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 are less stiff than water, which has a bulk modulus of about 22 000 bar.
Even if the concepts described in this book do not involve many considerations about fluid compressibility, it can be interesting to note how the bulk modulus of a fluid is in direct relation with the speed of sound within the fluid.
In fact, from the basic definitions of thermodynamics, the speed of sound c is defined as [13]
(2.8)
Considering the definition of Eq. (2.4), after noticing that ΔV/V = Δρ/ρ (the definition of fluid density, ρ, will be given in Section 2.5), the speed of sound results in
(2.9)
2.5 Fluid Density
Density is an important intensive property of a fluid, and it can be defined as the ratio between the mass and the volume at a given state. Considering reference condition as (p0, T0), the density ρ0 is given by
(2.10)
Typical values of density of different fluids at room temperature and atmospheric conditions are given in Table 2.3.
Following the same line of reasoning presented in Section 2.4, density variations can also be related to pressure and temperature. The isothermal bulk modulus and the isobaric cubic expansion coefficient defined in Section 2.4 can be used to quantify the dependence of density on pressure and temperature.
The variation with respect to pressure (assuming constant temperature, T0) is shown below:
However, the variation with respect to temperature (assuming constant pressure, p0) is as follows:
The chart of Figure 2.4 shows the typical variation of fluid density for three different oils. The plot results from Eq. (2.11) using the following values for the parameters of Eq. (2.12) (values taken from [14]):
From Figure 2.4, one can notice that the change in density over a large pressure variation is not negligible. In particular, for a mineral oil, the same mass of fluid is exposed to a volume reduction (or a density increase) of about 0.7% every 100 bar of pressure variation. Using the same parameters from Table 2.4, the plot of Figure 2.5 can be derived from Eq. (2.12) to represent the variation of fluid density with temperature.
Figure 2.4 Density variation with pressure for three oils (parameters listed in Table 2.4).
Figure 2.5 Density variation with temperature for three oils (parameters
