Bernoulli’s Equation

Bernoulli’s equation is an adaptation of the general law of conservation of energy to hydraulic systems.

The law of conservation of energy states that the amount of energy in a closed system remains constant. The form of the energy is irrelevant. Consequently, energy is not “generated” or “consumed,” but is always only converted from one form into another.

Bernoulli’s Equation: The Law of Conservation of Energy

A fundamental distinction is made between “potential” or “static” energy and “kinetic” or “dynamic” energy.

An example:
A boulder resting on the top of a mountain has a mass (m) and is located at a height (h). Furthermore, acceleration (a) acts upon it—in this case, the force of gravity (g).
Therefore, the description of potential energy is also written as “m · g · h.”

If the boulder now rolls down from the mountain peak, it converts the previously “stored” potential energy into kinetic energy. This is described as “1/2 m · v².” Thus, if the velocity and the height of the peak are known, the mass of the rock can be inferred, and vice versa.

The formula for the law of conservation of energy generally states:

m · h · g = 1/2 m · v²

The law of conservation of energy can be applied to practically all other energy systems. It is only in particle physics that it is no longer sufficient to explain the phenomena prevailing there.

Applied Law of Conservation of Energy for Hydraulic Systems

Bernoulli’s equation transfers the law of conservation of energy to hydraulic systems. For it to be applicable, two conditions must be met:

1. The system is completely filled with an incompressible fluid.
2. The flow in the system is friction-free.

These conditions exclude interfering factors.

For the example of a downpipe, Bernoulli’s equation is:

E = m/2 · v² + p · V + ϱ(rho) · h · g = Constant

E = Specific total energy in the closed system (J=Nm)
v = Flow velocity (m/s)
p = Pressure (N/m²)
V = Volume (m³)
ϱ (rho) = Density of the hydraulic fluid (kg/m³)
g = Gravitational acceleration (m/s²)
h = Height of the fall (m)

For technical reasons, the values g, h, and v can be replaced by other factors, such as the delivery rate of a drive pump.

Application of Bernoulli’s Equation

Essentially, this equation explains a curious effect that occurs when the cross-sections of pipelines change.

Along the streamline—the centerline of a fluid flow—pressure and velocity change along a line depending on the cross-sections of the tubes.

In a closed system (Condition 1) with incompressible fluid (Condition 2), the same volume per unit of time must always pass every point in the system. It follows that in a constriction, the velocity of the fluid increases, and it decreases when the cross-section widens.

Additionally, a remarkable effect occurs: although the velocity increases as the cross-section narrows, the pressure at this point drops. Likewise, it rises again when the cross-section widens in the next pipe section. This effect is explained by Bernoulli’s equation and can be derived from it.

In practice, the velocity and pressure behavior of fluids in closed hydraulic systems can be calculated exactly in this way.

This is very important for the design of wall thicknesses, seals, tightening torques, and many other factors. Hydraulic systems can thus be designed for optimized purposes and reinforced accordingly at critical points. In summary, Bernoulli’s equation is part of the basic knowledge of every hydraulics specialist.