Blade element theory

Blade element theory (BET) is a mathematical process originally designed by William Froude (1878),[1] David W. Taylor (1893) and Stefan Drzewiecki (1885) to determine the behavior of propellers.

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.

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.

The idea of analyzing the forces on elementary strips of propeller blades was first published by William Froude in 1878.

[1] It was also worked out independently by Drzewiecki and given in a book on mechanical flight published in Russia seven years later, in 1885.

[2] Again, in 1907, Lanchester published a somewhat more advanced form of the blade-element theory without knowledge of previous work on the subject.

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.

[3] The air flow around each element is considered two-dimensional and therefore unaffected by the adjacent parts of the blade.

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 thrust of the element is the component of the resultant force in the direction of the propeller axis (Fig.

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 sections and the angle Φ (or the advance per revolution, and consequently the pitch) as high as possible.

In the case of a wing moving horizontally, the air is given a downward velocity, as shown in Fig.

In fact, the general conclusion drawn from an exhaustive series of tests,[6] 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.

The elements of the blades at any particular radius form a cascade similar to a multiplane with negative stagger, as shown in Fig.

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.

chord airfoil, show peculiarities not found when the tests are run at a scale comparable with that of propeller elements.

The poor accuracy of the simple blade-element theory is very well shown in a report by Durand and Lesley,[8] 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: The divergencies 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.

In spite of all its inaccuracies the simple blade-element theory has been a useful tool in the hands of experienced propeller designers.

In choosing a propeller to analyze, it is desirable that its aerodynamic characteristics be known so that the accuracy of the calculated results can be checked.

(For sections having lower camber, CL should be corrected in accordance with the relation given in Fig.

these being the expressions for the total thrust and torque per blade per unit of dynamic pressure due to the velocity of advance.

In using Simpson's rule the radius is divided into an even number of equal parts, such as ten.

They also provide a check upon the computations, for incorrect points will not usually form a fair curve.

The thrust of the propeller in standard air is and the torque is The power absorbed by the propeller is or and the efficiency is The above-calculated performance compares with that measured in the wind tunnel as follows: The power as calculated by the simple blade-element theory is in this case over 11% too low, the thrust is about 5 % low, and the efficiency is about 8% high.

Some light may be thrown upon the discrepancy between the calculated and observed performance by referring again to the pressure distribution tests on a model propeller.

It is natural, then, that the calculated thrust and power of a propeller should be too low when based on airfoil characteristics for aspect ratio 6.

Many modifications to the simple blade-element theory have been suggested in order to make it more complete and to improve its accuracy.

Attribution This article incorporates text from this source, which is in the public domain: Weick, Fred Ernest (1899).