Derivations

Continuity equation

From the conservation of mass law, the below equation can be derived for an incompressible fluid:

Here:

• - Flow rate
• - Cross-sectional area
• - velocity

Bernoulli’s equation

For an incompressible fluid in steady flow, total head on a point is constant throughout a streamline. Can be derived from the conservation of energy law.

Here:

• - Datum head. Height to the point from a reference level.
• - Pressure head.
• - Velocity head. Kinetic energy per unit weight.

Note

• Head is the energy per unit weight.
• Piezometric head is equal to where is the piezometric pressure.

Steady flow momentum equation

For an incompressible fluid under steady flow:

Here:

• - Force exerted on the fluid within the control volume
• - Rate of change of momentum of the inflow fluid
• - Rate of change of momentum of the outflow fluid

Here:

• - rate of change of momentum
• - density
• - flow rate
• - velocity

Power

Here:

• - specific weight of the fluid
• - flow rate
• - total head