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Aviation History
1917
1917 - 0329.PDF
APRIL 5, 1917. THE SCREW PROPELLER IN AIR. By M. A. S. RIACH. (Continued SPECIAL CASES. I NOW propose to give a solution for the total efficiency of an airscrew having a certain type of blade outline under the hypothesis of inflow, and also a formula for the brake horse- power necessary to sustain a given weight without axial advance through the air. This latter constitutes the problem of the helicopter. For simplicity in the analysis in both the above cases, a form of blade outline analogous to that defined by Mr. A. R. from page 304.) So that if (b) = 1 ft., (P) - 6 ft., (N) - 2 blades, we have approximately:—(c,) is equal to J, i.e., the gap/chord ratio of the blade is 3 : 1 at this radius. Now return to the general efficiency formula for any type of airscrew blade, viz. :-— Cr \cy1. b . x~. sec*!. (1 —tan^, .tan 7,) .dx P Jf(l Fig. 5. Low as the "Rational Form" will be employed. The difference between Mr. Low's blade function and the one here adopted lies in the modification introduced as the result of the conception of an inflowing velocity ahead of the screw disc altering the value of the " gap " or distance between consecutive blade paths. Fig. 5 explains this point. The " gap " is seen to have a value of 2.v.x.sinA1 • • j•• 'v -'• "~\ - N ^ .„• .':'':•% :-i— ~:~.^-'<- where (N) denotes the number of blades of the airscrew. Hence, making the blade function / (x) equal to the " gap," we have :—• f, ^_ 2.x.x.sia Ax • • .••••• * ct.2.ir.x. sin Jtj Now substitute this value for the blade width (b) in the formula obtained from equating the thrusts, and we get :— and .-. b = ci.f (x) Tan A, =^^ 2.T.X. (f+Ci-cyi. tan yv) where P) denotes the effective pitch of the airscrew, i.e., the advance per revolution, and / = q. (1 -1- V^Vi). And the value of (£>) in terms of (c,) is obtained from the formula-:— cv2.it.x 2.v.x.c1.cyi+f.P s'~ -• ' N / and (C : Tcy]:tan ytf + [ (j is the value in this case of Blade width at radius (x) Gap at radius (x) Vervr roughly this formula for (6) boils down to :— RATIONAL BLAOL FOR/1 £mciE/VCY CMRT V\ = 2 .ir Cr ]cy,.b.xK sec Al (tan Ax + tan 7,) . dx] and substitutes for (Ay) and its functions from the formula already established for (tan .4,). I am, for simplicity and the sake of comparison, taking the lift coefficients and lift/drag ratios as remaining constant over the whole blade, and the value of (r0) as zero. This approximation does not invalidate the value of the results so obtained from the point of view of a guide to the suitable choice of the other parameters involved, such as blade width and pitch ratio. After integration and simplifying we obtain for the effi- ciency of an airscrew having a " Rational " form of blade outline the following formula :— :,- '-¥• - cvcyx. tan yi)~ tan tan sec2+ ,y, 7^ + ^J (,^, + cvcyt. sec2 yv +f. tan 7,) + 20. f*.Z- irwhere Z = _ n.d "It is interesting to notice that the formula contains both (c^) and cyi), i.e., the efficiency is made to depend both upon lift coefficient and blade width, since (cx) is a function of blade width. *» This is as might be expected from the initial hypotheses contained in the theory of inflow developed here. A graph of efficiency against pitch ratio (Z) is given in Fig. 6 from this formula for various values of (cj). It will be noticed that as (cx) increases, i.e., as (6) increases, the total blade efficiency decreases. In a recent paper on the subject, Mr. F. W. Lancnester gave the following formula for the brake horse-power neces- sary to sustain a given weight in the air. This formula was based directly upon the theory advanced by Mr. R. E. Froude. w Mr. Lanchester's formula was — tx 55O. \ W This assumes the inflow velocity to remain constant over the blade as well as the ratio (F; V ), which has the value of unity given to it by R. E. Froude. ./an &» 60 X 4V 20 net 9 ' n—-PITCH RATIO I Fir, 7 / X — \ G/fAPH b i-UO or Tm— ro / ML. > <f UNIT GAINST BMP. 0 /a - DlAtll • It $ i - - DIAMETER IN FEET- 329
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