Momentum theory

In <a href="/facts/Fluid_dynamics/y7uzSCfg">fluid dynamics</a>, momentum theory or disk actuator theory is a theory describing a <a href="/facts/Mathematical_model/CGhyuxgz">mathematical model</a> of an ideal actuator disk, such as a <a href="/facts/Propeller/0IN0JKXC">propeller</a> or <a href="/facts/Helicopter_rotor/J3BP6U2P">helicopter rotor</a>, by <a href="/facts/William_John_Macquorn_Rankine/zsX1bcVh">W.J.M. Rankine</a> (1865), <a href="/facts/Alfred_George_Greenhill/EnSoI5se">Alfred George Greenhill</a> (1888) and <a href="/facts/Robert_Edmund_Froude/bC31YPEj">Robert Edmund Froude</a> (1889).
The rotor is modeled as an infinitely thin disc, inducing a constant velocity along the axis of rotation. The basic state of a helicopter is <a href="/facts/Hover_(helicopter)/TyDgVhbu">hovering</a>. This disc creates a flow around the rotor. Under certain mathematical premises of the fluid, there can be extracted a mathematical connection between power, radius of the rotor, <a href="/facts/Torque/KcrcvNtV">torque</a> and induced velocity. <a href="/facts/Friction/d0108Rpu">Friction</a> is not included.
For a stationary open rotor with no outer duct, such as a helicopter in hover, the power required to produce a given thrust is:

P
        =
        
          
            
              
                T
                
                  3
                
              
              
                2
                ρ
                A
              
            
          
        
      
    
    {\displaystyle P={\sqrt {\frac {T^{3}}{2\rho A}}}}

where:

<ul><li>T is the <a href="/facts/Thrust/EM04CqV7">thrust</a></li>
<li>
 
 
 
 ρ
 
 
 {\displaystyle \rho }
 
 is the <a href="/facts/Density_of_air/7WqbHI6L">density of air</a> (or other medium)</li>
<li>A is the area of the rotor disc</li>
<li>P is power</li></ul>
A device which converts the translational energy of the fluid into rotational energy of the axis or vice versa is called a Rankine disk actuator. The real life implementations of such devices include marine and aviation <a href="/facts/Propeller/0IN0JKXC">propellers</a>, <a href="/facts/Windmill/D7rzFXT2">windmills</a>, <a href="/facts/Helicopter_rotor/J3BP6U2P">helicopter rotors</a>, <a href="/facts/Centrifugal_pump/lLTGaKYS">centrifugal pumps</a>, <a href="/facts/Wind_turbine/lUfWqfLg">wind turbines</a>, <a href="/facts/Turbocharger/xuabwHlB">turbochargers</a> and <a href="/facts/Agitator_(device)/1EIqEtqg">chemical agitators</a>.

Momentum theory open-in-new

Momentum theory