91Ó°ÊÓ

We are given that the free convection coefficient, \(\bar{h}\), is related to the temperature difference, \(\Delta T = T - T_{\infty}\), as follows: \[ \bar{h} = C\Delta T^{\frac{1}{4}} \] Using the results of Section 5.3.3, we know that the governing equation for cooling time is: \[ \dfrac{dT}{dt}=-\bar{h}\dfrac{A_{s}}{\rho V c_{p}}\left(T-T_{\infty}\right) \] Now, substituting the given expression for \(\bar{h}\): \[ \dfrac{dT}{dt}=-C\left(T-T_{\infty}\right)^{\frac{1}{4}}\dfrac{A_{s}}{\rho V c_{p}}\left(T-T_{\infty}\right) \] Now we can separate variables and integrate: \[ \int_{T_{i}}^{T_{f}}\dfrac{dT}{(T-T_{\infty})^{\frac{1}{4}}(T-T_{\infty})} = -C\dfrac{A_{s}}{\rho V c_{p}}\int_{0}^{\tau}dt \] With this setup done, we can now proceed with integration and solve for the cooling time, \(\tau\).

Given an aluminum alloy plate of thickness \(5\) mm, initially at \(225^{\circ} \mathrm{C}\) (temperature \(T_{i}\)), suspended in ambient air, at \(25^{\circ} \mathrm{C}\) (temperature \(T_{\infty}\)), we have to calculate the time required for the plate to reach \(80^{\circ} \mathrm{C}\) (temperature \(T_{f}\)), using the expression derived in Part (a) of this problem and the appropriate approximate free convection coefficient correlation. We can use the appropriate correlation for a square plate from Problem 9.11 to find the value of \(C\): \[ C = 0.68\left(\dfrac{g \beta}{\nu^{2}}\right)^{\frac{1}{4}} \] Now, we can use the expression derived in Part (a) to calculate the cooling time, \(\tau\).

Now that we have obtained a solution for the cooling time to reach \(80^{\circ} \mathrm{C}\), let's proceed with the third part of the question which is to plot the temperature-time history for the expression we derived. We will also need to compare this with the lumped capacitance method, which uses a constant free convection coefficient \(\bar{h}_{o}\). First, let's calculate the constant free convection coefficient \(\bar{h}_{o}\). This can be obtained using an appropriate correlation based on the average surface temperature \(\bar{T}=\left(T_{i}+T_{f}\right) / 2\): \[ \bar{h}_{o} = C\left[\left(\dfrac{T_{i}+T_{f}}{2}\right)-T_{\infty}\right]^{\frac{1}{4}} \] Substitute this into the lumped capacitance expression for temperature: \[ \dfrac{dT}{dt} = -\bar{h}_{o}\dfrac{A_{s}}{\rho V c_{p}}\left(T - T_{\infty}\right) \] Now, we have two expressions for temperature-time history: one for the method described in Section 5.3.3 and one using the lumped capacitance method. Plotting these two expressions will provide us with a comparison of the two approaches. This plot will show the differences in the cooling rate and time for the two methods, allowing for analysis of their relative strengths and weaknesses in modeling this specific cooling problem.