# 14.5 Conduction  (Page 4/8)

 Page 4 / 8

Conduction is caused by the random motion of atoms and molecules. As such, it is an ineffective mechanism for heat transport over macroscopic distances and short time distances. Take, for example, the temperature on the Earth, which would be unbearably cold during the night and extremely hot during the day if heat transport in the atmosphere was to be only through conduction. In another example, car engines would overheat unless there was a more efficient way to remove excess heat from the pistons.

How does the rate of heat transfer by conduction change when all spatial dimensions are doubled?

Because area is the product of two spatial dimensions, it increases by a factor of four when each dimension is doubled $\left({A}_{\text{final}}=\left(2d{\right)}^{2}=4{d}^{2}=4{A}_{\mathrm{initial}}\right)$ . The distance, however, simply doubles. Because the temperature difference and the coefficient of thermal conductivity are independent of the spatial dimensions, the rate of heat transfer by conduction increases by a factor of four divided by two, or two:

${\left(\frac{Q}{t}\right)}_{\text{final}}=\frac{{\text{kA}}_{\text{final}}\left({T}_{2}-{T}_{1}\right)}{{d}_{\text{final}}}=\frac{k\left({\text{4A}}_{\text{initial}}\right)\left({T}_{2}-{T}_{1}\right)}{{2d}_{\text{initial}}}=2\frac{{\text{kA}}_{\text{initial}}\left({T}_{2}-{T}_{1}\right)}{{d}_{\text{initial}}}=2{\left(\frac{Q}{t}\right)}_{\text{initial}}\text{.}$

## Summary

• Heat conduction is the transfer of heat between two objects in direct contact with each other.
• The rate of heat transfer $Q/t$ (energy per unit time) is proportional to the temperature difference ${T}_{2}-{T}_{1}$ and the contact area $A$ and inversely proportional to the distance $d$ between the objects:
$\frac{Q}{t}=\frac{\text{kA}\left({T}_{2}-{T}_{1}\right)}{d}\text{.}$

## Conceptual questions

Some electric stoves have a flat ceramic surface with heating elements hidden beneath. A pot placed over a heating element will be heated, while it is safe to touch the surface only a few centimeters away. Why is ceramic, with a conductivity less than that of a metal but greater than that of a good insulator, an ideal choice for the stove top?

Loose-fitting white clothing covering most of the body is ideal for desert dwellers, both in the hot Sun and during cold evenings. Explain how such clothing is advantageous during both day and night.

## Problems&Exercises

(a) Calculate the rate of heat conduction through house walls that are 13.0 cm thick and that have an average thermal conductivity twice that of glass wool. Assume there are no windows or doors. The surface area of the walls is $\text{120}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}$ and their inside surface is at $\text{18.}0º\text{C}$ , while their outside surface is at $5\text{.00º}\text{C}$ . (b) How many 1-kW room heaters would be needed to balance the heat transfer due to conduction?

(a) $1.01×{10}^{3}$ W

(b) One

The rate of heat conduction out of a window on a winter day is rapid enough to chill the air next to it. To see just how rapidly the windows transfer heat by conduction, calculate the rate of conduction in watts through a $3\text{.}{\text{00-m}}^{2}$ window that is thick (1/4 in) if the temperatures of the inner and outer surfaces are $5\text{.00ºC}$ and $-\text{10}\text{.}0º\text{C}$ , respectively. This rapid rate will not be maintained—the inner surface will cool, and even result in frost formation.

Calculate the rate of heat conduction out of the human body, assuming that the core internal temperature is $\text{37}\text{.}0º\text{C}$ , the skin temperature is $\text{34}\text{.}0º\text{C}$ , the thickness of the tissues between averages , and the surface area is $1\text{.}\text{40}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}$ .

84.0 W

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