CONDUCTION OF HEAT. The mathematical theory of conduction of heat was developed early in the 19th century by Fourier and other workers, and was brought to so high a pitch of excellence that little has remained for later writers to add to this department of the. subject. In fact, for a considerable period, the term " theory of heat " was practically synonymous with the mathematical treatment of a conduction. But later experimental researches have shown that the simple assumption of constant coefficients of conductivity and emissivity, on which the mathematical theory is based, is in many respects inadequate, and the special mathematical methods developed by J. B. J. Fourier need not be considered in detail here, as they are in many cases of mathematical rather than physical interest. The main object of
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the present article is to describe more recent work, and to discuss experimental difficulties and methods of measurement.
1. Mechanism of Conduction.—Conduction of heat implies transmission by contact from one body to another or between contiguous particles of the same body, but does not include transference of heat by the motion of masses or streams of matter from one place to another. This is termed convection, and is most important in the case of liquids and gases owing to their mobility. Conduction, however, is generally understood to include diffusion of heat in fluids due to the agitation of the ultimate molecules, which is really molecular convection. It also includes diffusion of heat by internal radiation, which must occur in transparent substances. In measuring conduction of heat in fluids, it is possible to some extent to eliminate the effects of molar convection or mixing, but it would not be possible to distinguish between diffusion, or internal radiation, and conduction. Some writers have supposed that the ultimate atoms are conductors, and that heat is transferred through them when they are in contact. This, however, is merely transferring the properties of matter in bulk to its molecules. It is much more probable that heat is really the kinetic energy of motion of the molecules, and is passed on from one to another by collisions. Further, if we adopt W. Weber's hypothesis of electric atoms, capable of diffusing through metallic bodies and conductors of electricity, but capable of vibration only in nonconductors, it is possible that the ultimate mechanism of conduction may be reduced in all cases to that of diffusion in metallic bodies or internal radiation in dielectrics. The high conductivity of metals is then explained by the small mass and high velocity of diffusion of these electric atoms. Assuming the kinetic energy of an electric atom at any temperature to be equal to that of a gaseous molecule, its velocity, on Sir J. J. Thomson's estimate of the mass, must be upwards of forty times that of the hydrogen molecule.
2. Law of Conduction.—The experimental law of conduction, which forms the basis of the mathematical theory, was established in a qualitative manner by Fourier and the early experimentalists. Although it is seldom explicitly stated as an experimental law, it should really be regarded in this light, and may be briefly worded as follows: "The rate of transmission of heat by conduction is proportional to the temperature gradient."
The " rate of transmission of heat " is here understood to mean the quantity of heat transferred in unit time through unit area of crosssection of the substance, the unit area being taken perpendicular to the lines of flow. It is clear that the quantity transferred in any case must be jointly proportional to the area and the time. The " gradient of temperature " is the fall of temperature in degrees per unit length along the lines of flow. The thermal conductivity of the substance is the constant ratio of the rate of transmission to the temperature gradient. To take the simple case of the " wall " or flat plate considered by Fourier for the definition of thermal conductivity, suppose that a quantity, of heat Q passes in the time T through an area A of a plate of conductivity k and thickness x, the sides of which are constantly maintained at temperatures 0' and 0". The rate of transmission of heat is Q/AT, and the temperature gradient, supposed uniform, is (0'O")/x, so that the law of conduction leads at once to the equationgradient is of the order of r° C. in too ft., but varies inversely with the conductivity of the strata at different depths.
3. Variable State: —A different type of problem is presented in those cases in which the temperature at each point varies with the time, as is the case near the surface of the soil with variations in the external conditions between day and night or summer and winter. The flow of heat may still be linear if the horizontal layers of the soil are of uniform composition, but the quantity flowing through each layer is no longer the same. Part of the heat is used up in changing the temperature of the successive layers. In this case it is generally more convenient to consider as unit of heat the thermal capacity c of unit volume, or that quantity which would produce a rise of one degree of temperature in unit volume of the soil or substance considered. If Q is expressed in terms of this unit in equation (I), it is necessary to divide by c, or to replace k on the righthand side by the ratio k/c. This ratio determines the rate of diffusion of temperature, and is called the thermometric conductivity or, more shortly, the diffusivity. The velocity of propagation of temperature waves will be the same under similar conditions in two substances which possess the same diffusivity, although they may differ in conductivity.
4. Emissivity.—Fourier defined another constant expressing the rate of loss of heat at a bounding surface per degree of difference of temperature between the surface of the body and its surroundings. This he called the external conductivity, but the term emissivity is more convenient. Taking Newton's law of cooling that the rate of loss of heat is simply proportional to the excess of temperature, the emissivity would be independent of the temperature. This is generally assumed to be the case in mathematical problems, but the assumption is admissible only in rough work, or if the temperature difference is small. The emissivity really depends on every variety of condition, such as the size, shape and position of the surface, as well as on its nature; it varies with the rate of cooling, as well as with the temperature excess, and it is generally so difficult to calculate, or to treat in any simple manner, that it forms the greatest source of uncertainty in all experimental investigations in which it occurs.
5. Experimental Methods.—Measurements of thermal conductivity present peculiar difficulties on account of the variety of quantities to be observed, the slowness of the process of conduction, the impossibility of isolating a quantity of heat, and the difficulty of exactly realizing the theoretical conditions of the problem. The most important methods may be classified roughly under three heads—(r) Steady Flow, (2) Variable Flow, (3) Electrical. The methods of the first class may be further subdivided according to the form of apparatus employed. The following are some of the special cases which have been utilized experimentally:
6. The "Wall" or Plate Method.—This method endeavours to realize the conditions of equation (I), namely, uniform rectilinear flow. Theoretically this requires an infinite plate, or a perfect heat insulator, so that the lateral flow can be prevented or rendered negligible. This condition can generally be satisfied with sufficient approximation with plates of reasonable dimensions. To find the conductivity, it is necessary to measure all the quantities which occur in equation (t) to a similar order of accuracy. The area A from which the heat is collected need not be the whole surface of the plate, but a measured central area where the flow is most nearly uniform. This variety is known as the " GuardRing " method, but it is generally rather difficult to determine the effective area of the ring. There is little difficulty in measuring the time of flow, provided that it is not too short. The measurement of the temperature gradient in the plate generally presents the greatest difficulties. If the plate is thin, it is necessary to measure the thickness with great care, and it is necessary to assume that the temperatures of the surfaces are the same as those of the media with which they are in contact, since there is no room to insert thermometers in the plate itself. This assumption does not present serious errors in the case of bad conductors, such as glass or wood, but has given rise to large mistakes in the case of metals. The conductivities of thin slices of crystals have been measured by C. H. Lees (Phil. Trans., 1892) by pressing them between plane amalgamated surfaces of metal. This gives very good contact, and the conductivity of the metal being more than too times that of the crystal, the temperature of the surface is determinate.
Q/AT = k (B' – 0" (/x = kdO/dx. (I)
This relation applies accurately to the case of the steady flow of heat in parallel straight lines through a homogeneous and isotropic solid, the isothermal surfaces, or surfaces of equal temperature, being planes perpendicular to the lines of flow. If the flow is steady, and the temperature of each point of the body invariable, the rate of transmission must be everywhere the same. If the gradient is not uniform, its value may be denoted by dO/dx. In the steady state, the product kd9/dx must be constant, or the gradient must vary inversely as the conductivity, if the latter is a function of 0 or x. One of the simplest illustrations of the rectilinear flow of heat is the steady outflow through the upper strata of the earth's crust, which may be considered practically plane in this connexion. This outflow of heat necessitates a rise of temperature with increase of depth. The corresponding
In applying the plate method to the determination of the conductivity of iron, E. H. Hall proposed to overcome this difficulty by coating the plate thickly with copper on both sides, and deducing the difference of temperature between the two surfaces of junction of the iron and the copper from the thermoelectric force observed by means of a number of fine copper wires attached to the copper coatings at different points of the disk. The advantage of the thermojunction for this purpose is that the distance between the surfaces of which the temperaturedifference is measured, is very exactly defined. The disadvantage is that the thermoelectric force is very small, about tenmillionths of a volt per degree, so that a small accidental disturbance may produce a serious error with a difference of temperature of only 1° between the junctions. The, chief uncertainty in applying this method appears to have arisen from variations of temperature at different parts of the surface, due to inequalities in the heating or cooling effect of the stream of water flowing over the surfaces. Uniformity of temperature could only be secured by using a high velocity of flow, or violent stirring. Neither of these methods could be applied in this experiment. The temperatures indicated by the different pairs of wires differed by as much as 1o%, but the mean of the whole would probably give a fair average. The heat transmitted was measured by observing the flow of water (about 20 gm./sec.) and the rise of temperature (about o.5°C.) in one of the streams. The results appear to be entitled to considerable weight on account of the directness of the method and the full consideration of possible errors. They were as follows:
Castiron, k=0.1490 C.G.S. at 3o° C., temp. coef.–0.00075. Pure iron, k =o. 1530 at 30° C., temp. cod.– 0.0003.
The disks were lo cms. in diam., and nearly 2 cros. thick, plated with copper to a thickness of 2 mm. The castiron contained about 3.5 % of 'carbon, 1.4% of silicon, and 0.5 % of manganese. It should be observed, however, that he obtained a much lower value for castiron, namely .105, by J. D. Forbes's method, which agrees better with the results given in § lo below.
7. Tube Method.—If the inside of a glass tube is exposed to steam, and the outside to a rapid current of water, or vice versa, the temperatures of the surfaces of the glass may be taken to be very approximately equal to those of the water and steam, which may be easily observed. If the thickness of the glass is small compared with the diameter of the tube, say onetenth, equation (I) may be applied with sufficient approximation, the area A being taken as the mean between the internal and external surfaces. It is necessary that the thickness x should be approximately uniform. Its mean value may be determined most satisfactorily from the weight and the density. The heat Q transmitted in a given time T may be deduced from an observation of the rise of temperature of the water, and the amount which passes in the interval. This is one of the simplest of all methods in practice, but it involves the measurement of several different quantities, some of which are difficult to observe accurately. The employment of the tube form evades one of the chief difficulties of the plate method, namely, the
uncertainty of the flow at the boundary Steam Inlet of the area considered. Unfortunately the method cannot be applied to good conductors, like the metals, because the difference of temperature between the surfaces may be five or ten times less than that between the water and steam in contact with them, even ifflow in this method are radial. The isothermal surfaces are coaxial cylinders. The areas of successive surfaces vary as their radii, hence the rate of transmission Q/AT varies inversely as the radius r, and is Q/2arlT, if l is the length of the cylinder, and Q the total heat, calculated from the condensation of steam observed in a time T. The outward gradient is dO/dr, and is negative if the central hole is heated. We have therefore the simple equation
–kdo/dr = Q/27rrlT. (2)
If k is constant the solution is evidently o=a log r+b, where a= –Q/27rklT, and b and k are determined from the known values of the temperatures observed at any two distances from the axis. This gives an average value of the conductivity over the range, but it is better to observe the temperatures at three distances, and to assume k to be a linear function of the temperature, in which case the solution of the equation is still very simple, namely,
o+1eo2=a log r+b, (3) where e is the temperaturecoefficient of the conductivity. The chief difficulty in this method lay in determining the effective distances of the bulbs of the thermometers from the axis of the cylinder, and in ensuring uniformity of flow of heat along different radii. For these reasons the temperaturecoefficient of the conductivity could not be determined satisfactorily on this particular form of apparatus, but the mean results were probably trustworthy to I or 2 %. They refer to a temperature of about 6o° C., and were
Castiron, 0.109; mild steel, 0•I19, C.G.S.
These are much smaller than Hall's results. The castiron contained nearly 3% each of silicon and graphite, and 1 % each of phosphorus and manganese. The steel contained less than I % of foreign materials. The low value for the castiron was confirmed by two entirely different methods given below.
9. Forbes's Bar Method.—Observation of the steady distribution of temperature along a bar heated at one end was very early employed by Fourier, Despretz and others for the comparison of conductivities. It is the most convenient method, in the case of good conductors, on account of the great facilities which it permits for the measurement of the temperature gradient at different points; but it has the disadvantage that the results depend almost entirely on a knowledge of the external heat loss or emissivity, or, in comparative experiments, on the assumption that it is the same in different cases. The method of Forbes (in which the conductivity is deduced from the steady distribution of temperature on the assumption that the rate of loss of heat at each point of the bar is the same as that observed in an auxiliary experiment in which a short bar of the same kind is set to cool under conditions which are supposed to be identical) is well known, but a consideration of its weak points is very instructive, and the results have been most remarkably misunderstood and misquoted. The method gives directly, not k, but k/c. P. G. Tait repeated Forbes's experiments, using one of the same iron bars, and endeavoured to correct his results for the variation of the specific heat c. J. C. Mitchell, under Tait's direction, repeated the experiments with the same bar nickelplated, correcting the thermometers for stemexposure, and also varying the conditions by cooling one end, so as to obtain a steeper gradient. The results of Forbes, Tait and Mitchell, on the same bar, and Mitchell's two results with the end of the bar " free " and " cooled," have been quoted as if they referred to different metals. This is not very surprising, if the values in the following table are compared:
the water is ener
I TABLE I.=Therm¢l Conductivity of Forbes's Iron Bar D (1•25 inches square). g
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C.G.S. Units.
8. Cylinder Method. —A variation of the tube method, which can be applied to metals and good conductors, depends on the employment of a thick cylinder with a small axial hole in place of a thin tube.
j The actual tempera
ture of the metal itself can then be observed by inserting thermometers or thermocouples at measured distances from the centre. This method
'~'watr has been applied by H. L. Callendar and J. T. Nicolson (Brit. Assoc. Report,
roseeam ~'J 1897) to cylinders of castiron and
T1lamometer i mild steel, 5 in. in diam. and 2 ft. long, with 1 in. axial holes. The surface of the central hole was heated by steam under pressure, and the total flow of heat was determined by observing the
amount of steam condensed in a given time. The outside of the cylinder
was cooled by water circulating round a spiral screw thread in a narrow
space with high velocity driven by a pressure of 120 lb per sq. in. A
very uniform surface temperature was thus obtained. The lines of
To $ wutor FIG. I.
Temp. Uncorrected for Variation of c. Corrected Variation of c.
Cent. for
Mitchell. Mitchell.
Forbes. Tait. Forbes. Tait.
Free. Cooled. Free. Cooled.
—
0° •207 .231 .197 .178 .213.. .238 .203 •184
Too° .157 .198 .178 .190 •168 •212 .190 .197
200° .136 .176 •16o .181 .152 .196 •178 •210
The variation of c is uncertain. The values credited to Forbes are those given by J. D. Everett on Balfour Stewart's authority. Tait gives different figures. The values given in the column headed
cooled " are those found by Mitchell with one end of the bar cooled. The discrepancies are chiefly due to the error of the fundamental assumption that the rate of cooling is the same at the same temperature under the very different conditions existing in the two parts of the experiment. They are also partly caused by the large uncertainties of the corrections, especially those of the mercury thermometers under the peculiar conditions of the experiment. The results of Forbes are interesting historically as having been the first approximately correct determinations of conductivity in absolute value. The same method was applied by R. W. Stewart (Phil. Trans., 1892), with the substitution of thermocouples (following Wiedemann) for mercury thermometers. This avoids the very uncertain correction for stemexposure, but it is doubtful how far
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 (i.0015 t) C.G.S. E. H. Hall using a similar method found for castiron at 5o° C. the value •1o5, but considers the method very uncertain as ordinarily practised.
to. 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 castiron, the conductivity of which was required for some experiments on the condensation of steam (Prot. 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 onequarter by lagging the bar like a steampipe to a thickness oft in. The heating vessel could be maintained at a steady temperature by highpressure steam. The other end was maintained at a temperature near that of the air by a steady stream of water flowing through a welllagged 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
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results obtained by this method at a temperature of 4o° 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. Angstrom's method described below.
11. GuardRing 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 thermojunctions with the mercury in the inner tube. The heatflow 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 eightyfive divisions. The contraction corresponding to the melting of 1 gramme of ice was assumed to be •o906 c.c., and was taken as being equivalent to 79 calories (I 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 „ Angstrom.
91 12. VariableFlow 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
893
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 steadyflow 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 flgw 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.
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 (dO/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'(db'/dx') —k"(dO"/dx") =c (area between curves)/(TT'), (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 •ooto 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 •oo6o 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
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applied to the study of variations of soiltemperatures by. harmonic analysis of the annual waves. But the theory is not strictly applicable, as the phenomena are not accurately periodic, and the state of the soil is continually varying, and differs at different depths, particularly in regard to its` degree of wetness. An additional difficulty arises in the case of observations made with long mercury thermometers buried in vertical holes, that the correction for the expansion of the liquid in the long stems is uncertain, and that the holes may serve as channels for percolation, and thus lead to exceptionally high values. The last error
894
diminished, so that at a depth of only 4 in. it was already reduced to about 6° F., and to less than a° at ro in. At the same time, the epoch of maximum or minimum was retarded, about 4 hours at 4 in., and nearly 12 hours at TO in., where the maximum temperature was reached between r and 2 A.M. The form of the wave was also changed. At 4 in. the rise was steeper than
the fall, at to in. the reverse was the case. This is due to the fact that the components of shorter period are more rapidly propagated. For instance, the velocity of propagation of a wave having a period of a day is nearly twenty times as great as that of a wave with a period of one year; but on the other hand the penetration of the diurnal wave is nearly twenty times less, and the shorter waves die out more rapidly.
14. A SimpleHarmonic or Sine Wave is the only kind which is propagated without change of form. In treating mathematically the propagation of other kinds of waves, it is necessary to analyse them into their simpleharmonic components, which may be treated as being propagated independently. To illustrate the main features of the calculation, we may suppose that the surface is subject to a simpleharmonic cycle of temperature variation, so that the temperature at any time t is given by an equation of the form
0 00=Asin 2rnt=Asin girt/T, (5) where Bo is the mean temperature of the surface, A the amplitude of the cycle, n the frequency, and T the period. In this simple case the temperature cycle at a depth x is a precisely similar curve of the same period, but with the amplitude reduced in the proportion ems, and the phase retarded by the fraction mx/2r of a cycle. The indexcoefficient m is d (rnc/i). The wave at a depth x is represented analytically by the equation
B Bo = Ae "'e sin (2rnt mx). (6)
A strictly periodic oscillation of this kind occurs in the working of a steam engine, in which the walls of the cylinder are exposed to regular fluctuations of temperature with the admission and release of steam. The curves in fig. 5 are drawn for a particular case, but they apply equally to the propagation of a simpleharmonic wave of any period in any substance changing only the scale on which they are drawn. The dotted boundary curves have the equation 0==, and show the rate of diminution of the amplitude of the temperature oscillation with depth in the metal. The wavelength in fig. 4 is o6o in., at which depth the amplitude of the variation is reduced to less than one fivehundredth part (e'r) of that at the surface, so that for all practical purposes the oscillation may be neglected beyond one wavelength At half a wavelength the amplitude is only Ord of that at the surface. The wavelength in any case is 2r/m.
The diffusivitcan be deduced from observations at different depths x' and x'', by observing the ratio of the amplitudes, which is e ''''w'''') for a simpleharmonic wave. The values obtained in this way for waves having a period of one second and a wavelength of half an inch agreed very well with those obtained in the same castiron by Angstrom's method (see below), with waves having a period of i hour and a length of 30 in. This agreement was a very satisfactory test of the accuracy of the fundamental law of conduction, as the gradients and periods varied so widely in the two cases.
15. Annual Variation.—A similar method has frequently been
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End of Article: CONDUCTION OF HEAT 

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