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Encyclopedia Britannica - Main :: DEM-DIO |
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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 0074reduced 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 Prrzm F luloAWMay 1895 Cams 01 zelsslV u%! Z09r /a Therm' A 11 111~111,~.1H11W/1 MM. MAIM MlIIIIN&IW I ini^^-~^r~^~^^^^^^u~^^^- ^vi^^^.^^^ ^=^^r~.~~^ ^/^ ^ 111.1~1Tlltiii~~Il NE^^^^^_~-^ g.iuli UHH1~ A; 11111!11! .1 HI _~ ^^^^.~~^.^~^ -fir,/ :u~^ ^^'^^ :1 ~^_^^ More. 6 Tues. 7 'red.& Thure. 9 Fre ' 10 0 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 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 (2ant-bx), where a2-b2 = 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 (4-in. 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 (1-0.00152 0) assuming c = .84476. Iron, k =0.1988 (1-0.00287 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 (1-0.0013 0), 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 i.ompared. In some experiments the bar was lagged with t 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 i %. Observations were taken at mean temperatures of toe C. and 54C., with the following results:Cast iron at Io2C., k/c=.1296, c=.858, k=.1113. 54C., 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 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/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 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 = cl'-(dS/dt-dS'/dt')/(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
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 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
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
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 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/dtsCdO/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 heat-loss 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
(RRo)/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
(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 thermo-couples. Neglecting the external heat-loss, 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 heat-loss 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 cad-blocks. 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 heat-loss. 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
The conductivity of liquids has been investigated by similar methods, generally variations of the thin plate or guard-ring 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
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