Concept Questions The block shown in the drawing has dimensions L0 ×2 L0 ×3 L0. In drawings A, B

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Concept Questions The block shown in the drawing has...

Concept Questions The block shown in the drawing has dimensions $L_{0} \times 2 L_{0} \times 3 L_{0}$. In drawings $\mathrm{A}, \mathrm{B},$ and $\mathrm{C},$ heat is conducted through the block in three different directions. In each case, the same temperature difference exists between the opposite surfaces through which the heat passes, and the time during which the heat flows is the same. (a) The cross-sectional area of the opposite surfaces in $\mathrm{C}$ is greater than that in A. Does this fact alone mean that the heat conducted in $\mathrm{C}$ is greater than that conducted in A? Provide a reason for your answer. (b) The length of material through which heat is conducted is greater in A than in B. Does this fact alone imply that the heat conducted in A is smaller than that conducted in B? Why? (c) Rank the heat conducted in each of the three cases, largest first. Problem The block in the drawing has a thermal conductivity of $250 \mathrm{~J} /\left(\mathrm{s} \cdot \mathrm{m} \cdot \mathrm{C}^{\circ}\right),$ and the length $L_{0}$ is $0.30 \mathrm{~m}$. For each case $-\mathrm{A}, \mathrm{B},$ and $\mathrm{C}-$ the temperature of the warmer surface is $35^{\circ} \mathrm{C}$ while that of the cooler surface is $19^{\circ} \mathrm{C}$. Determine the heat that flows in $5.0 \mathrm{~s}$ for each case. Be sure your answers are consistent with that in part (c) of the Concept Questions.

Key Concepts

Temperature Difference

The temperature difference between the two surfaces serves as the driving force for heat conduction. The greater the temperature difference, the higher the rate of heat transfer, assuming all other factors are constant.

Conduction Path Length

The conduction path length is the distance over which the heat must be transferred. A longer conduction path increases resistance to heat flow, thereby reducing the rate of heat transfer in accordance with Fourier’s Law.

Thermal Conductivity

Thermal conductivity is an intrinsic property of a material that measures its ability to conduct heat. It appears in Fourier's Law as a constant, determining how effectively heat is transmitted through the material under a given temperature gradient.

Fourier's Law of Heat Conduction

Fourier's Law states that the rate of heat transfer through a material is proportional to the cross-sectional area and the temperature difference, and inversely proportional to the length of the material. This fundamental principle is used to relate the measurable quantities in a conduction problem and forms the basis for calculating heat flow.

Cross-Sectional Area

The cross-sectional area through which heat flows influences the total amount of heat conducted. A larger area permits more heat to pass through for a given temperature difference and conduction path, making it a critical factor in problems involving heat transfer.

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