APPENDIX I
The application of a propeller, which has been found satisfactory in service or on tests, to a new design, where its conditions of operation are not identical, is based on the theory of mechanical similitude. The complete theory cannot always be satisfied by the designer, and where approximations are made it is necessary to examine the fundamental equations to appreciate the factors that have been neglected. '
The force exerted by a fluid upon any solid body moving through it is known to depend on the following quantities:
l—a linear dimension giving the size of the object in a series of geometrically similar objects.
? = the density of the fluid.
? = the viscosity of the fluid.
v = the relative velocity.
S = the velocity of sound in the fluid, which is a measure of its compressibility.
In fundamental units of mass, length and time the dimensions of these quantities are:
l=L,
?= M/L3
?= M/LT,
v= L/T,
S= L/T.
The force exerted F must have the dimensions of a force, or a mass times an acceleration. Thus.
F=ML/T2.
It is found experimentally that the force exerted by a fluid on a solid, neglecting viscosity and compressibility, is given by
F = K?l2v2,
where K is a constant depending on the shape of the body and its attitude. It is seen by reference to the dimensional relations above that this expression has the dimensions of a force.
To take account of viscosity and compressibility, additional factors may be introduced, provided they be of such form that the dimensional quality of the equation is unchanged. In other words, these factors must be pure arithmetical multipliers and dimensionless.
In the most general case, we may write F=l2v2f(?,S), where f is an unknown function of the two variables ? and S. This expression violates our fundamental dimensional relation. It may be corrected only by introducing ?, l and v in proper form into the function. As suggested by Lord Rayleigh,[1] let us write:
F=?l2v2f(?/?lv, S/v),
where f is an unknown function of the two variables indicated. It is convenient to introduce the quantity v, the kinematic viscosity of the fluid, defined by
v= ?/?,
and to rewrite our equation:
F=?l2v2f(v/vl, S/v),
It is seen that the quotients v/vl and are dimensionless and
hence any function of those quantities is dimensionless. Hence the whole expression for F is of the dimensions of a force and is correct in form.
*Technical Report of Advisory Committee for Aeronautics, 1909-10, pages 33-38
Since the form of function f is unknown, we may ignore its effect if the conditions of the problem can be arranged to keep v/vl and S/v constant, for then the function is a constant. To preserve S/v constant, since S, the velocity of sound, is constant for a given fluid, the relative velocity v must be kept constant. In testing model propellers, this is accomplished by running the model at the same speed of translation as the full-size propeller and at the same tip speed.
For speeds which do not approach the velocity of sound, the effect of compression is usually ignored. This is the case for ordinary aeroplane speeds. For an 8-foot propeller, turning 1300 revolutions per minute, the tip speed is nearly half the velocity of sound in air—1120 feet per second. In this case, some allowance might be made for compression, but in neglecting it no very great error is made. For rifle bullets, on the other hand, the compression of the air plays an important part in the resistance equation. For practical purposes of design of aeroplanes and propellers, it is customary to neglect the effect of changes in f(S/v).*
The term involving viscosity gives the frictional resistance, or viscous drag on the surface of the solid. For smooth fish-shaped bodies of approximate stream-line form, the motion is not turbulent and the greater part of the resistance is frictional. The same result is found for the resistance of fine ships in water.
Since viscosity and density are constant for a given fluid, to make constant, vl must be constant, or the speed must vary inversely as the size. Careful account of this fact is taken in design of hulls of dirigibles from tests on small models. The truth of the effect on frictional resistance of the fraction v/vl was verified experimentally by Osborn Reynolds in 1884, and since by Dr. Stanton ? in England and Dr. Prandtl in Germany.?
* F. H. Bramwell, “ Propellers,” Aeronautical Journal, p. 104, London, April, 1914.
?Technical Report of the Advisory Committee for Aeronautics, 1911-12, pages 41-44.
?Zeitschrift fiir Flugtechnik und Motorluftschiffahrt, May 16, 1914, Mitteilungen aus der Gottinger Modellversuchsanstalt, pages 140-145.
For unsymmetrical bodies such as an aeroplane wing or propeller blade, stream-line flow is replaced by discontinuity, and turbulence results, with the consequent generation of eddies and vortices. In this case, the resistance is largely due to setting up motion in fluid which was originally at rest. The momentum or kinetic energy imparted to the fluid is a cause of resistance and must obviously depend on its density, the volume of fluid affected and the speed imparted. Experiments show this resistance to be given by an expression of the form F=K?l2v2, with notation as above.
Frictional resistance is added to the above inertia resistance, of course, but in practice it is found that the effect of friction is so slight that it cannot be measured on ordinary laboratory instruments. Hence, the multipler K is nearly a constant .for the type of discontinuous motion considered. The error in neglecting the effect of viscosity may be as much as 5 per cent for a propeller blade, but probably not more.
If, then, we neglect friction, as being important only for symmetrical objects of easy form, and neglect compression of the air as being only of importance for velocities near and above that of sound in air, we are left with an expression for the force exerted by the air on an element of a propeller blade of the simple form
F=K?l2v2
The acceleration of gravity has purposely been left out of this discussion, as it is considered that the motive force is an external agent such as a gasoline motor, and hence quite independent of gravity. Also the object is completely submerged in the fluid and gives rise to no surface waves as is the case for a ship.
The consequence of acceptance of the above expression independent of gravity is that there is no need of a search for “corresponding speeds.” The forces for any size and speed can be predicted from the forces on a geometrically similar object in a similar attitude at some other speed. Naturally, we should make the reservation dictated by common sense that sizes and speeds considered should be of the same order of magnitude. By this restriction, errors made in ignoring viscosity and compressibility are reduced to a minimum.
Application to a Propeller Blade
Granted that the force on a solid object giving- rise to discontinuity and turbulence in a fluid medium is of the form
F — Kpl2v2,
we may consider the conditions under which this expression may be used for the study of propeller action.
Section of Propeller Blade, Showing Components of Velocity
The quantity v is the relative velocity of fluid and object. In the case of an element of a propeller blade, the relative motion between that element and the fluid at rest is composed of one component equal to the speed of advance V of the aeroplane, and a second component at right angles to it, equal to the linear velocity of rotation of the element.
Let N be revolutions per second and r the radius of the blade element considered. The relative velocity of the wind and blade element is then √V2+(2?Nr)2 and its direction makes an angle 9 with the trajectory equal to tan-2?Nr/V. The blade must then be given such a pitch or twist that the blade face receives this air at a small angle i which is analagous to the angle of incidence for an aeroplane surface. For marine propellers this angle is known as the slip angle. The force of the air on a blade element will then have a component in the line of flight giving thrust and a component at right angles producing torque. The relation between these two forces is characteristic of a given propeller working under given conditions. If a propeller of different size be desired to work under similar conditions, its rotational speed and velocity of advance must be adjusted to give the same slip angle on corresponding elements of the two propellers. Since the propellers may be made geometrically similar, the slip angle will be unchanged if the direction of the relative velocity of the air is unchanged. The direction of the relative velocity of each element is given by the angle ? defined above as
tan ?=2?Nr / V.
From conditions of similarity, it follows that if the relative velocity be the same for two corresponding blade elements, the same condition holds for any other two elements at corresponding radii. It is then necessary and sufficient to operate both propellers with the same relation between tip speed and speed of advance, or the quantity V/ND kept constant, where D is diameter.
The theory of dimensions can be used to obtain an expression for the torque and thrust of a propeller-blade element. Thus, let us assume that the thrust of a propeller is the sum of the forces exerted by the various blade elements. The thrust is then known to be affected by the following variables and by nothing else, ignoring viscosity and compression.
D = the diameter.
N=the revolutions per second.
V = the speed of advance of the aeroplane.
? = the density of the air.
? =the direction of relative motion of a blade element.
The expression for thrust must be a combination of these variables such that the result has the dimensions of a force, ML/T2.
From experience, we know that thrust is increased directly as density, as the area of the blades and as the square of the relative velocity. The latter is to be a function of the tip speed and proportional to ND. To write a dimensional expression we may try T— KpN2D4 in which we see that since N is of the dimensions 1/T, the expression for thrust reduces to
T=KxM/L3 x 1/T2 x L4/1= K ML/T2.
This has the correct dimensions of a force. The effect of slip or direction of relative motion has not yet been introduced. The direction of relative motion at any point was shown above to depend on
V/ND = L/T ? T/1 ? 1/L = 1.
It is hence dimensionless and can be introduced in any function into the general expression for thrust without disturbing the dimensional relations.
We then have in the most general case:
T= ?N2D4f(V/ND),
where f is an unknown function.
Similarly, the expression for torque may be deduced to be of the form
Q=?N2D4f2(V/ND).
The efficiency of the propeller, c, is the ratio of power expended by the thrust to power absorbed by the torque. Hence
E=TV/?NQ
But V is proportional to ND, and the dimensional form of e is found to be:
e=KTND/?NQ or K?N2D5f(V/ND) / ?N2D5f2 (V/ND)= ? (V/ND)
The efficiency of a propeller, then, is a function of the slip, as is known from experience with marine propellers.*
*D. W. Taylor, Naval Constructor, U. S. Navy. “ Recent Experiments at the U. S. Model Basin,” November, 1904. “ Model Basin Gleanings,” November, 1906. Society of Naval Architects and Marine Engineers, New York.
Mr. Taylor tested in water several series of propellers of different diameters. For similar propellers, he found the efficiency to be a function of the slip alone, with only small discrepancies which might be attributed to the experimental errors. The theory of similitude for water propellers was found to apply so long as cavitation did not enter into the phenomena.
The proposed propeller which is to be used on a new design may be used at the same efficiency as the prototype, provided the relation V/ND is kept unchanged.
The usual definition of slip for marine propellers is based on the rather arbitrary analogy between a propeller and a true screw. The face of the blade is given a helical surface, and no attention is usually paid to the back. The pitch of the screw is the pitch of the helix, and the slip s is defined by the expression s=1 – V/pN, where p is the pitch. The geometrical pitch p of a blade is a linear dimension of the propeller and will increase with the diameter for a series of similar propellers. Hence, since slip as above defined depends on V/pN, it may be said to depend on V/ND.
The virtual pitch of a screw with thick blades in general differs from the geometrical pitch and is only to be determined by experiment. Thus, if at speed V0 and revolutions N0 the propeller exert zero thrust, we say the virtual slip is zero, or
s=0=1-V0/pN0,
or
p=V0/N0,
the virtual pitch.
The difficulties of knowing the pitch, and hence the slip, of a propeller, are further complicated by the frequent use of a variable pitch, radial or axial.
By a rigid consideration of the conditions of similitude, it appears that there is no need to know the pitch as defined above. For geometrically similar propellers, it is sufficient only to know the ratio V/ND. For V/ND constant, the slip is constant whether real or virtual. It is proposed to call V/ND a slip function out of deference to the time-honored propeller theory, and hereafter to avoid mention or calculation of slip or pitch as usually considered. The action of a propeller is expressed by the following values:
Trust = T=?N2D4f(V/ND)
Torque = Q= ?N2D3f2(V/ND)
Trust horse-power =B.H.P.= ?QN= ?N2D4f(V/ND)
Efficiency =e= ?(V/ND)
In general, it is seen that if two of the above are observed, the remaining three can be calculated, as they are dependent variables.
For purposes of design, it is convenient to consider the thrust in order to make it equal to resistance of the ship or aeroplane. The efficiency is convenient to consider in order to adjust our conditions to provide the most favorable conditions for working. If the thrust is satisfactory and the efficiency high, the problem is solved, for the power required may then be calculated, since it is known that the torque of the propeller will equal the torque of the motor.
The thrust given by a member of a family of similar propellers is shown above to be represented by
T=?N2D4f(V/ND)
It is proposed to separate the density p and the non-dimensional multipliers into a characteristic coefficient to be called a thrust coefficient.
Let Kt=?f(V/ND)
Then T=KtN2D4
The thrust coefficient is determined for a series of values of V/ND or slip function by experiment. From the data of this single test, we can predict the thrust developed by any other similar propeller by using the thrust coefficient proper to a given value of slip function and substituting in the above equation. It is unnecessary to inquire about slip and pitch. The thrust coefficient is a function of the slip function, as is also the efficiency. It is necessary, then, to know the efficiency of a propeller at all values of the slip function, in order to predict the efficiency of another similar propeller.
At the University of Paris, a dynamometer car is propelled down a straight track by a full-size air propeller. Thrust and efficiency of a great number of propellers have been observed and published, and if a designer is fortunate enough to find one that fits his case he may copy it and proceed with confidence to disregard all propeller theory.
On the other hand, the designer is not often able to find a propeller that exactly fits his conditions. He may compromise then by running a propeller slower or faster than he would wish, and accept the loss of efficiency as unavoidable.
In aeroplane design, it is always necessary to fit a propeller that shall answer three independent requirements. First, the thrust must equal the resistance of the aeroplane at designed speed. Second, the torque of the propeller must equal the torque of the motor at full power; and third, since the.motor develops full power at hut one speed of revolution, the propeller must operate at those revolutions. It is, of course, possible to compromise with requirements two and three by the use of gearing or chains, but extra weight and complication are involved. A further restriction is that the propeller diameter must not be too great.
It is proposed to illustrate the theory of similitude, as developed above, by an application to a particular case.
Application of Propeller Similitude
In the aeroplane design outlined in the body of this paper, the resistance at 90 miles an hour is estimated to be 335 pounds. To be on the safe side, let us require our propeller to give a thrust of 380 pounds. At this speed the Salmson motor can develop 125 B. H. P. at 1300 R. P. M. Let our propeller then turn at this speed. Further, let us have our propeller work at its maximum efficiency under these conditions. We can count on better than 75 per cent efficiency and, hence, better than 94 thrust horsepower. The power required at this speed is only 81.
The University of Paris in its report for the year 1913 gives the results of tests at the St. Cyr Laboratory on three propellers in which the maximum efficiency reached 80 per cent in each case. We can do no better than attempt to apply one of these propellers to our design:
It is first necessary to convert the published data into the form of thrust coefficient on slip function. For convenience the work is done in metric units.
The following tables show the calculation:
Propeller No. 6
Diameter 2.142 m. Pitch 1.43-1.55 m. Tested at 1300 R. P. M. Two blades. Observed thrust and efficiency.
V/ ND
| Efficiency E | Trust at 1300 R.P.M T | T/ND4=Kt | |
.0 | .0 | 180 | .0187 | |
.1 | .18 | 169 | .0176 | |
.2 | .35 | 156.5 | .0163 | |
.3 | .51 | 142 | .0148 | |
.4 | .65 | 127-5 | .0133 | |
.5 | .78 | go | .00937 | |
.6 | .80 | 85-4 | .00888 | |
.7 | .70 | 43-2 | .00449 | |
.75 | .0 | 0 | 0
| |
| Propeller No. 8 |
| ||
Diameter 2.596 m. | Pitch 1.85-2.52 m. |
|
| |
Tested at 700 R. P. M. | M. Two blades. |
|
| |
V/ND | e | T
| Kt | |
.0 | .0 | 153 | .0229 | |
.1 | .15 | 151.5 | .0227 | |
.2 | .30 | 150.0 | .0225 | |
.3 | .44 | 145.0 | .0217 | |
.4 | .57 | 137-5 | .0206 | |
.5 | .67 | 128.5 | .01925 | |
.6 | .73 | 116.0 | .0174 | |
.7 | .77 | 102.5 | .0154 | |
.8 | .80 | 87.0 | .01305 | |
.9 | .78 | 71.8 | .01077 | |
Propeller No. 9
Diameter 2.90 m. Pitch 1.84-2.075 m.
Tested at 700 R. P. M. Two blades.
V/ND | e | T | Kt |
.0 | .0 | 180 | .0187 |
.1 | .18 | 171 | .0178 |
.2 | .36 | 157 | .0163 |
.3 | .51 | 142 | .0148 |
.4 | .64 | 128 | .0133 |
.5 | .73 | 110 | .01145 |
.6 | .79 | 88.2 | .00918 |
.65 | .80 | … | … |
.7 | .79 | 63 | .00656 |
As a check on the above, let us take the results of tests run at 600 revolutions per minute for the same propeller.
V/ND | T for 600 R. P. M. | Kt |
.0 | 130 | .0184 |
.1 | 123.5 | .0175 |
.2 | 113 | .0160 |
.3 | 103 | .0146 |
.4 | 92.3 | .0131 |
.5 | 79.3 | .01125 |
.6 | 63.7 | .00903 |
.7 | 45.5 | .00644 |
The Kt is found to be in close agreement from the two calculations, with a maximum error of about 1.5 per cent.
The above values of thrust coefficient and efficiency are plotted on Fig. VIII with slip function V/ND as abscissae.
To return to our problem, we note first that we are limited to 1300 revolutions per minute by the motor, or N= 21.66 R. P. S. The propeller diameter is therefore fixed for a given value of slip function. Thus, for s=V/ND, D=V/21.66S; we may then compute the diameter appropriate to a series of values of slip function.
s=V/ND | D(meters) |
.5 | 3.66 |
.6 | 3.05 |
.7 | 2.62 |
.8 | 2.29 |
.9 | 2.03 |
A curve of diameter in meters is plotted on Fig. VIII. Regardless of diameter, the propeller by the conditions of the problem must develop a thrust of 380 pounds or 172 kilograms.
The thrust
Also the speed of revolution is constant at N=21.66. The thrust at any slip function is:
T=KtN2D4 or Kt= T/N2D4 - 21.662D4
Hence, for a propeller of given diameter the thrust coefficient must be equal to a certain figure in order to produce the required thrust. Naturally, for smaller propellers the thrust coefficient must be higher to develop the same total thrust. Since the diameter is fixed at any value of slip function, and thrust coefficient is fixed for any diameter, it follows that the thrust coefficient required by our aeroplane can be plotted on slip function as abscissae. This is done on Fig. VIII. The intersection of the curve of thrust coefficient required with any curve of thrust coefficient for a propeller is a possible solution.
For example, the first intersection with the thrust coefficient curve for propeller No. 6 is at a value of V/ND= .66. For this slip function, the efficiency is 77 per cent. The diameter of the propeller must be 2.75 meters or 9 feet. The efficiency is nearly the maximum, but the diameter is too large. A 9-foot propeller would require a very high chassis, so that it might clear the ground. A high landing carriage is heavy, difficult to brace, and introduces a danger of capsizing when landing on rough ground.
It is obvious that propeller No. 8 will be more suitable. Here a high coefficient is given by the intersection with the curve of thrust coefficient required at V/ND = .795. The corresponding efficiency is 80 per cent, and the diameter 2.3 meters, or 7.54 feet. We then have combined the maximum efficiency with a reasonable diameter.
The motor, then, if run at 1300 R. P. M. developing 125 B. H. P., will utilize the full 80 per cent efficiency of the propeller and deliver 100 T. H. P. for a speed of advance of 90 miles an hour. At any lower or higher aeroplane speed the ratio V/ND is changed, and hence the efficiency falls away. To determine the thrust horse-power, available at reduced speed, it then is necessary to compute V/ND and pick off the corresponding efficiency from the curve of efficiency on slip function.
For example, for full throttle opening, or motor turning 1300 R. P. M., the B. H. P. is constant at 125 H. P. The thrust horsepower available is computed as follows:
Full Power, 125 B. H. P. 1300 R. P. M. = 21.667 R. P. S.
V, miles per hour | Meters per second | V/ND | e | T.H.P. |
45 | 20.1 | .402 | .57 | 71.2 |
50 | 22.3 | .446 | .625 | 78.0 |
60 | 26.8 | .536 | .695 | 86.8 |
70 | 31.3 | .626 | .740 | 92.5 |
80 | 35.8 | .716 | .775 | 96.8 |
90 | 40.2 | .804 | .80 | 100.0 |
100 | 44.7 | .893 | .78 | 97.5 |
The curve | of thrust | horse-power | available | was plotted on |
Fig. V in the body of this paper.
Fig. IX.—Curve of Power-Revolutions for Salmson Motor. Rated 120 H. P.
In a similar manner, any reduced throttle opening corresponds to a definite value of the B. H. P. This is determined from the power curve of the motor under consideration. For the Salmson motor the power-revolution curve is given on Fig. IX.
The calculation of power available for flight at reduced power is shown in the following tables:
Reduced Power, | 91 B. H. P. | 1000 R. P. M. | = 16.67 R. P. S. |
P, miles | P ND | e | T. H. P. |
45 | ?524 | .69 | . 62.8 |
50 | .582 | .72 | 65.5 |
60 | .70 | •77 | 70.0 |
70 | .815 | .80 | 72.8 |
80 | •934 | ? 75 | 68.0 |
Reduced Power, 82 B. H. P. 900 R. P. M. — 15 R. P. S,
V | P M7 | e | T. H. P. |
45 | .584 | .72 | 59 |
50 | .617 | .75 | 61.5 |
60 | .776 | .79 | 64.8 |
70 | .878 | .79 | 64.8 |
80 | 1.04 | .64 | 52.0 |
Reduced Power, | 73 B. H. P. | 800 R. P. M. | = 13.33 R. P. S. |
P | P TTD | e | T. H. P. |
45 | .658 | • 75 | 54-8 |
50 | •73 | •78 | 57-0 |
60 | .876 | •79 | 57-8 |
70 | 1.025 | .65 | 47-5 |
Reduced Power | 64 B. H. P. | 700 R. P. M. | =11.67 R.P.S. |
V | V/ND | e | T. H. P. |
|
|
| 43.5 |
| •75 | ?7° | 4° |
50 | • 83 | .80 | 51.2 |
60 | 1.00 | .65 | 41-5 |
Reduced Power, 55 B. H. P. 600 R. P. M. — 10 R. P. S.
V | V/ND | e | T. H. P. |
45 | .874 | .79 | 43.5 |
50 | .97 | .70 | 38.5 |
In each of the above cases, the power available for flight is plotted on Fig. VI above. It is to be noted that flying at reduced power may or may not be economical. If the propeller is designed for maximum efficiency at full speed and full power, running at
reduced power means running at reduced propeller efficiency, and no gain in radius of action can be counted on without a full knowledge of the variation of propeller efficiency.
In the above application of the theory of similitude attention has been directed to the aerodynamic side of the problem. Practically, it is necessary also to take account of the strength of the blade.
The centrifugal force at the root of a blade or at any section can easily be shown to be:
C.F. = W/g V2/R,
where W is the weight of blade outside the section, V the linear velocity of its mean point, and R the radius at its mean point.
Obviously, for geometrically similar propellers made of the same material,
W~D3
R~D,
V~ND;
hence,
C. F.~ N2D2
The area of the section will vary as D2, and hence the force per unit area or stress will vary as N2D2.
In a similar manner, it can be shown that the bending moment at that section due to the thrust of the blade outside that section is proportional to the torque of the propeller, or
Q~N2D5f (V/ND) ~N2D5 for f(V/ND)= constant.
Also the moment of inertia I of the section is proportional to D4, and the distance y from the center of gravity of the section to the most stressed fiber varies as D.
The well-known formula for a beam states that f=My/I, where f is the stress. In the blade, then, the stress due to bending will be:
f= Qy/I ~N2D2.
Hence, the combined stress due to centrifugal force and to bending is proportional to N2D2.
In the particular case in hand for the aeroplane, we use a diameter of 2.3 meters at 1300 R. P. M. The prototype had a diameter of 2.596 meters and was run at 700 R. P. M. The ratio
N12D12 /N22D22 =2.7
Our aeroplane propeller, if made of the same material as the model and geometrically similar, will be relatively weaker in the proportion of roughly one to three. The factor of safety in the original design is thus cut down to one-third. Since the centrifugal force and bending moment are important only for sections near the root of the blade, it is necessary to thicken up only those sections materially. However, such thickening of the root will have very little effect on the functioning of the propeller, as the major part of the work is done by the outer half of a blade.
It should be further noted that in very light propeller blades the distortion at high speeds is considerable and may have an important effect on efficiency. To preserve the same efficiency, a geometrically similar propeller must be run with N2D2 constant, so that the distortions, or the stresses which cause them, shall be the same. In our problem, we have neglected this consideration, and to this extent it is not valid to apply the method developed above to very thin blades run at high speed.
However, it is only fair in presenting a method to show the defects as well as the virtues. It is not contended that similitude gives an ultimate solution. But when used with discretion, it should furnish a fair solution, with every chance of being a better solution than that commonly resulting from the practice of the aeroplane inventor who tries different propellers on his machine and different motors until his perseverance is at last rewarded by getting his machine to leave the ground.