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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



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