In a production facility, 3-cm-thick large brass plates \(\left(k=110 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=8530 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=380 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.\), and \(\left.\alpha=33.9 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)\) that are initially at a uniform temperature of \(25^{\circ} \mathrm{C}\) are heated by passing them through oven maintained at \(700^{\circ} \mathrm{C}\). The plates remain in the oven for a period of \(10 \mathrm{~min}\). Taking the convection heat transfer coefficient to be \(h=80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the surface temperature of the plates when they come out of the oven. Solve this problem using analytical one-term approximation method (not the Heisler charts). Can this problem be solved using lumped system analysis? Justify your answer.

The Fourier number (Fo) is a dimensionless number that is crucial for analyzing heat conduction. It is defined as the ratio of diffusive/conductive heat transfer rate to the heat storage rate. In the context of the problem, we calculated the Fourier number using the formula \[\text{Fo} = \frac{\alpha t}{L^2}\] where \(\alpha\) represents the thermal diffusivity of the material, \(t\) is the time during which heat transfer takes place, and \(L\) is the characteristic length, which usually is the thickness of the object in question.

In practical terms, the Fourier number provides an estimate of how the temperature distribution within an object changes with time. A high Fourier number suggests that the heat transfer through the material is quick relative to the rate at which the material stores heat. In the given exercise, with a calculated Fourier number of 2.274, we observe a situation where considerable temperature change within the brass plate has occurred over time due to the prolonged exposure to the high oven temperature.

The Biot number (Bi) is another essential dimensionless number in the heat transfer analysis, especially when assessing the temperature gradients within a solid object. We calculate it using the formula \[\text{Bi} = \frac{hL}{k}\] where \(h\) is the convection heat transfer coefficient, \(L\) is the characteristic length as defined earlier, and \(k\) is the thermal conductivity of the material.

The Biot number helps to establish whether an object has uniform temperature within (lumped system analysis applies) or significant temperature variation across its volume. A small Biot number (Bi < 0.1) implies a uniform temperature distribution, suitable for lumped system analysis. Conversely, a larger Biot number indicates that temperature gradients within the object must be accounted for. In our example, the Biot number is calculated as 0.2182, which confirms the presence of temperature gradients within the brass plate, thereby invalidating the use of lumped system analysis for this problem.

The convection heat transfer coefficient \(h\) is significant in determining the rate at which heat is transferred by convection between a solid surface and a fluid in contact with it. It is a measure of the convection heat transfer ability of the fluid and surface in question, with a higher \(h\) indicating more efficient heat transfer.

In the context of our problem, the convection heat transfer coefficient affects the calculation of the Biot number and subsequently, the determination of whether lumped system analysis is applicable. The given value of \(h = 80 \mathrm{~W} / \mathrm{m}^2 \. \. \mathrm{K}\) significantly influences the temperature profile of the brass plates as they pass through the oven. Since convection is the mode of heat transfer to the plates, this coefficient plays a central role in the one-term approximation method used to approximate the surface temperature of the plates upon exiting the oven.