Example Problem

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Water can be used to cut metal when raised to a sufficiently high pressure using a high pressure pump and nozzle. 

By Flickr.com user "The Jamoker" (http://www.flickr.com/photos/jamoker/254610497/) [CC-BY-2.0], via Wikimedia Commons

Simple garden hoses generate higher velocities that help remove dirt and grime from our automobiles. The same fluid dynamics principles that are used with the garden hose can also be used to generate pressures high enough to cut metal using a modern water jet machine.


Problem

What is the exit velocity and inlet pressure for the water flowing through the nozzle?  Average fluid properties are given.  Frictional/shear stresses are negligible and the fluid is incompressible.

Solution

There are two unknowns, the inlet pressure and the exit velocity.   Therefore two equations are needed.  The possible equations include: 1) conservation of mass, 2) conservation of energy, and 3) momentum.  In this problem, the conservation of mass and Bernoulli’s equation (momentum) will be used. Note that the flow is steady state, the fluid is incompressible and inviscid since the frictional/shear stresses are negligible.  These three criteria make the use of Bernoulli’s equation acceptable. 

The conservation of mass equation for a single input-single output system can be written as

Since the fluid is incompressible, the densities cancel and the above equation become

Solving for the exit velocity yields

Bernoulli’s equation for a streamline starting at the inlet and ending at the exit is

Since there is no elevation change, this equation reduces to 

Solving for the inlet pressure yields

 

Learning Objectives:

Learn how to apply Bernoulli’s equation to solve practical fluid dynamics problems.