A uniform non-metallic annular cylinder of inner radius r1,...

A uniform non-metallic annular cylinder of inner radius $r_1$, outer radius $r_2$, length $l_0$ is maintained with its inner surface at $100^{\circ} \mathrm{C}$ and its outer surface at $0^{\circ} \mathrm{C}$. (a) What is the temperature distribution inside? (b) If it is then placed in a thermally insulated chamber of negligible heat capacity and allowed to come to temperature equilibrium, will its entropy increase, decrease or remain the same? Justify your answer. FIGURE CAN'T COPY.

A uniform non-metallic annular cylinder of inner radius $r_1$, outer radius $r_2$, length $l_0$ is maintained with its inner surface at $100^{\circ} \mathrm{C}$ and its outer surface at $0^{\circ} \mathrm{C}$.
(a) What is the temperature distribution inside?
(b) If it is then placed in a thermally insulated chamber of negligible heat capacity and allowed to come to temperature equilibrium, will its entropy increase, decrease or remain the same? Justify your answer.
FIGURE CAN'T COPY.

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Key Concepts

Steady-State Heat Conduction in Cylindrical Coordinates

This concept deals with solving the heat conduction equation in a system that has cylindrical symmetry. In steady state (where temperatures do not change in time), the temperature distribution is governed by Laplace’s equation in cylindrical coordinates. The solution requires applying appropriate boundary conditions (such as fixed temperatures on the inner and outer surfaces) and integrating the resulting differential equation to find a temperature profile that depends solely on the radial coordinate.

Entropy Change in Irreversible Processes

This concept relates to the second law of thermodynamics, which states that the total entropy of an isolated system can never decrease and generally increases during irreversible processes. When a system with a non-uniform temperature profile (i.e., a non-equilibrium state) is allowed to reach thermal equilibrium in an isolated environment, irreversible heat flow occurs, driving the system to a state of uniform temperature. This process dissipates energy as entropy and results in a net increase in the total entropy of the system.

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