Question

The finite difference formulation of steady two-dimensional heat conduction in a medium with heat generation and constant thermal conductivity is given by $$ \begin{gathered} \frac{T_{m-1, n}-2 T_{m, n}+T_{m+1, n}}{\Delta x^{2}}+\frac{T_{m, n-1}-2 T_{m, n}+T_{m, n+1}}{\Delta y^{2}} \\ +\frac{\dot{e}_{m, n}}{k}=0 \end{gathered} $$ in rectangular coordinates. Modify this relation for the threedimensional case.

Short answer

Answer: The three-dimensional finite difference equation for steady heat conduction with heat generation and constant thermal conductivity is given by: $$\frac{T_{m-1, n, p}-2 T_{m, n, p}+T_{m+1, n, p}}{\Delta x^{2}}+\frac{T_{m, n-1, p}-2 T_{m, n, p}+T_{m, n+1, p}}{\Delta y^{2}} + \frac{T_{m, n, p-1}-2 T_{m, n, p}+T_{m, n, p+1}}{\Delta z^{2}} + \frac{\dot{e}_{m, n, p}}{k} = 0$$

Step-by-step solution

Understand the given two-dimensional equation

The finite difference equation provided represents a two-dimensional heat conduction problem with heat generation and constant thermal conductivity. In this equation, the term \(\frac{T_{m-1, n}-2 T_{m, n}+T_{m+1, n}}{\Delta x^{2}}\) represents the heat conduction in the x-direction while \(\frac{T_{m, n-1}-2 T_{m, n}+T_{m, n+1}}{\Delta y^{2}}\) represents heat conduction in the y-direction. \(\frac{\dot{e}_{m, n}}{k}\) is the heat generation term divided by thermal conductivity.

Introduce the third dimension (z-direction)

As we want to modify this equation for the three-dimensional case, we need to introduce an additional variable, z, to account for heat conduction in the z-direction. To do this, we will add a term similar to the x and y directions, representing heat conduction in the z-direction: $$\frac{T_{m, n, p-1}-2 T_{m, n, p}+T_{m, n, p+1}}{\Delta z^{2}}$$

Formulate the three-dimensional finite difference equation

Now, we will combine the given two-dimensional equation with the new z-direction term to form the three-dimensional finite difference equation. $$\frac{T_{m-1, n, p}-2 T_{m, n, p}+T_{m+1, n, p}}{\Delta x^{2}}+\frac{T_{m, n-1, p}-2 T_{m, n, p}+T_{m, n+1, p}}{\Delta y^{2}} + \frac{T_{m, n, p-1}-2 T_{m, n, p}+T_{m, n, p+1}}{\Delta z^{2}} + \frac{\dot{e}_{m, n, p}}{k} = 0$$ Here, we have the three-dimensional finite difference formulation for steady heat conduction with heat generation and constant thermal conductivity.