Law of conservation of energy applied to hydraulic systems: Bernoulli’s equation

The general law of conservation of energy states that the amount of energy in a closed system of a hydraulic plant always remains the same. Energy can be converted, but it is not lost. The equation for this, known from mechanics, is:

½ m · v² = m · g · h

“g” stands for the acceleration due to gravity and can be replaced by a general “a” if a form of mass acceleration other than free fall is applied.

The law of conservation of energy is also applied in hydraulics. To this end, the Swiss mathematician and physicist Daniel Bernoulli initially postulated the following axioms for hydraulic systems in the 18th century:

1. The system is closed and completely filled with an incompressible fluid.
2. Friction can be neglected.

Based on these basic assumptions, Daniel Bernoulli, together with Giovanni Battista Venturi, developed the following principles using the example of a downpipe:

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

Law of conservation of energy explained simply:

“The total energy (E) in the hydraulic system is the square of the flow velocity (v2) times the mass divided by two, plus the internal pressure in the system (p) times the volume, plus the density of the hydraulic fluid (Rho) times the height of the fall (h) and the acceleration due to gravity (g).”

Here, too, g can be replaced or supplemented by any acceleration, for example by the delivery rate of a hydraulic pump. This equation explains how high differences in pressure and flow velocity can occur within a hydraulic system, for example in the case of narrowing or widening of pipe sections.

This hydraulic law of conservation of energy is also called the “energy equation” in this form. It can be converted into the “head equation” or the “pressure equation” using simple algebra. With this formula conversion, the designer has practically everything necessary at their disposal to be able to calculate a hydraulic system exactly. Bernoulli’s formula is therefore part of the basic knowledge for specialists in hydraulics.

The law of conservation of energy and Bernoulli’s equation in practice

In practice, knowledge of Bernoulli’s equations is the foundation, but in most cases it is not sufficient. Contrary to the basic axiom of negligible friction, this must certainly be taken into account in larger and complex systems. For this purpose, the extended Bernoulli energy equation for “viscous fluids” was developed. This adds the “pressure loss coefficient” (Zeta) to the first term of the energy equation. The general pressure loss depends on the change in pipe geometry.

If the structure of the inner wall of a hydraulic pipe is to be taken into account, the pipe friction factor Lambda is used. As in engineering mechanics, the law of conservation of energy in hydraulics is therefore only a theoretical approach. In practice, energy loss does occur through friction. This heats the hydraulic system along the entire flow direction and radiates into the environment. However, these heat losses are negligible for the design and calculation of the hydraulic system.