In a cylindrical pipe where area isn't constant, Equation...

In a cylindrical pipe where area isn't constant, Equation 16.5 takes the form $H=-k A(d T / d r),$ where $r$ is the radial coordinate measured from the pipe axis. Use this equation to show that the heat-loss rate from a cylindrical pipe of radius $R_{1}$ and length $L$ is $$H=\frac{2 \pi k L\left(T_{1}-T_{2}\right)}{\ln \left(R_{2} / R_{1}\right)}$$ where the pipe is surrounded by insulation of outer radius $R_{2}$ and thermal conductivity $k$ and where $T_{1}$ and $T_{2}$ are the temperatures at the pipe surface and the outer surface of the insulation, respectively. (Hint: Consider the heat flow through a thin section of pipe, with thickness $d r$, as shown in Fig. 16.16 Then integrate.) (FIGURE CANNOT COPY)

Key Concepts

Fourier's Law of Heat Conduction

This law states that the rate of heat transfer through a material is proportional to the negative gradient of temperature and the area through which the heat flows. It forms the basis for analyzing heat conduction problems by relating the local temperature gradient in a medium to the heat flux, making it fundamental for deriving expressions for heat transfer in systems with varying geometries.

Cylindrical Coordinates in Heat Transfer

When dealing with heat transfer in cylindrical geometries, such as pipes or cylinders, the area through which heat is conducted is not constant but varies with the radial coordinate. This leads to a modification of the heat conduction equations where the cross-sectional area is expressed as a function of the radial distance, typically involving a term like 2?rL, which accounts for the circular symmetry of the problem.

Differential Analysis and Integration

The technique involves considering an infinitesimally thin cylindrical shell within the larger geometry, where the heat flow can be expressed in differential form. By setting up a differential equation based on Fourier's Law and integrating it over the appropriate limits, one can obtain a complete description of the heat transfer rate across the entire insulation or material layer.

Logarithmic Temperature Profile in Cylindrical Systems

When integrating the differential form of heat conduction in cylindrical coordinates, the integration of a 1/r term naturally leads to a logarithmic function. This results in a temperature profile that varies logarithmically with the radial position, which is a characteristic outcome in problems dealing with heat flow through cylindrical layers such as insulated pipes.

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