Читать книгу Hydraulic Fluid Power - Andrea Vacca - Страница 29

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:

(2.4)

(2.5)

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 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)