The student from Problem 79 realizes that standing naked in...

The student from Problem 79 realizes that standing naked in a cold room will not give him the desired weight loss results since it is much less efficient than simply exercising. So he decides to "burn" calories through conduction. He fills the bathtub with $16^{\circ} \mathrm{C}$ water and gets in. The water right next to his skin warms up to the same temperature as his skin, $35^{\circ} \mathrm{C},$ but the water only $3.0 \mathrm{mm}$ away remains at $16^{\circ} \mathrm{C}$. At what rate (in kcal/h) would he "burn" calories? The thermal conductivity of water at this temperature is $0.58 \mathrm{W} /(\mathrm{m} \cdot \mathrm{K}) .$ [Warning: Do not try this. Sitting in water this cold can lead to hypothermia and even death. $]$

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Key Concepts

Temperature Gradient

The temperature gradient refers to the rate of change of temperature over a distance within a medium. It creates the driving force for heat conduction, with a steeper temperature gradient leading to a larger rate of heat transfer.

Fourier’s Law of Heat Conduction

Fourier’s law establishes that the rate of heat transfer through a material is proportional to the cross-sectional area and the temperature difference across the material, and inversely proportional to the distance over which the temperature changes. This principle is fundamental in understanding how heat is conducted in any medium.

Thermal Conductivity

Thermal conductivity is a property of a material that quantifies its ability to conduct heat. It is a critical factor in conduction problems because it directly affects the rate at which heat is transferred; higher thermal conductivity means quicker heat transfer.

Unit Conversion in Heat Transfer

In heat transfer calculations, converting between units such as watts (a measure of power in joules per second) and kilocalories per hour is essential for comparing energy transfer rates and understanding practical energy expenditures. Mastery of these conversions ensures consistency and accuracy in engineering and physics calculations.

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