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Aviation History
1917
1917 - 0275.PDF
MARCH 22, 1917. But, according to Prof. Henderson, the acceleration ahead of the actuator does not contribute to the thrust of the screw, but forms part of a conservative system, and that in con- sequence the thrust, work and efficiency are given by :— Thrust - M.V., Useful work/sec. = M. V..V Total work/sec. = \.M .\(V + F, + F2)2 — (V + F,)s] and therefore efficiency = -~ - Owing to the fact, however, that we are unable to determine the values of F, and F2 for each radius along the blade, the foregoing theories are insufficient for the practical design of airscrews, and in consequence one is forced to experiment and its corresponding empirical methods. THE BLADE ELEMENT THEORY. The most generally accepted theory of the airscrew at the present time is the one in which the reactions on the blades are calculated from the reactions on elementary strips of the blade, determined on the basis of experimental work carried out in wind channels on aerofoil sections. In this method certain empirical " correction factors " have been found necessary in order to bring calculated results into agreement with those obtained by experiment. It is usual to make allowance for these correction factors either at the com- mencement or at the end of the analysis in the practical design of airscrew blades. The method of analysis is briefly as follows :— Let Fig. i represent a section in plan and elevation of an airscrew blade cut off by two concentric radii of (x) and (% + dx) respectively. Let the angle made by the chord of the section with the direction of rotation, or disc of revolution, be denoted by (<p). Let the airscrew have an axial velocity of (V) in the direction of OY, and let it have a revolution speed of (n), so that the axial advance per revolution is (V/n). Let the angle of the effective helix be (A), and denote the angle (# - A) by (a), which may then be called the " apparent angle of attack." It is obvious from the figure that (a) is only the angle between the chord line and the relative wind so long as there is no acceleration of air going on in front of the propeller disc, for otherwise the true " angle of attack " of the section under consideration is less than (a), due to the inflowing velocity ahead of the screw. It is the variations in the analysis caused by an inflowing axial velocity which make it necessary to employ correction factors, although were it possible to predict the amount of such an inflow, the necessity for such empiricism would probably largely cease to exist. The necessary modification in the analysis to take account of the added axial velocity £|.head of the actuator is quite a simple matter, and the trouble is that the amount of such axial velocity is not known. We will therefore start by investigating the geometrical relations which exist between the various com- ponent forces assumed to be acting upon an element of an airscrew blade, and for this purpose we will introduce a modi- fication into the preceding statement of the general analysis by considering the effect of the addition of an axial velocity (Fj) ahead of the screw disc. Fig. i will now become modified as in Fig. 2, and the geometrical statement of the problem is then as follows :— Let Fig. 2 represent a section in plan and elevation of an airscrew blade cut by two concentric radii of (x) and (x + dx) respectively. Let the angle made by the chord of the section with the direction of rotation, or disc of revolution, be denoted by (<p). Let the airscrew have an axial velocity of (V) and a revolu- tion speed of (n), so that the axial advance per revolution is (V/n). Let there be an added axial velocity of inflow (Vx) im- mediately ahead of the actuator disc. Let the angle of the effective helix be (A) and denote as before the angle (<p —A) by (a) so that (a) represents the " angle of attack " of the section when (Vx) is equal to zero. But generally when (Vx) is not zero the real " angle of attack " is not (a) but (at) as in the figure, and then, for all positive values of (F,), (ax) is less than (a), and we may then denote the angle between the disc of revolution and the relative wind by (Ax). We then have at once the obvious relations :— <b = A "f- Of == A1 ~|~ ax and .". Ax = A + o, —ax =• $ — ax, and let (b) denote the width of the section at radius (x). Then we are in a position to at once write down the forces acting upon the element from a consideration of the geometry of the figure. Let Fig. 3 represent the forces acting upon the section at radius (x). 275 (Z,,) denotes the lift of the section considered as an aerofoil. (£>,) .. drag W* Then the resolved parts of (Lx) and (Dx) in an axial direction are :— Lx . cos A1 — Dx . sin Ax and the resolved parts of these two forces in a transverse direction parallel to the disc of revolution are £] . sin Ax +xDt . cos ^4, The first component represents the thrust of the blade element, and the second component represents the torque of vthe element divided by the radius. We then have the twofollowing equations for the thrust and torque of the blade element. Writing (dTx) for the thrust, and (dM ) for the torque we have ' dTx = L, . cos Ax — D, . sin At dMx — x . (L, . sin Ax + Dx . cos Ax) and the efficiency of the element is given by where tan 7, = tan (Ax •+• D,-*-^ Fig. 3. Now returning to the thrust and torque equations we see that since (Lx) denotes the elementary lift on the blade at a radius of (x), we may write L, •» cyx.p.b.dx.v2 = cyx.p.b.dx . (2.v.n.xf . sec2^ since v= 2.it.n.x. sec At Where (cy-i) denotes the absolute lift coefficient of the section at radius (x).- p denotes the mass/density of the fluid. (b) denotes the blade width of the section at radius (x). So that for the thrust of the element we have dTt - cyx.p.b.dx.it.ifi.ni.x'L. sec Ax. [1 - tan Ax. tan 7,] •" So that, denoting by (r0) and (r) the inside and outside blade limits, usually taken to be the boss and tip, we get by integration :— r-r T1=4.*2.n2.P.\cyl.b.x2. sec. Ax. [1—tan Ax. tan 71] . dx giving the total thrust on each blade of the airscrew under the conditions imposed by an axial velocity of inflow. And similarly for the total torque on each blade of an airscrew we have dMx=p.b.cyl.dx.xa.4.vi.n*. sec At. (tan Ax + tan 7,) and therefore the total torque on each blade is given by rr Mx-= 4.irs.n2.p.\cyx.b.x3. sec Ax. (tan Ax + tan 7,) . dx .'r0 and hence the brake horse-power (H) necessary to turn the airscrew is given by, denoting by (N) the number of blades,fr N.8.v>.n!i.i).\cy1.b.xs. sec. Ax. (tan At + tan 7^ . dx rr " fn 55O in lb., foot, second, units and where (p) has the value of (•00238). The total efficiency of the whole blade is obtained as follows from the ratio-total useful work done per second to total work put in per second. (V F.\cyx.b.x°. sec Av (1 — tan Ax. tan yx) . dx 2.ir.n.\cyx.b.xi. sec. Ax. (tan Ax + tan 7,) . dxJr 0Now none of these formula can be applied unless the value of (F,) for every radius (x) is known, so that (Ax) and its functions are also known. This brief summary of the geo- metrical analysis forms what may be termed the blade element aerofoil portion of the theory as enunciated in this paper, and is by way of being a mathematical tool for the investigation of problems of this nature, provided that the required values of the inflow velocity at each radius are known. (To be continued.)
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