NACA-TN-164

t NATIO?JALAW:S”’RY CWMINTTEE FOR AEROt4AUn~ ? NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS. NO. 164 + . b . . GENERAL THEORY OF WINDMILIJS. By Max M. Munk. —- OCtOber, 1923. -- NATIONAL ADVISORY COMMITTEE FOR AERONA?JTICS. l . . l * *. -. . fi l l - . TgCHNICAL Nom NO. 164. GENERAL THEORY OF WINDMILLS. By Max M, M-ink. summary . In this paper, pz’eparedfor the National Advisory Committee for Aeronautics, the application of the slip curve method to the design and analysis of windmills is discussed and is illustrated by an example. 1. 2. 3. 4* References. Analysis of W. F. Durand’s and E. P. Lesleyls Propeller Tests. By Max M. Munk. N.A.,C.A.Technical.Report No..175. ‘- Analysis of Dr. Schaffran~s Propeller Model Tests. By Max M. Munk. N.A*C.A. Technical Note No. 158. The Analysis of Free Flight propeller Tests and its Application to Design. By Max M. Mhnk. N.A.C.A. Te@nical Report No. 183. Wind Driven Propellers, By ~X M, Munk. N,A.G,A, Technical Memorandum No. 201. - .- Wind driven propellers are much used as small sources of ~wer in an aircraft, e.g., for radio instruments and fuel pumps. In principle, they are nothing but ordinary windmills on a small scale. They differ considerably from ordinary windmills, however, beoause they are constructed from another vZewpoint- In a windmill, the ,. , . . i ,- . < -2- problem is to obtain the greatest possible horsepower at the least .+.--= cost, it being a matter of indifference as to now much wind is utilized, for the wind costs nothing- In employing windmills on .— airplanes, however, allunnecessary retardation of the airplane must be avoided and disregarded. or v by In Papers analysis of and w are the efficiency of such windmills can by no means be -i -> 2, and 3, I presented a new method for the design a propeller. Certain nominal slipstream velocities computed from the thrust or the torque, respectively, the use of the followi~ equation: ThrustThrust coefficient CT = ~u Dz ?; - HorsepowerPower coefficient Cp = an Dz y-sf . 2 where D denotes the propeller diameter, V the velocity of flight and (1) (2) P the density of air. Then: the relative slip velocity computed from the thrust is . .---— -. —. 2 w“”v/v = 1 the relative slip connected with it ~p = + CT ‘1 or ~T= [l-+ -I velocity compiLted.from the horsepower is by the equation 3 2; +2(:)2 1~()‘Zv .. ? + . . * -i l c- -< ,. * a ., * ., -. . -3-” (3) the efficiency appeared to be .- The relative slip velocity when plotted agai,~stthe relative -- tip velocity U/V, where U is the tangential velocity of the pro- peller tip, gave a strai@t line within the working range and this made the method valuable. All this is discussed in detail in the papers referred to. used (4) (5) (6) (7) . This method, ori~lnally invented for propellers, can abO be to advantage in the design or the analysis of Relative slip velocity computed from the thrust: .---—--- J Thrust;=l- l-CT; CT=m2 L/ Relative slip velocity computed from the Efficiency: This 1 and 2. horsepower: equation follows from T = & by substituting equations __ The relation (6) is plotted in Fig. 2- From this curve -4-,& P . . . .. ‘- w can be taken if C* is known. As an illustration of the application of these equations the values of v/V and w/V the relative slip velocities referring to the thrust and that referring to the horsepower, are computed frOm ..._ an actual test (Ref. 4) and in Fig. 1 they a~e plotted against the relative tip velocity U/V. It appears that between U/V = 2 and 2.5 the slip curves so obtained are nearly straight and run parallel to and near each other ——.._ as with propellers. The slip curve for the power is the lower one, the space between the two curves represents the losses in addition to the theoretical slip stream loss. Mean U/V = 2 the slip curve for the thrust has a break and runs then straight again but with a smaller slope. The points computed from the power are less regu- larly arranged. The mill did not rotate bsyond u/v = 2.5 without being driven, in spite of the thrust having still the same direction. In the papers referred to I gave the following approximate form- +- ula the slip modulus m = ~ ~ ~, that is, the slope of the slip ourve 2,* s (8) . F m= 01 + 1*4 * ~u ~ where S is the entire blade area and (U/V)~ the intersecting point of the slip curve and the horizontal axis v~v = O, (U/V)O= 2.9 in this case. With the propellers this formula gave resul;s agreeing within a few percent of the observed value. The windmill investigat- ed had six blaQ-es, the diameter D ms =50 meters and the entire -L -5- ? b=de area was .055 sq.m. This gives m = .36. The curve computed . from the test gives m = .22. That is less than two-thirds of tine between l computd value, and a much larger disagreement/testand equation (~) than with the propellers. . I do not conclude from this, however, that the formula (4) is in itself less reliable for a windmill than it is for a propeller. The . propelle~ test had a Wch ~ma~~er blade area ratio and only ixO b blades. I interfered decreasing suppose that the six blades, comparatively close together, with each other, diminishing the produced lift and hence the value of the slip modulus. A propeller with six blades of the same shape would probably have a similar small slip modulus, . ‘ and equation (4) would not hold better for it either. . t Windmills used to have a larger relative blade area S/D2, ad therefore, wore studies, on the magnitude of the value of the Slip modulus m for such propeller shapes, are desirable. The value of m may be estimated, however, from the test quoted and from experi- “e. ments with propellers with very wide blades. Then the procedure to design a windmill is ~ite similar to the design of a propeller. One point of the slip curve is computed from the desired R-pCMC md .. the velocity of flight ~d from the desir~ or estimated diameter. That slip curve can then be drawn which gives the least change of L variation of revolutions,,similarly as shown in Eef. 4. The inter- section of the slip curve with the horizontal axis gives the pitch ~Ref. 1 and 4) and the slope of the curve gives.the area. This is ‘ discussed in the papers referred to, . -“ by actual trial, Only then will the and can only.be fully understood simplicity of the method appear. ,. l . . 0.5 0.4 -1 . f Q“3 0.1 . 0.0 . Fig. 1 i I I ftk ----1- 1 1 2 3 u/v Slip curves ..z-- — .+.. . , . .. . 1.0 0.9 0.8 0.7 0.6 # 0.5 0.4 0=3 O*2 0,1 1 I i 1 b t q’ .-+- ; h r I —.— ____ . . . ..— j I “! ‘ : i ! , 1 i} 1. I :: ~ I —— -— Cp Fig. 2 ~ against Cp — —- — —.