A long bar of AISI 316 stainless steel has an 8 cm ×8 cm square cross section. It is hot-rolle

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A long bar of AISI 316 stainless steel has an 8 cm ×8 cm...

A long bar of AISI 316 stainless steel has an $8 \mathrm{~cm} \times 8 \mathrm{~cm}$ square cross section. It is hot-rolled at $620^{\circ} \mathrm{C}$ and then cooled by cold air jets at $30^{\circ} \mathrm{C}$, giving a heat transfer coefficient of $400 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$. Determine the time required for the center temperature to decrease to $130^{\circ} \mathrm{C}$.

Key Concepts

Lumped Capacitance Method

This method is used to approximate the temperature response of a solid when the internal resistance to conduction is negligible compared to the external convective resistance (i.e., when the Biot number is small). It simplifies the analysis by assuming uniform temperature throughout the material and modeling the heat transfer as an exponential decay process.

Biot Number

The Biot number is a dimensionless parameter that compares the internal thermal resistance within a solid to the external convective resistance. It helps determine whether the temperature gradient inside the object is significant or if the lumped capacitance method can be applied. A low Biot number indicates that the temperature is nearly uniform in the body, simplifying the analysis.

Fourier Number

The Fourier number is a dimensionless time parameter in heat transfer, defined as the ratio of heat conduction rate to the rate of thermal energy storage. It is used to characterize the transient behavior of the temperature field, helping to quantify how quickly a material responds to changes in temperature. This parameter is crucial in comparing the experimental or calculated time to reach a given temperature with theoretical predictions.

Transient Heat Conduction

This concept deals with the study of how temperature within a material changes with both space and time, usually described by the heat equation. In problems like the one given, the focus is on how the interior temperature of the material evolves as heat is removed due to cooling, making transient analysis essential for determining the time required to reach a desired temperature.

Conduction Equation with Convection Boundary Conditions

When a solid loses heat to its environment, the conduction process inside the material is coupled with convection at its boundaries. This concept involves solving the heat equation subject to convective boundary conditions, which account for the heat transfer between the solid and the surrounding fluid. The convective heat transfer coefficient is a key parameter that relates to the cooling rate at the surface.

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