Editing Blade element theory (section)

=== Efficiency ===
[[File:Diagram BED.svg|400x400px|Fig 3. Diagram of efficiency|thumb]]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 

:<math>\begin{align} \eta & = \frac{dTV}{dQ2\pi n} \\ & = \frac{dR\cos(\phi+\gamma)V}{dR\sin(\phi+\gamma)2\pi nr} \\ & = \frac{\tan\phi}{\tan(\phi+\gamma)}. \end{align}</math> 

Now tan ''Φ'' is the ratio of the forward to the tangential velocity, and <math display="inline">\tan\gamma=\frac{D}{L}</math>. 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 <math display="inline">\frac{D}{L}</math> 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
[[File:Airflow BET.svg|400x400px|Fig 4. Airflow|thumb]]
<math display="block">\phi=45^\circ-\frac{\gamma}{2}</math>
[[File:Multiplane BET.svg|400x400px|Fig 5. Multiplane with negative stagger|thumb]]
The variation of efficiency with ''Φ'' is shown in Fig. 3 for two extreme values of ''γ''. The efficiency rises to a maximum at <math display="inline">45^\circ-\frac{\gamma}{2}</math> and then falls to zero again at <math display="inline">90^\circ-\gamma</math>. With an <math display="inline">\frac{L}{D}</math> of 28.6 the maximum possible efficiency of an element according to the simple theory is 0.932, while with an <math display="inline">\frac{L}{D}</math> 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 <math display="inline">\frac{L}{D}</math> 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 <math display="inline">\frac{L}{D}</math> of the airfoil sections and the angle ''Φ'' (or the advance per revolution, and consequently the pitch) as high as possible.