Page:Encyclopædia Britannica, Ninth Edition, v. 11.djvu/613

HEAT 579 whether it is tens or hundreds of times too small. Omit ting it then from the preceding statement, completing the correction by multiplying the 0023 by 2 (assuming the thickness of the plate to be 2| millimetres, as Peclet says it was between 2 and 3) to give Clement s result, and appending Angstrom s result, which we now know to be correct, we have the following statement for thermal con ductivity of copper in C. G. S. units : 0057, according to Cle ment. 178, ,, ,, Peclet. 1 I l, ,, ,, Angstrom. 76. The comparison of these results is highly instruc tive. Clement s result is two hundred times too small, and Pellet s five . times too small. Clement experimented by exposing one side of a plate of copper of a square metre surface and about two and a half millimetres thick to steam at 100 C., and the other side to water at 28 C. It was assumed that the difference of temperatures between the two sides was 72. The difference really was about 36 of a degree, as we know from the quantity of heat actually conducted through it in Clement s experiment, indicated by the amount of steam which he found to be condensed into water. In fact, the amount of steam condensed did not differ sensibly from what it would have been if the copper plate had been infinitely thin, or its substance of infinite thermal conductivity. It is important in engineering, and in many of the arts and manufactures involving thermal processes, and particularly in that one of them of greater everyday value to the human race than, all the others put together, cookery, to know that for copper or iron boilers, or steampipes, or pots or frying-pans, the transmission of heat from radiant burning coal or charcoal, or red or white hot fireclay or other solids, and from hot air in contact with them, on one side, to hot water or steam or oil or melted fat on the other side, or hot liquid or steam on one side and cool air on the other side, is for practical purposes sensibly the same as if the thermal conductivity of the metal were infinite, or its resistance to the transmission of heat nothing. The explanation is obvious to us now with the definite and sure knowledge regarding thermal conduc tivities of different substances and of matter in different conditions, solid, liquid, and gaseous, gained within the last twenty years. Angstrom, Forbes, F. Neumann, and Tait have given, each one of them with thoroughly sufficient experimental evidence to leave no room to doubt the sub stantial accuracy of his results, absolute values for the thermal conductivities of copper and iron. Clausius and Maxwell have given us thermal conductivities of air and other gases, from their splendid development of the kinetic theory, which are undoubtedly trustworthy as somewhat close approximations to the true values, and which it is quite possible are more accurate than we can hope to see ob tained from direct measurements of the conduction of heat through gases. J. T. Bottomley has given a trustworthy and somewhat closely accurate direct measurement of the . thermal conductivity of water. From the results of these experimenters work, reduced to uniform C. G. S. reckon ing in our table of thermal conductivities (Table VIII.), we see that the thermal conductivity of iron is 80 times, and that of copper 500 times that of water. The thermal conductivity of iron is 3500 times, and that of copper is 20,000 times that of air. Hence, although the water or air at the very interface of its contact with the metal is essentially at the same temperature as the metal, there must be great differences of temperature in very thin layers of the fluid close to the interface when there is large flux of ^ heat through the metal, and the temperature of the fluid as measured by any practicable thermometer, or 1 This result was relished by Peclet in 1853, in a work entitled Nouveaux documents relatifs au cha.uffa.ge et d la ventilation. inferred from knowledge of the average temperature of the whole fluid, or from the temperatures of entering and leaving currents of fluid, may differ by scores of degrees from the actual temperature of the solid at the interface. It is remarkable that Peclet, while perceiving that Clement s result was largely erroneous on this account, and improving the mode of experimenting by introducing a rotating mechanical stirrer to change very rapidly the fluid in contact with the solid, only multiplied Clement s con ductivity by 30 instead of by 200, which would have been necessary to annul the error. Notwithstanding his failure to obtain accurate results for metals, we have ventured to include his results for wood, and solids of lower con ductivity than wood, in our table, because we perceive that he was alive to the necessity for very energetic stirring of the liquid, and the mechanical means which he adopted for it, though utterly insufficient for the case of even the least conductive of the metals, were probably not so for wood and solids of lower conductivity than wood ; and because it is not probable that the complication of heat generated by the stirring (which Angstrom suggests as an objection to Pe clet s method) was in any case sufficient to produce a sensible influence upon the experimental results. 77. The first correct determinations of thermal conduc tivities were given by Forbes in his paper on the tempera ture of the earth, in the Transactions of the Royal Society of Edinburgh for 1846, as calculated from his observations of underground temperature at three localities in the neighbourhood of Edinburgh the trap rock of Calton Hill, the sand of the Experimental Garden, and the sand stone of Craigleith Quarry by an imperfect approximate method indicated by Poisson. A more complete analytical treatment of the observational results, analysed harmoni cally and interpreted by application of Fourier s formula (equation (19) of Math. App.) to each term separately by W. Thomson, gave results (quoted in Table VIII. below) for the conductivities, which differed but little from Forbes s approximate determinations. 78. It has always seemed to us that the best mode of experimenting on the conductivity of metals must, without doubt, be by an artificial imitation in a metallic bar, of the natural periodic variations of underground temperature, produced by periodically varied thermal appliances at one end of the bar. The effect of loss or gain of heat through the sides (or lateral surface) of the bar (ideally annullable by a coating of ideal varnish impermeable to heat) may be practically annulled by making the period of the variation small enough. Let k be the thermal conductivity of the substance and c its thermal capacity per unit bulk. Let e be the emissivity ( 71) of its surface. Let the bar be circular-cylindric, and r the radius of its cross section. At time t, let v be the mean temperature in a cross section at distance x from the end, and v the surface tempera ture at the circular boundary of this section, all temperatures being reckoned as differences from the temperature of the surrounding medium, called zero temporarily for brevity. The heat lost from the circumference of the bar between the cross sections x-dx and x + dx in time dt is cv . 2irrdxdt, and the heat conducted lengthwise, across the cross section x. in the- same time, is Trr 2 . Hence we dx readily find (see Math. App. below) as the equation of conduction of heat along the bar, very approximately if v differs very little from v (that is to say, if the temperature is very nearly uniform throughout each cross section), dv d / T dv 2e , m C-rr^-rlk-T-) V (1). dt dx dx) r To estimate v , let v" be the temperature at the centre of the cross section x, and let ( 1 denote the rate of decrease of tempera- V drj ture from within outwards in the substance of the bar close to its surface. We have clearly Esl ir g per ai tur the ^ e

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