## 3 Boundary Conditions and Governing Equations  
The study involved simulations using COMSOL 5.5, employing the finite element analysis technique. In order to achieve significant outcomes by solving equations, the following criteria were established:  
* Simulations were conducted under steady-state conditions. * All thermoelectric elements were of identical dimensions. * Heat transfer in the thermoelectric elements was radially maintained, with lateral parts remaining thermally insulated. * Electrical and thermal contact resistances, including those of the welding layers, were negligible. * Maximum output power from the thermoelectric module was achieved when the external resistance equaled the internal resistance. * The thermal and electrical properties of the thermoelectric materials were assumed to be constant and did not vary with temperature changes. For calculating the internal resistance within the tubular thermoelectric module, the following governing equations were used (as referenced in [41]):  
\[R_{pn}=R_{p}+R_{n}=N\left(\frac{(\rho_{p}+\rho_{n})ln\left(\frac{r_{2}}{r_{1} }\right)}{\Delta\theta W}+R_{C}\right) \tag{1}\]  
\[R_{C}=\rho_{C}\left(\frac{L_{C}}{A_{I}}+\frac{L_{C}}{A_{O}}\right) \tag{2}\]  
\[R_{TEG}=N\left(R_{pn}+R_{C}\right) \tag{3}\]  
In the governing equations mentioned above, various parameters and their relationships within the TTEG are defined as follows:  
\(R_{pn}\) represents the internal resistance of a single thermocouple pair (consisting of one P-type and one N-type thermoelectric element). \(R_{c}\) is the internal resistance of the connecting copper tube. \(R_{TEG}\) is the total internal resistance of the TTEG. \(R_{p}\) and \(R_{n}\), respectively, denote the internal resistances of the P-type and N-type thermoelectric elements. \(\rho_{p}\), \(\rho n\), and \(\rho_{c}\) are the electrical conductivities of the P-type and N-type thermoelectric elements and the connecting copper tube, respectively. \(r_{1}\) and \(r_{2}\) represent the inner and outer radii of the thermoelectric element. W is the width of the thermoelectric element. \(\Delta\theta\) is the angle of the annular thermoelectric element. \(L_{c}\) is the length of the connecting copper tube. \(A_{I}\) and \(A_{O}\) are the contact areas between the inner and outer connecting copper tubes and the thermoelectric element. \(N\) is the number of thermocouple pairs in the TTEG. Using the following control program, we can calculate the output power of the TEG [26]:  
\[V=I\times(R_{L}+R_{TEG}) \tag{4}\]  
\[P=I^{2}\times R_{L} \tag{5}\]  
In the control equations previously mentioned, the parameters are defined as follows: \(V\) represents the voltage across the thermoelectric module. \(I\) is the current flowing through the circuit. \(R_{L}\) denotes the external load resistance. \(P\) is the output power of the TEG. To determine the efficiency of the TEG, the following control equations can be used [40]:  
\[\dot{Q}_{h}=\dot{m}_{h}Cp_{h,side}(T_{h,in}-T_{h,out}) \tag{6}\]  
\[\dot{Q}_{h}=N\Bigg{[}\alpha\frac{\alpha(T_{h}-T_{c})}{\frac{R_{L}}{N}+R_{TEG}} T_{h}-\frac{1}{2}\Bigg{[}\frac{\alpha(T_{h}-T_{c})}{\frac{R_{L}}{N}+R_{TEG}} \Bigg{]}^{2}R_{TEG}+\kappa_{TEG}(T_{h}-T_{C})\Bigg{]} \tag{7}\]  
\[\dot{Q}_{c}=\dot{m}_{c}Cp_{c,side}(T_{c,out}-T_{c,in}) \tag{8}\]  
\[\dot{Q}_{c}=N\Bigg{[}\alpha\frac{\alpha(T_{h}-T_{c})}{\frac{R_{L}}{N}+R_{TEG} }T_{c}+\frac{1}{2}\Bigg{[}\frac{\alpha(T_{h}-T_{c})}{\frac{R_{L}}{N}+R_{TEG}} \Bigg{]}^{2}R_{TEG}+\kappa_{TEG}(T_{h}-T_{C})\Bigg{]} \tag{9}\]  
\[\eta_{TEG}=\frac{P}{\dot{Q}_{h}} \tag{10}\]  
In the control equations for the TEG, the terms are defined as follows:  
\(Q_{h}\) is the heat input at the hot side of the TEG. \(Q_{c}\) is the heat output at the cold side of the TEG. \(Q_{h}\) and \(Q_{c}\) are measured in watts (W). \(m_{h}\) and \(m_{c}\) represent the mass flow rates of the fluid at the hot and cold sides, respectively. \(C_{ph,side}\) and \(C_{pc,side}\) are the specific heat capacities of the fluid at the hot and cold sides.