Derivations | Sahithyan's Notes

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, the height to the point from a reference level
$\frac{P}{\rho g}$: Pressure head
$\frac{v^2}{2g}$: velocity head, the kinetic energy per unit weight

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

The momentum rate is:

\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