Similarly the position of the bucket in the down stroke when its valve opens can be determined.

The greatest height to which the water can be drawn in the barrel, provided it does not reach the bucket, is obtained by putting am-1=m=x; and therefore

x={(a+b)+/{4(a+b)2−(b−c)H};

and therefore a requisite condition is

(b−c)H<}(a+b)2, or BC<‡AB2/H;

otherwise the water would sink during the successive strokes; and the least value of x is thus (a+b).

Finally for the pump to draw, C must be below this level; and now, in full working order with the passages full of water, if h denotes the height of the discharge above the lower valve, and y the height of the bucket at any part of the stroke, the tension of the pump rod is DB(H+h-y)-Dß(H—a—y)=DB(a+h) lb;

so that the work done in one stroke is

DB(a+h)(b−c) ft lb;

the work required to lift the volume of water ẞß(b−c) ft3 through a+h ft.

261. Air Pumps.

In the ancient method of producing a vacuum, as invented by Otto von Guericke, 1650, the vessel to be exhausted (the Magdeburg hemispheres, for instance) was first filled with water, which was afterwards pumped out by a water pump.

The mechanical improvements of the pump made by Boyle, Hooke, and Hauksbee enabled them to dispense. with the water, and to construct the true air pump, as we have it nowadays.

Two suction pumps, side by side, actuated in opposite directions by racks on the piston rods engaging in a

OF HAUKSBEE, SMEATON, AND TATE.

367

toothed wheel between them, worked in a reciprocating motion by a handle, constitute Hauksbee's air pump; and two pumps are used, so that the atmospheric pressure on the tops of the pistons should balance them in any position.

The pumps draw the air through a pipe which terminates in the centre of a horizontal brass plate, upon which the glass jar or receiver, which is to be exhausted, has been placed, the lower edge of the glass having been ground and greased so as to make an air-tight contact with the brass plate.

262. Smeaton's air pump is essentially the lifting pump; he formed it by closing the top of Hauksbee's air pump with a cover, provided with a stuffing box for the piston rod and a valve opening outwards; the piston is thereby relieved from the pressure of the air during the greater part of the stroke, so that two pumps, balancing each other, are not required.

The lower fixed valve may also be dispensed with; and the piston valve too, if the pipe communicating with the receiver enters the side of the barrel at a distance from the bottom a little over the thickness of the piston.

These principles are illustrated in Tate's air pump, consisting of a double acting pump and the receiver (fig. 81); the piston is made long and provided with cannelures, by which leakage of air past it is prevented, in spite of the absence of packing; which may however be supplied by cupped leathers, as in figs. 11, 12, p. 23.

A valve at each end, consisting of a small flap of oiled silk covering from the outside a narrow slit, permits the escape of the air when compressed to the atmospheric pressure; no valve is required in the middle, as the

368

WORK REQUIRED TO

piston is worked just past the communication with the receiver.

263. Denoting by A the volume of the receiver and by B the volume of the barrel of the pump swept out by the passage of the piston in a single stroke; then, in the absence of any clearance, the air which occupied the volume A at the beginning will occupy the volume A+B at the end of the stroke; or denoting by pn-1 and the densities of the air in the receiver at the beginning and end of this the nth stroke

Pn

(A+B)pn=Apn-1,

264. Supposing the temperature constant, the piston valve opens in the return stroke when the air of density Pn occupying the volume B of the barrel is compressed to atmospheric density p; and therefore at a fraction p/p of the stroke, distances diminishing in G.P.

If ẞ denotes the cross section of the pump barrel in ft2, and kp the atmospheric pressure in lb/ft2, the force in lb required to move the pistons in Hauksbee's air pump, measured by the difference of the tensions of the rods, is

A

kpß-kpn-1B :-kpß+kpn-1B

A+Bx

A

B-Bx'

at a fraction of the nth stroke, until x=pn-1/p; after which the valve in the descending piston opens, and the effective tension of this rod is zero.

These tensions can be represented graphically by hyperbolas, as in the hydrometer of fig. 36, p. 113.

265. Work required to exhaust the receiver.

In the nth stroke, the ascending piston in Hauksbee's pump pushes back the atmosphere through a volume B, while the air underneath, originally at pressure kpn-1 and volume A, expands to volume A+B.

Therefore the work done by the ascending piston is, in ft-lb (§ 233),

kpB—kpn-14 log(1+2).

The descending piston yields to the atmospheric pressure kp through a volume B-Bpn-1/p, and at the same time compresses a volume B of air at pressure kpn-1 to a volume

G.H.

Bpn-1.

2A

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EFFECT OF CLEARANCE

Therefore the work done on the descending piston is

k(p— pn-1)B — kpn -1B log p/pn −1·

The work done in the nth stroke is the difference,

kp(A+B)Σ{p2−1(1−p+log p)− (1 − p)np2-1log p} = kp(A+B)(1 − p2+p2log p2)

Չ

if the air is exhausted to one-qth of the atmospheric density; this reduces, when the exhaustion is complete, to kp(A+B) ft-lb, the work required to force back the atmospheric pressure kp lb/ft2 through a volume A+B ft3, as is evident à priori.

A similar result holds for Smeaton's and Tate's air pump, where there is no clearance; the investigation when the effect of clearance is taken into account is left as an exercise.

266. Clearance.

Suppose the piston does not completely sweep out the cylinder, but leaves an untraversed space C, at the bottom of the barrel in Hauksbee's air pump; this space C is called the clearance (espace nuisible, schädlicher Raum).

A volume C of atmospheric air is now left in the barrel at the end of each stroke; and therefore the volume A+B of air at density p, is equivalent to a volume A at density p2-1 and a volume C at density p; so that

(A+B)pn= A pn-1+Cp.