CHAPTER XXX.

CONDUCTION OF HEAT.

328. Conduction.-When heat is applied to one end of a bar of metal, it is propagated through the substance of the bar, producing a rise of temperature, which is first perceptible at near and afterwards at remote portions. This transmission of heat is called conduction, and it differs notably from radiation (1), in being gradual instead of instantaneous; and (2), in exhibiting no preference for rectilinear transmission, the propagation of heat being as rapid through a crooked as through a straight bar.

328 A. Definition of Conductivity or Specific Conducting Power. If the application of heat to one end of the bar be continued for a sufficiently long time, and with great steadiness, the different portions of the bar will at length cease to rise in temperature, and will retain steadily the temperatures which they have acquired. We may thus distinguish two stages in the experiment: 1st, the variable stage, during which all portions of the bar are rising in temperature; and, 2d, the permanent state, which may subsist for any length of time without alteration. In the former stage the bar is gaining heat; that is, it is receiving more heat from the source than it gives out to surrounding bodies. In the latter stage the receipts and expenditure of heat are equal, and are equal not only for the bar as a whole, but for every small portion of which it is composed.

In the permanent state no further accumulation of heat takes place. All the heat which reaches an internal particle is transmitted by conduction, and the heat which reaches a superficial particle is given off partly by radiation and air-contact, and partly by conduction to colder neighbouring particles. In the earlier stage, on the contrary, only a portion of the heat received by a particle is thus disposed of, the remainder being accumulated in the particle, and serving to raise its temperature.

MEASUREMENT OF CONDUCTIVITY.

415

In order to obtain results depending on conduction free from complications arising from differences of specific heat (§ 340), we must, in all cases, wait for the permanent state. In the earlier stage great specific heat acts as an obstacle to rapid transmission, and a body of great specific heat would be liable to be mistaken for a body of small conductivity.

The measurement of the conductivity of a substance is still further simplified by making the flow of heat through it take place entirely in one definite direction (that is to say in parallel lines), avoiding all cross-currents. To this end it is necessary that all points in the same cross section should have the same temperature, a condition which is not strictly fulfilled in the bar above described, as the surface will be cooler than the interior. It is nearly fulfilled in the axial portions of the bar, and it is very nearly fulfilled in the central portions of a uniform plate whose breadth in all directions is a very large multiple of its thickness, when the whole of one face is maintained as nearly as practicable at one uniform temperature, and the other face at another uniform temperature. In the central portions of such a plate, the flow of heat will be perpendicularly through the plate; and when the permanent state has arrived, the amount of heat that passes in a unit of time through a cross section of area A, will be expressed by the formula

1

2

(1)

where x is the thickness of the plate, t1 and to the temperatures of its two faces, and k a coefficient depending on the material of the plate. This coefficient k is the conductivity of the material. It may be defined as the quantity of heat which flows in unit time through a cross section of unit area, when the thickness of the plate is unity, and one face is warmer by 1° than the other.1

1 The method of taking account of conductivity during the variable stage may be illustrated by considering the simplest case,—that in which the flow of heat is in parallel lines. Let x denote distance measured in the direction in which heat is flowing, v the temperature at the time t at a point specified by x, k the conductivity, and c the thermal capacity per unit volume (both at the temperature v). Then the flow of heat per unit time d v past a cross section of area A is -k A and the flow past an equal and parallel section further on by the small distance dx is greater by the amount

dx'

This latter expression therefore represents the loss of heat from the intervening prism Adx,

Forbes' experiments have shown that the conductivity of a substance is not the same at all temperatures. In view of this fact, k in the above formula denotes the average conductivity between the temperatures t, and t2. The variation of conductivity with temperature is, however, comparatively small.

1

329. Differences of Conductivity. The following experiments are often adduced in illustration of the different conducting powers of different solids.

Two bars of the same size but of different materials (Fig. 293) are placed end to end, and small wooden balls are attached by wax to their under-surfaces at equal distances. The bars are then heated at their contiguous ends, and, as the heat extends along them, the

wax melts, and the balls successively drop off. If the heating is continued till the permanent state arrives, it may generally be concluded that the bar which has lost most balls is the best conductor, especially if both bars have been coated with the same varnish, so as to make their radiating powers equal.

The well-known experiment of Ingenhousz is of the same kind. The apparatus consists of a copper box having a row of holes in one of its faces, in which rods of different materials can be fixed. The

and the resulting fall of temperature is the quotient of the loss by the thermal capacity cAdx, which quotient is

d v dť

This, then, is the fall of temperature per unit time, or is If the range of temperature is small enough to admit of our regarding k and c as constant, the equation

which applies approximately to the variations of temperature in the soil near the surface of the earth, x being in this case measured vertically. For the integral of this equation, see Trans. Roy. Soc. Edin. vol. xxii. part ii. p. 438.

CONDUCTING POWER OF METALS.

417

rods having previously been coated with wax, the box is filled with boiling water, which comes into contact with the inner ends of the

It is thus found that metals are unequally good conductors of heat, and that they may be arranged in the following order, beginning with the best conductors:-Silver, copper, gold, brass, tin, iron, lead, platinum, bismuth.

In both these experiments we must beware of attempting to measure conductivity by the quickness with which the melting advances. This quickness may be simply an indication of small specific heat.1

330. Conducting Power of Metals.-Metals, though differing considerably one from another, are as a class greatly superior in conductivity to other substances, such as wood, marble, brick. This explains several familiar phenomena. If the hand be placed upon a metal plate at the temperature of 10° C., or plunged into mercury at this temperature, a very marked sensation of cold is experienced. This sensation is less intense with a plate of marble at the same temperature, and still less with a piece of wood. The reason is that the hand, which is at a higher temperature than the substance to which it is applied, gives up a portion of its heat, which is conducted away by the substance, and consequently a larger portion of heat is parted with, and a more marked sensation of cold experienced, in the case of the body of greater conducting power.

If,

331. Davy Lamp. The conducting power of metals explains the curious property possessed by wire-gauze of cutting off a flame. for example, a piece of wire-gauze be placed above a jet of gas, the flame is prevented from rising above the gauze. If the gas be first allowed to pass through the gauze, and then lighted above, the flame is cut off from the burner, and is unable to extend itself to the undersurface of the gauze. These facts depend upon the conducting power

1

Strictly speaking, small specific heat per unit volume, not, as usual, per unit mass.

of metallic gauze, in virtue of which the heat of the flame is rapidly dissipated at the points of contact, the result being a diminution of

temperature sufficient to prevent ignition. This property of metallic gauze has been turned to account for various purposes, but its most useful application is in the safety-lamp of Sir Humphrey Davy. It is well known

that a gas called fire-damp is often given off in coal-mines. It is a compound of carbon and hydrogen, and is a large ingredient in ordinary coal-gas.

This fire-damp, when mixed with eight or ten times its volume of air, explodes with great violence on coming in contact with a lighted

body. To obviate this danger, Davy invented the safety-lamp, which is an ordinary lamp with the flame inclosed by wire-gauze. The explosive gases pass through the gauze, and burn inside the lamp, in such a manner as to warn the miner of their presence; but the flame is unable to pass through the gauze.

332. Various Applications.-The knowledge of the relative conducting powers of different bodies has several important practical applications.

In cold countries, where the heat produced in the interior of a house should be as far as possible prevented from escaping, the walls should be of brick or wood, which have feeble conducting powers. If they are of stone, which is a better conductor, a greater thickness is required. Thick walls are also useful in hot countries in resisting the power of the solar rays during the heat of the day.

We have already alluded (§ 224) to the advantage of employing fire-brick, which is a bad conductor, as a lining for stoves.