The passenger section of a jet airliner is in the shape of a...

The passenger section of a jet airliner is in the shape of a cylindrical tube with a length of 35.0 $\mathrm{m}$ and an inner radius of 2.50 $\mathrm{m}$ . Its walls are lined with an insulating material 6.00 $\mathrm{cm}$ in thickness and having a thermal conductivity of $4.00 \times 10^{-5} \mathrm{cal} / \mathrm{s} \cdot \mathrm{cm} \cdot^{\circ} \mathrm{C}$ . A heater must maintain the interior temperature at $25.0^{\circ} \mathrm{C}$ while the outside temperature is $-35.0^{\circ} \mathrm{C}$ . What power must be supplied to the heater? (Use the result of Problem $70 . )$

Key Concepts

Thermal Conduction

Thermal conduction is the process by which heat energy is transmitted through collisions between neighboring molecules or atoms, or via free electrons, without any overall movement of the material. It is governed by Fourier's law, which 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.

Fourier's Law

Fourier's law provides the fundamental relationship for conduction heat transfer. It expresses the heat flux as proportional to the temperature gradient and the material's thermal conductivity, which is a measure of how readily heat is conducted through a substance. This law forms the basis for calculating temperature distributions and required power inputs for heating or cooling systems.

Cylindrical Heat Transfer

Heat transfer through cylindrical geometries involves a modification of Fourier's law to account for the radial spread of heat. In such cases, the area through which heat is transferred varies with the radius, requiring specific forms of the conduction equation that integrate over the curved surface area. This concept is crucial when dealing with systems like pipes or, in this example, the passenger section of an airliner.

Thermal Insulation

Thermal insulation refers to the use of materials that reduce the rate of heat transfer, thereby maintaining differences in temperature between adjacent regions. The quality of insulation is characterized by its thickness and thermal conductivity; even small values of conductivity, when combined with sufficient thickness, can significantly reduce energy losses in a system.

Heat Transfer Rate and Power Calculation

The heat transfer rate or power required is determined by the amount of thermal energy that must be transferred per unit time to maintain a temperature difference between two regions. In the context of engineering systems like a heated cabin, power calculations involve understanding the geometry of the system, the thermal properties of the insulating materials, and the temperature difference that drives the conduction process.

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