The first hydraulic machines of the nineteenth century, such as the first hydraulic presses and hydraulic lifts, are based on this law.
The use of Pascal's law in an elementary hydraulic machine is shown in Figure 3.1. At level z*, the fluid pressure must be equal in both the vertical branches of the apparatus. Therefore:
It is evident that the geometrical ratio of the areas of the two pistons is related to the ratio of the force applied:
Figure 3.1 Basic hydraulic machine.
Equation (3.1) shows how it is possible to produce large loading forces using small geometrical areas and establishing high fluid pressures. This is based on power density, the main advantage of fluid power technology. Equation (3.2) is based on many hydraulic machines that require force multiplication, such as hydraulic brakes. For the practical applications of fluid power technology, the upper pressure limit (usually defined by the relief valve setting) is never given based on the fluid, but on the structural requirements of the components or of the machines. This also explains the current trend of increasing, where possible, the operating pressures of fluid power machines so that the power‐to‐weight ratio is reduced. A significant example for this is fluid power for aviation technology, where over the years the working pressure of the hydraulic actuation systems (flap and slat drives, landing gears, nose wheel steering, and many others) increased up to 210 bar, which is used in most commercial airliners. High performance military aircraft recently increased to 350 bar.
3.2 Basic Law of Fluid Statics
The Pascal's law presented in Section 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