measuring the diffusivity of a metal, since the conditions may be
widely varied and the correction for external loss of heat can be
made comparatively small. Owing, however, to the laborious nature
of the observations and reductions, the method does not appear to
have been seriously applied since its first invention, except in one
solitary instance by the writer to the case of cast-iron (fig. 2). The
equation of the method is the same as that for the linear flow with
the addition of a small term representing the radiation loss.
The heat per second gained by conduction by an element dx of the bar, of conductivity 𝑘 and cross section q, at a point where the gradient is 𝑑θ/dx, may be written qk(𝑑2θ/dx2)dx. This is equal to the product of the thermal capacity of the element, cqdx, by the rate of rise of temperature 𝑑θ/dt, together with the heat lost per second at the external surface, which may be written hpθdx, if p is the perimeter of the bar, and h the heat loss per second per degree excess of temperature θ above the surrounding medium. We thus obtain the differential equation
qk(𝑑2θ/dx)＝cdqθ/dt+hpθ,
which is satisfied by terms of the type
where 𝑎2−𝑏2＝hp/qk, and ab＝πnc/𝑘.
The rate of diminution of amplitude expressed by the coefficient 𝑎 in the index of the exponential is here greater than the coefficient 𝑏 expressing the retardation of phase by a small term depending on the emissivity h. If h＝0, 𝑎＝𝑏＝√(πnc/𝑘), as in the case of propagation of waves in the soil.
The apparatus of fig. 2 was designed for this method, and may serve to illustrate it. The steam pressure in the heater may be periodically varied by the gauge in such a manner as to produce an approximately simple harmonic oscillation of temperature at the hot end, while the cool end is kept at a steady temperature. The amplitudes and phases of the temperature waves at different points are observed by taking readings of the thermometers at regular intervals. In using mercury thermometers, it is best, as in the apparatus figured, to work on a large scale (4-in. bar) with waves of slow period, about 1 to 2 hours. Ångström endeavoured to find the variation of conductivity by this method, but he assumed 𝑐 to be the same for two different bars, and made no allowance for its variation with temperature. He thus found nearly the same rate of variation for the thermal as for the electric conductivity. His final results for copper and iron were as follows:—
Copper, 𝑘＝0·982 (1–0·00152 θ) assuming 𝑐＝·84476.
Iron, 𝑘＝0·1988 (1–0·00287 θ),,𝑐＝·88620.
Ångström’s value for iron, when corrected for obvious numerical errors, and for the probable variation of 𝑐, becomes—
but this is very doubtful as 𝑐 was not measured.
The experiments on cast-iron with the apparatus of fig. 2 were varied by taking three different periods, 60, 90 and 120 minutes, and two distances, 6 in. and 12 in., between the thermometers compared. In some experiments the bar was lagged with 1 in. of asbestos, but in others it was bare, the heat-loss being thus increased fourfold. In no case did this correction exceed 7 %. The extreme divergence of the resulting values of the diffusivity, including eight independent series of measurements on different days, was less than 1 %. Observations were taken at mean temperatures of 102° C. and 54°C., with the following results:—
Cast iron at 102°C., 𝑘/𝑐＝·1296, 𝑐＝·858, 𝑘＝·1113.
The variation of 𝑐 was determined by a special series of experiments. No allowance was made for the variation of density with temperature, or for the variation of the distance between the thermometers, owing to the expansion of the bar. Although this correction should be made if the definition were strictly followed, it is more convenient in practice to include the small effect of linear expansion in the temperature-coefficient in the case of solid bodies.
17. Lorenz’s Method.—F. Neumann, H. Weber, L. Lorenz and others have employed similar methods, depending on the observation of the rate of change of temperature at certain points of bars, rings, cylinders, cubes or spheres. Some of these results have been widely quoted, but they are far from consistent, and it may be doubted whether the difficulties of observing rapidly varying temperatures have been duly appreciated in many cases. From an experimental point of view the most ingenious and complete method was that of Lorenz (Wied. Ann. xiii. p. 422, 1881). He deduced the variations of the mean temperature of a section of a bar from the sum S of the E.M.F.’s of a number of couples, inserted at suitable equal intervals l and connected in series. The difference of the temperature gradients D/l at the ends of the section was simultaneously obtained from the difference D of the readings of a pair of couples at either end connected in opposition. The external heat-loss was eliminated by comparing observations taken at the same mean temperatures during heating and during cooling, assuming that the rate of loss of heat f(S) would be the same in the two cases. Lorenz thus obtained the equations:—
Heating, qk D/l＝cql dS/dt+f(S).
Cooling, qk D′/l＝cql dS′/dt′+f(S′).
Whence 𝑘＝cl2(𝑑S/dt−𝑑S′/dl)/(D−D′).
It may be questioned whether this assumption was justifiable, since the rate of change and the distribution of temperature were quite different in the two cases, in addition to the sign of the change itself. The chief difficulty, as usual, was the determination of the gradient, which depended on a difference of potential of the order of 20 microvolts between two junctions inserted in small holes 2 cms. apart in a bar 1·5 cms. in diameter. It was also tacitly assumed that the thermo-electric power of the couples for the gradient was the same as that of the couples for the mean temperature, although the temperatures were different. This might give rise to constant errors in the results. Owing to the difficulty of measuring the gradient, the order of divergence of individual observations averaged 2 or 3 %, but occasionally reached 5 or 10 %. The thermal conductivity was determined in the neighbourhood of 20° C. with a water jacket, and near 110° C. by the use of a steam jacket. The conductivity of the same bars was independently determined by the method of Forbes, employing an ingenious formula for the heat-loss in place of Newton’s law. The results of this method differ 2 or 3 % (in one case nearly 15 %) from the preceding, but it is probably less accurate. The thermal capacity and electrical conductivity were measured at various temperatures on the same specimens of metal. Owing to the completeness of the recorded data, and the great experimental skill with which the research was conducted, the results are probably among the most valuable hitherto available. One important result, which might be regarded as established by this work, was that the ratio 𝑘/𝑘′ of the thermal to the electrical conductivity, though nearly constant for the good conductors at any one temperature such as 0° C., increased with rise of temperature nearly in proportion to the absolute temperature. The value found for this ratio at 0° C. approximated to 1500 C.G.S. for the best conductors, but increased to 1800 or 2000 for bad conductors like German-silver and antimony. It is clear, however, that this relation cannot be generally true, for the cast-iron mentioned in the last section had a specific resistance of 112,000 C.G.S. at 100° C., which would make the ratio 𝑘/𝑘′＝12,500. The increase of resistance with temperature was also very small, so that the ratio varied very little with temperature.
18. Electrical Methods.—There are two electrical methods which have been recently applied to the measurement of the conductivity of metals, (𝑎) the resistance method, devised by Callendar, and applied by him, and also by R. O. King and J. D. Duncan, (𝑏) the thermo-electric method, devised by Kohlrausch, and applied by W. Jaeger and H. Dieselhorst. Both methods depend on the observation of the steady distribution of temperature in a bar or wire heated by an electric current. The advantage is that the quantities of heat are measured directly in absolute measure, in terms of the current, and that the results are independent of a knowledge of the specific heat. Incidentally it is possible to regulate the heat supply more perfectly than in other methods.
(𝑎) In the practice of the resistance method, both ends of a short bar are kept at a steady temperature by means of solid copper blocks provided with a water circulation, and the whole is surrounded by a jacket at the same temperature, which is taken as the zero of reference. The bar is heated by a steady electric current, which may be adjusted so that the external loss of heat from the surface of the bar is compensated by the increase of resistance of the bar with rise of temperature. In this case the curve representing the distribution of temperature is a parabola, and the conductivity 𝑘 is deduced from the mean rise of temperature (R−R0)/𝑎R0 by observing the increase of resistance R−R0 of the bar, and the current C. It is also necessary to measure the cross-section q, the length l, and the temperature-coefficient 𝑎 for the range of the experiment.
In the general case the distribution of temperature is observed by means of a number of potential leads. The differential equation for the distribution of temperature in this case includes the majority of the methods already considered, and may be stated as follows. The heat generated by the current C at a point x where the temperature-excess is θ is equal per unit length and time (t) to that lost by conduction −𝑑(qkdθ/dx)/dx, and by radiation hpθ (emissivity h, perimeter p), together with that employed in raising the temperature qcdθ/dt, and absorbed by the Thomson effect sC𝑑θ/dx. We thus obtain the equation—
| C2R0(1+𝑎θ)/l＝−𝑑(qkdθ/dx)/dx+hpθ+qcdθ/dt+sC𝑑θ/dx. | (8) |
If C＝0, this is the equation of Ångström’s method. If h also is zero, it becomes the equation of variable flow in the soil. If dθ/dt＝0, the equation represents the corresponding cases of steady flow. In the electrical method, observations of the variable flow are useful for finding the value of 𝑐 for the specimen, but are not otherwise required. The last term, representing the Thomson effect, is eliminated in the case of a bar cooled at both ends, since it is opposite in the two halves, but may be determined by observing the resistance of each half separately. If the current C is chosen so that C2R0𝑎＝hpl, the external heat-loss is compensated by the variation of resistance
