. The passenger section of a jet airliner has the shape of a...

. The passenger section of a jet airliner has 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 at $-35.0^{\circ} \mathrm{C}$. What power must be supplied to the heater if this temperature difference is to be maintained? (Use the result you obtained in Problem 70.)

. The passenger section of a jet airliner has 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 at $-35.0^{\circ} \mathrm{C}$. What power must be supplied to the heater if this temperature difference is to be maintained? (Use the result you obtained in Problem 70.)

Key Concepts

Steady State Heat Conduction

This concept refers to a condition where the temperature distribution within a material does not change with time, implying that the heat transfer rate is constant. This assumption allows for the use of simplified, steady state equations to determine power requirements in thermal management problems.

Insulation

Insulation materials reduce the rate of heat transfer between environments with different temperatures. Their effectiveness is determined by properties such as thermal conductivity and thickness, playing a critical role in maintaining a desired internal temperature.

Heat Transfer in Cylindrical Coordinates

In systems with cylindrical geometry, such as a jet airliner’s passenger section, heat conduction must be analyzed using a formulation adjusted for the curvature. This involves accounting for changes in the cross-sectional area as one moves radially outward, typically resulting in logarithmic relationships between the inner and outer radii.

Fourier's Law of Heat Conduction

This principle quantifies the rate at which heat energy is transferred through a material due to a temperature gradient. It forms the basis for calculating the power required to maintain a temperature difference by relating thermal conductivity, area, and the temperature gradient.

Thermal Conductivity

This is a material property that indicates how well a material conducts heat. Lower thermal conductivity means that the material resists heat flow, which is crucial when designing insulation and assessing the energy needed to overcome heat losses.

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