Elementary Text-book of Physics/Ch. III Part II

by William Arnold Anthony and Cyrus Fogg Brackett

CHAPTER II.

TRANSFER OF HEAT.

172. Transfer of Heat.—In the preceding discussions it has been assumed that heat may be transferred from one body to another, and that if two bodies in contact be at different temperatures, heat will be transferred from the hotter to the colder body. In general, if transfer of heat be possible in any system, heat will pass from the hotter to the colder parts of the system, and the temperature of the system will tend to become uniform. There are three ways in which this transfer is accomplished, called respectively convection, conduction, and radiation.

173. Convection.—If a vessel containing any fluid be heated at the bottom, the bottom layers become less dense than those above, producing a condition of instability. The lighter portions of the fluid rise, and the heavier portions from above, coming to the bottom, are in their turn heated. Hence continuous currents are caused. This process is called convection. By this process, masses of fluid, although fluids are poor conductors, may be rapidly heated. Water is often heated in a reservoir at a distance from the source of heat by the circulation produced in pipes leading to the source of heat and back. The winds and the great currents of the ocean are convection currents. An interesting result follows from the fact that water has a maximum density (§ 190). When the water of lakes cools in winter, currents are set up and maintained, so long as the surface water becomes more dense by cooling, or until the whole mass reaches 4°. Any further cooling makes the surface water lighter. It therefore remains at the surface, and its temperature rapidly falls to the freezing-point, while the great mass of the water remains at the temperature of its maximum density.

174. Conduction.—If one end of a metal rod be heated, it is found that the heat travels along the rod, since those portions at a distance from the source of heat finally become warm. This process of transfer of heat from molecule to molecule of a body, while the molecules themselves retain their relative places, is called conduction.

In the discussion of the transfer of heat by conduction it is assumed as a principle, borne out by experiment, that the flow of heat between two very near parallel planes, drawn in a substance, is proportional to the difference of temperature between those planes, or that the flow of heat across a plane is proportional to the rate of fall of temperature across that plane.

175. Flow of Heat across a Wall.—The simplest body in which the flow of heat can be studied is a wall of homogeneous material bounded by two parallel infinite planes, one of which is kept at the temperature $t'$ and the other at the temperature $t;$ we represent the distance between the planes or the thickness of the wall by $d.$ We suppose that the flow of heat across this wall has continued so long that it has become steady, or that the temperatures at all points have assumed flnal values. Manifestly the temperature at all points in any plane parallel with the faces of the wall is the same, and the same amount of heat passes across any one such plane as passes across any other. We conclude therefore by the fundamental principle assumed (§ 174) that the rate of change of temperature across each plane in the wall is the same, or that the change of temperature throughout the wall from one face to the other is uniform; the rate of change of temperature is therefore given by ${\frac {t'-t}{d}},$ where it has been assumed that $t'$ is the higher temperature. If $d'$ represent the distance of any plane in the wall from the hotter surface, the fall of temperature between it and the hotter surface is $(t'-t){\frac {d'}{d}},$ and the temperature of the plane is $t'-(t'-t){\frac {d'}{d}}\cdot$ The confirmation by experiment of this law of temperature distribution in a wall is a warrant for our assumption of the fundamental principle of the flow of heat.

176. Conductivity.—If, now, we consider a prism extending across the wall, bounded by planes perpendicular to the exposed surfaces, and represent the area of its exposed bases by $A,$ the quantity of heat which flows in a time $T$ through this prism may be represented by

$Q=K{\frac {t'-t}{d}}At,$

(64)

where $K$ is a constant depending upon the material of which the wall is composed. $K$ is the conductivity of the substance, and may be defined as the quantity of heat which in unit time flows through a section of unit area in a wall of the substance whose thickness is unity, when its exposed surfaces are maintained at a difference of temperature of one degree; or, in other words, it is the quantity of heat which in unit time flows through a section of unit area in a substance, where the rate of fall of temperature at that section is unity. In the above discussions the temperatures $t'$ and $t$ are taken as the actual temperatures of the surfaces of the wall. If the colder surface of the wall be exposed to air of temperature $T,$ to which the heat which traverses it is given up, $t$ will be greater than $T.$ The difference will depend upon the quantity of heat which flows, and upon the facility with which the surface parts with heat.

177. Flow of Heat along a Bar.—If a prism of a substance have one of its bases maintained at a temperature $t,$ while the other base and the sides are exposed to air at a lower temperature, the conditions of uniform fall of temperature no longer exist, and the amount of heat'which flows through the different sections is no longer the same: but the amount of heat which flows through any section is still proportional to the rate of fall of temperature at that section, and is equal to the heat which escapes from the portion of the bar beyond the section.

178. Measurement of Conductivity.—A bar heated at one end furnishes a convenient means of measuring conductivity. In Fig. 69 let $AB$ represent a bar heated at $A.$ Let the ordinates $aa',bb',cc'$, represent the excess of temperatures above the temperature of the air at the points from which they are drawn. These temperatures may be determined by means of thermometers inserted in cavities in the bar, or by means of a thermopile. Draw the curve $a'b'c'd'\dots$ through the summits of the ordinates. The inclination of this curve at any point represents the rate of fall of temperature at that point. The ordinates to the line $b'm,$ drawn tangent to the curve at the point $b',$ show what would be the temperatures at various points of the bar if the fall were uniform and at the same rate as at $b'.$ It shows that, at the rate of fall at $b',$ the bar would at $m$ be at the temperature of the air; or, in the length $bm,$ the fall of temperature would equal the amount represented by $bb'.$ The rate of fall is, therefore, ${\frac {bb'}{bm}}\cdot$ If $Q$ represent the quantity of heat passing the section at $b$ in the unit time, we have, from § 176,

$Q=K\times {\text{rate of fall of temperature}}\times {\text{area of section.}}$

$Q$ is equal to the quantity of heat that escapes in unit time from all that portion of the bar beyond $b.$ It may be found by heating a short piece of the same bar to a high temperature, allowing it to cool under the same conditions that surround the bar $AB,$ and observing its temperature from minute to minute as it falls. These observations furnish the data for computing the quantity of heat which escapes per minute from unit length of the bar at different temperatures. It is then easy to compute the amount of heat that escapes per minute from each portion, $bc,cd$, etc., of the bar beyond $b;$ each portion being taken so short that its temperature throughout may, without sensible error, be considered uniform and the same as that at its middle point. Summing up all these quantities, we obtain the quantity $Q$ which passes the section $b$ in the unit time. Then

$K={\frac {Q}{{\text{rate of fall of temperature at }}b\times {\text{area of section}}}}\cdot$

179. Conductivity diminishes as Temperature rises.—By the method described above, Forbes determined the conductivity of a bar of iron at points at different distances from the heated end, and found that the conductivity is not the same at all temperatures, but is greater as the temperature is lower.

180. Conductivity of Crystals.—The conductivity of crystals of the isometric system is the same in all directions, but in crystals of the other systems it is not so. In a crystal of Iceland spar the conductivity is greatest in the direction of the axis of symmetry, and equal in all directions in a plane at right angles to that axis.

181. Conductivity of Non-homogeneous Solids.—De la Rive and De Candolle were the first to show that wood conducts heat better in the direction of the fibres than at right angles to them. Tyndall, by experimenting upon cubes cut from wood, has shown that the conductivity has a maximum value parallel to the fibres, a minimum value at right angles to the fibres and parallel to the annual layers. Feathers, fur, and the materials of clothing are poor conductors because of their want of continuity.

182. Conductivity of Liquids.—The conductivity of liquids can be measured, in the same way as that of solids, by noting the fall of temperature at various distances from the source of heat in a column of liquid heated at the top. Great care must be taken in these experiments to avoid errors due to convection currents.

Liquids are generally poor conductors.

183. Radiation.—We have now considered those cases in which there is a transfer of heat between bodies in contact. Heat is also transferred between bodies not in contact. This is effected by a process called radiation, which will be subsequently considered.