A very long rod of 5-mm diameter and uniform thermal...

A very long rod of $5-\mathrm{mm}$ diameter and uniform thermal conductivity $k=25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ is subjected to a heat treatment process. The center, 30 -mm-long portion of the rod within the induction heating coil experiences uniform volumetric heat generation of $7.5 \times 10^{6} \mathrm{~W} / \mathrm{m}^{3}$. The unheated portions of the rod, which protrude from the heating coil on either side, experience convection with the ambient air at $T_{\infty}=20^{\circ} \mathrm{C}$ and $h=10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Assume that there is no convection from the surface of the rod within the coil. (a) Calculate the steady-state temperature $T_{o}$ of the rod at the midpoint of the heated portion in the coil. (b) Calculate the temperature of the rod $T_{b}$ at the edge of the heated portion.

A very long rod of $5-\mathrm{mm}$ diameter and uniform thermal conductivity $k=25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ is subjected to a heat treatment process. The center, 30 -mm-long portion of the rod within the induction heating coil experiences uniform volumetric heat generation of $7.5 \times 10^{6} \mathrm{~W} / \mathrm{m}^{3}$.
The unheated portions of the rod, which protrude from the heating coil on either side, experience convection with the ambient air at $T_{\infty}=20^{\circ} \mathrm{C}$ and $h=10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Assume that there is no convection from the surface of the rod within the coil.
(a) Calculate the steady-state temperature $T_{o}$ of the rod at the midpoint of the heated portion in the coil.
(b) Calculate the temperature of the rod $T_{b}$ at the edge of the heated portion.

Key Concepts

Energy Balance in Conduction Problems

An energy balance involves accounting for all the energy entering, leaving, and being produced within a system. In conduction problems, especially those involving internal heat generation and convective cooling, the conservation of energy principle is used to develop differential equations that describe the temperature profile within the material.

Boundary Conditions

Boundary conditions are the constraints applied to the edges of the domain in a thermal problem, such as specified temperatures, heat fluxes, or convective conditions. They provide the necessary additional information to uniquely determine the solution for temperature distributions in a conduction problem.

Convective Heat Transfer

This concept involves the transfer of heat between a solid surface and a moving fluid. The convective heat transfer coefficient (h) characterizes the rate of this heat transfer, which is a function of the fluid properties and the flow conditions, and is critical in problems where surfaces interact with the surrounding environment.

Thermal Conductivity

Thermal conductivity is an intrinsic property of a material that quantifies its ability to conduct heat. It plays a central role in heat conduction analyses by linking the heat flux to the temperature gradient, thereby governing the rate at which heat is transferred through the material.

Steady-State Heat Conduction

This concept refers to a thermal situation where the temperature distribution does not change with time, meaning all the energy entering a system is balanced by the energy leaving it. The mathematical models used in this context have no time-dependent terms, which simplifies the analysis to spatial variables only.

Volumetric Heat Generation

This is the notion of internal heat being generated within a material per unit volume. In many processes, such as induction heating or exothermic reactions, heat is produced uniformly or non-uniformly inside the body, and this internal heat source must be accounted for in energy balance calculations.

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