## 3 Mathematical Modeling  
A mathematical model is developed for the system that includes the heat removal rate of various components and interfaces. The proposed model is divided into three subdivisions such as: (i) a model for the thermoregulatory part; (ii) a model for the closed room system; (iii) a model for the primary TEC cooling system. For the mathematical analysis, the cold plate heat exchanger inlet and outlet temperature are considered as \(T_{CPI,1}\) and \(T_{CPO,1}\), respectively. The heat removal rate of the cold plate heat exchanger is calculated using the given formula for both the top and bottom sides:  
\[\dot{Q}_{H,TEC1}=\dot{Q}_{loss1}+\dot{m}_{w}C_{pw}(T_{CPI,1}-T_{CPO1}) \tag{1}\]  
where \(\dot{Q}_{H,TEC1}\), \(\dot{Q}_{loss1}\), \(\dot{m}_{w}\), \(C_{pw}\), \(T_{CPO,1}\), \(T_{CPI,1}\) are heat removal rate from hot side of the TEC module, total heat loss from top side, mass flow rate of water (=density of water \(\times\) volume flow rate of water) inside cold plate heat exchanger, specific heat of water, cold plate outlet temperature and cold plate inlet temperature for the top side of the TEC cooling system, respectively. Figure 1 presents schematically the overall energy flow in the system. The heat loss from the top side of the clot plate is calculated using the following equation:  
\[Ra_{L}=G_{r}P_{r} \tag{2}\]  
where for natural convection \(Ra_{L}\) (product of Grashof and Prandtl number), \(G_{r}\) and \(P_{r}\) are Rayleigh number, Grashof number, and Prandtl number, respectively. The Nusselt number can be obtained from:  
\[Nu=0.13 Ra_{L}^{1/3} \tag{3}\]  
where \(Nu\) is Nusselt number which is used to find the heat transfer coefficient. \[Nu=\frac{hL}{\lambda} \tag{4}\]  
Figure 1: Schematic diagram of show the energy flow of the system (**a**) TEC with cold plate and air channel, (**b**) chiller and closed room with copper heat exchanger. where \(h\), \(L\), and \(\lambda\) are heat transfer coefficient, length of the cold plate heat exchanger, and thermal conductivity, respectively. The heat loss from the top side of the surface can be calculated as:  
\[\dot{Q}_{loss1}=hA_{s}(T_{s}-T_{\infty}) \tag{5}\]  
where \(\dot{Q}_{loss}\), \(A_{s}\), \(T_{s}\) and \(T_{\infty}\) are total heat loss from the top surface, surface area of the cold plate heat exchanger, surface temperature of the cold plate, and ambient temperature of the room. However, to find the total heat removal rate of the bottom surface, it can be written as:  
\[\dot{Q}_{H,TEC2}=Q_{loss2}+\dot{m}_{w}C_{pw}(T_{CPO,2}-T_{CPI,2}) \tag{6}\]  
where \(\dot{Q}_{H,TEC2}\), \(\dot{Q}_{loss2}\), \(\dot{m}_{w}\), \(C_{pw}\), \(T_{CPO,2}\), \(T_{CPI,2}\) are heat removal rate from hot side of the TEC module at the bottom, total heat loss from bottom side, mass flow rate of water (= density of water \(\times\) volume flow rate of water) inside cold plate heat exchanger, specific heat of water, cold plate outlet temperature and cold plate inlet temperature for the bottom side of the TEC cooling system, respectively. Equation (6) is similar for the bottom surface; however, the Nusselt number for the bottom hot surface is:  
\[Nu=0.27{Ra_{L}}^{1/4} \tag{7}\]  
Therefore, the heat loss for the bottom hot surface is calculated using the same equation as Equation (5). However, the total heat removal rate is equal to the summation of the heat removal rate from the top and bottom surface, which is equal to the sum of Equations (1) and (6):  
\[\dot{Q}_{H,TEC}=\dot{Q}_{H,TEC1}+\dot{Q}_{H,TEC2} \tag{8}\]  
where \(\dot{Q}_{H,TEC}\) is total heat removal rate from the hot side of top and bottom surface of the cold plate heat exchanger. To find the heat removal rate from the cold side of the TEC module, the following equation is used:  
\[\dot{Q}_{C}=\dot{Q}_{H,TEC}-W_{in} \tag{9}\]  
where \(\dot{Q}_{C}\) is heat removal rate from the cold side of the TEC. Here, Win is the energy required by the TEC modules to provide the Peltier effect.