(a) The inside of a hollow cylinder is maintained at a...

(a) The inside of a hollow cylinder is maintained at a temperature $T_{a}$, and the outside is at a lower temperature, $T_{b}$ (Fig. $\mathrm{P} 19.45$ ). The wall of the cylinder has a thermal conductivity $k .$ Ignoring end effects, show that the rate of energy conduction from the inner surface to the outer surface in the radial direction is $$\frac{d Q}{d t}=2 \pi L k\left[\frac{T_{a}-T_{b}}{\ln (b / a)}\right]$$ Suggestions: The temperature gradient is $d T / d r .$ A radial energy current passes through a concentric cylinder of area $2 \pi r L$. (b) 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 $2.50 \mathrm{~m} .$ It 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?

(a) The inside of a hollow cylinder is maintained at a temperature $T_{a}$, and the outside is at a lower temperature, $T_{b}$ (Fig. $\mathrm{P} 19.45$ ). The wall of the cylinder has a thermal conductivity $k .$ Ignoring end effects, show that the rate of energy conduction from the inner surface to the outer surface in the radial direction is
$$\frac{d Q}{d t}=2 \pi L k\left[\frac{T_{a}-T_{b}}{\ln (b / a)}\right]$$
Suggestions: The temperature gradient is $d T / d r .$ A radial energy current passes through a concentric cylinder of area $2 \pi r L$. (b) 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 $2.50 \mathrm{~m} .$ It 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?

Key Concepts

Fourier's Law of Heat Conduction

Fourier's law states that the heat flux in a material is proportional to the negative of the temperature gradient, with the constant of proportionality being the thermal conductivity. This forms the basis for analyzing heat transfer through materials, regardless of geometry, and is essential for deriving formulas for heat conduction.

Cylindrical Coordinates in Heat Transfer

When heat conduction occurs in cylindrical geometries, the radial dependence must be taken into account. The cross-sectional area through which heat flows changes with radius, so the application of Fourier's law requires working in cylindrical coordinates, where the area term is a function of r (e.g., 2?rL). This geometric consideration is key to properly integrating the heat flux over the surface.

Integration over Variable Cross Section

In problems where the cross-sectional area for heat transfer is not constant, such as in a hollow cylinder, the differential form of Fourier's law must be integrated over the radial distance to determine the total rate of energy transfer. This process involves setting up an integral that accounts for the varying area and temperature gradient, leading to expressions that often include logarithmic terms.

Thermal Resistance Analogy

The concept of thermal resistance is analogous to electrical resistance, where the temperature difference drives the heat flow, similar to a voltage difference driving electrical current. In cylindrical systems, an effective thermal resistance can be derived from integrating the differential thermal resistances, which simplifies the understanding and calculation of heat transfer rates.

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