Consider a tube wall of inner and outer radii ri and ro,...

Consider a tube wall of inner and outer radii $r_{i}$ and $r_{o}$, whose temperatures are maintained at $T_{i}$ and $T_{o}$, respectively. The thermal conductivity of the cylinder is temperature dependent and may be represented by an expression of the form $k=k_{o}(1+a T)$, where $k_{o}$ and $a$ are constants. Obtain an expression for the heat transfer per unit length of the tube. What is the thermal resistance of the tube wall?

Key Concepts

Heat Conduction in Cylindrical Coordinates

This concept deals with the analysis of heat transfer through cylindrical geometries, where the governing equations are modified to account for the radial symmetry. In cylindrical coordinates, the heat flux and temperature gradients are expressed as functions of the radial coordinate, and the integration of these expressions over the geometry is necessary to find results such as the total heat transfer rate.

Fourier’s Law of Heat Conduction

Fourier’s law is the fundamental principle governing heat conduction, describing how the heat flux is proportional to the negative temperature gradient. In cylindrical geometries, this law is adapted to relate the radial heat flow to the temperature gradient along the radius, making it essential for setting up the differential equations that describe the problem.

Temperature-Dependent Thermal Conductivity

When the thermal conductivity of a material varies with temperature, the heat conduction problem becomes nonlinear. This concept requires modifying Fourier’s law to include a conductivity that is a function of temperature, which then affects the integration process when determining the heat transfer rate. It represents a more complex, realistic scenario compared to assuming constant conductivity.

Thermal Resistance in Conduction

Thermal resistance provides a measure of a material’s opposition to heat flow. In cylindrical systems, it is defined based on the integrated form of Fourier’s law over the geometry. The concept simplifies the analysis of heat transfer by allowing the treatment of the conduction path as an equivalent resistor, making it easier to calculate the overall heat transfer from one surface to another.

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