The Figure 11.4 shows a system of two concentric spheres of...

The Figure $11.4$ shows a system of two concentric spheres of radii $r_{1}$ and $r_{2}$ kept at temperature $T_{1}$ and $T_{2}$. The radial rate of flow of heat in a substance between the two concentric spheres is proportional to (A) $\frac{\left(r_{2}-r_{1}\right)}{\left(r_{1} r_{2}\right)}$ (B) $\operatorname{In}\left(\frac{r_{2}}{r_{1}}\right)$ (C) $\frac{r_{1} r_{2}}{\left(r_{2}-r_{1}\right)}$ (D) $\left(r_{2}-r_{1}\right)$

The Figure $11.4$ shows a system of two concentric spheres of radii $r_{1}$ and $r_{2}$ kept at temperature $T_{1}$ and $T_{2}$. The radial rate of flow of heat in a substance between the two concentric spheres is proportional to
(A) $\frac{\left(r_{2}-r_{1}\right)}{\left(r_{1} r_{2}\right)}$
(B) $\operatorname{In}\left(\frac{r_{2}}{r_{1}}\right)$
(C) $\frac{r_{1} r_{2}}{\left(r_{2}-r_{1}\right)}$
(D) $\left(r_{2}-r_{1}\right)$

Key Concepts

Steady-State Temperature Distribution

In steady-state conditions, the temperature distribution within a system does not change with time. In a spherically symmetric system, this leads to a temperature profile that can be derived by integrating the differential form of Fourier’s law, showing how geometric factors, such as radii in spherical shells, influence the rate of heat transfer.

Conduction in Spherical Coordinates

When dealing with spherical geometries, such as concentric spheres, the analysis of heat conduction requires expressing the temperature gradient in terms of spherical coordinates. This involves solving differential equations that account for the radial symmetry of the system, which leads to a temperature profile that typically varies with the inverse of the radial distance.

Heat Conduction

Heat conduction is the process of energy transfer within a material due to a temperature gradient. It involves the movement of thermal energy from regions of higher temperature to regions of lower temperature and is governed by the material's thermal conductivity. This concept is fundamental in understanding how heat flows through different media.

Fourier’s Law

Fourier’s law relates the rate of heat transfer (heat flux) to the spatial temperature gradient within a material. It states that the heat flux is proportional to the negative of the temperature gradient and is central to the analysis of conductive heat transfer in any system, providing the basis for solving conduction problems.

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