Hydraulic Fluid Power. Andrea Vacca
of esters and chlorides
a Refers to biodegradable fluids.
2.3 Physical Properties of Hydraulic Fluids
Properties of hydraulic fluids, such as density or viscosity, are usually tabulated (or expressed by analytical formulas) as functions of pressure and temperature. In order to justify the choice of pressure and temperature as independent variables, the Gibbs' phase law of thermodynamics should be considered:
The Gibbs' rule defines the number of degrees of freedom, f, necessary to describe the state of a substance in thermodynamic equilibrium. The formula uses the number of components of the substance (n) and the number of phases in equilibrium (κ). The number of degrees of freedom is the number of independent intensive variables that can be varied simultaneously and arbitrarily without determining one another. An intensive variable does not depend on the size of the considered system. For the case of a hydraulic oil, specific volume, density, pressure, temperature, and viscosity are examples of intensive variables.
If we consider the hydraulic fluid as a single component matter (i.e. a pure chemical) even if it is usually a mixture of different components, then from Eq. (2.1), for a single‐component system (n = 1) in the liquid phase (κ = 1), the number of degrees of freedom f is equal to two. This means that two intensive variables can be arbitrarily selected to completely determine the status of the fluid. Among all possible choices, choosing pressure and temperature is particularly convenient because:
pressure and temperature are relatively easy to measure, with respect to other intensive properties of the fluid; and
an engineer has a better ability or practical intuition to relate pressure and temperature to practical problems.
Two intensive variables fully define the status of a liquid. In hydraulic, pressure and temperature are the typical choice for the independent variables used to express the functional between fluid properties.
2.4 Fluid Compressibility: Bulk Modulus
From the consideration made in the previous paragraphs, the functional dependence of the volume occupied by a certain amount of hydraulic fluid has the following form:
(2.2)
The dependence of the volume on the variations of both pressure and temperature can be expressed made with a simple linear equation by considering the first‐order Taylor series expansion:
(2.3)
In other terms, the deviations from the initial volume V0 (at the reference conditions p0, T0) can be expressed in a linear form by defining the coefficients:
So
(2.6)
The parameter B is known as isothermal bulk modulus of the fluid, and it indicates the tendency of the fluid volume to vary under changes in pressure.
The negative sign in the definition of Eq. (2.4) implies an inverse type relationship, meaning that an increase of pressure causes a reduction in the fluid volume.
The reciprocal value 1/B is known as isothermal compressibility. In the fluid power field, however, the isothermal compressibility is not commonly used.
The parameter γ of Eq. (2.5) is known as isobaric cubic expansion coefficient (or simply volumetric expansion coefficient), and it expresses the tendency of the fluid volume to vary with temperature.
As it will be mentioned in the following chapter, pressure variations in the working fluid form the basis of the functioning of hydraulic systems. For this reason, the bulk modulus B is an important parameter that can be used to quantify the compressibility effects of the fluid. In the case of hydraulic systems, temperature effects on the fluid compressibility can be in most cases neglected. For this reason, the cubic expansion coefficient is a parameter rarely encountered when analyzing a fluid power system.
A practical definition for the bulk modulus is based on the finite form of Eq. (2.4), where finite differences are used instead of the differentials:
(2.7)
The nature of the processes used to measure the pressure and volume variations (for example, isothermal or adiabatic), as well as the way of experimentally evaluating volume variations (secant or tangent methods), results in slightly different definitions for the bulk modulus. It is out of the scope for this book to discuss these details. However, the reader could refer to specific literature on fluid properties, such as [1].
Typical values for the bulk modulus of hydraulic fluids range between 15 000 bar and 20 000 bar. It can be interesting to note that typical hydraulic fluids