DEPTH INOHE8.
Speed, 42 revolutions per minute; range, 200 at eurface. FIG. 5.
is best avoided by employing platinum thermometers buried horizontally. In any case results deduced from the annual wave must be expected to vary in different years according to the distribution of the rainfall, as the values represent averages depending chiefly on the diffusion of heat by percolating water. For this reason observations at different depths in the same locality often give very concordant results for the same period, as the total percolation and the average rate are necessarily nearly the same for the various strata, although the actual degree of wetness of each may vary considerably. The following are a few typical values for sand or gravel deduced from the annual wave in different localities:
Observer. Soil. Localit I Thermo Diffus
y meter. ivity.
Kelvin, r86o . Garden sand Edinburgh Mercury .0087
Neumann, 1863 Sandy loam .. •0136
Everett, r86o . Gravel Greenwich „ .0125
Angstrom, 1861 Sandy clay Upsala „ .0057
17 '0045
Angstrom , Coa rse sand 0094
Rudberg and 'oo6r
, instruments
The same soilforlace
Quetelet. ) re erent ,ears •0074
reduced different
Callendar, 1895 Garden sand Montreal Platinum .0036
Rambaut, 1900 Gravel Oxford „ .0074
The low value at Montreal is chiefly due to the absence of percolation during the winter. A. A. Rambaut's results were obtained with similar instruments similarly located, but he did not investigate the seasonal variations of diffusivity, or the effect of percolation. It is probable that the coarser soils, permitting more rapid percolation, would generally give higher results. In any case, it is evident that the transmission of heat by percolation would be much greater in porous soils and in the upper layers of the earth's crust than in the lower strata or in solid rocks. It is probable for this reason that the average conductivity of the earth's crust, as deduced from surface observations, is too large; and that estimates of the age of the earth based on such measurements are too low, and require to be raised; they would thereby be brought into better agreement with the conclusions of geologists derived from other lines of argument.
16. Angstrom's Method consists in observing the propagation of heat waves in a bar, and is probably the most accurate method for
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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 castiron (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 dO/dx, may be written gk(d20/dx2)dx. This is equal to the product of the thermal capacity of the element, cqdx, by the rate of rise of temperature dO/dt, together with the heat lost per second at the external surface, which may be written hpOdx, if p is the perimeter of the bar, and h the heat loss per second per degree excess of temperature 0 above the surrounding medium. We thus obtain the differential equation
qk (d29/dx2) = cgdo/dt+hp0,
which is satisfied by terms of the type
0=e " sin (2antbx),
where a2b2 = hp/qk, and ab = me/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=o, a=b= (2rnc/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 (4in. bar) with waves of slow period, about I to 2 hours. Angstrom 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 (10.00152 0) assuming c = .84476.
Iron, k =0.1988 (10.00287 0) , c= •88620. Angstrom's value for iron, when corrected for obvious numerical errors, and for the probable variation of c, becomes
Iron, k =0.164 (10.0013 0),
but this is very doubtful as c was not measured.
The experiments on castiron 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 i.ompared. In some experiments the bar was lagged with t in. of asbestos, but in others it was bare, the heatloss 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 i %. Observations were taken at mean temperatures of toe C. and 54°C., with the following results:
Cast iron at Io2°C., k/c=.1296, c=.858, k=.1113. 54°C., k/c=.1392, c=.823, k '.1144.
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 temperaturecoefficient 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/1 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 heatloss 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 = cl'(dS/dtdS'/dt')/(DD').
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 cros. apart in a bar I.5 ems. in diameter. It was also tacitly assumed that the thermoelectric 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 to %. The thermal conductivity was determined in the neighbourhood of 20° C. with a water jacket, and near to° 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 heatloss 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 o° C., increased with rise of temperature nearly in proportion to the absolute temperature. The value found for this ratio at o° C. approximated to 1500 C.G.S. for the best conductors, but increased to 1800 or 2000 for bad conductors like Germansilver and antimony. It is clear, however, that this relation cannot be generally true, for the castiron mentioned in the last section had a specific resistance of 112,O00 C.G.S. at loo° 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 thermoelectric 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 (RR°)/aR° by observing the increase of resistance RR° of the bar, and the current C. It is also necessary to measure the crosssection q, the length 1, and the temperaturecoefficient 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 temperatureexcess is 0 is equal per unit length and time (t) to that lost by conduction d(gkd0/dx)/dx, and by radiation hp0 (emissivity h, perimeter p), together with that employed in raising the temperature gcdO/dt, and absorbed by the Thomson effect sCdO/dx. We thus obtain the equation
C2Ro(i+aO)/l =d(gkdo/dx)/dx+hpe+gcdo/dt±sCdO/dx. (8)
If C =o, this is the equation of Angstrom's method. If h also is zero, it becomes the equation of variable flow in the soil. If d0/dt=o, 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 C2Roa=hpl, the external heatloss is compensated by the variation of resistance
896
with temperature. In this case the solution of the equation reduces to the form
9 = x(l – x)C2Ro/2lgk. (9) By a property of the parabola, the mean temperature is Irds of the maximum temperature, we have therefore
(R–Ro)/aRo =lC2Ro/I2LLk, (to) which gives the conductivity directly in terms of the quantities actually observed. If the dimensions of the bar are suitably chosen, the distribution of temperature is always very nearly parabolic, so that it is not necessary to determine the value of the critical current C2 = hpl/aRo very accurately, as the correction for external loss is a small percentage in any case. The chief difficulty is that of measuring the small change of resistance accurately, and of avoiding errors from accidental thermoelectric effects. In addition to the simple measurements of the conductivity (M'Gill College, 18951896), some very elaborate experiments were made by King (Proc. Amer. Acad., June 1898) on the temperature distribution in the case of long bars with a view to measuring the Thomson effect. Duncan (M'Gill College Reports, 1899), using the simple method under King's supervision, found the conductivity of very pure copper to be I•oo7 for a temperature of 33° C.
(b) The method of Kohlrausch, as carried out by Jaeger and Dieselhorst (Berlin Acad., July 1899), consists in observing the difference of temperature between the centre and the ends of the bar by means of insulated thermocouples. Neglecting the external heatloss, and the variation of the thermal and electric conductivities k and k', we obtain, as before, for the difference of temperature between the centre and ends, the equation
8,,,°z Bo =C2Rl/8qk=ECl/8qk=Elk'/8k, (II)
where E is the difference of electric potential between the ends. Lorenz, assuming that the ratio k/k'=aD, had previously given
02,.902=E2/4a, (12)
which is practically identical with the preceding for small differences of temperature. The last expression in terms of k/k' is very simple, but the first is more useful in practice, as the quantities actually measured are E, C, 1, q, and the difference of temperature. The current C was measured in the usual way by the difference of potential on a standard resistance. The external heatloss was estimated by varying the temperature of the jacket surrounding the bar, and applying a suitable correction to the observed difference of temperature. But the method (a) previously described appears to be preferable in this respect, since it is better to keep the jacket at the same temperature as the cadblocks. Moreover, the variation of thermal conductivity with temperature is small and uncertain, whereas the variation of electrical conductivity is large and can be accurately determined, and may therefore be legitimately utilized for eliminating the external heatloss.
From a comparison of this work with that of Lorenz, it is evident that the values of the conductivity vary widely with the purity of the material, and cannot be safely applied to other specimens than those for which they were found.
19. Conduction in Gases and Liquids.—The theory of conduction of heat by diffusion in gases has a particular interest, since it is possible to predict the value on certain assumptions, if the viscosity is known. On the kinetic theory the molecules of a gas are relatively far apart and there is nothing analogous to friction between two adjacent layers A and B moving with different velocities. There is, however, a continual interchange of molecules between A and B, which produces the same effect as viscosity in a liquid. Fastermoving particles diffusing from A to B carry their momentum with them, and tend to accelerate B; an equal number of slower particles diffusing from B to A act as a drag on A. This action and reaction between layers in relative motion is equivalent to a frictional stress tending to equalize the velocities of adjacent layers. The magnitude of the stress per unit area parallel to the direction of flow is evidently proportional to the velocity gradient, or the rate of change of velocity per cm. in passing from one layer to the next. It must also depend on the rate of interchange of molecules, that is to say, (I) on the number passing through each square centimetre per second in either direction, (2) on the average distance to which each can travel before collision (i.e. on the " mean free path "), and (3) on the average velocity of translation of the molecules, which varies as the square root of the temperature. Similarly if A is hotter than B, or if there is a gradient of temperature between adjacent layers, the diffusion of molecules from A to B tends to equalize the temperatures, or to conduct heat through the gas at a rate proportional to the temperature gradient, and depending also on the rate of interchange of molecules in the same way as the viscosity effect. Conductivity and viscosity in a gas should varyin a similar manner since each depends on diffusion in a similar way. The mechanism is the same, but in one case we have diffusion of momentum, in the other case diffusion of heat. Viscosity in a as was first studied theoretically from this point of view by J. Clerk Maxwell, who predicted that the effgct should be independent of the density within wide limits. This, at first sight, paradoxical result is explained by the fact that the mean free path of each molecule increases in the same proportion as the density is diminished, so that as the number of molecules crossing each square centimetre decreases, the distance to which each carries its momentum increases, and the total transfer of momentum is unaffected by variation of density. Maxwell himself verified this prediction experimentally for viscosity over a wide range of pressure. By similar reasoning the thermal conductivity of a gas should be independent of the density. This was verified by A. Kundt and E. Warburg (Jour. Phys. v. I18), who found that the rate of cooling of a thermometer in air between 150 mm. and 1 mm. pressure remained constant as the pressure was varied. At higher pressures the effect of conduction was masked by convection currents. The question of the variation of conductivity with temperature is more difficult.. If the effects depended merely on the velocity of translation of the molecules, both conductivity and viscosity should increase directly as the square root of the absolute temperature; but the mean free path also varies in a manner which cannot be predicted by theory and which appears to be different for different gases (Rayleigh, Proc. R.S., January 1896). Experiments by the capillary tube method have shown that the viscosity varies more nearly as 91, but indicate that the rate of increase diminishes at high temperatures. The conductivity probably changes with temperature in the same way, being proportional to the product of the viscosity and the specific heat; but the experimental investigation presents difficulties on account of the necessity of eliminating the effects of radiation and convection, and the results of different observers often differ considerably from theory and from each other. The values found for the conductivity of air at o° C. range from •000048 to •000057, and the temperaturecoefficient from •oo15 to '0028. The results are consistent with theory within the limits of experimental error, but the experimental methods certainly appear to admit of improvement.
The conductivity of liquids has been investigated by similar methods, generally variations of the thin plate or guardring method. A critical account of the subject is contained in a paper by C. Chree (Phil. Mag., July 1887). Many of the experiments were made by comparative methods, taking a standard liquid such as water for reference. A determination of the conductivity of water by S. R. Milner and A. P. Chattock, employing an electrical method, deserves mention on account of the careful elimination of various errors (Phil. Mag., July 1899). Their final result was k=•001433 at 2o° C., which may be compared with the results of other observers, G. Lundquist (1869), .00155 at 4o° C.; A. Winkelmann (1874), •00104 at 15° C.; H. F. Weber (corrected by H. Lorberg), •00138 at 4° C., and .00152 at 23.6° C.; C. H. Lees (Phil. Trans., 1898), .00136 at 25° C., and *00120 at 47° C.; C. Chree, •00124 at IS° C., and oo'36 at 19'50 C. The variations of these results illustrate the experimental difficulties. It appears probable that the conductivity of a liquid increases considerably with rise of temperature, althougl, the contrary would appear from the work of Lees. A large mas; of material has been collected, but the relations are obscured by experimental errors.
See also Fourier, Theory of Heat; T. Preston, Theory of Heat, cap. vii.; Kelvin, Collected Papers; O. E. Meyer, Die kinetische Theorie der Gase; A. Winkelmann, Handbuch der Physik.
(H. L. C.)
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