# Air Release and Cavitation Modeling with a Lumped Parameter Approach Based on the Rayleigh–Plesset Equation: The Case of an External Gear Pump

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## Abstract

**:**

## 1. Introduction

## 2. Lumped Parameter Approach for Modeling Fluid Power Systems

## 3. Cavitation Modeling Approach

#### 3.1. Homogeneous Mixture Modeling of Fluid Properties

#### 3.2. Dynamics of Homogeneous Mixtures in Control Volumes

#### 3.3. Closure Relations for Cavitation and Aeration

## 4. Hybrid Rayleigh–Plesset Equation (RPE) Model

#### 4.1. Basic Model

#### 4.2. Averaged Rayleigh–Plesset Equation

#### 4.3. Energy Equation for the Bubble Placed in an Infinite Fluid Medium

#### 4.4. Dimensional Analysis of the RP Equation

#### 4.5. Validation Tests and the Hybrid Modeling Approach

## 5. Implementation of the Proposed Model in HYGESim Tool

## 6. Experimental Setup

## 7. Simulation Results and Validation

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | Cross sectional area of the orifice in a plane normal to the flow direction |

a | Coefficient in power law relationship between dimensionless parameters |

b | Exponent in power law relationship between dimensionless parameters |

${b}_{g}$ | Width of a lateral gap |

${C}_{q}$ | Orifice coefficient |

c | Speed of sound |

${c}_{S}$ | Saturated concentration of the incondensable gases |

E | Bulk modulus of a phase/mixture |

${E}_{0}$ | Bulk modulus of the oil at a pressure ${p}_{0}$ |

${E}_{SS}$ | Steady state energy of a fluid bubble system |

$\mathbb{E}$ | Total energy of a fluid bubble system |

f | Mass fraction of a phase |

G | Contributions of energy from pressure and surface tension forces |

H | Henry’s constant |

h | Height of a lateral gap |

k | Kinetic energy of the fluid |

${k}_{ge}$ | Gas release relaxation parameter |

${k}_{gc}$ | Gas dissolution relaxation parameter |

${k}_{ve}$ | Vapor evaporation relaxation parameter |

${k}_{vc}$ | Vapor condensation relaxation parameter |

L | Length of a lateral gap |

${M}_{i}$ | Number of outflow connections in ith chamber |

m | Mass of a phase/mixture |

${N}_{i}$ | Number of inflow connections in ith chamber |

N | Bubble number density |

p | Pressure in a control volume |

${p}_{v}$ | Saturation pressure of the vapor component |

${p}_{0}$ | Saturation pressure of air |

${p}_{i}$ | Initial pressure |

${p}_{f}$ | Final pressure |

${p}_{\infty}$ | Pressure away from bubble interface |

${p}_{GE}$ | Partial pressure of the gas in equilibrium |

Q | Volumetric flow rate through an orifice |

${Q}_{b}$ | Flow through the backflow groove |

${Q}_{d}$ | Drain leakages |

${Q}_{l}$ | Lateral leakages |

${Q}_{r}$ | Radial leakages |

R | Bubble radius |

t | Time |

${t}^{*}$ | Dimensionless time |

T | Kinetic energy transported by the fluid |

v | Flow velocity |

V | Volume of a chamber/control volume |

W | Work done per unit flow of mass |

## Subscripts

B | Bubble |

g | Non condensable gas component |

v | vapor phase |

l | liquid phase |

$i,j,k$ | Summation indices |

$in$ | inlet/inflow |

$out$ | outlet/outflow |

$SS$ | Steady state |

## Greek Symbols

$\alpha $ | Void fraction |

$\epsilon $ | Viscous dissipation rate |

$\lambda $ | Polytropic gas constant |

$\mu $ | Dynamic viscosity |

$\nu $ | Kinematic viscosity |

$\rho $ | Density |

$\sigma $ | Coefficient of surface tension |

$\theta $ | Angular position |

## Abbreviations

HYGESim | Hydraulic Gear Simulator |

LHS | Left Hand Side |

RHS | Right Hand Side |

EGM | External Gear Machine |

CV | Control Volume |

TSV | Tooth Space Volume |

RPE | Rayleigh–Plesset Equation |

RP | Rayleigh–Plesset |

RMS | Root Mean Squared |

CFD | Computational Fluid Dynamics |

## Appendix A. Derivation of the Compressible Orifice Equation

## References and Note

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**Figure 1.**Schematic representation of the governing equations in the conventional lumped parameter approach: Pressure built up equation and the Orifice equations.

**Figure 2.**Structure of the simulation tool HYGESim [22].

**Figure 5.**Variation of the non-dimensionalized radius of a spherical bubble with initial radius ${R}_{0}=50$ $\mathsf{\mu}$m with parameters: ${\nu}_{l}=1.3\times {10}^{-6}$ m${}^{2}$/s, ${\rho}_{l}=1000$ kg/m${}^{3}$, ${p}_{v}=-0.7$ bar gage pressure, $\sigma =72.75\times {10}^{-3}$ kg/s${}^{2}$ and ${\dot{R}}_{0}=0$ m/s, subjected to forced pressure variation shown in Figure 4.

**Figure 6.**Instantaneous and averaged radius of the bubble for a sudden pressure drop from initial pressure of 0 bar gage pressure to a final pressure of −0.7 bar gage pressure far away from the bubble interface with the parameters: ${\nu}_{l}=1.3\times {10}^{-6}$ m${}^{2}$/s, ${\rho}_{l}=1000$ kg/m${}^{3}$, ${p}_{v}=-0.7$ bar gage pressure, $\sigma =72.75\times {10}^{-3}$ kg/s${}^{2}$ and ${\dot{R}}_{0}=0$ m/s—(

**a**) endpoints of the averaging interval (marked by solid points); (

**b**) comparison between instantaneous and averaged radius.

**Figure 7.**Variation of the Total Energy with time for a reference step pressure (Equation (24)) specified by ${p}_{i}=0\phantom{\rule{4.pt}{0ex}}\mathrm{bar}$ and ${p}_{f}=-0.69\phantom{\rule{4.pt}{0ex}}\mathrm{bar}$ gage pressure with the same fluid properties as that chosen for the case shown in Figure 5.

**Figure 8.**Variation of non-dimensional energy and energy dissipation rate for different values of ${\Pi}_{p}$ while keeping ${\Pi}_{S}=0.64\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{\Pi}_{\nu}=0.0114$ constant: (

**a**) variation of ${\Pi}_{E}$ with non-dimensionalized time ${t}^{*}$; (

**b**) variation of ${\Pi}_{\epsilon}$ with non-dimensionalized time ${t}^{*}$ and (

**c**) variation of ${\Pi}_{\epsilon}$ with ${\Pi}_{E}$.

**Figure 9.**Variation of non-dimensional energy dissipation rate with dimensionless energy for (

**a**) different values of ${\Pi}_{S}$ while keeping ${\Pi}_{p}=69\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{\Pi}_{\nu}=0.0114$ constant and (

**b**) different values of ${\Pi}_{\nu}$ while keeping ${\Pi}_{p}=69\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{\Pi}_{S}=0.64$ constant.

**Figure 10.**Comparison between the numerically computed solution to the Rayleigh–Plesset Equation and solution obtained using the proposed model with instantaneous non-dimensionalization parameters in (

**b**), based on a forced linear pressure variation at a distance from the bubble interface shown in (

**a**).

**Figure 11.**Comparison between the numerically computed solution to the Rayleigh–Plesset Equation and solution obtained using the proposed model with instantaneous non-dimensionalization parameters in (

**b**), based on a forced linear pressure variation at a distance from the bubble interface shown in (

**a**).

**Figure 12.**Simplified representation of the commercial EGM unit. The flow through the backflow groove, radial and lateral gaps are denoted, respectively, by ${Q}_{b},{Q}_{r}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{Q}_{l}$.

**Figure 13.**Simplified representation of an External Gear Machine along with the CVs and orifice connections.

**Figure 14.**Scaled view of two of the TSVs corresponding to the drive gear in Figure 13 showing the radial leakage connections (${Q}_{r}$), lateral leakage connection (${Q}_{l}$), backflow groove connection (${Q}_{b}$) and drain leakage connection (${Q}_{d}$).

**Figure 16.**Comparison between instantaneous TSV pressure profiles for the Asymptotic Growth Rate model, Hybrid Rayleigh–Plesset solution approach with experimental data for 1000 rpm speed and suction pressure of $-0.06$ bar gauge pressure.

**Figure 17.**Comparison between instantaneous TSV pressure profiles for the Asymptotic Growth Rate model, Hybrid Rayleigh–Plesset solution approach with experimental data for 1500 rpm speed and suction pressure of $-0.64$ bar gauge pressure.

**Figure 18.**Comparison between instantaneous TSV pressure profiles for the Asymptotic Growth Rate model, Hybrid Rayleigh–Plesset solution approach with experimental data for 2000 rpm speed and suction pressure of $-0.72$ bar gauge pressure.

**Figure 19.**Comparison between delivery pressure ripple for the Asymptotic Growth Rate model, the Hybrid Rayleigh–Plesset Solution approach with experimental data, for an operating speed of 1000 rpm and a suction pressure of −0.06 bar gauge pressure. The modeled pressure fluctuations are within +3.5% and −0.2% while the measured data is within +2.5% and −3.1% of the reference value.

**Figure 20.**Comparison between delivery pressure ripple for the Asymptotic Growth Rate model, the Hybrid Rayleigh–Plesset Solution approach with experimental data, for an operating speed of 1500 rpm and a suction pressure of $-0.64$ bar gauge pressure. The modeled pressure fluctuations are within +0.6% and −2.7% while the measured data is within +2.8% and −3.2% of the reference value.

**Figure 21.**Comparison between delivery pressure ripple for the Asymptotic Growth Rate model, the Hybrid Rayleigh–Plesset Solution approach with experimental data, for an operating speed of 2000 rpm and a suction pressure of $-0.72$ bar gauge pressure. The modeled pressure fluctuations are within +7.5% and −0.0% while the measured data is within +4.0% and −4.5% of the reference value.

**Figure 22.**Comparison between delivery flow rates for the Asymptotic Growth Rate model and the Hybrid Rayleigh–Plesset Solution approach, for an operating speed of 1000 rpm.

Feature | Description |
---|---|

Number of teeth | 12 |

Displacement | 11.2 cc |

Hydraulic oil | ISO VG 46 mineral oil |

Suction pressure | <2 bar |

Maximum pressure | 250 bar |

Operating pressure | 500–2500 rpm |

**Table 2.**Description of the sensors used in the experiment [12].

Sensor | Type | Manufacturer | Range | Accuracy (% F.S.) |
---|---|---|---|---|

T1 | Resistive | TERSID^{®} (Milan, Italy) | (−50 °C, 200 °C) | 1.00% |

P1, P3 | Strain gauge | WIKA^{®} (Bavaria, Germany) | 0–40 bar | 0.25% |

P2 | Strain gauge | WIKA^{®} (Bavaria, Germany) | 0–400 bar | 0.25% |

P4 | Piezoelectric | KISTLER^{®} (Winterthur, Switzerland) | 0–1000 bar | 0.8% |

(quartz, 140 kHz) | ||||

P5 | Piezoresistive | ENTRAN^{®} (Ludwigshafen, Germany) | 0–350 bar | 0.5% |

(450 kHz) | ||||

Q1 | Flow meter | VSE^{®} (Neuenrade Germany) VS1 | 0.05–80 L/min | 0.3% measured value |

**Table 3.**Mean flow rate comparison between the Hybrid Rayleigh–Plesset equation model and the Asymptotic Growth Rate model.

Operating | Suction | Delivery | Hybrid RPE Model | Asymptotic Growth Rate |
---|---|---|---|---|

Speed (rpm) | Pressure (bar) | Pressure (bar) ($\mathit{Q}/{\mathit{Q}}_{\mathit{exp}}$) | Flow Rate | Model Flow Rate ($\mathit{Q}/{\mathit{Q}}_{\mathit{exp}}$) |

1000 | −0.06 | 100 | 1.06 | 1.03 |

1500 | −0.64 | 100 | 1.02 | 0.93 |

2000 | −0.72 | 150 | 1.0 | 1.0 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shah, Y.G.; Vacca, A.; Dabiri, S.
Air Release and Cavitation Modeling with a Lumped Parameter Approach Based on the Rayleigh–Plesset Equation: The Case of an External Gear Pump. *Energies* **2018**, *11*, 3472.
https://doi.org/10.3390/en11123472

**AMA Style**

Shah YG, Vacca A, Dabiri S.
Air Release and Cavitation Modeling with a Lumped Parameter Approach Based on the Rayleigh–Plesset Equation: The Case of an External Gear Pump. *Energies*. 2018; 11(12):3472.
https://doi.org/10.3390/en11123472

**Chicago/Turabian Style**

Shah, Yash Girish, Andrea Vacca, and Sadegh Dabiri.
2018. "Air Release and Cavitation Modeling with a Lumped Parameter Approach Based on the Rayleigh–Plesset Equation: The Case of an External Gear Pump" *Energies* 11, no. 12: 3472.
https://doi.org/10.3390/en11123472