Air in

Motion.

by its weight alone. This is evidently analogous to the hydraulic problem of water flowing out of a veffel.

And here we muft be contented with referring our readers to the folutions which have been given of that problem, and the demonftration that it flows with the velocity which a heavy body would acquire by falling from a height equal to the depth of the hole under the furface of the water in the veffel. In whatever way we attempt to demonftrate that propofition, every ftep, nay, every word, of the demonftration applies equally to the air, or to any fluid whatever. Or, if our readers should wifh to fee the connection or analogy of the cafes, we only defire them to recollect an undoubted maxim in the science of motion, that when the moving force and the matter to be moved vary in the same proportion, the ve locity will be the fame. If therefore there be fimilar veffels of air, water, oil, or any other fluid, all of the height of a homogeneous atmosphere, they will all run through equal and fimilar holes with the fame velocity; for in whatever proportion the quantity of matter moving through the hole be varied by a variation of denfity, the preffure which forces it out, by acting in circumftances perfectly fimilar, varies in the fame proportion by the fame variation of denfity.

We must therefore affume it as the leading propofition, that air rushes from the atmosphere into a void with the velocity which a heavy body would acquire by falling from the top of a homogeneous atmosphere.

It is known that air is about 840 times lighter than water, and that the preffure of the atmosphere fupports water at the height of 33 feet nearly. The height therefore of a homogeneous atmosphere is nearly 33X840, or 27720 feet. Moreover, to know the ve. locity acquired by any fall, recollect that a heavy body by falling one foot acquires the velocity of 8 feet per fecond; and that the velocities acquired by falling thro' different heights are as the fquare roots of the heights. Therefore, to find the velocity correfponding to any height, expreffed in feet per fecond, multiply the fquare root of the height by 8. We have therefore in the prefent inftance V=827220,=8X166,493, 1332 feet per fecond. This therefore is the velocity with which common air will rush into a void; and this may be taken as a standard number in pneumatics, as 16 and 32 are standard numbers in the general science of mechanics, expreffing the action of gravity at the furface

of the earth.

It is eafy to fee that greater precifion is not neceffary in this matter. The height of a homogeneous atmofphere is a variable thing, depending on the temperature of the air. If this reafon feems any objection against the ufe of the number 1332, we may retain 8/H in place of it, where H expreffes the height of a homoge neous atmosphere of the given, temperature. A variation of the barometer makes no change in the velocity, nor in the height of the homogeneous atmosphere, becaufe it is accompanied by a proportional variation in the denfity of the air. When it is increased, for inftance, the denfity is also increased; and thus the expelling force and the matter to be moved are changed in the fame proportion, and the velocity remains the fame. N. B. We do not here confider the velocity which the air acquires after its iffuing into the void by its continual expanfion. This may be afcertained by

the 39th prop. of Newton's Principia, b. i. Nay, which Air in appears very paradoxical, if a cylinder of air, communi. Motion cating in this manner with a void, be compressed by a pifton loaded with a weight, which preffes it down as the air flows out, and thus keeps it of the fame denfity, the velocity of efflux will ftill be the fame however great the preffure may chance to be: for the firft and imme diate effect of the load on the pifton is to reduce the air in the cylinder to fuch a denfity that its elafticity fhall: exactly balance the load; and because the elafticity of air is proportional to its denfity, the denfity of the air will be increased in the fame proportion with the load,. that is, with the expelling power (for we are neglecting at prefent the weight of the included air as too inconfi derable to have any fenfible effect.) Therefore, fince the matter to be moved is increased in the fame proportion with the preffure, the velocity will be the fame as before.

285

which

It is equally eafy to determine the velocity with which And the the air of the atmosphere will rush into a space contain- velocity ing rarer air. Whatever may be the denfity of this air, with whi its elafticity, which follows the proportion of its denfity, into a fpace will balance a proportional part of the preffure of the containing atmosphere; and it is the excefs of this laft only which rarer air, is the moving force. The matter to be moved is the fame as before. Let D be the natural density of the air, and the denfity of the air contained in the vessel into which it is fuppofed to run, and let P be the 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 = Now the moving force in the prefent inftance is P-, or P-5 Laftly, let V be the velocity of air rufhing into a void, and the velocity with which it will rush into this rarefied air.

P♪

P♪

D

It remains to determine the time t expreffed in feconds, in which the air of the atmosphere will flow into this veffel from its ftate of vacuity till the air in the veffel has acquired any propofed denfity ♪.

For this purpose let H, expreffed in feet, be the height through which a heavy body muft fall in order to acquire the velocity V, expreffed alfo in feet per fecond. This we fhall 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,

of the air be D.

Since the quantity of aerial matter contained in a veffel depends on the capacity of the veffel and the denfity 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 s. In order to obtain the rate at which it fills, we must take the fluxion of this quantity C. 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 know ledge of the velocity, and the area of the orifice, and the denfity. The velocity is V, or 8H, at the first inftant; and when the air in the veffel has acquired the denfity, that is, at the end of the time, the velocity is 8√/H/1 or 8/H 8/H/D

or 8√H√D_♪

VD

D'

The rate of influx therefore (which may be conceived as measured by the little mafs of air which will enter during the time with this velocity) will be &/HOD✓D, or 8√/HOD√D—t, mul

The motion ceases when the air in the veffel has acquired the denfity of the external air; that is, when C C =D, or when t=4/HO√√D, = ×✔D, 4√/HO' Therefore the time of completely filling the veffel is C 4/HO

Let us illuftrate this by an example in numbers. Suppofing then that air is 840 times lighter than water, and the height of the homogeneous atmosphere 27720 feet, we have 4/H666. 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 shall make enters is an inch square, or of a square foot. Then

1,

8"

T13666'

or

1152"

666 or 1,7297". If the hole is only To of a fquare inch, that is, if its fide is of an inch, the time of completely filling the hogfhead 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 reafons obvious to all who have ftudied hydraulics. In like manner we can tell the time necessary for bringing the air in the veffel to of its ordinary density. The only variable part of our fluent is the coefficient VD, or V1. Let & be

then VV, and —=; and the time is 861" very nearly when the hole is of

an inch wide.

Air in

Motion,

Plate®

Let us now fuppofe that the air in the veffel ABCD (fig. 64.) is compreffed by a weight acting on the CCCCV cover AD, which is moveable down the vessel, and is 287, thus expelled into the external air.

P.

The velo city of air

The immediate effect of this external preffure is to with the comprefs the air and give it another denfity. The additional denfity D of the external air correfponds to its preffure impulfe of Let the additional preffure on the cover of the a weight veffel be p, and the denfity of the air in the veffel down the moving be d. We fhall have P: P+p=D:d; and therefore vessel d-D p=Px Then, because the preffure which exD pels the air is the difference between the force which compreffes the air in the vessel and the force which compreffes the external air, the expelling force is p. And because the quantities of motion are as the forces which fimilarly produce them, we fhall have P:PX=MV: mv; where M and m express the quantities of matter expelled, V expreffes the velocity with which air rushes into a void, and v expreffes the velocity fought. But because the quantities of aerial matter which iffue from the fame orifice in a moment are as the denfities and velocities jointly, we shall have MV: mv=DVV : dvv, =DV1 : dv3. Therefore P:p =DV' : dv.. Hence we deduce.

d-D

d-D D

We may have another expreffion of the velocity without confidering the denfity. We had P: P+p=D:d: DxP+, and d―D= DXP+-D, therefore. d= and d-D-DXP+P

P

and

d-D

d

P : therefore v=Vx ̧ P+p

P+p

288

which is a very fimple and convenient expreffion. Hitherto we have confidered the motion of air as The effect produced by its weight only. Let us now confider the of the air's effect of its elasticity. elafticity

Let ABCD (fig. 64.) be a veffel containing air of confidered any denfity D. This air is in a state of compreffion; and if the compreffing force be removed, it will expand, and its clafticity will diminish along with its density.

Its

Air in its denfity, and V its velocity, and D the denfity of Motion, the air in CFGH. And, after the time, let the density of the air in ABCD be 7, its velocity v, and the denfity of the air in CFGH be . The expelling PD force from ABCD will be P

293 When air

in bellows.

Plate

as

Therefore

at the firft inftant, Pq Pr bea e QV: qu3, which gives

and at the end of the time it will be PD Pq P

we fhall have P.

T=VXQ andthemotion (7), andthe motion will ceafe whens-q.

B

Let A be the capacity of the first veffel, and B that of the second. We have the fecond equation A(Q-9)+BD. AQ+BD=Aq+Bs, and therefores = Substituting this value of ò in the former value of v, we have Q[B (9—D)—A(Q—9)] v=VX which gives q B (Q-D) P the relation between the velocity v and the denfity q. In order to afcertain the time when the air in ABCD has acquired the denfity q, it will be conve nient to abridge the work by fome fubftitutions. There fore make Q (B+A)=M, BQD+BQ'=N, BQBD=R and Mm. m. Then, proceeding as before, we ✔M2—N· obtain the fluxionary equation 8✔HOq

N

AVR AQ-Aq=-Aq whence i X. 9 8√HO√ √q--mq of which the fluent, completed fo that to when qQ AVR ~{m+√(Q2mQ ist=8 /HOM× Log ( q-im+(g*mq)

Some of these questions are of difficult folution, and is expelled they are not of frequent ufe in the more important and by force, ufual applications of the doctrines of pneumatics, at leaft in their prefent form. The cafes of greatest use 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 piston. This laft cafe merits a particular confideration; and, fortunately, the investigation is extremely easy. Let AD fig. 64. be confidered as a pifton moving CCCCV. downward with the uniform velocity f, and let the area of the piston ben times the area of the hole of efflux, then the velocity of efflux arifing from the motion of the pifton will be nf. Add this to the velocity V produced by the elafticity of the air in the first question, and the whole velocity will be V+nf. It will be the fame in the others. The problem is alfo freed from the confideration of the time of efflux. For this depends now on the velocity of the piston. It is ftill, however, a very intricate problem to ascertain the relation between the time and the density, even though the piston 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 denfity of the air in the veffel varies in a very intricate manner, as alfo its refiftance or reaction on the piston. For this reafon, a pifton which moves uniformly by means of an external force will never make an uniform blast by fuc

ceffive ftrokes; it will always be weaker at the beginning Air in of the ftroke. The beft way for fecuring an uniform blaft Motion. is to employ the external force only for lifting up the pifton, 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 correfponding ela iticity exactly balance the weight of the pifton. After this the pifton will defcend equably, and the blast will be uniform. We fhall have occafion to confider this more particulary under the head of PNEUMATICAL Machines. Thefe obfervations and theorems will ferve

to determine the initial velocity of the air in all im

portant cafes of its expulfion. The philofopher will learn
the rate of its efflux out of one veffel into another;
the chemist will be able to calculate the quantities of
the different gafes which are employed in the curious
experiments of the ingenious but unfortunate Lavoi-
fier on Combustion, and will find them extremely diffe-
rent from what he fuppofed; the engineer will learn
how to proportion the motive force of his machine to
the quantity of aerial matter which his bellows muit
fupply. But it is not enough, for this purpose, that
the air begin to iffue in the proper quantity; we must
of its fubfequent paffage.
fee whether it be not affected by the circumstances

to 100.

294

fimilar to

All the modifications of motion which are observed Passage of in water conduits take place alfo in the paffage of air air through through pipes and holes of all kinds. There is the Pipes, &c. fame diminution of quantity paffing through a hole in the motion a thin plate that is obferved in water.. We know of water in that (abating the fmall effect of friction) water if-conduits. fues 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 fhall find the quan tity of water deficient nearly in the proportion of 63 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. Alfo the motion of water is greatly impeded 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, and this in proportion to the fquares of the velocities. Thus, if a machine working a pump caufes it to give a certain number of ftrokes in a minute, it will deliver a deter mined 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 (a thing which frequently happens at the valves), the water must 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 will appear very odd, and is never fufpected by engineers), if no part of the paffage is narrower than the barrel of the pump, but on the contrary a part much wider, and if the conduit be again contracted to the width of the barrel, an additional force must be applied to the pifton to drive the water through this paffage, which would not have been neceffary 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.

Air in The fame thing takes place in the motion of air, and Motion. therefore all contractions and dilatations must be carefully avoided, when we want to preferve the velocity Air fuffers unimpaired.

295

the fame retardation along pipes as water, and the

to this.

Air alfo fuffers the fame retardation in its motion along pipes. By not knowing, or not attending to that, engineers of the first reputation have been prodigiously difappointed in their expectations of the quantity of neceffity of air which will be delivered by long pipes. Its extreme attending mobility and lightness hindered them from fufpecting that it would fuffer any fenfible retardation. Dr Papin, a moft ingenious man, propofed this as the moft effectual method of transferring the action of a moving power to a great diftance. Suppose, for inftance, that it was required to raife water out of a mine by a water-machine, and that there was no fall of water nearer than a mile's distance. He employed this water to drive a pifton, which fhould comprefs the air in a cylinder communicating, by a long pipe, with another cylinder at the mouth of the mine. This fecond cylinder had a piston in it, whofe rod was to give motion to the pumps at the mine. He expected, that as foon as the pifton at the water-machine had compreffed the air fufficiently, it would cause the air in the cylinder at the mine to force up its pifton, and thus work the pumps. Doctor Hooke made many objections to the method, when laid before the Royal Society, and it was much debated there. But dynamics was at this time an infant fcience, and very little understood. Newton had not then taken any part in the bufinefs of the fociety, otherwife the true objections would not have escaped his fagacious mind. Notwithstanding Papin's great reputation as an engineer and mechanic, he could not bring his fcheme into ufe in England; but afterwards, in France and in Germany, where he fettled, he got fome perfons of great fortunes to employ him in this project; and he erected great machines in Auvergne and Weftphalia for draining mines. But, fo far from being effective machines, they would not even begin to move. He attributed the failure to the quantity of air in the pipe of communication, which must be condensed before it can condenfe the air in the remote cylinder. This indeed is true, and he fhould have thought of this earlier. He therefore diminished the fize of this pipe, and made his water-machine exhauft inftead of condenfing, and had no doubt but that the immenfe velocity with which air rushes into a void would make a rapid and effectual communication of power. But he was equally disappointed here, and the machine at the mine ftood still as before.

Near a century after this, a very intelligent engineer attempted a much more feasible thing of this kind at an iron-foundery in Wales. He erected a machine at a powerful fall of water, which worked a fet of cylinder bellows, the blow pipe of which was conducted to the distance 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 Imooth as poffible, it would hardly blow out a candle. The failure was afcribed to the impoffibility of making the pipe air-tight. But, what was furprifing, above ten minutes elapfed after the action of the piftons in the bellows before the leaft wind could be perceived at the end of the pipe; whereas the engineer expected an interval of 6 feconds only.

Motion.

295

No very diftinct theory can be delivered on this fub- Air in ject; but we may derive confiderable affiftance in underftanding the causes of the obftruction to the motion of water in long pipes, by confidering what happens No duitinet to air. The elafticity of the air, and its great com-theory on preffibility, have given us the diftin&teft notions of flui- this subject. dity in general, fhowing 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. We fhall therefore take this opportunity to give a view of the subject, which did not occur to us when treating of the motion of water in pipes, referving a further difcuffion to the articles RIVER, WATER-Works.

296 The writers on hydrodynamics have always confider- How fluids

ed the obftruction to the motion of fluids along canals are ob- in of any kind, as owing to fomething like the friction by "ructed which the motion of folid bodies on each other is ob-along ca ftructed; but we cannot form to ourselves any diftin&t nais notion of refemblance, or even analogy between them. The fact is, however, that a fluid running along a cànal has its motion obftructed; and that this obftruction is greatest in the immediate vicinity of the folid canal, and gradually diminishes to the middle of the ftream. 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 mo tion. The parts in physical contact with the fides and bottom are retarded by these immoveable bodies. The particles of the next ftratum of fluid cannot preserve their initial velocities without overpaffing the particles of the firft ftratum; and it appears from the fact that they are by this means retarded. They retard in the fame manner the particles of the third ftratum, and fo on to the middle ftratum or thread of fluid. It appears from the fact, therefore, 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, as we shall fee by the experiments which have been made on the fubject; and as our experiments on the compreffion of air fhow us the particls of air ten times nearer to each other in some cases than in others (viz. when we fee air a thousand times denfer in these cases), and therefore force us to acknowledge that they are not in contact; it is plain that this obftruction has no analogy to friction, which fuppofes roughness or inequality of furface. No fuch inequality can be fuppofed in the furface of an aerial particle; nor would it be of any service in explaining the obftruction, fince the particles do not rub on each other, but pafs each other at some small and imper ceptible diftance.

We must therefore have recourfe to fome other mode of explication. We fhall apply this to air only in this place; and, fince it is proved by the uncontrovertible experiments of Canton, Zimmerman, 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 phenomenon in hydraulics.

The most highly polished body which we know muft be conceived as having an uneven furface when we compare it with the small spaces in which the corpufcular forces are exerted; and a quantity of air moving