A hot brass plate is having its upper surface cooled by impinging jet of air at temperature of \(15^{\circ} \mathrm{C}\) and convection heat transfer coefficient of \(220 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The \(10-\mathrm{cm}\) 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 \(\alpha=33.9 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) ) has a uniform initial temperature of \(650^{\circ} \mathrm{C}\), and the bottom surface of the plate is insulated. Determine the temperature at the center plane of the brass plate after 3 minutes of cooling. Solve this problem using analytical oneterm approximation method (not the Heisler charts).

Step by step solution

01

Calculate the Biot number

The Biot number (Bi) is a dimensionless quantity that represents the ratio of the convection heat transfer resistance to conduction heat transfer resistance. It's calculated as: Bi = \(\frac{hL_{c}}{k}\), where: \(h = 220 \mathrm{~W}/\mathrm{m}^{2}\cdot\mathrm{K}\) (convection heat transfer coefficient), \(L_{c} = \frac{L}{2} = \frac{10 \times 10^{-2} \mathrm{m}}{2}\) (the characteristic length, which is considered as half the thickness of the plate for this case), \(k = 110 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) (thermal conductivity of brass).

02

Calculate the Fourier number

The Fourier number (Fo) is a dimensionless quantity used to analyze transient heat conduction problems. It's calculated as: Fo = \(\frac{\alpha t}{L_{c}^{2}}\), where: \(\alpha = 33.9 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) (thermal diffusivity), \(t = 3 \times 60 \mathrm{~s}\) (time after cooling).

03

Calculate the standardized temperature

The standardized temperature (\(\theta^{\ast}\)) is used to simplify the temperature calculations and is given by: \(\theta^{\ast}\) = \(\frac{T - T_{\infty}}{T_{i} - T_{\infty}}\), where: \(T\) is the temperature we want to find, \(T_{\infty} = 15^{\circ} \mathrm{C}\) (air temperature), \(T_{i} = 650^{\circ} \mathrm{C}\) (initial plate temperature). Then, \(\theta^{\ast}\) can be calculated using the one-term approximation method as: \(\theta^{\ast} \approx \frac{2\text{exp}\left(-\text{Bi}\cdot\text{Fo}\right)}{\pi \text{Bi}}\cdot\text{sinh}\left(\pi\sqrt{\text{Fo}}\right)\)

04

Calculate the temperature at the center plane

Now we can solve for the temperature at the center plane using the formula: \(T = T_{\infty} + (T_{i} - T_{\infty})\cdot \theta^{\ast}\) Follow these steps to determine the temperature at the center plane of the brass plate after 3 minutes of cooling using the analytical one-term approximation method.

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