A long cylindrical rod of diameter 200 mm with thermal conductivity of 0.5 W / m ·K experience

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A long cylindrical rod of diameter 200 mm with thermal...

A long cylindrical rod of diameter $200 \mathrm{~mm}$ with thermal conductivity of $0.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ experiences uniform volumetric heat generation of $24,000 \mathrm{~W} / \mathrm{m}^{3}$. The rod is encapsulated by a circular sleeve having an outer diameter of $400 \mathrm{~mm}$ and a thermal conductivity of $4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The outer surface of the sleeve is exposed to cross flow of air at $27^{\circ} \mathrm{C}$ with a convection coefficient of $25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. (a) Find the temperature at the interface between the rod and sleeve and on the outer surface. (b) What is the temperature at the center of the rod?

A long cylindrical rod of diameter $200 \mathrm{~mm}$ with thermal conductivity of $0.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ experiences uniform volumetric heat generation of $24,000 \mathrm{~W} / \mathrm{m}^{3}$. The rod is encapsulated by a circular sleeve having an outer diameter of $400 \mathrm{~mm}$ and a thermal conductivity of $4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The outer surface of the sleeve is exposed to cross flow of air at $27^{\circ} \mathrm{C}$ with a convection coefficient of $25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(a) Find the temperature at the interface between the rod and sleeve and on the outer surface.
(b) What is the temperature at the center of the rod?

Key Concepts

Cylindrical Heat Conduction

This concept involves analyzing heat transfer in geometries where the radial direction is prominent, such as in long cylindrical rods or pipes. It requires the use of differential equations in cylindrical coordinates to determine temperature distributions, accounting for the radial symmetry and the associated boundary conditions. The principles are crucial when evaluating conduction through different layers of a composite cylindrical system.

Volumetric Heat Generation

Volumetric heat generation refers to the internal production of heat per unit volume within a material, often due to chemical reactions, electrical resistance, or other internal sources. In cylindrical systems, this generation introduces a source term in the heat equation, which must be considered to accurately determine temperature distributions and thermal gradients within the material.

Thermal Resistance in Composite Cylinders

This concept deals with the method of representing the heat transfer pathway as a series of thermal resistances, especially when multiple materials are involved. Each layer of material (for instance, a rod and an outer sleeve) offers a certain resistance to conductive heat transfer. Summing these resistances provides a simplified way to compute the overall temperature drop across the layers, which is essential for analyzing multi-material systems.

Convection Heat Transfer

Convection heat transfer describes the process by which heat is transferred from a surface to a fluid moving over it, characterized by the convection heat transfer coefficient. This concept is important when evaluating the cooling (or heating) of external surfaces of solid bodies, as it defines the boundary condition for the heat transfer problem and influences the overall temperature distribution in the system.

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