), for \(T_{h}\) = 50 \({}^{\circ}\)C and the following parameters were obtained: \(R_{m}\) = 0.985 \(\Omega\), \(K_{m}\) = 0.946 K/W and \(S_{m}\) = 0.048 V/K. Using Equation (11), in the first situation, \(T_{a}\) = 298.15 K (25 \({}^{\circ}\)C), the TEC drive current (I) was defined as 5A, and it was assumed that \(R_{1}\) = \(R_{2}\) = 0.1 K/W and \(T_{1}\) = 293.15 K (20 \({}^{\circ}\)C) at which the pump source can work stably. For the two groups of TEMs with \(R_{3}\) = 0.04 K/W, the cooling output of each TEM was \(Q_{c1}\) = 39.5 W, and for the other two groups of TEMs with \(R_{3}\) = 0.03 K/W, the cooling output of each TEM was \(Q_{c2}\) = 40 W. Consequently, the total cooling output (\(Q_{c}\)) of the 16 TEMs was about 636W.