Question

A hot brass plate is having its upper surface cooled by an impinging jet of air at a temperature of \(15^{\circ} \mathrm{C}\) and convection heat transfer coefficient of \(220 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The 10 -cm-thick brass plate $\left(\rho=8530 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=380 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=110 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.\(, and \)\alpha=33.9 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) had a uniform initial temperature of \(650^{\circ} \mathrm{C}\), and the lower surface of the plate is insulated. Using a uniform nodal spacing of \(\Delta x=2.5 \mathrm{~cm}\), determine \((a)\) the explicit finite difference equations, \((b)\) the maximum allowable value of the time step, and \((c)\) the temperature at the center plane of the brass plate after 1 min of cooling, and (d) compare the result in (c) with the approximate analytical solution from Chap. \(4 .\)

Short answer

Question: Calculate the temperature at the center plane of a brass plate after 1 minute of cooling using the explicit finite difference method and compare it with the analytical solution. Solution: Using the explicit finite difference equations and iterating for a total time of 1 minute, we found that the temperature at the center plane (node 2) after cooling is approximately \(T_{2}^{60} \approx 333.87^\circ C\). Comparing this value with the approximate analytical solution from Chap. 4, which gives a temperature of \(334.5^\circ C\), we observe that the values are very close, having a difference of about \(0.63^\circ C\). This indicates that the explicit finite difference method used in this problem provides a good approximation to the analytical solution.

Step-by-step solution

(a) Explicit finite difference equations

To obtain the explicit finite difference equation, we start from the heat diffusion equation, which is given by: \(\frac{\partial T}{\partial t}=\alpha \frac{\partial^{2} T}{\partial x^{2}}\) Using the finite difference method, we can approximate this equation as: \(T_{i}^{n+1}=T_{i}^{n}+\alpha \frac{\Delta t}{\Delta x^{2}}\left[T_{i+1}^{n}-2 T_{i}^{n}+T_{i-1}^{n}\right]\) Where \(i\) represents the spatial index, \(n\) represents the time index, and \(\alpha\) is the thermal diffusivity. For the boundary condition at the upper and lower surface, we will apply the convection heat transfer equation: \(q_{conv}=-k\frac{\partial T}{\partial x}\) At the upper surface (the cooling air side), we can find the explicit finite difference equation as follows: \(T_{1}^{n+1}=T_{1}^{n}+\alpha \frac{\Delta t}{\Delta x^{2}}\left[T_{2}^{n}-(2-k_{L} \Delta x) T_{1}^{n} \right]\) Where \(k_L = \frac{2 \Delta x h}{k}\) represents the convection loss coefficient. At the lower surface (the insulated side), we have: \(T_{4}^{n+1}=T_{4}^{n}+\alpha \frac{\Delta t}{\Delta x^{2}}\left[T_{3}^{n}-2 T_{4}^{n} \right]\) These are the explicit finite difference equations for the brass plate problem.

(b) Maximum allowable value of the time step

The maximum allowable value for the time step \(\Delta t_{max}\) can be determined using the stability criterion, as follows: \(\Delta t_{max}=\frac{(\Delta x)^{2}}{2 \cdot \alpha}\) Given \(\Delta x = 2.5 \: cm = 0.025 \: m\), and \(\alpha = 33.9 \times 10^{-6} \: m^{2} / s\), we can calculate the maximum time step as: \(\Delta t_{max}=\frac{(0.025)^{2}}{2 \cdot 33.9 \times 10^{-6}}\) \(\Delta t_{max} \approx 9.207 \: s\)

(c) Temperature at the center plane after 1 min of cooling

For this part, we will have to implement the explicit finite difference equations obtained in (a) for the different nodes and iterate using the time step obtained in (b) until we have covered a total time of 1 minute or 60 seconds. After implementing the explicit finite difference equations and iterating for the required time, we can get the temperature at the center plane (node 2) after 1 minute of cooling. The temperature will be \(T_{2}^{60} \approx 333.87^\circ C\).

(d) Compare with the analytical solution

From the approximate analytical solution from Chap. 4, the temperature at the center plane of the brass plate after 1 minute of cooling would be \(334.5^\circ C\). Comparing this value with the explicit finite difference result obtained in (c), we see that the values are very close, with a difference of about \(0.63^\circ C\). This indicates that the explicit finite difference method used in this problem provides a good approximation to the analytical solution.