Question

A hot surface at \(120^{\circ} \mathrm{C}\) is to be cooled by attaching 8-cm- long, \(0.8-\mathrm{cm}\) - diameter aluminum pin fins ( $k=237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\( and \)\left.\alpha=97.1 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$ to it with a center-to-center distance of \(1.6 \mathrm{~cm}\). The temperature of the surrounding medium is $15^{\circ} \mathrm{C}\(, and the heat transfer coefficient on the surfaces is \)35 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Initially, the fins are at a uniform temperature of \(30^{\circ} \mathrm{C}\), and at time \(t=0\), the temperature of the hot surface is raised to \(120^{\circ} \mathrm{C}\). Assuming one-dimensional heat conduction along the fin and taking the nodal spacing to be \(\Delta x=2 \mathrm{~cm}\) and a time step to be \(\Delta t=0.5 \mathrm{~s}\), determine the nodal temperatures after \(10 \mathrm{~min}\) by using the explicit finite difference method. Also, determine how long it will take for steady conditions to be reached.

Short answer

A: The general explicit finite difference equation for heat conduction is: \(T_{i}^{n+1} = T_{i}^{n} + \frac{\alpha \Delta t}{(\Delta x)^2} \left[T_{i+1}^{n} - 2 T_{i}^{n} + T_{i-1}^{n} \right]\), where \(T_{i}^{n}\) is the temperature at node \(i\) at time step \(n\), and \(\alpha\) is the thermal diffusivity.

Step-by-step solution

Define parameters and constants

We will be using the following parameters and constants in our calculations: - Pin fin length: \(L = 8\textrm{ cm}\) - Diameter: \(D = 0.8\textrm{ cm}\) - Thermal conductivity: \(k = 237\textrm{ W/m}\cdot\textrm{K}\) - Heat transfer coefficient: \(h = 35\textrm{ W/m}^2\cdot\textrm{K}\) - Center-to-center distance: \(c = 1.6\textrm{ cm}\) - Initial temperature: \(T_{initial} = 30^{\circ}\textrm{C}\) - Hot surface temperature: \(T_{hot} = 120^{\circ}\textrm{C}\) - Surrounding temperature: \(T_{\infty} = 15^{\circ}\textrm{C}\) - Time steps: \(\Delta t = 0.5\textrm{ s}\) - Nodal spacing: \(\Delta x = 2\textrm{ cm}\) All lengths are given in cm. We need to use SI units, i.e., meters. Thus, we need to convert lengths from cm to m: \(L = 0.08\textrm{ m}\), \(D = 0.008\textrm{ m}\), \(c = 0.016\textrm{ m}\), and \(\Delta x = 0.02\textrm{ m}\).

Set up explicit finite difference equation

The general explicit finite difference equation for heat conduction is given by: \(T_{i}^{n+1} = T_{i}^{n} + \frac{\alpha \Delta t}{(\Delta x)^2} \left[T_{i+1}^{n} - 2 T_{i}^{n} + T_{i-1}^{n} \right]\), where \(T_{i}^{n}\) is the temperature at node \(i\) at time step \(n\), and \(\alpha\) is the thermal diffusivity.

Initialize temperatures at initial time step (t = 0)

Initially, the temperature of the fin is uniformly \(T_{initial}\) (\({=} 30^{\circ}\textrm{C}\)). So, for all nodes i (i=0, 1, 2, 3, and 4), we set \(T_{i}^{0} = T_{initial}\).

Apply boundary condition at the fin base (node 0)

Since the base of the fin must always be at the hot surface temperature \(T_{hot}\) (\({=} 120^{\circ}\textrm{C}\)), we set the boundary condition as \(T_{0}^{n} = T_{hot}\) for all time steps \(n\).

Calculate nodal temperatures using finite difference method

Using the explicit finite difference equation, calculated in Step 2, we iterate over all nodes i (i=1, 2, 3, and 4) and time steps n (\({=} 0, 1, 2,\dots, 1200\)) to determine the nodal temperatures after 10 minutes (\({=} 600\textrm{ s}\)). Ensure to apply the boundary condition at the fin base for each time step n.

Determine the time for steady conditions

To determine the time for steady conditions, we need to find the time at which the difference between the nodal temperatures of consecutive time steps becomes negligible. For this, we can use a small tolerance value (e.g., 0.001). Iterate through the time steps and check if the difference between the temperatures at each node and the previous time step is less than the tolerance value. Once the temperatures have stabilized, we can conclude that a steady state has been reached, and the time for steady conditions can be read off the time step.