Blade element theory (BET) is a mathematical process originally designed by William Froude (1878), David W. Taylor (1893) and Stefan Drzewiecki (1885) to determine the behavior of propellers. It involves breaking a blade down into several small parts then determining the forces on each of these small blade elements. These forces are then integrated along the entire blade and over one rotor revolution in order to obtain the forces and moments produced by the entire propeller or rotor. One of the key difficulties lies in modelling the induced velocity on the rotor disk. Because of this the blade element theory is often combined with momentum theory to provide additional relationships necessary to describe the induced velocity on the rotor disk, producing blade element momentum theory. At the most basic level of approximation a uniform induced velocity on the disk is assumed:
Alternatively the variation of the induced velocity along the radius can be modeled by breaking the blade down into small annuli and applying the conservation of mass, momentum and energy to every annulus. This approach is sometimes called the Froude–Finsterwalder equation.
If the blade element method is applied to helicopter rotors in forward flight it is necessary to consider the flapping motion of the blades as well as the longitudinal and lateral distribution of the induced velocity on the rotor disk. The most simple forward flight inflow models are first harmonic models.
Simple blade element theory
While the momentum theory is useful for determining ideal efficiency, it gives a very incomplete account of the action of screw propellers, neglecting among other things the torque. In order to investigate propeller action in greater detail, the blades are considered as made up of a number of small elements, and the air forces on each element are calculated. Thus, while the momentum theory deals with the flow of the air, the blade-element theory deals primarily with the forces on the propeller blades. The idea of analyzing the forces on elementary strips of propeller blades was first published by William Froude in 1878. It was also worked out independently by Drzewiecki and given in a book on mechanical flight published in Russia seven years later, in 1885.This fact, which is not generally known in English-speaking countries, was called to the author’s attention by Prof. F. W. Pawlowski of the University of Michigan. Drzewiecki’s first French paper on his theory was published in 1892. He wrote in all seven papers on aircraft propulsion which were presented to l’Academie des Sciences, l’Association Technique Maritime, and Le Congrès International d’Architecture et de Construction Navale, held on July 15, 1900. He finally wrote a book summing up all of his work called "Théorie Générale de l’Hé1ice Propulsive," published in 1920 by Gauthier-Villars in Paris. Again, in 1907, Lanchester published a somewhat more advanced form of the blade-element theory without knowledge of previous work on the subject. The simple blade-element theory is usually referred to, however, as the Drzewiecki theory, for it was Drzewiecki who put it into practical form and brought it into general use. Also, he was the first to sum up the forces on the blade elements to obtain the thrust and torque for a whole propeller and the first to introduce the idea of using airfoil data to find the forces on the blade elements.
In the Drzewiecki blade-element theory the propeller is considered a warped or twisted airfoil, each segment of which follows a helical path and is treated as a segment of an ordinary wing. It is usually assumed in the simple theory that airfoil coefficients obtained from wind tunnel tests of model wings (ordinarily tested with an aspect ratio of 6) apply directly to propeller blade elements of the same cross-sectional shape.Drzewiecki suggested that the airfoil characteristics could be obtained from tests on special model propellers.
The air flow around each element is considered two-dimensional and therefore unaffected by the adjacent parts of the blade. The independence of the blade elements at any given radius with respect to the neighbouring elements has been established theoretically and has also been shown to be substantially true for the working sections of the blade by special experiments made for the purpose. It is also assumed that the air passes through the propeller with no radial flow (i.e., there is no contraction of the slipstream in passing through the propeller disc) and that there is no blade interference.
Aerodynamic forces on a blade element Consider the element at radius r, shown in Fig. 1, which has the infinitesimal length dr and the width b. The motion of the element in an aircraft propeller in flight is along a helical path determined by the forward velocity V of the aircraft and the tangential velocity 2πrn of the element in the plane of the propeller disc, where n represents the revolutions per unit time. The velocity of the element with respect to the air Vr is then the resultant of the forward and tangential velocities, as shown in Fig. 2. Call the angle between the direction of motion of the element and the plane of rotation Φ, and the blade angle β. The angle of attack α of the element relative to the air is then .
Applying ordinary airfoil coefficients, the lift force on the element is:
Let γ be the angle between the lift component and the resultant force, or . Then the total resultant air force on the element is:
The thrust of the element is the component of the resultant force in the direction of the propeller axis (Fig. 2), or
and since
For convenience let
and
Then
and the total thrust for the propeller (of B blades) is:
Referring again to Fig. 2, the tangential or torque force is
and the torque on the element is
which, if , can be written
The expression for the torque of the whole propeller is therefore
The horsepower absorbed by the propeller, or the torque horsepower, is
and the efficiency is
Efficiency Because of the variation of the blade width, angle, and airfoil section along the blade, it is not possible to obtain a simple expression for the thrust, torque, and efficiency of propellers in general. A single element at about two-thirds or three-fourths of the tip radius is, however, fairly representative of the whole propeller, and it is therefore interesting to examine the expression for the efficiency of a single element. The efficiency of an element is the ratio of the useful power to the power absorbed, or
Now tan Φ is the ratio of the forward to the tangential velocity, and . According to the simple blade-element theory, therefore, the efficiency of an element of a propeller depends only on the ratio of the forward to the tangential velocity and on the of the airfoil section.
The value of Φ which gives the maximum efficiency for an element, as found by differentiating the efficiency with respect to Φ and equating the result to zero, is
The variation of efficiency with Φ is shown in Fig. 3 for two extreme values of γ. The efficiency rises to a maximum at and then falls to zero again at . With an of 28.6 the maximum possible efficiency of an element according to the simple theory is 0.932, while with an of 9.5 it is only 0.812. At the values of Φ at which the most important elements of the majority of propellers work (10° to 15°) the effect of on efficiency is still greater. Within the range of 10° to 15°, the curves in Fig. 3 indicate that it is advantageous to have both the of the airfoil sections and the angle Φ (or the advance per revolution, and consequently the pitch) as high as possible.
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, 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.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.
In order to eliminate 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.