(III) A cylindrical pipe has inner radius R1 and outer...

(III) A cylindrical pipe has inner radius $R_{1}$ and outer radius $R_{2} .$ The interior of the pipe carries hot water at temperature $T_{1} .$ The temperature outside is $T_{2}\left( < T_{1}\right)$ . (a) Show that the rate of heat loss for a length $\ell$ of pipe is $$\frac{d Q}{d t}=\frac{2 \pi k\left(T_{1}-T_{2}\right) \ell}{\ln \left(R_{2} / R_{1}\right)}$$ where $k$ is the thermal conductivity of the pipe. $(b)$ Suppose the pipe is steel with $R_{1}=3.3 \mathrm{cm}, R_{2}=4.0 \mathrm{cm},$ and $T_{2}=18^{\circ} \mathrm{C} .$ If the pipe holds still water at $T_{1}=71^{\circ} \mathrm{C},$ what will be the initial rate of change of its temperature? (c) Suppose water at $71^{\circ} \mathrm{C}$ enters the pipe and moves at a speed of 8.0 $\mathrm{cm} / \mathrm{s} .$ What will be its temperature drop per centimeter of travel?

(III) A cylindrical pipe has inner radius $R_{1}$ and outer
radius $R_{2} .$ The interior of the pipe carries hot water at
temperature $T_{1} .$ The temperature outside is $T_{2}\left( < T_{1}\right)$ .
(a) Show that the rate of heat loss for a length $\ell$ of pipe is
$$\frac{d Q}{d t}=\frac{2 \pi k\left(T_{1}-T_{2}\right) \ell}{\ln \left(R_{2} / R_{1}\right)}$$
where $k$ is the thermal conductivity of the pipe. $(b)$ Suppose
the pipe is steel with $R_{1}=3.3 \mathrm{cm}, R_{2}=4.0 \mathrm{cm},$ and
$T_{2}=18^{\circ} \mathrm{C} .$ If the pipe holds still water at $T_{1}=71^{\circ} \mathrm{C},$ what
will be the initial rate of change of its temperature?
(c) Suppose water at $71^{\circ} \mathrm{C}$ enters the pipe and moves at a
speed of 8.0 $\mathrm{cm} / \mathrm{s} .$ What will be its temperature drop per
centimeter of travel?

Key Concepts

Energy Balance in Fluid Flow

When considering a flowing fluid that is exchanging heat with its surroundings, an energy balance is necessary to determine the change in temperature of the fluid. The rate at which heat is lost from the system is compared to the amount of thermal energy carried by the fluid per unit length, incorporating concepts like specific heat and mass flux, to determine the temperature drop along the flow path.

Logarithmic Thermal Resistance

In cylindrical systems, the resistance to heat flow is not linear with distance but instead depends on the natural logarithm of the ratio of outer to inner radii. This logarithmic term arises from the integration of the conduction equation in a radial geometry and represents the cumulative resistance to heat flow from the inner surface to the outer surface.

Fourier's Law of Heat Conduction

Fourier's Law states that the rate at which heat is conducted through a material is proportional to the negative temperature gradient and the area through which the heat flows. This fundamental principle is used to calculate how energy is transferred by conduction in various geometries by relating the thermal conductivity, the cross-sectional area, and the temperature difference across a distance.

Conduction in Cylindrical Coordinates

When heat conduction occurs in cylindrical objects, such as pipes, the geometry necessitates a treatment in cylindrical coordinates. The area available for heat transfer is not constant but increases with the radius. As a result, integration of Fourier’s law over the radial distance leads to a logarithmic relationship between the inner and outer radii, which is characteristic of cylindrical heat conduction problems.

Thermal Conductivity

Thermal conductivity is a material property that quantifies its ability to conduct heat. Higher thermal conductivity means that the material can transfer thermal energy more effectively. It appears explicitly in the equations derived from Fourier’s Law and determines the rate of heat loss or gain in conductive systems.

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