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 maintace 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 $30 .$ min?

Key Concepts

Energy Transfer in Conduction

The energy conducted through a material over a certain time interval can be calculated by multiplying the steady-state heat transfer rate by the time duration. This quantification involves applying Fourier’s Law along with the computed temperature differences and material resistances, allowing the calculation of total energy transferred over specified time periods such as minutes or hours.

Thermal Resistance

Thermal resistance is a concept that quantifies how a material or geometry resists the flow of heat. Analogous to electrical resistance in circuits, it is defined as the ratio of temperature difference across the material to the heat transfer rate. In conduction problems involving rods, the resistance depends on the material’s thermal conductivity and the rod’s dimensions, making it a key factor in systems with rods in series.

Composite Conduction Through Materials in Series

In a system where two different materials are joined in series, the overall temperature profile is determined by matching the heat flow across both materials. The continuity of heat flow requires that the same amount of energy passes through each rod per unit time, leading to a balance of thermal resistances. This concept is essential in finding the junction temperature between different rods in steady-state conditions.

Fourier's Law of Thermal Conduction

This law states that the rate of heat transfer through a material is proportional to the negative gradient of temperature and the area through which the heat flows. It serves as the fundamental principle for analyzing steady-state heat conduction, enabling the calculation of heat flux in rods or other homogeneous media.

Thermal Conductivity

Thermal conductivity is an intrinsic property of a material, indicating its ability to conduct heat. Materials with high thermal conductivity, like aluminum, allow heat to flow quickly, while those with low thermal conductivity, such as iron in certain conditions, impede the flow. This property is crucial for determining how temperatures distribute in composite rod systems.

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