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Or since, Art. 72, p = ^- , ^- K 2 a? ; and Art. 98, X = \f t [-] + A*,/ ( - ), we

2 M \s/ \s/

have, remembering that M is constant,

/

V

fl VTT ( . c . /c\ , /c\) 1

= - -r c M ~ + mX i/ ~ M- m ^2/2 1 - If ~

2 I W \sj) p

(99)

in which

A depends only on the shape of the tube and is ^ for a flat tube,

/C\ 60

f I - } is of the order - when - is zero and is finite when - is infinite,

' W s s s

fi(-) and/s (-) are unity when - is zero and zero when - is infinite,

\s / \s/ s s

352 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

and

/, ( - | and / 4 (- ) are zero when - is zero and unity when - is infinite; all

J \s) \s) s s

the functions varying continuously between the limits here ascribed.

Also \! depends on the nature of the surface but not upon the nature of

the gas, while A, and X 3 may depend both upon the gas and the surface.

Putting

T /c\ VTT f . /c >

1 (~) = ~9~ '~

* ' I ..

and

(100)

we have for the general form of the equation of transpiration

P

SECTION X. VERIFICATION OF THE GENERAL EQUATION OF

TRANSPIRATION.

103. In this section the general equation obtained in Section IX. is

applied to the particular cases of transpiration which have been the subject

of experiments. It will thus appear how I was led to infer the results, and

thence to make the experiments.

A summary of the experimental results has already been given in Art. 9,

but for the immediate purposes the results may be stated as follows :

Experimental results.

I. The law of corresponding results at corresponding densities, shown

by the fitting of the logarithmic homologues. (See Arts. 28 and 40.)

II. The gradual manner in which the results varied as the density

increased, shown by the continuous curvature of the curves which express

these results. (See Figures 8, 9, and 12.)

III. The uniformity in direction in which both the time of transpiration

under pressure (see Tables XIV. to XVII.) and the ratio of the thermal

differences of pressure to the mean pressure (see Tables III. to XIII.) vary as

the density increased.

33] IN THE GASEOUS STATE. 353

IV. The fact, sufficiently proved by Graham, that, cceteris paribus, the

times of transpiration are proportional to the ratio of the differences of

pressure to the mean pressure, the difference of pressure being small.

V. The fact, to a certain extent taken for granted, that the ratio of the

thermal differences of pressure to the mean pressure is, cceteris paribus,

proportional to the ratio of the difference of temperature to the absolute

temperature, this ratio being small.

VI. The continual approximation towards constancy of the time of

transpiration under pressure as the density diminished. (See Tables XIV.

to XVII., and Figure 12, page 298.)

VII. The relation between the ultimate values of the times of transpi-

ration for different gases (air and hydrogen) for small densities; the times

are proportional to the square roots of their atomic weights. (See Art. 42.)

VIII. The fact that the times of transpiration for the same gas in

capillary tubes, and at considerable densities, are inversely as the density

and independent of the temperature. Maxwell* and Graham f.

IX. The difference in the variation of the times of transpiration for

different gases, shown by the fact that the logarithmic curves for hydrogen

cannot be made to fit those for air. (Figures 8, 9 and 12.)

X. The approximation towards a constant value of the ratio which the

thermal differences of pressure bear to the mean pressure as the density

diminishes, whatever be the gas or plate ; the ratio is that of the difference

of the square roots of the absolute temperatures to the square root of the

absolute temperature.

XI. The approximation, as the density increases, to a linear relation

between the thermal differences of pressure and the reciprocal of the density.

XII. The difference between the law of variation of the thermal

differences of pressure for different gases, as shown by the non-agreement

of the logarithmic homologues for air and hydrogen. (Figures 8 and 9.)

XIII. The transpiration of a varying mixture of gases through a porous

plate. Investigated by Graham.

104. In order to bring out the agreement of the experimental results

with those deduced from the equation, we put

V

^r for the time of transpiration,

_ _

b - for the difference of pressure on the two sides of the plate,

dx

b for the difference of temperature on the two sides of the plate.

dx

* Phil. Trans., 1866, pp. 249268, also note to Art. 94. t lb., 1849, pp. 349362.

O. K. 23

354 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

The suffix s will be used to distinguish quantities relating to the stucco

plate, and m to distinguish those relating to meerschaum.

x, y are the coordinates of a point on any one of the curves on Fig. 11,

page 297, or Figure 8, page 286 A, which are the logarithmic homologues of

the experimental curves.

105. The experimental result I. follows from the general form of

equation (101).

For, putting, as in the experiment on transpiration under pressure, ^- = 0,

and M and - -~- constant, equation (101) becomes

.. ...(102).

p dx

The times of transpiration are proportional to ^ for the same tube or

plate, and if V be a factor depending on the number and size of the openings

V

through the plate, we have the time of transpiration equal to pr.

IX

Putting

y = log^, * = logi ........................ (103),

and indicating the quantities referring to particular plates by s and m, we

have

+ log c s = log j

+ logc m = log^

(104).

Whence taking the coefficients A, m, X x , \ 2 , to be the same for stucco as

for meerschaum (see Appendix, note 4), it follows from equation (102) that

when-* = ^

Sg S m .

v

c v < 105 >-

2/=^-log|^j

Hence we see that the curves expressing the relation between the

logarithms of the reciprocals of the mean ranges, and the logarithms of the

times of transpiration, must have the same shape for different plates, such as

stucco and meerschaum. And, moreover, that the difference between the

33] IN THE GASEOUS STATE. 355

abscissae of corresponding points for the different plates is the logarithm of

the ratio of the coarseness of the plates whatever may be the nature of the

gas.

In the experiments we have an exactly similar agreement between the

curves expressing the log. of the densities, and the log. of the times.

Hence the only point of difference between the results deduced from the

equation, and those derived from the experiments is, that the one depends on

- and the other upon p the temperature being constant. Whereas it appears

S

not only as in Art. 93, but in whichever way we examine s, that however s

may vary with the molecular mass and with the temperature, it must be

inversely proportional to the density.

Therefore the fitting of the logarithmic curves is a direct inference from

the form of the general equation (101).

We also see that the common difference in the abscissae of the curves

deduced from the equation is the logarithm of the ratio of the diameters of

the interstices ; and hence we infer that the difference in the abscissas of the

experimental curves for meerschaum and stucco gives the ratio of the mean

diameters of the interstices in these plates. (See Appendix, note 4.)

The common difference in the ordinates is, according to the equation, the

V c

logarithm of the ratio "L*; and although V m and V s are unknown, the

experiments verify the theory in as much as they show that the common

difference is independent of the nature of the gas the same difference being

obtained with hydrogen as with air and depends entirely on the plates.

The fitting of the curves which express the logarithms of the thermal

differences of pressure follows in a precisely similar manner from equation

(101).

In these experiments O = and - -y- and M were constant, so that

equation (101) becomes

/-\ l J,v* / s>\ Hf /-7

(106).

\.8J p

And putting

* = log-,

o

232

356 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

we have as in the previous case, supposing the coefficient in F l and F 2 to be

f* (*

the same for stucco as for meerschaum (see Appendix, note 4), where - =

And since T and M are the same for both plates

Pin _ PIH

PS PS'

Hence in this case, according to the general equation (106), the common

difference in the ordinates of corresponding points is the logarithm of the

ratios of corresponding densities, while the difference in the abscissae is the

logarithm of the ratio of the coarseness of the plates, which is the reciprocal

of the ratios of the mean ranges. If, therefore, as has just been assumed, the

densities are proportional to the mean ranges, the common difference of the

ordinates should be the same as that of the abscissae, and the same for these

curves as for those of transpiration under pressure.

Thus we have excellent opportunities of verifying the conclusion that s

varies inversely as p, and the indication as to the manner in which c enters

into the relation between dp and dr.

This verification is complete, for although there is a slight discrepancy

between the common difference for the ordinates and that for the abscissae,

this, as has been explained in Art. 30, was in all probability owing to certain

discrepancies in the difference of temperature maintained on the two sides

of the plates (see Appendix, note 4). And even if unexplained these

discrepancies are small enough to be neglected.

The actual differences are as follows :

Thermal Transpiration. Transpiration.

Plates. Abscissae. Ordinates. Abscissae.

Meerschaum No. 3, and Stucco No. 1 '698 775

2 -745 -890 -819

Thus the dependence of transpiration on the ratio - first revealed by the

s

theory, as expressed in equation (101), has been completely verified by the

experiments of transpiration under pressure, and on thermal transpiration.

And it must be noticed that while the verification has been obtained both

for hydrogen and air, the experiments on either gas suffice for complete

verification. And thus the exact agreement of the common differences both

33] IN THE GASEOUS STATE. 357

of ordinates and abscissae for the two gases (although the absolute ordinates

differ widely, and the shapes of the curves differ considerably) not only affords

a double verification, but precludes the possibility of accidental coincidence.

It is further to be noticed, both with respect to the foregoing comparison

of the theoretical with the experimental results, and also with respect of such

further comparisons as will be made, that the reasoning admits of being

reversed ; arid instead of deducing the experimental results from the equation,

it might have been shown that a similar equation is the necessary outcome

of the experimental results. Indeed, this has been already done, and it is

only out of regard to the length of this paper that I refrain from including

the inverse reasoning.

106. The experimental results II. and III. follow at once from the fact

s*

that the various functions of - in equation (101) increase or decrease con-

c c

tinuously between the values - = and - = oo .

s s

Results IV. and V. also follow so directly from equation (101) as to

require no comment.

Results VI. and VII. refer to transpiration under pressure when - is small.

Under these circumstances, since -y- = 0, equation (99) becomes

VTT 1 dp . .

- (108),

M

and taking, as in the experiments, T and - -~- constant, we have for the same

p CLCO

plate

V'

which is result VI.

And assuming, as in Art. 98, that m\ is independent of M or any property

of the gas, we have

noc-3=

and therefore the times of transpiration of the different gases through the

same plate are proportional to the square roots of the molecular weights,

which is experimental result VII.

358 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

This result, therefore, verities the conclusion arrived at in Art. 98, that

when the tube is small compared with s, the effect of the impacts at the

surface is independent of the nature of the gas.

Result VIII. relates to transpiration under pressure when - is large.

Then we have from equation (99)

_n = = _^ J A + mx ,)l* (io9).

//c 2 T 2 \ s ipdx

V M

1 CtT) i ^

Therefore since r, M, and - -f are to be taken as constant ; when -

p dx s

becomes sufficiently large

1

f ) Q .._

n '

that is

ft ocp;

and this is result VIII.

c .

s

G

In order to compare different gases we have, when - is sufficiently large,

..(110).

M s p

Therefore

This gives the relative values of s for different gases ; as, for instance, air

and hydrogen. Graham found that the times of transpiration of these gases

through a capillary tube are in the ratio 2'04. The ratio of the square roots

of the molecular weights is 3 '8. Hence at equal pressures and equal tem-

peratures the mean range for hydrogen is to the mean range for air as 3 - 8 is

to 2-04.

It appears, however, at once from the equation, that these ratios are not

constant unless c/s is very large. As c/s diminishes, the term involving X 2

becomes important, and it is to this term we must look for the explanation of

the result IX. the marked non-correspondence of the curves for hydrogen

and air. If Xj depends on the nature of the gas then this difference in shape

is accounted for, which confirms the conclusion of Art. 98, that when the tube

is large compared with s the effect of the impacts at the surface will probably

depend on the nature of the eras.

* o

33] IN THE GASEOUS STATE. 359

107. Result X. refers to the thermal differences of pressure when - is

small.

In this case H = 0. while - , -7- , and M are constant.

T dx

Equation (99) becomes

1 dp _ I m' 1 dr _ m' 1 d VT

p dx 2 m T dm m \/~r dx

The exact relation between m and m would appear, as explained in

Art. 101, to depend on the shape of the section of the tube, and to be some-

2

where between 1 and 1 H , its respective values for a flat and round tube.

7T

This view, however, is based on the assumption that the molecules are

uniformly distributed as regards direction, whereas it appears probable, from

reasoning similar to that of Art. 98, that the molecules tend to assume a

direction normal to the surface, and in this case for a tube of curvilinear

section the value of would be reduced.

m

According to the experiments, it appears that as the density diminishes,

m'

- approaches to unity ; but owing to the impossibility of measuring the

exact difference on the two sides of the plate this determination is not

very definite.

n

Result XI. refers to the thermal difference of pressure when - is large.

1 dr

In this case H = 0, while - , ~r , and M are constant.

r dx

Equation (99) becomes

/ . c \ 1 dp \ s f _f fc\ \ 1 dr

(A - + raX,, - j = <T~ / ~ + ^2 1 ~ -j-

\ s J p dx ZTT c \ \sj / rdx

in which f[-} has some finite value.

J \s/

In the limit, therefore, we may neglect m\^, and we have

c\ . , \ldr (UH)

And since s x - and c is constant

P

Idp ldr

p dx p* r dx

which is result XL

360 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

Since the coefficient of - -p in equation (113) involves X;,, which (Art. 98)

depends on the nature of the gas, this equation indicates that different results

would be obtained with different gases.

And this appears still more in the case of intermediate pressures when

mX 2 on the left of equation (112) is important.

These conclusions are according to result XII., which therefore affords

additional proof of the correctness of the conclusions in Art. 98 respecting

the value of X 2 -

108. I have now shown how I was led to predict the experimental

results, and how in every particular the experiments have verified the theory,

both as regards transpiration under pressure and the thermal differences of

pressure. This concludes the application of the theory to those experimental

results of transpiration which were revealed by the theory.

There remains, however, an important class of transpiration phenomena of

which, as yet, no mention has been made. These are the phenomena of

transpiration when the gas on the two sides of the plate differs in molecular

constitution.

Transpiration by a variation in the molecular condition of the gas.

108 A. These phenomena are well known, and were experimentally

investigated by Graham, but hitherto, I believe, no complete theoretical

explanation of them has been given. The diffusion of one gas into another

has been explained by Maxwell ; but what has not been explained, so far as

I know, is, that there should result a current from the side of the denser to

that of the lighter gas. Indeed, from the manner in which these phenomena

have been for the most part described, it would appear that the importance

of this current has been overlooked ; for, owing to the fact that a larger

volume of the lighter gas passes, the phenomena are generally described as if

the current were from the lighter to the denser gas.

These phenomena of transpiration, like those already considered, may be

shown to follow directly from the theory. But as has been already mentioned

in Art. 73, in order to completely adapt the equations of transpiration to the

case of two or more gases, it would be necessary to commence by considering

the case of two or more systems of molecules having different molecular

weights, after the manner adopted by Maxwell*. Such an adaptation of the

equations is too long to be included in this paper ; but it may be seen from

* Phil. Trans., 1867.

33] IN THE GASEOUS STATE. 361

the equations, as they have already been deduced, that these particular

phenomena would, and in some cases do, follow.

Suppose that the gas on the two sides of the plate is at the same pressure

and temperature, but that there is a difference in molecular constitution as

air and hydrogen. Thus when the condition has become steady there will be

a gradual variation of the molecular condition of the gas through the plate ;

in this case r is constant and p is constant, but the mean value of M varies.

If we take M t and M 2 (as the molecular masses of the two systems of

molecules), and consider a case in which M^ differs but very slightly from M 2 ,

equation (98) becomes

dM

................... (114)

dx

where M is the mean mass of the molecules, or if p l and p 2 are the densities

of the two ases

Whence, putting N l = & , , ^ 2 = &- ,

rf _N 1 M 1

And since the pressure and temperature are constant

where N is constant throughout the gas.

Therefore

_Mi-M,jv_dp.

dx

and (114) becomes

, 1 dp

(115).

C I C\

If - is small, then F. 2 ( - } = - f'Xt,

s \sj 4

Hence in this case

/* rl r\

(116).

4

And this is in exact accordance with Graham's law, which is that the rate

362 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

of transpiration is proportional to the difference in the square roots of the

densities of the gas. For

dx

and since M 1 - M z is small

or

This form of equation is obtained by neglecting the difference of Jf, and

M 2 ; but by taking into account the two systems of molecules throughout the

investigation, an equation similar to (117) would have been obtained without

any such assumption.

Thus we see that the general equation of transpiration may be made

to include not only the cases of transpiration under pressure and thermal

transpiration, but also the well-known phenomena of transpiration caused by

the difference in the molecular constitution of the gas. And in this case, as

in that of transpiration under pressure, the equation reveals laws connecting

the results obtained with plates of different coarseness and different densities

of gas, which doubtless admit of experimental verification.

This completes the explanation of the phenomena of transpiration through

porous plates.

SECTION XL APPLICATION TO APERTURES IN THIN PLATES AND

IMPULSION. CONDITION OF THE GAS.

\

109. When the gas within a vessel is in a uniform condition, excepting

in so far as it is disturbed by a steady flow of gas or of heat, from what,

compared with the size of the vessel, may be considered as a small space,

such as a small aperture in the side of the vessel or a small hot body within

the vessel, the effect of such steady flow will be to cause a varying condition

throughout the gas. The lines of flow, whether of heat or of gas, will diverge

from the exceptional space, and the surfaces of equal pressure and temperature

will be everywhere perpendicular to the lines of flow of matter and heat

respectively. Except in the case of absolute symmetry, the lines of flow will

not be straight, nor will the directions of the lines of flow in the immediate

region of any point focus in a point.

33]

IN THE GASEOUS STATE.

363

But in the immediate neighbourhood of any point P, the direction of the

lines of flow must be such that the directions of the lines of flow parallel to

some plane, xy, will converge to some point C, while the directions of the

lines of flow parallel to the perpendicular plane xz will converge to some

point C' in CP, which it will be seen is taken parallel to the axis of x.

Whence putting PG r y and PC' = r z , the surface of equal pressure or

temperature at P will be a surface perpendicular to x, and having r y and r z

as its principal radii of curvature in the planes of xy and xz respectively.

The simplest cases are those in which either the two radii are equal or one is

infinite, and these are the cases which will for the most part be considered.

It will at once be seen that at any point within gas in the condition just

described, p, a, u 2 + v 2 + W and U 2 + V 2 + W 2 are functions of r y , r z .

Also, remembering that the axis of x is taken in the direction of the lines

of flow at P, the point considered, we see that at P, V, W, v, w, are severally

, dU dU d*V d*W dV dW

zero, as are also , - , -7-

dy dz

' df>

d 2 U

dz 2 ' dx ' dx '

1 dU 1 u \

df

d 2 U

r y dx r y 2

1 dU 1

dz 2

dV _

dy ~

r z dx r z 2

U

V

\ .

dW _

dz

d 2 V .

dxdy

d*W _

U

1 ' dU_l_ u

r y dx ry*

1 dU 1 rr

- 11.

r . '

(118).

dxdz r z dx r z

Also putting f(pa) for any function of p and a, -y~ /(/>) and ^

are

zero, while

df

dx

(119).

dz 2

dx

The equations of steady condition.

110. Equations (118) and (119), together with equations (43) to (47),

enable us to obtain from equation (57) the equations of steady condition.

364 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

For steady density

d ( I TT S dpO\\ /i9m

\r y r z ( P U- j=-p){*?P .................. ( 12 )-

dx ( V VTT ax 1}

For steady momentum putting, as before, -~ =-j- (<r x (Mu)}

*_*AWfl + 1)1=0 ..... (121).

dx VTT dx ( \r y r z / )

For steady pressure

- j= s

v TT

These equations (120), (121), (122), might be treated in a manner similar

to that in which the corresponding equations for the case of the tube were

treated in Section IX., but for various reasons another method commends

itself.

In the first place we cannot in this case ignore the condition of steady

pressure, for there can be no lateral adjustment of temperature as in the case

of the tube. (See Art. 91.) The physical meaning of this is, that in this

case the condition of the gas cannot be supposed to vary uniformly even along

the lines of flow. It must vary after a fixed law, and this fact restricts the

conditions under which the equations can be considered to hold to points

s is \ 2

where - is so small that f - j may be neglected. So that any general result

obtained from these equations would only apply to points at considerable

distances from the foci C and G'.

Again, these equations as they stand include the case in which the flow

of the gas may be caused by a considerable difference of pressure, as, for

example, transpiration through a small aperture under pressure, whereas if

s 2 tlcL-

we exclude this case we may, by neglecting such terms as j- , very greatly

simplify the equations without affecting their application to the cases which

it is our principal object to explain.

These two cases are as follow

1. The flow of gas through a small orifice in a thin plate when the mean

pressure of the gas is the same on both sides of the plate, the flow being

caused by a difference in temperature on the two sides of the plate, or a

difference in the molecular condition of the gas.

2. The excess of pressure which the gas exerts on a small body when

2 M \s/ \s/

have, remembering that M is constant,

/

V

fl VTT ( . c . /c\ , /c\) 1

= - -r c M ~ + mX i/ ~ M- m ^2/2 1 - If ~

2 I W \sj) p

(99)

in which

A depends only on the shape of the tube and is ^ for a flat tube,

/C\ 60

f I - } is of the order - when - is zero and is finite when - is infinite,

' W s s s

fi(-) and/s (-) are unity when - is zero and zero when - is infinite,

\s / \s/ s s

352 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

and

/, ( - | and / 4 (- ) are zero when - is zero and unity when - is infinite; all

J \s) \s) s s

the functions varying continuously between the limits here ascribed.

Also \! depends on the nature of the surface but not upon the nature of

the gas, while A, and X 3 may depend both upon the gas and the surface.

Putting

T /c\ VTT f . /c >

1 (~) = ~9~ '~

* ' I ..

and

(100)

we have for the general form of the equation of transpiration

P

SECTION X. VERIFICATION OF THE GENERAL EQUATION OF

TRANSPIRATION.

103. In this section the general equation obtained in Section IX. is

applied to the particular cases of transpiration which have been the subject

of experiments. It will thus appear how I was led to infer the results, and

thence to make the experiments.

A summary of the experimental results has already been given in Art. 9,

but for the immediate purposes the results may be stated as follows :

Experimental results.

I. The law of corresponding results at corresponding densities, shown

by the fitting of the logarithmic homologues. (See Arts. 28 and 40.)

II. The gradual manner in which the results varied as the density

increased, shown by the continuous curvature of the curves which express

these results. (See Figures 8, 9, and 12.)

III. The uniformity in direction in which both the time of transpiration

under pressure (see Tables XIV. to XVII.) and the ratio of the thermal

differences of pressure to the mean pressure (see Tables III. to XIII.) vary as

the density increased.

33] IN THE GASEOUS STATE. 353

IV. The fact, sufficiently proved by Graham, that, cceteris paribus, the

times of transpiration are proportional to the ratio of the differences of

pressure to the mean pressure, the difference of pressure being small.

V. The fact, to a certain extent taken for granted, that the ratio of the

thermal differences of pressure to the mean pressure is, cceteris paribus,

proportional to the ratio of the difference of temperature to the absolute

temperature, this ratio being small.

VI. The continual approximation towards constancy of the time of

transpiration under pressure as the density diminished. (See Tables XIV.

to XVII., and Figure 12, page 298.)

VII. The relation between the ultimate values of the times of transpi-

ration for different gases (air and hydrogen) for small densities; the times

are proportional to the square roots of their atomic weights. (See Art. 42.)

VIII. The fact that the times of transpiration for the same gas in

capillary tubes, and at considerable densities, are inversely as the density

and independent of the temperature. Maxwell* and Graham f.

IX. The difference in the variation of the times of transpiration for

different gases, shown by the fact that the logarithmic curves for hydrogen

cannot be made to fit those for air. (Figures 8, 9 and 12.)

X. The approximation towards a constant value of the ratio which the

thermal differences of pressure bear to the mean pressure as the density

diminishes, whatever be the gas or plate ; the ratio is that of the difference

of the square roots of the absolute temperatures to the square root of the

absolute temperature.

XI. The approximation, as the density increases, to a linear relation

between the thermal differences of pressure and the reciprocal of the density.

XII. The difference between the law of variation of the thermal

differences of pressure for different gases, as shown by the non-agreement

of the logarithmic homologues for air and hydrogen. (Figures 8 and 9.)

XIII. The transpiration of a varying mixture of gases through a porous

plate. Investigated by Graham.

104. In order to bring out the agreement of the experimental results

with those deduced from the equation, we put

V

^r for the time of transpiration,

_ _

b - for the difference of pressure on the two sides of the plate,

dx

b for the difference of temperature on the two sides of the plate.

dx

* Phil. Trans., 1866, pp. 249268, also note to Art. 94. t lb., 1849, pp. 349362.

O. K. 23

354 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

The suffix s will be used to distinguish quantities relating to the stucco

plate, and m to distinguish those relating to meerschaum.

x, y are the coordinates of a point on any one of the curves on Fig. 11,

page 297, or Figure 8, page 286 A, which are the logarithmic homologues of

the experimental curves.

105. The experimental result I. follows from the general form of

equation (101).

For, putting, as in the experiment on transpiration under pressure, ^- = 0,

and M and - -~- constant, equation (101) becomes

.. ...(102).

p dx

The times of transpiration are proportional to ^ for the same tube or

plate, and if V be a factor depending on the number and size of the openings

V

through the plate, we have the time of transpiration equal to pr.

IX

Putting

y = log^, * = logi ........................ (103),

and indicating the quantities referring to particular plates by s and m, we

have

+ log c s = log j

+ logc m = log^

(104).

Whence taking the coefficients A, m, X x , \ 2 , to be the same for stucco as

for meerschaum (see Appendix, note 4), it follows from equation (102) that

when-* = ^

Sg S m .

v

c v < 105 >-

2/=^-log|^j

Hence we see that the curves expressing the relation between the

logarithms of the reciprocals of the mean ranges, and the logarithms of the

times of transpiration, must have the same shape for different plates, such as

stucco and meerschaum. And, moreover, that the difference between the

33] IN THE GASEOUS STATE. 355

abscissae of corresponding points for the different plates is the logarithm of

the ratio of the coarseness of the plates whatever may be the nature of the

gas.

In the experiments we have an exactly similar agreement between the

curves expressing the log. of the densities, and the log. of the times.

Hence the only point of difference between the results deduced from the

equation, and those derived from the experiments is, that the one depends on

- and the other upon p the temperature being constant. Whereas it appears

S

not only as in Art. 93, but in whichever way we examine s, that however s

may vary with the molecular mass and with the temperature, it must be

inversely proportional to the density.

Therefore the fitting of the logarithmic curves is a direct inference from

the form of the general equation (101).

We also see that the common difference in the abscissae of the curves

deduced from the equation is the logarithm of the ratio of the diameters of

the interstices ; and hence we infer that the difference in the abscissas of the

experimental curves for meerschaum and stucco gives the ratio of the mean

diameters of the interstices in these plates. (See Appendix, note 4.)

The common difference in the ordinates is, according to the equation, the

V c

logarithm of the ratio "L*; and although V m and V s are unknown, the

experiments verify the theory in as much as they show that the common

difference is independent of the nature of the gas the same difference being

obtained with hydrogen as with air and depends entirely on the plates.

The fitting of the curves which express the logarithms of the thermal

differences of pressure follows in a precisely similar manner from equation

(101).

In these experiments O = and - -y- and M were constant, so that

equation (101) becomes

/-\ l J,v* / s>\ Hf /-7

(106).

\.8J p

And putting

* = log-,

o

232

356 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

we have as in the previous case, supposing the coefficient in F l and F 2 to be

f* (*

the same for stucco as for meerschaum (see Appendix, note 4), where - =

And since T and M are the same for both plates

Pin _ PIH

PS PS'

Hence in this case, according to the general equation (106), the common

difference in the ordinates of corresponding points is the logarithm of the

ratios of corresponding densities, while the difference in the abscissae is the

logarithm of the ratio of the coarseness of the plates, which is the reciprocal

of the ratios of the mean ranges. If, therefore, as has just been assumed, the

densities are proportional to the mean ranges, the common difference of the

ordinates should be the same as that of the abscissae, and the same for these

curves as for those of transpiration under pressure.

Thus we have excellent opportunities of verifying the conclusion that s

varies inversely as p, and the indication as to the manner in which c enters

into the relation between dp and dr.

This verification is complete, for although there is a slight discrepancy

between the common difference for the ordinates and that for the abscissae,

this, as has been explained in Art. 30, was in all probability owing to certain

discrepancies in the difference of temperature maintained on the two sides

of the plates (see Appendix, note 4). And even if unexplained these

discrepancies are small enough to be neglected.

The actual differences are as follows :

Thermal Transpiration. Transpiration.

Plates. Abscissae. Ordinates. Abscissae.

Meerschaum No. 3, and Stucco No. 1 '698 775

2 -745 -890 -819

Thus the dependence of transpiration on the ratio - first revealed by the

s

theory, as expressed in equation (101), has been completely verified by the

experiments of transpiration under pressure, and on thermal transpiration.

And it must be noticed that while the verification has been obtained both

for hydrogen and air, the experiments on either gas suffice for complete

verification. And thus the exact agreement of the common differences both

33] IN THE GASEOUS STATE. 357

of ordinates and abscissae for the two gases (although the absolute ordinates

differ widely, and the shapes of the curves differ considerably) not only affords

a double verification, but precludes the possibility of accidental coincidence.

It is further to be noticed, both with respect to the foregoing comparison

of the theoretical with the experimental results, and also with respect of such

further comparisons as will be made, that the reasoning admits of being

reversed ; arid instead of deducing the experimental results from the equation,

it might have been shown that a similar equation is the necessary outcome

of the experimental results. Indeed, this has been already done, and it is

only out of regard to the length of this paper that I refrain from including

the inverse reasoning.

106. The experimental results II. and III. follow at once from the fact

s*

that the various functions of - in equation (101) increase or decrease con-

c c

tinuously between the values - = and - = oo .

s s

Results IV. and V. also follow so directly from equation (101) as to

require no comment.

Results VI. and VII. refer to transpiration under pressure when - is small.

Under these circumstances, since -y- = 0, equation (99) becomes

VTT 1 dp . .

- (108),

M

and taking, as in the experiments, T and - -~- constant, we have for the same

p CLCO

plate

V'

which is result VI.

And assuming, as in Art. 98, that m\ is independent of M or any property

of the gas, we have

noc-3=

and therefore the times of transpiration of the different gases through the

same plate are proportional to the square roots of the molecular weights,

which is experimental result VII.

358 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

This result, therefore, verities the conclusion arrived at in Art. 98, that

when the tube is small compared with s, the effect of the impacts at the

surface is independent of the nature of the gas.

Result VIII. relates to transpiration under pressure when - is large.

Then we have from equation (99)

_n = = _^ J A + mx ,)l* (io9).

//c 2 T 2 \ s ipdx

V M

1 CtT) i ^

Therefore since r, M, and - -f are to be taken as constant ; when -

p dx s

becomes sufficiently large

1

f ) Q .._

n '

that is

ft ocp;

and this is result VIII.

c .

s

G

In order to compare different gases we have, when - is sufficiently large,

..(110).

M s p

Therefore

This gives the relative values of s for different gases ; as, for instance, air

and hydrogen. Graham found that the times of transpiration of these gases

through a capillary tube are in the ratio 2'04. The ratio of the square roots

of the molecular weights is 3 '8. Hence at equal pressures and equal tem-

peratures the mean range for hydrogen is to the mean range for air as 3 - 8 is

to 2-04.

It appears, however, at once from the equation, that these ratios are not

constant unless c/s is very large. As c/s diminishes, the term involving X 2

becomes important, and it is to this term we must look for the explanation of

the result IX. the marked non-correspondence of the curves for hydrogen

and air. If Xj depends on the nature of the gas then this difference in shape

is accounted for, which confirms the conclusion of Art. 98, that when the tube

is large compared with s the effect of the impacts at the surface will probably

depend on the nature of the eras.

* o

33] IN THE GASEOUS STATE. 359

107. Result X. refers to the thermal differences of pressure when - is

small.

In this case H = 0. while - , -7- , and M are constant.

T dx

Equation (99) becomes

1 dp _ I m' 1 dr _ m' 1 d VT

p dx 2 m T dm m \/~r dx

The exact relation between m and m would appear, as explained in

Art. 101, to depend on the shape of the section of the tube, and to be some-

2

where between 1 and 1 H , its respective values for a flat and round tube.

7T

This view, however, is based on the assumption that the molecules are

uniformly distributed as regards direction, whereas it appears probable, from

reasoning similar to that of Art. 98, that the molecules tend to assume a

direction normal to the surface, and in this case for a tube of curvilinear

section the value of would be reduced.

m

According to the experiments, it appears that as the density diminishes,

m'

- approaches to unity ; but owing to the impossibility of measuring the

exact difference on the two sides of the plate this determination is not

very definite.

n

Result XI. refers to the thermal difference of pressure when - is large.

1 dr

In this case H = 0, while - , ~r , and M are constant.

r dx

Equation (99) becomes

/ . c \ 1 dp \ s f _f fc\ \ 1 dr

(A - + raX,, - j = <T~ / ~ + ^2 1 ~ -j-

\ s J p dx ZTT c \ \sj / rdx

in which f[-} has some finite value.

J \s/

In the limit, therefore, we may neglect m\^, and we have

c\ . , \ldr (UH)

And since s x - and c is constant

P

Idp ldr

p dx p* r dx

which is result XL

360 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

Since the coefficient of - -p in equation (113) involves X;,, which (Art. 98)

depends on the nature of the gas, this equation indicates that different results

would be obtained with different gases.

And this appears still more in the case of intermediate pressures when

mX 2 on the left of equation (112) is important.

These conclusions are according to result XII., which therefore affords

additional proof of the correctness of the conclusions in Art. 98 respecting

the value of X 2 -

108. I have now shown how I was led to predict the experimental

results, and how in every particular the experiments have verified the theory,

both as regards transpiration under pressure and the thermal differences of

pressure. This concludes the application of the theory to those experimental

results of transpiration which were revealed by the theory.

There remains, however, an important class of transpiration phenomena of

which, as yet, no mention has been made. These are the phenomena of

transpiration when the gas on the two sides of the plate differs in molecular

constitution.

Transpiration by a variation in the molecular condition of the gas.

108 A. These phenomena are well known, and were experimentally

investigated by Graham, but hitherto, I believe, no complete theoretical

explanation of them has been given. The diffusion of one gas into another

has been explained by Maxwell ; but what has not been explained, so far as

I know, is, that there should result a current from the side of the denser to

that of the lighter gas. Indeed, from the manner in which these phenomena

have been for the most part described, it would appear that the importance

of this current has been overlooked ; for, owing to the fact that a larger

volume of the lighter gas passes, the phenomena are generally described as if

the current were from the lighter to the denser gas.

These phenomena of transpiration, like those already considered, may be

shown to follow directly from the theory. But as has been already mentioned

in Art. 73, in order to completely adapt the equations of transpiration to the

case of two or more gases, it would be necessary to commence by considering

the case of two or more systems of molecules having different molecular

weights, after the manner adopted by Maxwell*. Such an adaptation of the

equations is too long to be included in this paper ; but it may be seen from

* Phil. Trans., 1867.

33] IN THE GASEOUS STATE. 361

the equations, as they have already been deduced, that these particular

phenomena would, and in some cases do, follow.

Suppose that the gas on the two sides of the plate is at the same pressure

and temperature, but that there is a difference in molecular constitution as

air and hydrogen. Thus when the condition has become steady there will be

a gradual variation of the molecular condition of the gas through the plate ;

in this case r is constant and p is constant, but the mean value of M varies.

If we take M t and M 2 (as the molecular masses of the two systems of

molecules), and consider a case in which M^ differs but very slightly from M 2 ,

equation (98) becomes

dM

................... (114)

dx

where M is the mean mass of the molecules, or if p l and p 2 are the densities

of the two ases

Whence, putting N l = & , , ^ 2 = &- ,

rf _N 1 M 1

And since the pressure and temperature are constant

where N is constant throughout the gas.

Therefore

_Mi-M,jv_dp.

dx

and (114) becomes

, 1 dp

(115).

C I C\

If - is small, then F. 2 ( - } = - f'Xt,

s \sj 4

Hence in this case

/* rl r\

(116).

4

And this is in exact accordance with Graham's law, which is that the rate

362 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

of transpiration is proportional to the difference in the square roots of the

densities of the gas. For

dx

and since M 1 - M z is small

or

This form of equation is obtained by neglecting the difference of Jf, and

M 2 ; but by taking into account the two systems of molecules throughout the

investigation, an equation similar to (117) would have been obtained without

any such assumption.

Thus we see that the general equation of transpiration may be made

to include not only the cases of transpiration under pressure and thermal

transpiration, but also the well-known phenomena of transpiration caused by

the difference in the molecular constitution of the gas. And in this case, as

in that of transpiration under pressure, the equation reveals laws connecting

the results obtained with plates of different coarseness and different densities

of gas, which doubtless admit of experimental verification.

This completes the explanation of the phenomena of transpiration through

porous plates.

SECTION XL APPLICATION TO APERTURES IN THIN PLATES AND

IMPULSION. CONDITION OF THE GAS.

\

109. When the gas within a vessel is in a uniform condition, excepting

in so far as it is disturbed by a steady flow of gas or of heat, from what,

compared with the size of the vessel, may be considered as a small space,

such as a small aperture in the side of the vessel or a small hot body within

the vessel, the effect of such steady flow will be to cause a varying condition

throughout the gas. The lines of flow, whether of heat or of gas, will diverge

from the exceptional space, and the surfaces of equal pressure and temperature

will be everywhere perpendicular to the lines of flow of matter and heat

respectively. Except in the case of absolute symmetry, the lines of flow will

not be straight, nor will the directions of the lines of flow in the immediate

region of any point focus in a point.

33]

IN THE GASEOUS STATE.

363

But in the immediate neighbourhood of any point P, the direction of the

lines of flow must be such that the directions of the lines of flow parallel to

some plane, xy, will converge to some point C, while the directions of the

lines of flow parallel to the perpendicular plane xz will converge to some

point C' in CP, which it will be seen is taken parallel to the axis of x.

Whence putting PG r y and PC' = r z , the surface of equal pressure or

temperature at P will be a surface perpendicular to x, and having r y and r z

as its principal radii of curvature in the planes of xy and xz respectively.

The simplest cases are those in which either the two radii are equal or one is

infinite, and these are the cases which will for the most part be considered.

It will at once be seen that at any point within gas in the condition just

described, p, a, u 2 + v 2 + W and U 2 + V 2 + W 2 are functions of r y , r z .

Also, remembering that the axis of x is taken in the direction of the lines

of flow at P, the point considered, we see that at P, V, W, v, w, are severally

, dU dU d*V d*W dV dW

zero, as are also , - , -7-

dy dz

' df>

d 2 U

dz 2 ' dx ' dx '

1 dU 1 u \

df

d 2 U

r y dx r y 2

1 dU 1

dz 2

dV _

dy ~

r z dx r z 2

U

V

\ .

dW _

dz

d 2 V .

dxdy

d*W _

U

1 ' dU_l_ u

r y dx ry*

1 dU 1 rr

- 11.

r . '

(118).

dxdz r z dx r z

Also putting f(pa) for any function of p and a, -y~ /(/>) and ^

are

zero, while

df

dx

(119).

dz 2

dx

The equations of steady condition.

110. Equations (118) and (119), together with equations (43) to (47),

enable us to obtain from equation (57) the equations of steady condition.

364 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

For steady density

d ( I TT S dpO\\ /i9m

\r y r z ( P U- j=-p){*?P .................. ( 12 )-

dx ( V VTT ax 1}

For steady momentum putting, as before, -~ =-j- (<r x (Mu)}

*_*AWfl + 1)1=0 ..... (121).

dx VTT dx ( \r y r z / )

For steady pressure

- j= s

v TT

These equations (120), (121), (122), might be treated in a manner similar

to that in which the corresponding equations for the case of the tube were

treated in Section IX., but for various reasons another method commends

itself.

In the first place we cannot in this case ignore the condition of steady

pressure, for there can be no lateral adjustment of temperature as in the case

of the tube. (See Art. 91.) The physical meaning of this is, that in this

case the condition of the gas cannot be supposed to vary uniformly even along

the lines of flow. It must vary after a fixed law, and this fact restricts the

conditions under which the equations can be considered to hold to points

s is \ 2

where - is so small that f - j may be neglected. So that any general result

obtained from these equations would only apply to points at considerable

distances from the foci C and G'.

Again, these equations as they stand include the case in which the flow

of the gas may be caused by a considerable difference of pressure, as, for

example, transpiration through a small aperture under pressure, whereas if

s 2 tlcL-

we exclude this case we may, by neglecting such terms as j- , very greatly

simplify the equations without affecting their application to the cases which

it is our principal object to explain.

These two cases are as follow

1. The flow of gas through a small orifice in a thin plate when the mean

pressure of the gas is the same on both sides of the plate, the flow being

caused by a difference in temperature on the two sides of the plate, or a

difference in the molecular condition of the gas.

2. The excess of pressure which the gas exerts on a small body when

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