Augmented Bernoulli Equation

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The friction losses in a pipe will result in a pressure drop between the inlet and outlet.  This pressure drop is not recoverable and work must be expended to regain the pressure.  Long pipelines employ pumping stations to raise the pressure when it drops too low.

By Artinge (Own work) [CC-BY-SA-3.0], via Wikimedia Commons

There were a number of assumptions used in deriving the Bernoulli equation. One of these was that the flow had no viscous losses. An augmented Bernoulli equation is frequently used to include various head loss or head creation terms. A pump is a device that creates head, while friction losses decrease head.

The “head” form of Bernoulli’s equation is


 

Recall that Bernoulli’s equation was derived using the assumption that there were no viscous shear stresses (ie. the flow is inviscid).  The Bernoulli equation is generally modified to include viscous losses after the fact.  One such modification is friction losses in pipe flow.  The Darcy friction factor states that the head loss is equal to 


Where the friction factor, f, can be computed using a Moody diagram.  The augmented Bernoulli equation that accounts for the frictional head loss is


 

Note that the friction loss term appears on the right hand side of the equation (the outlet side).

 

Learning Objectives

How friction losses are incorporated into the Bernoulli equation.