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Aviation History
1910
1910 - 1001.PDF
DECEMBER 3, 191c Jygfif] THE PROBLEM OF THE HELICOPTER. (Concluded from page 978.) AT this point, in order to avoid misunderstanding in the treatment of a subject that has infinite possibilities of confusion, we take the opportunity of recapitulating the gist of our preceding arguments in order that they may stand out clear and distinct from those that follow. Thus far we have dealt exclusively with that aspect of the problem presented by what may be termed the " disc area theory." This theory assumes the propeller to be a device for the purpose of creating and maintaining a uniform slip stream velocity throughout its entire disc area, and it assesses the merit of practical results obtained from actual screws by equating them to this theoretical basis on the grounds that the basis in question represents the highest possible degree of efficiency. By efficiency is meant the ratio of power to static thrust, and we would again emphasise that this particular article is entirely confined to an investigation of the theory of the efficiency of static thrust sere AS as thus understood. This ratio is not a true ratio in the scientific sense, for its factors have different dimensions, thrust and power being equations of different degree. But, like the disc area theory itself, it is a point of view very commonly adopted in practice, and if only for that reason would have to be taken into account. This brings us to a point at which we can turn our a'tention to another aspect of the theory of the screw that we wish to put forward in this article. It mus-t be obvious that the blade of a propeller is only energising a small portion of slip stream at any given instant of time, and let the revolutions be as high as they may, nevertheless each individual blade will still be incapable of operating upon the whole of the disc area simultaneously. On first thoughts, this would appear to be an argument completely opposed to the disc area theory, and in principle it is, of course, quite different, but there are practical considerations that cause the disc area theory to present an eminently useful point of view, and one of its useful purposes is to indicate the possibility of limitations in any theory that is solely concerned with the independent action of each blade separately. It is, at any rate, plausible that the disturbance over the disc area, which occupies a fixed position in space, may ultimately attain to such an approxi mately uniform value as to render any further increase in the number of blades or revolutions to be a highly inefficient means of attaining increased effect. It is, therefore, the fact that the disc area con stitutes a limited field of operations that justifies the fundamental theory based thereon, and also renders it necessary to bear in mind the possible limitation of applying what is commonly termed the " aeroplane analogy " to the case of the static thrust screw. In principle, it is obviously proper to regard the • blade of a screw as an aeroplane, and it is extremely interesting to compare this aspect of the problem presented by the heli copter with the corresponding aspect of the problem presented by the aeroplane. Thus, if we regard the blade of a pro peller as an aeroplane and argue on the premises founded by our recent articles entitled '' Can we fly faster for less power ? " we find that so far as each blade is concerned, the condition of least resistance, that is to say least thrust per unit of load, obtains when the power expended on skin friction is equal to the power expended on load and that for the stated coefficient of skin friction (ooooiSV2) the angle of least resistance is 5°. Converting this into the terminology of a propeller we must consider the thrust applied to the blade as a function of the torque applied to the shaft, while the load supported by the blade becomes the thrust of the screw. By this analogy, we may say that the least torque per unit of thrust obtains when the work done on skin friction is equal to the energy in the slip stream. The torque in question must, of course, be considered as a value expressed in pounds applied with a certain leverage from the axis of rotation representing a suitable mean value of the blade radius. On this assumption the equation of pounds torque to pounds thrust becomes a direct measure of efficiency, inasmuch as the numerator and the denominator of the fraction represented by the ratio are both of the same dimensions, whereas the ordinary method of equating pounds thrust to horse-power in the shaft is not a true measure of efficiency at all, for the dimensions of the two factors in the ratio are not of the same degree. It is, however, obvious that the idea of efficiency in connection with a helicopter is unquestionably associated with the ratio of thrust to horse-power, and the situation thus presented leads to one or two other very interesting considerations, in which this dis crepancy between the dimensions of the factors thus equated becomes the keynote. In all the foregoing deductions as to what is most efficient in the design of a helicopter it will be found that the inevitable conclusion is always the same, and may be summed up in the words "go slow." The advantage of a large diameter is due to the fact that it reduces the velocity of flight on the part of the blade by reducing the revolutions for a given thrust. These deductions are precisely the same as were arrived at in the above mentioned article on the aeroplane, where, it will be remem bered, one of the most important conclusions was that for a given effective angle the thrust required per unit of load supported remained independent of velocity, consequently the power required per unit of load supported varies directly as the flight speed. Applying this theory to the propeller blade we may say that for a given effective angle the torque per unit of thrust is independent of revolutions and that the power per unit of thrust is therefore directly proportional to revolutions. The so-called static thrust efficiency of the screw, l>eing an expression of the thrust per unit of power, is therefore inversely proportional to the revolutions for a given blade angle, which once again brings us to the same conclusion that everything must be done to enable the speed of the blades to be slow if we would have a high efficiency. Varying the speed of flight wiih a constant angle causes a variation in the load theoretically proportional to the square of the speed, so that for a constant thrust from a fixed diameter with vaiying revo lutions it is necessary to change the angle. It will lie remembered, however, that the ratio of thrust per unit of load is dependent on angle although independent of velocity, consequently the torque for a given load will be greater whenever the angle is other than that of least resistance. Moreover, and this is the important point, the increment in question will more than neutralise any anticipated gain of efficiency, from the power-thrust point of view, resulting from reduced revolutions, supposing the angle to have been increased for the purpose of reducing the speed. It is, of course, true that increasing the angle will reduce the revolutions at which a given thrust will be produced, and were the ratio of torque to thrust constant for all angles, such decrease in speed would be a sheer gain in efficiency. This would lead to the use of the steepest effective blade angle when a given thrust was required from a given diameter*—a result that is distinctly opposed to the theory of least resistance, and the analogy between the aeroplane and propeller blade. In this connection the important point to bear in mind is that the thiust is proportional to the square of the revolutions, and only directly proportional to the tangent of the angle ; when the angle is increased the reduction in speed required to maintain the same load is, therefore, comparatively small, and the increment in torque thrust ratio neutralises its advantage. The mathematics of this point involves a comparison of the formula; for lift and thrust of aeroplanes. The formula for lift is V- tan fl P = 0 200 whence V J--300J'fl V tan $ so that if the lift remains constant with an increasing angle the flight speed is reduced :— V oc -j tan 0 vx where x is the multiple representing the increment of the angle. The ratio of thrust per unit of load is given by the expression : — T _ tanJfl ) -0072 "p^j ~ 2 tan & which has a minimum value = tan ft, and is 01 such an order that any increment of tan /3 above the value of least resistance (where 3 = 50) produces a ratio :— l'« > Jx tan 0 where x is again the multiple increment of tan 0. Now the power required to sustain the load is a product of thrust and speed, and as a comparison of the above expressions shows that for a given load supported by a variable angle the thrust increases in greater ratio than the reduction in speed, it follows that the net result is a loss when the angle is increased above its value of least resistance. On the other hand, if this loss of efficiency is small over a small range of angles then a variable pitch screw should be capable of producing thrusts approximately directly proportional to the power applied to the shaft, provided always that the effective velocity of the slip-stream is not increased either by an increase in the angle or revolutions. If the velocity of slip-stream is increased, then the increment in thrust would only obey the law ("^/KfP, as already explained in the first part of this article. We can thus see how the disc area theory and the aeroplane analogy are inter-connected. So long as the revolutions are low the blade may be capable of obeying aeroplane laws, but when the revolutions are high limiting values are imposed by disc area considerations. • In the subheading of the first part of this article, this condition was inadvertent^- given as one of greatest efficiency. 999
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