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 H P^ 6 
 
AIR-SCREWS 
 
 An Introduction to the Aerofoil Theory of 
 Screw Propulsion 
 
 BY 
 
 M. A. S. RIACH 
 
 ) i 
 
 ASSOCIATE FELLOW OF THE AERONAUTICAL SOCIETY 
 
 ** In mediis qure rigore omni vacant resistantiiS corporum 
 stint in duplicata ratione velocitatum." Newton. 
 
 NEW YORK 
 
 D. APPLETON AND COMPANY 
 MCMXVI 
 
PRINTED IN CHEAT liRITAIN 
 
PREFACE 
 
 WITH the coming of the Aeroplane the quantitative study of 
 screws working in air has assumed a great importance. 
 
 Formerly in the design of screw propellers for marine work 
 experiments with models were carried out and the performance 
 of the full size screw calculated from them, and it was not 
 until 1882 that Drzewiecki first drew attention to a possible 
 more powerful method of design obtained by considering each 
 element along the blade as independent and behaving in the 
 same manner as if moving through the fluid in a straight line. 
 
 This method has since assumed great importance in the 
 practical design of air-screw blades, and the results obtained 
 seemed to justify the utilization of this theory as at least 
 approximately correct provided certain limits are not exceeded. 
 
 In the present work the theory has been assumed to be 
 absolutely correct, and the results obtained have been carried 
 to their logical conclusions. This has been done for various 
 reasons. 
 
 It does not make for completeness in any argument if the 
 results of the initial hypothesis are not carried to their ultimate 
 logical conclusions, and although in the present instance the 
 results so obtained may not be completely borne out in practice, 
 yet, in giving an insight into the applications of the theory, 
 and in establishing at any rate an approximate method for 
 dealing with the many cases arising out of the performances 
 of aircraft, the conclusions arrived at will, it is hoped, not be 
 without interest. 
 
 In any case the practical application of some of the more 
 extreme results should not be made without due caution, and 
 
 358026 
 
iv PREFACE 
 
 in fact they are to be regarded as of an extremely tentative 
 nature. This caution is necessary, for " There is no more 
 common error than to assume that, because prolonged and 
 accurate mathematical calculations have been made, the appli- 
 cation of. the result to some fact of nature is absolutely certain. 
 The conclusion of no argument can be more certain than the 
 assumption from which it starts " (Whitehead, " Introduction 
 to Mathematics "). Mathematics are too often apt to be 
 regarded as capable of " creating " results, quite independently 
 of any initial hypotheses, when they are nothing more than a 
 very useful tool. 
 
 I have endeavoured to present the subject of air-screw 
 design in as simple a manner as possible, so that the ordinary 
 non-mathematical reader may be able to follow the train of 
 reasoning, at any rate as far as its qualitative nature is 
 concerned. 
 
 It may be that the first chapter is unnecessarily drawn out, 
 but it appears to me that in any investigation of this kind 
 the first essential is to be able to clearly ''visualise" what is 
 being done, the mere application of analytical processes being 
 but a secondary matter. 
 
 I have introduced graphical methods wherever it seemed to 
 be necessary, or where it was impossible to obtain solutions 
 without them. At the same time the results for the design of 
 an air-screw to fulfil any specified outside conditions have 
 been put into such a form that it is hoped that the design 
 will be able to be correctly carried out by the rules given, 
 even if the analytical processes have not been able to be 
 followed by the reader. 
 
 With regard to the possible errors involved in the applica- 
 tion of the results given in the text, these should not be found 
 to materially affect any but the last chapters of the book. 
 The chapters on " static " thrust, efficiency of air-screws from 
 (V) equal to zero up to the velocity of flight, and on direct 
 lifting systems, are admittedly of a speculative chara'cter. 
 
PREFACE v 
 
 It did not seem that a work of this kind could be regarded 
 as complete without some reference to the stresses occurring in 
 an air-screw blade, and accordingly a chapter on centrifugal 
 and bending stresses has been included. 
 
 It is hoped that the book will be found to be not without 
 interest to engineers desiring an introduction to the theory of 
 air-screws, while at the same time it may perhaps conduce to 
 a more scientific study of the subject, in place of what has 
 been aptly described as the " make it 4 by 2 " methods so 
 dear to the heart of the " practical " man. 
 
 I take this opportunity of thanking Mr. H. Bolas, of the 
 Air Department, Admiralty, for Ids valuable criticisms upon 
 the proofs. My thanks are also due to Mr. T. E. Ritchie for 
 his help in reading through the proofs, and to Mr. A. King for 
 his assistance in the compiling of the various diagrams. 
 
 The Controller of His Majesty's Stationery Office has 
 permitted me to quote from certain of the Technical Reports of 
 the Advisory Committee for Aeronautics. 
 
 M. A. S. R. 
 HENDON. 
 
CONTENTS 
 
 INTRODUCTION. 
 
 PAGE 
 
 PRESSURE ON AEROFOILS . ... 1 
 
 CHAPTER I. 
 THE PITCH OF AN AIR- SCREW ....... 6 
 
 CHAPTER II. 
 THE FORCES ACTING ON AN AIR- SCREW BLADE . . . .14 
 
 CHAPTER III. 
 BLADE SHAPE AND EFFICIENCY ....... 25 
 
 CHAPTER IV. 
 BLADE SECTIONS, AND WORKING FORMULAE . . . . .47 
 
 CHAPTER V. 
 " LAYING OUT " THE AIR-SCREW . . . . . . .72 
 
 CHAPTER VI. 
 STRESSES IN AIR- SCREW BLADES ... .77 
 
 CHAPTER VII. 
 STATIC THRUST .... 
 
viii CONTENTS 
 
 CHAPTER VIII. 
 
 PAGE 
 
 EFFICIENCY OF AN AIR- SCREW AT DIFFERENT SPEEDS OF TRANSLATION 91 
 
 CHAPTER IX. 
 DIRECT LIFTING SYSTEMS ........ 109 
 
 APPENDIX I. 
 NOTE ON THE INFLUENCE OF "ASPECT RATIO" .... 1'zi 
 
 APPENDIX II. 
 NOTE ON THE EFFECT OF THE INDRAUGHT IN FRONT OF AN 
 
 AlR-SCREW . . 123 
 
 INDEX . . .125 
 
AIE-SGEEWS 
 
 INTRODUCTION. 
 
 PKESSURE ON AEROFOILS. 
 
 THE problem of determining the form of air flow generated by 
 an aerofoil, moving through still air, is one that has so far 
 defied mathematical analysis. This problem, which applies 
 equally to water and other fluids, is one for which a solution 
 has only been obtained when the aerofoil is a flat plate of 
 infinite span.* The analytical results however do not conform 
 with those obtained by experiment, and it would seem that the 
 still more complex problem of the curved surface is beyond 
 the reach of present-day analysis. 
 
 The mathematical theory makes the pressure on a flat 
 surface vary as the square of the general velocity of the stream, 
 and experiment has shown that between fairly large limits in 
 velocity this holds good. 
 
 Briefly, if an aerofoil be exposed to a moving current of air 
 or, what is the same thing, if an aeroplane wing be moving 
 through still air, the resultant air pressure on the aerofoil may 
 be expressed by the formula (R = &.S.V 2 ), where (S) denotes 
 the area of the aerofoil surface, and (V) the velocity of flight 
 of the aerofoil relative to the air. (k) is a constant the 
 numerical value of which depends upon the units employed in 
 the measurements of the quantities (R), (S), and (V). 
 
 * A further development has recently been obtained by Professor 
 G. H. Bryan and Mr. E. Jones. "Discontinuous Fluid Motion past a 
 Bent Plane, with Special Eeference to Aeroplane Problems." By G. H. 
 Bryan, Sc.D., F.E.S., and Eobert Jones, M.A. (Proceedings of the Eoyal 
 Society, Vol. 91, No. A 630). 
 
 B 
 
2, : : . : AIE-SCEEWS 
 
 It can easily be seen that (k) has thus the dimensions of 
 density. If we introduce the density of the air into the 
 equation, we can write (E = c./o.S.V 2 ), where (p) denotes 
 atmospheric density, and the constant (c) is then non- 
 dimensional. 
 
 The resultant pressure (E) is composed of two quantities, 
 the pressure on the under surface of the aerofoil and the 
 negative pressure on the top surface of the aerofoil. This 
 latter as a rule forms the principal portion of (E) and in many 
 cases is from three to four times as large as the pressure on the 
 under surface. 
 
 (E) can be split up into two components, measured normal 
 to, and tangential with, the line of flight of the aerofoil. 
 These two components are termed the lift and drag components 
 respectively, and the value of their ratio is termed the lift- 
 drag ratio of the aerofoil. 
 
 Since (c) has been shown to be non-dimensional, the two 
 constants in the expressions for the lift and drag will also be 
 non-dimensional, and accordingly we may write 
 
 L = c^.p.S.V 2 , 
 D = c x .p.S.V 2 , 
 
 where the suffixes ( y ) and ( x ) affixed to (c) denote vertical and 
 horizontal components respectively. 
 
 (c y ) and (c x ) are known as the absolute lift coefficient and 
 absolute drag coefficient respectively. 
 
 From the above it can be seen that the ratio ^- is equal 
 
 to the ratio , and so, when estimating the value of the lift- 
 c x 
 
 drag ratio of an aerofoil, it is unnecessary to know the value of 
 the actual pressures concerned; it is sufficient to know the value 
 of each of the coefficients. 
 
 f* 
 
 The value of this ratio - is of great importance in aeroplane 
 
 c x 
 
 design, and the suitability of an aerofoil shape as a wing section 
 depends largely upon this value. 
 
 It is always desirable to make this ratio as large as possible. 
 
INTRODUCTION 3 
 
 This also applies to the theory of air-screw design based on the 
 aerofoil analogy. 
 
 In records of experimental results, the reciprocal of this 
 ratio is sometimes used. 
 
 n 
 
 The ratio -^ can also be expressed as cot 7, where 7 denotes 
 
 c x 
 
 the gliding angle of the aerofoil. 
 
 The numerical values of (c y ) and (c x ) are found to vary with 
 variations in the angle of the chord incidence of the aerofoil 
 with the direction of motion, and their values are usually given 
 over a large range of angles. 
 
 For many forms of aerofoil, the angle corresponding with 
 
 the highest value of is found to be in the neighbourhood of 4. 
 
 c x 
 
 The value of (c) for normal incidence, e.g. " broad side 
 on," of a flat surface of infinite span is (*44) on the modern 
 mathematical theory, the actual value found from experiment 
 being ( 64). 
 
 There is another quantity upon which the values of (c y ) 
 and (c x ) depend to a certain extent. This quantity is the ratio 
 of the lengths of the span to the chord of an aeroplane wing. 
 It is commonly known as the " Aspect Katio." 
 
 Model aerofoils used for purposes of testing in a wind 
 tunnel usually have a value of (6) for this ratio. As a rule 
 the higher the value of the aspect ratio, the higher the 
 
 value of . 
 
 C* 
 
 It can hardly be said that there is anything essentially new 
 in the development of a theory of air-screws from the analogy 
 presented with the motion of an aerofoil in a straight line. 
 
 This subject was first investigated by Drzewiecki in 1882 
 and has since been developed by Lanchester.* 
 
 The fundamental hypothesis underlying the whole theory 
 here set forth is that : 
 
 (1) Each infinitesimal element along the blade of an air- 
 screw may be treated as a separate aerofoil possessing the same 
 characteristics as those which an aerofoil, having the same shape 
 
 * "Aerial Flight," by F. W. Lanchester. 
 
 B 2 
 
4 AIR-SCREWS 
 
 and haviDg an aspect ratio equal to the aspect ratio of the 
 whole air-screw, would possess ; 
 
 (2) The velocity of each blade element, compounded of the 
 translational velocity and the circumferential velocity, at the 
 point in question, may be treated as causing no appreciable 
 variation from the V 2 law, so that the infinitesimal pressures 
 on every element of the blade may be considered as satisfying 
 the relation K = c.p.S.V 2 . 
 
 Messrs. F. H. Bramwell and A. Fage, of the National 
 Physical Laboratory, say with reference to the application of 
 these assumptions : * 
 
 " There are at present two systems mainly employed in the 
 design of propellers. The first, which is generally used in 
 the design of marine propellers, consists in making small 
 variations from existing successful designs : it is necessary 
 that no very great variation should be made at any one time. 
 
 " The second, which has so far been used almost exclusively 
 for the design of aerial propellers, attempts to predict the 
 performance of a propeller from a consideration of the forces 
 on elementary strips of the blade. This method, if sensibly 
 correct, is far more powerful than the older one, as it affords 
 a means of introducing new features irrespective of whether 
 the variation from existing types is small or large. 
 
 " The initial assumptions underlying this method, which has 
 been developed by Lanchester and Drzewiecki, are that the 
 forces on the blades are due directly to the velocities of the 
 various sections relative to still air, these velocities being com- 
 pounded of the translational velocity and the circumferential 
 velocity at the point in question, and also that the sections 
 may be treated independently. . . . 
 
 " The final conclusion arrived at is that although the method 
 is not strictly correct, yet in the hands of a careful designer it 
 is probably by far the best method that can be used for the 
 design of propellers in the present state of knowledge on the 
 subject. . . . 
 
 " The question of practical importance, however, is whether 
 
 * "Technical Eeport of the Advisory Committee for Aeronautics, 
 1912-13." 
 
INTRODUCTION 5 
 
 the theory affords a sufficient basis for purposes of design. 
 When examined from this point of view it is founcf that, if 
 the range of comparison be limited to that usual in flying 
 machines, the experimental and calculated results are in 
 sufficiently good agreement, and that so long as tho conditions 
 under which the propeller is working are not varied too widely, 
 the theory may be satisfactorily applied. The occasional failure 
 of propellers to satisfy the conditions of design may be due to 
 an overstepping of these limits. In most cases the differences 
 between the calculated and experimental results are not 
 sufficiently large for their effects to be observed in the flying 
 of the completed aeroplane. . . . 
 
 " On the other hand, it is more probable that most of the 
 discrepancies are due to the centrifugal forces on the air in 
 contact with the blade ; this must alter the character of the 
 flow round the blade very considerably, and it is perhaps a 
 matter for surprise that the agreement between the calculated 
 and experimental values is as close as it is, and not that they 
 do not agree exactly." 
 
AIE-SCEEWS 
 
 CHAPTER I. 
 
 THE PITCH OF AN AIR-SCREW. 
 
 IT is fairly obvious that when an air-screw is moving through 
 the aii 1 with some definite translational velocity the distance it 
 traverses in each revolution will be constant, providing the 
 line of flight be horizontal. 
 
 FIG. l. 
 
 Now if we consider any portion of the blade at a distance 
 of (x) feet say from the centre of the boss of the air- screw, we 
 shall find that as the air-screw as a whole moves forward, the 
 portion of the blade under consideration moves up some 
 particular helicoidal path due to the air-screw rotating about 
 its axis at the boss centre. 
 
THE PITCH OF AN AIR-SCREW 
 
 The steepness of the helix traversed by the portion of the 
 blade at radius (x) will depend upon the value of the distance 
 traversed translationally by the air-screw in each revolution. 
 
 It will also depend upon the value of (#), that is upon the 
 distance of the portion of the blade considered from the centre 
 of the boss of the air-screw. 
 
 This may perhaps be more clearly visualized if we imagine 
 
 FIG. 2. 
 
 a cylinder having a radius of (x) and a depth of (P), as in 
 
 Kg. (1). 
 
 Or it may be demonstrated by taking a rectangular piece of 
 paper and drawing a diagonal line as in Fig. (2). 
 
 Then if the paper be rolled so as to form a cylinder having 
 both ends open, the diagonal line will represent the path or 
 helix traversed by the point under consideration ; the diameter 
 of the cylinder so formed will represent twice the distance of 
 the point considered from the centre of the cylinder (i.e. the 
 centre of the boss of the air-screw). 
 
 The circumference of the base of the cylinder will represent 
 
8 
 
 AIE-SCEEWS 
 
 the path that would be traced out, by the point considered, in 
 one revolution, if the air-screw was not moving forward at all, 
 but merely revolving on its axis ; and the length of this circum- 
 ference can be seen to be equal to 
 
 (Tr).(the diameter of the cylinder), 
 
 that is (2.7T.X.), where (x) is the distance of the walls of the 
 cylinder from the centre of the cylinder, corresponding to the 
 
 K*) 
 
 2TTX 
 
 FIG. 3. 
 
 distance of the portion of the air-screw blade considered from 
 its boss centre. 
 
 And the depth of the cylinder will then represent the 
 distance advanced through translationally by the point, and 
 therefore by the whole air-screw, at each revolution. 
 
 If the paper cylinder be now flattened out as in (Fig. 3), it 
 at once becomes apparent that the helix traversed by the point 
 in each revolution of the air-screw is the hypotenuse of a right- 
 angled triangle, and therefore that its length is equal to 
 2 "^ b Euclid L 4 Hr 
 
THE PITCH OF AN AIR-SCREW 9 
 
 Let us denote the angle which the helix line makes with 
 the base by (A). Then if (V) ft./sec. be the translational 
 velocity and (n) the number of revs. /sec. of the air-screw, we 
 see that the distance advanced in each revolution translationally 
 
 y 
 
 is feet. 
 
 This distance we shall here denote by (P), and (P) 
 
 is then defined as being the effective pitch of the air-screw. 
 
 We have so far only considered the path traced out by one 
 portion of the blade at a radius of (x). Suppose that we have 
 a large number of paper cylinders, each having the same depth 
 but of varying diameters. Then their respective diagonals 
 
 will represent the respective helicoidal paths traced out by the 
 various portions of the blade. It is at once obvious that to 
 completely portray the paths traced out by every portion of 
 the blade we should require an infinite number of such 
 cylinders. 
 
 We can, however, do this very easily for a limited number 
 of different parts of the blade by drawing a right-angled 
 triangle having a base equal to 
 
 (2.7r).(half the diameter of the air-screw), 
 
 and having straight lines drawn from the vertex of the triangle 
 to various points along the blade, as in Fig. (4). 
 
 The height of the triangle is then, as before, equal to (P), 
 
10 AIR-SCEEWS 
 
 the effective pitch, and the angles formed by the various lines 
 at the base of the triangle represent the helix angles of the 
 respective portions of the blade considered. 
 
 We may denote these angles by (A 1} A 2 , A 3 , ) corre- 
 sponding to radii from the boss centre of (x lt x 2 , x 3 ). 
 
 The helix angle of the blade tip, that is the angle at a 
 radius equal to half the diameter of the air-screw, may be 
 denoted by (0). 
 
 It can be seen from the figure that since all the helicoidal 
 paths of the various portions of the blade meet in a point, they 
 must all satisfy the relation 
 
 tan A = -., 
 
 p / p \ 
 
 , so that A = tan- 1 ( ^ ), 
 
 2.7T.X \2.7T.xJ 
 
 giving the value of the helix angle (A) in degrees for any 
 radius (x) from the boss centre of the air-screw, providing the 
 value of (P) be known. 
 
 Now suppose that we have an air-screw, and that we 
 measure the actual chord angles of the blade to the disc of 
 revolution at various distances from the boss centre. 
 
 Let us denote these various chord angles by (fa, fa, fa ), 
 
 corresponding to radii of (x l} x 2 , x 3 ) from the centre of 
 
 the boss. 
 
 In some types of air-screws, these angles are designed so 
 that they all satisfy the relation 
 
 (<) being any chord angle measured along the blade.* 
 
 Air-screws of such a type are sometimes said to be of 
 " constant pitch," although this term is somewhat misleading. 
 It must not be confused with the "effective pitch" already 
 defined. 
 
 The above condition of an air-screw having a "constant 
 pitch " is however in the nature of a restriction, and we shall 
 
 y 
 
 * (P) does not necessarily here denote the value of , but usually has 
 
 a larger value than the effective pitch of the air-screw. 
 
THE PITCH OF AN AIE-SCEEW 11 
 
 therefore start with the assumption that the chord angles 
 
 ($i 4>z, <f>3 ) have no specified connection, but that they 
 
 may be anything whatever. 
 
 Now suppose our air-screw, having the chord angles 
 
 (<i> $2> $3 ) as defined, to have a translational velocity 
 
 of (V) feet/sec, and a speed of revolution of (ri) revs. /sec., then 
 the distance advanced through in the direction of translation 
 
 y 
 per revolution is - - feet, and has already been defined as being 
 
 the effective pitch of the air-screw. 
 
 Suppose that we keep (n) constant, and give to (V) the 
 successive values of (V 1} V 2 , V 3 ). 
 
 Then the distances advanced through at each revolution 
 
 . n , V T V 2 V 3 
 will be , , 
 
 n n n 
 
 Thus it is possible to have an infinite number of values 
 for the effective pitch of any given air-screw, correspond- 
 ing to an infinite number of values of the translational 
 velocity (V). 
 
 It is obvious therefore that the effective pitch of any air- 
 screw is not necessarily a fixed quantity, but depends upon the 
 values of (V) and (n). 
 
 It is usual however to associate the pitch of a screw with 
 the screw itself, and thus to imagine it to be a fixed quantity 
 for any given screw. 
 
 In order to determine some analogous expression, in the 
 case of an air-screw, which is a constant quantity for any given 
 type of air-screw, we may define a particular value of the ratio 
 y 
 
 - at which there is 
 n 
 
 (1) no resultant thrust on the blades in the direction of 
 
 translation ; 
 
 (2) no resultant torque on the blades in a direction 
 
 normal to the direction of translation, and there- 
 fore tangential to the disc of revolution of the 
 blades ; 
 
 (3) no " average" reaction on the blades. 
 
12 
 
 AIE-SCKEWS 
 
 The three values of 'corresponding to these three cases 
 
 will be constants for any given type of air-screw. 
 
 The quantitative determination of these values of the 
 effective pitch may be found from the results of the analysis 
 to be proved later. 
 
 We may denote these values of (P) by (Pj), (P 2 ), and (P 3 ), 
 
 or by P),P), and (I). 
 \n/i W/2 \n/ 3 
 
 The "experimental mean pitch" of an air-screw has been 
 
 FIG. 5. 
 
 defined by Mr. F. H. Bramwell as being the value of the ratio - 
 
 TI 
 
 for which there is no thrust on the blades in the direction of 
 translation. 
 
 This definition corresponds to (1) already given. 
 
 Suppose then that our air-screw receives a velocity of trans- 
 lation of (V) feet/sec., and therefore has some definite value of 
 y 
 
 , the effective pitch. 
 n 
 
THE PITCH OF AN AIK-SCREW 13 
 
 , Denoting the helix angles along the blade at radii of 
 
 (x lf a? a , a? 3 , ) by (A lf A 2 , A 3 , ......) and the chord 
 
 angles by (<j> lt $ 2 , </>3, ) we may represent the paths of 
 
 the various portions of the blade considered by Fig. (5). 
 
 We have made the chord angles in each case greater than 
 the corresponding helix angles for the sake of clearness. 
 
 Then it is obvious that, since each of the blade elements 
 considered is moving up the hypotenuse of one of the corre- 
 sponding right-angled triangles, the actual angles at which 
 these blade elements move to their respective helicoidal paths 
 
 are (<i - Aj), (< 2 - A 2 ), (< 3 - A 3 ), , and these angles 
 
 are analogous to the chord angles of incidence of an aeroplane 
 
 wing in flight. They are here denoted by (a l} a 2 , a 3 , ) 
 
 and are termed the "angles of attack" of the various blade 
 elements considered. 
 
 Since the angles (< lf <f> 2 , < 3 , ) are perfectly arbitrary, 
 
 these angles of attack are likewise arbitrary and may be made 
 anything that is convenient. 
 
14 AIK-SCEEWS 
 
 CHAPTEE II. 
 
 THE FORCES ACTING ON AN AIR-SCREW BLADE. 
 
 Now it has already been stated that if an aerofoil be moving 
 in a straight line at a velocity of (V), the two components, 
 normal to and tangential with its line of flight, of the resultant 
 pressure exerted upon it by the air can be written 
 
 L = c y ./>.S.V 2 , 
 D = ^.p.S.V 2 , 
 
 and we have the obvious relation already mentioned, 
 L c y 
 
 D = ;r = cot * 
 
 (7) being the gliding angle of the aerofoil at the particular 
 angle of incidence to its line of flight considered. 
 
 (7) is the angle which the direction of the resultant 
 air-pressure on an aerofoil makes with the direction of the 
 vertical component of (R), that is the lift. 
 
 It will be noticed that the blade sections of an air-screw 
 are similar to those used on aeroplane wings, and hence it at 
 once raises the question: Cannot we treat the sections along 
 an air-screw blade as if they were aeroplane wings moving at 
 angles of incidence of (a lf a 2 > s> ) ? 
 
 It is upon this very assumption that the whole of the 
 theory of air-screw design is based, as already explained in the 
 Introduction. 
 
 It is also obvious that, in order to be able to correctly 
 follow out the consequences of this assumption, we must treat 
 each of the portions of the blade as being of infinitely small 
 oreadth in the direction of the radius (x). 
 
FOBCES ACTING ON AN AIE-SOEEW BLADE 15 
 
 We may then without error sum up the forces on all these 
 infinitely narrow blade strips, and hence obtain a correct 
 quantitative determination of the characteristics of the 
 air-screw.* 
 
 This is of course the ordinary mathematical process of 
 integration. 
 
 Let us then consider the forces acting upon a strip of blade 
 
 FIG. 6. 
 
 at a radius of (x) from the boss centre of the air-screw. 
 Fig. (6) shows the two views of the blade, in side elevation 
 and plan. 
 
 Since the element of blade is moving up the hypotenuse of 
 the right-angled triangle, the two components of the resultant 
 
 * This is of course an assumption of the theory. 
 
16 AIE-SCKEWS 
 
 air-pressure upon it will be normal to and tangential with the 
 hypotenuse of the triangle respectively. 
 
 These infinitesimal pressures may be denoted by (dL) 
 and (dD). 
 
 Let the width of the blade at radius (x) be denoted by (&). 
 Then, applying the formulae for air-pressure, we have 
 
 dL = Cy.p.b.dx.v 2 , 
 and dD = c x .p.b.dx.v 2 . 
 
 And the value of (v) is obviously equal to 
 
 for, as already shown, the quantity ^P 2 4-4.7r.V is the length 
 of the helix traversed by the element of blade at each revolution 
 of the air-screw, and hence the distance traversed by the element 
 per second is (n) times this amount, where (n) is equal to the 
 number of revolutions of the air-screw per second. 
 
 Whence the velocity of the blade element is n. x/PM-^Tr 2 ^ 2 . 
 So that we have 
 
 dL = p.ri*.Cy.1>. (P 2 + 4.7r 2 . 2 ). dx, 
 dD = p.n' 2 .c x .b. (P 2 + 4.7r 2 .z 2 ). dx. 
 
 Now we are not immediately concerned with (dL), but with 
 its components normal to and tangential with the disc of 
 revolution of the blade. 
 
 These are 
 
 (dL. cos A) and (dL. sin A) respectively. 
 
 Again, consider the value of (dD), the drag of the element. 
 We have 
 
 dD = P .n 2 .c x .b. (P 2 + 4.7i 2 .# 2 ). dx, 
 
 and this may also be split up into two components normal to 
 and tangential with the disc of revolution of the blade. 
 These components are 
 
 ( dD. sin A) and (dD. cos A) respectively. 
 
 Now the thrust on the element is measured by the com- 
 ponents of all the forces acting on the element in a direction 
 
FOBCES ACTING ON AN AIR-SCEEW BLADE 17 
 
 normal to the disc of revolution of the blade, that is, parallel 
 with the line of advance of the air-screw. 
 The forces so acting are seen to be 
 
 (dL. cos A) and ( dD. sin A), 
 
 so that the resultant force on the element in a direction normal 
 to the disc of revolution of the air-screw is (dL. cos A dD. sin A), 
 and this may be denoted by (dT), the elementary thrust on the 
 element. 
 
 In a similar manner it can be shown that the remaining 
 forces make up a total force in a direction tangential to the 
 disc of revolution of the air-screw, and of amount 
 
 (dL. sin A 4- dD. cos A). 
 
 This force comprises the drag or resistance exerted by the 
 element to circular motion, and is analogous to a friction 
 brake applied to the rim of a revolving wheel. It tends to 
 retard the motion. 
 
 It can be seen that the product of the above quantity and 
 the distance (x) of the element from the boss centre measures 
 the torque of the element. 
 
 Thus, denoting the torque by (dM.), we have 
 
 dM = x. (dL. sin A + dD. cos A), 
 
 and we have already shown that the thrust on the element may 
 be expressed by 
 
 dT = (dL. cos A-dD. sin A). 
 
 From these two equations and from the equation already 
 obtained for the lift on the aerofoil, we can solve all the 
 problems presented in the design of air-screws. 
 
 We have 
 
 dT = dL. cos A dD. sin A, 
 
 and we know that 
 
 c x drag dD 
 
 tan i = =: "Eft- = dL' 
 
18 AIPt-SCREWS 
 
 whence, substituting, we obtain 
 
 dTl = p.n 2 .Cy.b. (2.7r.x P. tan 7).\/P 2 -f-4.7r 2 ..?j 2 . dx, 
 and therefore 
 
 T = p.n 2 . A (2.7r.a;-P. tan 7)VF+47T 2 ^?. dx, 
 Jr, 
 
 giving the total thrust on each blade of the air-screw. 
 
 (r) denotes the length of each blade from the boss centre. 
 
 (r ) denotes the length from the boss centre to the extreme 
 portion of the blade where the " streamlining " of the sections 
 ceases. As a rule, in determining approximate values for the 
 integrals, we may take (? 1 ) as equal to zero without appreciable 
 error. 
 
 We have also 
 
 dM = x. (dL. sin A+d~D. cos A) 
 = p.tf.Cy.l.x. (P + 2.7r.x. tan 
 
 and therefore 
 
 M = .?i 2 . \C.b.x. P + 2.7r.#. tanVP 2 + 4.7T 2 . 2 . dx. 
 
 And since (M.2.7rji) is proportional to the B.H.P. required 
 to turn each blade of the air-screw, we can at once determine 
 the necessary value of the blade width at each radius to satisfy 
 any given set of conditions. 
 
 We are now in a position to write down the efficiency of 
 the whole blade, that is, of the air-screw. 
 
 The work done per revolution by the air-screw is the 
 product of the total thrust exerted and the distance through 
 which it is exerted, that is, the quantity having the value of 
 
 Y 
 
 - and defined as the effective pitch of the air-screw. 
 n 
 
 Let (N) denote the number of blades of the air-screw. 
 Then we have 
 
 work done^by air-screw per revolution = (N.T.P.), 
 and 
 
 work done by motor per revolution in 
 
 turning the air-screw at (n) revs./sec. = (N.M.2.7T), 
 
FOKCES ACTING ON AN A IE-SCREW BLADE 19 
 
 and therefore the efficiency of the whole air-screw is given by 
 N.T.P. 
 
 V = 
 
 and /. T = 
 
 N.M.2.7T. 
 P. I l.Cy. (2.7T.-P. tan 7)VFT4P r . 2 . dx 
 
 Jr 
 
 5.7T. b.Cy.x. 
 
 Jr n 
 
 .7r 2 .a 2 . dx 
 
 This gives the general formula for the efficiency of any type 
 
 V 
 of air-screw, for any specified value of . 
 
 We may now endeavour to determine the values of the 
 pitch, corresponding to no thrust on the blades, to no torque 
 on the blades, and to no resultant " average " reaction on the 
 blades. 
 
 The first of these quantities is called the " Experimental 
 Mean Pitch," and we shall now endeavour to determine by 
 calculation the value of this quantity for any given set of 
 
 conditions. 
 
 y 
 This value of the ratio is usually determined for any 
 
 given air-screw experimentally in a wind-tunnel. 
 
 The method to determine this theoretically is as follows. 
 
 We have already shown that the expression for the 
 thrust on each blade of any given type of air-screw may 
 be written as 
 
 T = p.n 2 . \Cy.l. (2.7T.&-P. tan 7). 
 Jr 
 
 . dx, 
 
 Y 
 
 where P = . 
 n 
 
 Now if T = 0, then since (p.n*) is finite we have 
 
 fr 
 
 = Cy.l. (2.7r.a;-P. tan 7). \/P 2 +4.7r a .a; 2 . dx, 
 
 Jr 
 
 C 2 
 
20 AIR-SCREWS 
 
 fr _ _ 
 
 i.e. 2.7T. I C.l).x. \/P 2 -{-4.7r 2 . 2 . dx 
 
 fr _ _ 
 
 2.7T. I Cy.l).x. \/P 2 -{-4.7r 2 . 2 . 
 Jr 
 
 = P. c y .t>. 
 Jr 
 
 tan 7. \P 2 + 4.7r 2 . 2 . dor. 
 
 And this may be solved graphically as follows. 
 
 Take successive values of (P) from some value greater than 
 what would correspond to the effective pitch of the air-screw 
 upwards, and plot the two graphs 2.7r.%.6.#.\/P 2 -{-4.7r 2 .a; 2 and 
 P.Cy.6. tan 7. \/P 2 + 4.7r 2 .a3 2 against (x) between (?) and (r) for 
 each of the values of (P) taken. Take the areas of the two 
 figures thus obtained in each case. When the two areas in any 
 case are numerically equal, then the value of (P) taken for this 
 case is the value of the " experimental mean pitch " of the air- 
 screw. 
 
 It is of course apparent that the values of (c y ) and (tan 7) 
 will vary with each value of (P) taken. 
 
 We can, however, determine their respective values in the 
 following manner. 
 
 Let us denote the successive arbitrary values of (P) taken 
 by (P', P", P'", ...... ), and let us suppose that P' < P" < P'" < 
 
 Moreover, let us denote the chord angles along the blade of 
 
 the air-screw by ($!, < 2 , $3 ). Then if (A/, A 2 ', A 3 ' 
 
 ) be the helix angles at radii (x it x 2 , x 3 ) 
 
 y 
 respectively, for a value of equal to the first value of (P) 
 
 taken, namely (P'), the value of each of these helix angles at 
 any radius (x) will be given by the relation 
 
 A' = tan- 
 
 2.7T.O?./ 
 
 And therefore the values of the respective angles of attack 
 of the sections at these radii will be given by 
 
 a = (<j> A') = < tan" 1 
 for the particular value (P f ) considered. 
 
FOKCES ACTING ON AN AIE-SCEEW BLADE 21 
 
 Similarly, it may be shown that the angles of attack of the 
 sections for the other values of (P) taken, namely (P", P"' 
 ), will be given by 
 
 '" 
 a'" = - tan- 1 - 
 
 Now measure up the forms of the sections of the blade at, 
 say, (8) different radii. Then, if the sections so obtained are 
 ones that have been tested as aerofoils in a wind-tunnel, we 
 can at once write down their characteristics for any value of 
 the angle of attack (a). 
 
 We may thus plot a series of graphs, for each of the 
 arbitrary values of (P) taken, of (c y ) and (tan 7) for the 
 different radii taken along the blade. 
 
 If we then construct a series of such graphs for (c y ) and 
 (tan 7) for all the values of (P) taken, namely (P, P", P'" 
 
 ), we shall be in a position to determine the value of 
 
 the experimental mean pitch. 
 
 And we have to plot the two graphs already given for all 
 the values of (P) taken. 
 
 In order to be able to plot these two curves in any case, 
 we must draw out a curve of the actual blade widths against 
 radii (#). 
 
 It then only remains to obtain the two areas enclosed by 
 these two curves and the (x) axis. The value of (P) taken, 
 which makes these two areas equal, is the value of the experi- 
 mental mean pitch, corresponding to no thrust on the blade. 
 
 y 
 
 In a similar manner we can determine the value of for 
 
 n 
 
 which there is no torque on the whole blade. 
 
 V 
 
 To determine the value of -- for which there is a zero 
 
 n 
 
 value of the " average " reaction over the whole blade, it is first 
 necessary to investigate the value of this resultant pressure at 
 any radius (x), and so compute the value over the whole blade 
 of the air-screw. 
 
22 
 
 AIR-SCREWS 
 
 We shall then be in a position to estimate the value 
 y 
 of - - for which this " average " resultant pressure over the 
 
 whole blade is zero. 
 
 In order to obtain the value of this resultant pressure (R) 
 we proceed as follows. 
 
 Consider a blade section at radius (#), and having a helix 
 angle of (A), Fig. (7). 
 
 The resultant air-pressure (R) upon the section is usually 
 
 T 
 
 FIG. 7. 
 
 split up into two components called the lift and drag of the 
 section. Let (7) denote the gliding angle of the section. 
 Then 
 
 Lift = R. cos 7, 
 and 
 
 Drag = R. sin 7, 
 whence 
 
 Lift 
 
 = cot 7. 
 
 Now we have already shown that the resultant thrust on 
 the section, in a direction parallel to the line of advance of the 
 whole air-screw, is given by 
 
 T = L. cos A-D. sin A, 
 
FOECES ACTING ON AN AIE-SCKEW BLADE 23 
 
 where (L) and (D) denote the lift and drag components of (R) 
 respectively. 
 But 
 
 L = R. cos 7, 
 and 
 
 D = R. sin 7, 
 whence 
 
 L. cos A D. sin A = R. cos 7. cos A R. sin 7. sin A, 
 
 and therefore 
 
 T = R. cos (A + 7), 
 whence 
 
 R = T. sec (A + 7) for every point along the blade. 
 
 It is also obvious that 
 
 1 
 
 sec (A +7) = - T : 7^ 
 
 cos A. cos 7 sin A. sm 7 
 
 2.7T.OJ. cos 7 P. sin 7' 
 whence 
 
 T. \/P 2 +4.7r 2 .a; 2 
 
 ~ cos 7. (2.7r.a-P. tan 7)' 
 and this in strictness should be written 
 
 ~ cos 7. (2.ir".i-P. tan 7)' 
 since the pressures considered are infinitesimals/ 
 
 And dT = p.n 2 J>.Cy. (2.7r.aj-P. tan 7). 
 so that 
 
 ^R = p.?i a .6.c y . (P 2 +4.7r a .aj 2 ). sec 7. ^, 
 whence 
 
 R = p.n\ Ib.Cy. (P 2 + 4.7r 2 .^ 2 ). sec 7. dx, 
 
 Jro 
 
 and if this be now plotted against (x), we shall obtain the 
 " Load Grading Curve " for the whole blade. 
 
 Since (tan 7) is usually nearly constant between (r) and 
 
24 AIE-SCEEWS 
 
 (r ), and also of small amount, we may write (sec 7) equal to 
 unity, and our expression for the total resultant force over the 
 whole blade then becomes 
 
 fr 
 
 R = p.w?. I b.Cy. 
 
 Jr 
 
 dx, 
 
 and the curve obtained when integrating this expression 
 graphically will not appreciably differ from the one obtained by 
 inserting the rather troublesome (sec 7) under the integral sign. 
 
 Having then obtained the value of (R) for each blade of the 
 air-screw, we may proceed to determine the value of (P) for 
 which (B) has a zero value, in the same manner as for the case 
 of a zero value of the thrust on each blade. 
 
 y 
 
 In thus estimating this value of it will be prudent to 
 
 insert the (sec 7) in our expression for (R). 
 We have then 
 
 R = p.n 2 . I b.Cy. (P 2 + 4.7T 2 .# 2 ). sec 7. dx, 
 
 Jr 
 
 and since (R) = 0, and (p.n 2 ) is finite, we obtain 
 
 (r 
 
 = I hep (P 2 -f 4.7r 2 . 2 ). sec 7. dx. 
 
 J^o 
 
 / c 2 c. 2 
 Now (sec 7) = \/ I -f- 2, and -^ is always a positive quantity. 
 
 Cy Cy 
 
 Hence (sec 7) is always a positive quantity, and therefore 
 (cy) must be negative, when (R) is equal to zero, for at least 
 some portion of the blade. 
 
25 
 
 CHAPTER III. 
 
 BLADE SHAPE AND EFFICIENCY. 
 
 WE return now to a consideration of the formula already 
 deduced for the efficiency of any type of air-screw blade. 
 
 It is obvious that, if the sections at every radius from the 
 boss centre along the blade were of such a form (if it were 
 
 s* 
 
 possible) as to possess no drag, so that the value of would be 
 
 c x 
 
 infinite, the efficiency of the whole air-screw would be unity. 
 That this is so may be seen by making (tan 7) equal to zero 
 in the expression already obtained for the efficiency of any 
 type of blade. 
 We have 
 
 P. I Cy.b. (2.7r.x-l\ tan 7). \/W+^r\x\ dx 
 
 n - J r 
 
 V- r jr 
 
 2.7r. c y .b.x. (P + 2.TT.X. tail 7). \/P 2 4- 4.7^.0?. dx 
 
 Jr 
 
 and now, making (tan 7) = 0, we get 
 P. I Cy.b.2.7r x. 
 
 .7T. 
 
 . dx 
 
 Unfortunately, however, (tan 7) never has the value of zero, 
 although the smaller its value the higher the efficiency and 
 vice versa. 
 
 It is obvious, since (&) is a function of the radius (x), that 
 the value of (77) will vary for different forms of blade outline, 
 
26 AIPv-SCREWS 
 
 ? 
 
 and it therefore becomes necessary^to find some form which 
 will give as large a value as possible to (?;) consistent with 
 structural considerations. 
 
 Let us consider the efficiency of any blade element at 
 radius (x) from the boss centre. 
 
 We have already shown that the nett thrust of the element, 
 in a direction normal to the air-screw's disc of revolution, is 
 given by 
 
 dT = dL. cos A (1 -tan A. tan 7), 
 
 and that the nett drag of the element, in a direction tangential 
 to the disc of revolution of the air-screw, may be written 
 
 c?R* = dlj. sin A-f-^D- cos A 
 
 = dL. sin A (1 + cot A. tan 7), 
 
 and therefore the work done by the element per revolution of 
 the air-screw is 
 
 dT.P = P.</L. cos A (1 -tan A. tan 7), 
 
 and the work done by the motor in turning the element per 
 revolution of the air-screw is 
 
 dP\,.2.Tr.x = 2.7r.x.d~L. sin A (1-fcot A. tan 7), 
 whence the efficiency of the element is given by 
 
 _ P. cot A. (1 - tan A. tan 7) _ tan A 
 ^ A ~ 2.7r.x. (1+cot A. tan 7) ~ tan (A + 7)' 
 
 where the suffix ( A ) in (?7 A ) signifies the efficiency of an element 
 at a helix angle of (A). 
 
 We might also write this as (rj x ) denoting the efficiency of 
 an element at a radius of (x). So that (T; A ) = (rj x ). 
 
 If we require the efficiency of the element in terms-of the 
 radius (x), we have 
 
 P. (2.7T.0-P. tan 7) 
 Vx ~ Z.TT.X. (P + 2.7T. tan 7. x)' 
 
 We have thus shown that the efficiency varies at different 
 points along the blade, and hence we can plot a curve of efficien- 
 cies against values of (x). Such a curve is shown in Fig. (8). 
 
 * Not to be confused with the resultant pressure on an element, 
 denoted by the same symbol. 
 
BLADE SHAPE AND EFFICIENCY 27 
 
 The maximum point of efficiency along the blade may be 
 found by putting 
 
 The point of maximum efficiency is found to be at a value 
 
 of (A) = 45-? that is, in the neighbourhood of 43. 
 
 ^ -* 
 
 Since tan A = [ ^ ), we can write 
 
 j.TT.XJ 
 
 , giving the value of (x) for the 
 
 2.7T. tan 
 maximum point of efficiency along the blade. 
 
 i' a' 3' 4' 
 
 Radius. (x) 
 FlG. 8. 
 
 The point of maximum efficiency so obtained is only strictly 
 true when (7) is supposecf to remain constant over the whole 
 blade, that is, for every radius. 
 
 ft 
 
 But in practice the values of - x - for the various blade sections 
 
 Cy 
 
 will not be found to vary very greatly, and hence the point of 
 maximum efficiency will not fall very far short of that given 
 by the formula. 
 
 It is obvious from the foregoing that the most efficient blade 
 would be one in which the whole of the blade surface was 
 concentrated at the point of maximum efficiency. The blade 
 width (&) would then become infinite and the length of the 
 blade in the direction of the radius (x) would be infinitesimal. 
 
'28 AIB-SCKEWS 
 
 As, however, such a form of blade is impossible to construct 
 we are forced to adopt a compromise between the width of the 
 blade and the diameter of the air-screw. 
 
 Let us first consider the point along the blade at which 
 the efficiency reaches a maximum value. We may regard (7) 
 as a variable, and we will denote it by tan" 1 ty(x), so that 
 tan 7 = -ty(x) ; then we have 
 
 2.TT.X. 
 
 7 7O 
 
 and for a maximum value of (%),-^ = 0, and , ^ is negative. 
 
 ClX CLX 
 
 So that 
 
 and this expression gives the value* of (x) for which (rj x ) is a 
 maximum. 
 
 In order to evaluate the above it is necessary to know the 
 form of the function ty(x), i.e. (tan 7). 
 
 If (tan 7) is constant over the whole blade, then ^r'(x) 
 vanishes, and the above expression reduces to 
 
 P 
 
 !.7r.tan(45-|) 
 
 We thus obtain the original relation 
 A = 45 - 
 
 for the angle at which the point of maximum efficiency occurs 
 along the blade. 
 
 Now we have not as yet prescribed any particular value 
 to (&), the width of the blade at radius (x). 
 
 In the majority of air-screw blades the value of (Z>) varies 
 for different values of (x), i.e. for different radii along the 
 blade. 
 
 Now it is obvious that the blade should, from considerations 
 of overall efficiency, be wider at some radii than at others. 
 Hence in designing our blade we may plot a curve of propor- 
 
BLADE SHAPE AND EFFICIENCY 29 
 
 tional blade widths against radii (x\ so that the true or actual 
 blade width for each value of (x) will be equal to the width 
 shown on the curve multiplied by some constant. 
 
 We may thus write the blade width (&) as equal to the 
 product of a constant and a function of (x), or 
 
 b = C.f( X ). 
 
 Now we have already assumed that (7) is a variable and 
 may have different values for different values of (x), and we 
 
 /> 
 
 have denoted the ratio of by ^(x) y so that 
 
 Cy 
 
 tan 7 = ty(x\ and therefore 
 
 7 = tan" 1 Tfr(x), as already given. 
 
 /i 
 
 But if (7) and therefore is a variable or function of (x), 
 
 Cy 
 
 so also is (c y \ the absolute lift coefficient of the section at 
 radius (x). Hence we may also write this in the form 
 
 It has already been shown that 
 
 P. 2.7T 
 
 which when plotted against radii (x) will give a curve of 
 efficiency for each value of (x). 
 
 Now suppose we regard the curve so obtained as our pro- 
 portional blade width curve, so that at each point along the 
 blade the actual blade width is proportional to the efficiency 
 at that point. We shall then ensure at any rate that our blade 
 is widest at its most efficient point. This will not of course 
 necessarily give the most efficient form of blade outline, but 
 the efficiency of such a type of blade will be greater than that 
 of one possessing the same characteristics in other directions 
 but having a uniform blade width throughout. 
 
 If we turn to the " efficiency curve " already given, we shall 
 notice the following. 
 
 (1) As (x) increases from zero up to about (J) the length 
 of the blade, the efficiency rises very rapidly. 
 
30 AIR-SCREWS 
 
 (2) As (x) increases, from the value corresponding to the 
 point of maximum efficiency, to its maximum value (i.e. the 
 length of the blade), the efficiency steadily decreases, although 
 somewhat slowly compared with its initial rise from zero up 
 to its maximum value. 
 
 (3) Hence, after the point of maximum efficiency has been 
 reached, the blade becomes less and less efficient as we proceed 
 to the tip. Hence the useful work done by the blade elements 
 decreases towards the blade tip. 
 
 (4) This at once suggests the advisability of making the 
 actual blade widths proportionally less at the outside radii than 
 they would be if made exactly proportional to the efficiency 
 curve. The second curve in Fig. (9) would then be a more 
 
 efficient shape than the efficiency curve. The maximum ordinate 
 is now further to the right than in the efficiency curve. 
 
 There are however other considerations bearing upon the 
 problem of the most efficient blade outline besides those 
 already given. 
 
 If we pursue still further the analogy with an aeroplane 
 wing, it at once becomes apparent that there may be a limit to 
 the useful blade width possible in an air-screw. 
 
 In the case of an aeroplane having two or more superposed 
 surfaces (e.g. as in a biplane) it is well known that the lift of 
 the lower wing is affected quite appreciably by the " wash " of the 
 top wing, and that the smaller the vertical distance between 
 the two surfaces the greater is the loss in lift on the lower wing. 
 
 This vertical distance between the wings of an aeroplane is 
 
BLADE SHAPE AND EFFICIENCY 
 
 31 
 
 known as the " gap," and the ratio between this distance and 
 the width of each wing in the line of flight is known as 
 
 the " |f , " ratio. This ratio has usually a value of between 
 chord 
 
 (8) and (1*2) and is often equal to unity in standard types of 
 aeroplanes. 
 
 The value of this vertical distance or " gap " in the case of 
 the blades of an air-screw is equal to 
 
 P. cos A 
 
 for an air-screw having (N) blades. 
 
 (fl 
 
 FIG. 10. 
 It is the value of the vertical distance between any two 
 
 consecutive helicoidal paths after ^ th of a revolution and after 
 
 2 
 
 ^ ths of a revolution respectively. 
 
 This may be illustrated if we have recourse once again to 
 the paper cylinder. 
 
 Consider an air-screw having (4) blades. And draw on the 
 rectangular piece of paper (7) parallel lines at equal distances 
 apart, Fig. (10). 
 
32 
 
 AIK-SCREWS 
 
 Now fold the paper into a cylinder having a depth as before 
 of (P), Fig. (11). Then the lines so drawn will represent the 
 helicoidal paths traced out by the same point on each of the (4) 
 blades of the air-screw. We notice that the vertical distance 
 or " gap " between any two such consecutive paths is equal to 
 
 P. cos A 
 
 If then we make the width of the air-screw blade at any 
 
 FIG. 11. 
 
 radius (x) proportional to the vertical distance between any 
 two consecutive helicoidal paths, we introduce an allowance for 
 possible blade interference, Fig. (12). 
 So that we have 
 
 P. cos A 
 
 whence 
 
 I a 
 
 P. cos A 
 
 "IT" 
 
BLADE SHAPE AND EFFICIENCY 
 CL 
 
 33 
 
34 
 
 AIR-SCEEWS 
 
 An air-screw having this type of blade outline has been 
 called by Mr. A. E. Low the " Eational " blade. We shall 
 refer to this type of blade by that name in what follows. 
 There are other types of air-screw blades, and we shall now 
 consider two of these further types. 
 
 The first type considered is one which would at first sight 
 seem to be the simplest possible type of blade, the blade of 
 uniform width. Here obviously (&) is independent of (x), 
 since it is the same for all values of the radii. 
 
 FIG. 13. 
 Hence (&) = constant = c.f(x), whence /(,>;) = 1, and therefore 
 
 (*)=(). 
 
 And (c) has always the value of the ratio 
 
 actual blade width at radius (x) 
 scale blade width at radius (x) ' 
 
 In this case we can take the scale blade width as being 
 equal to unity. 
 
 A blade having a constant chord width has been called by 
 Mr. A. E. Low the " Normale " blade. We shall use this name 
 when referring to this type of air-screw blade. 
 
 The second form of blade considered is one which from a 
 constructional point of view forms the limiting curve of 
 construction in which all the lamina pass through the boss. 
 
 If in any type of air- screw the blade widths at any point 
 come outside this curve, it is not possible to construct such a 
 blade without using offsets on the laminae. 
 
 Such a form of blade is shown in Fig. (13). 
 
BLADE SHAPE AND EFFICIENCY 35 
 
 The form of blade outline is seen to be given by 
 
 b = B. cosec (A + a), 
 whence 
 
 c.f(x) = B. cosec (A + a), 
 and therefore 
 
 c = B, and f(x) 
 and therefore 
 
 " PTcos a + 2.Tr.x. sin a 
 and this expression does not greatly differ from 
 
 the one obtained by making (a) = 0. This, as will be seen later, 
 greatly simplifies the subsequent evaluation of the quantities 
 characteristic of such a type of air-screw blade. 
 
 This form of blade may be termed the " Constructional 
 Limit " type. 
 
 We shall now apply the formulae already deduced for 
 any type of air-screw blade to the four blade types already 
 considered, namely : 
 
 (1) The " Efficiency Curve" type of blade outline. 
 
 (2) The " Bational" type of blade outline. 
 
 (3) The " Normale " type of blade outline. 
 
 (4) The " Constructional Limit " type of blade outline. 
 
 The three expressions for Thrust, Torque, and Efficiency 
 have been shown to be given by 
 
 T = p.n 2 . Icy.l. (2.ir,oj-P. tan 7). \/P 2 + 4.7r 2 .a3 2 . dx 9 
 Jr 
 
 which may be written as 
 
 T = p ,n*.c. \f(x)4(x). (2.7r,9;-P.^( t ,') ). \/F 
 
 Jr 
 
 giving the Thrust exerted by each blade of the air-screw. 
 
 D 2 
 
36 AIR-SCREWS 
 
 The Torque on each blade is given by 
 
 M = p.n 2 . \Cy.l.x. (P + 2.1T.B. tan 7). \/P 2 + 4.7r 2 . 2 . dx, 
 Jr 
 
 which may be written as 
 M = p.n\c. \f(x). 4>(x). x, (P + 2.ir.x. ^(x) ). \/P 2 +4.?r 2 ^ dx. 
 
 Jr 
 
 The total Efficiency of each blade, and therefore of the whole 
 air-screw, is given by 
 
 . \c y .b. (2.7T.E- 
 
 P. tan 7). \/P 2 + 4.7r 2 ,-e 2 . dx 
 
 2.7T. \c y .b.x. (P + 2.7r.. tan 7). \/P 
 
 J^o 
 
 which may be written as 
 
 P. /(#). ^(a?). (2.ir.a; - P. ^0) )- 
 
 2.7T. /(a;), 
 
 and we shall now apply these general results to the four types 
 of blade outline considered. 
 
 "Efficiency Curve" type of blade outline. 
 
 // \ P. (2.-n-.a?-P.^(aQ) 
 re J ( ' ~ 2.7r.x. (P + 2.7r.aj; ^(x) )' 
 
 so that the expression for the Thrust on each blade becomes 
 
 r 
 
 ... . \/P 2 -f-4.7r 2 ^ 2 . (2.ir.a;-P. ^)) 2 . dx 
 
 2.7T 
 
BLADE SHAPE AND EFFICIENCY 37 
 
 The Torque on each blade is given by 
 
 M 
 
 = ^?. fj(a>). (2.7r,*-P. 
 
 J/7r >o 
 
 The total Efficiency of the whole blade is given by 
 
 rr 
 
 I (f)(x). (2.7T.OJ-P. ^0)) ! 
 
 2.7T. \<t>(x). (2.7r.# P. tfrfa)). vP 2 + 4.7r' 2 .a5 2 . dx 
 Rational " type of blade outline. 
 
 N. 
 
 so that the expression for the Thrust on each blade becomes 
 
 2 Po^ 2 c f r 
 T = - -' i '. \x. 6(x). (2.TT.X P. ty(x)\ dx. 
 
 N Jr. 
 
 The Torque on each blade of the air-screw is given by 
 
 J r o 
 
 And the total Efficiency of the whole blade is given by 
 P. lx.<t>(x). (2. TT.X- P. ^(x)).dx 
 
 2.7T. <x?. $(x). (P4-2.ir.ic. ^(x)). dx 
 
 It will be noticed that in the case of these expressions for 
 the "Kational" blade, the making of the blade width at any 
 radius proportional to the " gap " has led to a great simplification 
 being introduced into the results of the analysis. 
 
38 AIR-SCREWS 
 
 This becomes at once apparent when we consider the 
 particular case of uniform section over the whole blade. 
 
 </>(V) and ^r(x) are then constants, and the three expressions 
 given above can very easily be evaluated. 
 
 In this case the value of (c), which is the value of the 
 
 " - ' ratio of the blades, is usually chosen beforehand, 
 gap 
 
 and a good value for this constant appears to be about one- 
 third to one-quarter. Mr. A. R. Low suggests the rule : 
 
 1 < m < 4, where (m) is the reciprocal of (c). 
 
 If (c) be evaluated from outside considerations of B.H.l*. 
 available, etc., and therefore from the formula given for the 
 Torque, it is possible to see whether there is likely to be 
 any appreciable interfering action between the blades due 
 to forms of blade sections and therefore necessary blade 
 widths used, etc. 
 
 " Normale " type of blade outline. 
 
 Here (b) = (c), and f(x) = 1, so that the Thrust on cacti 
 blade is given by 
 
 T = p.n*.c. 
 The Torque on each blade is given by 
 
 _ _ 
 M = p.tf.c. x. <(>>). (P + 2.7r.#. ^-(a?)). \/P 2 + 4.7r 2 .^. dx. 
 
 Jr 
 
 And the total Efficiency of the whole blade is given by 
 
 V = 
 
 />, 
 
 Jr 
 
 .TT 2 .^. dx 
 
BLADE SHAPE AND EFFICIENCY 39 
 
 " Constructional Limit" type of blade outline. 
 
 \/PM-~4 7T 2 X 1 
 
 Here /(a?) = p ' , so that the Thrust on each blade 
 
 is given by 
 
 The Torque on each blade is given by 
 M = P 
 
 And the total Efficiency of the whole blade is given by 
 
 ' \ fr\ 
 
 Jr.' 
 
 As can be seen from the above three expressions for Thrust, 
 Torque, and Efficiency, the evaluation of each expression is a 
 simple matter if we take <$>(%) and ty(x) as being constants, 
 that is if we assume the air-screw to have a uniform section 
 throughout. The approximation thus introduced does not affect 
 the accuracy of the expressions to such a material extent as 
 might be expected. The main difficulty in the prediction of 
 the performance of an air-screw lies in the shape of the blade 
 outline not being as a rule readily capable of being expressed 
 by some simple function of the radius (x). 
 
 We shall now determine a few values for the expressions 
 already deduced for Thrust, Torque, and Efficiency, in the four 
 cases already considered, when the blade section is assumed to 
 be uniform throughout. This makes cf>(x) and ty(x) constants, 
 and we shall refer to these functions when considered as being 
 constants by (c y ) and (tan 7) respectively. 
 
 We shall assume that (r ) = 0. 
 
 For the present we shall confine our attention to the 
 expressions for the efficiency of two only of the types of blade 
 considered. 
 
40 AIR-SCREWS 
 
 " Rational " type of blade outline. 
 We have 
 
 Cr 
 
 P. \x. (2.7T.X P. tan 7). dx 
 
 Jr = P. (4.7r.r-?).P. tan 7) 
 
 97 = f ~ - /. -n , Qirrr tan ^ 
 
 !.7T. p. 
 J'o 
 
 And now, if we write (Z) equal to the ratio of the effective 
 
 p 
 pitch to the diameter of the air-screw, i.e. -r, we obtain 
 
 = 2.Z. (2.7T-3. tan 7 .Z) 
 77 = TT. (4.Z + 3.7T. tan 7) * 
 
 The substitution of (Z) for the ratio of pitch to diameter, 
 p 
 -y, puts the expression into a more convenient form. 
 
 Mr. A. K. Low has deduced practically the same expression 
 for the efficiency of this type of blade, his formula being 
 
 _ ' 
 
 2.w 
 
 V ~ o -\irin 
 
 where his (w) is equal to ^ (as used here), and his (M') and 
 
 Li 
 
 (M'"), which are expressions for the average values of (tan 7) 
 over the blade, are equal to (tan 7), supposed to be constant 
 over the whole blade (as used here). 
 
 If we make these substitutions, the two expressions for the 
 efficiency of this type of blade become identical.* 
 
 We can introduce a still further simplification into the 
 "Rational" blade efficiency formula by taking the value of 
 (tan 7) as 1/12. This is a very fair average value for the 
 usual type of blade section employed in practice. 
 
 * See Paper on " Air- screws " read before the Aeronautical Society 
 in April, 1913. 
 
BLADE SHAPE AND EFFICIENCY 41 
 
 We have then 
 
 _ 2.Z.(8w-Z) 
 
 77 ~~ 7T. (16.Z+7T)' 
 
 This curve may now be plotted against values of (Z). Such 
 a curve is shown in Eig. (14). 
 
 FIG. 14. 
 
 Constructional Limit " type of blade outline. 
 
 We have 
 
 ?? = 
 
 P. f(2.7r^-P. tan 7) (P 2 + 4.7r 2 .^ 2 ). dx 
 
 2.7T. x.(P + 2.ir.x. tan 
 
 Jr 
 
 5.P.(6.7r.P 2 .^-12.P 3 . 
 
 .7r' 2 ^ 2 . dx 
 
 .^ 2 . tan 7) 
 
 .P 2 .^. tan 
 
 .7r 3 .rf 3 . tan 7)' 
 
42 AIE-SCEEWS 
 
 and now, as before, making the substitution (Z) = -y, we obtain 
 
 Cv 
 
 5.Z. (G.T^Z^J^.Z^an 7-f3.7r 3 -4.77 2 .Z. tan7) 
 77 ~ TT. 30.Z 3 + 20^rrZ 2 . tan 
 
 And if we make (tan 7) = 1/12, this reduces to 
 5.Z. (18.7r.Z 2 -3.Z 3 + 9.7r 3 -7T 2 .Z) 
 
 = 7T. (90.Z 3 + 5.7T.Z-+ 45.77 2 .Z + 3.7T 3 )' 
 
 and this curve may likewise be plotted against (Z). The curve 
 of this function is shown in Fig. (14). 
 
 It will be noticed from the two formulae deduced for both 
 the " Eational " and " Constructional Limit " efficiencies, that 
 neither expression contains (d), the diameter of the air-screw. 
 
 This is as might be expected, since (ij) is non-dimensional, 
 and hence the expression for the same should only involve 
 ratios such as (Z). 
 
 If we make (tan 7) equal constant, and therefore assume 
 that (cy) is constant over the blade, we may obtain quantitative 
 expressions immediately evaluable for the case considered of 
 the " Eational " blade shape. These are then as follows : 
 
 -d 2 . (2.7r.d 3. P. tan 7) 
 "12 
 
 \r M c-Tf-P-Y-^-Cy-n 2 . (4.P + 3.7T. tan 7. d) 
 48 
 
 , . 2.C.P.7T.^ , . . 
 
 and, since (b) = =, we may obtain the above 
 
 N.\/P 2 -f4.7r 2 .a; 2 
 
 expressions for the Thrust and Torque in terms of the width of 
 the air-screw blade at the tip, i.e. when (x) is equal to (?). 
 We may denote this tip blade width by (b r ). 
 These expressions then become 
 
 _ 5 r .9i a .p.^VFT7r' 5 Q 2 .' (2.7r.^-3.P. tan 7) 
 12 
 
 .n' 2 .d 2 . (4.P-f 3.?r. tan 7. <j). 
 
BLADE SHAPE AND EFFICIENCY 43 
 
 and the value of ( r ), the width of the air- screw blade at the 
 tip, is given by 
 
 C.P.TT.d 
 
 In deciding upon the best form of blade outline for the 
 air-screw, -an ideal condition would appear to be obtained when 
 the velocity in the slip stream is everywhere parallel to the 
 axis and uniform, that is when the thrust at any radius is 
 proportional to that radius. The ideal thrust grading diagram 
 would thus be a straight line passing through the origin at the 
 boss centre. 
 
 We can investigate the form of blade outline necessary to 
 secure such a condition from the results already obtained in 
 the general case. 
 
 It has already been shown that for any form of blade 
 outline whatever, the thrust at any radius is given by 
 
 (IT = c.M 2 xx.2.7r.x-?.<>rx. z +4 : .'Tr*.x 2 . dx 
 
 and the condition specified is that 
 
 dT = m.x.dx, 
 whence 
 
 m.x = c.pM z .f(x)4(x).(2.Tr.x^f. 
 and therefore 
 
 __ 
 c.p.n*.<t>(x).(2.7r.x - P.^(a;)). v/P 2 -f 4.7r 2 .^' 
 
 which gives the necessary form of blade outline to satisfy these 
 conditions. 
 
 Whence it is only necessary to plot a curve of 
 
 to obtain the required blade outline for any specified outside 
 conditions. 
 
44 AIR SCREWS 
 
 It will be noticed that when (#) has the value of 
 
 that is at a distance of about one inch from the boss centre, the 
 blade outline curve f(x) becomes discontinuous, being of the 
 form shown in Fig. (14A). 
 
 Thus in estimating the characteristics of such a type of 
 
 FIG. 14A. 
 
 blade it is impossible to integrate from the value (r ) = 0, since 
 the curve becomes discontinuous at the value -J^ of the 
 
 2.7T 
 
 radius. 
 
 It becomes necessary therefore, when evaluating the 
 expressions characteristic of such a type of blade, to take for 
 
BLADE SHAPE AND EFFICIENCY 45 
 
 the value of (r ) a value greater than _i^i?2. Since this value 
 
 2.7T 
 
 is usually very small, it will be sufficient to take for (r ) a 
 value of anything from say (-^Q) to (J) of the length of 
 each blade. 
 
 It is interesting to see how the efficiency of this type of 
 blade compares with those types already investigated. 
 
 The efficiency of any type of blade is given by 
 
 ( r ________ 
 
 P. f(x). $(x). (2.7r.-P. Tjr(#))VP 2 4-4.7r a .sc' 2 . dx 
 
 = Jr ' 
 
 r 2 - v/ ; 
 .j 
 
 .7r 2 .ic 2 . dx 
 
 and taking the value of 
 
 ,, s _ 
 ~ 
 we obtain 
 
 Tr 
 
 L") I /yi //O'* 
 L.It//. C6tXy 
 
 Jn 
 
 p 
 
 and this becomes, after making the substitutions = (Z), and 
 
 Cb 
 
 (r ) = -Q, and ^(x) = tan 7 = constant, 
 
 = _ 3.77 2 .Z 
 
 
 and if tan 7 = , 
 12 
 
 " 
 
 . log, 
 
46 AIR-SCREWS 
 
 It would appear that the curve of efficiency plotted against 
 values of (Z) for this type of blade does not greatly differ from 
 those already given. The efficiency for the respective values 
 of (Z) is slightly higher than that obtained for the " Rational " 
 blade under similar conditions of blade section and angles 
 of attack. 
 
47 
 
 CHAPTER IV. 
 
 BLADE SECTIONS, AND WORKING FORMUL/E. 
 
 IN air-screws manufactured for and used upon existing types 
 of air-craft it will almost invariably be found that the blade 
 section changes from the tip to the boss, and this is in fact a 
 necessity, from considerations of strength due to the stresses in 
 the blades caused by the thrust exerted by the air-screw and 
 the centrifugal action due to the rotation of the whole air-screw 
 about its axis at the boss centre. 
 
 Aerodynamically the use of a varying form of section along 
 the blade usually somewhat tends to decrease the efficiency of 
 the air-screw as a whole, although this probably does not 
 amount to much. That is to say that the efficiency of any 
 type of air-screw blade having a section varying from the boss 
 to the tip is less than that of a blade of similar outline but 
 having a uniform section throughout, provided that such a 
 section is of such a form as to have a value of (tan 7) smaller 
 than the average value of (tan 7) on the other blade, and 
 equal to the smallest value of (tan 7) on any section there 
 employed. 
 
 As a rule, for sections of similar shape, the thinner the 
 
 fil 1 f* 1C Tl f*^S 
 
 section, i.e. the less the value of the - r -, ratio, the lower 
 
 the value of (tan 7), and hence the greater the overall efficiency 
 of the whole blade employing such a section. 
 
 This only applies, however, between certain fairly well 
 
 , ,. thickness ,. 
 defined limits, for, after a certain value or the g -j ratio 
 
 is reached, the section becomes less and less efficient, that is has 
 a larger and larger value of (tan 7), until the limit is reached, 
 
48 A IE-SCREWS 
 
 when the " camber " of the section altogether disappears and the 
 section ultimately becomes a flat plate for which the value of 
 
 the YJ ratio is quite small, being of the order of (*7) : (1) at 
 .Drag 
 
 the most efficient angle of incidence. 
 
 As a rule, in designing an air-screw to fulfil any given 
 outside conditions, the forms of the blade sections are chosen 
 
 ChoYd Angle of Incidence (ctegrte*) 
 
 FIG. 15. 
 
 so as to effect as far as possible a compromise between 
 considerations of aerodynamical efficiency and the necessary 
 strength of the various portions of the blade. 
 
 The aerodynamical considerations of overall blade efficiency 
 have, however, already been dealt with, and the expression for 
 the efficiency of any type of blade shape put into a suitable 
 form for purposes of air-screw design, so that we may proceed 
 
BLADE SECTIONS, AND WOEKING FORMULAE 49 
 
 to select our sections along the blade without stopping to 
 consider whether it will be possible to express analytically 
 the characteristics of the same in estimating the Thrust, Torque, 
 and Efficiency of the whole blade, providing of course that the 
 forms of the blade sections chosen are such that they conform 
 to sections of which the characteristics are known from tests 
 
 Chord A ng/e of Incidence fdtyrreij 
 FlG. 16. 
 
 carried out in a wind-channel for the same when considered 
 as aerofoils moving in a straight line. 
 
 A somewhat typical series of such sections are given in the 
 (< Technical Report of the Advisory Committee for Aeronautics 
 for 1911-1912," and their characteristics are shown plotted in 
 Figs. (15), (16), (17), (18), (19), (20), (21). 
 
 It will be noticed that the maximum value of the ^p- 
 
 Drag 
 
50 
 
 AIR-SCKEWS 
 
 ratio, and hence the minimum value of (tan 7), occurs in all the 
 sections at an angle of approximately (4) degrees. 
 
 We may of course use any " angle of attack" we please for 
 each section along the blade, but it is advisable to use the angle 
 corresponding to the least value of (tan 7) fur each section 
 considered, as this gives a better overall efficiency for the whole 
 blade, since the efficiency of any section is equal to 
 
 tan A 
 tan (A +7)' 
 
 Chord Angle of Inc. i d e. nee 
 
 FIG. 17. 
 
 and hence the smajler the value of (tan 7), and hence of (7), 
 the greater the efficiency at that section. 
 
 We now proceed then to a consideration of the design of 
 any type of air-screw to fulfil any given specified set of 
 conditions. 
 
BLADE SECTIONS, AND WORKING FORMULAE 51 
 
 It is usual to start with the following data, which are 
 supposed to be fixed from outside considerations. 
 
 (1) (V) The horizontal velocity of the air-craft. 
 
 (2) (?i) The speed of revolution of the air-screw in 
 
 revs. /second. 
 
 Chord Anyle of Incidence. 
 
 FIG. 18. 
 
 (3) (d) The diameter of the air-screw. This is 
 
 usually fixed from considerations of 
 ground clearance, etc., and is usually 
 made as large as the design of the air- 
 craft will permit. This seems to be 
 fairly standard practice at present. 
 Experimental research is required before 
 this point can be definitely settled. 
 
 E 2 
 
52 AIR-SCEEWS 
 
 (4) <p(x) These functions of (%) and (tan 7) depend 
 
 and upon the form of the sections employed 
 
 fy(x) at various radii along the blade and their 
 
 respective " angles of attack," and may 
 
 be plotted when the form and position 
 
 of these relative to the boss centre are 
 
 known. 
 
 Chc-rd Anyle of Incide nc e Idc y 
 
 FIG. 19. 
 
 (5) (H) The B.H.P. of the motor multiplied by 
 the efficiency of the transmission (if 
 any) supplied to each air-screw (if there 
 are more than one). 
 
 We now proceed to draw out the blade shape of the air- 
 screw, making ordinates on the curve represent proportional 
 
BLADE SECTIONS, AND WORKING FORMULA 53 
 
 widths. Here experience and a study of the particular type of 
 air-craft to which the air-screw is to be fitted will guide us. 
 
 Portions of the air-craft coming within the air-screw's disc 
 of revolution must be taken into account when designing the 
 blade shape. 
 
 An air-screw fitted to a rotary motor having a large cowl, 
 for instance, will not probably require so large a blade width 
 
 Chord Anyle of Inc i de nc t 
 
 FIG. 20. 
 
 for best efficiency at the inside radii near the boss as would 
 otherwise be advisable if the air-screw were working free from 
 any such obstructions in the air-flow. 
 
 In the theory of air-screw design as set forth in this book 
 no quantitative notice is taken of such stationary parts of the 
 air-craft, and to do so would be in most cases an exceedingly 
 difficult problem. 
 
54 
 
 AIR-SCKEWS 
 
 A study of the efficiency curve for the blade will also help 
 us in the designing of a suitable blade outline. 
 
 Bearing then these considerations in mind we commence the 
 designing of our air-screw as follows : 
 
 (1) Plot the efficiency curve 
 P. 2/ 
 
 J Incidence (deyrees) 
 
 FIG. 21. 
 
 We do not as yet know the value of the function ^r(,r), 
 but for a first approximation we may take this as 
 
 being constant and equal to say T ^. 
 
 Z 
 
 In slow running air-screws this curve of blade efficiency 
 will be found to be almost a parallel line with the 
 (x) axis, except close up to the boss, where it runs 
 down to the origin very rapidly. 
 
BLADE SECTIONS, AND WORKING FORMULA 55 
 
 In high speed air-screws the change in slope of the 
 efficiency curve will be more marked, and this 
 brings us to a consideration of the best blade shape 
 under these conditions. 
 
 As an example of a good blade outline for an air-screw, the 
 four-bladed air-screw as used on the Royal Aircraft Factory 
 aeroplanes may be cited. This screw revolves at a speed of 
 900 r.p.m., which is if anything rather on the slow side. 
 
 We may however take it as a fairly good general rule that 
 in designing an air-screw blade, the maximum ordinate on the 
 blade outline curve, that is the point where the blade is widest, 
 should be somewhat nearer the tip of the blade than the point 
 of maximum efficiency as given by the efficiency curve.* 
 
 FIG. 22. 
 
 It is not proposed here to discuss the theoretically best form 
 for the blade outline function f(x), as to do so would be a very 
 difficult matter when treated generally.! Experience is a good 
 guide in choosing a suitable shape for the form of this function. 
 
 Taking then the blade outline as being something of the 
 
 * Recent tests on aerofoils with elliptical-shaped ends have shown a 
 marked improvement, in ratio, over similar aerofoils with square- 
 cut ends, provided that the section of blade in the former case is 
 everywhere geometrically similar. This indicates the superiority, from 
 efficiency point of view, of blades with tapered tips, apart from any other 
 considerations. 
 
 f A general treatment of the variation in (77) due to variation of /(#), 
 and the determination of the form of f(x) giving to (77) a maximum 
 value, would require the application of the Calculus of Variations, and 
 is beyond the scope of the present work. 
 
56 AIE-SCKEWS 
 
 form shown in Fig. (22), we may proceed to the determination 
 of the complete design of the air-screw. 
 
 /o\ T ^ fi ^ thickness ,. , . 
 
 (2) It is first necessary to fix the r T ratios of the 
 
 sections along the blade, and here again practical experience 
 will help us. It is undesirable to make the blade sections too 
 thin, especially near the boss, as this leads to undue flexibility 
 in the blade, with corresponding losses in efficiency. 
 
 In an air-screw having a diameter of (8) or (9) feet, the 
 following proportioning of the sections along the blade would 
 seem to be fairly standard practice : 
 
 At a radius of (1) foot from the boss centre, 
 
 Section No. 7, Fig. (21). 
 
 (2) feet 6, Fig. (20). 
 
 (3) 5,Fig.(19). 
 
 (4) ' 4 Fig. (18). 
 
 Having then selected a suitable series of aerofoil sections 
 and having spaced them along the blade, we may proceed 
 to the determination of the two functions </>(#) and ^(#) 
 characteristic of such sections. 
 
 The respective " angles of attack " of the various sections 
 employed will of course be the angles corresponding to the 
 least value of (tan 7) in each case, since this will give the 
 highest efficiency for any specified spacing of the sections and 
 form of blade outline. 
 
 In the case considered, these " angles of attack " are taken 
 as being constant and equal to (4) degrees, since, as already 
 stated, and as a reference to the curves given in Figs. (18, 
 19, 20, 21) will show, the respective minimum values of (tan 7) 
 occur approximately at the same angle of incidence in each 
 case, namely (4) degrees. 
 
 In this connection it does not seem to be unreasonable to 
 assume that sections lying between any two selected sections 
 along the blade will have characteristics intermediate between 
 those of the two respective outside sections considered, and 
 
BLADE SECTIONS, AND WOEKING FORMULAE 57 
 
 quantitatively will approximate to the values of the character- 
 istics shown on a smooth curve drawn between the various 
 points along the blade corresponding to the characteristics of 
 the selected sections at those points. 
 
 We shall then obtain a series of points for the values of 
 (c y ) and (tan 7) for the various radii chosen, and if smooth 
 curves be drawn through these points we shall obtain the two 
 curves denoted by <(^) and ty(x). 
 
 Fig. (23) gives these two curves for the sections and spacing 
 of the sections considered and for a uniform " angle of attack " 
 of (4) degrees. 
 
 We are now able to read off from the two graphs plotted 
 of <f>(x) and -fy(x) the respective values of these two functions 
 for any value of (x) considered, that is for any point along the 
 blade. 
 
 (3) It now only remains to determine the true value of the 
 blade width for every point along the blade. 
 
 The efficiency of the whole blade can be found at once, since 
 we know the values of the two functions </>(x) and ty(x) for 
 each value of (x) considered. 
 
 The value of (&), the true blade width at any radius (#), can 
 be obtained in the following manner. 
 
 We already know that 
 
 b = c.flx), 
 
 where f(x) denotes the scale blade width for each value of (x) 
 considered, and the value of f(x) is thus the value of the 
 ordinate on the curve of proportional blade widths already 
 drawn. So that we have only to find the value of (c) in order 
 to be able to completely determine the value of (&) for any 
 value of the radius (x). 
 
 It has already been shown that the Torque (M) of each 
 blade of the air-screw may be expressed as 
 
 M = c.p.n\ \x.f(x). $(x). (P + 2.7T,*. f (^))VPH-"4;7r^; dx. 
 
 Jr 
 
 And further (N.M.2.7r.w) is proportional to the horse-power 
 
58 
 
 AIE-SCEEWS 
 
 o * 
 
 ' / vW 
 
 
 
 y <X/ 
 
 to 
 
 
 
 V 
 
 
 I / 
 
 V 
 
 U. 
 
 CO 
 
 o? 
 
 
 i 
 
 & 
 
BLADE SECTIONS, AND WOEKING FORMULAE 59 
 
 available to turn the whole air-screw, where (N) denotes the 
 number of blades. 
 
 Hence if we use Ib./ft./sec. units we get that 
 
 IN . ivi.-j.7T. n . 
 
 ^TK = J~ij 
 
 ooO 
 
 where (H) is the available B.H.P. after allowing for losses in 
 transmission (if any transmission is employed). 
 
 We also have the value of (p) as being equal to ( 00238) in 
 Ib./ft./sec. units. 
 
 So that we can at once write 
 
 = H 
 
 whence 
 
 550. H 
 
 (r 
 
 2.7T.n S .N.p. X.f($). <(. 
 
 Jr 
 
 and this gives the necessary value of (c) for any given outside 
 conditions. 
 
 It will be noted that all dimensions must now be measured 
 in feet. 
 
 We proceed then to the evaluation of the blade width 
 constant (c). 
 
 Since we have not obtained the functions <j>(x) and ty(x) in 
 an algebraic form, that is we do not know the equations to 
 these two curves, we cannot evaluate the definite integral 
 
 I, 
 
 x.f(x). </>(. 
 
 algebraically, and hence must employ a graphical method 
 
 throughout. This also applies to the evaluation of (/;), the 
 efficiency of the whole blade. 
 
 In order then to obtain the value of (c) it is first 
 
60 . AIE-SCEEWS 
 
 necessary to determine graphically the value of the definite 
 integral 
 
 \X.f(x). $(X). (P-f2.7T.JE. ^)).V^ + ^7T\X\ dx, 
 
 h 
 
 and we may proceed to do this as follows. 
 Plot the graph of the function 
 x. f(x). $(x] 
 
 to some convenient scale against values of (x), the radius from 
 the boss centre in feet. 
 
 y 
 We already know the value of (P) ; it is ; and both (V) 
 
 and (n) are known to start with, since they are assumed to be 
 fixed from outside considerations. 
 
 The value of f(x) for any radius (x) is known, since it is the 
 value of the ordinate on the scale blade width already drawn. 
 The value of f(x) is to be measured in feet, that is to say we 
 must take some convenient scale on the graph paper to represent 
 so many inches equal to one foot. 
 
 The respective values of </>(x) and ^r(x) are determined at 
 once from a reference to the two curves already plotted of these 
 functions for any value of (x). 
 
 Having then taken a sufficient number of values of (x), and 
 having determined the corresponding values of the function the 
 graph of which we are plotting, a smooth curve drawn through 
 the points so obtained will give the curve required. 
 
 Now draw two ordinates from the (x) axis at the 
 points (r ) and (r) respectively until same cut the curve just 
 plotted. 
 
 Then the area of the enclosed figure so obtained, that is the 
 figure contained by the curve, the two extreme ordinates at (r ) 
 and (r), and the (x) axis, is the value of the definite integral 
 required. 
 
 The areas of closed figures of this kind are most easily 
 obtained by means of a planimeter. 
 
 It is to be noted here in this connection that the actual area 
 of the figure so obtained will be in, say, square inches, and 
 
BLADE SECTIONS, AND WOKKING FORMULAE 61 
 
 hence that it must be multiplied by some constant in order to 
 find the real value of the definite integral. 
 
 The value of this constant will depend, of course, upon the 
 scale employed in the plotting of the graph. 
 
 Thus, if the ordinate on the curve corresponding to any 
 value of (x) has a value obtained from the formula 
 
 T f(l-\ d\(t>\ (~P 4- 9 TT V ^(^\ 
 
 ^J vV'V^V / \ -T^.TT.X. Y v*v 
 
 of such and such an amount, this value will probably not be 
 able to be represented on the graph paper used, since it may be 
 very much too large when taken to the same scale as that of 
 the (x) axis, and hence the scale of " heights " or ordinates will 
 probably have to be made much smaller than the scale used for 
 the (x) axis. 
 
 Having then obtained the value of this area in the required 
 dimensions, we may determine the value of (c) at once. 
 
 It is given by 
 
 550. H 
 
 /> ^ , , 
 
 2.7r,n 3 .N.p. (area of figure obtained)' 
 
 where (p) has the value of (-00238) as already given. 
 
 We have then that the true or necessary blade width at 
 each radius (x) is the value of the scale blade width at that 
 radius multiplied by the value of the constant (c) already 
 found. 
 
 Or 
 
 I = cf(x). 
 
 The value of f(x) for any radius (x) will, of course, be 
 measured off the curve of this function already drawn, and 
 its corresponding real value found by reference to the scale 
 employed in drawing the curve. 
 
 Thus, if 1 inch on the curve ordinate represents 1 foot as 
 the actual scale blade width, a value measured at any radius 
 of, say, ( ' 75) inch as the ordinate of the curve at that radius 
 would represent an actual scale blade width of ( 75) foot, that 
 is (9) inches. 
 
 And further, if the value of (c) was found to be, say, (1 2), 
 then the true or actual blade width at this point would be 
 
62 AIR-SCREWS 
 
 equal to (9 X 1*2) inches, that is (10*8) inches. And this 
 would be the width of the blade at that radius to be used in 
 the construction of the air-screw. 
 
 We may obtain the value of the total efficiency of each 
 blade, and hence of the whole air-screw, in a similar manner 
 as follows. 
 
 Plot the two curves 
 
 4.7r^ (1) 
 
 and 
 
 h4.^ 2 (2), 
 
 which is the one already plotted, and which therefore it is 
 unnecessary to replot. 
 
 Take the area of the figure enclosed by (1), the extreme 
 ordinates at (r ) and (r), and the axis of (x). 
 
 The area enclosed by (2) has already been obtained from the 
 evaluation of (c). 
 
 Divide the area enclosed by (1) by the area enclosed by (2), 
 
 p 
 and multiply the result by ^ . The answer will be the 
 
 efficiency of the air-screw. 
 
 It will not, of course, be necessary to trouble about scale 
 constants, etc., in determining the value of the efficiency of the 
 whole blade, as, provided that the two graphs (1) and (2) are 
 drawn to the same scale, it will only be necessary to divide 
 their actual areas one by the other in whatever dimensions 
 these two areas are obtained, provided, of course, that each area 
 is measured in the same dimensions. 
 
 In the determination of (c), and hence of the real blade 
 width for each radius, it will usually be found that the true 
 blade widths for each point along the blade will be somewhat 
 smaller than those actually employed in practice on a similar 
 form of air-screw. This is due to the fact that the calculated 
 Torque (M) is higher than the actual Torque found in practice, 
 and hence that the necessary blade widths at each radius will 
 have to be larger than those given by the theory. 
 
 Of course no one for a moment supposes that the theory of 
 
BLADE SECTIONS, AND WORKING FORMULA 63 
 
 air-screw design based on an analogy with aerofoils moving in 
 a straight line is absolutely exact. It is merely a very useful 
 theory, the results of which conform very closely with those 
 actually obtained by experiment. 
 
 It is difficult to estimate the amount of this difference 
 between the calculated and actual Torques, without actual 
 quantitative experimental results, but it would appear that the 
 
 actual value of (c) is between -==- and ..- times the calculated 
 
 / o 7o 
 
 value as given by the theory. 
 
 So that after having obtained the calculated value of (c), it 
 will be prudent to augment this value by say 33 %, that is 
 
 multiply the value of (c) obtained from the formula by -=- . 
 
 7 o 
 
 It will be useful here to have some independent check upon 
 the working, so that any arithmetical slips in the evaluation of 
 quantities such as (c) may be as far as possible avoided. 
 
 We may obtain a rather approximate check of this kind 
 by reference to the " Rational " blade outline form already 
 considered. 
 
 If, in the expressions deduced for the characteristics of this 
 form of blade, we assume that both (c y ) and (tan 7) are constant 
 over the blade, and that (r ) is equal to zero, and further if 
 
 we take the value of (tan 7) to be ^r, then we obtain the 
 
 \2t 
 
 following expression for the necessary blade width constant (c). 
 
 52800. H 
 
 and since as already shown 
 
 c = 
 
 where (b r ) denotes the tip blade width, we can obtain the 
 necessary value of (b r ) by substitution, thus 
 
 52800. H 
 
 And this then provides a useful approximate check upon 
 
64 AIR-SCREWS 
 
 the previous work in the determination of the true blade width 
 for each radius (#). 
 
 It will be noticed from this and previous expressions for 
 blade width, that doubling the number of blades of an air-screw 
 merely halves the respective widths of same at corresponding 
 radii. 
 
 This theory does not in fact take any notice of possible 
 improved efficiency by the use of more than the customary two 
 blades for an air-screw.* 
 
 We may also obtain an approximate value for the efficiency 
 of the whole blade by means of the same " Rational " form of 
 blade outline. 
 
 This has already been shown to be given by 
 
 2.Z. (2.7T - 3. tan 7. Z) 
 TT. (4.Z + 3.7T. tan 7 ) ' 
 
 which reduces to the simpler form 
 
 2.Z. (8.7T-Z) 
 
 77 == 7T. (7T + 16.Z) 
 
 when (tan 7) has the value of . 
 
 Now there will be, corresponding to some particular value 
 of (Z), a value of (rf) which will be a maximum, and this will 
 therefore give the theoretically best value of (Z) and therefore 
 of (n) to use for any given set of conditions, since the values 
 of (V) and (d) will usually be fixed from outside considerations. 
 
 We may obtain the value of this maximum efficiency and 
 the value of (Z) for which it is a maximum as follows. 
 
 The condition for a maximum value of (77) is 
 
 ^L- 
 dZ ~ U ' 
 
 and this gives 
 
 Z l = ~. [\/8 + 9. tan 2 ~r7-3. tan 7], 
 
 giving the requisite value of (Z) for a maximum value of (77). 
 
 * Except in so far as the greater the number of blades employed the 
 higher the aspect ratio of each, and hence the greater the efficiency of 
 the air-screw due to increase of aspect ratio. 
 
BLADE SECTIONS, AND WORKING FORMULA 65 
 
 This expression is seen to approximate to the value !^_f 
 
 as (tan 7) approaches zero. 
 
 As a rule this value is in the neighbourhood of (2) and 
 hence we may say that, for a near approximation, (77) has a 
 maximum value when (Z) is equal to (2). 
 
 P V 
 
 Now (Z) = - = - and hence if (Y) and '(d) are fixed by 
 
 Cl> 1l'.(L 
 
 outside considerations, the value of (n) for which the efficiency 
 of the air-screw is a maximum is given by 
 
 Zl = ^d ~ 2 ' 
 whence 
 
 (n L ) = y-j revs. /sec. 
 
 So that if 
 
 (V) = 100 feet/sec., 
 (d) = 10 feet, 
 
 the speed at which the air-screw, or air-screws, should be run in 
 order to obtain the maximum efficiency would be (5) revs. /sec., 
 that is (300) revs./min. 
 
 This speed is of course abnormally slow in the light of 
 present-day practice, although the speed of revolution of the 
 air-screws in some of the Wright aeroplanes is as slow as 
 (450) revs. /ruin. 
 
 A curve of efficiency for values of the ^i - ratio (Z) 
 
 Diameter 
 
 has already been given for values of (Z) occurring in practice. 
 
 Mr. H. Bolas gives a formula for the efficiency of an air-" 
 screw of good shape * (see " Technical Report of the Advisory 
 
 * This formula is only an approximate one. It is 
 
 " 1 4- Ic. tan y. cot 6 
 
 - Z 
 
 Z 4- 7T./t. tan y' 
 
 and if tan y = , A' = *7, we get 
 
 Z 
 
 ~ Z + '188' 
 which may then be plotted against (Z). 
 
66 AIK-SCKEWS 
 
 Committee for Aeronautics, 1911-12 "), and this has also been 
 plotted against values of (Z). The two curves of efficiency are 
 shown in Fig. (14), and their close general resemblance will be 
 noticed. 
 
 It would seem that at any rate to a first approximation 
 the efficiency of almost any modern type of air-screw may be 
 obtained from the formula or graph given for the " Eational " 
 shape. 
 
 As a rule, in practice it is found that the actual recorded 
 efficiencies of air-screws are higher than those given by the 
 theory. 
 
 Since (Z) may be taken as equal to (2) for a maximum 
 value of (??), the maximum value of (77) will be obtained by 
 substituting this value for (Z) in the efficiency formula. We 
 
 shall take (tan 7) as being equal to ^ as before, and then 
 
 we have the value of (?;) as ('84). Hence the maximum 
 overall efficiency of a,n air-screw having a " Eational" blade 
 
 shape is (84%) when (tan 7) = ^ and (Z) = (2). 
 
 Suppose that we wish to design an air-screw for an 
 aeroplane having a speed (Y) of (100) feet/sec., a rotational 
 speed (11) of (20) revs./sec., a diameter (d) of (9) feet, and 
 having a motor capable of developing an effective horse-power 
 (H) of (100). 
 
 Then 
 
 1} and hence (P) - Y - = 5 feet. 
 
 ( /Z' ) .. \J S Ji> 
 
 (d) = 9 
 (H) = 100 
 
 Let the spacing of the sections along the blade be such as 
 already given with a uniform " angle of attack " of (4), and 
 hence let the curves of $(#) and ^(x) be such as given in 
 Kg. (23). 
 
 Suppose that the form of /(#), the blade outline, is that 
 given in Fig. (22). 
 
BLADE SECTIONS, AND WOEKING FORMULAE 67 
 
 Then we have to determine (1) The actual blade widths at 
 
 every point along the 
 blade. 
 
 (2) The total efficiency of the 
 whole air-screw. 
 
 We shall neglect the B.H.P. consumed in turning the 
 portion of the air-screw from the boss centre to the inside 
 radius (r ), as by doing so we shall be on the safe side, since 
 our blade widths will now tend to come out larger than if we 
 allowed for the B.H.P. used near the boss. In any case the 
 amount of this B.H.P. is very small. 
 
 We proceed to determine the value of (c) in the manner 
 given by plotting the graph of 
 
 x.f(x). <(<'-') (P- 
 
 taking values of f(z), <(#), and ty(a') from the curves already 
 plotted of these functions. 
 
 The graph is shown in Fig. (24), and the actual area (as 
 originally drawn) is (50*25) sq. ins., between the two extreme 
 radii. 
 
 Now the scale of ordinates for this curve is taken to be for 
 
 convenience of the horizontal scale, so that the true area 
 
 o 
 
 enclosed by the curve will be (8 X 50*25) sq. ins., and, since 
 the horizontal scale employed is (2) ins. equal to one foot, this 
 area corresponds to (100*5) sq. feet. 
 
 Hence, applying the formula for (c), we get 
 
 550. H 
 
 2.7r.7i 3 .N./o. (Area of figure) 
 
 where (N) = (2). It will be noticed that if we employ (4) 
 blades instead of (2), the value of (c), and hence the value of 
 the blade width at any radius, is halved. 
 
 If we now multiply the value (2 3) for (c) obtained above 
 
 by say -^ K , so as to allow for differences between the 
 
 7o 
 
 F 2 
 
68 
 
 AIR-SCREWS 
 
 calculated and actual Torques, we obtain (3-07) as a more 
 exact value for this constant. 
 
 We can now of course obtain the true blade widths at each 
 radius by multiplying their respective values as given on the 
 
 i 
 
 s 
 
 5colc o/ 5colr Blade Width 
 Some as Horizontal 5 c a I e . 
 
 scale blade width curve by (3 -07) when the air-screw has 
 (2) blades. 
 
 We can also check this value by applying the " Rational " 
 blade constant formula and this then gives 
 
 52800. H^ 
 
 Tr.n*.p.c y .d?.N. (16.P + TTJ!) V 
 
BLADE SECTIONS, AND WORKING FOEMUL^E 69 
 
 = (-486) foot, which is the value of the width of the blade at 
 the tip, when (N) = (2). The value of (c y ) has been taken to 
 be (*36). This value of the blade tip width is about what 
 might be expected for this type of blade, where the blade 
 increases in width towards the tip. 
 
 We may now determine the value of (77), the total efficiency 
 of the whole blade. We plot the curve 
 
 and take the area of the figure enclosed by it, the two extreme 
 radii at (r ) and (?'), and the (,/;) axis. 
 
 The area of this curve as originally drawn is approximately 
 (47 6) sq. ins. Eig. (24). 
 
 The value of the total efficiency of the whole blade is then 
 equal to 
 
 47-6 \ / P \ /47-6\ / 5 \ 
 
 = 
 
 since (?) is equal to (5) feet. 
 
 The efficiency of an air-screw of this type would therefore 
 be approximately 75 3 %. 
 
 The blade sections of an air-screw at the inside radii near 
 the boss have to be made thick from considerations of strength. 
 We may, however, choose suitable shapes for the outside radii 
 
 from sections which have a high value of the , ratio as 
 
 Drag 
 
 aerofoils. 
 
 A few typical examples of such suitable shapes are given 
 in Eigs. (25), (26), (27), (28).* 
 
 * See " Technical Eeport of the Advisory Committee for Aeronautics, 
 1912-13." 
 
AIR-SCKEWS 
 
 DCS.CN /*e.fo). 
 
 FIG. 25. 
 
 CAorrf An,/* o/ tnci-Jtct.(d*yi) 
 
 FIG. 26. 
 
BLADE SECTIONS, AND WORKING FORMULAE 71 
 
 FIG. 27. 
 
 FIG. 28. 
 
72 
 
 AIR-SCREWS 
 
 CHAPTEE V. 
 
 " LAYING OUT " THE AIR-SCREW. 
 
 IN commencing to lay out the blade sections of an air-screw, 
 it is first necessary to determine the chord angles of the blade 
 at several radii. Having done this-, the plan and elevation of 
 the blade may be drawn out, consideration being given to the 
 fact that as far as possible the two following conditions should 
 be satisfied : 
 
 (1) The centre of area of the sections should lie on the 
 
 blade axis. 
 
 (2) The respective positions of the centres of pressure of 
 
 the sections should be so arranged about the blade 
 axis as to eliminate as far as possible all twist on the 
 blade. The loading on the blade may be taken as 
 being uniplanar. 
 
 FIG. 29. 
 
 These two conditions may be occasionally somewhat 
 antagonistic. 
 
 A symmetrical plan form is undesirable. A good plan form 
 is shown in Fig. (29). 
 
"LAYING OUT" THE AIR-SCKEW 73 
 
 We can obtain the true chord angles ((/>) for each radius 
 f the blade considered from the relation 
 
 (j) v = a x + tan 
 where (P) has the value of as already denned. 
 
 The values of (a x ) may vary along the blade, although 
 usually it will be found that the values of the " angles of 
 attack " are approximately constant and in the neighbourhood 
 of 4 degrees. 
 
 If we draw, Fig (30), a vertical line to represent the value 
 y 
 of and a horizontal line to represent (radii X 2.?r) in the same 
 
 units, we may, by drawing in the various hypotenuses, obtain 
 the inclination of the helix paths for any element along the 
 blade. And if these helix angles (A) be augmented by the 
 respective " angles of attack " at these points, we shall obtain 
 the true chord angles for the various radii considered. 
 
 If, further, the widths of the blade at these radii be drawn 
 in to scale along the chord angle lines, we may at once proceed 
 to lay out the plan and elevation of the whole blade.* 
 
 Sections near the boss may be thickened up if necessary by 
 adding a convex lower surface. In such sections the calculated 
 chord angles may have to be departed from. This is not of 
 great importance, although the actual chord angles of such 
 sections should not be less than their respective helix angles 
 at these radii. 
 
 Modern air-screw blades are built up of several separate 
 laminations of wood. French walnut is usually chosen as 
 the most suitable material from which to construct the 
 air-screw. 
 
 The lamina? may be easily laid out when the chord angles 
 
 * Strictly the blade widths and sections at each radius when obtained 
 should lie on cylindrical sections coaxial with the air-screw, and not on 
 plane sections at right angles to a fixed arbitrary line in the blade. The 
 difference, however, is small at all but the smallest radii, where it is of 
 least importance. 
 
AIR-SCREWS 
 
LAYING OUT" THE AIE-SCBEW 
 
76 
 
 AIE-SCEEWS 
 
 at the several radii considered have been determined. The 
 method is indicated in Fig. (31). 
 
 The plotting of the contours along the blade is obtained 
 from a consideration of the plan form of blade and the con- 
 struction of the laminae. ' A specimen contour plotting is 
 shown in Fig. (."32). 
 
 In the laying out of the air-screw the various curves should 
 as far as possible be run into the boss with smooth curves, the 
 
 
 FIG. 31. 
 
 size of the boss being fixed from considerations of blade width 
 and type of air-screw mounting used. 
 
 The thickening up of blade sections by means of a convex 
 under surface would not appear to affect their aerodynamic 
 properties to any great extent. Such thickening up may 
 sometimes be necessary from considerations of strength.* 
 
 * The employment of a convex undersurface appears to slightly 
 decrease the value of (c y ), and hence necessitates the utilisation of a 
 wider blade section. This is sometimes done when a stronger section is 
 
 required. The 
 
 Lift 
 Drag 
 
 ratio is only very slightly affected. 
 
77 
 
 CHAPTEE VI. 
 
 STRESSES IN AIR-SCREAV BLADES. 
 
 Centrifugal Stresses. 
 
 WHEN an air-screw is rotating about its axis at the boss centre, 
 the various elements which go to make up each blade are forced 
 to follow a circular path. The forces necessary to make these 
 portions of the blade follow such circular paths are directed 
 towards the point about which the air-screw is rotating (i.e. 
 the boss centre), and are of amount equal to 
 
 (weight of portion of blade considered) . (average velocity 
 of portion considered) 2 divided by (g) . (average 
 distance of portion from the boss centre). 
 
 Hence the reactionary forces with which the portions of the 
 blade " pull" on any section considered are of the same amount, 
 and constitute a stress in the material of which the blades of 
 the air-screw are made. 
 
 Let Fig. (33) represent a blade of an air-screw, and let AA' 
 be a section of same at a radius of (X) feet from the boss 
 centre. Then the centrifugal stress 011 the section AA' is 
 composed of the " total pull " exerted by the portion of the 
 blade from A A' to the tip, divided by the area of the section 
 at AA'. 
 
 Let (?) denote the overall length of each blade in feet. 
 And consider the portion of the blade from AA' to the blade 
 tip of length (r-X.) feet. 
 
 Consider the centrifugal pull on AA' due to a small element 
 of the blade cut off by the two radii (#) and (x + dv) 
 respectively. Then (X) remains constant, while (./) varies from 
 the value (X) to the value (r). 
 
78 
 
 AIR-SCREWS 
 
 Then centrifugal pull on A A' due to the element considered 
 may be written , 
 
 /F _ (weight of element) (velocity of element) 2 
 $ 
 
 And let (w) = weight in Ibs. of one cubic foot of the material 
 of which the blades of the air-screw are made, and (iv) is then 
 assumed to remain constant for all values of (x). 
 
 FIG. 33. 
 
 And let (a) = area of element considered in sq. ins. 
 
 And therefore 
 in sq. ft. 
 
 = area ^ sec ^ on f element considered 
 
 a.dx 
 
 And therefore volume of element = -Vr cu. ft. 
 And therefore weight of element = TJJ- Ibs. 
 
 And circumferential velocity of element is equal to 
 (2.7r.x.n) ft./sec. 
 
STEESSES IN AIE-SCEEW BLADES 79 
 
 So that we may write 
 
 giving the centrifugal pull on AA' due to the element con- 
 sidered. 
 
 Hence the total pull on AA' is given by 
 
 E = ' a.x.dx Ibs. 
 
 Whence the stress due to the centrifugal pull at AA' is 
 riven by 
 
 a.x.dx Ibs./sq. in., 
 
 where (c^) is equal to the area of the section at A A' in 
 sq. ins. 
 
 Now () denotes the area in sq. ins. of the section at 
 radius (#) from the boss centre, and consequently the value 
 of (a) may vary with (x). It is, however, a simple matter to 
 
 E 
 
 evaluate graphically the expression for , if necessary. 
 
 We have then 
 Stress due to centrifugal action at any section distance (X) 
 
 F 
 
 from the boss centre = Ibs./sq. in., 
 
 where 
 
 P =T -I' 
 
 = x 
 
 We may at once estimate the tensile stress, due to 
 centrifugal action, near the boss of the air-screw, if we 
 
 O 7 
 
 assume that 
 
 (1) The blade width is uniform, and 
 
 (2) The blade section is constant, except near the boss. 
 
80 AIR-SCREWS 
 
 Then we have fit once that 
 
 F 7r-.n\w. 
 
 ! "' 3Q.ff.a-i alx 
 since () is constant 
 
 And this formula holds good for any value of (X) providing 
 the initial assumptions (1) and (2) are satisfied. 
 
 Since near the boss (X) may be taken equal to zero, the 
 formula becomes 
 
 F 7r 2 .n 2 .w.a.r 2 .. , 
 - = ^-r- - Ibs./sq. in., 
 ! 7'2.g.a l 
 
 and this gives the value of the tensile' stress at or near the boss 
 due to the centrifugal pull of the whole blade. 
 
 An example will make the application of this result clearer. 
 
 Let (n) = 20 revs./sec., (r) = 4 ft., (w) = 35 Ibs./cu. ft., 
 (a) = 7 sq. ins., and (i) = 16 sq. ins. 
 
 Then, taking the value of (?r 2 ) as being equal to 10, we get 
 
 F 10.400.35.7.16 
 
 T ~T2.3216" ^ lbs./sq. in. 
 
 Hence the amount of the tensile stress at or near the boss, 
 due to the centrifugal pull of the whole blade, is equal to 
 425 Ibs./sq. in. This result is quite in the usual order of 
 practical work. 
 
 Stresses due to bending. 
 
 The stresses in the blades due to bending are due to the 
 resultant air pressure exerted upon each element of the blade. 
 
 Consider any section as in Fig. (34). 
 
 And let (Y c ) = the distance in inches of the extreme ordinate 
 of the section from the neutral axis passing through the centre 
 of area of the section. 
 
 And let (Y f ) = the distance in inches of the chord line, 
 assuming the sections to be flat underneath, from the neutral 
 axis. 
 
STEESSES IN AIK-SCEEW BLADES 81 
 
 Then the extreme values of the tensile and compressive 
 stresses occur at the layers of the material most remote from 
 the neutral axis. 
 
 So that we have 
 
 Maximum value of compression stress at any section in 
 Ibs./sq. in. = compressive stress at outside fibres 
 
 = M.Y C 
 I 
 
 And similarly, 
 
 Maximum value of tension stress at same section in 
 Ibs./sq. in. = tension stress at outside fibres 
 
 I ' 
 
 FIG. 34. 
 
 where (M) = bending moment at section considered, and 
 
 (I) = moment of inertia of the area of the section about 
 
 the neutral axis, 
 and (Y c ) and (Y) have already been defined. 
 
 If the values of ^-? and '=-* - be then worked out for 
 
 various values of the radius (X), we can determine the values 
 
 G 
 
82 
 
 AIR-SCREWS 
 
 of the max. compressive and max. tensile stresses due to bending 
 and centrifugal force. 
 
 For we have 
 Max. compressive stress at any section, due to bending and 
 
 centrifugal force = ~ - centrifugal stress at section. 
 
 And similarly, 
 Max. tensile stress at the same section, due to bending and 
 
 centrifugal force = 
 
 M.Y, 
 
 4- centrifugal stress at section. 
 
 FIG. 35. 
 
 And if this be done for several sections of the radius (X} 
 we shall obtain the " maximum maximorum " stress at some 
 radius, which stress must not exceed the safe working stress of 
 the material used in the construction of the air-screw. 
 
 We have then to determine the Bending Moment (M x ) at 
 any distance (X) from the boss centre, that is, at any section 
 considered along the blade. 
 
 Let Fig. (35) represent a side elevation of the air-screw 
 blade. 
 
 Consider a section at A A' at a distance (X) from the boss 
 
STEESSES IN AIR-SCREW BLADES 83 
 
 centre. Then the Bending Moment at AA' is equal to the 
 sum. of all the B.M.'s at the sections from A A' to the blade tip. 
 
 Let (x) be any radius from the boss centre, lying between 
 AA' and the blade tip, and let (dE) represent the resultant air- 
 pressure at this section. The value of (dE) has already been 
 determined in terms of the radius (x). 
 
 Then B.M. at AA' due to the force (dE) = (dE) . (x - X), 
 and therefore 
 the total B.M. at AA' at a radius (X) from the boss centre 
 
 Ja? = r 
 dE. (x - X). 
 r = X 
 
 Now 
 
 (dR) = p.tf.b.cy. (P 2 + 4.7T 2 .a? 2 ). sec 7. dx, 
 so that 
 
 ^x - X) = p.n*. \(x - X). b.Cy. (P 2 + 4.7r 2 .^ 2 ). sec 7. dx 
 
 X Jx = X 
 
 = B.M. at AA' at a distance of (X) from the 
 
 boss centre. 
 
 We are now able to calculate the value of the B.M. on any 
 section at any distance from the boss centre. 
 
 Since we may without appreciable error take the value of 
 (sec 7) as equal to unity, the above becomes 
 
 ; x = p.n 2 . 
 
 x X 
 
 and, since (R) is usually measured in Ibs., (x) and (X) will 
 be measured in feet, when (p) = '00238, and therefore the 
 value of (M x ) will be obtained in Ibs. /ft. 
 
 But, since (Y c ), (Y t ) and (I) are usually measured in inches, 
 (M x ) must also be measured in inches in order to give the 
 stresses in lbs./sq. in. 
 
 Hence we must multiply the value of (M x ), obtained from 
 the formula already given, by (12). 
 
 So that we then have 
 B.M. at (X) in Ibs./in. = (12). (M x ), which is equal to 
 
 * f(L- 
 Jx = X 
 
 G 2 
 
84 AIK-SCEEWS 
 
 and we may evaluate this at once if we assume that the 
 conditions (1) and (2) already given are satisfied, so that (b) 
 and (Cy) are constants over the blade. 
 
 To estimate the Bending Moment at or near the boss, we 
 put (X) equal to zero, and the formula then becomes 
 
 B.M. at (X) in Ibs./in. = e>.p.n\l).c y .r\ (P 2 -f 2.7r 2 .r 2 ), 
 
 and if we take as an example that 
 
 p = -00238, n = 20, r = 4, b = -75, c y = -2, P = 5, ?r 2 = 10, 
 
 we have 
 
 B.M. in Ibs./in. at or near the boss is equal to 
 
 (6) ( 00238) (400) ( 75) ( - 2) (16) (25 + 20x16) 
 = 4728 Ibs./in. 
 
 Now we have the maximum value of Compression stresses 
 at (X) ft. from the boss centre, due to Bending Moment and 
 
 centrifugal force in Ibs./sq. in. = 12. c centrifugal stress 
 
 
 = ~->.?i 2 . (*-X).&.<v.(P i +4.irUB).&! 
 
 ir*.n?.w (* r j 
 - -^ a.x.dx. 
 36.0.i J x 
 
 And similarly the maximum value of the Tension stresses 
 at (X) feet from the boss centre due to Bending Moment and 
 
 centrifugal force in Ibs./sq. in. = 12. -' - + centrifugal stress 
 
 12.Mx.Y t 
 
 I x 
 
 = 12. Y ^ 2 PL X x 5c 
 
 1 ^ ir*n*w r 
 
 + T-T I < 
 
 36.tf.ftn ' Iv 
 
 t/ * y/-A 
 
 r 
 
 a.x. ax. 
 x 
 
 And we may ^now obtain the values of both the maximum 
 Tension and maximum Compression stresses at any distance 
 (X) feet from the boss centre. 
 
STKESSES IN AIE-SCEEW BLADES 85 
 
 If then the values of these stresses be worked out for 
 several values of (X), we shall obtain a " maximum maximorum" 
 value for the Tension and Compression stresses at some 
 particular value of (X). 
 
 The greatest of these two values so obtained must not 
 exceed the safe working stress of the material of which the 
 blades of air-screws are made. An approximate value for the 
 safe working load of walnut is 2000 Ibs./sq. in. 
 
86 AIR-SCEEWS 
 
 CHAPTER VII. 
 
 STATIC THRUST. 
 
 THE supposed "effectiveness" of an air-screw is sometimes 
 thought to depend upon the " pull " it exerts when revolving 
 on the ground, that is when the aeroplane to which it is 
 attached is being held back prior to a flight. 
 
 Now although what may be termed the " Static " thrust of 
 an air-screw cannot be considered as necessarily being a 
 criterion of efficiency, yet it is interesting to see how far it is 
 possible to predict quantitative values for this thrust when 
 considered in the light of the aerofoil theory. 
 
 It is quite evident that the analogy still holds in this case, 
 for we may consider each section along the blade as moving 
 with a definite velocity and hence exerting a definite thrust, 
 although the air-screw is acting like a fan, since it has no 
 velocity in a direction normal to the disc of revolution of the 
 blades.* 
 
 The helix angles (A) are thus equal to zero, and the 
 " angles of attack " (a) must therefore be considered as being 
 equal to the actual measured chord angles at the various radii. 
 
 Thus (P) = 0, and (a) = (</>). 
 
 The general expression for the thrust on an air-screw blade 
 
 y 
 for any value of is given by 
 
 ( r 
 
 T = c.p.n*. f(x). <j)(x). (2.TT.X-P. ^(x) ). \XP"-f 4.ir a .a; a . dx. 
 
 J r 
 
 * Owing to some of the speeding up of air occurring before the 
 air-screw is reached, the theory cannot be applied directly without very 
 appreciable error. This error is not so great in the case of stationary 
 air-screws working in an enclosing channel. 
 
STATIC THKUST 87 
 
 And, since in this case (P) = 0, this becomes 
 
 T = 4.7r 2 .c.p^ 2 . P/0). $(x). x\ dx, 
 
 Jr 
 
 And this can be evaluated for any form of air-screw blade 
 considered. 
 
 Let us suppose however that the blade widths are uniform 
 from boss to tip, and that the section is also uniform. 
 
 We may regard (r ) = 0, so that we get 
 
 T = 4.7r 2 .p.&.?t 2 .c v . I y?. dx 
 
 and .-. 
 
 which gives an approximate value for the " Static " thrust on 
 each blade. 
 
 The value of (c y ) will depend upon the form of the blade 
 sections at the outside radii, and the " average v values of ($), 
 the chord angles of the blade to the air-screw's disc of 
 revolution. 
 
 Approximate values for (c y ) can be obtained from reference 
 to tests carried out on sections similar in form to those 
 employed at the outside radii of the blade and at angles of 
 the same amount as the " average " chord angles. 
 
 (c y ) often has a value of about ( 4). 
 
 Ic is useful, however, to obtain this " Static " thrust expres- 
 sion in terms of the horse-power of the motor. 
 
 Since 
 
 550.H.5 550.H.7; 
 V P.w, 
 
 in Ib./ft./sec. units, we obtain 
 
 = __ 
 
 S.ir.n.d. tan 7 
 
 The value of (tan 7) depends upon the values of the outside 
 
 chord angles. 
 
88 AIR-SCEEWS 
 
 If we take (tan y) = -, we get 
 N.T = 
 
 1 
 3' 
 
 700.H 
 
 n.d 
 
 This formula, while being very simple, is of course only a 
 very approximate one. 
 
 1000 
 
 8 CO 
 
 GO 
 
 400 
 
 20C 
 
 H.P. - 
 
 CuYve 
 
 400 
 
 800 /ROC 
 
 Revs, / 
 
 FIG. 36. 
 
 /600 
 
 8000 
 
 MOO 
 
 If as an example 
 
 H = 40, n = 2Q,d = 8, we get 
 
 XT 700 x 40 
 
 20x8 = ' a verv 
 
 In any case, however, it would appear that we can put the 
 expression for the " Static " thrust into the form 
 
 N.T = 
 
 X.H 
 
 n.d' 
 
 where (X) has a value in the neighbourhood of (1000). 
 
STATIC THEUST 
 
 Suppose 
 
 H = 40, n = 8, d = 8, \ = 1000 ; then 
 
 XTT 1000x40 
 N ' T= 8x8 
 
 We notice that the " Static " thrust increases inversely as 
 the rate of revolution of the air-screw, so that we should expect 
 a slow-running, i.e. geared-down, air-screw to have a higher 
 
 26OC 
 
 Sfahc Tef(V"c). 
 
 
 ROOD 
 
 <n 
 -Q 
 ^ 
 ^ /500 
 
 i/l 
 
 3 
 
 >. 
 -C 
 
 K 
 
 
 TArusf - 5/>rcd Cur^e 
 Projbe//er, 
 
 / 
 
 
 
 
 
 / 
 
 
 
 
 
 
 / 
 
 
 500 
 
 
 
 / 
 
 / 
 
 
 
 
 _^ 
 
 / 
 
 
 
 
 o 
 
 400 80O '200 /600 t. 000 2*00 
 
 ??evs. per Min. 
 
 FIG. 37. 
 
 " Static " thrust than one which was coupled direct to the 
 motor, for the same horse-power. 
 
 The " Static " thrust of an air-screw on an aeroplane is 
 usually measured by attaching a spring balance to the rear 
 portion of the machine and attaching a rope from the spring 
 balance to some fixed support. 
 
 We see also from the general formula for the static thrust 
 of an air-screw that the static thrust varies as the square of the 
 
90 AIR-SCBEWS 
 
 rotational speed, and that the necessary B.H.P. required to turn 
 the air-screw varies as the cube of the rotational speed. 
 
 This is borne out approximately by experimental tests on 
 air-screws, as Figs. (36), (37) will show. These Figures are 
 taken from the results of tests carried out at the National 
 Physical Laboratory by Mr. F. H. Brain well and Mr. A. Fage.* 
 
 We notice from these experimental curves that for a static 
 thrust of approximately 1050 Ibs. on an 8 ft. diameter air- 
 screw having a rotational speed of 1600 revs, per min., the 
 B.H.P. necessary is approximately 270. Hence if we apply 
 the formula deduced for the static thrust we may determine 
 an approximate value for the constant (X). 
 
 We have 
 
 "X TT 
 Total static thrust of air-screw = X.T = 
 
 n.d 
 
 whence 
 
 X.270.60 
 1600.8 ' 
 
 and therefore 
 
 (X) = 830. 
 
 This value is rather smaller than the value proposed, from 
 reference to actual static tests carried out on aeroplane air- 
 screws, for this constant, namely 1000. 
 
 * See " The Aeroplane," By A. Fage, A.E.C.Sc. (Charles Griffin & Co.). 
 
91 
 
 CHAPTER VIII. 
 
 EFFICIENCY OF AN AIR-SCREW AT DIFFERENT SPEEDS OF 
 TRANSLATION. 
 
 PRESENT day aircraft, whether aeroplanes or dirigibles, have as 
 a rule two distinct limiting speeds of flight. In the dirigible 
 the minimum speed will of course be zero, but in the aeroplane 
 this is not so, and the minimum possible speed of flight for 
 any given type may be calculated approximately when the 
 characteristics of the machine are known. 
 
 Now between these two limiting speeds, that is between the 
 minimum and maximum climbing speeds, there will be an 
 infinite range of speeds at which the aeroplane may fly. That 
 is to say that, if an aeroplane has a maximum velocity of say 
 100 feet per second and a minimum velocity of say 50 feet 
 per second, then between these two outside speeds the aeroplane 
 has an infinite 'number of different velocities at which it 
 may fly. 
 
 Now it will usually be found that in the case of an aeroplane 
 the speed at which it is able to climb fastest, that is the speed 
 at which the reserve thrust horse-power of the motor is greatest, 
 will lie between its maximum and minimum speeds and will 
 usually be nearer the latter. 
 
 If we examine the thrust horse-power curve of an aeroplane 
 it will usually be found that the ordinates on the curve have 
 at some value of the abscissae (in this case velocity) a minimum 
 value, and the velocity corresponding to this minimum point 
 will be the velocity at which the thrust horse-power required 
 for horizontal flight is a minimum, and hence the reserve thrust 
 horse-power is a maximum, that is the velocity of ascent will 
 be greatest at this value of the velocity. 
 
92 AIE-SCEEWS 
 
 Now this is only true when the efficiency of the air-screw 
 is supposed to remain constant throughout the interval between 
 the maximum and minimum speeds of flight. In practice, 
 however, this is never so, since the air-screw is usually designed 
 for the maximum flight velocity, and hence for any other value 
 of the velocity the efficiency of the air-screw falls off. 
 
 Hence the curve of available thrust horse-power is now no 
 longer a straight line, and therefore the maximum difference 
 between the ordinates of this curve and the curve of thrust 
 horse-power required for horizontal flight will no longer neces- 
 sarily occur at the velocity corresponding to the minimum 
 value of the thrust horse-power required for horizontal flight, 
 but will usually be found to occur at a value of the flight 
 velocity slightly greater than this value. 
 
 Now in order to be able to determine this point on the 
 curve and hence the maximum rate of climb possible and also 
 the speed of flight corresponding to this maximum climb, it is 
 necessary to know how the efficiency of the air-screw varies 
 for different values of (V), the velocity of flight. Experimental 
 results of tests on air-screws have been obtained for various 
 types and curves of efficiency plotted against values of (V). 
 
 It is interesting, however, to attempt to predict the amount 
 of this variation in efficiency from the results already obtained 
 on the assumption of the aerofoil analogy. 
 
 We have already obtained an expression for the efficiency 
 
 y 
 of any type of air-screw at any value of the effective pitch , 
 
 tv 
 
 that is at any value of the velocity of advance (V). 
 
 Hence we may, providing we possess the necessary informa- 
 tion with regard to the values of the lift and j of the sections 
 
 drag 
 
 at various angles of incidence, determine the value of (r;), the 
 efficiency of the air-screw, under varying sets of conditions and 
 for various values of the translational velocity. 
 
 Now it has already oeen shown that the results obtained 
 from the " Rational^' blade form in many cases approximate 
 very closely to standard types of air-screw blades, and we shall 
 therefore use this form of blade in the quantitative determina- 
 
EFFICIENCY AT DIFFEKENT SPEEDS 93 
 
 tion of the values of (7;) for various values of the velocity of 
 advance (V). 
 
 This modification is introduced for convenience and simplicity 
 of working out the numerical examples, which become very 
 tedious when treated from the most general expression for 
 efficiency. 
 
 We have then the efficiency of an air-screw of this type is 
 given by 
 
 P. ( X. ftx). (2.7T.X - P. ^(X) ). dx 
 Jr 
 
 J,r 
 x*. $0). (P + 2.TT.X. ^r(x) ). dx 
 > 
 
 for any value of (P) = , and we proceed to discuss the varia- 
 
 tion in the value of (77) for variations in the value of (V). 
 
 Now it is not proposed to attempt to evaluate algebraically 
 
 y 
 the expression for (77) for any value of , since to do so would 
 
 71 
 
 necessitate the determination of at least approximate equations 
 
 y 
 for the two functions <f>(x) and ty(x) for each value of - 
 
 considered. 
 
 We shall therefore employ a graphical method in this 
 investigation. 
 
 It will be necessary to determine the value of (77), the 
 efficiency of the air-screw, for several values of the advance per 
 
 revolution . and when these values have been obtained a 
 n 
 
 smooth curve drawn through the points will give at any rate 
 a near approximation to the value of the air-screw's efficiency 
 at any value of the velocity (V) considered. 
 We have then to plot the two graphs of 
 
 and 
 
 Y 
 
 for each value of taken, then determine the two areas 
 
 n 
 
94 AIK-SCKEWS 
 
 enclosed by each respectively, the extreme ordinates at radii of 
 (?' ) and (r), and the (x) axis, divide the area thus enclosed by 
 the first curve by the area enclosed by the second curve, and 
 
 V 
 
 multiply the result by ^ , where (V) has the particular 
 
 value of the velocity of advance chosen. The value of (P) in 
 
 y 
 
 each case will then of course be , (V) having the particular 
 
 r fi 
 
 value already defined. 
 
 Since we know the form of the blade sections at various 
 radii we may, by reference to tests carried out in a wind- 
 channel on sections of similar form, determine the probable 
 values of (c y ) and (tan 7), that is of </>(x) and ^(#), of the 
 sections at these radii when we know the respective values of 
 the angles of attack of the sections considered. 
 
 We may determine the values of these angles of attack for 
 
 y 
 
 each value of (V), and hence of , chosen as follows. 
 
 iii 
 
 We must first determine the values of the chord angles (</>) 
 at the radii considered, and we can do this either by direct 
 measurement of these angles on the air-screw, or, if we know 
 
 Y 
 
 for what value of - - the air-screw is designed, it is only 
 id 
 
 necessary to determine the values of (</>) for any radius chosen 
 from the relation 
 
 tan 
 
 -' 
 
 where (P) has the value of the designed effective pitch, that is 
 the value of the velocity of the aircraft (usually the maximum 
 velocity) for which the air-screw is designed, divided by the 
 value of (n) at this velocity. 
 
 (a x ) is the value of the " angle of attack " of a section at 
 radius (x), for the velocity of the aircraft for which the air- 
 screw is designed. 
 
 (a x ) may be termed the "initial angle of attack" of any 
 section at radius (x). 
 
 Having then found the value of (fa), the chord angle for 
 
EFFICIENCY AT DIFFERENT SPEEDS 
 
 95 
 
 any radius (x), we may proceed to determine the new value of 
 (a x ) for the various values of (V) considered. 
 
 Since the effective pitch has now a different value from its 
 "initial" designed value, we may denote it by (P'), and the 
 new " angles of attack " at each radius (x) by (a x r ). We then 
 have at once that 
 
 ; = <^- tan-' (-, 
 
 and we are now in a position to make the following table for 
 each value of *(V) taken. 
 
 Section No. 
 
 7 
 
 6 
 
 5 
 
 4 
 
 Eadius (x) feet . 
 
 1 foot 
 
 2 feet 
 
 3 feet 
 
 4 feet 
 
 Chord angle at) 
 radius (x) . ./ 
 
 42 30' 
 
 25 3 42' 
 
 18 51' 
 
 15 15' 
 
 Angle of attack) 
 at radius (x) / 
 
 10 
 
 8 
 
 7 o 
 
 6 13' 
 
 Abs. lift coeff. at) 
 radius (x) . J 
 
 338 
 
 525 
 
 475 
 
 425 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 radius (x) . 
 
 
 ^8 
 
 
 
 3-4 
 
 10-5 
 
 12 
 
 The above table has been taken as an example. It is for an 
 air-screw designed for a velocity of advance (V) of 100 feet 
 per sec., and a speed of revolution (n) of 20 revs, per second. 
 The value of the effective pitch is thus 5 feet, and the table 
 given is for the case considered when the velocity of advance 
 is 80 feet per second, so that the new value of the effective 
 pitch is 4 feet. The sections are those given in the " Technical 
 Eeport of the Advisory Committee for Aeronautics, 1911-12," 
 page 76, and already referred to in a previous chapter. 
 
 The values of <f>(x) and -^(x) are taken from the curves 
 plotted as a result of tests in a wind-channel on these sections 
 at the National Physical Laboratory. 
 
 Similar tables to the one given may now be obtained for 
 y 
 
 several other values of from zero up to and beyond the 
 
 n 
 
96 AIK-SCBEWS 
 
 designed velocity of flight. From each of these tables we may 
 plot two curves, the curves respectively of 
 
 and 
 
 &.#X). (P + 2.TTJCW) ), 
 
 and the two areas enclosed by these two curves may then be 
 
 found in each case. We may thus calculate the efficiency of 
 
 y 
 
 the air-screw for each value of - chosen, and the values so 
 
 n 
 
 obtained can then be plotted against their respective values of 
 (V), giving a curve of efficiency for any value of the velocity of 
 advance of the air-screw considered. 
 
 It will be found that when (V) = 0, then (?;) = 0, and that 
 when (V) has values appreciably greater than the designed 
 value of (V) the value of (rj) rapidly falls off. 
 
 An example will make the application of this method 
 clearer. 
 Let 
 
 (V) the designed or initial value of the flight velocity 
 of the aircraft = 100 ft. per second. 
 
 (n) the speed of revolution of the air-screw at this 
 value of (V) = 20 revs, per second. 
 
 Then (P) the initial value of the effective pitch = 5 feet. 
 (d) the diameter of the air-screw = 8 feet. 
 
 (a x ) the initial angles of attack of the sections along 
 the blade = 4 = constant over blade. 
 
 Let the sections along the blade be those already referred to 
 and let them be spaced as follows : 
 
 At a radius of 1 foot, section no. 7. 
 2 feet, 6. 
 
 O K 
 
 o ,, ,, ,, o. 
 
 4 4. 
 
 We may then form the following table : 
 
EFFICIENCY AT DIFFERENT SPEEDS 97 
 
 y 
 
 - = 5 feet. 
 
 Section No. 
 
 7 
 
 6 
 
 5 
 
 4 
 
 ^ 
 
 42 30' 
 
 25 42' 
 
 18 51' 
 
 15 15' 
 
 a x 
 
 4 
 
 4 
 
 4 
 
 4 
 
 (j>(x) 
 
 416 
 
 384 
 
 374 
 
 344 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 Y(x) 
 
 10-2 
 
 11-4 
 
 11-8 
 
 12-8 
 
 X 
 
 Ifoot 
 
 2 feet 
 
 3 feet 
 
 4 feet 
 
 From this table plot the two graphs of 
 
 and 
 
 taking values of <(#) and ^r(x) for each value of (x) taken from 
 the above table. (P) has of course the value of (5). 
 
 These two curves are shown plotted in Fig. 38. 
 
 The efficiency of the air-screw under these conditions is 
 
 seen to be 75%. 
 
 Y 
 Now take successive values of , of say 1 ft., 2 ft., 3 ft., 4 ft., 
 
 fv 
 
 6 ft., and 7 ft., and in each case form the corresponding table. 
 These tables are given below in order. 
 
 - = 1 foot. 
 n 
 
 Section No. 
 
 7 
 
 6 
 
 5 
 
 4 
 
 0* 
 
 42 30' 
 
 25 42' 
 
 18 51' 
 
 15 15' 
 
 a x 
 
 33 28' 
 
 21 9' 
 
 15 49' 
 
 13 
 
 #() 
 
 
 44 
 
 41 
 
 53 
 
 *(*) 
 
 
 
 1 
 2-2 
 
 1 
 
 3-1 
 
 1 
 
 7 
 
 X 
 
 Ifoot 
 
 2 feet 
 
 3 feet 
 
 4 feet 
 
 II 
 
98 
 
 AIR-SCREWS 
 
 G9- - 
 
 {A-reo rncloed by TopCurve = 
 w M Lower . = 
 
 13 38 
 20 63 
 
 ... -n = _ . - = .75 
 
 1 2.TT SO 63 
 
 35 
 
 15 
 
 2' 
 
 R a d i u s(x)/ 
 FIG. 38. 
 
EFFICIENCY AT DIFFERENT SPEEDS 
 
 99 
 
 Y 
 
 - = 2 feet. 
 
 n 
 
 Section No. 
 
 7 
 
 6 
 
 5 
 
 4 
 
 *. 
 
 42 30' 
 
 25 42' 
 
 18 51' 
 
 15 15' 
 
 a x 
 
 25 
 
 16 40' 
 
 12 48' 
 
 10 42' 
 
 *(*) 
 
 48 
 
 39 
 
 425 
 
 575 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 YW 
 
 2 
 
 2 -.5 
 
 4-3 
 
 10 
 
 03 
 
 1 foot 
 
 2 feet 
 
 3 feet 
 
 4 feet 
 
 Y 
 
 - = 3 feet. 
 
 Section No. 
 
 7 
 
 6 
 
 5 
 
 4 
 
 0* 
 
 42 30' 
 
 25 42' 
 
 18 51' 
 
 15 15' 
 
 o 
 
 17 
 
 12 18' 
 
 9 49' 
 
 8 27' 
 
 *() 
 
 38 
 
 38 
 
 56 
 
 5 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 y(aO 
 
 2-3 
 
 3'7 
 
 9 
 
 11 
 
 * 
 
 1 foot 
 
 2 feet 
 
 3 feet 
 
 4 feet 
 
 Y 
 
 - = 6 feet. 
 
 n 
 
 Section No. 
 
 7 
 
 6 
 
 5 
 
 4 
 
 0, 
 
 42 30' 
 
 25 42' 
 
 18 51' 
 
 15 15' 
 
 a x 
 
 - 1 11' 
 
 oir 
 
 113' 
 
 148' 
 
 *() 
 
 25 
 
 26 
 
 275 
 
 25 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 VW 
 
 7'5 
 
 9 
 
 10-5 
 
 11 
 
 a? 
 
 Ifoot 
 
 2 feet 
 
 3 feet 
 
 4 feet 
 
 H 2 
 
100 
 
 AIR-SCKEWS 
 
 y 
 
 = 7 feet. 
 n 
 
 Section No. 
 
 7 
 
 6 
 
 5 
 
 4 
 
 0* 
 
 42 30' 
 
 25 42' 
 
 18 51' 
 
 15 15' 
 
 a x 
 
 - 5 30' 
 
 - 3 20' 
 
 - 1 27' 
 
 - 21' 
 
 MX) 
 
 06 
 
 12 
 
 17 
 
 15 
 
 , 
 
 1 
 
 1 
 
 1 
 
 1 
 
 Y(x) 
 
 1-5 
 
 b-5 
 
 6 
 
 7 
 
 X 
 
 Ifoot 
 
 2 feet 
 
 3 feet 
 
 4 feet 
 
 From these tables plot the curves given in Figs. 39, 40, 41, 
 42, 43, and 44. 
 
 The respective efficiencies are seen to be as follows. 
 
 The value of (n) has been supposed' to remain constant 
 throughout. This assumption does not affect in any way the 
 generality of the method. 
 
 y 
 
 At a speed of translation of 20 ft./sec., - = 1 ft., (rj) = 15-5% 
 
 11 
 
 40 =2,, = 32-6% 
 
 60 =3,, = 54-5% 
 
 80 = 4 = 67% 
 
 120 =6,, = 76-7% 
 
 140 = 7,, =62-7% 
 
 A curve of efficiency against values of (V) may now be 
 plotted, Fig. (45). 
 
 Y 
 
 It will be noticed that when = 6 ft., i.e. at a velocity of 
 
 translation of 120 ft. per second, the efficiency of the air-screw 
 is higher than the efficiency at the speed for which it is 
 designed. This is due to the fact that the value of (Z), 
 
 i.e. , is greater than the initial value, and, although the -^~ 
 a drag 
 
 of the individual sections is less than in the initial case, yet 
 
EFFICIENCY AT DIFFERENT SPEED'S 101 
 
 osed >^ To\a Curve s. 2.S-82 q 
 
 . Lowv . = E3-5 - 
 
 FIG. 39. 
 
102 
 
 AiE-SCEEWS 
 
 FIG. 40. 
 
EFFICIENCY AT DIFFEKENT SPEEDS 103 
 
 50 
 
 45 
 
 10 
 
 GO- 
 
 JAv. 
 
 AYCQ enclosed bj Top Curve. = 86 '14 
 
 . Lower . = 8 66 
 
 a' 
 
 FIG. 41. 
 
104 
 
 AIK-SCBEWS 
 
 I Aro enc/ostfd by Top Curve - E4-34 s<j, mv 
 . Lowev % - 3 /4 
 
 - . _ _ V Area enclosed by Tof> Curve 
 
 p TT |Q /\v a e nclose^^^Tl^^v^M^^^I we 
 
 g> 4-34 
 
 TT S3 I 4 
 
 67 
 
 "Radius (x) 
 
 FIG. 42. 
 
EFFICIENCY AT DIFFERENT SPEEDS 105 
 
 50 
 
 35 
 
 fl*o enclosed byT o .Curv O55 S^ .oS.l 
 . Lower . f6-84 ,. . ) 
 
 & .-. T 
 
 " si? 
 
 -767 
 
 R Qd/ 
 
 FIG. 43. 
 
106 
 
 AIR-SCEEWS 
 
 {b-rca enclosed by Top Curve = 7. 
 . Lower . = 13 
 
 .54 
 4 
 
 g.TT 
 
 Rod i u <, (x) / 
 FIG. 44. 
 
EFFICIENCY AT DIFFEEENT SPEEDS 107 
 
 OR)'"'- 
 
 FIG. 45. 
 
108 AIK-SCEEWS 
 
 the higher value of (Z) makes a preponderance in the efficiency 
 
 y 
 
 of the whole blade. At a value of - - =7 ft., however, we 
 
 n 
 
 notice that the efficiency is very much smaller in spite of the 
 still higher value of (Z). 
 
 Of course if the air-screw was designed initially for a value 
 
 V 
 
 of - = 6 ft., the efficiency would be greater than that now 
 
 y 
 
 given at this value of , providing of course that the spacing 
 
 / TL 
 
 and forms of the sections, etc., remained the same as in the 
 initial case here considered. 
 
 This curve of efficiency against values of translational 
 velocity (Y) appears to conform to curves obtained by 
 experiment on similar types of air-screws. 
 
109 
 
 CHAPTEE IX. 
 
 DIRECT LIFTING SYSTEMS. 
 
 WHEN" a force (P) moves its point of application through a 
 distance (s), in the direction of the force, it is said to do (P.,s) 
 units of work. 
 
 If a weight of (W) Ibs. be lifted through a distance of 
 (x) feet the work done against gravity is (W.x) ibot-lbs., sup- 
 posing the distance (x) to be negligible in comparison with the 
 earth's radius. 
 
 Now suppose that a body of weight (W) Ibs. is being lifted 
 against gravity at a uniform speed of (Y) feet/sec., then the 
 work done per second is equal to (W.V) ibot-lbs., and the 
 horse-power necessary to keep the body moving at this velocity 
 
 is equal to - 550 . 
 
 And suppose the B.H.P. of the motors installed in the 
 machine to be denoted by (H), and the efficiency of the lifting 
 screws to be (rj), then the Effective Thrust Horse-Power of the 
 motors is equal to (H.??), and this is the horse-power available 
 for keeping the machine in motion. 
 
 So that we have 
 
 11-77 = 550 ' 
 whence 
 
 V = ,i?' feet/sec., 
 W 
 
 or 
 
 W.V 
 
 : 550^' 
 
110 AIK-SCKEWS 
 
 Now suppose that (V) = 0, so that the machine remains 
 stationary in the air, then 
 
 H ------ 
 
 = 550' rj = : V 
 
 and if (77) is not zero, the B.H.P. required to sustain the 
 weight (W) vanishes. 
 
 But from the general expression for the efficiency of any 
 type of air-screw, we notice that when the effective pitch, 
 that is the advance per revolution of the air-screw, is zero 
 the efficiency of the air-screw is also zero, so that (H) 
 is not necessarily a vanishing quantity, for we obtain the 
 expression 
 
 W_ Q ^ 
 
 ~ 550*0 " 0' 
 
 which is indeterminate, and may be zero or a finite quantity. 
 
 We can however obtain the value of (H) for the case 
 (V) = from a consideration of the more general case when 
 (V) is not zero. 
 
 And if, having obtained the value of (H) for finite values 
 of (V), we consider (V) and therefore the effective pitch to 
 become very small and ultimately vanish, we shall obtain the 
 required value of (H) for the case when the machine remains 
 stationary in the air. 
 
 Suppose then that a body is moving vertically upwards 
 with a uniform velocity of (V) feet/sec., under the action of 
 screws which revolve at (n) re vs. /sec. Then the effective pitch 
 
 y 
 of such screws will be - feet, and if we know the diameter (d) 
 
 71* 
 
 of each screw we can find an expression for the efficiency of 
 the screws in terms of the quantities already known. 
 
 We will suppose for the sake of simplicity that we are 
 using a shape of air-screw blade similar to that already defined 
 as the " Eational " blade. Then, if we further suppose that 
 (cy) and (tan 7) are constants over the blade, and that (tan 7) 
 
 has the value of =-^, we can at once write down an expression 
 
DIRECT LIFTING SYSTEMS 111 
 
 for the efficiency of the whole air-screw under the conditions 
 specified. 
 
 The above assumptions are made for the sake of simplicity 
 and convenience, and do not affect the general results to any 
 very great extent. The " Rational " form of blade outline is 
 such a simple one to deal with analytically that the great 
 simplification in the work obtained by using this form of blade 
 is a justification for not treating the problem of direct lift in 
 its more general form, when the blade outline is considered as 
 some arbitrary undetermined function of the radius (x). 
 
 The expression for the efficiency is therefore given by 
 
 2.Z.(8.7r-Z) 
 ^.(TT + IG.Z)' 
 
 P V 
 
 and, since (Z) = -7 = we obtain 
 
 Cv ??/ . Ct> 
 
 And this expression is the value of the efficiency of the 
 lifting screws under the conditions specified. 
 
 We are now in a position to estimate the necessary B.H.P. 
 of the motors for the velocity (V) upwards. We have already 
 shown that the required B.H.P. is given by 
 
 II - W V 
 
 " 550* 7,' 
 
 and 
 
 2. V. (S.TT.n.d -V) 
 
 71 = 
 
 so that 
 
 1100' (S.Tr.n.a- \) 
 
 
 
 W TT nd 
 And now, if (V) = 0, then (H) = ^^- t 
 
 which gives the B.H.P. necessary for " hovering." 
 
112 AIR-SCEEWS 
 
 This was the case which before we were unable to evaluate 
 owing to the indeterminateness of the expression-.,. 
 Thus as an example suppose that 
 
 W = 1100 Ibs., 
 n = 20 revs. /sec., 
 d = 8 feet, 
 
 then the B.H.P. necessary to keep the machine hovering in the 
 air is given by the value of (H), which in the above example is 
 found to be equal to (63) B.H.P. approx. 
 
 It can be seen from the formula that when (V) is not zero, 
 the B.H.P. required is greater than what it was for the case 
 when (V) = (0), and as (V) increases so does (H) increase. 
 
 The value of (H) is strictly the value of the B.H.P. of the 
 motors multiplied by the efficiency of the transmission to the 
 lifting screws. 
 
 It will be noticed that when 
 
 (V) = 8.7r.n.d 
 
 the value of (H) becomes infinite, which forms therefore the 
 limiting value of (V). This limiting value of (V) is caused 
 by the fact that when (V) has the value given above, (Z) is 
 equal to (8.77-.), and this corresponds to a zero value of the 
 efficiency of the air-screws. 
 
 Now in order that the screws may be capable of lifting 
 the whole machine, it is necessary that their combined 
 effective thrusts shall be greater than the total weight of the 
 machine (W). 
 
 This is equivalent to saying that the air-screws must be 
 capable of sustaining a greater weight than (W) when (V) = 0, 
 for if it be supposed that the machine has an extra weight (w) 
 attached to it, and that the screws are just supporting the 
 combined weights of (W) and (w) in the air, then it is obvious 
 that if the weight (w) be detached from the machine it will 
 fall, and the machine will then be subjected to an accelerating 
 force greater than its weight (W), and hence that it w T ill 
 commence to rise. 
 
DIEECT LIFTING SYSTEMS 113 
 
 In order therefore that our helicopter may be capable of 
 rising off the ground, the effective horse-power of the motors 
 must satisfy the relation 
 
 TJ- W.ir.n.d 
 *> 880CT- 
 
 The quantitative determination of the acceleration produced 
 by the reserve horse-power is somewhat difficult to arrive at. 
 
 There are however other considerations bearing upon the 
 subject of vertical ascent in the air. 
 
 We have already shown that 
 
 H W 
 
 ~ 1100 
 
 y = Tr.n.d.(8800.TI-W.7r.n.d) 
 
 whence 
 
 which gives the velocity of ascent through the air. 
 We notice at once that when 
 
 W.Tr.n.d 
 8800 
 
 the value of (Y) is zero, and the machine remains stationary. 
 This is the condition already established for "hovering" flight. 
 Iii order, therefore, that (V) may be positive, it is necessary 
 that 
 
 W.ir.n.d 
 8800 ' 
 
 and this condition brings us to a consideration of the requisite 
 values of (n) and (d), which values so far have been assumed 
 to be anything whatever. 
 
 It is of course obvious that in order to obtain a good 
 efficiency for the lifting screws, the value of (Z) should approxi- 
 mate to the value at which the efficiency of the screws is a 
 maximum. But since it will be found that the velocity of 
 ascent, and hence the effective pitch, may be small compared 
 
114 AIPt-SCEEWS 
 
 to the diameter of the air-screw, unless this diameter be itself 
 small, it will be more economical to use possibly a large 
 number of separate air-screws, in which the ratio of effective 
 pitch to diameter is such as to entail at least a moderately 
 good efficiency. This may of course make the diameter of 
 each individual air-screw quite small. The reason for using a 
 number of separate air-screws is that the necessary width of 
 blade would be inordinately large in the case of one or two 
 lifting screws only, and hence the value of (c) would be such 
 as to entail an enormous amount of interfering action between 
 the blades of each air-screw. 
 
 Suppose then that there are (Q) separate air-screws, each 
 
 W 
 
 exerting a thrust of -~- Ibs., then the horse-power available for 
 
 turning each screw is ^-. 
 
 And let each air-screw (all of which are assumed to be 
 identical) have (N) blades. 
 Then we have 
 
 Total thrust exerted by each air-screw = (N.T) Ibs., 
 
 and 
 
 Total weight required to be supported by each air-screw 
 
 = Q lbS " 
 
 whence 
 
 W 
 (N.T) =-- Q, 
 
 Now we have for simplicity assumed that the blade shape 
 for each air-screw is that defined as the "Eational" blade 
 shape. And we have further assumed that the section of the 
 blade is uniform throughout the entire blade, except perhaps 
 near the boss, which does not affect the argument to any 
 appreciable extent.- 
 
 We can then at once apply the formula already obtained 
 for the thrust of an air-screw having this shape. 
 
DIEECT LIFTING SYSTEMS 115 
 
 We have 
 
 c.^^p.c y .l\d' 2 .(2.7r.d-3.P^nry) 
 
 12 
 whence 
 
 / V \ 
 
 ~r T c.n.TT.p.V.d^.Cu. I 2.7T.d 3. - . tan 7 ) 
 W V n '/ 
 
 Q : ~l2~ ' 
 
 and /. 
 
 12.W 
 
 Q = 
 
 c.p.7T.V.d 2 .c y . (2.7r.d.n 3.V. tan 7)' 
 
 In this formula for the determination of (Q) all the factors 
 are known with the exception of (V), the velocity of ascent. 
 We have, however, already obtained a formula for (V), viz. 
 
 ir.n.d.(88QQ.'E-W.Tr.n.d) 
 
 ~ ' 
 
 so that we can at once determine the least necessary value of 
 (Q) for any chosen value of (c). 
 Suppose that 
 
 W = 1000 Ibs., 
 
 n = 20 revs. /sec., 
 
 d = 10 feet, 
 H = 100 effective horse power, 
 
 and let 
 
 1 
 C = 3' 
 
 Cy = "4, 
 
 tan 7 = j^, 
 
 then, since we are using lb./ft./ sec - units, (p) will have the 
 value of (-00238), and hence we get the value of (Q) as 
 given by 
 
 12x1000 
 
 y = -i -- ~r~ 1? : 
 
 i X 00238 XTTX 15 x 10 2 x '4^2.^.10.20 - ~ 
 
 I 2 
 
116 AIR-SCREWS 
 
 / 
 
 We thus obtain the least value of (Q) required in order 
 that the initial conditions may be satisfied. 
 
 We notice that if (c) be less than -^ the necessary value 
 
 o 
 
 of (Q) will be greater than (6). 
 
 If we solve formally to obtain (Q) in terms of (H), we get 
 
 * = 
 
 and it can be seen from this that as (d) decreases in value (Q) 
 increases. It is interesting to consider for what value of (d) 
 (V) has a maximum value, and hence to obtain the necessary 
 A'alue of (Q) for this. 
 
 To find the value of (d) for which (V) is a maximum, we put 
 
 Id = ' 
 
 and this gives 
 
 _ (712-5).H 227.H 
 
 w.v.^r WM 
 
 as the value giving (V) a maximum value. 
 
 To obtain the value of the maximum value of (V), we 
 substitute the value of (d) obtained from the above in the 
 general formula for (Y), and get 
 
 462.H , 
 v max = w feet/sec., 
 
 and this gives the maximum velocity upwards under the best 
 possible conditions. 
 
 The value of the number (462) depends upon the value of 
 (tan 7) taken. 
 
 We also notice that the efficiency of the lifting screws 
 
 under these conditions is equal to ^-^, that is approximately 
 
 (84) per cent. 
 
 Thus a machine, weighing complete with engines, screws 
 
DIRECT LIFTING SYSTEMS 117 
 
 fuel, etc., (1000) Ibs., and having motors capable of developing 
 an effective horse-power of (100), would have an upward 
 velocity under the best conditions as already given of (46 2) 
 feet per second, or roughly (2800) feet per minute. This rate 
 of climbing is about double that of the fastest aeroplane scouts 
 at present in existence. 
 
 We can now obtain the necessary value of (Q) for a 
 maximum value of (V). 
 
 We get 
 
 QV *JiJj.j\*-v )- ./i./fc 
 7 ~r T-f* J 
 
 and since 
 
 227 H 
 
 d = ^ r ' : for a maximum value of (Y), 
 W.n 
 
 where (n) is arbitrary, we can find the necessary value of (Q) 
 for any arbitrary chosen value of (d). 
 This then gives 
 
 o = (_'i99)ao- 4 ).w_ 3 
 
 whence the larger the value of (d) the smaller the value of (Q). 
 Applying this result to the previous example, where 
 
 W = 1000 Ibs., 
 _ 1 
 
 <'<!/ = * 3 > 
 
 H - 100, 
 d = 10 feet, 
 
 we get 
 
 Q = 26-2. 
 
 So that we shall require at least (27) separate helices in 
 order that (c) may have a value not greater than -,. 
 
 Thus the blades would have to be between two and three 
 
118 AIR-SCREWS 
 
 feet wide and the air-screw would have to rotate at a speed 
 given by the relation 
 
 227.H 
 W.d' 
 so that 
 
 (n) = 2* 27, which is approximately (136) revs./rnin. 
 
 It is a simple matter to determine the necessary value of 
 the blade widths at any radius (x) along the blade. We have 
 
 2. C .P.^.. 
 
 giving the value of (b) for any radius (x). 
 
 At the tip of the blade (x) = (r), and the value of (b) is then 
 given by 
 
 C.l\7T.d 
 
 b r = - 
 
 Applying this last result to the example given above 
 
 1 46-2 
 
 - X 
 
 we get 
 
 So that if N = 2, that is, if each air-screw has (2) blades 
 we get 
 
 b r = (2 '65) feet (approx.), 
 
 and this is the necessary value of the blade width at the tip 
 under the conditions specified. 
 
 As stated in the Preface the results given by the theory in 
 the case of Direct Lifting Systems and for Static Thrust must 
 not be accepted without due caution. 
 
 Owing to the fact that in both these cases the translational 
 velocity of the machine is usually zero or very small, and 
 hence the ratio of the slip-stream velocity to this velocity 
 rather high, the angles of attack of the various blade elements 
 
DIEECT LIFTING SYSTEMS 119 
 
 are not the same as given by the theory, and consequently the 
 results obtained from it by calculation may be quite fallacious. 
 
 It would appear, however, that, providing the value of (Z) 
 is sufficiently large and in the neighbourhood of that obtaining 
 in standard types of air-screws as used upon aeroplanes (that is 
 having a value of say between *4 and unity), the results as 
 given by the theory in Chap. IX. should be found to be 
 sufficiently true, at any rate as a basis for practical design and 
 further investigation. 
 
 This would probably necessitate a very high rate of ascent, 
 or at any rate the utilization of the value of (Q) giving a 
 maximum value to (V), in order to obtain the requisitely large 
 value for (Z). 
 
 The calculated value of the B.H.P. required for " hovering," 
 given on page 112, is for these reasons probably much too small. 
 
 Mr. F. W. La.nchester, in a recent paper,* gives a formula 
 for the determination of the least requisite B.H.P. for a 
 stationary helicopter, his expression being 
 
 where (A) is the area of the propeller disc, (W) is the weight 
 sustained, and (p) the density of the fluid, in the case of air 
 
 approx. , or 078. 
 
 If we apply this to the example quoted, we obtain (135) as 
 a minimum value for (H). This is probably a much more 
 accurate figure. 
 
 * "The Screw Propeller," by F. W. Lanchester, M.Inst.C.E., read 
 before the Institution of Automobile Engineers in April 1915. 
 
121 
 
 APPENDIX I 
 
 NOTE ON THE INFLUENCE OF "ASPECT EATIO." 
 
 IN the Introduction a brief reference was made to the ratio of 
 the span to the chord of a wing, commonly known as the 
 " Aspect Batio." It is fairly certain that the characteristics of 
 an aerofoil vary with variation of aspect ratio, although any 
 exact quantitative determination of the alteration of lift and 
 
 , corresponding to a given change in the value of the 
 drag' 
 
 aspect ratio of a wing, would appear to be impossible at 
 present. 
 
 At the same time it would appear that an increase in the 
 
 , lift , 
 value of the -^ - always accompanies an increase in aspect 
 
 ratio. 
 
 The graph given in Fig. 46 is plotted from the result of 
 tests on a wing carried out at the National Physical Labora- 
 tory,* and serves as an indication of the kind of change to be 
 expected. The difficulty of correctly anticipating the amount 
 of this change in any case appears to be largely due to the 
 form of the wing tips employed. 
 
 This question of aspect ratio should be taken into account 
 as far as possible when designing an air-screw, as experimental 
 tests have shown that high efficiencies may usually be expected 
 from screws having blades of a high aspect ratio. 
 
 In fact it is apparent that, without attempting to formulate 
 any exact connection between blade efficiency and aspect ratio, 
 a high speed air-screw might conceivably have a better 
 efficiency owing to the necessary comparative narrowness of 
 
 * " Technical Eeport of the Advisory Committee for Aeronautics, 
 1911-12." 
 
122 
 
 AIR-SCREWS 
 
 the blades than one designed for the same conditions but to 
 rotate at a slower and otherwise more economical speed. 
 
 It is obvious, therefore, that this question cannot be 
 altogether neglected either in aeroplane or air-screw design, 
 and, in view of the comparatively meagre information at 
 present available on the subject, further experimental research 
 in this direction would appear to be required. 
 
 / 2 3 4 
 
 & 10 
 
 FIG. 46. 
 
123 
 
 APPENDIX II 
 
 NOTE ON THE EFFECT OF THE INDRAUGHT IN FRONT 
 OF AN AIR-SCREW. 
 
 IT was stated in the Preface that the theory here outlined was 
 not in any sense to be regarded as bearing any more than a 
 fairly close relationship to the actual conditions surrounding 
 the working of a screw in air. It would indeed be well nigh an 
 impossibility to formulate a theory which would adequately 
 deal with all the various complex factors entering into the 
 problem of screw propulsion in fluids, and the most that 
 scientific analysis can do is to build up some kind of a working 
 hypothesis which may reasonably be expected to give results 
 sufficiently true for the purpose of practical design. 
 
 In fact, in any investigation of this kind, certain factors 
 bearing upon the problem may have to be ignored owing to the 
 difficulty of accurately representing their effects without the 
 too continuous employment of experimental data in the form 
 of checks upon the theory. 
 
 One such factor, of which no quantitative notice has been 
 taken in the previous work, is the indraught of air in front 
 of an airscrew, the effect of which is to modify the conditions 
 under which any element of blade has been assumed to be 
 working. The main modifications introduced by such an 
 indraught appear to be a decrease in the " angle of attack " and 
 an increase in the relative air velocity of each element along 
 the blade, thus altering to some extent the values of the air 
 reactions upon the same, although it is evident that these two 
 effects tend to partially neutralise each other. Quantitatively, 
 however, it appears to be very difficult to readjust the foregoing 
 theory so as to include this effect of indraught, without very 
 much greater experimental evidence than is at the present 
 time available. Some experiments have been carried out by 
 G. Eiffel * to determine the magnitude of this incoming air, 
 and he shews that the ratio of the indraught velocity to the 
 
 velocity of translation is a function of the = - ratio only. 
 
 diameter 
 
 * " Nouvelles Recherches sur la Resistance de 1'Air et 1'Aviation," 1914. 
 
124 AIB-SCBEWS 
 
 If we assume that the analogy with aerofoils still holds for 
 every element of the blade, it is a simple matter to shew that 
 this ratio cannot exceed a certain limit directly depending 
 upon the value of the " angle of no lift " for the blade section 
 considered. The value of this ratio for air-screws of the type 
 
 as at present employed, and having 5 _ ratios of from 
 
 - diameter 
 
 ( 5) to ( * 7), appears to lie in the neighbourhood of ( 6) to ( 7) 
 for the effective portion of the blade. 
 
 Further, if we assume that the correction factor already 
 given for the calculated torque of a blade to be capable of 
 being applied to each element along the blade between the 
 limits of integration, we obtain a value of about ( 4) for this 
 ratio as sufficient to account for the difference between the 
 values of the torque as calculated and found by experiment. 
 
 The value of the correction factor taken is (-^ 
 
 \7o J 
 
 For a value of ( 4) for this ratio, the " angles of attack " of 
 the elements near the blade tip are found to be between (3) 
 and (4) less than the original values assigned to them when 
 no account was taken of the indraught. This would make the 
 real values of these angles of attack nearly zero for blades 
 designed on the aerofoil theory. 
 
 Eiffel's experiments do not, however, appear to substantiate 
 this view, the values of this ratio for various speeds of trans- 
 lation appearing not to exceed (-1). In any case, without 
 further experimental evidence it is quite useless to attempt to 
 fix any definite value for this ratio. 
 
 This question of the ratio of the indraught velocity to the 
 translational velocity is one which becomes of fundamental 
 importance in the case of screws having very small or zero 
 values of the effective pitch to diameter ratios, and were it 
 possible to obtain an accurate determination of the amount of 
 this ratio in such cases, the problem of the helicopter already 
 discussed would be much easier of a representative solution 
 than has so far been found to be the case, owing to the 
 limitations of the present theory. 
 
125 
 
 INDEX 
 
 " Aerial Flight," 3 
 Aerodynamical efficiency, 
 
 48 
 
 Aerofoil, 1 
 
 Aeronautical Society, 40 
 " Aeroplane," 90 
 Air-flow, 1 
 Air pressure, 16 
 " Air-screws," Paper on, 40 
 "Angles of attack," 13, 56 
 Angles of incidence, 13 
 " Aspect Ratio," 3, 121 
 " Average " reaction, 11,21 
 
 Bending moment, 81, 82 
 Bent plane, 1 
 B.H.P., 18, 52, 59, 119 
 Biplane, analogy of, 30 
 Blade elements, 13 
 Blade sections, 14, 47 
 Blade shape, 25 
 Blade tip angle, 10 
 Blade width constant, 59 
 
 Bolas, H., 65 
 
 Bramwell, F. H., 4, 12, 90 
 
 Bryan, G. H., 1 
 
 (c), value of, for normal 
 
 incidence, 3 
 
 Calculus of Variations, 55 
 Camber, 48 
 Centre of area of sections, 
 
 72 
 
 Centre of pressure of sec- 
 tions, 72 
 
 Centrifugal forces, 5, 77 
 Centrifugal pull, 77 
 Chord angles, 10, 13 
 4 ' Constant Pitch," 10 
 "Constructional Limit" 
 
 type of blade, 35 
 Contours, plotting of, 76 
 Convex under-surface, 76 
 Correction factor on blade 
 
 width, 63 
 Cylinder, 7 
 
126 
 
 AIR-SCKEWS 
 
 Diameter of air-screw, 51 
 
 Fluid motion, discon- 
 tinuous, 1 
 
 Formula for air-pressure, 1 
 Formulae, working. 47 
 
 Direct lifting systems, 109, 
 
 118 
 
 Dirigible, 91 
 Discontinuous blade out- | Fundamental hypothesis, 3 
 
 line curve, 44 
 Discontinuous fluid mo- "Gap," 31 
 
 tion, 1 | "Gap/chord" ratio, 31 
 
 Drag, 2 j Gliding angle, 3, 14 
 
 Drag coefficient, absolute, 2 i Graphical methods, iv, 60, 
 Drzewiecki, iii, 3, 4 
 
 "Effective Pitch," 9, 10 
 Efficiency curve, 27 
 
 Helicoidal path, 6 
 I Helicopter, 113, 119 
 
 " Efficiency curve " blade j Helix, 7, 8 
 
 shape, 29, 35 j Helix, angle of, 9 
 
 Efficiency, maximum point I Helix angles, 13 
 
 of, 27 
 
 Efficiency of an air-screw, 
 
 19, 91 
 Efficiency of an element, 26 
 
 Helix, length of, 8 
 Hovering flight, 113, 119 
 
 Ideal thrust grading dia- 
 
 Eiffel, G., 123 gram, 43 
 
 Elliptical shaped ends, 55 Indraught in front of an 
 
 Empirical multiplying fac- 
 
 tor, 63 
 Equations, 17 
 
 air-screw, 123 
 Infinitesimals, 23 
 Integration, 15 
 Experimental Mean Pitch, ! Interference, 30 
 
 12, 19,21 
 Experimental research, 51 | J nes > & * 
 
 Fage, A., 4, 90 
 Flat plate, 1 
 
 Laminae, 34, 73 
 Lanchester, F. W., 3, 4, 119 
 
INDEX 
 
 127 
 
 Laying out blade, 72 
 Lift coefficient, absolute, 2 
 Lift/drag ratio, 2 
 Lift of an aerofoil, 2 
 " Load Grading Curve," 23 
 Low, A. R., 34, 38, 40 
 
 Marine propellers, 4 
 
 Marine work, iii 
 
 Mathematical theory, 1 
 
 Mathematics, iv 
 
 " Maximum maximorum," 
 82, 85 
 
 Maximum ordinate, posi- 
 tion of, 27, 30 
 
 Moment of inertia, 8 1 
 
 National Physical Labora- 
 tory, 4, 90 
 
 Negative pressure, 2 
 
 Neutral axis, 80 
 
 "Normale" blade shape, 
 34, 35 
 
 Outside fibres, 81 
 
 Pitch of an air-screw, 6 
 Pitch of zero " average " 
 
 Reaction, 11 
 Pitch of zero Thrust, 1 1 
 Pitch of zero Torque, 1 1 
 Pitch ratio, 40 
 
 Pressure, negative, 2 
 Pressure, positive, 2 
 
 " Rational " blade efficiency 
 
 formula, 40 
 " Rational " blade shape, 
 
 34, 35 
 Resultant air pressure, 1, 
 
 14, 21, 22 
 
 Resultant Thrust, 1 1 
 Resultant Torque, 11 
 Royal Society, Proceedings 
 
 of, 1 
 
 Screws, marine, 4 
 Slip-stream, 43 
 Slip-stream velocity, 119 
 Speeding up of air, 86 
 Static Thrust, 86, 118 
 Stresses, 47 
 Stresses, bending, 80 
 Stresses, centrifugal, 77 
 Symmetrical plan form, 72 
 
 " Technical Report of the 
 Advisory Committee for 
 Aeronautics, 1911-12," 
 49, 65 
 
 " Technical Report of the 
 Advisory Committee for 
 Aeronautics, 1912-13," 
 4, 69 
 
128 AIK-SCREWS 
 
 " The Screw Propeller," 119 i Velocity, minimum, 91 
 Thickness/chord ratio, 47, ; Velocity of blade element, 
 
 56 16 
 
 Thrust of a blade, 18, 19 i Vertical ascent, 113 
 Thrust of an element, 1 6 
 
 Thrust, Static, 86 Walnut, safe working load 
 
 Torque of a blade, 18 of, 85 
 
 Torque of an element, 17 "Wash" of top wing, 
 Translational velocity, 9, 30 
 
 118 Whitehead, A. N., iv 
 
 I Wind-tunnel, 19 
 
 Velocity, climbing, 91 Working formulae, 47 
 
 Velocity, maximum, 91 ; Wright Aeroplane, 65 
 
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