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DEPTH -INOHE8. See also:Speed, 42 revolutions per See also:minute; range, 200 at eurface. FIG. 5. is best avoided by employing See also:platinum thermometers buried horizontally. In any See also:case results deduced from the See also:annual See also:wave must be expected to vary in different years according to the See also:distribution of the rainfall, as the values represent averages depending chiefly on the See also:diffusion of See also:heat by percolating See also:water. For this See also:reason observations at different depths in the same locality often give very concordant results for the same See also:period, as the See also:total percolation and the See also:average See also: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 See also:sand or See also:gravel deduced from the annual wave in different localities: Observer. See also:Soil. Localit I Thermo- Diffus- y See also:meter. ivity. See also:Kelvin, r86o . See also:Garden sand See also:Edinburgh See also:Mercury .0087 See also:Neumann, 1863 Sandy See also:loam .. •0136 See also:Everett, r86o . Gravel See also:Greenwich „ .0125 See also:Angstrom, 1861 Sandy See also:clay See also:Upsala „ .0057 17 '0045 Angstrom , Coa rse sand 0094 Rudberg and 'oo6r , See also:instruments The same soilforlace See also:Quetelet. ) re erent ,ears •0074 reduced different Callendar, 1895 Garden sand See also:Montreal Platinum .0036 Rambaut, 1900 Gravel See also:Oxford „ .0074 The See also:low value at Montreal is chiefly due to the See also:absence of percolation during the See also:winter. A. A. Rambaut's results were obtained with similar instruments similarly located, but he did not investigate the seasonal See also: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 See also:earth's crust than in the See also:lower strata or in solid rocks. It is probable for this reason that the average conductivity of the earth's crust, as deduced from See also:surface observations, is too large; and that estimates of the See also: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 See also:argument. 16. Angstrom's Method consists in observing the See also:propagation of heat waves in a See also: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 _~ ^^^^.~~^.^~^ -See also:fir,/ :u~^ ^^'^^ :1 ~^_^—^ More. 6 Tues. 7 'red.& Thure. 9 Fre ' 10 0 measuring the diffusivity of a See also:metal, since the conditions may be widely varied and the correction for See also: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 See also:cast-See also:iron (fig. 2). The See also:equation of the method is the same as that for the linear flow with the addition of a small See also:term representing the See also:radiation loss. The heat per second gained by See also:conduction by an See also:element dx of the bar, of conductivity k and See also:cross See also: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 See also:medium. We thus obtain the See also:differential equation qk (d29/dx2) = cgdo/dt+hp0, which is satisfied by terms of the type 0=e " See also:sin (2ant-bx), where See also:a2-b2 = hp/qk, and ab = me/k. The rate of diminution of See also:amplitude expressed by the coefficient a in the See also: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 See also:steam pressure in the heater may be periodically varied by the See also:gauge in such a manner as to produce an approximately See also:simple See also: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 See also:regular intervals. In using mercury thermometers, it is best, as in the apparatus figured, to See also:work on a large See also:scale (4-in. bar) with waves of slow period, about I to 2 See also: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 See also: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 See also: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 See also:asbestos, but in others it was See also: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 See also:independent See also: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 See also:special series of experiments. No allowance was made for the variation of See also: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 See also: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. See also:Weber, L. Lorenz and others have employed similar methods, depending on the observation of the rate of See also:change of temperature at certain points of bars, rings, cylinders, cubes or See also: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 See also:complete method was that of Lorenz (Wied. See also: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 See also: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 See also: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 See also:chief difficulty, as usual, was the determination of the gradient, which depended on a difference of potential of the See also:order of 20 microvolts between two junctions inserted in small holes 2 cros. apart in a bar I.5 See also:ems. in See also:diameter. It was also tacitly assumed that the thermo-electric See also: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 See also: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 See also:Forbes, employing an ingenious See also:formula for the heat-loss in See also:place of See also:Newton's See also: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 See also:electrical conductivity were measured at various temperatures on the same specimens of metal. Owing to the completeness of the recorded data, and the See also:great experimental skill with which the See also: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 See also:good conductors at any one temperature such as o° C., increased with rise of temperature nearly in proportion to the See also: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 See also:bad conductors like See also:German-See also:silver and See also: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,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. See also: Incidentally it is possible to regulate the heat See also:supply more perfectly than in other methods. (a) In the practice of the resistance method, both ends of a See also: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 See also:curve representing the distribution of temperature is a See also:parabola, and the conductivity k is deduced from the mean rise of temperature (R-R°)/aR° by observing the increase of resistance R-R° of the bar, and the current C. It is also necessary to measure the cross-section q, the length 1, and the temperature-coefficient a for the range of the experiment. In the See also: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 See also: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 See also: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 See also: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 See also: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 See also:solution of the equation reduces to the See also:form 9 = x(l – x)C2Ro/2lgk. (9) By a See also: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 See also: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 thermo-electric effects. In addition to the simple measurements of the conductivity (M'Gill See also:College, 1895-1896), some very elaborate experiments were made by King (Proc. Amer. Acad., See also:June 1898) on the temperature distribution in the case of See also: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 (See also:Berlin Acad., See also: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=See also: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 See also: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 See also: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 See also: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 See also:gas are relatively far apart and there is nothing analogous to See also: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. Faster-moving 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 See also:act as a See also:drag on A. This See also:action and reaction between layers in relative See also:motion is See also:equivalent to a frictional stress tending to equalize the velocities of adjacent layers. The magnitude of the stress per unit See also: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 See also:free path "), and (3) on the average velocity of See also:translation of the molecules, which varies as the square See also: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 See also: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 See also:molecule increases in the same proportion as the density is diminished, so that as the number of molecules See also:crossing each square centimetre decreases, the distance to which each carries its momentum increases, and the total See also:transfer of momentum is unaffected by variation of density. Maxwell him-self 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. See also:Kundt and E. Warburg (Jour. Phys. v. I18), who found that the rate of cooling of a thermometer in See also: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 (See also:Rayleigh, Proc. R.S., See also:January 1896). Experiments by the capillary See also: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 See also:account of the See also: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 temperature-coefficient from •oo15 to '0028. The results are consistent with theory within the limits of experimental See also: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 See also:plate or guard-See also:ring method. A critical account of the subject is contained in a See also:paper by C. Chree (Phil. Mag., July 1887). Many of the experiments were made by See also:comparative methods, taking a standard liquid such as water for reference. A determination of the conductivity of water by S. R. See also: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. See also: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 See also:Fourier, Theory of Heat; T. See also:Preston, Theory of Heat, cap. vii.; Kelvin, Collected Papers; O. E. See also:Meyer, See also:Die kinetische Theorie der Gase; A. Winkelmann, Handbuch der Physik. (H. L. Additional information and CommentsThere are no comments yet for this article.
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