$(20$ pts$.)$ A solid with node points identified is shown in the figure below. The solid has a depth $(L)$ into the page as indicated in the figure, and has uniform node spacing $(x$ and $y),$ also indicated on the figure. The solid experiences uniform volumetric generation $(q).$ Note the insulation shown in the figure. The horizontal surface between Nodes $1$ and $3$ is insulated, as is the horizontal surface between Nodes $7$ and $8.$ All uninsulated surfaces, except the horizontal surface between Nodes $5$ and $6,$ are exposed to a fluid $(h,T_{})$ as indicated in the figure. The horizontal surface between Nodes $5$ and $6$ experiences an imposed heat flux $(q^{\text{'}\text{'}})$ and is NOT exposed to the fluid. Using an appropriate sketch, and an energy balance approach as described in the course, develop the steady$-$state node equation for $T_5.$ Keep the node equation for $T_5$ in symbolic form $($no numerical values$).$ No algebraic manipulation is required.

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