Moment of Momentum Equation
Introduction: The moment of momentum equation is also referred to as the angular momentum equation. The moment of momentum equation for a fixed and nondeforming control volume can be derived using by taking a material derivative of the angular momentum of a particle (omitted here), which will give
Where r is the position vector (about a reference point), V is the absolute velocity of the fluid, and n is the outward unit normal vector. For steady flow, the first term on the right hand side of the moment of momentum equation vanishes. The term V·n will be positive when the flow is out of the control volume while it is negative when the flow is into the control volume (see another section for more discussion). The directions of the quantities r×F and r×V are determined by the right hand rule described previously.
The moment of momentum equation can be stated in words as follows:
The moment of momentum equation can be applied to analyze rotating systems, such as turbines, sprinklers, and fans. In analyzing such systems, the vector form of the equation is often resolved into scalar form, and only the direction along the axis of rotation is considered. Take the lawn sprinkler as shown in the figure for example. For steady flow, the moment of momentum equation reduces to
But notice, there are two possible velocities with this formulation. Vrel is the velocity relative to the rotating nozzle, as if one is standing on the nozzle head. Next, Vabs is the absolute velocity as viewed from the non-rotating frame of reference (i.e. the ground). So why two velocities? Due to the conservation of mass, the mass flow rate, dm/dt is constant, and thus ρVrel must also be constant. Vrel is constant for a non-accelerating system. The Vabs is the rate the mass is being really moving as compared to the ground and this drives the rotating arm. It is critical that the relationship between Vabs and and Vrel is understood and used correctly. They are related as
Where Vtip is usually in the opposite direction, and thus negative. Generally, the basic steady state angular moment equation is written with separate terms for fluid coming in and going out (velocity is assumed perpendicular to the inlet or outlet area, and area does not change) giving
This form is for fixed, nondeforming control volumes. For rotating system, the absolute velocity must be used. Along the axis of rotation, the above equation can also be written in scalar form for outlet only systems as simply
Where Tshaft is the torque of the shaft, Vθ is the tangential velocity of the fluid, and dm/dt (=ρAVrel) is the total mass flow out of the control volume. Notice, this is a scalar equation, and the minus sign means that the shaft torque opposes the rotating arms. This equation assumes no fluid is coming into the system except at the center of rotation (i.e. a sprinkler type system).