Hydraulic Fluid Power. Andrea Vacca

Hydraulic Fluid Power

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) rho Subscript t o t Baseline equals alpha Subscript g Baseline rho Subscript g Baseline plus alpha Subscript v Baseline rho Subscript v Baseline plus left-parenthesis 1 minus alpha Subscript g Baseline minus alpha Subscript v Baseline right-parenthesis rho Subscript l

where α_g and α_v are, respectively, the volume fraction of the air and of the vapor:

(2.24) alpha Subscript g Baseline equals StartFraction upper V Subscript g Baseline Over upper V Subscript t o t Baseline EndFraction semicolon alpha Subscript v Baseline equals StartFraction upper V Subscript v Baseline Over upper V Subscript t o t Baseline EndFraction

Similarly, for the viscosity,

(2.25) mu Subscript t o t Baseline equals alpha Subscript g Baseline mu Subscript g Baseline plus alpha Subscript v Baseline mu Subscript v Baseline plus left-parenthesis 1 minus alpha Subscript g Baseline minus alpha Subscript v Baseline right-parenthesis mu Subscript l

Also, for the bulk modulus, a similar expression can be found:

(2.26)

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 V_g. 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:

(2.27) upper B Subscript t o t Baseline equals minus upper V Subscript t o t Baseline left-parenthesis StartFraction italic d p Over italic d upper V Subscript t o t Baseline EndFraction right-parenthesis

Schematic illustration of the compression of a mixture of liquid and undissolved gas.

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

(2.28) p Subscript g Baseline upper V Subscript g Baseline Superscript gamma Baseline equals const

where γ is the polytropic constant². By differentiating the expression of Eq. (2.28),

(2.29) italic d p Subscript g Baseline upper V Subscript g Baseline Superscript gamma Baseline plus p Subscript g Baseline gamma upper V Subscript g Baseline Superscript gamma minus 1 Baseline italic d upper V Subscript g Baseline equals 0

from which, by applying the general definition for the bulk modulus (Eq. (2.4)),

(2.30) upper B Subscript g Baseline equals gamma p Subscript g

In order to evaluate the equivalent density, viscosity, and bulk modulus for the fluid mixture, by using the expressions presented in the previous paragraphs, it is necessary to first know the amount on undissolved gases. The simpler method for doing such estimation is based on the equilibrium assumption, according to which the gas and the vapor are released or dissolved instantaneously. In reality, these processes are not instantaneous, as they are characterized by time dynamics on the order of tens of milliseconds [25, 26]. It is documented that the air release or vaporization processes happen more rapidly than the opposite dissolving processes [23].

According to the equilibrium assumption, the amount of undissolved air or vapor can be evaluated according to the Henry–Dalton law (Eq. (2.22)), with the graphical representation of Figure 2.13. The saturation line represents the equilibrium states according to Eq. (2.22). For example, considering a hydraulic fluid with 9% of dissolved air at atmospheric pressure (p_o ≅ 1 bar, state 0), if the pressure is brought to a lower pressure (state 1), the amount of air that can be dissolved is reduced (V_{air, d, o} < V_{air, d, 1}), meaning that a certain amount of air, V_{air, r, 1}, is released:

(2.31)

In the process that brings the V_{air, r, 1} from the saturation conditions (p_sat = p₀) to the actual pressure p₁, the calculated volume is subject to an expansion that can be approximated as polytropic; therefore the final expression that can be used to estimate the volume of undissolved air is

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