3.1 is related to the basic equation of fluid statics:
Equation (3.3) implies that for a liquid at rest, the pressure linearly increases with depth, due to the effect of gravity. The integration of the differential Eq. (3.3) provides an expression for the pressure difference between two points at different elevation:
The concept expressed by the above Eq. (3.4) is illustrated in Figure 3.2.
Many hydraulic machines are often referred to as “hydrostatic machines” because their basic functioning can be evaluated using only static equations (not involving fluid velocity), as described in Section 3.1. Indeed, in almost all hydraulic components, the effect of fluid pressure is significantly higher than that of fluid velocity, thus making the latter negligible.
Figure 3.2 Fluid pressure increases with depth.
Figure 3.3 Elevation difference in the hydraulic circuit of a mobile application.
Despite this, the basic law of fluid statics is almost always omitted from the analysis of a fluid power system1. Neglecting elevation effects on fluid pressure is reasonable in all cases in which minimal variation of fluid pressure does not affect the operation of the systems. To better explain this concept, Figure 3.3 shows an example of a circuit of a mobile machine.
A reasonable value for h can be 3 m. Considering a fluid density of 870 kg/m3, which is typical for a hydraulic oil, the pressure difference between two points with the maximum elevation difference is
(3.5)
which is considerably lower than the typical operating pressure of the system, which is easily above 100 bar.
The elevation difference in typical hydrostatic circuits can be neglected when calculating the system pressures. The effects of elevation become critical only when in parts of the system the pressure is close to the saturation conditions, such as at the suction of hydraulic pumps.
3.3 Volumetric Flow Rate
A hydraulic circuit comprises a network of components such as pumps, valves, cylinders, and filters, which are connected through fluid conveyance elements, such as hoses or pipes. The flow rate through the connecting hoses or pipes is a recurring parameter used in systems analysis. This section provides a high‐level review of the concept of flow rate starting from the generic case of flow through a pipe, represented in Figure 3.4.
Before entering into more specific considerations for the hydraulic case, it is important to recall the definition of scalar product. Figure 3.4 shows the generic case of a pipe where a particle of fluid is traveling with a defined velocity along the direction of the pipe. The figure also represents a generic surface area defined by the section of the pipe with a generic plane. As shown in the figure, both are vector quantities: vector
From the scalar product of Eq. (3.6), the convention for the flow crossing a surface can be defined: an in‐flow is associated with a negative scalar product, while an out‐flow has a positive scalar product.
Figure 3.4 Flow through a pipe: velocity vector and surface area vector. (a) outflow; (b) inflow.
The scalar product in Eq. (3.6) is particularly important for the definition of volumetric flow rate across a section. In general:
Figure 3.4 represents the case of a single fluid particle, Eq. (3.7) integrates this relationship to the whole section of the pipe. The velocity of each particle on the section is different and its profile depends on the flow regime. For laminar flow conditions, where viscous effects prevail over fluid inertia effect, the velocity profile resembles the parabolic distribution illustrated in Figure 3.5a.
For turbulent flow conditions (Figure 3.5b), the velocity profiles are fuller in the middle of the pipe while still satisfying the no‐slip condition (zero velocity in proximity of the wall). The dimensionless Reynolds number is typically used to distinguish the flow regime conditions:
