The heat flow between two concentric cylinders of radii r1...

The heat flow between two concentric cylinders of radii $r_{1}$ and $r$ at temperatures $T_{1}$ and $T$, respectively, is, from Eq. (2.14), $$ \dot{Q}=\frac{2 \pi k L\left(T_{1}-T\right)}{\ln \left(r / r_{1}\right)} \quad \text { or } \quad T=T_{1}-\frac{\dot{Q}}{2 \pi k L} \ln \frac{r}{r_{1}} $$ which can be interpreted as the temperature due to a line source of strength $\dot{Q}$ at $r=0$, expressed in terms of the temperature $T_{1}$ at a reference radius $r_{1}$. Derive the shape factor $S$ given by item 8 of Table $3.2$ by superimposing the temperature fields of a line source $Q$ below the ground and a line $\sin \mathrm{k}-\underline{Q}$ above the ground, in a mirror image position as shown. Define an excess temperature $T-T_{2}$, where $T_{2}$ is the ground surface temperature, and show that the isotherms are concentric circles with origins at $x=0, y=a(1+c) /(1-c)$, and radii $2 c^{1 / 2} a /(1-c)$, where $c$ is a constant parameter. Also show that the ground surface is an isotherm.

The heat flow between two concentric cylinders of radii $r_{1}$ and $r$ at temperatures $T_{1}$ and $T$, respectively, is, from Eq. (2.14),
$$
\dot{Q}=\frac{2 \pi k L\left(T_{1}-T\right)}{\ln \left(r / r_{1}\right)} \quad \text { or } \quad T=T_{1}-\frac{\dot{Q}}{2 \pi k L} \ln \frac{r}{r_{1}}
$$
which can be interpreted as the temperature due to a line source of strength $\dot{Q}$ at $r=0$, expressed in terms of the temperature $T_{1}$ at a reference radius $r_{1}$. Derive the shape factor $S$ given by item 8 of Table $3.2$ by superimposing the temperature fields of a line source $Q$ below the ground and a line $\sin \mathrm{k}-\underline{Q}$ above the ground, in a mirror image position as shown. Define an excess temperature $T-T_{2}$, where $T_{2}$ is the ground surface temperature, and show that the isotherms are concentric circles with origins at $x=0, y=a(1+c) /(1-c)$, and radii $2 c^{1 / 2} a /(1-c)$, where $c$ is a constant parameter. Also show that the ground surface is an isotherm.

Key Concepts

Analysis of Isotherms

Isotherms are curves or surfaces of constant temperature. In the study of heat conduction, analyzing the shape and position of isotherms provides insight into how heat distributes itself within a medium. Through techniques like superposition of multiple source fields, one can demonstrate that the resulting temperature field leads to isotherms that form specific geometric shapes—in this case, concentric circles with defined origins and radii. This analysis is crucial when verifying that certain surfaces, such as the ground surface in the problem, act as isothermal boundaries.

Method of Images

The method of images is a mathematical technique used to simplify boundary-value problems in potential theory and heat conduction. It involves replacing boundaries with fictitious sources (images) placed in mirror positions relative to the boundary. This approach effectively enforces the boundary conditions, such as maintaining a constant temperature along a surface, by canceling out undesired contributions from the original source, thus allowing for the construction of a symmetric temperature distribution.

Shape Factor

The shape factor, in the context of heat transfer, quantitatively relates the geometry of the system to the rate of heat transfer. It is a parameter that encapsulates the influence of the configuration of heat conductors (or sources) on the conduction process. Deriving the shape factor involves the integration of the temperature fields, often using methods such as superposition and symmetry, to relate the thermal resistance of the system to its geometric characteristics.

Heat Conduction in Cylindrical Coordinates

This concept deals with the analysis of heat transfer processes in geometries that exhibit cylindrical symmetry. In such systems, the governing differential equations, often derived from Fourier’s law of heat conduction, lead to solutions that typically involve logarithmic temperature profiles. This is particularly important when considering heat flow between cylindrical surfaces, where the geometry directly influences the spatial variation of the temperature field.

Superposition Principle

The superposition principle is a fundamental concept in linear systems which states that the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. In heat conduction problems, this allows one to construct complex temperature fields by analytically adding the effects of simpler sources, which is especially useful when constructing solutions that satisfy boundary conditions or particular geometries.

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