Question

Consider steady two-dimensional heat transfer in a square cross section $(3 \mathrm{~cm} \times 3 \mathrm{~cm})$ with the prescribed temperatures at the top, right, bottom, and left surfaces to be $100^{\circ} \mathrm{C}, 200^{\circ} \mathrm{C}, 300^{\circ} \mathrm{C}\(, and \)500^{\circ} \mathrm{C}$, respectively. Using a uniform mesh size \(\Delta x=\Delta y\), determine \((a)\) the finite difference equations and \((b)\) the nodal temperatures with the GaussSeidel iterative method.

Short answer

Answer: The key steps in solving the steady two-dimensional heat transfer problem are: 1) Determining mesh size and node points, 2) Applying the finite difference equations, 3) Applying boundary conditions, 4) Applying the Gauss-Seidel iterative method, and 5) Determining the nodal temperatures.

Step-by-step solution

Determine Mesh Size and Node Points

Use the given uniform mesh size, which is \(\Delta x = \Delta y\). Since the dimensions are \(3\,\mathrm{cm} \times 3\, \mathrm{cm}\), choose the convenient mesh size, such as: \(\Delta x = \Delta y = 1\, \mathrm{cm}\). Then, the square can be divided into 9 nodes labeled as \((1,1)\) to \((3,3)\) and arranged in a 3x3 matrix.

Apply the Finite Difference Equations

Use the finite difference equation for steady two-dimensional heat transfer, given as: \(\frac{T_{i+1,j} - 2T_{i,j} + T_{i-1,j}}{(\Delta x)^2} + \frac{T_{i,j+1} - 2T_{i,j} + T_{i,j-1}}{(\Delta y)^2} = 0\) Since \(\Delta x = \Delta y\), the equation becomes: \(T_{i+1,j} - 2T_{i,j} + T_{i-1,j} + T_{i,j+1} - 2T_{i,j} + T_{i,j-1} = 0\) The above equation can be simplified to: \(4T_{i,j} = T_{i+1,j} + T_{i-1,j} + T_{i,j+1} + T_{i,j-1}\)

Apply Boundary Conditions

Use the given boundary temperatures to replace the corresponding terms in the finite difference equation, starting from the corners and moving inwards. For example, when \(i=1, j=1\), the equation is: \(4T_{1,1} = T_{2,1}+500+T_{1,2} + 300\) Similarly, determine the other boundary conditions while taking note of the values across each side.

Apply the Gauss-Seidel Iterative Method

Using the Gauss-Seidel iterative method, update the temperature values at each node point iteratively. Start with initial guesses for the internal nodal temperatures, like \(T_{ij}=0\). Update the internal temperatures using the boundary conditions and the finite difference equation. \(T^{k+1}_{i,j} = \frac{1}{4} (T^{k}_{i+1,j} + T^{k+1}_{i-1,j} + T^{k}_{i,j+1} + T^{k+1}_{i,j-1})\) Repeat this process until the convergence criterion is reached or the maximum number of iterations is completed. The convergence criterion can be based on the difference between successive iterations or the error in the solution's accuracy.

Determine Nodal Temperatures

After applying the Gauss-Seidel iterative method and reaching the desired convergence level, find the nodal temperatures corresponding to the converged values of \(T_{i,j}\) and output the results. In conclusion, following these steps will allow determining the finite difference equations and finding the nodal temperatures using the Gauss-Seidel iterative method for the steady two-dimensional heat transfer in a square cross-section.