D

preffure of the atmosphere, and therefore equal to the force, which impels it into a void; and let be the force with which this rarer air would run into a void. We have D:♪=P:~, and -=Ꮲ . Now the moving force in the prefent inftance is P-s, or P-D. Laftly, let V be the velocity of air rushing into a void, and the velocity with which it will rush into this rarefied air.

It is a theorem in the motion of fluids, that the preffures are as the squares of the velocities of effus. Therefore P: P- P♪

-V: v. Hence we de

It remains to determine the time 7 expressed in feconds, in which the air of the atmosphere will Bow into this veffel from its ftate of vacuity till the air in the veffel has acquired any propofed denty. For this purpose let H, expreffed in feet, be the height through which a heavy body mult fall, in order to acquire the velocity V, expreffed allo in feet per fecond. This we shall exprefs more briefly in future, by calling it the height producing the velocity V. Let C reprefent the capacity of the veffel, expreffed in cubic feet and O the area or fection of the orifice, expreffed in fuperficial or square feet; and let the natural denfity

of the air be D.

As the quantity of aerial matter contained in a veffel depends on the capacity of the veffel and the density of the air jointly, we may exprefs the air which would fill this veffel by the fymbol CD when the air is in its ordinary state, and by C when it has the denfity 8. In order to obtain the rate at which it fills, we must take the fluxion of this quantity Cs. This is C; for C is a conftant quantity, and is a variable or flowing quantity. But we alfo obtain the rate of influx by our knowledge of the velocity, and the area of the orifice, and the density. The velocity is V, or /H, at the first inftant; and when the air of the vellel has acquired the density, that is, at the

HO Therefore the time of completely filling
C
the veffel is.

4/HO.

bers. Suppofing then that air is 840 times lighter
We may illuftrate this by an example in num-
than water, and the height of the homogeneous
atmosphere 27720 feet, we have 4/H=666. Let
us further fuppofe the veffel to contain 8 cubic
feet, which is nearly a wine hogfhead, and that
the hole by which the air of the ordinary denfity,
which we fhall make 1, enters, is an inch square,
or of a fquare foot. Then the time in fe-
conds of completely filling it will be

1152"

66.6

8"

666, or

or 1,7297". If the hole is only vos of a fquare inch, that is, if its fide is of an inch, the time of completely filling the hogshead will be 173" very nearly, or fomething lefs than three minutes.

If we make the experiment with a hole cut in a thin plate, we fhall find the time greater nearly in the proportion of 63 to 100, for reasons obvious to all who have ftudied hydraulics. In like manner, we can tell the time neceflary for bringing the air in the veffel to of its ordinary density. The only variable part of our fluent is the coefficient →√D—s, or √s. Let & be, then √

, and 1-V; and the time is 861" very nearly, when the hole is of an inch wide..

Let us now fuppose the air in the vessel ABCD g. 52. pl. 280.) is compreffed by a weight acting on the cover AD, which is moveable down the veffel, and is thus expelled into the external air. The immediate effect of this external preffure is to comprefs the air, and give it another denfity. The denfity D of the external air correfponds to

1

its

its preffure P. Let the additional preffure on the
cover of the veffel be p, and the denfity of the air
in the veffel bed. We shall have P: P+p=D:d;
and therefore p=Px-D Then, because the
preffure which expels the air is the difference be-
tween the force which compreffes the air in the
veffel and the force which compreffes the external
air, the expelling force is p. And becaufe the quan-
tities of motion are as the forces which fimilarly
d-D
produce them, we shall have P: PX- =MV:
mu; where M and m exprefs the quantities of
matter expelled, V expreffes the velocity with
which air rushes into a void, and expreffes the
velocity fought. But because the quantities of
aerial matter which iffue from the fame orifice in
a moment are the denfities and velocities jointly,
we shall have MV: mv=DVV : dvv, DV1: dv2.
Therefore P:p=DV1: dv. Hence we de-
duce v=V

The fluent of this is t= fluent must be fo taken that may beo when d

=p there- D. Therefore the correct fluent will be /=

The confequence of this reafoning is, that if this fmall portion of the veffel be removed, and thus a paffage be made into a void, the air will begin to flow out with the fame velocity with which it would flow when impeli d by its weight+BD=Aq+B, and therefore! alone, or with the velocity acquired by falling

Let A be the capacity of the first vef:) and B that of the fecond. We have the fecond equation 40

from the top of a homogeneous atmosphere, or Subftituting this value of 3 in the former valara “

we have v=V X

9 B(Q-D)

which gives the relation between the velocity v and the denfity q.

Some of thefe queftions are of difficult folution, and they are not of frequent ufe in pneumatics. The cafes of greateft ufe are when the air is expelled from a veffel by an external force, as when bellows are worked, whether of the ordinary form or confifting of a cylinder fitted with a moveable pifton. This laft cafe merits a particular confideration, and the inveftigation is extremely easy.

Q[B(q—D)—A(Q−q)] by all contractions of its paffage. Thefe oblige it to accelerate its velocity, and therefore require an increase of preffure to force it through them, in proportion to the fquares of the velocities. Thus, if a machine working a pump caufes it to give a certain number of Atrokes in a minute, it will deliver a determined quantity of water in that time. Should it happen that the paffage of the water is contracted to one half in any part of the machine (which often happens at the valves), the water muft move through this contraction with twice the velocity that it has in the reft of the paffage. This will require four times the force to be exerted on the pifton. Nay, which appears very odd, and is never fufpected by engineers, if no part of the paffage is narrower than the barrel of the pump, but a part much wider, and if the conduit be again contracted to the width of the barrel, an additional force muft be applied to the pifton to drive the water through this paffage, which would not have been necessary if the paffage had not been widened in any part. It will require a force equal to the weight of a column of water of the height neceffary for communicating a velocity, the fquare of which is equal to the difference of the fquares of the velocities of the water in the wide and the narrow part of the conduit. The fame thing takes place in the motion of air, and therefore all contractions and dilatations must be carefully avoided, when we want to preferve the velocity unimpaired.

Let AD, fig. 52. be a pifton moving downward with the uniform velocity f, and let the area of the pifton ben times the area of the hole of efflux, thes the velocity of efflux arifing from the motion of the pifton will be nf. Add this to the velocity V produced by the elasticity of the air in the firft quation, and the whole velocity will be V+nf. It will be the fame in the others. The problem is also freed from the confideration of the time of elax; for this depends now on the velocity of the pifton. It is ftill, however, a very intricate problem to afcertain the relation between the time and the denfity, even though the pifton is moving uniformly; for at the beginning of the motion the air is of common denfity. As the pifton defcends, it both expels and compreffes the air, and the denity of the air in the veffel varies in a very intricate manner, as alfo its refiftance or reaction on the piston. For this reason, a piston which moves mformly by an external force will never make an form blaft by fucceffive ftrokes; it will always be weaker at the beginning of the ftroke. The beft way for fecuring an uniform blaft is to employ the external force only for lifting up the pton, and then to let the pifton defcend by its own weight. In this way it will quickly fink down, compreffing the air, till its denfity and correfpond. ing elafticity exactly balance the weight of the piton. After this the pifton will defcend equably, and the blaft will be uniform. These obfervations and theorems will determine the initial velocity of the air in all important cafes of its expulfion. The philofopher will learn the rate of its efflux out of one veffel into another; the chemift will be able to calculate the quantities of the different gafes employed in the curious experiments of the ingenious but unfortunate LAVOISIER on combuftion; and the engineer will learn how to proportion the motive force of his machine to the quantity of aerial matter which his bellows muft fupply.

All the modifications of motion in water conduits take place alfo in the paffage of air through pipes and holes of all kinds. There is the fame diminution of quantity paffing through a hole in a thin plate that is obferved in water. Abating the small effect of friction, water iffues with the velocity acquired by falling from the furface: and yet if we calculate by this velocity and by the area of the orifice, we find the quantity of water deficient nearly in the proportion of 63 to 1oo. This is owing to the water preffing towards the orifice from all fides, which occafions a contraction of the jet. The fame thing happens in the efflux of air. The motion of water is alfo greatly impeded Voz. XVIII. PART !.

AIR alfo fuffers the fame retardation in its motion along pipes. By not attending to this, engineers of the first reputation have been prodigiously disappointed in their expectations of the quantity of air which will be delivered by long pipes.. Its extreme mobility and lightnefs hindered them from fufpecting that it would fuffer any fenfible retardation. Dr PAPIN, a moft ingenious man, proposed this as the moft effectual method of transferring the action of a moving power to a great diftance. Notwithstanding his great reputation, he could not get his fcheme patronised in England; but in France and Germany he got fome perfons of fortune to affift him in this project; and he erected great machines in Auvergne and Weftphalia for draining mines. But, so far from being effective, they would not even begin to move. He attributed the failure to the quantity of air in the pipe of communication, which indeed muft be condenfed before it can condenfe the air in the remote cylinder. He therefore diminished the fize of his pipe, and made his water-machine exhauft, inftead of condenfing. But he was equally difappointed, and the machines at the mines stood still as before.

Near a century after this, a very intelligent engineer attempted a much more feafible engine at an iron foundery in Wales. He erected a machine at a powerful water-fall, which worked a fet of cylinder bellows, the blow-pipe of which was conducted to the diftance of a mile and a half, where it was applied to a blaft furnace. But notwithstanding every care to make the conducting pipe very air-tight, of great fize, and as smooth as poffible, it would hardly blow out a candle. The failure was afcribed to the impofüblity of making

C

the

18

PNEUMATIC S

the pipe air-tight. No very diftinct theory can be delivered on this fubject; but we may derive confiderable affiftance in understanding the caufes of the obstruction to the motion of water in long pipes, by confidering what happens to air. The elafticity of the air and its great compreffibility, have given us the most diftinct notions of fluidity in general; showing us, in a way that can hardly be controverted, that the particles of a fluid are kept at a distance from each other, and from other bodies, by the corpufcular forces. See WATER-WORKS.

The writers on hydrodynamics have always confidered the obftruction to the motion of fluids along canals of any kind, as owing to fomething like the friction by which the motion of fid bodies on each fide is obftructed; but we cannot form any diftinct notion of resemblance, or even analogy between them. The fact is, that a fluid running along the canal has its motion obftructed; and that this obftruction is greateft in the immediate vicinity of the folid canal, and gradually diminishes to the middle of the stream. It appears, therefore, that the parts of fluids can no more move among each other than among folid bodies, without fuffering a diminution of their motion. The parts in phyfical contact with the fides and bottom are retarded by thefe immoveable bodies. The particles of the next ftratum of fluid cannot preferve their initial velocities without overpaling the particles of the first ftratum; and it appears from the fact that they are by thefe means retarded. They retard in the fame manner the particles of the third ftratum, and fo on to the middle ftratum or thread of Buid. The fact, therefore, is, that this fort of friction is not a confequence of rigidity alone, but that it is equally competent to fluids. Nay, fince it is a matter of fact in air, and is even more remarkable there than in any other fluid, and as our experiments on the compreffion of air fhow us the particles of air ten times nearer to each other in fome cafes than in others, and Icco times denfer, and thus force us to acknowledge that they are not in contact, it is plain that this obstruction has no analogy to friction, which fuppofes inequality of furface. No fuch inequality can be fuppofed in the furface of an aerial particle; nor would it be of any fervice in explaining the obftruction, fince the particles do not rub on each other, but pafs each other at fome fmall imper. ceptible diftance.

We must therefore have recourfe to fome other
mode of explication. We shall apply this to air
only in this place; and, fince it is proved by the
incontrovertible experiments of CANTON, ZIM-
MERMAN, and others, that water, mercury, oil,
&c. are alfo compreffible and perfectly elaftic, the
argument from this principle, which is conclufive
in air, muft equally explain the fimilar pheno-
menon in hydraulics.

The most highly polished body which we know
must be conceived as having an uneven furface,
when
we compare it with the mall spaces in
which the corpufcular forces are exerted; a
quantity of air moving in a polifhed pipe may be
compared to a quantity of fmall hot fliding down
a channel with undulated fides and bottom. The
row of particles immediately contiguous to the

fides will therefore have an undulated motion; but this undulation of the contiguous particles of air will not be fo great as that of the furface along which they glide; for not only every motion requires force to produce it, but also every change of motion. The particles of air refift this change from rilineal to an undulating motion; and, being elastic, that is repelling each other and other bodies, they keep a little nearer to the furface as they are pafling over an eminence, and their path tion of the next row of particles will be less than is lefs incurvated than the furface. The undulathat of the fift; that of the third row lefs than that of the fecond, and fo on, as represented in fig. 54. pl. CCLXXXI. And thus, while the mafs of air has a progreffive motion along the pipe or canal. a line parallel to the direction of the canal is the each particle is defcribing a waving line, of which axis, cutting all these undulations. This axis of each undulated path will be ftraight or curved as the canal is, and the excurfions of the path on each fide of its axis will be lefs and lefs as the axis of Let us now fee what fenfible effect this will have; the path is nearer to the axis of the canal. for all the motion which we here speak of is im perceptible. It is demonftrated in mechanics, that if a body moving, with any velocity be deflected from its rectilineal path by a curved and perfectly fmooth channel, to which the rectilineal path is a tangent, it will proceed along this channel with undiminished velocity. Now, the path in the prefent cafe, may be confidered as perfectly smooth, fince the particles do not touch it. There should not, therefore, be any diminution of the velocity. Let us grant this of the abfolute velocity of the particle; but what we obferve is the velocity of a fenfible object, and let us attend to two fuch the mafs. Let us suppose a single atom to be particles, one at the fide, and the other in the middle; although we cannot perceive the undulations of thefe particles during their progreffive motions, we fee the progreffive motions them felves. Let us fuppofe then that the middle particle has moved without any undulation whatever, and that it has advanced ten feet. The lateral particle will also have moved ten feet; but this has not been in a ftraight line. It will not be fo far advanced, therefore, in the direction of the canal; it will be left behind, and will appear to have been retarded particles will be more and more retarded (appa in its motion; and in like manner each thread of rently only), as it recedes farther from the axis of the canal, or what is ufually called the thread of the ftream. And thus the obferved fact is a neceffary confequence of the nature of a compreffible or elaftic fluid; and without fuppoling any diminution a diminution of the velocity of the fenfible threads in the real velocity of each particle, there will be whole quantity of air which pafles along it during of the general ftream, and a diminution of the a given time.

Let us now fuppofe a parcel of air impelled along a pipe, which is perfectly fimooth, out of a larger veffel, and iffuing from this pipe with a cer tain velocity. It requires a certain force to change its velocity in the vellel to the greater velocity there which it has in the pipe. This is clearly demonftrated. How long foever we fuppofe this pipe,

there will be no change in the velocity, or in the
furce to keep it up. But let us fuppofe that about
the middle of this pipe there is a part of it which
has fuddenly got an undulated furface, however
imperceptible. Let us further fuppofe that the
En velocity of the middle thread is the fame as
before. In this cafe, the fum total of the motions
of all the particles is greater than before, because
the abfolute motions of the lateral particles is
greater than that of the central particles, which
we lappole the fame as before. This abfolute in-
creafe of motion cannot be without an increase of
propelling force: the force acting now, therefore,
mt be greater than the force acting formerly.
Therefore, if only the former force had continued
to at, the fame motion of the central particles
could not have been preferved, or the progreffive
motion of the whole ftream must be diminished.
And thus we fee that this internal infenfible un-
dubatory motion becomes a real obftruction to the
lexible motion which we obferve, and occafions
at expenfe of power.

The confequence of extending this obftructing Jurface further along the canal, muft evidently be at augmentation of the motion produced, if the central velocity be ftill kept up; for the particles which are now in contact with the fides do not continue to occupy that fituation: the middle purocles moving fafter forvard get over them, and their turn come next the fide: and as they are y moving equally faft, but not in the direction which they are now to be forced, force is nedeary for changing the direction alfo; and this is is addition to the force neceflary for producing the ondulations fo minutely treated of. The conequence of this must be, that an additional force will be neceffary for preferving a given progreffive mation in a longer obtruding pipe, and that the motion produced in a pipe of greater length by a Sve force will be lefs than in a fhorter one, and the eux will be diminished.

Another circumftance has an influence here. Nothing is more irrefragably demonftrated than the neceffity of an additional force for producing x through ady contraction, even though it nald be fucceeded by a dilatation of the paffage. Now both the inequalities of the fides and the undelation of the motions of each particle are equiient to a fucceffion of contractions and dilata although each of thefe is next to infinitely imal, their number is alfo next to infinitely great, and therefore the total effect may be fenfible. Hitherto we have fuppofed, that the abfolute exity of the particies was not diminished: this We did, having affumed that the interval of each andnlation of the fides was without inequalities, But both thefe affumptions were gratuitous. We here no realon for excluding angular afperities. The most certainly often produce real diminubos in the velocity of the contiguous particles; and this muft extend to the very axis of the canal, and produce a diminution of the fum total of mo10 and topreferve the fame sensible progreffive motion, a greater force must be employed. This all that can be meant by faying that there is a relitace to the motion of air through long pipes. What has been faid on this fubject is fuflicient to explain the prodigious and unexpected obftruc

tion to the paffage of air through long and narrow pipes. We may draw an important maxim from it, viz. that all pipes of communication should be made as wide as circumftances will permit; for it is plain that the obstruction depends on the internal furface, and the force to overcome it must be in proportion to the mafs of matter which is in motion. The firft increases as the diameter of the pipe, and the laft as the fquare. The obftruction must therefore bear a greater proportion to the whole motion in a fmall pipe than in a large one. ́

In a very compreffible fluid, fuch as air, each particle may be confidered as a folitary body, actuated by a projectile and a tranfverfe force, ariling from the action of the adjoining particles. Its motion must depend on the adjustment of thefe forces, in the fame manner as the ellistical notion of a planet depends on the adjuftment of the force of projection, with a gravitation inversely proportional to the fquare of the diftance from the focus. The tranfverfe force in the present cafe has its origin in the preffure on the air which is propelling it along the pipe; this, by fqueezing the particles together, brings their mutual repul fion into action. Now it is the property of a perfect fluid, that a preffure exerted on any part of it is propagated equally through the whole fluid; therefore the tranfverfe forces which are excited by this preffure are proportional to the preffure itfelf; and we know that the preffures exerted on the furface of a fluid, fo as to expel it through any orifice, or along any canal, are proportional to the fquares of the velocities which they produce. Therefore, in every point of the undulatory motion of any particle, the tranfverfe force by which it is deflected into a curve is proportional to the fquare of its velocity. When this is the cafe, a body would continue to defcribe the fame curve as before; but, by the very com. preffion, the curvatures are increased, supposing then to remain fimilar. This would require an increase of the tranfverfe forces; but this is not to be found; therefore the particle will not defcribe a fimilar curve, but one which is lefs incurvated in all its parts; confequently the progreffive velocity of the whole, which is the only thing perceivable by us, will not be so much diminished; that is, the obftructions will not increafe fo faft as they would otherwise do, or as the fquares of the velocities. This reafoning is equally applicable to all fluids, and is abundantly confirmed by experiments in hydraulics.

Air in motion is a very familiar object of observation, and it is interefting. In all languages it has got a name. We call it WIND; and it is only upon reflection that we confider air as wind in a quiefcent ftate, and that wind is air in motion.

It is of importance to know the VELOCITY of WIND; but no unexceptionable method has been contrived for this purpofe. The best seems to be by measuring the space paffed over by the fhadow of a cloud; but this is extremely fallacious. For, though we fuppofe that the cloud has the velocity of the air in which it is carried along, this is not an exact measure of the current on the furface of the earth; we may be almoft certain that it is greater; for air, like all other fluids, is reC 2

tarded