## 3 Mathematical Modeling
### FEM-Based Steady-State Thermal Model  
Steady-state heat transfer in and around any rectangular bus bar shown three-dimensionally in Figure 1 is defined by the following two-dimensional second-order partial differential equation [23; 30]:  
\[\
abla\cdot(-k_{t}\
abla T)=\frac{\partial}{\partial x}\left(-k_{t}\frac{ \partial T}{\partial x}\right)+\frac{\partial}{\partial y}\left(-k_{t}\frac{ \partial T}{\partial y}\right)=Q_{tg,v} \tag{35}\]  
where x and y are the Cartesian spatial coordinates in m; \(k_{t}\) is the thermal conductivity of the bus bar material in W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\); T is the unknown nodal temperature in K; and \(Q_{tg,v}\) is the volume power of heat sources defined by Equation (2) in W\(\cdot\)m\({}^{-3}\). Each side, top, or bottom surface of any rectangular bus bar from Figure 1 is represented by a combination of  
\[\overrightarrow{n}\cdot(-k_{t}\
abla T)=\varepsilon\sigma_{SB}(T^{4}-T_{a}^{ 4})=h_{r}(T-T_{a}) \tag{36}\]  
* radiation boundary condition; and \[\overrightarrow{n}\cdot(-k_{t}\
abla T)=h_{c}(T-T_{a})\] (37)
* convection boundary condition [23; 30]. In Equations (36) and (37), \(\overrightarrow{n}\) is the normal vector oriented outwards in relation to the side, top, or bottom surface of the bus bar; T is the unknown temperature of the side, top, or bottom surface of the bus bar in K; h\({}_{r}\) is the heat transfer coefficient due to radiation from the bus bar surfaces defined by Equation (4) in W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\); h\({}_{c}\) is the heat transfer coefficient due to free or vibration-assisted convection from the side, top, or bottom surface of the bus bar in W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\); and T\({}_{a}\) is the temperature of the ambient air contacting the bus bar surfaces in K. The radiation boundary condition in each of the FEM-based models can be specified either by the thermal emissivity \(\varepsilon\) or by the radiation heat transfer coefficient h\({}_{r}\). The effect of horizontal transverse vibrations on free convection from any rectangular bus bar is included in the corresponding FEM-based model using the convection heat transfer coefficient h\({}_{c}\) in the following manner: h\({}_{c}=h_{cS}\)--for its side surfaces, h\({}_{c}=h_{cT}\)--for its top surface, and h\({}_{c}=h_{cB}\)--for its bottom surface. In addition, the combination of radiation and convection boundary conditions ensures that Equation (35) is non-linear, regardless of the fact that here the temperature difference \((T_{s}-T_{a})\) should be constant and equal to 30 K.  
Bus bar ampacities are also determined in COMSOL 4.3 by applying the FEM to two-dimensional computational domains corresponding to the rectangular bus bars from Figure 1, for the materials given in Table 1 and the service conditions specified in Section 2. In this regard, the values for \(Q_{tg,v}\) are increased or decreased gradually from prescribed initial values to the values corresponding to the continuously permissible temperature T\({}_{cp}=70\)\({}^{\circ}\)C. The values of \(Q_{tg,v}\) obtained in this way are compared with those generated by the BUSBAR.m program. Then, these \(Q_{tg,v}\) values and Equations (2) and (5) are used to calculate the corresponding bus bar ampacities (for I = I\({}_{cp}\)). The I\({}_{cp}\) values are also compared with those obtained using the BUSBAR.m program. The FEM-based verification of the proposed analytical model is carried out in the Heat Transfer Module of COMSOL 4.3 [23] by means of the experimental data from [21]. In addition to the material properties and service conditions taken from the relevant literature, the Heat Transfer Module also uses as input data the volume power of heat sources, radiation heat transfer coefficient, and convection heat transfer coefficients generated by the updated version of the BUSBAR.m program.