A cylindrical copper rod of base area A and length L is insulated on its lateral surface. One en

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A cylindrical copper rod of base area A and length L is...

A cylindrical copper rod of base area $\mathrm{A}$ and length $L$ is insulated on its lateral surface. One end of the rod is in contact with a wall at temperature $T_{\mathrm{H}}$. The other end is in contact with a wall at a lower temperature $T_{\mathrm{C}}$. At steady state, the rate at which energy is conducted into the rod from the hot wall is $$ \dot{Q}_{\mathrm{H}}=\frac{\kappa \mathrm{A}\left(T_{\mathrm{H}}-T_{\mathrm{C}}\right)}{L} $$ where $\kappa$ is the thermal conductivity of the copper rod. (a) For the rod as the system, obtain an expression for the time rate of entropy production in terms of A, $L, T_{\mathrm{H}}, T_{\mathrm{C}}$, and $\kappa$. (b) If $T_{\mathrm{H}}=327^{\circ} \mathrm{C}, T_{\mathrm{C}}=77^{\circ} \mathrm{C}, \kappa=0.4 \mathrm{~kW} / \mathrm{m} \cdot \mathrm{K}, \mathrm{A}=0.1$ $\mathrm{m}^{2}$, plot the heat transfer rate $\dot{Q}_{\mathrm{H}}$, in $\mathrm{kW}$, and the time rate of entropy production, in $\mathrm{kW} / \mathrm{K}$, each versus $L$ ranging from $0.01$ to $1.0 \mathrm{~m}$. Discuss.

A cylindrical copper rod of base area $\mathrm{A}$ and length $L$ is insulated on its lateral surface. One end of the rod is in contact with a wall at temperature $T_{\mathrm{H}}$. The other end is in contact with a wall at a lower temperature $T_{\mathrm{C}}$. At steady state, the rate at which energy is conducted into the rod from the hot wall is
$$
\dot{Q}_{\mathrm{H}}=\frac{\kappa \mathrm{A}\left(T_{\mathrm{H}}-T_{\mathrm{C}}\right)}{L}
$$
where $\kappa$ is the thermal conductivity of the copper rod.
(a) For the rod as the system, obtain an expression for the time rate of entropy production in terms of A, $L, T_{\mathrm{H}}, T_{\mathrm{C}}$, and $\kappa$.
(b) If $T_{\mathrm{H}}=327^{\circ} \mathrm{C}, T_{\mathrm{C}}=77^{\circ} \mathrm{C}, \kappa=0.4 \mathrm{~kW} / \mathrm{m} \cdot \mathrm{K}, \mathrm{A}=0.1$ $\mathrm{m}^{2}$, plot the heat transfer rate $\dot{Q}_{\mathrm{H}}$, in $\mathrm{kW}$, and the time rate of entropy production, in $\mathrm{kW} / \mathrm{K}$, each versus $L$ ranging from $0.01$ to $1.0 \mathrm{~m}$. Discuss.

Key Concepts

Steady-State Systems

A steady-state system is one in which the macroscopic properties, such as temperature and heat flux, remain constant with time despite ongoing processes. In the context of thermal conduction between two reservoirs at different temperatures, a steady state implies a constant heat flow, even though there is a persistent entropy production. Understanding steady-state conditions is critical for analyzing energy transfers and determining performance characteristics of systems under continuous operation.

Fourier's Law of Heat Conduction

Fourier's law is a fundamental principle that quantifies the rate of heat transfer through a material. It states that the heat flux passing through a material is proportional to the negative of the temperature gradient and the material's thermal conductivity. This concept is essential for understanding how heat moves through solids and is used to derive expressions for the conduction rate in systems such as rods connecting two temperature reservoirs.

Entropy Production in Irreversible Processes

In any irreversible process, including heat conduction between bodies at different temperatures, entropy is produced. The rate of entropy production quantifies the degree of irreversibility and is calculated by considering the heat flows into or out of a system divided by the temperature at which those flows occur. This concept, derived from the second law of thermodynamics, provides insights into the efficiency and dissipation within thermal systems.

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