Bernoulli's Principle
Table of Contents
The formula for Bernoulli’s Principle (also known as Bernoulli’s Equation) states that for two points along the streamline of an incompressible, inviscid, steady-flow fluid, the sum of the pressure, kinetic energy per unit volume, and potential energy per unit volume remains constant. It has the following forms:
Pressure Form #
$$ P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2 $$
Where
- \( \small \small P_1 \) and \( \small \small P_2 \) are the pressures at points 1 and 2,
- \( \small \small \rho \) is the density of the fluid,
- \( \small \small v_1 \) and \( \small \small v_2 \) are the velocities at points 1 and 2,
- \( \small \small g \) is the acceleration due to gravity, and
- \( \small \small h_1 \) and \( \small \small h_2 \) are the heights at points 1 and 2.
It can be also be written by stating that the sum of the quantities remain constant:
$$ P + \frac{1}{2} \rho v^2 + \rho g h = \text{const} $$
Energy Form #
$$ \frac{P}{\rho} + \frac{1}{2} v^2 + g h = \text{const} $$
$$ \frac{P_1}{\rho} + \frac{1}{2} v_1^2 + g h_1 = \frac{P_2}{\rho} + \frac{1}{2} v_2^2 + g h_2 $$
Head Form #
$$ \frac{P}{\gamma} + \frac{v^2}{2g} + h = \text{const} $$
$$ \frac{P_1}{\gamma} + \frac{v_1^2}{2g} + h_1 = \frac{P_2}{\gamma} + \frac{v_2^2}{2g} + h_2 $$
Where
- \( \small \small \gamma \) is the specific weight of the fluid.