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+12𝑒θ2＝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:—
| Temp. Cent. |
Uncorrected for Variation of 𝑐. | Corrected for Variation of 𝑐. | ||||||
| Forbes. | Tait. | Mitchell. | Forbes. | Tait. | Mitchell | |||
| Free. | Cooled. | Free. | Cooled. | |||||
| 0° | ·207 | ·231 | ·197 | ·178 | ·213 | ·238 | ·203 | ·184 |
| 100° | ·157 | ·198 | ·178 | ·190 | ·168 | ·212 | ·190 | ·197 |
| 200° | ·136 | ·176 | ·160 | ·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 thermo-couples (following Wiedemann) for mercury thermometers. This avoids the very uncertain correction for stem-exposure, but it is doubtful how far
