an insulated couple, inserted in a hole in the bar, may be trusted to
attain the true temperature. The other uncertainties of the method
remain. R. W. Stewart found for pure iron, 𝑘＝·175 (1 − ·0015 𝑡)
C.G.S. E. H. Hall using a similar method found for cast-iron at
50° C. the value ·105, but considers the method very uncertain as
ordinarily practised.
10. Calorimetric Bar Method.—To avoid the uncertainties of surface loss of heat, it is necessary to reduce it to the rank of a small correction by employing a large bar and protecting it from loss of heat. The heat transmitted should be measured calorimetrically, and not in terms of the uncertain emissivity. The apparatus shown in fig. 2 was constructed by Callendar and Nicolson with this object. The bar was a special sample of cast-iron, the conductivity of which was required for some experiments on the condensation of steam (Proc. Inst. C.E., 1898). It had a diameter of 4 in., and a length of 4 ft. between the heater and the calorimeter. The emissivity was reduced to one-quarter by lagging the bar like a steam-pipe to a thickness of 1 in. The heating vessel could be maintained at a steady temperature by high-pressure steam. The other end was maintained at a temperature near that of the air by a steady stream of water flowing through a well-lagged vessel surrounding the bar. The heat transmitted was measured by observing the difference of temperature between the inflow and the outflow, and the weight of water which passed in a given time. The gradient near the entrance to the calorimeter was deduced from observations with five thermometers at suitable intervals along the bar. The results obtained by this method at a temperature of 40° C. varied from ·116 to ·118 C.G.S. from observations on different days, and were probably more accurate than those obtained by the cylinder method. The same apparatus was employed in another series of experiments by A. J. Angström’s method described below.
Fig. 2.
11. Guard-Ring Method.—This may be regarded as a variety of the plate method, but is more particularly applicable to good conductors, which require the use of a thick plate, so that the temperature of the metal may be observed at different points inside it. A. Berget (Journ. Phys. vii. p. 503, 1888) applied this method directly to mercury, and determined the conductivity of some other metals by comparison with mercury. In the case of mercury he employed a column in a glass tube 13 mm. in diam. surrounded by a guard cylinder of the same height, but 6 to 12 cm. in diam. The mean section of the inner column was carefully determined by weighing, and found to be 1·403 sq. cm. The top of the mercury was heated by steam, the lower end rested on an iron plate cooled by ice. The temperature at different heights was measured by iron wires forming thermo-junctions with the mercury in the inner tube. The heat-flow through the central column amounted to about 7·5 calories in 54 seconds, and was measured by continuing the tube through the iron plate into the bulb of a Bunsen ice calorimeter, and observing with a chronometer to a fifth of a second the time taken by the mercury to contract through a given number of divisions. The calorimeter tube was calibrated by a thread of mercury weighing 19 milligrams, which occupied eighty-five divisions. The contraction corresponding to the melting of 1 gramme of ice was assumed to be ·0906 c.c., and was taken as being equivalent to 79 calories (1 calorie＝15·59 mgrm. mercury). The chief uncertainty of this method is the area from which the heat is collected, which probably exceeds that of the central column, owing to the disturbance of the linear flow by the projecting bulb of the calorimeter. This would tend to make the value too high, as may be inferred from the following results:—
| Mercury, | 𝑘＝0·02015 | C.G.S. | Berget. |
| ,, | 𝑘＝0·01479 | ,, | Weber. |
| ,, | 𝑘＝0·0177 | ,, | Ångström. |
12. Variable-Flow Methods.—In these methods the flow of heat is deduced from observations of the rate of change of temperature with time in a body exposed to known external or boundary conditions. No calorimetric observations are required, but the results are obtained in terms of the thermal capacity of unit volume 𝑐, and the measurements give the diffusivity 𝑘/𝑐, instead of the calorimetric conductivity 𝑘. Since both 𝑘 and 𝑐 are generally variable with the temperature, and the mode of variation of either is often unknown, the results of these methods are generally less certain with regard to the actual flow of heat. As in the case of steady-flow methods, by far the simplest example to consider is that of the linear flow of heat in an infinite solid, which is most nearly realized in nature in the propagation of temperature waves in the surface of the soil. One of the best methods of studying the flow of heat in this case is to draw a series of curves showing the variations of temperature with depth in the soil for a series of consecutive days. The curves given in fig. 3 were obtained from the readings of a number of platinum thermometers buried in undisturbed soil in horizontal positions at M‘Gill College, Montreal.
Fig. 3.
The method of deducing the diffusivity from these curves is as follows:—The total quantity of heat absorbed by the soil per unit area of surface between any two dates, and any two depths, 𝑥′ and 𝑥″, is equal to 𝑐 times the area included between the corresponding curves. This can be measured graphically without any knowledge of the law of variation of the surface temperature, or of the laws of propagation of heat waves. The quantity of heat absorbed by the stratum (𝑥′ 𝑥″) in the interval considered can also be expressed in terms of the calorimetric conductivity 𝑘. The heat transmitted through the plane 𝑥 is equal per unit area of surface to the product of 𝑘 by the mean temperature gradient (𝑑θ/𝑑𝑥) and the interval of time, T−T′. The mean temperature gradient is found by plotting the curves for each day from the daily observations. The heat absorbed is the difference of the quantities transmitted through the bounding planes of the stratum. We thus obtain the simple equation—
| 𝑘′(𝑑θ′/𝑑𝑥′) −𝑘″(𝑑θ″/𝑑𝑥″)＝𝑐 (area between curves)/(T−T′), | (4) |
by means of which the average value of the diffusivity 𝑘/𝑐 can be found for any convenient interval of time, at different seasons of the year, in different states of the soil. For the particular soil in question it was found that the diffusivity varied enormously with the degree of moisture, falling as low as ·0010 C.G.S. in the winter for the surface layers, which became extremely dry under the protection of the frozen ice and snow from December to March, but rising to an average of ·0060 to ·0070 in the spring and autumn. The greater part of the diffusion of heat was certainly due to the percolation of water. On some occasions, owing to the sudden melting of a surface layer of ice and snow, a large quantity of cold water, percolating rapidly, gave for a short time values of the diffusivity as high as ·0300. Excluding these exceptional cases, however, the variations of the diffusivity appeared to follow the variations of the seasons with considerable regularity in successive years. The presence of water in the soil always increased the value of 𝑘/𝑐, and as it necessarily increased 𝑐, the increase of 𝑘 must have been greater than that of 𝑘/𝑐.
13. Periodic Flow of Heat.—The above method is perfectly general, and can be applied in any case in which the requisite observations can be taken. A case of special interest and importance is that in which the flow is periodic. The general characteristics of such a flow are illustrated in fig. 4, showing the propagation of temperature waves due to diurnal variations in the temperature of the surface. The daily range of temperature of the air and of the surface of the soil was about 20° F. On a sunny day, the temperature reached a maximum about 2 p.m. and a minimum about 5 a.m. As the waves were propagated downwards through the soil the amplitude rapidly
