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Aviation History
1909
1909 - 0391.PDF
JULY 3, 1909. PROPELLER MATHEMATICS FOR NOVICES.* By JOHN SQUIRES, M.E., Chief of Physical Laboratory, E. R. Thomas Motor Co. To begin with, I am obliged to comment upon the danger of giving data on propellers, which same holds good with aeronautical data in general, because of the peculiarly hazardous risks involved, without explaining the method by which the data was attained, unless offered by someone whose past performance is a guarantee of his being in the right, as otherwise the amateur has nothing to check against, and perforce, being compelled to accept it as gospel if he uses it at all. His apparatus goes or stays in due proportion to the amount of mis- information built into it. The above is intended, not as a diatribe, but merely as a warning to the amateur, as the expert is bound to offer such data as he has been able to accumulate either by experiment or hard knocks, and the editor plays his part in full in bringing it to the attention of readers. If the method used is explained, one can usually determine for himself as to whether the data ought to be accurate or not. In the present article, however, it is only my intention to lay down the effects of the four purely mathematical variants involved in propeller design, all of which effects can be demonstrated by calculation and without mechanical experiment. I may say that I have verified all the conclusions set forth below by actual experiment; a great many pro- pellers having been built as well as the necessary apparatus to test them. It is well to state at this point that there are two distinct types of propellers, only one of which will be dealt with here, and for the present purpose we will consider them as delivering their work in slip, that is, not moving through the air, but delivering their thrust at a fixed, immovable point. One of these types produces thrust by moving its blades as nearly as possible over undisturbed air, and we will call this the "gliding-blade type." The other, how- ever, exerts its work in moving the greatest possible area of air, at a velocity corresponding to its pitch and speed of revolution, thus getting its effect as though the thrust were accomplished by the effect of air blowing against the propeller, and we will call this the " air-moving type." Obviously the gliding principle of the first type demands it to be of large diameter, which again causes it to be frail in proportion to the work required of it, if weight is to be avoided, and the very fact of its being designed for undisturbed air places it at a disadvantage in rough weather. The air-moving type, however, can be built small, compact, is easily applied, and can be built very strong without greatly added weight, and this is the type which is considered mathematically in this article. In my early experiments I, of course, did not know which was the best type of propeller, and was con- sequently put to many pains" to try the various types designed by other experimenters. Some I found ex- tremely deficient and others good. That is, it is possible to measure a propeller and tell under what conditions it will do its best work, and from these good ones gradually began to develop the laws governing the design of a propeller for theoretically perfect efficiency. (A) For instance, it is possible to design a propeller * From American Aeronautics. which will exert its work throughout the- whole area swept through by the blades. Consequently, perfect efficiency is the effect of air blowing against this area at a uniform velocity throughout, equal to the pitch speed of the propeller, and minus the head resistance and surface friction of the hub and. blades. Of course, as my data began to accumulate some formulae increased in accuracy and some had to be dis- carded altogether, and many entirely new ones were obtained, until in the last propellers built and tested' remarkably high efficiencies became the rule. Owing to the surprising effect that a change in some of the variants produce, I have thought it well, as stated^ above, to explain the principles governing these mathe- matical variants, for the reason that the whole design of the propeller, cross-section, and outline shapes of blades,- the number of blades, &c, is so different between a design for low speed of revolution and one for high that it is extremely difficult for a person even fairly familiar with propellers to believe that both will do the same work with the same efficiency. (B) Let me say here that there is no standard cross- sectional shape nor type of blade as regards normal pro- jected shape, that is equally efficient under more than' moderately diversified working conditions. Consequently, as stated in (A), it being possible to get full swept-area propulsive effect, this is only true with' blades of the correct cross-sectional and normal projected shapes, and (C) with a sufficient number of blades. If in plotting the thrust-curve the thrust units be con^ sidered as abscissae and the blade units as ordinates, it is evident that as more blades are added the line becomes horizontal, and when this condition is reached, a sufficient number of blades is indicated, and more would only serve to use up power in blade surface friction, and it is- at this point that we begin to obtain data on blade surface friction. Knowing that it is possible to get full swept-area pro- pulsive effect, the proportional value of the four purely mathematical variants which affect the work output and power consumption must be intelligently determined before it is possible to make even a rational guess as to what is required as a whole. The value of the first of these is usually determined by outside conditions : such, for instance, as weight, and is the power at command ; and the second, which hinges on the first, is the speed of revolution. Then comes the area, which is largely governed by construction con- ditions, and last comes pitch, which necessarily, under the above circumstances, is somewhat at the mercy of the other three, although equally important with the speed of revolution, owing to the fact of pressure caused by moving air varying with the square of velocity (Vs). Thus it will be seen that variations of either the speed of revolution or the pitch is bound to vary the work with the third power, or cube, because each of these is a second power, or square, to begin with. Therefore, it is my intention to lay down the effect of each of these variants in the abstract, as though they were purely selective, and show the effects of varying each of them singly as well as in combination with variations of the other three, in relation to the effect on power, and on the thrust per unit of power, which is the 393
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