Propeller theory

Taking an arbitrary radial section of a blade at r, if revolutions are N then the rotational velocity is ${\displaystyle \scriptstyle 2\pi Nr}$. If the blade was a complete screw it would advance through a solid at the rate of NP, where P is the pitch of the blade. In water the advance speed, ${\displaystyle \scriptstyle V_{a}}$, is rather lower. The difference, or slip ratio, is:

${\displaystyle {\mbox{Slip}}={\frac {NP-V_{a}}{NP}}=1-{\frac {J}{p}}}$

where ${\displaystyle \scriptstyle J={\frac {V_{a}}{ND}}}$ is the advance coefficient, and ${\displaystyle \scriptstyle p={\frac {P}{D}}}$ is the pitch ratio, with ${\displaystyle \scriptstyle D}$ being the diameter of the propeller.

The forces of lift and drag on the blade, dA, where force normal to the surface is dL:

${\displaystyle {\mbox{d}}L={\frac {1}{2}}\rho V_{1}^{2}C_{L}dA={\frac {1}{2}}\rho C_{L}[V_{a}^{2}(1+a)^{2}+4\pi ^{2}r^{2}(1-a')^{2}]b{\mbox{d}}r}$

where:

${\displaystyle {\begin{aligned}V_{1}^{2}&=V_{a}^{2}(1+a)^{2}+4\pi ^{2}r^{2}(1-a')^{2}\\{\mbox{d}}D&={\frac {1}{2}}\rho V_{1}^{2}C_{D}{\mbox{d}}A={\frac {1}{2}}\rho C_{D}[V_{a}^{2}(1+a)^{2}+4\pi ^{2}r^{2}(1-a')^{2}]b{\mbox{d}}r\end{aligned}}}$

These forces contribute to thrust, T, on the blade:

${\displaystyle {\mbox{d}}T={\mbox{d}}L\cos \varphi -{\mbox{d}}D\sin \varphi ={\mbox{d}}L(\cos \varphi -{\frac {{\mbox{d}}D}{{\mbox{d}}L}}\sin \varphi )}$

where:

${\displaystyle {\begin{aligned}\tan \beta &={\frac {{\mbox{d}}D}{{\mbox{d}}L}}={\frac {C_{D}}{C_{L}}}\\&={\frac {1}{2}}\rho V_{1}^{2}C_{L}{\frac {\cos(\varphi +\beta )}{\cos \beta }}b{\mbox{d}}r\end{aligned}}}$

As ${\displaystyle \scriptstyle V_{1}={\frac {V_{a}(1+a)}{\sin \varphi }}}$,

${\displaystyle {\mbox{d}}T={\frac {1}{2}}\rho C_{L}{\frac {V_{a}^{2}(1+a)^{2}\cos(\varphi +\beta )}{\sin ^{2}\varphi \cos \beta }}b{\mbox{d}}r}$

From this total thrust can be obtained by integrating this expression along the blade. The transverse force is found in a similar manner:

${\displaystyle {\begin{aligned}{\mbox{d}}M&={\mbox{d}}L\sin \varphi +{\mbox{d}}D\cos \varphi \\&={\mbox{d}}L(\sin \varphi +{\frac {{\mbox{d}}D}{{\mbox{d}}L}}\cos \varphi )\\&={\frac {1}{2}}\rho V_{1}^{2}C_{L}{\frac {\sin(\varphi +\beta )}{\cos \varphi }}b{\mbox{d}}r\end{aligned}}}$

Substituting for ${\displaystyle \scriptstyle V_{1}}$ and multiplying by r, gives torque as:

${\displaystyle {\mbox{d}}Q=r{\mbox{d}}M={\frac {1}{2}}\rho C_{L}{\frac {V_{a}^{2}(1+a)^{2}\sin(\varphi +\beta )}{\sin ^{2}\varphi \cos \beta }}br{\mbox{d}}r}$

which can be integrated as before.

The total thrust power of the propeller is proportional to ${\displaystyle \scriptstyle TV_{a}}$ and the shaft power to ${\displaystyle \scriptstyle 2\pi NQ}$. So efficiency is ${\displaystyle \scriptstyle {\frac {TV_{a}}{2\pi NQ}}}$. The blade efficiency is in the ratio between thrust and torque:

${\displaystyle {\mbox{blade element efficiency}}={\frac {V_{a}}{2\pi Nr}}\cdot {\frac {1}{\tan(\varphi +\beta )}}}$

showing that the blade efficiency is determined by its momentum and its qualities in the form of angles ${\displaystyle \scriptstyle \varphi }$ and ${\displaystyle \scriptstyle \beta }$, where ${\displaystyle \scriptstyle \beta }$ is the ratio of the drag and lift coefficients.

This analysis is simplified and ignores a number of significant factors including interference between the blades and the influence of tip vortices.

The thrust, T, and torque, Q, depend on the propeller's diameter, D, revolutions, N, and rate of advance, ${\displaystyle V_{a}}$, together with the character of the fluid in which the propeller is operating and gravity. These factors create the following non-dimensional relationship:

${\displaystyle T=\rho V^{2}D^{2}\left[f_{1}\left({\frac {ND}{V_{a}}}\right),f_{2}\left({\frac {v}{V_{a}D}}\right),f_{3}\left({\frac {gD}{V_{a}^{2}}}\right)\right]}$

where ${\displaystyle f_{1}}$ is a function of the advance coefficient, ${\displaystyle f_{2}}$ is a function of the Reynolds' number, and ${\displaystyle f_{3}}$ is a function of the Froude number. Both ${\displaystyle f_{2}}$ and ${\displaystyle f_{3}}$ are likely to be small in comparison to ${\displaystyle f_{1}}$ under normal operating conditions, so the expression can be reduced to:

${\displaystyle T=\rho V_{a}^{2}D^{2}\times f_{r}\left({\frac {ND}{V_{a}}}\right)}$

For two identical propellers the expression for both will be the same. So with the propellers ${\displaystyle T_{1},T_{2}}$, and using the same subscripts to indicate each propeller:

${\displaystyle {\frac {T_{1}}{T_{2}}}={\frac {\rho _{1}}{\rho _{2}}}\times {\frac {V_{a1}^{2}}{V_{a2}^{2}}}\times {\frac {D_{1}^{2}}{D_{2}^{2}}}}$

For both Froude number and advance coefficient:

${\displaystyle {\frac {T_{1}}{T_{2}}}={\frac {\rho _{1}}{\rho _{2}}}\times {\frac {D_{1}^{3}}{D_{2}^{3}}}={\frac {\rho _{1}}{\rho _{2}}}\lambda ^{3}}$

where ${\displaystyle \lambda }$ is the ratio of the linear dimensions.

Thrust and velocity, at the same Froude number, give thrust power:

${\displaystyle {\frac {P_{T1}}{P_{T2}}}={\frac {\rho _{1}}{\rho _{2}}}\lambda ^{3.5}}$

For torque:

${\displaystyle Q=\rho V_{a}^{2}D^{3}\times f_{q}\left({\frac {ND}{V_{a}}}\right)}$

${\displaystyle ...}$