PROBLEM STATEMENT

A tunnel system for transporting gases is designed to facilitate the efficient movement of gaseous substances, such as natural gas, hydrogen, or carbon dioxide, through an extensive network of underground conduits. As a mechanical engineering consultancy, your company has been selected to participate in a multidisciplinary project. Your responsibility is to conduct a preliminary analysis of the temperatures at the densely located nodes of a tunnel system model with three channels for gas transport $($see Figure $1).$ The back of the tunnel is insulated to prevent heat loss. The tunnel has a thermal conductivity of $\backslash ($ k $\backslash )$ and is equipped with a heating element that generates $\backslash (\backslash$dot$\{$q$\}\backslash )$ of energy to uniformly heat the gases in the channels, while the surface is exposed to ambient air. Note: $\backslash (\backslash$Delta x$=\backslash$Delta y$=30\backslash$mathrm$\{$~mm$\}\backslash )$

Nodal number increases from left to right starting from Node $1.$

Figure $1$: Ideal model of three tunnel system