Hydraulic Fluid Power. Andrea Vacca

alt="upper V Subscript a i r comma r comma 1 Baseline equals upper V Subscript s a t Baseline left-parenthesis 1 minus StartFraction p 1 Over p Subscript s a t Baseline EndFraction right-parenthesis left-parenthesis StartFraction p Subscript s a t Baseline Over p 1 EndFraction right-parenthesis Superscript StartFraction 1 Over gamma EndFraction"/>

Figure 2.13 Graphical representation of the dissolved and undissolved air in a hydraulic oil. The volume in the x axis is referred to the reference pressure.

      Source: Adapted from Nervegna [14].

      This expression can then be used to evaluate the volume fraction α_g in the above expressions.

      As previously discussed, in the majority of cases, hydraulic components do not see instances of vapor cavitation; therefore it can be usually assumed that α_v = 0. However, in certain conditions, vapor cavitation can occur. If the reader has interest in these cases, for the evaluation of the parameter α_v, it is recommended to refer to more specialized literature, such as [25, 26].

      Example 2.2 Volumetric flow rate of a hydraulic pump

      In the field of hydraulics, it is very common to express flow rates as volumetric flow rates. As it will be discussed in the next chapter, there are two main reasons for that. First, it is straightforward to evaluate the motion velocity of a hydraulic actuator if the volumetric flow through it is known. Second, it is quite common to use volumetric flow meters for the measurement of the flow rate.

      This example describes what is the effect of fluid compressibility on the volumetric flow rate.

      A hydraulic pump supplies flow to a hydraulic cylinder that lifts a certain load. Due to this load, the pump outlet is pressurized at 100 bar. The inlet flow, from a tank open to atmosphere, is 100 l/min. Determine the volumetric flow exiting the pump. The density of the fluid is 870 kg/m³; bulk modulus of the fluid is 18 000 bar.

      Consider the following cases:

      1 The pump inlet pressure is the same as the tank pressure.

      2 The pump inlet pressure is −0.3 bar gauge pressure.

The Bunsen coefficient for the oil is 9%.

       Given:

      The pump inlet flow, Q₁ = 100 l/min; the pump outlet pressure p₂ = 100 bar (gauge pressure); the Bunsen coefficient of the oil, α_a = 9 % ; the density of the fluid, ρ = 870 [kg/m³]; the bulk modulus of the fluid B = 18 000 bar.

      The inlet pump pressure in two cases: (a) p₁ = p_T = 1 bar (absolute pressure); (b) p₁ = 0.7 bar (absolute pressure).

       Find:

      The outlet pump flow rate, Q₂, for the two cases (a) and (b).

      Solution

       Case (a)

      There is no pressure loss from the tank to the pump inlet. Therefore, assuming that the fluid is at saturation condition in the tank, there is no undissolved air at the pump inlet. This means that α_g = 0 and the fluid is entirely liquid.

      Assuming steady‐state flow conditions, the same mass flows at the same rate at the inlet and outlet of the pump, so that

      Considering that

      we have

      Due to the effect of the fluid compressibility, the volumetric flow rate at the pump outlet is lower than the volumetric flow rate at the inlet by 0.56%.

Case (b)

      In this case, a certain amount of dissolved air is present at the pump inlet, being p₁ < p_T = p_SAT. The amount of air at the inlet port 1 can be evaluated as

      where p_sat = p_T, considering the process as isothermal (γ = 1):

      Therefore, the density of the fluid at the inlet section is

      In the above expression, it is considered that the density of the air at standard conditions is ρ_g = 1.225 kg/m³.

      The density at the pump outlet can be calculated considering that at high pressure (p₂ = 100 bar) all the fluid is liquid:

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