as determined by the Henry–Dalton relation of Eq. (2.22). The rest of the air is released from the liquid and is in free state. In this condition, the fluid is a mixture of liquid and air, in the form of bubbles.
Figure 2.11 Equilibrium states for a liquid considering gas solubility.
For hydraulic systems, the condition in which incondensable gases are released is generally referred to as gaseous cavitation or aeration.
The gaseous cavitation should be treated differently from the entrained air aspect described in Section 2.7.1, which is sometimes referred to as pseudo‐cavitation.
Below the vapor pressure, pVAP, the air is completely released, and the hydraulic fluid is in the vapor form. This condition for the hydraulic fluid is usually referred to as vapor cavitation.
Typical hydraulic oils are always a mixture of different components; therefore there is not a defined value of pVAP, but rather an interval [pVAP, L, pVAP, H] of pressure throughout which the vaporization of the fluid occurs. Typical values of pVAP, L and pVAP, H for mineral oils range between 15 000 and 30 000 Pa (absolute pressure) [23].
All the abovementioned cavitation phenomena can be detrimental to the operation of a hydraulic system. More than the formation of the vapor or of the gas bubbles, it is usually the opposite process (that occurs in an abrupt manner), which causes erosion damages of the mechanical parts. For this reason, designers of hydraulic components normally put the highest attention in preventing cavitation phenomena. More specifically, the selection of hose diameters and length should avoid the fluid pressure to fall below vapor pressure and, if possible, below saturation pressure.
However, gaseous cavitation is often unavoidable, particularly in the lines connecting the reservoir (where the fluid is at saturation conditions) to the pump supply port. This is due to the frictional losses that cause the pressure to decrease as the fluid travels into the line. For this reason, the connection between the pump and the reservoir must be designed to limit these pressure losses as much as possible so that the pump can operate under a minimal (sometimes negligible) aeration condition.
Moreover, in certain parts of the hydraulic system, such as in hydraulic control valves or hydraulic pumps, there are sometimes violent flow restrictions where the fluid accelerates to high velocities, causing the static pressure to fall below the saturation pressure (see Bernoulli's equation, Chapter 3). All these reasons should give an idea why the cavitation is a very common issue in hydraulic control systems.
2.7.3 Equivalent Properties of Liquid–Air Mixtures
In presence of entrained air, or when vapor or air is released, the fluid becomes a mixture, and the equivalent density and bulk modulus significantly decrease with respect to the pure liquid condition.
Simple formulas can be derived based on the continuum fluid assumption. In this approach the different phases (gas and liquid) are considered to be the same media without a distinct separating interface [24]. Under this assumption, the fluid density can be calculated as a weighted average of the single densities:
(2.23)
where αg and αv are, respectively, the volume fraction of the air and of the vapor:
(2.24)
Similarly, for the viscosity,
(2.25)
Also, for the bulk modulus, a similar expression can be found:
An interpretation of the expression (2.26) is shown in Figure 2.12: according to the continuum assumption, all the undissolved gas particles can be treated as a single bubble with the overall volume Vg. With that in mind, it is possible to apply the definition of the bulk modulus (Eq. (2.4)) to the overall system of the two components, considering its total volume change under a certain pressure difference as the sum of the two contributions of the gas volume change and of the liquid volume change:
Figure 2.12 Compression of a mixture of liquid and undissolved gas.
With simple analytical passages, Eq. (2.26) can be derived from Eq. (2.27). The bulk modulus of the gas phases can be evaluated by considering an ideal gas behavior, according to which the pressure and volume variations follow the expression
where
