Читать книгу Hydraulic Fluid Power - Andrea Vacca - Страница 41

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:

Therefore, from the expression

we have

It is therefore possible to observe how, in this (gaseous) cavitation condition, the reduction in outlet flow is about 4.3%, much more pronounced with respect to the case (a), where there was no cavitation.

A final remark can be made on the evaluation of the equivalent (or effective) bulk modulus. For the typical operating pressure of hydraulic control systems, the elasticity of the material is also not negligible. Consider again the case of Figure 2.12 while also including the elasticity of the walls according to the bulk modulus of the material (similar to the Young modulus definition):

(2.33)

The equivalent bulk modulus for the system becomes

(2.34)