# 1911 Encyclopædia Britannica/Conduction of Heat

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​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 ​ 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 non-conductors, 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 cross-section 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 θ′ and θ″. The rate of transmission of heat is Q/AT, and the temperature gradient, supposed uniform, is (θ′−θ″)/𝑥, so that the law of conduction leads at once to the equation
| Q/AT＝𝑘(θ′ −θ″)/𝑥＝𝑘𝑑θ/𝑑𝑥. | (1) |
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 𝑑θ/𝑑𝑥. In the steady state, the product 𝑘𝑑θ/𝑑𝑥 must be constant, or the gradient must vary inversely as the conductivity, if the latter is a function of θ 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 gradient is of the order of 1° C. in 100 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 (1), it is necessary to divide by 𝑐, or to replace 𝑘 on the right-hand 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—(1) 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 (1), 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 (1) 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 “Guard-Ring” 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 100 times that of the crystal, the temperature of the surface is determinate.
​ 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 thermo-electric 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 thermo-junction for this purpose is that the distance between the surfaces of which the temperature-difference is measured, is very exactly defined. The disadvantage is that the thermo-electric force is very small, about ten-millionths 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 10 %, 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 0·5°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:—
Cast-iron, 𝑘＝0·1490 C.G.S. at 30° C., temp. coef.–0·00075. Pure iron, 𝑘＝0·1530 at 30° C., temp. coef.–0·0003.
The disks were 10 cms. in diam., and nearly 2 cms. thick, plated with copper to a thickness of 2 mm. The cast-iron 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 cast-iron, namely ·105, by J. D. Forbes’s method, which agrees better with the results given in § 10 below.
Fig. 1. 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 one-tenth, equation (1) 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 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 if the water is energetically stirred.
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. The actual temperature of the metal itself can then be observed by inserting thermometers or thermo-couples at measured distances from the centre. This method has been applied by H. L. Callendar and J. T. Nicolson (Brit. Assoc. Report, 1897) to cylinders of cast-iron and 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 ℔ per sq. in. A very uniform surface temperature was thus obtained. The lines of flow 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/2π𝑟𝑙T, 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 𝑑θ/𝑑𝑟, and is negative if the central hole is heated. We have therefore the simple equation
| −𝑘𝑑θ/𝑑𝑟＝Q/2π𝑟𝑙 T. | (2) |
If 𝑘 is constant the solution is evidently θ＝𝑎 log 𝑟+𝑏, where 𝑎＝−Q/2π𝑘𝑙 T, 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,
| θ+1/2𝑒θ$^2$＝𝑎 log 𝑟+𝑏, | (3) |
where 𝑒 is the temperature-coefficient 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 temperature-coefficient of the conductivity could not be determined satisfactorily on this particular form of apparatus, but the mean results were probably trustworthy to 1 or 2%. They refer to a temperature of about 60° C., and were—
Cast-iron, 0·109; mild steel, 0·119, C.G.S.
These are much smaller than Hall’s results. The cast-iron contained nearly 3 % each of silicon and graphite, and 1 % each of phosphorus and manganese. The steel contained less than 1 % of foreign materials. The low value for the cast-iron 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 nickel-plated, correcting the thermometers for stem-exposure, 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 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 thermo-couples (following Wiedemann) for mercury thermometers. This avoids the very uncertain correction for stem-exposure, 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 (1 − ·0015 𝑡) C.G.S. E. H. Hall using a similar method found for cast-iron at 50° C. the value ·105, but considers the method very uncertain as ordinarily practised.
10. 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 cast-iron, the conductivity of which was required for some experiments on the condensation of steam (Proc. 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 one-quarter by lagging the bar like a steam-pipe to a thickness of 1 in. The heating vessel could be maintained at a steady temperature by high-pressure steam. The other end was maintained at a temperature near that of the air by a steady stream of water flowing through a well-lagged 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 results obtained by this method at a temperature of 40° 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. Angström’s method described below.
Fig. 2.
11. Guard-Ring 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 thermo-junctions with the mercury in the inner tube. The heat-flow 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 eighty-five divisions. The contraction corresponding to the melting of 1 gramme of ice was assumed to be ·0906 c.c., and was taken as being equivalent to 79 calories (1 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 | ,, | Ångström. |
12. Variable-Flow 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 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 steady-flow 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 flow 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.
Fig. 3.
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 (𝑑θ/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′(𝑑θ′/𝑑𝑥′) −𝑘″(𝑑θ″/𝑑𝑥″)＝𝑐 (area between curves)/(T−T′), | (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 ·0010 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 ·0060 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 ​ diminished, so that at a depth of only 4 in. it was already reduced to about 6° F., and to less than 2° at 10 in. At the same time, the epoch of maximum or minimum was retarded, about 4 hours at 4 in., and nearly 12 hours at 10 in., where the maximum temperature was reached between 1 and 2 a.m. The form of the wave was also changed. At 4 in. the rise was steeper than the fall, at 10 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.
Fig. 4.
14. A Simple-Harmonic 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 simple-harmonic 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 simple-harmonic cycle of temperature variation, so that the temperature at any time 𝑡 is given by an equation of the form—
| θ−θ$_0$ = A sin 2π𝑛𝑡 = A sin 2π𝑡/T, | (5) |
where θ$_0$ 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 𝑒$^{𝑚𝑥}$, and the phase retarded by the fraction 𝑚𝑥/2π of a cycle. The index-coefficient 𝑚 is √(π𝑛𝑐/𝑘). The wave at a depth 𝑥 is represented analytically by the equation
| θ−θ$_0$ = A𝑒$^{−𝑚𝑥}$ sin (2π𝑛𝑡−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 simple-harmonic wave of any period in any substance changing only the scale on which they are drawn. The dotted boundary curves have the equation θ =±𝑒$^{−𝑚𝑥}$, and show the rate of diminution of the amplitude of the temperature oscillation with depth in the metal. The wave-length in fig. 4 is 0·60 in., at which depth the amplitude of the variation is reduced to less than one five-hundredth part (𝑒$^{−2π}$) of that at the surface, so that for all practical purposes the oscillation may be neglected beyond one wave-length At half a wave-length the amplitude is only 1/23rd of that at the surface. The wave-length in any case is 2π/𝑚.
The diffusivity can be deduced from observations at different depths 𝑥′ and 𝑥″, by observing the ratio of the amplitudes, which is 𝑒$^{𝑚(𝑥′−𝑥″)}$ for a simple-harmonic wave. The values obtained in this way for waves having a period of one second and a wave-length of half an inch agreed very well with those obtained in the same cast-iron by Ångström’s method (see below), with waves having a period of 1 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.
Speed, 42 revolutions per minute; range, 20° at surface. Fig. 5.
15. Annual Variation.—A similar method has frequently been applied to the study of variations of soil-temperatures 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 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:—
Table II.—Diffusivity of Sandy Soils. C.G.S. Units.
| Observer. | Observer. | Soil. | Locality. | Thermo- meter. | Diffusivity. | Diffusivity. |
| Kelvin, 1860 | Kelvin, 1860 | Garden sand | Edinburgh | Mercury | ·0087 | ·0087 |
| Neumann, 1863 | Neumann, 1863 | Sandy loam | .. | ” | ·0136 | ·0136 |
| Everett, 1860. | Everett, 1860. | Gravel | Greenwich | ” | ·0125 | ·0125 |
| Ångström, 1861 | Ångström, 1861 | Sandy clay | Upsala | ” | ·0057 | ·0057 |
| ” | ” | ” | ” | ” | ·0045 | ·0045 |
| Ångström | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,$ | Coarse sand | ” | ” | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$ | ·0094 |
| Rudberg | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,$ | The same soil, place and instruments | The same soil, place and instruments | The same soil, place and instruments | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$ | ·0061 |
| Quetelet | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,$ | reduced for different years | reduced for different years | reduced for different years | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$ | ·0074 |
| Callendar, 1895 | Callendar, 1895 | Garden sand | Montreal | Platinum | ·0036 | ·0036 |
| Rambaut, 1900 | Rambaut, 1900 | Gravel | Oxford | ” | ·0074 | ·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. Ångström’s Method consists in observing the propagation of heat waves in a bar, and is probably the most accurate method for ​ measuring the diffusivity of a metal, since the conditions may be widely varied and the correction for external loss of heat can be made comparatively small. Owing, however, to the laborious nature of the observations and reductions, the method does not appear to have been seriously applied since its first invention, except in one solitary instance by the writer to the case of cast-iron (fig. 2). The equation of the method is the same as that for the linear flow with the addition of a small term representing the radiation loss.
The heat per second gained by conduction by an element dx of the bar, of conductivity 𝑘 and cross section q, at a point where the gradient is 𝑑θ/dx, may be written qk(𝑑$^2$θ/dx$^2$)dx. This is equal to the product of the thermal capacity of the element, cqdx, by the rate of rise of temperature 𝑑θ/dt, together with the heat lost per second at the external surface, which may be written hpθdx, if p is the perimeter of the bar, and h the heat loss per second per degree excess of temperature θ above the surrounding medium. We thus obtain the differential equation
qk(𝑑$^2$θ/dx)＝cdqθ/dt+hpθ,
which is satisfied by terms of the type
where 𝑎$^2$−𝑏$^2$＝hp/qk, and ab＝πnc/𝑘.
The rate of diminution of amplitude expressed by the coefficient 𝑎 in the index of the exponential is here greater than the coefficient 𝑏 expressing the retardation of phase by a small term depending on the emissivity h. If h＝0, 𝑎＝𝑏＝√(πnc/𝑘), as in the case of propagation of waves in the soil.
The apparatus of fig. 2 was designed for this method, and may serve to illustrate it. The steam pressure in the heater may be periodically varied by the gauge in such a manner as to produce an approximately simple harmonic oscillation of temperature at the hot end, while the cool end is kept at a steady temperature. The amplitudes and phases of the temperature waves at different points are observed by taking readings of the thermometers at regular intervals. In using mercury thermometers, it is best, as in the apparatus figured, to work on a large scale (4-in. bar) with waves of slow period, about 1 to 2 hours. Ångström endeavoured to find the variation of conductivity by this method, but he assumed 𝑐 to be the same for two different bars, and made no allowance for its variation with temperature. He thus found nearly the same rate of variation for the thermal as for the electric conductivity. His final results for copper and iron were as follows:—
Copper, 𝑘＝0·982 (1–0·00152 θ) assuming 𝑐＝·84476. Iron, 𝑘＝0·1988 (1–0·00287 θ),,𝑐＝·88620.
Ångström’s value for iron, when corrected for obvious numerical errors, and for the probable variation of 𝑐, becomes—
but this is very doubtful as 𝑐 was not measured.
The experiments on cast-iron with the apparatus of fig. 2 were varied by taking three different periods, 60, 90 and 120 minutes, and two distances, 6 in. and 12 in., between the thermometers compared. In some experiments the bar was lagged with 1 in. of asbestos, but in others it was bare, the heat-loss being thus increased fourfold. In no case did this correction exceed 7 %. The extreme divergence of the resulting values of the diffusivity, including eight independent series of measurements on different days, was less than 1 %. Observations were taken at mean temperatures of 102° C. and 54°C., with the following results:—
Cast iron at 102°C., 𝑘/𝑐＝·1296, 𝑐＝·858, 𝑘＝·1113.
The variation of 𝑐 was determined by a special series of experiments. No allowance was made for the variation of density with temperature, or for the variation of the distance between the thermometers, owing to the expansion of the bar. Although this correction should be made if the definition were strictly followed, it is more convenient in practice to include the small effect of linear expansion in the temperature-coefficient in the case of solid bodies.
17. Lorenz’s Method.—F. Neumann, H. Weber, L. Lorenz and others have employed similar methods, depending on the observation of the rate of change of temperature at certain points of bars, rings, cylinders, cubes or spheres. Some of these results have been widely quoted, but they are far from consistent, and it may be doubted whether the difficulties of observing rapidly varying temperatures have been duly appreciated in many cases. From an experimental point of view the most ingenious and complete method was that of Lorenz (Wied. Ann. xiii. p. 422, 1881). He deduced the variations of the mean temperature of a section of a bar from the sum S of the E.M.F.’s of a number of couples, inserted at suitable equal intervals l and connected in series. The difference of the temperature gradients D/l at the ends of the section was simultaneously obtained from the difference D of the readings of a pair of couples at either end connected in opposition. The external heat-loss was eliminated by comparing observations taken at the same mean temperatures during heating and during cooling, assuming that the rate of loss of heat f(S) would be the same in the two cases. Lorenz thus obtained the equations:—
Heating, qk D/l＝cql dS/dt+f(S). Cooling, qk D′/l＝cql dS′/dt′+f(S′). Whence 𝑘＝cl$^2$(𝑑S/dt−𝑑S′/dl)/(D−D′).
It may be questioned whether this assumption was justifiable, since the rate of change and the distribution of temperature were quite different in the two cases, in addition to the sign of the change itself. The chief difficulty, as usual, was the determination of the gradient, which depended on a difference of potential of the order of 20 microvolts between two junctions inserted in small holes 2 cms. apart in a bar 1·5 cms. in diameter. It was also tacitly assumed that the thermo-electric power of the couples for the gradient was the same as that of the couples for the mean temperature, although the temperatures were different. This might give rise to constant errors in the results. Owing to the difficulty of measuring the gradient, the order of divergence of individual observations averaged 2 or 3 %, but occasionally reached 5 or 10 %. The thermal conductivity was determined in the neighbourhood of 20° C. with a water jacket, and near 110° C. by the use of a steam jacket. The conductivity of the same bars was independently determined by the method of Forbes, employing an ingenious formula for the heat-loss in place of Newton’s law. The results of this method differ 2 or 3 % (in one case nearly 15 %) from the preceding, but it is probably less accurate. The thermal capacity and electrical conductivity were measured at various temperatures on the same specimens of metal. Owing to the completeness of the recorded data, and the great experimental skill with which the research was conducted, the results are probably among the most valuable hitherto available. One important result, which might be regarded as established by this work, was that the ratio 𝑘/𝑘′ of the thermal to the electrical conductivity, though nearly constant for the good conductors at any one temperature such as 0° C., increased with rise of temperature nearly in proportion to the absolute temperature. The value found for this ratio at 0° C. approximated to 1500 C.G.S. for the best conductors, but increased to 1800 or 2000 for bad conductors like German-silver and antimony. It is clear, however, that this relation cannot be generally true, for the cast-iron mentioned in the last section had a specific resistance of 112,000 C.G.S. at 100° C., which would make the ratio 𝑘/𝑘′＝12,500. The increase of resistance with temperature was also very small, so that the ratio varied very little with temperature.
18. Electrical Methods.—There are two electrical methods which have been recently applied to the measurement of the conductivity of metals, (𝑎) the resistance method, devised by Callendar, and applied by him, and also by R. O. King and J. D. Duncan, (𝑏) the thermo-electric method, devised by Kohlrausch, and applied by W. Jaeger and H. Dieselhorst. Both methods depend on the observation of the steady distribution of temperature in a bar or wire heated by an electric current. The advantage is that the quantities of heat are measured directly in absolute measure, in terms of the current, and that the results are independent of a knowledge of the specific heat. Incidentally it is possible to regulate the heat supply more perfectly than in other methods.
(𝑎) In the practice of the resistance method, both ends of a short bar are kept at a steady temperature by means of solid copper blocks provided with a water circulation, and the whole is surrounded by a jacket at the same temperature, which is taken as the zero of reference. The bar is heated by a steady electric current, which may be adjusted so that the external loss of heat from the surface of the bar is compensated by the increase of resistance of the bar with rise of temperature. In this case the curve representing the distribution of temperature is a parabola, and the conductivity 𝑘 is deduced from the mean rise of temperature (R−R$^0$)/𝑎R$^0$ by observing the increase of resistance R−R$^0$ of the bar, and the current C. It is also necessary to measure the cross-section q, the length l, and the temperature-coefficient 𝑎 for the range of the experiment.
In the general case the distribution of temperature is observed by means of a number of potential leads. The differential equation for the distribution of temperature in this case includes the majority of the methods already considered, and may be stated as follows. The heat generated by the current C at a point x where the temperature-excess is θ is equal per unit length and time (t) to that lost by conduction −𝑑(qkdθ/dx)/dx, and by radiation hpθ (emissivity h, perimeter p), together with that employed in raising the temperature qcdθ/dt, and absorbed by the Thomson effect sC𝑑θ/dx. We thus obtain the equation—
| C$^2$R$_0$(1+𝑎θ)/l＝−𝑑(qkdθ/dx)/dx+hpθ+qcdθ/dt+sC𝑑θ/dx. | (8) |
If C＝0, this is the equation of Ångström’s method. If h also is zero, it becomes the equation of variable flow in the soil. If dθ/dt＝0, the equation represents the corresponding cases of steady flow. In the electrical method, observations of the variable flow are useful for finding the value of 𝑐 for the specimen, but are not otherwise required. The last term, representing the Thomson effect, is eliminated in the case of a bar cooled at both ends, since it is opposite in the two halves, but may be determined by observing the resistance of each half separately. If the current C is chosen so that C$^2$R$_0$𝑎＝hpl, the external heat-loss is compensated by the variation of resistance ​with temperature. In this case the solution of the equation reduces to the form
| θ＝x(𝑙−x)C$^2$R$_0$/2lqk. | (9) |
By a property of the parabola, the mean temperature is 2/3rds of the maximum temperature, we have therefore
| (R−R$_0$)/𝑎R$_0$＝𝑙C$^2$R$_0$/12qk, | (10) |
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 C$^2$=hpl/𝑎R$_0$ 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 College, 1895–1896), 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 1·007 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 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
| θ$_{max}$−θ$_0$＝C$^2$R𝑙/8qk＝EC𝑙/8qk＝E$^2$𝑘′/8𝑘, | (11) |
where E is the difference of electric potential between the ends. Lorenz, assuming that the ratio 𝑘/𝑘′ =𝑎θ, had previously given
| θ$^2$$_{max}$−θ$_0$$^2$＝E$^2$/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, 𝑙, 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 end-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, 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. 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 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, (1) 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 vary in 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 gas was first studied theoretically from this point of view by J. Clerk Maxwell, who predicted that the effect 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. 118), 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 θ$^{3/4}$, 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 0° C. range from ·000048 to ·000057, and the temperature coefficient from ·0015 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 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 20° C., which may be compared with the results of other observers, G. Lundquist (1869), ·00155 at 40° C.; A. Winkelmann (1874), ·001104 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 18° C., and ·00136 at 19·5° 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, although the contrary would appear from the work of Lees. A large mass 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.)