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<!DOCTYPE html>
<html>
<!--------------------------------------------------------------
-- file : prop_book_10.html
-- purpose : Page for website Piet Lammertse
----------------------------------------------------------------
-- history :
-- 2025-01-03 new
--------------------------------------------------------------->
<!-- FORMATTING ---------------------------------------------->
<head>
<link rel="stylesheet" href="plammertse.css">
<title> Piet Lammertse on propeller theory </title>
</head>
<body>
<!-- NAVIGATION BAR MENU -------------------------------------->
<ul>
<li> <a href="index.html"> Home </a> </li>
<li> <a href="prop_book_09.html">◄ Previous </a> </li>
<li> <a style="background-color:#0000C0;"
href="prop_book_00_TOC.html">
Contents ▼</a> </li>
<li> <a href="prop_book_11.html"> Next ►</a> </li>
</ul>
<!-- TEXT BLOCK HEADER ---------------------------------------->
<p style="margin-left : 320px"> <i> ROTATION </i> </p>
<hr style="margin : 0 0;">
<!-- CHAPTER 10 ----------------------------------------------->
<h2> 10   Optimizing the rotational loss </h2>
<!-- SECTION 10.1 --------------------------------------------->
<h4> 10.1   Marginal loss </h4>
<p> We will now find the thrust distribution which minimizes
the rotational loss for a given overall thrust, by the
method of "constant marginal loss".
</p>
<p> 
The rotational momentum loss ratio (9.3)
and the efficiency (9.4) vary along the blade,
since they depend on the local pitch angle <i>φ</i>.
Intuitively, we can move thrust from a radius
with low efficiency to a radius with high efficiency,
making the whole propeller more efficient.
We should keep doing this until all locations have the
same efficiency, so no further gains are possible.
</p>
<p> 
By inspection of (9.3) and (9.7), for light loading
we can make
the efficiency the same everywhere by scheduling
<i>b</i> (<i>x</i>)
with cos<sup>2</sup><i> φ</i>.
We will find below that this is in fact the optimal solution.
</p>
<p> 
But having the same efficiency everywhere is not formal proof
of optimality.
The formal proof requires that we make the <i>marginal</i>
efficiency the same everywhere.
This is not the same thing.
</p>
<p> 
A small change in the ring thrust d<i>T</i>,
or better a small change in the net ring power
d<i>T . V</i>,
will cause a small change in the ring power loss.
The ratio between this extra power loss and the extra
net power is called the <i>marginal loss</i>.
</p>
<p> 
As long as the derivative of d<i>P</i><sub>m</sub>
with respect to d<i>T</i> is not the same everywhere,
we can gain efficiency by taking in­cre­ments
of thrust away from locations with high marginal loss,
and moving thrust to locations with low marginal loss.
We will stop moving thrust only when the derivative
has reached the same value everywhere,
i.e. when the <i>marginal</i> loss
has the same value everywhere.
We will call this constant marginal loss ratio <i>C</i> :
</p>
<!-- EQUATION ------------------------------------------------->
<table> <tr> <td>
<math>
<mstyle displaystyle="true">
<mfrac>
<mrow> <mn>d</mn>
<mo>(</mo>
<mn>d</mn> <msub> <mi>P</mi> <mn>m</mn> </msub>
<mo>)</mo>
</mrow>
<mrow> <mn>d</mn>
<mo>(</mo>
<mn>d</mn> <mi>T</mi> <mn> . </mn> <mi>V</mi>
<mo>)</mo>
</mrow>
</mfrac>
<mspace/> <mo>=</mo> <mspace/>
<mi>C</mi>
</math>
<td style="width:50px; text-align:right"> (10.1)
</td> </tr> </table>
<!-- END EQUATION --------------------------------------------->
<p> The inner "d" 's indicate ring values,
the outer "d" 's
are the symbols for the derivative.
</p>
<p> 
Since we are not varying flight conditions
like <i>ρ</i> or <i>V</i>,
or any propeller geometrical parameters
like <i>R</i> or <i>X</i>,
then by (3.9) we can write the same equation
in non-dimensional form :
</p>
<!-- EQUATION ------------------------------------------------->
<table> <tr> <td>
<math>
<mstyle displaystyle="true">
<mfrac>
<mrow> <mn>d</mn>
<mo>(</mo>
<mn>d</mn> <msub> <mi>P</mi> <mn>m</mn> </msub>
<mn>′</mn>
<mo>)</mo>
</mrow>
<mrow> <mn>d</mn>
<mo>(</mo>
<mn>d</mn> <mi>T</mi> <mn> ′</mn>
<mo>)</mo>
</mrow>
</mfrac>
<mspace/> <mo>=</mo> <mspace/>
<mi>C</mi>
</math>
<td style="width:50px; text-align:right"> (10.2)
</td> </tr> </table>
<!-- END EQUATION --------------------------------------------->
<p> The ring thrust d<i>T</i> ′
is the variable driving the change.
We could express d<i>P</i><sub>m</sub>′
as a function of d<i>T</i> ′,
and take the derivative of d<i>P</i><sub>m</sub> ′
with respect to
d<i>T</i> ′ directly.
</p>
<p> 
But it is easier to use the local value
of the induction ratio <i>b </i>(<i>x</i>)
as the variable driving both d<i>T</i> ′
and d<i>P</i><sub>m</sub>′.
</p>
<p>
The marginal loss then becomes :
</p>
<!-- EQUATION ------------------------------------------------->
<table> <tr> <td>
<math>
<mstyle displaystyle="true">
<mfrac>
<mrow> <mn>d</mn>
<mo>(</mo>
<mn>d</mn>
<msub> <mi>P</mi> <mn>m</mn> </msub> <mn>′</mn>
<mo>)</mo>
<mo>/</mo> <mn>d</mn> <mi>b</mi>
</mrow>
<mrow>
<mn> </mn>
<mn>d</mn>
<mo>(</mo>
<mn>d</mn> <mi>T</mi> <mn> ′</mn>
<mo>)</mo>
<mn> </mn>
<mo>/</mo> <mn>d</mn> <mi>b</mi>
</mrow>
</mfrac>
<mspace/> <mo>=</mo> <mspace/>
<mi>C</mi>
</math>
<td style="width:50px; text-align:right"> (10.3)
</td> </tr> </table>
<!-- END EQUATION --------------------------------------------->
<p> We will look at the light loading case first.
This will take the factor of
( 1 + ½ <i>b</i> )
out of the d <i>P</i><sub>m</sub>′ of (9.4).
</p>
<p> Taking the derivative with respect to <i>b</i> yields :
<!-- EQUATION ------------------------------------------------->
<table> <tr> <td>
<math>
<mstyle displaystyle="true">
<mfrac>
<mn>d</mn>
<mrow> <mn>d </mn> <mi>b</mi> </mrow>
</mfrac>
<mn>(</mn>
<mn>d</mn> <msub> <mi>P</mi> <mn>m</mn> </msub>
<mn> ′ </mn>
<mn>)</mn>
<mspace/> <mo> = </mo> <mspace/>
<mstyle displaystyle="true">
<mfrac>
<mrow>
<mn>2 </mn> <mi>x</mi>
<mo>.</mo>
<mn>d</mn> <mi>x</mi>
</mrow>
<mrow>
<msup> <mi>X</mi> <mn>2</mn> </msup>
</mrow>
</mfrac>
<mo> . </mo>
<mstyle displaystyle="true">
<mfrac>
<mi>b</mi>
<mrow>
<msup> <mn>cos</mn> <mn>2</mn> </msup>
<mi>φ</mi>
</mrow>
</mfrac>
</math>
<td style="width:50px; text-align:right"> (10.4)
</td> </tr> </table>
<!-- END EQUATION --------------------------------------------->
<p> Taking the derivative of the
d<i>T</i> ′ of (7.14) yields :
<!-- EQUATION ------------------------------------------------->
<table> <tr> <td>
<math>
<mstyle displaystyle="true">
<mfrac>
<mn>d</mn>
<mrow> <mn>d </mn> <mi>b</mi> </mrow>
</mfrac>
<mn>(</mn>
<mn>d</mn> <msub> <mi>T</mi> <mn>m</mn> </msub>
<mn> ′ </mn>
<mn>)</mn>
<mspace/> <mo> = </mo> <mspace/>
<mstyle displaystyle="true">
<mfrac>
<mrow>
<mn>2 </mn> <mi>x</mi>
<mo>.</mo>
<mn>d</mn> <mi>x</mi>
</mrow>
<mrow>
<msup> <mi>X</mi> <mn>2</mn> </msup>
</mrow>
</mfrac>
</math>
<td style="width:50px; text-align:right"> (10.5)
</td> </tr> </table>
<!-- END EQUATION --------------------------------------------->
<p>
The results are so simple because the derivative of
½ <i>b</i><sup> 2</sup> is <i>b</i>,
and the derivative of <i>b</i> is 1.
</p>
<p> Subsitituting (10.4) and (10.5) into (10.3) yields :
</p>
<!-- EQUATION ------------------------------------------------->
<table> <tr> <td>
<math>
<mstyle displaystyle="true">
<mfrac>
<mi>b</mi>
<mrow>
<msup> <mn>cos</mn> <mn>2</mn> </msup>
<mi>φ</mi>
</mrow>
</mfrac>
<mspace/> <mo>=</mo> <mspace/>
<mi>C</mi>
</math>
<td style="width:50px; text-align:right"> (10.6)
</td> </tr> </table>
<!-- END EQUATION --------------------------------------------->
<a name="(10.7)"></a>
<p> For optimality, the marginal loss <i>C</i> must be a constant.
As a result the axial induction ratio <i>b</i>
will have to vary along the blade :
</p>
<!-- EQUATION ------------------------------------------------->
<table> <tr> <td>
<math>
<mi>b</mi> <mn>(</mn> <mi>x</mi> <mn>)</mn>
<mspace/> <mo>=</mo> <mspace/>
<mi>C</mi>
<mo> . </mo>
<msup> <mn>cos</mn> <mn>2</mn> </msup>
<mi>φ</mi>
</math>
<td style="width:50px; text-align:right"> (10.7)
</td> </tr> </table>
<!-- END EQUATION --------------------------------------------->
<!-- SECTION 10.2 --------------------------------------------->
<a name="10.2"></a>
<h4> 10.2   The shape of the reduction factor </h4>
<p>The rotationally optimal induction (10.7)
contains a factor of
cos<sup> 2 </sup><i>φ</i>.
We will meet this factor in numerous places
in the theory of propellers.
We will give it a name, although we will also often
write it out in full :
</p>
<!-- EQUATION ------------------------------------------------->
<table> <tr> <td>
<math>
<mi>g</mi> <mn>(</mn> <mi>x</mi> <mn>)</mn>
<mspace/> <mo>≡</mo> <mspace/>
<msup> <mn>cos</mn> <mn>2</mn> </msup>
<mi>φ</mi>
s</math>
<td style="width:50px; text-align:right"> (10.8)
</td> </tr> </table>
<!-- END EQUATION --------------------------------------------->
<p> By (10.7),
<i>g</i> ( <i>x</i> )
is a reduction factor
on the local induction,
relative to an ideal actuator disk
of uniform induction <i>C</i>.
The factor is always smaller than 1,
since the cosine is always smaller than 1.
</p>
<p> 
With (2.7) we can also express
cos<sup> 2</sup><i>φ</i> directly in <i>x</i> :
</p>
<!-- EQUATION ------------------------------------------------->
<table> <tr> <td>
<math>
<mi>g</mi> <mn>(</mn> <mi>x</mi> <mn>)</mn>
<mspace/> <mo> = </mo> <mspace/>
<msup> <mn>cos</mn> <mn>2</mn> </msup>
<mi>φ</mi>
<mspace/> <mo> = </mo> <mspace/>
<mstyle displaystyle="true">
<mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mn>1</mn> <mo>+</mo>
<msup> <mi>x</mi> <mn>2</mn> </msup>
</mrow>
</mfrac>
<mspace/> <mo> = </mo> <mspace/>
<mn> 1 </mn>
<mo> − </mo>
<mstyle displaystyle="true">
<mfrac>
<mn>1</mn>
<mrow> <mn>1</mn> <mo>+</mo>
<msup> <mi>x</mi> <mn>2</mn> </msup>
</mrow>
</mfrac>
</math></td>
<td style="width:50px; text-align:right"> (10.9)
</td>
</tr> </table>
<!-- END EQUATION --------------------------------------------->
<a name="(10.1)"></a>
<p> For <i>x</i> = 1, we have
<i>g</i> ( <i>x</i> ) = ½.
This fits with the cos<sup> 2</sup><i>φ</i> formulation,
since the local pitch angle at <i>x</i> = 1 is
<i>φ</i> = 45°,
and cos (45°) = 1/√2.
For large <i>x</i>, the fraction goes to 1
by a difference of 1<i> / x </i><sup>2</sup>.
</p>
<p> 
Figure 10.1 shows the shape of
cos<sup> 2</sup><i> φ</i>
versus <i>x</i>.
On a finite propeller blade, <i>x</i> will vary
from <i>x</i> = 0 at the root
to <i>x</i> = <i>X</i>
at the tip.
</p>
<!-- FIGURE --------------------------------------------------->
<img width="600" alt="Cosine squared versus x";
style="margin-left :120px;
margin-top : -2px;"
src="images/prop_cos2_plot_x.gif">
<p style="margin-left:260px">
<i> Figure 10.1 :  
The optimal induction factor cos<sup>2</sup> φ.</i>
</p>
<!------------------------------------------------------------->
<!-- SECTION 10.3 --------------------------------------------->
<h4> 10.3   The optimal overall efficiency </h4>
<p>
Having found the marginal loss <i>C</i>
for the lightly loaded propeller,
we are also interested in the <i>actual</i> overall loss ratio.
Substituting the rotationally optimal distribution of (10.7)
into the local loss ratio (9.6),
the cos<sup> 2</sup> <i>φ</i> terms cancel,
to yield :
</p>
<!-- EQUATION ------------------------------------------------->
<table> <tr> <td>
<math>
<mstyle displaystyle="true">
<mfrac>
<mrow> <mn>d</mn> <msub> <mi>P</mi> <mn>m</mn> </msub>
<mn> ′</mn>
</mrow>
<mrow> <mn>d</mn> <mi>T</mi> <mn> ′</mn>
</mrow>
</mfrac>
<mspace/> <mo> = </mo> <mspace/> </mo>
<mn>½ </mn> <mi>C</mi>
</math>
</td>
<td style="width:50px; text-align:right"> (10.10)
</td> </tr> </table>
<!-- END EQUATION --------------------------------------------->
<p> This actual local loss ratio is exactly <i>half</i> the
marginal loss ratio <i>C</i> of (10.2).
This is not a coincidence.
The momen­tum loss of (9.4), for light loading where
( 1 + ½ <i>b</i> ) = 1,
is just a quadratic in <i>b</i>, and we know that the local slope of
a quadratic ( in this case, with the thrust <i>b</i> )
is exactly twice its local value.
</p>
<p>  Since the local momentum loss ratio is the same for
every ring, the overall loss ratio will be the same :
</p>
<!-- EQUATION ------------------------------------------------->
<table> <tr> <td>
<math>
<mstyle displaystyle="true">
<mfrac>
<mrow> <msub> <mi>P</mi> <mn>m</mn> </msub>
<mn> ′</mn>
</mrow>
<mrow> <mi>T</mi> <mn> ′</mn>
</mrow>
</mfrac>
<mspace/> <mo> = </mo> <mspace/> </mo>
<mstyle displaystyle="true">
<mfrac>
<mrow> <mn>d</mn> <msub> <mi>P</mi> <mn>m</mn> </msub>
<mn> ′</mn>
</mrow>
<mrow> <mn>d</mn> <mi>T</mi> <mn> ′</mn>
</mrow>
</mfrac>
<mspace/> <mo> = </mo> <mspace/> </mo>
<mn>½ </mn> <mi>C</mi>
</math>
</td>
<td style="width:50px; text-align:right"> (10.11)
</td> </tr> </table>
<!-- END EQUATION --------------------------------------------->
<p> With (10.6) in (9.7), the overall momentum efficiency becomes :
</p>
<!-- EQUATION ------------------------------------------------->
<table> <tr> <td>
<math>
<msub> <mi>η</mi> <mn>m</mn> </msub>
<mspace/> <mo>≡</mo> <mspace/>
<mstyle displaystyle="true">
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn> <mo>+</mo> <mn>½ </mn> <mi>C</mi>
</mrow>
</mfrac>
</math> </td>
<td style="width:50px; text-align:right"> (10.12)
</td> </tr> </table>
<!------------------------------------------------------------->
<p> 
We now have an expression for the optimal momentum efficiency
of the propeller which looks a lot like the efficiency
(4.18) of a uniform actuator disk
which had a <i>constant</i> induction of
<i>b</i> ( <i>x</i> ) = C.
</p>
<p> 
It would look for a moment as if the rotation did not
harm the efficiency of the propeler at all.
But that is a bit misleading.
For by (10.5), for the same efficiency
( i.e. for the same overall loss ratio
½ <i>C</i> ),
the <i>thrust</i> of this propeller
will be less than that of the actuator disk.
</p>
<!-- SECTION 10.2 --------------------------------------------->
<h4> 10.4   Heavy loading </h4>
<p> We will briefly look at the case of heavy loading.
Picking up the derivatives (10.4) and (10.5), using
(7.13) and (7.17) for "heavy" loading, and omitting
the terms of 2<i>x</i>.d<i>x / X</i><sup> 2</sup>
which wil drop out later anyway, we note :
</p>
<!-- EQUATION ------------------------------------------------->
<table> <tr> <td>
<math>
<mstyle displaystyle="true">
<mfrac>
<mn>d</mn>
<row> <mn>d</mn> <mi>b</mi>
</mrow>
</mfrac>
<!-- <mn> (</mn> -->
<mn> (</mn>
<mn> 1 + ½ </mn> <mi>b</mi>
<mn>)</mn>
<mo>.</mo> <mn>½ </mn>
<msup> <mi>b</mi> <mn>2</mn> </msup>
<!-- <mn> )</mn> -->
<mspace/> <mo> = </mo> <mspace/> </mo>
<mi>b</mi> <mo>+</mo>
<mn>¾ </mn>
<msup> <mi>b</mi> <mn>2</mn> </msup>
</math>
</td>
<td style="width:50px; text-align:right"> (10.13)
</td> </tr> </table>
<!-- END EQUATION --------------------------------------------->
<!-- EQUATION ------------------------------------------------->
<table> <tr> <td>
<math>
<mstyle displaystyle="true">
<mfrac>
<mn>d</mn>
<row> <mn>d</mn> <mi>b</mi>
</mrow>
</mfrac>
<!-- <mn> (</mn> -->
<mn> (</mn>
<mn> 1 + ½ </mn> <mi>b</mi>
<mn>)</mn>
<mo>.</mo>
<mi>b</mi>
<!-- <mn> )</mn> -->
<mspace/><mspace/><mspace/><mspace/>
<mo> = </mo> <mspace/> </mo>
<mn>1</mn> <mo>+</mo>
<mi>b</mi>
</math>
</td>
<td style="width:50px; text-align:right"> (10.14)
</td> </tr> </table>
<!-- END EQUATION --------------------------------------------->
<p> The ratio between the two is not the easy <i>b</i> of (10.6),
but it is close.
</p>
<p> 
However, in (9.4) there is also an additional term
1 <i>/ </i>cos<sup>2</sup><i> φ</i>.
Looking at figure 8.2 and equation (8.6), the local pitch
angle <i>φ</i>
will also vary with <i>b</i>.
So we also need to take the derivative
of 1<i>/ </i>cos<sup>2 </sup><i>φ</i>
with respect to <i>b</i>, and apply the chain rule in (10.4).
</p>
<p> 
Although this can all be done, and approximations are available,
the result will destroy the simplicity of the op­timal
induction (10.7) and make all subsequent equations opaque,
without adding significant accuracy or opti­mality
to the results.
</p>
<p> 
This is the reason why the literature on classical
propeller theory invariably applies the "light loading"
assumption in all optimization problems, and so will we.
</p>
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<br> <br>
<div class="footer"> P. Lammertse, 2025 </div>
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