Bernoulli

Bernoulli's principle

Bernoulli's principle is often used in fluid dynamics and it states that an increase in velocity of a fluid results in a decrease of the static pressure and or a decrease in the potential energy. The sum of the static pressure, kinetic energy and potential energy must be constant along a streamline:

With $p$ $[\frac{N}{m^2}]$ equal to the pressure of the fluid, $\rho$ $[\frac{kg}{m^3}]$ to the density of the fluid, $v$ $[\frac{m}{s}]$ to the velocity of the fluid, $g$ $[\frac{m}{s^2}]$ to the gravitation constant and $h$ $[m]$ to the height of the point along the streamline.

Assumptions for the equation

• The stream is steady, the velocity and density at any point can't change over time.
• The flow is incompressible, the density must have the same value at any point of the stream.
• Friction is negligible

Use of the equation

Equation (1) is mostly used to calculate the pressure or the velocity at which a fluid is flowing at a point along the streamline. In order to do this, there must be only 1 unknown in the equation which then can be solved for.

Example problem

In figure 1 a streamline with a constant diameter can be seen, where Bernoulli's principle can be applied. Let's say that the pressure of the fluid in point C needs to be calculated.

The following values are known:

• $p_{A} = p_{D} = p_{atm} = 1*10^5[Pa]$ (The pipe is open ended)
• $v_{A} = v_{B} = v_{C} = v_{D}$ (More in mass flow rate)
• $h_{A} = 1[m]$ and $h_{C} = 0.5[m]$
• $\rho = 1000[\frac{kg}{m^{3}}]$
• $g = 9.81[\frac{m}{s^{2}}]$

Figure 1. Visualization of a Bernoulli streamline

The pressure in point C can be calculated by using equation (1) and filling it in for 2 points of which 1 point is point C and the other point is one where all values of the equation are known:

$p_{A} + \frac{1}{2}\rho v^2_A+\rho gh_A = p_{C} + \frac{1}{2}\rho v^2_C+\rho gh_C$

The equation can be rewritten without $\frac{1}{2}\rho v^2$ since this is constant everywhere:

$p_{A} + \rho gh_A = p_C+\rho gh_C$

$p_C=p_A - \rho g(h_A - h_C)$

The known values can be filled in:

$p_C=1*10^5-1000*9.81(1-0.5)$

Thus $p_C = 95095[Pa]$

Now there is shown how Bernoulli's principle is applied to simple streamlines, and that calculations can be done using this principle. See the next reading for more information about streamlines.