A 10 cm cube of Inconel-X alloy is removed from an oven at 810^∘ C and quenched in coolant at 4

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A 10 cm cube of Inconel-X alloy is removed from an oven at...

A $10 \mathrm{~cm}$ cube of Inconel-X alloy is removed from an oven at $810^{\circ} \mathrm{C}$ and quenched in coolant at $40^{\circ} \mathrm{C}$, giving a heat transfer coefficient of $800 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$. Calculate the time required for (i) the center temperature to undergo $95 \%$ of its decrease to the equilibrium value. (ii) a $95 \%$ fractional energy loss. Compare the two results and comment.

A $10 \mathrm{~cm}$ cube of Inconel-X alloy is removed from an oven at $810^{\circ} \mathrm{C}$ and quenched in coolant at $40^{\circ} \mathrm{C}$, giving a heat transfer coefficient of $800 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$. Calculate the time required for
(i) the center temperature to undergo $95 \%$ of its decrease to the equilibrium value.
(ii) a $95 \%$ fractional energy loss.
Compare the two results and comment.

Key Concepts

Energy Balance in Transient Processes

Energy balance in transient processes involves accounting for the decrease in the system’s internal energy as it cools, integrating the temperature distribution over the entire volume of the body. Unlike the center temperature, which only reflects local behavior, the fractional energy loss provides a measure of the overall reduction in stored thermal energy and may follow a different time scale, highlighting the difference between pointwise temperature evolution and global energy dissipation.

Eigenvalue Analysis and Characteristic Time Constants

Eigenvalue analysis is a mathematical approach used to solve the transient conduction problem by decomposing the temperature field into a series of modes, each decaying exponentially with a characteristic time constant. The primary (lowest) eigenvalue often dominates the long?term response, and characteristic time constants can be defined such that a specified fraction of the total temperature change (e.g., 95%) occurs over a period determined by these constants.

Biot Number

The Biot number is a dimensionless parameter that compares the internal resistance to heat conduction within a solid to the external resistance to heat transfer by convection. It is important in determining whether the temperature within the solid can be assumed uniform (small Biot number) or if spatial temperature gradients must be considered (larger Biot number), thereby influencing the choice of analytical or numerical solution methods.

Convection Heat Transfer

Convection heat transfer refers to the process by which heat is removed or added at the surface of a body through a fluid medium, governed by Newton’s law of cooling. It defines the boundary conditions for the solid's surface, where the heat transfer coefficient quantifies the rate of energy exchange between the solid and the surrounding fluid, crucial for accurately modeling the overall cooling process.

Transient Heat Conduction

This concept deals with the time?dependent change in temperature within a solid as heat is transferred from one part of the material to another. In transient heat conduction problems, the temperature distribution evolves over time until a steady state is reached, and solutions often involve mathematical models such as the use of partial differential equations and eigenfunctions to capture spatial and temporal variations.

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