Energetic interpretation and forms of the relationship

Three equivalent forms

Bernoulli’s equation can be written in three forms, each suited to a particular measurement context:

Form	Expression	Unit
Pressure	P + (1/2)ρv² + ρgz = constant	Pa
Height (head)	P/(ρg) + v²/(2g) + z = constant	m
Volumetric energy	identical to the pressure form	J/m³

Physical meaning of the terms

• P: static pressure, associated with the fluid’s pressure energy.
• (1/2)ρv²: dynamic pressure, associated with the kinetic energy of the moving fluid.
• ρgz: positional pressure, associated with gravitational potential energy.

Bernoulli’s equation therefore expresses the conservation of a fluid’s volumetric mechanical energy along a streamline, in the absence of friction and of work exchange with the external environment (no pump, no turbine along the path under consideration).

Practical example

In a horizontal pipe (constant z) that narrows, the continuity equation (conservation of flow rate, S*v = constant for an incompressible fluid) dictates that v must increase at the point where the pipe narrows. Bernoulli’s principle then dictates that P must decrease at this point: this is precisely the principle utilised in the Venturi tube, which is discussed in the following chapter.

Pitfall to avoid

Do not confuse static pressure P, measured perpendicular to the flow, with total pressure P + (1/2)ρv², measured in the direction of the flow (for example, using a Pitot tube). These two quantities do not have the same physical meaning, and confusing them will distort any calculation of velocity.