Editing Blade element theory (section)

=== Example ===
[[File:Ordinates BET.svg|thumb|400x400px|Fig 6. Ordinates of standard propeller section based on R.A.F.-6.]]
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. It is also desirable that the analysis be made of a propeller operating at a relatively low tip speed in order to be free from any effects of compressibility and that it be running free from body interference. The only propeller tests which satisfy all of these conditions are tests of model propellers in a wind tunnel. We shall therefore take for our example the central or master propeller of a series of model wood propellers of standard Navy form, tested by Dr. W. F. Durand at [[Stanford University]].<ref>{{Cite book|last=Durand|first=W. F.|title=Tests on Thirteen Navy Type Model Propellers|publisher=N.A.C.A .T.R. 237|year=1926|at=propeller model C}}</ref> This is a two-bladed propeller 3 ft. in diameter, with a uniform geometrical pitch of 2.1 ft. (or a pitch-diameter ratio of 0.7). The blades have standard propeller sections based on the R.A.F-6 airfoil (Fig. 6), and the blade widths, thicknesses, and angles are as given in the first part of Table I. In our analysis we shall consider the propeller as advancing with a velocity of 40 m.p.h. and turning at the rate of 1,800 [[Revolutions per minute|r.p.m.]][[File:Two sections BET.svg|thumb|400x400px|Fig 7. Two flat sections facing each other face-to-face.]]For the section at 75% of the tip radius, the radius is 1.125 ft., the blade width is 0.198 ft., the thickness ratio is 0.107, the lower camber is zero, and the blade angle ''β'' is 16.6°. 

The forward velocity
<math display="block">\begin{align}
V &= 40\  \mathrm{m.p.h.}\\
 & = \frac{40\times88}{60} \\
& = 58.65\ \text{ft./sec.},
\end{align}</math>
[[File:Convex lower chamber BET.svg|thumb|400x400px|Fig 8. Correction to lift coefficient for the convex lower chamber. (NOTE: For a Section with Lower Chamber, <math display="inline">C_L=C_{L(flat face)} -\Delta C_L</math>)]]
and 
<math display="block">\begin{align} n & = \frac{1800}{60} \\ & = 30\ \text{r.p.s.} \end{align}</math>

The path angle 

:<math>\begin{align} \phi& = \arctan\frac{V}{2\pi rn} \\ & = \arctan\frac{58.65}{2\pi\times1.125\times30} \\ & = 15.5^\circ \end{align}</math>

The angle of attack is therefore

:<math>\begin{align} \alpha & = \beta-\phi \\ & = 16.6^\circ-15.5^\circ \\ & = 1.1^\circ \\ \end{align}</math>

From Fig. 7, for a flat-faced section of thickness ratio 0.107 at an angle of attack of 1.1°, ''γ'' = 3.0°, and, from Fig. 9, ''C<sub>L</sub>'' = 0.425. (For sections having lower camber, ''C<sub>L</sub>'' should be corrected in accordance with the relation given in Fig. 8, and ''γ'' is given the same value as that for a flat-faced section having the upper camber only.)
[[File:Thrust and torque grading curves BET.svg|400x400px|Fig 9. Thrust and torque grading curves.|thumb]]
Then 

:<math>\begin{align} K & = \frac{C_Lb}{\sin^2\phi\cos\gamma} \\ & = \frac{0.425\times0.198}{0.2672^2\times0.999} \\ & = 1.180, \end{align}</math> 

and,

:<math>\begin{align} T_C & = K\cos(\phi+\gamma) \\ & = 1.180\times\cos18.5^\circ \\ & = 1.119. \\ \end{align}</math> 

Also,

:<math>\begin{align} Q_C & = Kr\sin(\phi+\gamma) \\ & = 1.180\times1.125\times\sin18.5^\circ \\ & = 1.421. \end{align}</math>

The computations of ''T<sub>c</sub>'' and ''Q<sub>c</sub>'' for six representative elements of the propeller are given in convenient tabular form in Table I, and the values of ''T<sub>c</sub>'' and ''Q<sub>c</sub>'' are plotted against radius in Fig. 9. The curves drawn through these points are sometimes referred to as the torque grading curves. The areas under the curve represent
<math display="block">\int_{0}^{R} T_c dr</math>
and 
<math display="block">\int_{0}^{R} Q_c dr,</math>
these being the expressions for the total thrust and torque per blade per unit of dynamic pressure due to the velocity of advance. The areas may be found by means of a planimeter, proper consideration, of course, being given to the scales of values, or the integration may be performed approximately (but with satisfactory accuracy) by means of [[Simpson's rule]]. 

In using Simpson's rule the radius is divided into an even number of equal parts, such as ten. The ordinate at each division can then be found from the grading curve. If the original blade elements divide the blade into an even number of equal parts it is not necessary to plot the grading curves, but the curves are advantageous in that they show graphically the distribution of thrust and torque along the blade. They also provide a check upon the computations, for incorrect points will not usually form a fair curve. 
{| class="wikitable"
|+Table I. — Computations for Propeller Analysis with Simple Blade- element Theory
!D = 3.0 ft.
p = 2.1 ft.
! colspan="6" |Forward velocity = 40 m.p.h. = 58.65 ft. /sec.
Rotational velocity = 1,800 r.p.m. = 30 r.p.s.
|-
|{{math|''r''/''R''}}
|0.15
|0.30
|0.45
|0.60
|0.75
|0.90
|-
|r (ft.)
|0.225
|0.450
|0.675
|0.900
|1.125
|1.350
|-
|b (ft.)
|0.225
|0.236
|0.250
|0.236
|0.198
|0.135
|-
|h<sub>v</sub>/b
|0.190
|0.200
|0.167
|0.133
|0.107
|0.090
|-
|h<sub>l</sub>/b
|0.180
|0.058
|0.007
|000
|000
|000
|-
|{{mvar|β}}(deg.)
|56.1
|36.6
|26.4
|20.4
|16.6
|13.9
|-
|{{math|2''πrn''}}
|42.3
|84.7
|127.1
|169.6
|212.0
|254.0
|-
|<math display="inline">\tan\phi=\frac{V}{2\pi rn}</math>
|1.389
|0.693
|0.461
|0.346
|0.277
|0.231
|-
|{{math|Φ}} (deg.)
|54.2
|34.7
|24.7
|19.1
|15.5
|13.0
|-
|<math>\alpha=\beta-\phi \ \mathrm{(deg.)}</math>
|1.9
|1.9
|1.7
|1.3
|1.1
|0.9
|-
|{{mvar|γ}} (deg.)
|3.9
|4.1
|3.6
|3.3
|3.0
|3.0
|-
|<math>\cos\gamma</math>
|0.998
|0.997
|0.998
|0.998
|0.999
|0.999
|-
|C<sub>L</sub>
|0.084
|0.445
|0.588
|0.514
|0.425
|0.356
|-
|{{math|sin Φ}}
|0.8111
|0.5693
|0.4179
|0.3272
|0.2672
|0.2250
|-
|<math display="inline">K = \frac{C_Lb}{\cos\gamma \sin^2\phi}</math>
|0.0288
|0.325
|0.843
|1.135
|1.180
|0.949
|-
|{{math|Φ+''γ''}} (deg.)
|58.1
|38.8
|28.3
|22.4
|18.5
|16.0
|-
|{{math|cos(''γ''+Φ)}}
|0.5280 
|0.7793
|0.8805
|0.9245
|0.9483
|0.9613
|-
|<math display="inline">T_c = K\cos(\gamma+\phi)</math>
|0.0152
|0.253
|0.742
|1.050
|1.119
|0.912
|-
|{{math|sin(''γ''+Φ)}}
|0.8490
|0.6266
|0.4741
|0.3811
|0.3173
|0.2756
|-
|<math display="inline">Q_c=Kr\sin(\gamma+\phi)</math>
|0.0055
|0.0916
|0.270
|0.389
|0.421
|0.353
|}
If the abscissas are denoted by ''r'' and the ordinates at the various divisions by ''y''<sub>1</sub>, ''y''<sub>2</sub>, ..., ''y''<sub>11</sub>, according to Simpson’s rule the area with ten equal divisions will be 
<math display="block">\int_{0}^{R} F(r) \, dr = \frac{\Delta r}{3}[y_1 + 2 (y_3 + y_5 + y_7 + y_9) + 4 (y_2 + y_4 + y_6 + y_8 + y_{10}) + y_{11}].</math> 

The area under the thrust-grading curve of our example is therefore 
:<math>\begin{align} \int_{0}^{R} T_cdr & = \frac{0.15}{3}[0+2(0.038+0.600+1.050+1.091)+4(0+0.253+0.863+1.120+0912)+0] \\ & = 0.9075, \end{align}</math> 

and in like manner
<math display="block">\int_{0}^{R} Q_c \, dr=0.340.</math> 

The above integrations have also been made by means of a planimeter, and the average results from five trials agree with those obtained by means of Simpson’s rule within one-fourth of one per cent.  

The thrust of the propeller in standard air is 

:<math>\begin{align} T & = \frac{1}{2}\rho V^2B\int_{0}^{R} T_cdr \\ & = \frac{1}{2}\times0.002378\times58.65^2\times2\times0.9075 \\ & = 7.42\ \mathrm{lb.}, \end{align}</math>

and the torque is

:<math>\begin{align} Q & = \frac{1}{2}\rho V^2B\int_{0}^{R} Q_cdr \\ & = \frac{1}{2} \times 0.002378\times58.65^2\times2\times0.340 \\ & = 2.78\ \mathrm{lb.ft.} \end{align}</math>

The power absorbed by the propeller is

:<math>\begin{align} P & = 2\pi nQ \\ & = 2\times\pi\times30\times2.78 \\ & = 524\ \mathrm{ft.lb. /\ sec}. \end{align}</math>

or

:<math>\begin{align} HP & = \frac{524}{550} \\ & = 0.953, \end{align}</math> 

and the efficiency is 

:<math>\begin{align} \eta & = \frac{TV}{2\pi nQ} \\ & = \frac{7.42\times58.65}{524} \\ & = 0.830. \end{align}</math> 

The above-calculated performance compares with that measured in the wind tunnel as follows:
{| class="wikitable"
!
!Calculated
!Model test
|-
!Power absorbed, horsepower
|0.953
|1.073
|-
!Thrust, pounds
|7.42
|7.77
|-
!Efficiency
|0.830
|0.771
|}
[[File:Aerodynamic data of Aerofoil BET.svg|thumb|400x400px|Fig 10. - (''From R. and M. 681.'') <u>Legend:</u> '''•''' Direct measurement of forces on an aerofoil of aspect ratio 6 with square ends; '''o''' Calculated from the pressure distribution over the median section of the aerofoil of aspect ratio 6; '''x''' Calculated from the pressure distribution over the section C of an aerofoil shaped as an airscrew blade but without a twist]]
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. Of course, a differently calculated performance would have been obtained if propeller-section characteristics from tests on the same series of airfoils in a different wind tunnel had been used, but the variable-density tunnel tests are probably the most reliable of all.

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.<ref name=":0" /> In these tests the pressure distribution over several sections of a propeller blade was measured while the propeller was running in a wind tunnel, and the three following sets of tests were made on corresponding airfoils: 
{{ordered list | list-style-type = lower-alpha
| Standard force tests on airfoils of aspect ratio 6. 
| Tests of the pressure distribution on the median section  of the above airfoils of aspect ratio 6. 
| Tests of the pressure distribution over a special airfoil made in the form of one blade of the propeller, but without twist, the pressure being measured at the same sections as in the propeller blade. 
}}
The results of these three sets of airfoil tests are shown for the section at three-fourths of the tip radius in Fig. 10, which has been taken from the report. It will be noticed that the coefficients of resultant force ''C<sub>R</sub>'' agree quite well for the median section of the airfoil of aspect ratio 6 and the corresponding section of the special propeller-blade airfoil but that the resultant force coefficient for the entire airfoil of aspect ratio 6 is considerably lower. 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.