508 MECHANICAL EQUIVALENT OF HEAT. 417. The constant factor which, according to the First Law of Thermodynamics, converts units of heat which disappear into the equivalent units of work performed is called the Mechanical Equivalent of Heat, and is denoted by J. According to Joule's experiments, as revised recently by Rowland and Griffiths, 1 B.T.U. =779 ft-lb (at Manchester); 1 large calorie = 427 kg-m (at Paris); 1 small calorie = 419 × 107 ergs = 4.19 joules. The reciprocal of J is the Heat Equivalent of Work ; it is generally denoted by A. In the Thermodynamical equations, unless expressly stated otherwise, we adopt one of two systems of units : (i) The British (F.P.S) system of the foot, pound, second, and Fahrenheit scale, and the gravitation measure of force; measuring volume v in ft3, pressure p in lb/ft2, work in ft-lb, heat H in B.T.U.; and thus take J=779. (ii.) The C.G.S. system of the centimetre, gramme, second, and Centigrade scale, and the absolute measure of force; measuring volume v in cm3, pressure p in barads (dynes/cm2), work in ergs, heat in small calories or therms; and take J=4-19 x 107, A = 2.386 × 10-8. As an application, consider the theory of the Injector on Thermodynamical Principles; then if W lb of water is injected by Slb of steam against a pressure head of h ft of water, and if the water injected is raised in temperature from F1 to F2, and if H denotes the total heat of one lb of steam at the boiler temperature F; then the heat which disappears in the Injector is, in B.T.U., where 1 SH-S(F2-32) — W(F1⁄2 — F1), THERMODYNAMICS OF THE INJECTOR. 509 and if the water is lifted h。 ft, and the frictional losses are denoted by L ft-lb, the work done is (W+S)h+Wh2+L. Therefore, by the First Law of Thermodynamics, (W+S)h+Wh。+L=J{SH−S(F2 −32) — W(F2— F1)}. Suppose for example that the boiler pressure is 100 lb/in2, and F=328; suppose also F1=50, F2 = 120; then, neglecting h。 and L, we find W/S=16, about. 1 418. The simplest thermodynamic machine is a gun or cannon; it is a single-acting engine which completes its work in one stroke, and does not work in a continuous series of cycles like most steam engines. When the gun is fired, the shot is expelled by the pressure of the powder gases; the pressure is represented on a (p, v) diagram (§ 197) by the ordinate MP of the curve CPD, OM representing to scale the volume of the powder gases when the base of the shot has advanced from A to M; the curve CPD starts from a point C, such that the ordinate AC represents the pressure when the shot begins to move (fig. 105). The area AMPC then represents the work done by the powder (per unit area of cross section of the bore) when the base of the shot has advanced from A to M, the area ABDC representing the total work done by the powder as the base of the shot is leaving the muzzle B. If OM represents cubic inches and MP represents tons per square inch, then the areas represent inch-tons of work, reducible to foot-tons by dividing by 12. Suppose the calibre of the gun is d inches and the shot weighs W lb; and that it acquires velocity v f/s at M; then equating the kinetic energy and the work done, Wv2/g=2240 x 1πd2 × area AMPC÷12. 510 GRAPHICAL REPRESENTATION OF WORK This supposes the bore is smooth; but if it is rifled with a pitch of b feet, the angular velocity at M is 2v/b; so that if the radius of gyration of the shot about its axis is k feet, the kinetic energy is replaced by To allow for the friction of the bore an empirical deduction, say of ton/in2, is made from the pressure represented by MP. 419. Such a diagram is called the Indicator Diagram of the shot; and if the gun is free to recoil, there is a similar indicator diagram for the gun, representing the pressure on the base of the bore at corresponding points of the length of recoil. The recoil can be measured at any instant by Sebert's velocimeter; the travel of the shot is measured by electric contacts at equal intervals along the bore, and the corresponding pressures are recorded by crusher gauges (§ 10) fixed in the side of the gun; the muzzle velocity is found from electric records outside the gun, and thence is inferred the average pressure in the bore, represented by the ordinate AH, such that the rectangle AB, AH is equal to the area ABDC. A comparison of these different records affords an independent check on the work done by the powder gases, inferred from the experiments of Noble and Abel, and enables us to assign the pressure deduction due to the friction of the bore. As in fig. 42, the curve AQE is drawn, such that its ordinate MQ represents to scale the work done by the powder or the kinetic energy acquired by the shot, each proportional to the area AMPC; and therefore the IN THE BORE OF A GUN. 511 velocity at M may be represented by the ordinate Mv of the curve AvV, 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. 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, pv-pov, the work done on the shot is (§ 233) Povo-pv Po 7-1 γ P1 (0-20), inch-tons, where v in3 denotes the volume of the powder chamber, and v in3 the total volume of the bore. Thus if Po and v are given, the work is a maximum when Ро 1 y(v/v)-1=1, or v。=v(1/2)7-1; 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. |