Question

A cylindrical aluminum fin with adiabatic tip is attached to a wall with surface temperature of \(300^{\circ} \mathrm{C}\), and it is exposed to an ambient air condition of \(15^{\circ} \mathrm{C}\) with convection heat transfer coefficient of \(150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The fin has a uniform cross section with diameter of \(1 \mathrm{~cm}\), length of $5 \mathrm{~cm}\(, and thermal conductivity of \)237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. Assuming steady one-dimensional heat transfer along the fin and the nodal spacing to be uniformly \(10 \mathrm{~mm},(a)\) obtain the finite difference equations for use with the Gauss-Seidel iterative method, and \((b)\) determine the nodal temperatures using the Gauss-Seidel iterative method, and compare the results with the analytical solution.

Short answer

Answer: The convergence criteria in the Gauss-Seidel method for the cylindrical fin problem is typically defined as the difference between temperatures in subsequent iterations being less than a small value, such as 0.001.

Step-by-step solution

Governing Heat Equation for Cylindrical Fin

The heat conduction equation for a cylindrical fin with a uniform cross-section is given by: $$\frac{\mathrm{d}}{\mathrm{d}x}\left( kA\frac{\mathrm{d}T}{\mathrm{d}x} \right) - hP(T - T_{\infty}) = 0 $$ where \(k\) is the thermal conductivity, \(A\) is the cross-sectional area, \(T\) is the temperature, \(x\) is the distance along the fin, \(h\) is the convection heat transfer coefficient, \(P\) is the perimeter, and \(T_{\infty}\) is the ambient temperature.

Finite Difference Discretization

We can discretize the governing equation using the central difference method for the second derivative term and the first derivative term. This gives us the finite difference equation: $$kA\left(\frac{T_{i+1} - 2T_{i} + T_{i-1}}{\Delta{x}^2}\right) - hP(T_{i} - T_{\infty}) = 0 $$ Now, we will write the nodal equations for each node to be used with the Gauss-Seidel iterative method.

Nodal Equations for Gauss-Seidel Iterative Method

Let's write the nodal equations for the Gauss-Seidel method: Node 0 (wall): $$T_0 = T_{w} = 300^{\circ}\mathrm{C}$$ Node \(i\): $$\left(2kA + hP\Delta{x}\right)T_{i} - kA(T_{i+1} + T_{i-1}) = hP\Delta{x} T_{\infty} $$ Node N (adiabatic tip): $$\frac{kA}{\Delta{x}}(T_{\text{N-1}} - T_{\text{N}})=0$$

Gauss-Seidel Iterative Method for Nodal Temperatures

To obtain the nodal temperatures, we will use the Gauss-Seidel iterative method by solving the nodal equations and updating temperatures in each iteration until convergence. The convergence criteria can be defined as the difference between temperatures in subsequent iterations being less than a small value (e.g., 0.001). The Gauss-Seidel method will result in a set of temperatures for all nodes, which can be compared to the analytical solution.

Comparing Results with Analytical Solution

The analytical solution for the cylindrical fin problem is given by: $$T(x) = T_{\infty} + (T_{w}-T_{\infty})\cosh\left(\frac{m(L-x)}{A_c}\right) \Big/ \cosh\left(\frac{mL}{A_c}\right) $$ where \(m = \sqrt{\frac{hP}{kA}}\) and \(A_c\) is the tip cross-sectional area. We can compare the nodal temperatures obtained from the Gauss-Seidel method with the analytical solution for different nodes along the length of the fin. This will give us an idea of the accuracy and convergence of our numerical solution.