# Modelling and Validation of Cavitating Orifice Flow in Hydraulic Systems

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^{2}

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## Abstract

**:**

## 1. Introduction

^{®}CFX.

^{®}. Zhou et al. [4,5] developed a lumped parameter model for the study of gaseous cavitation in hydraulic orifices, obtaining good results from the experimentation. Shah et al. [6] proposed a model for the study of both gaseous and vapor cavitation by applying it to gear pumps; the same authors [7,8] used the identical methodology on gerotor pumps. Rundo et al. [9] presented an article on lubrication pumps which considers the air dissolution dynamics, reaching a great correspondence with experimental results.

## 2. CFD Model

_{l}is the liquid mass fraction, f

_{g}the air mass fraction, and f

_{v}the vapor mass fraction. Furthermore, the liquid phase is modelled as a variable composition mixture consisting of oil and dissolved air.

#### 2.1. Gaseous Cavitation

_{A}of Equation (1) needs to be split into two components:

_{g,g}and one for non-condensable gases in dissolved liquid phase, f

_{g,l}.

_{d}and R

_{a}are the mass rate for desorption and absorption, respectively. C

_{d}and C

_{a}are empirical coefficients to be determined that influence the speed with which the associated phenomenon occurs.

_{g}is below the equilibrium pressure of p

_{equil}. Likewise, absorption occurs when the partial pressure is above the equilibrium pressure. The equilibrium pressure is assumed as the sum of the vapor pressure and the gas partial pressure, calculated as the product between the Henry constant and the molar mass of the dissolved air.

#### 2.2. Vapor Cavitation

_{B}represents the bubble radius and p

_{v}is the vaporization pressure.

_{c}, or vaporization, F

_{v}.

## 3. Lumped Parameter Approach

^{®}environment, assumes that at pressure levels higher than the saturation pressure, the entire quantity of gas is completely dissolved in the liquid [12]. Thus, the gas does not contribute in volume but only in mass in the equation of state of the fluid and therefore it follows the compressibility law of the liquids. Below the saturation pressure, according to the Dalton–Henry law, part of the gas begins to separate and therefore to increase the total volume of the fluid. Three different pressure ranges are considered.

- p > p
_{SAT}

- p
_{VAPL}< p < p_{SAT}

_{SAT}and that this process ends at the lower vapor pressure p

_{VAPL}. Since the lower vaporization pressure is a very low value, as reported in Table 1, once p

_{VAPL}is reached, it is assumed that all gas has been completely released. The free gas fraction follows Henry’s law.

_{SAT}and p

_{VAPL}.

_{VAPH}the oil begins to vaporize, and the fluid is modelled as a uniform mixture of free air, oil vapor, and liquid oil. In this pressure range, the vapor to liquid mass fraction is described with a polynomial function.

- p < p
_{VAPL}

_{17}H

_{36}[13].

_{T}and the downstream pressure is p

_{S}, the ideal fluid velocity at the throat is calculated as

_{sat}the integral is analytically solvable and leads to

## 4. Experimental Activity

## 5. CFD Simulations

^{®}CFX software. The gas cavitation model is not present in CFX, but it was implemented through the CFX Expression Language (CEL). CEL is a scripting language inside the main CFD code that allows operations on additional user-defined variables to be performed.

_{c}and F

_{v}parameters.

_{equil}, while it is unbounded during absorption.

## 6. Lumped Parameter Model Results

## 7. Conclusions

^{®}CFX in which an air release and absorption model is introduced, the other is a lumped parameter fluid model.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

f | Mass fraction |

ρ | Density |

U | Velocity |

R_{B} | Bubble radius (m) |

σ | Surface tension (Pa) |

p | Pressure (Pa) |

${\dot{m}}_{fg}$ | Interphase mass transfer rate (kg/s) |

B | Bulk modulus (Pa) |

T | Temperature (K) |

${\tilde{R}}_{G}$ | Universal gas constant |

R | Mass rate |

$\tilde{m}$ | Oil vapor molecular mass (average) |

p_{T} | Total pressure (Pa) |

p_{S} | Static pressure (Pa) |

x | Gas to liquid volume fraction |

γ | Isentropic exponent |

C_{d} | Orifice flow coefficient |

c | Fluid velocity (m/s) |

Subscript | |

l | Liquid |

g | Gas |

v | Vapor |

d | Desorption |

a | Absorption |

g,g | Gas released |

g,l | Gas dissolved |

_{0} | Reference condition |

_{SAT} | Saturation condition |

## References

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**Figure 7.**Comparison between experimental mass flow rate and CFD simulation results with varying parameter sets.

**Figure 10.**Air volume fraction distribution in the domain. Pressure drop 0.256 bar, Set A parameters.

**Figure 12.**Vapor volume fraction distribution in the domain. Pressure drop 0.43 bar, Set A parameters.

**Figure 13.**Comparison between experimental results and both CFD and lumped parameter approach results.

**Figure 15.**Comparison between ideal speed and speed of sound: (

**a**) total upstream pressure of 1.05 bar; (

**b**) intersection of the two curves for total upstream pressure of 0.75 bar.

Variable | Value | Unit |
---|---|---|

x | 9 | % |

p_{SAT} | 1 | bar |

p_{VAPH} | 0.2 | bar |

p_{VAPL} | 0.0001 | bar |

Variable | Sensor | Main Features |
---|---|---|

p_{u} | Pressure Transducer | 0–10 barA ± 0.5%FS |

pd | Pressure Transducer | 0–10 barA ± 0.5%FS |

${\dot{V}}_{delivery}$ | Flow Meter | 0.1–80 L/min ± 0.3% measured value |

ω | Speed sensor | accuracy class 0.05 |

Geometric Relationship | Value |
---|---|

$\raisebox{1ex}{$D1$}\!\left/ \!\raisebox{-1ex}{$d$}\right.$ | 1.8 |

$\raisebox{1ex}{$D1$}\!\left/ \!\raisebox{-1ex}{$D2$}\right.$ | 1.3 |

$\raisebox{1ex}{$L1$}\!\left/ \!\raisebox{-1ex}{$d$}\right.$ | 4.9 |

$\raisebox{1ex}{$L2$}\!\left/ \!\raisebox{-1ex}{$d$}\right.$ | 12.9 |

Min Element Size | Max Element Size | Average Element Quality | Average Aspect Ratio | Average Orthogonal Quality | Average Skewness |
---|---|---|---|---|---|

$2.5\cdot {10}^{-5}$ | $3.0\cdot {10}^{-4}$ | $3.6\cdot {10}^{-1}$ | 7.2 | $9.6\cdot {10}^{-1}$ | $5.0\cdot {10}^{-3}$ |

Vapor Pressure | R_{b} | R_{nuc} | F_{v} | F_{c} | C_{d} | C_{a} | |
---|---|---|---|---|---|---|---|

Set A | 0.2 bar | 10 μm | 5 ⋅ 10^{−4} | 0.9 | 0.01 | 2 | −0.1 |

Set B | 0.2 bar | 10 μm | 5 ⋅ 10^{−4} | 50 | 0.01 | 2 | −0.1 |

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**MDPI and ACS Style**

Casoli, P.; Scolari, F.; Rundo, M.
Modelling and Validation of Cavitating Orifice Flow in Hydraulic Systems. *Sustainability* **2021**, *13*, 7239.
https://doi.org/10.3390/su13137239

**AMA Style**

Casoli P, Scolari F, Rundo M.
Modelling and Validation of Cavitating Orifice Flow in Hydraulic Systems. *Sustainability*. 2021; 13(13):7239.
https://doi.org/10.3390/su13137239

**Chicago/Turabian Style**

Casoli, Paolo, Fabio Scolari, and Massimo Rundo.
2021. "Modelling and Validation of Cavitating Orifice Flow in Hydraulic Systems" *Sustainability* 13, no. 13: 7239.
https://doi.org/10.3390/su13137239