Question

A circular fin of uniform cross section, with diameter of \(10 \mathrm{~mm}\) and length of \(50 \mathrm{~mm}\), is attached to a wall with a surface temperature of \(350^{\circ} \mathrm{C}\). The fin is made of material with thermal conductivity of \(240 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), it is exposed to an ambient air condition of \(25^{\circ} \mathrm{C}\), and the convection heat transfer coefficient is \(250 \mathrm{~W} / \mathrm{m}^{2}\). \(\mathrm{K}\). Assume steady one-dimensional heat transfer along the fin and the nodal spacing to be uniformly \(10 \mathrm{~mm}\). (a) Using the energy balance approach, obtain the finite difference equations to determine the nodal temperatures. (b) Determine the nodal temperatures along the fin by solving those equations, and compare the results with the analytical solution. (c) Calculate the heat transfer rate, and compare the result with the analytical solution.

Short answer

Question: Using the energy balance approach and finite difference method, determine the nodal temperatures along a circular fin and calculate the heat transfer rate. Compare these results with the analytical solution for a 1D steady circular fin problem. Solution: - Step 1: Determine the fin's basic parameters (diameter, length, thermal conductivity, wall temperature, ambient air temperature, convection heat transfer coefficient, nodal spacing). - Step 2: Define the energy balance equation (q_cond = q_conv). - Step 3: Discretize the energy balance equation to obtain the finite difference equation for nodal temperature (T_i). - Step 4: Solve the finite difference equations to obtain the nodal temperatures along the fin (T_1, T_2, ..., T_n). - Step 5: Calculate the heat transfer rate (q) using the temperature gradient at the base of the fin (node 1). - Step 6: Compare the numerical results (nodal temperatures and heat transfer rate) with the analytical solution for the 1D steady circular fin problem to validate your calculations.

Step-by-step solution

Determine the basic parameters for the fin

To start, we need to identify the important parameters for the circular fin problem: - Diameter: \(D = 10 \mathrm{~mm} = 0.01 \mathrm{~m}\) - Length: \(L = 50 \mathrm{~mm} = 0.05 \mathrm{~m}\) - Thermal conductivity: \(k = 240~\mathrm{W} / \mathrm{m} \cdot \mathrm{K}\) - Wall temperature: \(T_w = 350^{\circ} \mathrm{C}\) - Ambient air temperature: \(T_\infty = 25^{\circ} \mathrm{C}\) - Convection heat transfer coefficient: \(h = 250~\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\) - Nodal spacing: \(\Delta x = 10 \mathrm{~mm} = 0.01 \mathrm{~m}\)

Define the energy balance equation

To determine the finite difference equations for the nodal temperatures, we will use the energy balance equation based on the heat conduction and convection processes: \(q_{cond} = q_{conv}\) Where \(q_{cond}\) represents the rate of heat conduction along the fin, and \(q_{conv}\) represents the rate of heat convection leaving the fin. Using Fourier's law for heat conduction and Newton's law of cooling for heat convection, we get: \(-kA\frac{dT}{dx} = hP(T - T_\infty)\)

Discretize the energy balance equation

Now we will discretize the energy balance equation for the 1D finite difference method, by discretizing the temperature gradient in space: \(-kA\frac{T_{i+1} - T_{i-1}}{2\Delta x} = hP(T_i - T_\infty)\) Where \(T_{i+1}\), \(T_i\), and \(T_{i-1}\) are the temperatures at nodes \(i+1\), \(i\), and \(i-1\), respectively, with node spacing \(\Delta x\). Now, arranging the equation to obtain the finite difference equation for nodal temperature \(T_i\): \(T_{i+1} - T_{i-1} = \frac{2\Delta x hP(T_i - T_\infty)}{kA}\)

Solve the finite difference equations

To solve the finite difference equations for the nodal temperatures, we can derive the discretized equations for each node, and then solve the system of linear equations using available numerical techniques, e.g., Gaussian elimination or matrix inversion. After solving the system of equations, we will obtain the nodal temperatures at various nodes along the fin (\(T_1, T_2, \cdots, T_n\)).

Calculate heat transfer rate

To calculate the heat transfer rate along the fin, we can use the following formula: \(q = kA\frac{dT}{dx}\) By calculating the temperature gradient (\(dT/dx\)) at the base of the fin (node 1), we can determine the heat transfer rate: \(q = kA\frac{T_1 - T_\infty}{\Delta x}\)

Compare the results with the analytical solution

Finally, to validate and compare the results obtained in steps 4 and 5, we can use the analytical solution for the 1D steady circular fin problem, which can be found in textbooks and literature. Analytical solutions can be applied for: - Nodal temperatures along the fin. - Heat transfer rate. Now compare the nodal temperatures and heat transfer rate obtained from the numerical finite difference method with their corresponding analytical solutions to assess the correctness and accuracy of the calculations.