Question

The explicit finite difference formulation of a general interior node for transient heat conduction in a plane wall is given by $$ T_{m=1}^{i}-2 T_{m}^{i}+T_{m+1}^{i}+\frac{e_{m}^{i} \Delta x^{2}}{k}=\frac{T_{m}^{i+1}-T_{m}^{i}}{\tau} $$ Obtain the finite difference formulation for the steady case by simplifying the preceding relation.

Short answer

Based on the provided solution for finite difference formulation for steady-state heat conduction in a plane wall, the final equation we derived is: $$ T_{m-1}^{i} - 2T_{m}^{i} + T_{m+1}^{i} + \frac{e_{m}^{i} \Delta x^{2}}{k} = 0 $$

Step-by-step solution

Identify Steady-State Condition

For the steady-state case, the temperature does not change with time. So, the change in temperature between two successive time steps will be zero: $$ T_{m}^{i+1} - T_{m}^{i} = 0 $$

Substitute the Steady-State Condition into the Transient Equation

Replace the \((T_{m}^{i+1} - T_{m}^{i})\) term in the given transient equation with zero, as identified in Step 1: $$ T_{m=1}^{i} - 2T_{m}^{i} + T_{m+1}^{i} + \frac{e_{m}^{i} \Delta x^{2}}{k} = \frac{0}{\tau} $$

Simplify the Equation

Now, let's simplify the equation by removing the fraction: $$ T_{m-1}^{i} - 2T_{m}^{i} + T_{m+1}^{i} + \frac{e_{m}^{i} \Delta x^{2}}{k} = 0 $$ This is the finite difference formulation for steady-state heat conduction in a plane wall.