IN THE BORE OF A GUN.

511

velocity at M may be represented by the ordinate Mv of the curve Av V, where Mv is proportional to ✅MQ.

Thus if, as in the pneumatic gun, we may take the pressure as uniform and represented by the line HK of average pressure, then AQE will be a straight line, and AvV a parabola; in this case the gun may be made of uniform thickness, calculated by § 290, and great economy in weight is secured.

Fig. 105.

If the curve CPD is taken as a straight line sloping downwards, then AQE is a parabola and AvV an ellipse; if sloping upwards, AvV is a hyperbola.

If the pressure curve is assumed to be an adiabatic, pʊv=pv, the work done on the shot is (§ 233)

where v in3 denotes the volume of the powder chamber, and v in3 the total volume of the bore.

Thus if

when

Po

and v are given, the work is a maximum

1

y(v/v)7-1=1, or v。=v(1/y)ï−1;

reducing, when y=1, to v=v/e.

512

GRAPHICAL REPRESENTATION OF WORK

Fig. 105 represents a 6 inch gun, firing a projectile weighing 100 lb, with a charge of 13 lb of cordite, giving a muzzle velocity of about 2200 f/s.

The length of travel of the shot being 16 ft, and the pitch of the rifling 15 ft, this implies an average pressure of 7 tons/in2.

The initial pressure AC is found to be about 8 tons/in2, rising to a maximum of 16 tons/in2, and falling to 4 tons/in2 at the muzzle.

420. In the cylinder of a steam engine (fig. 106) the steam is admitted alternately to act on each side of the piston, and the operations continue periodically in cycles, when the piston actuates the crank of a revolving shaft by means of a connecting rod, as in an ordinary or locomotive steam engine.

IN THE CYLINDER OF A STEAM ENGINE. 513

We represent, as in the gun, the volume of the steam and its pressure by OM and MP, and now OA represents the clearance at the end of the cylinder (§ 266).

As the piston moves from A to E, the point of cut off of the steam, the pressure is supposed to be equal to the full boiler pressure, represented by the line CF, so that the work done on the piston from A to E is represented by the rectangle AE, AC.

Communication with the boiler is now cut off; and the steam, which at boiler pressure filled the length OE of the cylinder, does work by expansion; so that if FPD is the pressure curve, the area EBDF represents the work done by the expansion of the steam.

Steam is now being admitted in the same manner to the other end of the cylinder (of which the indicator diagram C'F'D' may be drawn below OB), while the steam first admitted is allowed to escape; either into the atmosphere, as in a locomotive engine, when the pressure MP is taken as the gauge-pressure; or else into a condenser, which may be supposed at nearly zero pressure absolute, or at a negative gauge pressure of one atmosphere; and these operations continue periodically.

421. An instrument, called Watt's Indicator, is employed to record the pressure at any point of the stroke, and thence the work done by the steam; it consists of a small cylinder communicating with one end of the engine cylinder (fig. 106).

The Indicator cylinder is closed by a light piston held down by a light spiral spring, the piston rod actuating a pencil which draws a line on a piece of paper wrapped round a brass drum; the moving parts are made as light as possible, to diminish the effect of inertia.

514

WATT'S INDICATOR.

As the engine moves round and the piston reciprocates, the drum is made to revolve through a proportionately reduced distance of the piston travel by a thread attached to a point of the reciprocating machinery, the thread being kept tight by a spiral spring in the drum.

The spring of the Indicator piston gives a displacement proportional to the pressure, so that the pencil traces on the paper a reduced copy of the curve CFD, giving in addition the curve of diminished pressure as the steam is being exhausted, and drawing periodically the same closed cycle, the area of which represents the work done by the steam on one side of the piston in a single stroke; a similar Indicator giving the work done on the other side of the piston.

We may take fig. 106 to represent the diagram drawn by the Indicator, the pencil moving in a fixed line ON, while the paper moves in a perpendicular direction through a proportionally reduced distance; and the ideal Indicator Diagram would be the closed curve ACFDB; the real diagram has the corners more or less rounded, as shown by the dotted line.

The area is either read off by a Planimeter, or else calculated by Simpson's Rule, and the mean pressure P lb/in2 is thence inferred; and now the indicated horsepower of the one end of the cylinder is given by

PLAN 33,000,

where I ft denotes the length of stroke, A in2 the piston area, and N the revs/min.

The Indicator may be made double acting, like the engine, and now the indicator diagram ACFDC'F'D' will be made up by the superposition of the two separate diagrams, end for end.

ELASTICITY OF A GAS.

515

422. The behaviour of a given quantity of gas with varying volume, pressure, and temperature may be studied by supposing it to fill the space OE behind the piston at pressure EF, as in a gas engine; and then determining the curve of pressure FPD as the gas expands from the volume OE=v, and pressure EF=po to any other volume OM=v, at which the pressure is MP=p; we thus discuss the (p, v) diagram, connecting p or MP and v or OM.

If the gas is compressed from OM to OM', or from v to v-Av by reversing the motion of the piston, then Av is the diminution of volume, and the ratio Av/v is called. the cubical compression.

DEFINITION. The elasticity of a fluid under given conditions is the (limiting) ratio of the small increment of pressure to the cubical compression produced.

Thus if the pressure rises from MP to M'P', or from p to p-Ap, then the elasticity is

a positive quantity because p increases as v diminishes. On the diagram,

if the tangent at P meets ON in V.

Thus along an isothermal hyperbola (§ 198)
NV=ON, so that the elasticity is p.

Along an adiabatic

pvconstant,

NV=y. ON, and the elasticity is yp (§ 233).