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Aviation History
1940
1940 - 0276.PDF
98 FEBRUARY T, 1940 FIG.3a or gas is at rest, that is, in the same condition as at influx: — v = V. (4) There is a phase during which the air or gas is brought to rest relatively to the aeroplane, that is to sav, its velocity component in the direction of flight = V relatively to datum. Using the same notation as previously, the force (drag) during the scooping up of the air in the induction system, due to the communication of momentum, is (m/s) V, and the energy per second = FV is therefore (m/s) V2. But this air is now travelling with a velocity V and the energy per second (m/s) V2 is therefore , so that a quantity of energy (a dif- ference) equal to this is not accounted for; by postulate (2) it cannot have disappeared. If we suppose the pipe, shown straight in Fig. 2a (see page 495, December 14, 1939), to be bent through a right angle as in Fig. 3a, the course of the air flow is diverted from the direction A A' to B' B. The momen- tum communicated is the same as before, — (m/s) V, for the air passing along the limb B' B has the same momentum in the direction of flight as though it had been brought to rest relatively to the aeroplane. But in addition to the energy per second = -—— due to the velocity V in the direction of flight, we have now to take into account an equal amount of energy due to the motion of the air in the limb B' B at right angles to the direction of flight, with velocity =V; so now the energy account balances. The two equal vectors indicating the velocity of the limb B' B at right angles to itself (due to the motion of the aeroplane) O P, and the velocity of the air within that limb O Q, each equal to V, have for their resultant the velocity vector OR — VX^a and this is relative to datum; hence the energy per second is, (m/s) x (VJ2)- . , . ,„ -—— —-—— = (m/s) V2 so that, as before, the whole 2 of the energy accounted for. In Fig. 3b, two further limbs, C and D, ha\re been added to represent the ex- haust efflux system. A- We may imagine any- thing to happen as the air passes from B to C, provided that no mechanical loss takes place. For ex- ample, at B we may suppose the air be made to drive a tur- bine, and at C the air be impelled by a fan or blower of some kind, but if there is no loss of energy it wilt come to the same thing if we suppose the limb C to be a direct continuation of the limb B' B, as shown dotted in the figure. Now we will suppose an observer aboard the aeroplane in a closed chamber through which the pipe BC passes, and • let him record the the mass per second and the velocity of the air passing; then the mass per second will be (m/s) — u n- changed. This velo- city we will call v, which in this special ca'-•*- (although the ob- server has no means of knowing it) is equal to V. Assuming that th;; observer has means of measuring the work done, .as would be the case if a blower or impeller of some kind were used at C to impart the velocity v; this blower would be doing work at a rate (m/s) v- (m/s) V2. . . , , , • - = This is the worx performed (per sec.) in the act of propulsion, but the energy released is twice this, since the velocity relatively to datum in the limb B' B C is V x V2- which is the energy available for propulsion; that is to say, the propulsion efficiency is 200 per cent. In eccordarce with postulate 1, the undisturbed air is taken as datum, both as concerns velocity andenergy. The air in the limb B C partakes of the motion of the aeroplane in its flight, (postulate 4). The velocity of the air along the limb B C is inci- dental to the method of representa- tion ; it is a simple method of complying with postulate 2. Although the conduit system figured does not represent or even resemble an actual layout, it serves to show clearly the source of the hidden energy that gives rise to the unexpectedly high efficiency. It is the energy due to the mass projected having originated as part of the aeroplane rind as such being endowed with energy due t;> the velocity of flight, namely the horizontal velocity vector in figs. -5a and 3b. The theory of Torricelli is, strictly speaking, only applicable in the case of an incompressible fluid. But since the general analysis given in article I applies to fluids compressible and incompressible alike, the theory should yield concordant results. This is so. Nowadays we do not drag in "g" in equations in which it really plays no part; the pressure velocity relation /2P • V= .»/ — IIP Fig. 4 is a diagrammatic representation of a rocket, showing 1 chamber containing fluid whose density is p, maintained vt pressure = p, or rather p is a pressure difference, internal is given by" in which p is pressure, and p density. 71 - PRESSURE p - DENSITY AREA • A AREA • a FIG.4. The principle, de.monstrably true for the Borda nozzle, holds equally for any form of nozzle, whether it be a simple orifice or a tapered nozzle like that of a fire hose, but no known mathematical treatment is capable of giving a general solution. (Compare " Aerodynamics " Lanchester, 95—96. Also Proc. Inst. N. Architects, " Theory of Propulsion," 1915.) minus external. The aperture, area A, is provided with a re* entrant or Borda nozzle, through which efflux takes place. When a fluid under pressure flows through an aperture, there is always some contraction, i.e., reduction of area. In the case of the Borda nozzle the degree of contraction is capable of exact calculation, the momentum per second of the issuing jet being equated with the resultant force due to the relief of pressure on an area equal to that of the aperture. Thus, referring to Fig. 4. The mass per second of the issuing jet = (m/s) = apv .'. momentum per sec. = F (force) = (m/s) v = apv2 = ap x — = 2ap; but F = Ap .'. A = 2a. That is to say, the cross-sectional area of the jet is half thatof the nozzle. This is a well-known result. The energy per second imparted to the efflux jet = (m/sJ v (apv).(2P) 2 2p ^ And energy per second effective in propulsion = ApV = 2apV .".Efficiency = *__ =—— (See article I, equation (4).) When in the act of passing the proof of the above for Press, I have received a letter of one more critic, Mr. Medley J. Thomson, who goes over the same ground once more. As far as I can see the only new point he lias introduced is that he has no objection to my expression 2V;v on condition that 1 do not call it efficiency ; this is most kind ! Unfortunately, in my first article (which perhaps Mr. Thomson has not read or may have forgotten) I defined the meaning of efficiency in the manner used universally in discussing the efficiency of the s:rrc-w propellor or more generally of propulsion (page d, Flight Nov. 16th, 1939), and in discussing my thesis no other defini- tion is permissible. The discussion was confined to the steady state of motion, i.e.. V is taken as constant. And the charac- teristic which distinguishes efllux propulsion is that the body propelled is parting with its substance, and with its energy. Now it we had been making an energy account from the time an aeroplane starts from rest to the time it comes to rest we should have found that some of the work done during acceleration had disappeared when it came to rest, and this energy is that which being expended in the propulsion of the aeroplane (or what is lelt ot it) gives rise to the abnormal efficiency. But we are not drawing up such an energy account, we are dealing with a steady state of flight at velocity = V that is as given, and any other assumption would invalidate the treatment in its subsequent applications. F. W. LANCHESTER. * Pcrived thus :—v — \ 2M» : P ~ *<' -!- £, "r * ~- P<'K '. •'• v '
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