# Momentum Equation

## momentum_equation.jpg

The forces and torque acting on the centrifugal pump impeller can be determined using the momentum equation.  The choice of control volumes is particularly important in this case since the impeller is rotating and the control volume must rotate with it.

"Centrifugal 2". Licensed under CC BY-SA 3.0 via Wikimedia Commons.

The Momentum Equation is one of three fundamental equations used to describe the motion of fluids. The other two are Conservation of Mass and Conservation of Mass. The momentum equation tells us how forces acting on a fluid system change the momentum of the flow.

Newton’s Second Law states that the sum of the forces acting on a system is equal to the change in momentum.  For a particle, this may be written as

Reynolds Transport Theorem can be used to make this equation applicable to a control volume representation of a fluid system.

The Reynolds Transport Theorem states that the time rate of change of the extensive variable B is given by the equation

The momentum equation for a control volume is obtained by letting B equal the momentum.  The associated intensive variable, b, is obtained by dividing B by the volume.

Therefore, the momentum equation for a control volume is

Note that the momentum equation is actually three equations – one for each coordinate direction.  In terms of the Cartesian coordinate directions x, y, and z the three equations are

Also note that                    is a scalar quantity.

Special Forms of Momentum Equations

The momentum equations can be simplified when the flow doesn’t change with time (i.e. steady state), and when the inlets and outlets are normal to the flow direction.  Under these conditions the equations reduce to

Example Problem

A pipe 90 degree elbow has an inside diameter of 5 inches and a centerline radius of 6 inches. The inlet velocity of the water flowing through the elbow is 20 ft/sec.  What are the x-y components of the force exerted on the elbow by the fluid?

The first step in the solution process is to draw the control volume for the fluid.  Note the control volume is the fluid inside the elbow and not the elbow surrounding the fluid.

The mass flow rate is constant since the fluid is incompressible and there is only one inlet and outlet, each of which has the same cross sectional area.

Force in the X-direction

The force in the x-direction is obtained from the x-direction momentum equation.   There is no x-component of velocity at the exit.

Force in the Y-direction

The force in the y-direction is obtained from the y-direction momentum equation.  There is no y-component of velocity at the inlet.

Forces Acting on Elbow

The momentum equations give the force components acting on the fluid control volume.  The forces acting on the elbow are of equal magnitude but opposite direction.

## Learning Objectives:

How to compute the forces acting on a control volume using the momentum equation.