Question

Consider steady one-dimensional heat conduction in a pin fin of constant diameter \(D\) with constant thermal conductivity. The fin is losing heat by convection to the ambient air at \(T_{\infty}\) with a heat transfer coefficient of \(h\). The nodal network of the fin consists of nodes 0 (at the base), 1 (in the middle), and 2 (at the fin tip) with a uniform nodal spacing of $\Delta x$. Using the energy balance approach, obtain the finite difference formulation of this problem to determine \(T_{1}\) and \(T_{2}\) for the case of specified temperature at the fin base and negligible heat transfer at the fin tip. All temperatures are in \({ }^{\circ} \mathrm{C}\).

Short answer

Question: Determine the approximate temperatures at nodes 1 and 2 for a pin fin experiencing steady one-dimensional heat conduction with constant thermal conductivity and constant diameter, given the fin's base temperature, \(T_0\), and the ambient temperature, \(T_{\infty}\). Short Answer: First, discretize the fin into three nodes. Then, apply the energy balance approach to create equations for \(T_1\) and \(T_2\). Approximate \(T_2\) as close to \(T_{\infty}\) and solve for \(T_1\). Finally, use the calculated value of \(T_1\) to solve for \(T_2\). Remember that this is an approximation, and for more accurate values, perform iterations with the updated values of \(T_1\) and \(T_2\).

Step-by-step solution

Discretize the fin using the energy balance approach

Discretize the fin into three nodes (0 at the base, 1 in the middle, and 2 at the fin tip) with uniform spacing \(\Delta x\). The energy balance at each node would take the form: At node 1: \(-kA\frac{T_1-T_0}{\Delta x}+hP(\Delta x)(T_1-T_{\infty})-kA\frac{T_2-T_1}{\Delta x}=0\) At node 2: \(-kA\frac{T_2-T_1}{\Delta x}+hP(\Delta x)(T_2-T_{\infty})=0\) Here, \(k\) is the thermal conductivity, \(A\) is the cross-sectional area of the fin, \(P\) is the perimeter of the fin, and \(T_{\infty}\) is the ambient temperature.

Rearrange the equations for \(T_1\) and \(T_2\)

From the energy balance at node 1, we can rearrange the equation for \(T_1\): \(T_1=\frac{kA(T_0+T_2)}{\Delta x^2}+\frac{hP(\Delta x)(T_\infty)}{2kA/\Delta x+hP(\Delta x)}\) From the energy balance at node 2, we can rearrange the equation for \(T_2\): \(T_2=\frac{kA T_1}{\Delta x^2}+\frac{hP(\Delta x)(T_\infty)}{kA/\Delta x}\)

Solve for \(T_1\) using the specified temperature at the fin base and negligible heat transfer at the fin tip

We are given \(T_0\) and told to assume negligible heat transfer at the fin tip. Therefore, we can approximate \(T_2\) as close to \(T_{\infty}\). Substitute these values into the equation for \(T_1\) and solve for \(T_1\). It's important to note that this is an approximation, and \(T_2\) should be calculated using the full equation for better accuracy.

Solve for \(T_2\) using the calculated value of \(T_1\)

Now that we have an estimate for \(T_1\), we can substitute this value into the equation for \(T_2\). Solve for \(T_2\) to find the temperature at the fin tip. Remember that this is an approximation, and for an accurate value of \(T_2\), we should iterate with the updated value of \(T_1\). By following these steps, we can obtain the temperatures at nodes 1 and 2 for the pin fin using the energy balance approach and finite difference formulation. The results give us insight into the heat transfer behavior of the fin and can be used to better understand its performance.