Editing Blade element theory (section)

{{Short description|Mathematicall process by William Froude}}

'''Blade element theory''' ('''BET''') is a [[Mathematics|mathematical]] process originally designed by [[William Froude]] (1878),<ref name=":1">{{Cite journal |last=Froude |first=William |date=11 April 1878 |title=The Elementary Relation between Pitch, Slip, and Propulsive Efficiency |url=https://babel.hathitrust.org/cgi/pt?id=iau.31858019833668&seq=81 |journal=Inst. Naval Architects |volume=19 |pages=47 |via=Hathi Trust}}</ref> [[David W. Taylor]] (1893) and [[Stefan Drzewiecki]] (1885) to determine the behavior of [[propeller]]s. It involves breaking a [[blade]] down into several small parts then determining the [[Force|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 [[Moment (physics)|moments]] produced by the entire [[propeller]] or [[Rotorcraft|rotor]]. One of the key difficulties lies in modelling the induced [[Velocity-addition formula|velocity]] on the [[Disk loading|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:
:<math>v_i = \sqrt{\frac{T}{A} \cdot \frac{1}{2 \rho}}.</math>

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]], [[Conservation of momentum|momentum]] and [[Conservation of energy|energy]] to every [[Annulus (mathematics)|annulus]]. This approach is sometimes called the [[William Froude|Froude]]–[[Sebastian Finsterwalder|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.