Derivations

Continuity equation

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

$$Q= Av$$

Here:

• $Q$ - Flow rate
• $A$ - Cross-sectional area
• $v$ - 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.

$$\text{Total head }H= z + \frac{P}{\rho g} + \frac{v^2}{2g}$$

Here:

• $z$ - Datum head. Height to the point from a reference level.
• $\frac{P}{\rho g}$ - Pressure head.
• $\frac{v^2}{2g}$ - Velocity head. Kinetic energy per unit weight.

Note

• Head is the energy per unit weight.
• Piezometric head is equal to $\frac{P^*}{\rho g}$ where $P^*$ is the piezometric pressure.

Steady flow momentum equation

For an incompressible fluid under steady flow:

$$F_S = \dot{M_o}-\dot{M_i}$$

Here:

• $F_S$ - Force exerted on the fluid within the control volume
• $\dot{M_o}$ - Rate of change of momentum of the inflow fluid
• $\dot{M_i}$ - Rate of change of momentum of the outflow fluid

$$\dot{M} = \rho Q v$$

Here:

• $\dot{M}$ - rate of change of momentum
• $\rho$ - density
• $Q$ - flow rate
• $v$ - velocity

Power

$$P=\gamma Q H$$

Here:

• $\gamma$ - specific weight of the fluid
• $Q$ - flow rate
• $H$ - total head