BET Power Required

Depending on the units of torque, one can divide Equation 16 by 550 to get horsepower (assuming torque is in English units of 𝑓𝑡 ∗ 𝑙𝑏𝑠) or leave as is for Power in Watts (assuming torque is in SI units of 𝑁 ∗ 𝑚).

Recall the assumptions that were made while developing the BET relationships. In particular:

1. 2-Dimensional flow assumed, therefore interaction of the airflow between blade elements is ignored

19

3. Increased velocity in front of the propeller (inflow velocity) is incorrect due to airfoil data only being published for a rectangular or slightly tapered wing (not accounting for

taper/twist used in propeller designs) (Weick)

4. Interference between the propeller blades is ignored

5. Tip loss is ignored – thrust and torque values are calculated higher for the elements near the tip then what is shown by experimentation (Lock & Bateman)

As with the simple Momentum Theory, BET also overestimates the thrust produced and therefore provides an unrealistically high thrust estimate. Therefore the real value of the simple BET is for a relatively quick approximation (although the computations are more involved then momentum theory) or for qualitatively comparing the performance of 2 or more propellers. There are many variations to the momentum and blade element theories including accounting for some of the effects ignored by the simple BET however the momentum and blade element theories will always be limited by the basic assumption that each blade element acts independently from one another and therefore has no effect on the adjacent elements.

2.4 – Vortex Theory (Wald)

Once Prandtl developed the lifting line theory of wings, the modern vortex propeller theory was soon to follow. After all, the simple blade element theory had already treated the propeller problem by dissecting each propeller blade into 2-dimensional airfoils. Vortex theory expounded on the idea of treating propellers as rotating wings by considering the propeller blade as a lifting surface with a circulation associated with the bound vorticity and a vortex sheet that is

continuously shed from the trailing edge. The BET uses 2-dimensional aerodynamics (airfoils) whereas vortex theory uses 3-dimensional aerodynamics (vortex system). (Wald, 2008)

As early as 1919, it was realized that in order to truly solve the propeller problem the induced velocities along the blades had to be considered. Additionally, it was theorized that an optimum

20

loading must exist for a propeller with given conditions (RPM, Velocity, Diameter, Blades) analogous to the elliptical loading case on a wing. Betz developed the vortex theory which shows that the load distribution for lightly loaded propellers with minimum induced loss is such that shed vorticity forms rigid helicoidal vortex sheets moving backward behind the propeller. Around the same time, Prandtl developed a mathematical method for calculating the loading based on an infinite number of blades and then applying a tip correction factor. Prandtl’s approximation is sufficient when the advance ratio is small and the number of blades is large. As the blades decrease or the advance ratio increases, the approximation deteriorates. (Wald, Theodorsen)

In 1929, Goldstein solved the potential flow and the distribution of circulation for a helicoidal vortex system for small advance ratios. Goldstein’s analysis assumed the propellers were lightly loaded. This assumption was later shown to be unnecessary when Theodorsen proved that the Goldstein functions are applicable directly to all loadings, as long as the reference is made to the helicoidal sheet surface far behind the propeller and not to the surface of the propeller itself. Additionally, Theodorsen added to the theory of propellers by using the analysis of the trailing helicoidal vortex sheets to determine the conditions at the propeller and thus the necessary geometry of the propeller. (Wald, Theodorsen)

In 2006, Wald used the previous works of Goldstein and Theodorsen as well as the mathematical solutions provided by Tibery and Wrench (Tibery & Wrench, 1964) to treat the design of

propellers for minimum induced loss. His analysis corrected some of the errors in previous work and expanded the coverage for the Goldstein circulation function tables for helicoidal vortex sheets. Additionally, Wald considered cases for interference effects from fuselages, nacelles, and spinners. The vortex theory as presented by Wald is given below although QPROP and QMIL do not take full advantage of the theory as described by Wald.

21

Wald used Theodorsen’s theory which suggested that one can find the conditions at the propeller blades by first focusing on the necessary conditions in the trailing vortex system. Once the conditions in the trailing vortex system are determined, one can then compute the circulation on the propeller blades and determine the physical characteristics (chord and beta distributions, as well as the number of blades) of the propeller that would create the conditions in the vortex system.

Figure 11: Velocity Components at a Blade Element (Wald)

Figure 11 shows the local rotor-induced velocities (𝑣𝑡, 𝑣𝑎). Note that the lifting line assumption, as developed originally by Prandtl, states that if the blades are of sufficiently small chord then the induced velocity does not vary significantly along the chord. Therefore, one can assume that the lift on the blade element is related to the local angle of attack and the local relative velocity as in 2-D airfoil theory. Caution is needed here however, because if the chord is relatively wide (as is the case for MIL propellers designed for relatively high thrust production at relatively slow speeds) then the variation of induced velocity along the chord must be accounted for. (Wald)

Some important assumptions:

1. Interference of nacelles, fuselages, etc are ignored (these may be handled through variations of the fundamental vortex theory)

22

2. Blades are relatively narrow such that there is no variation of induced velocity along the chord.

3. Blades are equally spaced about the hub; therefore vortex lines representing each blade do not induce any velocity on any of the other lines. In other words, the effect on each blade due to the bound vorticity on the other blades is ignored and only trailing vorticity contributes to the resultant velocity at the blade.

Wald showed that the light loading restriction assumed by Betz can actually be relieved. He did this by splitting the propeller load into increments of load that lie directly on the vortex sheet behind the propeller as shown in Figure 12. He then showed that the variation of thrust and the variation of torque at each of these incremental locations must vanish if the propeller has an optimum radial distribution of load (MIL condition).

Figure 12: Load Increments on the Trailing Vortex Sheets (Theodorsen)

Additionally, Wald showed that the condition for optimum loading is:

𝑟 tan 𝜙 =𝑟(𝑉∞+ 𝑣𝑎(𝑟)) Ω𝑟 − 𝑣𝑡(𝑟)

= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡