THE SAFETY VALVE.

13

Where the pressure of a fluid is exerted over a circular area or piston, it is often convenient to estimate the pressure in pounds per circular inch, written as lb/O in, or lb/O"; and many pressure gauges attached to hydraulic machinery are graduated in this manner; a pressure of p lb/in2 being πр or '7854p lb/○ in.

Then the thrust on a circular area d inches in diameter is obtained by multiplying this pressure in lb/O" by d2.

It is important in steam boilers that the area of escape from the safety valve should be sufficiently large, so as to allow the steam to escape as fast as it is generated; according to a rule given by Rankine the area of the valve in in2 should be 0.006 times the number of lb of water evaporated per hour.

If the orifice of the safety valve is d ins diameter at the top and conical, the semi-vertical angle of the conical plug being a, then a lift of x ins of the valve will

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THE PRESSURE GAUGE.

give an annular area of internal diameter d- 2x tan a ins, and therefore of area πx tan a(d — x tan a) in2.

But if we consider the valve as a flat disc, of d ins diameter, a lift of x ins will give πdx in2 area of escape sideways.

10. The Pressure Gauge.

To measure pressures continually without blowing off at the Safety Valve, the simplest and most efficient instrument is Bourdon's Pressure Gauge (fig. 5).

Fig. 5.

This consists essentially of a tube AB, bent into the arc of a circle, closed at one end A, and communicating at the other end B with the vessel containing the fluid whose pressure is to be measured.

The cross section of the tube AB is flattened or elliptical, the longer diameter standing at right angles to the plane of the tube AB, thus .

THE PRESSURE GAUGE.

15

The working of the instrument depends upon the principle, discovered accidentally by its inventor M. Bourdon (Proc. I. C. E., XI., 1851), that as the pressure in the interior increases and tends to make the elliptic cross section more circular, the tube AB tends to uncurl into an arc of smaller curvature and greater radius; and the elasticity of the tube AB brings it back again to its original shape as the pressure is removed.

The end B being fixed, the motion of the free end A is communicated by a lever and rack to a pointer on a dial, graduated empirically by the application of known test pressures.

By making the tube AB of very thin metal, and the cross section a very flattened ellipse or double segment, the instrument can be employed to register slight variations of pressure, such as those of the atmosphere; it is then called Bourdon's Aneroid Barometer.

But when required for registering steam pressures, reaching up to 150 or 200 lb/in2, the tube is made thicker; and when employed for measuring hydraulic pressures of 750 to 1000 lb/in2, or even in some cases to 5 or 10 tons/in2, the tube AB must be made of steel, carefully bored out from a solid circular bar, and afterwards flattened into the elliptical cross section, and bent into a circular arc.

Pressures in artillery due to gunpowder reach up to 35,000 or 40,000 lb/in2, and more, say up to 20 tons/in2; or from about 2,500 to 3,000 atmospheres, or kg/cm2; such high pressures require to be measured by special instruments called crusher gauges, depending on the amount of crushing of small copper cylinders by the

pressure.

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THE EQUALITY OF FLUID PRESSURE

11. The Equality of Fluid Pressure in all directions. We may now repeat the Definition of a Fluid given in Maxwell's Theory of Heat, chap. V.;

Definition of a Fluid.

"A fluid is a body the contiguous parts of which, when at rest, act on one another with a pressure which is perpendicular to the plane interface which separates those parts."

From the definition of a Fluid we deduce the important THEOREM. "The pressures in any two directions at a point of a fluid are equal."

Let the plane of the paper be that of the two given directions, and draw an isosceles triangle whose sides are perpendicular to the two given directions respectively, and consider the equilibrium of a small triangular prism of fluid, of which the triangle is the cross section (fig 6).

Let P, Q be the thrusts perpendicular to the sides and R that perpendicular to the base. Then since these three forces are in equilibrium, and since R makes equal angles with P and Q, therefore P and Q must be equal.

But the faces on which P and Q act are also equal; therefore the pressures, or thrusts per unit area, on these faces are equal, which was to be proved.

Generally for any scalene triangle abc, the thrusts or forces P, Q, R acting through the middle points of the sides and perpendicular to the sides are in equilibrium if proportional to their respective sides, so that the pressure is the same on each face; and a similar proof will hold if a tetrahedron or polyhedron of fluid is taken.

If we consider the equilibrium of any portion of the fluid enclosed in a polyhedron when the pressure of the fluid is uniform, we are led to the theorem in Statics that

IN ALL DIRECTIONS.

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"Forces acting all inwards or all outwards through the centres of gravity of the faces of a polyhedron, each proportional to and perpendicular to the face on which it acts, are in equilibrium."

12. The Transmissibility of Fluid Pressure. The Hydraulic Press.

Any additional pressure applied to the fluid will, if the fluid is an incompressible liquid, be transmitted equally to every point of the liquid: this principle of the "Transmissibility of Pressure" was enunciated by Pascal (Equilibre des liqueurs, 1653), and applied by him to the invention of

The Hydraulic Press.

This machine consists essentially of two communicating cylinders, filled with liquid, and closed by pistons (fig. 7); then if a thrust P lb is applied to one piston, of area B square feet, it will be balanced by the thrust W lb applied to the other piston of area A square feet such that

P/B=W/A,

the pressure of the liquid being supposed uniform and equal to P/B or W/A, lb/ft2; and by making the ratio of A/B sufficiently large, the mechanical advantage W/P can be increased to any desired amount.