An aluminum rod and an iron rod are joined end to end in...

An aluminum rod and an iron rod are joined end to end in good thermal contact. The two rods have equal lengths and radii. The free end of the aluminum rod is maintained at a temperature of $100^{\circ} \mathrm{C}$, and the free end of the iron rod is maintained at $0^{\circ} \mathrm{C}$. (a) Determine the temperature of the interface between the two rods. (b) If each rod is $15 \mathrm{~cm}$ long and each has a cross-sectional area of $5.0 \mathrm{~cm}^{2}$, what quantity of energy is conducted across the combination in $30 \mathrm{~min}$ ?

Key Concepts

Thermal Resistance in Composite Rods

In composite systems where different materials are joined in series, each material offers a thermal resistance based on its conductivity, length, and cross-sectional area. The overall heat transfer is influenced by the sum of these individual resistances, and determining the interface temperature involves equating the heat flux through each segment under steady-state conditions.

Steady-State Heat Conduction and Energy Transfer

Steady-state conditions imply that the temperature distribution does not change with time, meaning the heat flux is constant throughout the system. Under these conditions, the net energy transfer over a given time can be calculated by multiplying the constant rate of conduction (derived from Fourier's law) by the time interval. This concept is essential for predicting the amount of heat energy transmitted through the system.

Fourier's Law of Heat Conduction

Fourier's Law states that the rate of heat transfer through a material is proportional to the negative gradient of temperatures and the area through which the heat flows. In practical terms, it means that the heat flux moves from regions of higher temperature to lower temperature and is governed by the material's thermal conductivity, cross-sectional area, and the length over which the conduction happens.

Thermal Conductivity

Thermal conductivity is a property of a material that measures its ability to conduct heat. Materials with high thermal conductivity, like aluminum, facilitate rapid heat transfer, while those with lower conductivity, like iron, impede the flow of thermal energy. This characteristic is central to determining temperature distribution and energy transfer rates in composite systems.

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