Question

A heated 6-mm-thick Pyroceram plate $\left(\rho=2600 \mathrm{~kg} / \mathrm{m}^{3}\right.\(, \)c_{p}=808 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=3.98 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\(, and \)\left.\alpha=1.89 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$ is being cooled in a room with air temperature of \(25^{\circ} \mathrm{C}\) and convection heat transfer coefficient of \(13.3 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The heated Pyroceram plate had an initial temperature of \(500^{\circ} \mathrm{C}\), and it is allowed to cool for \(286 \mathrm{~s}\). If the mass of the Pyroceram plate is \(10 \mathrm{~kg}\), determine the heat transfer from the Pyroceram plate during the cooling process using the analytical one-term approximation method.

Short answer

The heat transfer from the Pyroceram plate during the cooling process is 3,826,200 J.

Step-by-step solution

Calculate the Biot Number (Bi)

The Biot number (Bi) is a dimensionless number that characterized the ratio of internal conductive resistance to external convective resistance. It's calculated as: $$Bi=\frac{hL_{c}}{k}$$ Where: \(h\) = convection heat transfer coefficient \((13.3 \,\mathrm{W/m^2K})\) \(L_{c}\) = characteristic length (thickness of the plate, \(0.006 \,\mathrm{m}\)) \(k\) = thermal conductivity of Pyroceram \((3.98\,\mathrm{W/mK})\) Now, calculate the Biot number: $$Bi=\frac{13.3(0.006)}{3.98} = 0.02$$

Calculate the Fourier Number (Fo) times time (t)

The Fourier number (Fo) is a dimensionless number characterized by the ratio of the heat conduction rate to the heat storage rate. Multiply the Fourier number by the cooling time to obtain the product: $$Fo\cdot t=\frac{\alpha t}{L_{c}^{2}}$$ Where: \(\alpha\) = thermal diffusivity of Pyroceram \((1.89\times 10^{-6}\,\mathrm{m^2/s})\) \(t\) = cooling time \((286\,\mathrm{s})\) \(L_{c}\) = characteristic length \((0.006\,\mathrm{m})\) Now, calculate the product Fo and t: $$Fo\cdot t=\frac{1.89\times 10^{-6}(286)}{(0.006)^{2}}=2.694$$

Obtain the one-term approximation dimensionless temperature (\(\Theta\)) Using Bi and Fo

The one-term approximation dimensionless temperature is obtained from the chart or table provided in textbooks where the X-axis represents the Biot number (Bi) and the Y-axis represents the product of Fourier number (Fo) and time (t): $$\Theta=1.0$$

Calculate the actual temperature change and total heat transfer

Calculate the actual temperature change of the plate using the initial temperature (\(T_{i}\)), room temperature (\(T_{\infty}\)), and dimensionless temperature (\(\Theta\)): $$\Delta T=(T_{i}-T_{\infty})\Theta$$ Next, use the mass (\(m\)), specific heat (\(c_p\)), and actual temperature change to find the total heat transfer (\(Q\)): $$Q=mc_{p}\Delta T$$ Plug in the values: $$\Delta T=(500-25)1.0=475\,\mathrm{K}$$ $$Q=(10)(808)(475)=3,826,200\,\mathrm{J}$$ The heat transfer from the Pyroceram plate during the cooling process is 3,826,200 J using the one-term approximation method.