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Bernoulli's equation is one of the most important equations in fluid mechanics.

Bernoulli's equation establishes the concept of energy conservation within a flow.

In textbooks on basic fluid mechanics, Bernoulli's equation is derived using two possible methods: one considers the conservation of mass and the momentum equation applied to a differential CV and another – perhaps more intuitively – starts from the principle of energy conservation. The reader can refer to [15] for more details. In this chapter, the classic form of the Bernoulli's equation is presented without discussing its derivation, in order to highlight its implications in hydraulic systems:

(3.22)

This equation is valid under steady‐state conditions, for incompressible and inviscid (frictionless) flows. Each term of the equation has units of energy per unit mass (J/kg) and summarizes three possible ways in which a fluid can store energy:

(3.23)

Equation (3.22) is used to describe the relation between pressure and fluid velocity in a flow stream:

(3.24)

As previously discussed, in hydraulic systems, the operating pressure is so high that even large differences in elevation are mostly negligible in fluid power machines. However, large variations in fluid velocities and in pressure can be found within hydraulic components. Neglecting the elevation contribution, Eq. (3.24) states that changes in fluid pressure in a fluid stream correspond to a quadratic change in fluid velocity.

In the presence of fluid contractions, where the fluid velocity increases, because of the conservation of mass, the pressure decreases. On the contrary, in case of expansions, the velocity decreases and the pressure rises. This is also illustrated in Figure 3.9.

Sharp contractions or expansions are often present within hydraulic components. It is therefore important to note the trend in the variation of fluid pressure and velocity as the flow crosses a certain geometric configuration. Equations (3.22) and (3.24) are valid for the ideal case of frictionless fluid (frictional effects due to viscosity negligible); and, strictly speaking, for flow particles along the same velocity streamline (an example of particle streamlines is represented in Figure 3.9). These ideal conditions are realistic for most contractions, where the frictional effects do not have much influence on the velocity profile. However, at expansion, when the flow is on an adverse pressure gradient, frictional effects are more relevant, and conditions can be far from the ideal ones. If the geometrical area increases too rapidly, the boundary layer portion of the flow can grow in an unstable way and causes flow separation effects [15]. In simple terms, with reference again to Figure 3.9, the relations discussed above are valid for the convergent section up to section 2, while the actual pressure recovery at section 3 is much more limited (p₃ < p₁) and it depends on the how gradual is the area increase from section 2 to section 3.

Figure 3.9 Venturi tube and representation of streamlines.