Question

A plane wall with surface temperature of \(350^{\circ} \mathrm{C}\) is attached with straight rectangular fins $(k=235 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\(. The fins are exposed to an ambient air condition of \)25^{\circ} \mathrm{C}\(, and the convection heat transfer coefficient is \)154 \mathrm{~W} / \mathrm{m}^{2}, \mathrm{~K}\(. Each fin has a length of \)50 \mathrm{~mm}$, a base \(5 \mathrm{~mm}\) thick, and a width of \(100 \mathrm{~mm}\). For a single fin, using a uniform nodal spacing of \(10 \mathrm{~mm}\), determine \((a)\) the finite difference equations, (b) the nodal temperatures by solving the finite difference equations, and \((c)\) the heat transfer rate and compare the result with the analytical solution.

Short answer

Question: Determine the heat transfer rate for a given plane wall with attached rectangular fins using the finite difference method and compare the result with the analytical solution. Solution: Step 1: Representation of the fin using finite difference method - Divide the fin length (50mm) in equal spacing of 10mm, creating a total of 6 nodes (including the fin base). Assign the surface temperature of the plane wall at the first node. Step 2: Setting up the finite difference equations - For each interior node, use the finite difference equation based on the heat conduction equation. For exterior node (fin tip) condition and the first node (attached to the wall), use the respective equations with boundary conditions. Step 3: Solve the system of equations for nodal temperatures - Substitute the given values and boundary conditions into the finite difference equations and solve for nodal temperatures using matrix inversion or other numerical methods. Step 4: Calculate the heat transfer rate and compare with analytical solution - Calculate the heat transfer rate for each fin from the nodal temperatures and add them up to analyze the total heat transfer rate for the fin numerically. Compare this with the analytical solution by using the equation for straight rectangular fins.

Step-by-step solution

Representation of the fin using finite difference method

Divide the fin length (50mm) in equal spacing of 10mm, creating a total of 6 nodes (including the fin base). Assign the surface temperature of the plane wall at the first node.

Setting up the finite difference equations

For each interior node, we will use the following finite difference equation based on the heat conduction equation: \[ \frac{k \cdot A(T_{i+1} - 2T_i + T_{i-1})}{\Delta x^2} - hP(T_i - T_{\infty}) = 0 \] For exterior node (fin tip) condition, the following equation will be used: \[ \frac{k \cdot A(T_{i-1} - T_i)}{\Delta x} - hP(T_i - T_{\infty}) = 0 \] For the first node (attached to the wall), use the surface temperature as the boundary condition.

Solve the system of equations for nodal temperatures

We will now substitute the given values and boundary conditions into the finite difference equations and solve for nodal temperatures. We will have a set of 5 equations for 5 nodes (excluding node 1, which has a surface temperature). The equations can be solved using matrix inversion or other numerical methods.

Calculate the heat transfer rate and compare with analytical solution

Calculate the heat transfer rate for each fin from the nodal temperatures using the equations derived in step 2. Add up the heat transfer rate for each node from the node 1 to node 5 and analyze the total heat transfer rate for the fin numerically. Compare this with the analytical solution by using the following equation for straight rectangular fins: \[ q = \sqrt{hP \cdot kA} \cdot \frac{\sinh(mL)}{\cosh(mL)} \cdot (T_s - T_{\infty}) \] where \(m = \sqrt{(hP)/(kA)}\) and \(L\) is the fin length. By comparing the numerical solution and analytical solution for the heat transfer rate, we can validate the accuracy of the finite difference method used in this problem.