66

The values of the ratio of "pi-po" to "W" are given in a separate column, and may be compared with the values of the ratio for the same dissipation of watts with discharge operating alone. The latter have been calculated from a smooth graph constructed from the values in Table I.

The agreement between corresponding numbers in the two columns is striking. In fact such a close agreement was somewhat unexpected, since the strength of the wind is not the same in the two cases which are compared. Such differences as exist between the observed and calculated values are in general in the direction which neglect of wind variation would cause. These results show conclusively that the pressure effect is purely a heating effect.

Table II. (c) shows values obtained for the rise in pressure produced by heat alone. It will be seen that owing to the absence of the electric wind the values of the ratio are much higher.

This is also shown in curves A, B, and C, which are typical of a number which have been obtained. Curve A is the pressure time curve when 0.091 watt was supplied by

discharge alone. Curve B, that for combined supply of 0.094 by discharge and 0.091 by a heating current in the wire. Curve C, up to the point D, for 0.094 watt supplied by current alone.

At the time marked by the point D, in curve C, a discharge current supplying 094 watt was switched on. The cooling effect of the wind is well shown by the drop in pressure, the final value of which was practically that which would have been reached after combined action from zero time.

Lastly, mention must be made of an experiment by Warner, by which he claims to have shown that instead of a heating effect at the instant of starting of the discharge from an unheated wire, the gas near the wire actually cools. This he deduced from the movement of the spot of a galvanometer connected to a thermo-junction, placed in the tube at a distance of about 4 mm. from the axial wire.

The authors repeated this experiment, and found that the deflexion observed is not due to a thermo-electric current, but to an electrostatic effect when the thermo-junction takes up the potential of the air. Thus this deflexion is also observed when the galvanometer is not included in the circuit containing the thermo-couple, but is merely in electrical contact with it. The cooling of a red-hot wire which Warner quotes as occurring when discharge starts from it, is undoubtedly caused by the electric wind which is thus set up.

Summary.

When glow discharge starts between a cylinder and an axial wire, in a closed tube at atmospheric pressure, a sudden rise of pressure is observed. It has been argued by some that this cannot be due to heat generated in the discharge, and an alternative theory, that it is due to ionization, has been advanced.

In the above paper the authors criticise the argument for the ionization theory, and the interpretation of the experimental results upon which it is based. They also verify quantitatively that the effect is purely thermal in origin.

Physics Department,

University of Bristol.

Jan. 7, 1918.

Phil. Mag. S. 6. Vol. 35. No. 207. March 1918.

XXX. Some Problems of Evaporation. By HAROLD JEFFREYS, M.A., D.Sc., Fellow of St. John's College, Cambridge.

HE problem of evaporation is practically one of gaseous

the density of the air at any point that is due to water vapour, V will vary from time to time and place to place according to the equation

where k is the effective coefficient of diffusion, t the time, and d/dt denotes the total differential following a particle of the fluid. In general V, as above defined, will be referred to as the concentration.

Then if u, v, w be the components of velocity and V be supposed expressed as a function of x, y, z, and t,

dV av av ат av

+u +v

dt де dx ду

(2)

The boundary conditions are that the air in contact with a liquid surface is saturated, so that V is there equal to the concentration in saturated air at the temperature of the liquid, and that at a great distance from any liquid V tends to a finite value.

The equation (1) is identical in form with those that determine the transference of heat and momentum; and when the transference is due entirely to turbulence the quantity k has the same value in all three cases †. In contact with a solid or liquid surface, on the other hand, the velocity of the medium is zero, and k diminishes to the value it has when there is no turbulence: that is to say, in the evaporation problem k is equal to the coefficient of diffusion of the vapour; in the thermal problem it is the thermometric conductivity of air; and in the equations of motion it is the kinematic viscosity of air. These three quantities are of the same order of magnitude, but are not equal. When the air is at rest k is a constant in all cases and the problem is simply that of solving the equation

where denotes the Laplacian operator.

* Communicated by the Author.

+ G. I. Taylor, "Eddy Motion in the Atmosphere," Phil. Trans.

215 A. (1915).

When the air is in motion, k at a distance from a boundary is almost wholly due to turbulence, and is practically independent of position. The velocity near a boundary rapidly increases from zero to about half its amount at a considerable distance; this transition is accomplished in a thin layer of shearing, whose thickness in centimetres is estimated at 40/U, where U is the velocity in centimetres per second at a considerable distance *. In outdoor problems it is therefore usually of the order of a millimetre or smaller. The concentration and temperature also change rapidly within this layer, and in most cases it will be justifiable to assume that at the outer boundary of it each is constant and midway between the values at the surface and at a great distance. Outside this layer transference of heat and vapour will take place according to equation (3), where k is now put equal to the eddy viscosity.

If the dimensions of the liquid surface are of order 1, and the time needed for any considerable change of condition over it is of order 7, we see that the terms like ǝV/at, uə Vəx, and kə2V/Əx2 are relatively of orders

1/T, U/l, and k/l2.

Taking k to be of the order of 103 cm./sec., which is a somewhat small estimate for outdoor problems, T=1 second, and U=400 cm./sec., we see that the first term is small compared with the last provided is less than 10 cm., and for slower variations this term will be small for still larger values of l. Again, we see that the second term is small compared with the last, provided l is less than 2 cm. ; otherwise it must be taken into account. In indoor problems U is probably not far from zero, k=0·24 cm.2/sec., and the whole of d/dt can therefore be neglected for a surface 1 cm. across provided the saturation near it does not change considerably in 8 seconds.

When is sufficiently small to satisfy both these conditions the equation of transference reduces to V2V=0, subject to the same boundary conditions as before. Without loss of generality we can take the concentration at a great distance to be zero. For if it is actually Va, V-Va will satisfy the same differential equation and will be constant over all the wetted surfaces, and hence if a problem is solved for Va=0 the solution for any other value of Va can at once be found by merely writing V-Va for V throughout.

Now, V2V=0 is the equation of steady diffusion and

* Private communication from Major G. I. Taylor.

is also satisfied by the potential in electrostatic problems; it follows that, provided the initial change is not too abrupt and the dimensions are not too great, the value of V at any point is the same as the electric potential at that point when the wetted surface is regarded as a conductor charged to potential Vo, where V, is the concentration at the edge of the layer of shearing when the air is moving, and the saturation concentration at the boundary when it is at rest. The charge needed for this is CVo, where C is the electrostatic capacity of the conductor. Now the rate of transference outwards from the boundary is

where av denotes the element of the outward normal, p is the density of air, and the integral is taken over the boundary.

But in the electrostatic problem, if a denotes the density of the electric charge on the surface,

This determines the rate of evaporation, which is shown, other things being equal, to be proportional to the linear dimensions, since the electrostatic capacity varies in this way for bodies of the same shape.

The above is practically Stefan's* solution, which has been experimentally proved to be correct subject to the conditions stated †. When the velocity is large enough to need to be taken into account, a general solution is no longer possible, but for a large wetted surface the curvature may be neglected, and the problem reduces to that of a wind blowing over a flat surface. This is treated in the next

section.

* Wien. Akad. Ber. lxxxiii. Abteil 2, p. 613 (1881).

† H. T. Brown & F. Escombe, Phil. Trans. 193 B. pp. 223–291.