# Thermal Modelling of External Gear Machines and Experimental Validation

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## Abstract

**:**

## 1. Introduction

## 2. Thermal-HYGESim Model

#### 2.1. Fluid Dynamic Module

#### 2.1.1. Pressure and Temperature Evaluation in CVs

#### 2.1.2. Mass Flow between the CVs

#### 2.1.3. Enthalpy Flow between the CVs

#### 2.2. Loading and Gear Micromotion Module

#### 2.2.1. Effect of Fluid Pressure

#### 2.2.2. Effect of Friction at the Tooth Tip

#### 2.2.3. Effect of Friction at the Lateral Leakage Interface

#### 2.2.4. Effect of Friction at the Drain Leakage Interface

#### 2.2.5. Contact and Friction Force at the Meshing Interface

#### 2.2.6. Power Losses Due to Friction

#### 2.2.7. Gear Micromotion

#### 2.3. Geometrical Module

## 3. Results and Experimental Validation

#### 3.1. Simulation Setup

#### 3.2. Key Results from Thermal-HYGESim Model

- Lateral leakage flow from the lagging TSV, i.e., from ${V}_{i-1}$ to ${V}_{i}$ in Figure 13c. The fluid in ${V}_{i-1}$ has higher pressure and temperature, and thus, has higher specific enthalpy as compared to the fluid in ${V}_{i}$. Thus, the lateral leakage flow carries the high specific enthalpy fluid to a low specific enthalpy environment raising the temperature of the latter. This effect is mathematically captured by the terms $\sum}_{i}{\dot{H}}_{i}-h{\displaystyle \sum}_{i}{\dot{m}}_{i$ in the temperature rise equation (Equation (2)). From Figure 13b, the leakage enthalpy flow ${\dot{H}}_{i-1,i}$ is higher than the product of the specific enthalpy of the fluid in ${V}_{i}$ and the leakage mass flow rate (${h}_{i}\xb7{\dot{m}}_{i-1,i}$). Thus, the aforementioned terms in the temperature rise equation yield a positive value promoting $dT/dt$.
- Friction heating due to the meshing of the gear teeth (black curve in Figure 13b). In the temperature rise equation (Equation (2)), this factor appears as a positive $\dot{Q}$ promoting $dT/dt$.

#### 3.3. Heat Loss to the Environment

^{2}$\Rightarrow {\dot{Q}}_{loss}=1.24$ W. This is negligible compared to the circumferential enthalpy flow $=\dot{m}h=2674$ watts (at $1000$ rpm). Similarly, the heat flux in the axial direction (at the lateral lubrication interface) is ${q}^{\u2033}\approx 4.6\times {10}^{-3}$ W/mm

^{2}. Thus, the heat losses over the area of lateral and drain leakage interface (for single tooth) are $6.4\times {10}^{-2}$ W and $4.7\times {10}^{-2}$ W, respectively. These values are insignificant compared to the levels of enthalpy flow rates at these interfaces (Figure 15).

#### 3.4. Experimental Validation of Thermal-HYGESim Model

#### 3.5. Significance of Thermal-HYGESim Model

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$b$ | Width of the leakage gap |

${C}_{f}$ | Orifice flow coefficient |

${c}_{p}$ | Specific heat of the fluid |

$e$ | Unit vector |

$F,f$ | Force |

$\dot{H}$ | Enthalpy flow rate |

$h$ | Specific enthalpy of the fluid |

$\hslash $ | Leakage gap height |

${K}_{T}$ | Isothermal bulk modulus |

$L$ | Length of the leakage gap |

$l$ | Distance of the line of action of forces from the gear center |

$M$ | Moment |

$m$ | Mass |

$\dot{m}$ | Mass flow rate |

$N$ | Shaft speed |

$n$ | Number of ports in a generic CV |

$p$ | Pressure |

$\dot{Q}$ | Heat flow rate |

${q}^{\u2033}$ | Heat flux |

$r$ | Radius |

$T$ | Temperature |

$t$ | Time |

$u$ | Internal energy |

$V$ | Volume |

$v$ | Relative velocity of the two surfaces at the leakage interface |

$\dot{W}$ | Rate of work done |

$x,y$ | Cartesian directions |

Subscripts | |

$f$ | Friction |

$d,dr$ | Drain |

$g$ | Gear |

$i,j$ | Iterators |

$in$ | Inlet |

$out$ | Outlet |

$m$ | Meshing |

$p$ | Pressure |

$r$ | Root |

$t$ | Tip |

$l$ | Lateral |

Greek letters | |

${\alpha}_{p}$ | Isobaric thermal expansion coefficient |

$\delta $ | Conicity error |

$\u03f5$ | Concentricity error |

${\eta}_{v}$ | Volumetric efficiency |

$\theta $ | Angular position |

$\mu $ | Viscosity of the fluid |

$\rho $ | Density of the fluid |

$\tau $ | Shear stress |

$\mathsf{\Omega}$ | Area |

$\omega $ | Rotational velocity of the gear |

Abbreviations | |

CV | Control Volume |

EGM | External Gear Machine |

EHL | Elastohydrodynamic lubrication |

TSV | Tooth Space Volume |

## Appendix A. Derivation of Pressure Build up Equation

## Appendix B. Derivation of Temperature Rise Equation

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**Figure 2.**Operation of EGMs and leakage flows. (

**a**) front view, where the color of the fluid volume qualitatively indicates the level of fluid pressure in the pumping action; (

**b**) side view, showing the drain leakage path in the reference EGM.

**Figure 3.**Effect of temperature on viscosity for the fluid taken as reference in this research (ISO VG 46 mineral oil).

**Figure 6.**Schematic of a hydraulic resistor connecting two lumped volumes at pressures ${p}_{i}$ and ${p}_{j}$ and temperatures ${T}_{i}$ and ${T}_{j}$.

**Figure 7.**Forces acting on the gears. Force from fluid pressure on one TSV is shown: ${f}_{x}$ and ${f}_{y}$ are the forces and ${\mathsf{\Omega}}_{yz}$ and ${\mathsf{\Omega}}_{xz}$ are the projection areas of the TSV in the cartesian directions. ${F}_{c}$ and ${F}_{f}$ are contact and friction forces, respectively.

**Figure 9.**Differential slices for the evaluation of shear stresses in lateral leakage and drain leakage interfaces. Green arrows show the flow directions.

**Figure 11.**Temperatures at the inlet, outlet and drain CVs of the EGM obtained from the simulation setup in Figure 10 with $N=1000$ rpm.

**Figure 12.**(

**a**) Pressure and temperature variation in a TSV of the drive gear over two shaft revolutions; (

**b**) Gap height over the tip of the leading tooth of a TSV of the drive gear. For $\theta <73\xb0$ and $\theta >292\xb0$, the tooth tip does not form a leakage interface with the casing, so, the gap height is not defined for that range.

**Figure 13.**(

**a**) TSV temperature in the meshing region; (

**b**) Plot of different terms in the temperature rise equation (Equation (2)). ${\dot{H}}_{i-1,i}$ and ${\dot{m}}_{i-1,i}$ are the leakage enthalpy and mass flow rates, respectively, from ${V}_{i-1}$ to ${V}_{i}$. ${h}_{i}$ is the specific enthalpy of the fluid in ${V}_{i}$; (

**c**) Illustration of the lateral leakage flow (green arrow) from TSV, ${V}_{i-1}$, to the TSV under investigation, ${V}_{i}$. The color of CVs are the qualitative indicators of the fluid pressure.

**Figure 14.**(

**a**) CAD of the reference EGM (end cover removed to show the interior domain) and the mounting plate used in the steady state thermal analysis. (

**b**) Heat flux at the interior surfaces obtained from the results of thermal simulation in ANSYS.

**Figure 15.**Enthalpy flow rates at the lateral and drain leakage interfaces for one tooth over its full revolution. The results are obtained from the simulation of the reference EGM at $N=1000$ rpm, ${p}_{out}=150$ bar, ${T}_{in}=35\mathbb{C}$.

**Figure 18.**Illustration of (

**a**) conicity error and (

**b**) concentricity error in gears: ${O}_{S}$ and ${O}_{G}$ are shaft and gear centers respectively.

**Figure 19.**Comparison of the volumetric efficiency predicted by the simulations (Thermal HYGESim and traditional isothermal HYGESim) and evaluated from flow measurements in the experiments for all six units.

**Figure 20.**Comparison of the outlet temperature predicted by the simulations and the outlet temperature measured in the experiments for all six units.

**Figure 21.**Three sets of simulations: (

**a**) Isothermal simulation setup with ${T}_{sim}={T}_{tank}$; (

**b**) Isothermal simulation setup with ${T}_{sim}=0.5\left({T}_{tank}+{T}_{out}\right)$; (

**c**) Thermal simulation setup with ${T}_{in}={T}_{tank}$. The symbols in red are the outputs of the simulations.

**Figure 22.**Comparison of the results in terms of the volumetric efficiency from three sets of simulations, considering the case of unit 3 of Table 3.

EGM Parameter | Value |
---|---|

Geometric displacement | 3.25 cc/rev |

Teeth | 11 |

Gear diameter (nominal) | 29.58 mm |

Gear face width (nominal) | 9.5 mm |

Bearing type | Needle |

Sensor | Model | Specifications | Accuracy |
---|---|---|---|

Pressure transducer | Wike type S-10 | 0–400 bar | $\pm 0.5\%$ of span |

Flow meter | VSE VS 02 | 0.1–120 L/min | $\pm 0.3\%$ |

Thermocouple | Omega K-type | −5–200 $\mathbb{C}$ | $\pm 1\mathbb{C}$ |

Speed sensor | HBM T10FS | 0–15,000 rpm | $\pm 3$ rpm |

**Table 3.**Effective gap heights at the tooth tips of the gears in the six units used in the experimental tests.

Unit | Drive Gear Conicity Error (µm) | Driven Gear Conicity Error (µm) | Drive Gear Concentricity Error (µm) | Driven Gear Concentricity Error (µm) | Effective Gap Height (µm) |
---|---|---|---|---|---|

1 | 16.5 | 11.5 | 1 | 0.5 | 48.3 |

2 | 15 | 10 | 10 | 10 | 48.9 |

3 | 1.5 | 5 | 20 | 20.5 | 49.2 |

4 | 20 | 20.5 | 1 | 0.5 | 46.7 |

5 | 0.5 | 0.5 | 10.5 | 10 | 39.5 |

6 | 0.5 | 0.5 | 1 | 4.5 | 30.8 |

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## Share and Cite

**MDPI and ACS Style**

Rituraj, R.; Vacca, A.; Morselli, M.A.
Thermal Modelling of External Gear Machines and Experimental Validation. *Energies* **2020**, *13*, 2920.
https://doi.org/10.3390/en13112920

**AMA Style**

Rituraj R, Vacca A, Morselli MA.
Thermal Modelling of External Gear Machines and Experimental Validation. *Energies*. 2020; 13(11):2920.
https://doi.org/10.3390/en13112920

**Chicago/Turabian Style**

Rituraj, Rituraj, Andrea Vacca, and Mario Antonio Morselli.
2020. "Thermal Modelling of External Gear Machines and Experimental Validation" *Energies* 13, no. 11: 2920.
https://doi.org/10.3390/en13112920