# Relation to Energy Equation

## relation_to_ee.jpg

The differences between static and dynamic pressures are shown in the figure.  As the velocity increases the static pressure decreases. The total pressure is constant at each station.

By my own work (Own work) [GFDL, CC-BY-SA-3.0 or CC-BY-SA-2.5-2.0-1.0], via Wikimedia Commons

There are various ways at which to arrive at Bernoulli’s equation. You can start by applying the momentum equation along a streamline as was done previously. You can also arrive at Bernoulli’s equation from the Pressure-Energy Equation or Euler’s Equation which are discussed in the publication, “Fundamentals of Gas Dynamics”. You can also arrive at Bernoulli’s equation using the Conservation of Energy and Mass Equations.

Bernoulli’s equation was developed by applying the momentum equation along the tangential direction of a streamline.  As such, Bernoulli’s equation states that the quantity

is equal to a constant at all points along a streamline.

As shown in the step-by-step derivation block, under certain conditions, the above quantity is also equivalent to the energy equation.  The single inlet –single outlet version of the energy equation uses average properties obtained by integrating over the inlet and outlet surface areas.   While the Bernoulli equation uses point values located on a specific streamline.

## Learning Objectives:

Learn that Bernoulli’s equation can be developed from different starting points and thus satisfies all of the fundamental equations: Conservation of Energy, Conservation of Mass, and Time Rate of Change of Momentum.