Editing Blade element theory (section)

=== Limitations ===
According to momentum theory, a velocity is imparted to the air passing through the propeller, and half of this velocity is given to the air by the time it reaches the propeller plane. This increase of velocity of the air as it passes into the propeller disc is called the inflow velocity. It is always found where there is pressure discontinuity in a fluid. In the case of a wing moving horizontally, the air is given a downward velocity, as shown in Fig. 4., and theoretically half of this velocity is imparted in front of and above the wing, and the other half below and behind.  

This induced downflow is present in the model wing tests from which the airfoil coefficients used in the blade-element theory are obtained; the inflow indicated by the momentum theory is therefore automatically taken into account in the simple blade-element theory. However, the induced downflow is widely different for different aspect ratios, being zero for infinite aspect ratio. Most model airfoil tests are made with rectangular wings having an arbitrarily chosen aspect ratio of 6, and there is no reason to suppose that the downflow in such a test corresponds to the inflow for each element of a propeller blade. In fact, the general conclusion drawn from an exhaustive series of tests,<ref name=":0">{{Cite book| last=Fage|first=A.| title=A Consideration of Airscrew Theory in the Light of Data Derived from an Experimental Investigation of the Distribution of Pressure over the Entire Surface of an Airscrew Blade, and also over Airfoils of Appropriate Shapes| last2=Howard|first2=R. G.|publisher=British R. and M. 681|year=1921}}</ref> in which the pressure distribution was measured over 12 sections of a model propeller running in a wind tunnel, is that the lift coefficient of the propeller blade element differs considerably from that measured at the same angle of attack on an airfoil of aspect ratio 6. This is one of the greatest weaknesses of the simple blade-element theory. 

Another weakness is that the interference between the propeller blades is not considered. The elements of the blades at any particular radius form a cascade similar to a multiplane with negative stagger, as shown in Fig. 5. Near the tips where the gap is large the interference is very small, but in toward the blade roots it is quite large. 

In actual propellers, there is a tip loss which the blade-element theory does not take into consideration. The thrust and torque forces as computed by means of the theory are therefore greater for the elements near the tip than those found by experiment.<ref>An Analysis of the Family of Airscrews by Means of the Vortex Theory and Measurements of Total Head, by C. N. H. Lock, and H. Bateman, British R. and M. 892, 1923.</ref> 

In order to eliminate [[Scale (geography)|scale effect]], the [[wind tunnel]] tests on model wings should be run at the same value of [[Reynolds number]] (scale) as the corresponding elements in the propeller blades. Airfoil characteristics measured at such a low scale as, for example, an air velocity of 30 m.p.h. with a 3-in. chord airfoil, show peculiarities not found when the tests are run at a scale comparable with that of propeller elements. The standard propeller section characteristics given in Figs. 11, 12, 13, and 14 were obtained from high Reynolds-number tests in the [[Variable Density Tunnel]] of the [[NACA]], and, fortunately, for all excepting the thickest of these sections there is very little difference in characteristics at high and low Reynolds numbers. These values may be used with reasonable accuracy as to scale for propellers operating at tip speeds well below the speed of sound in air, and therefore relatively free from any effects of [[compressibility]]. 

The poor accuracy of the simple blade-element theory is very well shown in a report by [[William F. Durand|Durand]] and Lesley,<ref>Comparison of Model Propeller Tests with Airfoil Theory, by William F. Durand, and E. P. Lesley, N.A.C.A .T.R. 196, 1924.</ref> in which they have computed the performance of a large number of model propellers (80) and compared the computed values with the actual performances obtained from tests on the model propellers themselves. In the words of the authors: <blockquote>The divergencies<!-- [sic]? --> between the two sets of results, while showing certain elements of consistency, are on the whole too large and too capriciously distributed to justify the use of the theory in this simplest form for other than approximate estimates or for comparative purposes. </blockquote>The airfoils were tested in two different wind tunnels and in one of the tunnels at two different air velocities, and the propeller characteristics computed from the three sets of airfoil data differ by as much as 28%, illustrating quite forcibly the necessity for having the airfoil tests made at the correct scale. 

In spite of all its inaccuracies the simple blade-element theory has been a useful tool in the hands of experienced propeller designers. With it a skilful designer having a knowledge of suitable empirical factors can design propellers which usually fit the main conditions imposed upon them fairly well in that they absorb the engine power at very nearly the proper revolution speed. They are not, however, necessarily the most efficient propellers for their purpose, for the simple theory is not sufficiently accurate to show slight differences in efficiency due to changes in pitch distribution, plan forms, etc.