Conduction of heat through powders and its dependence …rspa.royalsocietypublishing.org/content/royprsa/113/764/459.full.pdfThe study of the thermal conductivity of powders in gases

Conduction of Heat through Powders and its Dependence on the Pressure and Conductivity of the Gaseous Phase.

By J. A b e r d e e n , M.Sc., and T. H. L a b y , M.A., Sc.D., F.Inst.P., Professor of Natural Philosophy in the University of Melbourne.

(Communicated by H. L. Callendar, F.R.S.—Received October 27, 1926.)

Introduction.The object of our investigation has been to study the conduction of heat

through a light powder and to find how it depends upon the pressure and thermal conductivity of the gas in which the powder is immersed. A solution of this question is part of the solution of the problem of the conduction of heat through a certain class of “ solid ” heat insulators—a class which includes those of lowest thermal conductivity. The class of insulator referred to are solids dispersed in gases, or gases dispersed in solids, and consists of three kinds of substances, (1) fibrous substances ( e.g., wool, eiderdown, asbestos), (2) cellular substances (e.g., cork, pumice stone) and (3) powders (e.g., lamp-black, powdered cork, silox or monox). It might be expected that substances so different as those mentioned would have very different thermal conductivities. Actually their conductivities range from about 8 to 11 times 10-5 cal. cm.-1 deg.-1 sec.-1. As there is nothing common to the solid part of these substances, their conductivities, it would seem probable, are determined mainly by the factor which is common to them all, that is, the gaseous part, which is air.

Our experiments have been made with a very light powder known as monox or silox, and the conductivity of this powder when immersed in air, in carbon dioxide, and in hydrogen at various pressures has been determined. We find that there is a linear relation between the conductivity of the powder and the logarithm of the pressure of the gas in which it is immersed, so that if k is the measure of the conductivity of the powder, that of the gas in which it is immersed, and jj the measure of the gas pressure, then

k = \kQ logio - napproximately, where n is a constant for a given gas.

We were not acquainted, when we began our experiments, with any investigation which showed how the thermal conductivity of powders depends upon the gas surrounding the powder, and our work had been in progress some time

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460 J. Aberdeen and T. H . Laby.

when we found that M. Smoluchowski* had made a very interesting study of the conduction of heat through powders and granular substances. He gives a theory for conduction through powders, and the relation k = ijc «0 log,, (1 + sp).

where zis a constant characteristic of the gas. Our experimental methods and results differ, however, in certain respects from those of Smoluchowski, as is explained later. Quite recently we have found that Sir James Dewar in 1898 made some qualitative experiments on the subject. Both of these papers appear to state the physical principle underlying the Stanley patent, which is mentioned later.

In the passage of heat between parallel walls at different temperatures separated by a gas, the heat is transferred by (1) convection, (2) radiation, and (3) conduction. We will consider briefly and in a general manner the effect, upon each of these processes of heat transfer, of introducing a finely divided powder between the parallel walls.

(1) The Convection currents in the gas will decrease (a) by reason of the subdivision of the gas into small volumes and the consequent decrease in the forces which set up the currents, and (b) by reason of the increased friction which accompanies the great increase in the area of the solid surfaces in contact with the gas. The combined effect of both factors will reduce the heat carried by convection to a small fraction of the whole heat transferred.

(2) Radiation.—Consider two infinite plane parallel walls at temperatures 02 and Oj. The radiation passing from the hotter to the colder is proportional to 024 — Oj4. The effect of introducing between the walls a thin lamina whose surfaces have the same emissive power as the surfaces it separates will be to reduce the heat transfer by radiation to one-half, for the following reasons :— Let the temperature which the lamina takes up be 0X, then the rates at which it absorbs and emits radiation are equal, that is, 024 — Qx4 = 0X4 — 0X4, and the radiation, which crosses from the hot to the cold wall, is reduced to (QJ — 0!4)/(024 — 0J4) of its value in the absence of the lamina, and this fraction

is equal to one-half .f I t is evident, then, that the introduction of any substance between the walls which is equivalent to introducing a number of such laminae will considerably reduce the amount of heat which is transferred by radiation.

(3) Conduction.—It is known that there are two types of gas conduction between parallel walls depending on the value of the ratio of the mean free

* M. M aryan Smoluchowski, c Bui. intern. Acad. Sciences Cracovie,’ vol. 5a, p. 129 (1910) and vol. 8a, p. 548 (1911). This author’s name is given in his earlier papers on the conduction of heat in gases as Smoluchowski R. von Smolan, and as de Smolan.

f Experim ental evidence supporting this is given by Dewar, 6 Proc. Roy. In s t.,’ vol. 15, p. 824 (1898).



Conduction of Heat through 461

path of the gas molecule to the distance between the walls (for which we write A/c?) and that when X/d is small, for a range of pressures, the conductivity of the gas is independent of the pressure ; but that when X/d is greater than unity, the conductivity of the gas is proportional to its pressure.* The value of X /dmay become greater than unity either because the pressure is diminished, which increases X, or because d is diminished. In a finely divided powder and in cellular and fibrous substances the mean distance between solid surfaces will become comparable in magnitude to the molecular mean free path of the gas filling the space between those surfaces. The effect of this is to make the conductivity of the gas proportional to its pressure at higher pressures than those which would be required in the absence of the powder.

In addition to the passage of heat through the gas there may be conduction through the solid itself. The small area of contact between the individual particles of a powder will considerably reduce the amount of heat transferred by conduction through the solid. In the case of cellular and fibrous substances the conduction will be along fine filaments or thin sheets. Such condition is very unfavourable to the passage of heat.

All the above considerations imply that powders, cellular and fibrous substances should be good insulators, and that by decreasing the pressure or the conductivity of the gas in which they are immersed the insulating power should be increased. That they are good insulatorsf is well known, but the reasons for a number of them possessing nearly the same conductivity appear not to have been stated except in so far as Smoluchowski’s theory does so. Except in this author’s paper and in one by Sir James Dewar,J we have not met with any reference in purely scientific literature to the improvement in thermal insulation of the substances in question which is obtained by a reduction of the

* K undt and Warburg, ‘ Pogg. Ann.,’ vol. 155, p. 337 (1878). Smoluchowski, ‘ Ann. d. Physik,’ vol. 64, p. 101 (1898), and loc. cit. Knudsen, ‘ Ann. d. Physik,’ vol. 34, p. 593 (1911). Soddy and Berry, ‘ Roy. Soc. Proc.,’ vol. 84, p. 576 (1911). Baule, ‘ Ann. d. Physik,’ vol. 44, p. 145 (1914). Langmuir, ‘ Phys. Rev.,’ vol. 8, p. 149 (1916), and ‘ J . Am. Chem. Soc.,’ vol. 37 (1915).

t The following are values of the therm al conductivities of the substances in question :—

Eiderdown, 11 -10-5 cal. cm.-1 deg.-1 sec.-1 (R ) ; wool, 11 • 10-5 a t 100° C. (R ) ; fine granulated cork, 8 - 4 . 10-6 to 11 • 10-5 (G ); cork slab, 11 • 10~5 (G) ; slag wool, 10-10- 5 (G). R = Randolph, ‘ Trans. Am. Electrochem. Soc.,’ vol. 21, p. 545 (1912). G = Griffiths,‘ Department of Scientific and Industrial Research, F irst Report on H eat Insulators ’ (1924). Other references are Lamb and Wilson, ‘ Roy. Soc. Proc.,’ A, vol. 65, p. 283 (1899). Nussclt,‘ Forsch. Ver. d. d. Ing.,’ vol. 53, p. 1750, and p. 1808 (1909).

% Dewar, loc. cit.

2 iVOL. CXIII.— A.



462 J. Aberdeen and T. H. Laby.

pressure of the gas in which they are immersed, or the substitution for air of a gas of lower conductivity, e.g., carbon dioxide.

The use of a powder in air at reduced pressure is applied, however, in the Stanley vacuum flask. In the patent specification* of that flask it is stated that the use of a powder in its walls enables high insulation to be obtained when the order of magnitude of the pressure of the residual gas is one hundred times as great as the gas pressure in a Dewar flask, and consequently when a powder is placed between the walls, a sufficiently high vacuum can be obtained in a metal flask. The relative efficiency of the insulation of the Dewar and Stanley flasks is considered later.

Principle of the Experiment.

Experimental arrangements for the study of the conductivity of the best thermal insulators have been devised by Wilson, Nusselt, Randolph, Griffiths, and others. References to the papers of these authors have been given. Griffiths has studied the conditions required for accuracy when the conductivity of coarsely divided substances is being studied, and has determined the conductivities of a number of the best insulators used technically.

The study of the thermal conductivity of powders in gases at low pressures presents two difficulties, namely, (1) the measurement of conductivities as small as one-tenth of that of air, and (2) the attainment, in an apparatus suited for such thermal measurements, of the necessary vacuum.

We have used two apparatus for the conductivity experiments. In both the powder was placed between co-axal cylinders (see fig. 1, p. 465). The outer cylinder was cooled with water at 0° C. and the inner cylinder was electrically heated. If the heat flow is radial, the quantity of heat which flows from the inner,to the outer cylinder, II cal. sec.-1, is given by H = 2 (02 — Oj)/log (R/r), where l cm. is the effective length of the inner cylinder, R jr the ratio of the radii of the cylinders, and (02 — 61)0 C. the difference of their temperatures, and k cal. cm.-1 sec.-1 deg.-1 is the conductivity of the medium between them. The heat flow will be radial if the temperature along each cylinder is uniform. To ensure this is the most important consideration in the design of the conductivity apparatus. A rapid flow of iced water over the external surface of the outer cylinder made its internal surface at a uniform temperature which would be a fraction of a degree above that of the external surface. Only a part of the length of the inner cylinder can be made uniform in temperature, as its ends must be in solid connection with the cold outer cylinder.

* U.S. patent, W. Stanley, September 2, 1913. No. 1.071,817.



Conduction of Heat through Poivders. 463

A consideration of the factors which determine the distribution of temperature along the inner cylinder shows (1) how that temperature may be made most uniform, and (2) that in the apparatus we have used there is a length of the inner cylinder at a sensibly uniform temperature.

Let S cm.2 be the effective* cross section of the inner cylinder, K cal. sec.-1 deg. 1 cm.-1 its effective conductivity, l cm. its length, R ohm cm.-1 the resistance per unit length of the manganine heating wire. When the temperature of an element of the cylinder (between the dotted lines in the figure given) of length

ex e o2i i ' !

t 1 * *x=0 sc+ x+8x x - l

§x cm. is steady, the heat which it gains by conduction and by electric heating is equal to the heat which it loses by conduction to the outer cylinder. That cylinder is at 0° C. The equation expressing this heat balance is

r72fiKS --- 8x +dxl

C2R S x_ 2rc&0 8xJ log (R

where k is the measure of the conductivity of the medium between the cylinders, 0 deg. the temperature at a distance of x cm. from one end, and C ampere the heating current.

This equation may be written<m

where A = 2tc&/KS log (R and B The solution of this equation is

- A0 = - B,

C2R/JKS.

0 = B/A + D e ^ Ee~ ^ . *.As the two ends of the inner cylinder are differently connected to the

outside cylinder, their temperatures will be different. When = 0, we put 0 = 0 ! and when x — l,0 = 02. To obtain D and E we have the equations 0i = B/A + D -j- E and 02 = B/A + Dc a . i _j_ ^ e-UK. q -which give

0 = B/A + (02 - B /A )r( '-^ A + (0! _

approximately, as e~ly/A is very small since l = 48, and A = 0 ’1 at least in the majority of the experiments.

* The inner cylinder is a thin-walled glass tube through which are threaded the heating wire and six other fine wires (see fig. 1).

2 I 2




If A (that is, 2rafc/KS log (R/r)) is large, the temperature of the inner cylinder, 6° C., increases with a; rapidly and asymptotically to the value B/A, that is, C2R log (R/r)l2nkJ, at which temperature it is over a large proportion of its length. As in these experiments k is sometimes very small, it is difficult to construct an apparatus in which for the smallest values of k, 2izk/'KS log (R/r) is large. To overcome this difficulty and to ensure that the middle part of the inner cylinder should be at a uniform temperature—

(1) The cylinder was made long (48 cm.)* compared to its diameter (1 cm.).(2) The temperature of the cylinder was adjusted in each experiment to be

about that of the air. As the glass, which connects the inner and outer cylinder is in contact with brass and vulcanite, which are in contact with the air (see fig. 1), the cooling of the inner by the outer cylinder is diminished. That is, (0X — B/A) is small.

(3) The other end of the inner cylinder was isolated, except for a thin copper wire connection to the outer cylinder. The two ends of this wire are at approximately the same temperature. That is, (02 — B/A) is small.

The distribution of temperature along the inner cylinder will be least uniform for the smallest value of A which arises in the experiments. Taking B/A as 20°, the observed temperature of the middle of the cylinder in some experiments, the equation for 0 is

0 = 20 + (02 — 20)e-('-*)VA + (0X — 20)e-^A

= 20 — 20 . e-(J-*)7£ — 20 . e-WA

if 0X = 02 = 0, which is the smallest value that 0X and 02 can possibly have. If k = 10~5, A = 0-12, and the temperature distribution is given by the following table :—

x.cm. 6 12 18 20 22 24 266° C. 17-55 19-70 19-96 19-98 19-99 19-99 19-99

Thus, even in the very unfavourable conditions assumed, the temperature does not vary by more than 0-02° in 20*00° over a length of 8 cm. of the inner cylinder. By placing two thermometers near the middle of the inner cylinder of the first apparatus, this was experimentally verified. If the medium separating the cylinders is air or a better conductor, a still longer length of the inner cylinder is at a sensibly uniform temperature.

As the heat flow is radial over the middle portion of the cylinders, the expression H = 2nkl(Q2 — 0i)/ log (R jr) is applicable. The heat was measured by observing the current, C ampere, which flowed through the manganine heating wire, and

* The dimensions in brackets are those of the second apparatus.



Conduction of Heat through Powders. 465

the potential drop, E volt, over a length, l cm., of the middle part of that wire. Writing S' for 2ttZ/log (R jr),“ the space factor”, the expression for is k = EC/JS'(02— Oj) =0-002154 EC/(02- Bj),

since J = 4-182 watt. sec. calorie-1, and S' = 27t. 20/log, (31/10) = 111.

It was not possible to measure R and r very accurately. It is important to notice that even if the wrong value of S' is adopted, that does not make the relative conductivities which we obtained incorrect, and it is the relative conductivities which are important, for our object was to find for a given density of packing of the powder how its conductivity varies with the density and conductivity of the gas surrounding it. The absolute conductivity of the whole may depend onIother factors too.

The value given above for the space factor was checked, however, by measuring the conductivity of air and carbon dioxide, as is described later.

The second apparatus which we have used is shown in figs. 1 and 2.

It consists of co-axal glass tubes A and B.48 cm. long, connected at C, but the outer cylinder can be opened at its other end.The tube B is 1 cm. external diameter, and A is 3 • 1 cm. internal diameter. A manganine wire (No. 28 s.w.g.) is stretched along the axis of the tube B, one end being connected to a platinum wire, which passes through the closed end of a glass tube B, the other end being connected to a device E which stretches the wire. The platinum wire is connected to the cap L through a flexible wire J and a platinum seal. Two copper potential leads are attached to the manganine wire at points 20 cm. apart. The platinum thermometer coil P is wound on the surface of the inner glass tube, through which it is sealed. The resistance of the leads to this thermometer are compensated by exactly similar leads at Q. The four terminals

Apparatus. (Not to scale.) The glass tube A is 3-1 cm. internal diameter, 48 cm. long ; the tube B, 1 cm. diameter.



466 J. A berdeen and T. H . Laby.

of the platinum thermometer and three of the terminals of the heating wire are soldered to copper posts T supported by a vulcanite block, which is attached

Ice box

B a ro m eter

F ig . 2— (Not to scale).

to the glass by a short brass tube (of thinner tubing than shown in the drawing). This tube, being in the air, protects the inner cylinder from being cooled by the outer cylinder, so the tube acts as a temperature shield.

The outer tube A is cooled by a water jacket H. Water is circulated by a motor driven pump (see fig. 2) over ice contained in an ice-box, then through rubber tubing to M past the thermometer 0, through the jacket H, past a second thermometer R,, back to the ice-box. To insulate this wrater circulation thermally, the conduction apparatus is enclosed in a wooden box (see fig. 2) filled with kapok. A window enables the lower thermometer to be read. The readings of the two thermometers showed that the tube A was maintained at a steady and uniform temperature less than one-tenth of a degree above 0° C.

The arrangements for obtaining a vacuum and introducing the gas are shown in fig. 2. The conductivity apparatus is connected by ground-glass joints with mercury seals to a phosphorus pentoxide drying bulb, a barometer tube, a



Conduction of Heat through 467

coeoanut charcoal tube, a “ hyvac ” motor-driven oil pump, and finally to a glass bulb in which the gas to be used is stored.

Measurements.

The platinum thermometer, which measured the temperature 02° C., had a fundamental interval of 1-48 ohms. Its resistance at 0° C., ohm, and at 100° C., fioo ohm, were, found by filling the conductivity apparatus with hydrogen* and circulating ice-water or steam in the jacket surrounding the outer cylinder. To measure the resistance of the thermometer, a Callendar- Griffiths’ bridge was used in the earlier experiments, and a Muller bridgef in the later ones. Callendar’s equations t 100 (rt — r0)/(>'ioo — »o)> and (3 — £ = $0(6 — 100) 10 were used to calculate 0, where rt ohm is the resistance of the thermometer at the platinum temperature t°, or at 0° C. on the hydrogen scale. $ was determined for a piece of the wire which was used in the thermometer.

The power, EC watt, supplied to the heating wire was measured by pointer instruments.

As previously mentioned, the value of the space factor, 111 cm., was checked by measuring the conductivity of the air and carbon dioxide. With each of these gases at a pressure of about 50 mm. of mercury—a pressure at which convection currents are found to be inappreciable—the power to maintain a temperature difference of about 18° C. was found. In another experiment with the apparatus evacuated with charcoal and liquid air and the mercury manometer sealed off, the power for about the same temperature difference was again measured. In this case the heat is principally carried by radiation : a little may be carried by gas conduction. The difference of these powers per degree gives the conductivity. For example, in three pairs of experiments with air the mean readings were—conduction and radiation = 0-05126 watt per degree, radiation = 0-02456, giving for the conductivity of air at 10° C.

&io = 0-002154 (0-05126 — 0-02456) = 5-75 X 10-5.

The three experiments with carbon dioxide gave = 3 • 3 X 10 5 as a mean value at 9° C.

Assuming for air a temperature coefficient of 0-003 deg.-1, 5-58 X 10-5.

* Used on account of its high conductivity, t The bridge used had five decades of 0-001 to 10 ohms.



4G8 J. A berdeen and T. H . Laby.

This is within the range of values of other observers who have made absolute determinations, as is shown by the following values of k0:—

Weighted mean* of 14 determinations 5-22 X 10~5.Hercus and Labyf 5-40 X 10-5.Gregory and Archer J 5*83 x I0-5.

The last observers used a hot-wire method which they have devised to eliminate convection currents and end corrections.

The value we find for C02 also agrees with that found by previous workers, and so the space factor we used is confirmed.

Details of an Experiment.

With the stopper G of fig. 1 removed, the apparatus is filled with the powder. The stopper is reinserted and sealed with sealing wax. The powder is heated to as high a temperature as the presence of this wax allows with the oil pump in operation. The apparatus is left evacuated for 24 hours, the tap to the pump being turned off so that the phosphorus pentoxide may absorb the water vapour. A dry gas (air, hydrogen, or carbon dioxide) is admitted till the pressure is atmospheric.

The conductivity of the powrder with the gas at this pressure is then found by setting the ice-water in circulation, and adjusting the heating current until the temperature of the inner cylinder, 02° C., is about that of the air. When this temperature is quite steady, it is measured, as is the power, E.C. watt, spent in the middle 20 cm. of the heating wire. The gas pressure is reduced, the heating current readjusted until the temperature, 02° C., is again steady at its former value. A smaller value of 02 was used with hydrogen than for the other gases. The following are examples of observations :—

Date. Gas. Press.,mm.

Current,amp.

P.D.,volt.

Power,watt.

r**tohm.

0.20 c. t X 105

6.2.25 Air 757 1-72 0-558 0-960 4-140 21-66 9-556.2.25 8 1 1 2 0-368 0-412 4-145 21-98 4-046.2.25 9 9 2 cm.* sp. 0-526 0-176 0-0925 4-143 21-84 0-91

21.2.25 Hydrogen 7 60 2-34 0-75 1 • 75 3-956 9-34 40-5

* Alternative length of an induction coil spark connected to electrodes in the gas surrounding the powder.

f re. ohm is the resistance of the platinum thermometer at the temperature 0 >° C.

* T. H. Laby and Miss E. A. Nelson, ‘ In ternational Critical Tables.’ t 1 Roy. Soc. Proc.,’ A, vol. 95, p. 206 (1918).%‘ Roy. Soc. Proc.,’ A, vol. 110, p. 117 (1926).



Conduction of Heat through Po wders. 469

The Powder and Gases used.The powder we have used is known as “ monox ” or £' silox.” It is an

extremely fine light brown powder. It is probably a mixture of silicon, silicon monoxide, and silica. Potter gives 2-24 gm. cm.-3 as the average density of the particles of the powder. The powder if poured in air into a vessel is very light (density 0-04 gm. cm.-3), and is about 1/56 of the density of the grains ■constituting the powder.*

The powder occludes gases, especially at low temperatures. We were not able to heat it strongly in the conductivity apparatus, but it was strongly heated before being put in and then dried in a vacuum as described.

The gases used in addition to air were hydrogen and carbon dioxide as supplied in steel cylinders for industrial purposes.

The conductivities which we have found for the powder when immersed in hydrogen, air, or carbon dioxide at various pressures are given in the following tables:—

Thermal Conductivity of Silox Powder immersed in Various Gases.(Unit of conductivity : cal. cm.-1 sec.-1 deg.-1.)

Second Apparatus.

Experiment. Pressure. log p- Conductivity.

Air.2.2.25 ............................ |

mm.760 2-88 9-76 X ID-3

6.2.25 757 2-87 9-55 X IQ-52.2.25 ............................1 385 2-59 8-37 x to -5<>.2.25 ............................! 137 2 -14 7-64 X 10- 52.2.25 ............................ 83 1-91 6-66 X Id-3<5.2.25 ............................1 41 1-61 6 07 X 1C-52.2.25 ............................j 10 1-00 3-52 X 10-56.2.25 ............................ 8 0-90 4-04 X 10 56.2.25 ............................ 2 0-30 2-54 X 10-56.2.25 ............................ 0-5 - 0 - 3 0 1-85 X 10-5

c o 2.17.2.25 ............................ 712 2-85 6-70 X 10-518.2.25 ............................ 683 2-83 6-75 X 10-514.2.25 ............................ 669 2-83 6-80 X 10-520.2.25 ............................ 132 2-12 5-68 X lO-517.2.25 121 2-08 5 57X lO-514.2.25 ............................ 119 2 07 5-73 X HI-517.2.25 ............................ 55 1-74 4-90 X 10-314.2.25 ............................ 51 -5 1 - 71 4-89 X lO-514.2.25 15-5 1 • 19 3-73 X lO-517.2.25 14 1 1 5 3-68 X lO-520.2.25 8 0-90 3-29 x lO-518.2.25 ............................ 5 0-70 3 02 X lO-514.2.25 ............................ 4-5 0-65 2-62 X lO-5

* All these data are from Potter. ‘ Trans. Am. Electroch. Soc.,’ vol. 12. p. 215(1907). See also later volumes of this journal.



470 J. A berdeen and T. H . Laby.

Thermal Conductivity of Si)ox Powder immersed in Various Gases—(continued). (Unit of conductivity : cal. cm.-1 sec.-1 deg.-1.)

Second App

Experiment. Pressure. Log Conductivity.

Hydroge21.2.25 ' .... ................

mm.760 2-88 40-5 x 10-5

23.2.25 ............................. 755 2-88 38-0 x 10"521.2.25 ............................. 750 2-87 37-7 x 10~523.2.25 ............................. 136 2 1 3 29-7 x lO"521.2.25 ............................. 129 2 1 1 29-3 X H r 521.2.25 ............................. 41 1-61 20-1 x 10-521.2.25 ............................. 40 1-60 20-5 X lO -523.2.25 ............................. 39 1-59 20-5 X 10-521.2.25 ............................. 5 0-70 8-27 X H r 523.2.25 ............................. 3-2 0-50 7-52 X H r 5

Low pressures after air.3.1.25 ............................. over 4 cm. spk.* 0-71 X lO -53.1.25 ............................. over 4 cm. spk. 0-72 x U r 56.2.25 ............................. 2 cm. spk. 0-91 X 10~5

* This is the alternative spark length of an induction coil connected to electrodes in the gas; surrounding the powder. The vacuum was obtained by immersing the charcoal tube (see fig. 2) in liquid air. p in these experim ents would be less than 10~3.

The following values for the conductivity were obtained with the first apparatus. While the thermal measurements obtained with it are probably as reliable as those of the second, it was difficult to keep vacuum tight and to dry the gas in it as well as in the second apparatus.

Conductivities obtained with First Apparatus.

Experiment.|

Pressure. |1

Log p. Conductivity.

A ir. mm.23.4.24 ........................... 765 2-88 9-73 X 10 "523.4.24 348 2-54 9-02 x u r 523.4.24 ............... 31 1-49 5-66 X 10-523.4.24 7 0-85 3-50 X 10-&

c o 211.1.24 ........................... 766 2-88 7-57 X 10-524.4.25 ............... 755 2-88 7-13 X 10~524.4.24 ............... 540 2-73 6-90 X 10”524.4.25 ............... 305 2-48 6-54 X 10-s24.4.25 150 2-18 6-04 X 10-5

3.4.24 ............... 45 1-65 5-02 X 10 "5

Hv3.4.24 ........................... 760 2-88 47-4 X 10~53.4.25 ............... 45 1-65 21-4 X 10~5




As the packing of the powder will be different in the two apparatus, exactly the same conductivities at a given pressure are not to be expected. While the values for air at one atmosphere agree, the values for C02 and H2 at 760 mm. are higher for the first apparatus. Probably the main difficulty is to get the gas pure (in particular free from moisture) when in the interstices of the powder.

IQxlO5

8

6

4

2 k

~\ lo9l0-p 0 T 2 3

F ig . 3.—Silox immersed in Different Gases.

In fig. 3 k is plotted against log10 p as found with the second apparatus. The points for the powder immersed in a given gas fall on a straight line for a range of pressures of about 3 to 760 mm. The deviation of the points from the straight line is not greater than the differences in A; at a given pressure. In the case of air three observations were taken at pressures less than 2 mm. These fall on the dotted line, and not on the straight line, which represents the observations above 2 mm. In the figure a heavy line is drawn over the range for which a straight line fits the observations. Further experiments are required, at pressures less than a millimetre, with carbon dioxide and hydrogen. Apparently at such pressures the conductivity is higher than the straight line law would give.

In the first table of conductivities it will be seen that for the highest vacuum we were able to attain the conductivity was found to be 0-71 and 0-72 X 10 “5 cal. cm.-1 sec.-1 deg.-1. We will take this as the conductivity of the powder itself, and subtract it from the observed conductivities in each gas, in order to



472 J. A berdeen and T. H. Laby.

find the effect of the presence of a gas alone. When this is done, the equations of the straight lines are—Hydrogen ... 10% = 13*7 logio p + 0-06 = 13*7 logio

Air ..............10% = 2-79 log10 p — 0-36 = 2-79 logi01 * u O

Carbon dioxide 10% = 2*06 logi0 p — 0-34 = 2-06 logio

The slopes of these straight lines are approximately proportional to the conductivities of the gas, as the following table of conductivities and numbers proportional to the slopes shows :—

Gas. Hydrogen. Air. Carbon dioxide.

*0 33-75/9° 5-77/10° 3-29/9°Ratio of slopes . . . . 28-29 5-77 4-26

Fig. 4.—Silox immersed in Hydrogen.




The values given for ktare those found with the second apparatus using the same supplies of hydrogen and carbon dioxide as was used with the powders. The agreement may not appear very close, but it is to be remembered that in the interstices of the powder it is difficult to prevent the hydrogen from being mixed with some air and water vapour which lower the conductivity of the hydrogen. Similar contamination of the carbon dioxide raises its conductivity. There seems little doubt, then, that the conductivity of the silox is determined by the gas in which it is immersed, a fact not only of theoretical but of technical interest.

The equations given above can be written with sufficient accuracy in the form

k = p 0 log10

where k0 is the measure of the conductivity of the gas in which the powder is immersed, and n is a constant for each gas. It is to be remembered that in this expression k is the measure of the conductivity of the powder minus its value in a vacuum.

Smoluchowski’s R e s u l t s .Before considering our results further it will be convenient to state briefly

those obtained by Smoluchowski.* He has determined the conductivities of several powders in air, and some of them in carbon dioxide and hydrogen too. He used (1) granular powders of iron, zinc, copper oxide, and quartz, and (2) spongy ones of lamp-black, cork, diatomaceous earth, and lycopodium.

Smoluchowski’s apparatus consisted of concentric cylinders, the innerf being the bulb of a mercury thermometer whose rate of cooling was observed with the outer in ice. This method is subject to two defects : (1) When the conductivity to be determined is small, the cooling along the thermometer stem is a large part of the whole loss of heat, (2) the cooling method requires the specific heat and density of the medium under Investigation to be known, except when the conductivity is small. Thus the method is only available to determine relative conductivities of substances of constant specific heat and density whose conductivities are not very small nor very large. In spite of these restrictions, Smoluchowski was able to use the cooling method, as in finding the effect of varying the gas pressure and conductivity the specific heat

* Loc. cit.t His inner cylinder was 10 mm. in diameter, and 42 mm. lo n g ; ours was 10 mm. in

diameter and 480 mm. long. The need for this greater length with poor conductors is set out earlier in this paper.




and density of the powder do not change appreciably. Nevertheless, the defects of his method have to be kept in mind in considering his results.

Smoluchowski* gives a theory of conduction in a powder. In the outline of it given below steps are omitted which will be found in his two papers on the subject. Assuming a powder made up of perfect conducting spheres of radius a cm., in contact and arranged in cubical order, he calculates the flow of heat through the gas separating the spheres. The heat, H calorie, conducted per second between two spheres at a difference of temperature A0° C. is, assuming Fourier’s law H = — ko$ dQ/dx,

H = *oAelJ 2a (1 — cos </>'<f>)

= — cnzk0AQ log, (1 — cos </>),

where k 0.is the measure of the conductivity of the gas. This would mean that an infinite quantity of heat would be conducted near the point of contact (</> = 0). To meet this difficulty Smoluchowski introduces an important assumption by means of which he had accounted for observations he had made on the conduction in gas at low pressures.

We know, he states, that at the surface of a solid in a gas there exists a discontinuity of temperature, whose influence can perhaps be evaluated by imagining each surface to be displaced, 8 = (3 A (cm.), where X cm. is the mean free path of the gas molecule, and (3 depends on the properties of the gas and is nearly unity for air. This reasoning is only correct for plane surfaces. Taking now the shortest distance between the spheres as 28 cm., the relation given above becomes

H = *0 A 6 p nJ o 28 + 2 (1 — cos</>)

rW27t ko a A0 — d cos <f)

J o S/a + 1 — cos </>"

= — k0 ci A6 log, (1 + a/8).The expression is only correct for small values of </>; nevertheless, taking as the limit of the integral is permissible if 8/a is small, otherwise the relation is a rough approximation. It is to be expected, then, that it will fail at low gas pressures when X, and therefore 8, increases and becomes larger than a.

For a cubical arrangement of the spheres A0 = 2a ddjdx, and the conductivity of the powder so far as it is due to the gas is given by

k = \nkolog, (1 + a/8) = | tt&o log, (1 + ap/80po), where 80p0 = p8.

* ‘ Phil. Mag.,’ vol. 46, p. 192 (1898); ‘ Wied. Ann.,’ vol. 64, p. 101 (1898); ‘ Wien. Akad. S itzb.,’ I I a, vol. 107, p. 304 (1898), and vol. 108, p. 7 (1899).




In a second paper Smoluchowski discusses the conduction through a powder in a vacuum. We have seen earlier in this paper that the introduction of a lamina between parallel walls halves the heat transferred by radiation. The introduction of spheres of diameter 2 a cm. between walls d cm. apart will, on certain assumptions, reduce the measure of the radiation from R to R2a/d, and the latter will be very small. He also show’s that if the spheres are assumed to make perfect thermal contact over an area, which he calculates by Hertz’s theory, the resulting calculated conductivity is 1000 times greater than he found with zinc powder prepared by distillation, which consisted of spheres of fairly uniform size. He concluded that good thermal contact does not exist between the grains. He found by experiment (but his apparatus did not enable him to measure it correctly), as w’e have found, some conduction through powders in a vacuum, but the conduction through the gas in the interstices of the powder given by the expression k = k0 log* (1 + a/8) on his theory is the important part of the conduction at pressures of about one atmosphere.

The important result of his experimental work is the verification of this relation, and therefore of the assumption made in deriving it, that there is a discontinuity of temperature at the surface of separation of a gas and a solid when heat is passing from one to the other. For example, he measured the conductivity of a zinc powder of approximately constant grain diameter (0-028 mm.) in air, in C02, and in H2, at pressures from 22 to 760 mm., and roughly verified the relation k = ck() log (1 + a/8) in respect to the variation of k with A'o and with p. He found 8 = 1 - 2 X for air and C02, and 8 = 2X for PI2, where X cm. = mean free path of the gas molecule.

The relation we have found to fit our observations is ^k0 loge p /n ; it therefore approximates to his, which may be written k — A logt, (1 -f- e if zp is large compared with 1 ; but he finds for granular powders, as we do forsilox, that his formula holds for pressures as small as or smaller than 1 mm. While our observations in this sense agree with Smoluchowski’s theory, the density of the powder we have used is too small for the grains of which it is constituted to be in contact, as is assumed in his theory.

Smoluchowski determined the conductivity of light powrders (lamp-black, diatomaceous earth, cork, and lycopodium), but his results in no way agree with those we have obtained for silox. His conductivities are much smaller than ours and his values of k when plotted against log p give an J- shaped curve, not the straight line which we get. With the short inner cylinder (a thermometer bulb 42 mm. long) of his apparatus the loss of heat along the thermometer stem would introduce an error for which his correction may be too large.



47(5 J. Aberdeen and T. H . Laby.

A Powder in Vacuum as Insulation for Liquid Oxygen Vessels.A powder such as silox appears to possess advantages for insulating liquid

oxygen vessels. Means for storing and carrying that liquid are of importance in mine rescue work, in aviation, and for medical purposes. The insulation of such vessels is considered in the Report of the Oxygen Research Committee,* which deals in detail with the physics of vacuum vessels and the construction of double-walled metal vessels. The report, however, does not refer to the insulating power of powders in a vacuum, to Dewar’s, Smoluchowski’s, and to Knudsen’s work on the conduction in gases at low pressures, or to the Stanley flask. Large Dewar vessels of glass not being practicable for the purposes referred to above, metal ones have been substituted, the inner metal surfaces of the wall being polished. The vacuum in these vessels slowly deteriorates, and charcoal is used to re-establish the vacuum when the flask contains liquid oxygen. If the space between the walls were filled with a light powder such as silox or absorbent silica, this powder itself would play the part of the charcoal, give effective insulation, and make the polishing of the metal unnecessary. The heat insulation of such vessels is compared, in the following table, with that of Dewar vessels tested by the Oxygen Research Committee, which in its report states the amount of liquid oxygen which evaporates from metal and glass flasks.

Rate of Heat Inflow per Unit Area into Liquid Oxygen Vessels.

Vessel. Capacity. Wallthickness.

Heat inflow per unit area (temp, diff., 290° C.).

litre. cm.Dewar of glass .... _ 1 — 4 to 8 X 10-4 cal. sec.-1 cm.-2.

99 ' 99 ----- 2 — - 1'8 X 10-4 cal. sec.-1 cm.-2.,, metal 3 — 3 to 6 X 10-4 cal. sec.-1 cm.-2.9 9 9 9 ----- 5 — *3 X 10-4 cal. sec.-1 cm.-2.

Silox powder in a vacuumC " —12-5

14 X 10-4 cal. sec.-1 cm.-2. 5-6 X 10-4 cal. sec.-1 cm.-2.

* Corrected for heat loss up neck of flask.

The heat inflow in the above table is expressed in cal. sec.-1 per cm.2 of the internal wall area when the difference of temperature between the inside and outside of the flask is 200° C. The conductivity of silox powder in a vacuum assumed is 0-7 X 10-5 cal. cm.-1 sec.-1 deg.-1, which we have observed at 10° C. I t is probable that the conductivity would be less with the powder

* D epartm ent of Scientific and Industrial Research, H.M.S.O. (1924).



Conduction of Heat through Powders4 77

at a mean temperature of — 90 0 C., as the conducting power of the residual gas will decrease with temperature. It is to be expected, then, that a vessel with a wall of 2-5 cm. of silox powder in a vacuum will give as good insulation as some metal Dewar flasks and one with a thicker wall than 2-5 cm. better insulation than most metal flasks. The accidental admission of air to the vacuum would not reduce the insulation of the former as much as it would for a Dewar vessel.

Summary.1. The thermal conductivity of a very light powder silox in air, in C02, and in

112, at pressures from 1 to 760 mm., can be expressed by = -|&o l°gio pfn> where k0 is the measure of the conductivity, and p of the pressure of the gas in which the powder is immersed, and n a constant for each gas.

2. An outline of Smoluchowski’s theory of conduction in powders, which leadsto the relation k = cf0 logc (1 + e p),is given.

3. Our observations on a light powder are not in agreement with those of Smoluchowski.

4. The conductivity of silox in the best vacuum we were able to attain is 0-7 X 10-5 cal. cm."1 sec.-1 deg.-1.

5. An apparatus capable of measuring thermal conductivities of powders in gases as small as this has been designed.

6. The value of a light powder in a vacuum for insulating liquid oxygen vessels is pointed out.

We are indebted to Prof. J. H. Michell, F.R.S., for having elucidated a difficulty we had in understanding a part of Smoluchowski’s theory of conduction in powders. A grant for apparatus was received from the Victorian Government Research Fund, Melbourne, September, 1926.

2 KVOL. CXIII.— A.



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