## 3 Mathematical Modeling  
### Analytical Steady-State Thermal Model  
If the law of conservation of energy is applied to any rectangular bus bar (i.e., any volume element of length L) from Figure 1, then the energy equation for steady-state heat transfer in an indoor environment (where there is no solar heating) becomes [22; 25]:  
\[{\rm Q_{fg,\ v}}=[2{\rm h_{cS}}{\rm S_{S}}+{\rm h_{cT}}{\rm S_{T}}+{\rm h_{cB }}{\rm S_{B}}+{\rm h_{r}}{\rm S_{o1}}]({\rm T_{s}}-{\rm T_{a}}) \tag{1}\]  
where  
\[{\rm Q_{fg,\ v}}=\frac{{\rm k_{s}}{\rm k_{p}}\rho_{\rm dc}{\rm I^{2}}}{{\rm S ^{2}}} \tag{2}\]  
\[\rho_{\rm dc}=\rho_{\rm e20}[1+\alpha_{\rho}({\rm T_{s}}-293.157)] \tag{3}\]  
\[{\rm h_{r}}=\sigma_{\rm SB}\varepsilon\left({\rm T_{s}^{2}+T_{a}^{2}}\right)( {\rm T_{s}}+{\rm T_{a}}) \tag{4}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Aluminum Alloy** & **k\({}_{\rm t}\) (W\(\cdot\)m\({}^{-1}\)\(\cdot\)K\({}^{-1}\))** & **\(\rho_{\rm e20}\) (\(\Omega\)\(\cdot\)m)** & **\(\alpha_{\rho}\) (K\({}^{-1}\))** \\ \hline
5052-O & 138.0 & 4.930 \(\times\)10\({}^{-8}\) & 0.00383 \\
6101-T61 & 218.5 & 2.998 \(\times\) 10\({}^{-8}\) & 0.00383 \\ \hline \end{tabular}
\end{table} 
Table 1: Thermal and electrical properties of the materials used. \(\rm V=\rm WHL\) is the volume of the bus bar in \(\rm m^{3}\), \(\rm S_{S}=\rm HL\) is the area of one side surface of the bus bar in \(\rm m^{2}\), \(\rm S_{T}=\rm WL\) is the area of the top surface of the bus bar in \(\rm m^{2}\), \(\rm S_{B}=\rm WL\) is the area of the bottom surface of the bus bar in \(\rm m^{2}\), \(\rm h_{r}\) is the heat transfer coefficient due to radiation between the bus bar surface and the indoor environment in \(\rm W\cdot m^{-2}\cdot K^{-1}\), \(\rm S_{o1}=2\rm S_{S}+\rm S_{T}+\rm S_{B}\) is the surface area of the bus bar in \(\rm m^{2}\), \(\rm\rho_{dc}\) is the d.c. resistivity of the bus bar material in \(\rm\Omega\cdot m\), \(\rm\sigma_{SB}=\)5.67 \(\times\) 10\({}^{-8}\) W\(\cdot\)m\({}^{-2}\cdot K^{-4}\) is the Stefan-Boltzmann constant, \(\rm I\) is the bus bar RMS current in A (where RMS stands for Root-Mean Square value), and \(\rm S=\rm WH\) is the geometric cross-sectional area of the bus bar in \(\rm m^{2}\). If there is no solar heating in an indoor environment, it means that the solar absorptivity is zero and/or the solar irradiance is zero. It should be emphasized here that the nominal and geometric cross-sectional areas are the same for any rectangular bus bar. When the bus bar temperature \(\rm T_{s}\) is equal to the associated continuously permissible temperature \(\rm T_{cp}\), the RMS current I becomes equal to the ampacity \(\rm I_{cp}\). Accordingly, from Equations (1) and (2), the bus bar ampacity \(\rm I_{cp}\) can be expressed as [22]:  
\[\rm I_{cp}=\frac{\left[(2\rm h_{c}S_{S}+\rm h_{c}F_{T}+\rm h_{c}B_{S}+\rm h_{r }S_{01})\big{(}T_{cp}-T_{a}\big{)}\right]^{1/2}S}{\left(\rm k_{s}k_{p}\rho_{dc} V\right)^{1/2}} \tag{5}\]  
where \(\rm\rho_{dc}\) and \(\rm h_{r}\) should be calculated using Equations (3) and (4) for \(\rm T_{s}=\rm T_{cp}\). Moreover, based on Equation (1), the unknown bus bar temperature \(\rm T_{s}\) can be estimated as follows [22]:  
\[\rm T_{s}=\frac{Q_{bg,v}V}{2\rm h_{c}S_{S}+\rm h_{c}F_{T}+\rm h_{c}B_{S}+\rm h _{r}S_{01}}+T_{a} \tag{6}\]  
Thermodynamic and physical properties of ambient air are needed to calculate the coefficients due to free, vibration-assisted, or forced convection. These convection coefficients are determined at the film temperature \(\rm T_{film}=(T_{s}+T_{a})/2\). For the ambient air at the film temperature \(\rm T_{film}\), the density \(\rm\rho_{a}\) in kg\(\cdot m^{-3}\), specific heat at constant pressure \(\rm c_{t,a}\) in J\(\cdot\)kg\({}^{-1}\cdot K^{-1}\), dynamic viscosity \(\rm\mu_{a}\) in kg\(\cdot m^{-1}\cdot s^{-1}\), thermal conductivity \(\rm k_{t,a}\) in W\(\cdot m^{-1}\cdot K^{-1}\), or Prandtl number \(\rm Pr\) is obtained by means of cubic spline interpolation and two arrays of discrete values from the pair of corresponding input data files.