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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.

      3.5.1 Generalized Bernoulli's Equation

      For the analysis of pipe flow problems, the basic Bernoulli's equation can be extended to its generalized form:

(3.25)

The generalized Bernoulli's equation (Eq. (3.25)) is written between two reference sections of a pipe flow stream: an upstream section (generally indicated with subscript 1), and a downstream section (subscript 2). At the second member, the equation includes terms denoting the presence of an energy loss, h_l, and of an energy generation, h_p. The term for energy loss represents the loss due to fluid shear between the control sections 1 and 2. The term for energy generation (h_p) includes the energy provided to the flow, for example, when pumps are present. This equation will also be used to introduce the hydraulic pumps in Chapter 6.

Here, the generalized Bernoulli's law is now applied to the whole pipe (delimited by the two control sections); the Bernoulli's equation discussed in the previous paragraph is valid only along a streamline. This implies a different evaluation for the kinetic energy. In Eq. (3.25), this kinetic energy contribution is calculated by considering the average velocity on the area, thus introducing a kinetic energy coefficient (α) to consider the actual velocity profile (Figure 3.10). Without going into the specifics of how α is calculated, which can be found, for example, in [15], one can just consider:

      (3.26)

      The energy loss per unit mass, h_l, is typically referred in fluid mechanics as head loss. It comprises two terms:

      (3.27)

The major loss, h_major, includes the friction effects generated in all the sections of the pipe (between the reference sections 1 and 2) with constant sectional areas and flow in fully developed conditions. On the other hand, the minor loss, h_minor, includes other frictional losses caused by singularities such as sectional discontinuities of fittings, bends, and entrances.

Figure 3.10 Laminar (a) and turbulent (b) velocity profiles in a pipe, with same average velocity.

      The head loss h_l expresses the energy loss per unit mass in a defined flow section. It comprises two terms, the major loss (portions with constant sectional areas) and the minor loss (singularities).

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