# CFD Analysis on the Heat Dissipation of a Dry-Lubricated Gear Stage

^{*}

## Abstract

**:**

## 1. Introduction

^{3}< Re < 10

^{5}. In order to calculate the heat dissipation of gears, Nusselt correlations for cylinders rotating in fluid are often considered. Changenet et al. [17] adapted the approach by Lebeck [16] for calculating the heat transfer of gears rotating in oil.

## 2. Methods and Materials

#### 2.1. Object of Investigation

#### 2.2. Operating Conditions

_{oil}, the pinion bulk temperature T

_{M}

_{1}, and the wheel bulk temperature T

_{M}

_{2}for the circumferential speeds v

_{t}= {2.0; 5.0; 10.0} m/s at no-load. The material properties are considered as constant per circumferential speed. For both gear geometries gear-C and gear-LL, the same thermal boundary conditions are considered to focus on the influence of the air flow on the heat transfer.

_{air}is set to T

_{oil}. The influence of the circumferential speed is investigated by varying v

_{t}= {2.0; 5.0; 10.0} m/s. The rotational direction of the gears is visualized in Figure 1.

#### 2.3. Numerical Modeling and Calculation

^{®}Fluent 2020/R1 (ANSYS, Inc., Canonsburg, PA, USA) is used. For detailed information on the fundamentals of computational fluid dynamics, the reader refers to the specialized literature, e.g., [29,30].

#### 2.3.1. Geometry and Mesh Models

_{gear-C}(a) and Mesh

_{gear-LL}(b).

_{gear-C}has an element number of approx. 1.9 M due to the predominant discretization with hexahedral and prismatic cells. Mesh

_{gear-LL}has an element number of approx. 5.2 M due to tetrahedral elements in the gear zones and full model approach. Detailed information on the mesh models is given in Table 3.

#### 2.3.2. Governing Equations

#### 2.3.3. Finite Volume Method

#### 2.3.4. Turbulence Model

#### 2.3.5. Wall Modeling

^{+}of less than one. The dimensionless distance y

^{+}is defined by the absolute distance from the wall $y$, the friction velocity ${u}_{\tau}$, and the kinematic viscosity $\nu $:

^{+}greater than one can be used [30]. Furthermore, the latest wall models, such as the Menter–Lechner wall model, allow a y

^{+}-independent calculation of the flow in areas close to the wall. In this study, the Menter–Lechner wall-function as implemented in Ansys@ Fluent [36] is applied. In addition, a wall adhesion condition is assigned to the walls.

#### 2.3.6. Heat Transfer

_{wall}and reference temperature T

_{ref}, see Equation (5).

_{wall}and T

_{ref}, the fluid properties $\rho $ and ${c}_{p}$, the dimensionless temperature ${T}_{c}^{*}$, and dimensionless velocity u*.

#### 2.3.7. Simulation Procedure

_{gear-C}and Mesh

_{gear-LL}are developed, as described in Section 2.3.2. Third, the CFD models CFD

_{gear-C}and CFD

_{gear-LL}are derived based on the numerical modeling approach presented in Section 2.3.2, Section 2.3.3, Section 2.3.4, Section 2.3.5 and Section 2.3.6. A Newtonian fluid behavior is considered.

_{wall}refers to the temperatures T

_{M1}or, rather, T

_{M2}and the reference temperature T

_{re}

_{f}refers to the air temperature T

_{air}.

^{−5}and used for all equations. All calculations are carried out at the Leibnitz Rechenzentrum (LRZ) using 112 cores (Intel Xeon E5-2690 v3 (Intel Corporation, Santa Clara, CA, USA), clock rate 2.6 GHz, and working memory 64 GB). Within a simulated physical time of t

_{sim}= 0.5 s, a quasi-stationary state is reached with respect to the convergence of the calculated HTC within the transient CFD simulations.

## 3. Results

#### 3.1. Analysis of the Isothermal Flow Field

_{t}= {2.0; 5.0; 10.0} m/s by vectors that are colored based on the local flow velocity. Figure 5 shows the results for gear-C (a) and for gear-LL (b). It can be seen that the rotating gears drag air in the circumferential direction, especially by the teeth. The air flow around the gears also has a component in the radial direction. The results feature clear similarities to the fluid flow in a dip-lubricated gearbox investigated by Liu et al. [3] and Hildebrand et al. [22], showing oil flowing in the circumferential as well as radial direction due to inertia forces. In the gear meshing zone, the individual air flows of the pinion and wheel encounter each other, which results in an axial deflection and flow.

_{x}in the x-direction and v

_{z}in the z-direction below the wheel of gear-C and gear-LL for v

_{t}= {2.0; 5.0; 10.0} m/s. The velocities are evaluated on an evaluation line that is shown on the left exemplarily for gear-C. The vertical position on the evaluation line is represented by y’. The flow velocity v

_{x}is zero at the gearbox housing bottom and equal to the circumferential speed of the wheels at the tip circle. For both gears, v

_{x}decreases strongly in proximity of the tip circle. Then, v

_{x}is relatively constant. At v

_{t}= 10 m/s, v

_{x}shows a higher value for gear-C compared to gear-LL. Hence, gear-C generates a stronger flow field in the radial direction. The flow velocity v

_{z}is negative in proximity of the tip circle, which means a flow toward the rear of the gearbox. Near the gearbox bottom, v

_{z}is positive, meaning a flow toward the front of the gearbox. The flow velocity v

_{z}is, over wide ranges, higher for gear-LL compared to gear-C, which indicates a more pronounced conveying effect of the helical gear.

#### 3.2. Analysis of the Heat Transfer Coefficient

#### 3.2.1. Local Heat Transfer Coefficient

_{t}= {2.0; 5.0; 10.0} m/s. It is visualized by the local coloring of the gear surfaces. Figure 7a refers to gear-C and in Figure 7b to gear-LL.

#### 3.2.2. Influence of Flow Characteristics

_{t}= 10.0 m/s.

#### 3.2.3. Surface-Averaged Heat Transfer Coefficient

_{t}= {2.0; 5.0; 10.0} m/s. They show an increase with v

_{t}. The tooth surfaces exhibit the highest values, followed by the side surfaces and the shaft surfaces. Thus, higher heat transfer coefficients are present for surfaces that lie further out on the gear in the radial direction. The comparison of gear-C and gear-LL shows, for the shaft and side surfaces, comparable surface-averaged HTC values for all circumferential speeds. In contrast, the surface-averaged HTC values for the tooth surfaces of gear-LL are significantly higher. This is in agreement with Section 3.2.2, according to which the helical gear-LL shows a more pronounced incident flow of the teeth.

_{t}= {2.0; 5.0; 10.0} m/s. For gear-LL, higher surface-averaged HTC values result for a given circumferential speed compared to gear-C. Figure 11 also shows the calculated HTC values based on the Nusselt correlation according to Becker [15], see Equations (2) and (3). The tip diameter was chosen as the reference length in accordance with Changenet et al. [17]. The underlying Reynolds number is within the given validity range of 10

^{3}< Re < 10

^{5}. The analytical results predict the magnitude and trends over the circumferential speed in good agreement with the simulation results. With increasing circumferential speed, larger deviations can be noticed. Based on the Nusselt correlation, comparable HTC values are calculated for gear-C and gear-LL: The surface-averaged HTC value of gear-LL for the different circumferential speeds is only about 2.5% lower than that of gear-C, due to the smaller tip diameters of gear-LL. This demonstrates that the influence of the gear geometry and fluid flow on the HTC cannot be easily addressed by simple correlation analysis.

_{t}= 2.0…10.0 m/s, the surface-averaged HTC for the considered dry-lubricated gears can be classified in the range of $\overline{h}$ = 20…50 W/(m

^{2}K). This correlates with the magnitude of the HTC for solids and forced convection in gases specified by the VDI Heat Atlas [38]. The results of Lu et al. [14] showed that the HTC between gears and oil for a circumferential velocity of about v

_{t}= 30 m/s is of the magnitude of $\overline{h}$ = 400 W/m

^{2}K.

## 4. Conclusions

- A circulating flow is formed in the gearbox due to the displacement of air by the gears and backflow to the gears.
- Spur gears show a distinct radial displacement and a symmetrical axial backflow of air to the teeth. Helical gears show a distinct axial air flow due to conveying effects.
- High heat transfer coefficients are particularly present on surfaces that interact strongly with air, e.g., on the leading tooth flanks.
- Surface-averaged heat transfer coefficients show higher values for helical gears compared to spur gears.
- The simplified Nusselt correlation can predict the order of magnitude of the simulated heat transfer coefficient as well as its trend over the circumferential speed.
- The specific influence of gear geometry and the fluid flow on the heat transfer coefficient cannot be addressed by simple correlation analysis.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Notation | ||

$\stackrel{\rightharpoonup}{\mathrm{f}}$ | Forces | N |

$\stackrel{\rightharpoonup}{\mathrm{n}}$ | Vector orthogonal to surface | - |

${\mathrm{q}}_{\mathsf{\varphi}}$ | Source or sink of $\mathsf{\varphi}$ | - |

$\stackrel{\rightharpoonup}{\mathrm{v}}$ | Fluid velocity | m/s |

a | Center distance | mm |

b | Tooth width | mm |

D | Reference length | m |

d_{a} | Tip diameter | mm |

h | Heat transfer coefficient│Energy | W/(m²K)│J |

m_{n} | Normal module | mm |

Nu | Nusselt number | - |

Pr | Prantl number | - |

Re | Reynolds number | - |

S | Surface | m^{2} |

t | Time | s |

T | Temperature | K |

T_{M} | Bulk temperature | K |

u* | Dimensionless velocity | - |

u_{τ} | Friction velocity | m/s |

V | Volume | m^{3} |

v_{t} | Circumferential speed | m/s |

x | Coordinate axis | m |

y | Distance from wall│Coordinate axis | m│m |

y^{+} | Dimensionless distance | - |

z | Tooth number│Coordinate axis | -│m |

α | Pressure angle | ° |

β | Helix angle | ° |

λ | Thermal conductivity | W/(mK) |

ν | Kinematic viscosity | mm^{2}/s |

ρ | Density | kg/m^{3} |

$\mathsf{\varphi}$ | Generic quantity | - |

Indices | ||

wall | Wall | |

ref | Reference | |

oil | Oil | |

x | Direction of x coordinate axis | |

y | Direction of y coordinate axis | |

z | Direction of z coordinate axis | |

sim | Simulation | |

1 | Pinion | |

2 | Wheel |

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**Figure 5.**Isothermal flow fields in 3D view and top view for v

_{t}= {2.0; 5.0; 10.0} m/s based on a steady-state calculation for gear-C (

**a**) and gear-LL (

**b**).

**Figure 6.**Flow velocities v

_{x}and v

_{y}below the wheel of gear-C and gear-LL for v

_{t}= {2.0; 5.0; 10.0} m/s.

**Figure 7.**Heat transfer coefficient h between the gear surfaces and air for v

_{t}= {2.0; 5.0; 10.0} m/s in 3D and top view for gear-C (

**a**) and for gear-LL (

**b**).

**Figure 8.**Axial flow velocity (z-direction) on a plane congruent to the side surface for gear-C (

**a**) and for gear-LL (

**b**) at circumferential speed v

_{t}= 10.0 m/s.

**Figure 9.**Detailed view on axial flow velocity (z-direction) and HTC between gear surface and air for the wheels of gear-C (

**a**) and gear-LL (

**b**) for v

_{t}= 10.0 m/s.

**Figure 10.**Surface-averaged HTC $h$ for shaft, side, and tooth surfaces for the wheel of gear-C and gear-LL for circumferential speeds v

_{t}= {2.0; 5.0; 10.0} m/s.

**Figure 11.**Surface-averaged HTC $\overline{h}$ for gear-C and gear-LL for circumferential speeds v

_{t}= {2.0; 5.0; 10.0} m/s in comparison to HTC based on Nusselt correlation according to Becker [15].

gear-C | |||||||
---|---|---|---|---|---|---|---|

a in mm | z_{1|2} | m_{n} in mm | α_{n} in ° | β in ° | b_{1|2} in mm | d_{a1|2} in mm | |

Pinion (1) | 91.5 | 16 | 4.5 | 20.0 | 0 | 14 | 82.5 |

Wheel (2) | 24 | 118.4 | |||||

gear-LL | |||||||

a in mm | z_{1|2} | m_{n} in mm | α_{n} in ° | β in ° | b_{1|2} in mm | d_{a1|2} in mm | |

Pinion (1) | 91.5 | 24 | 2.75 | 30.0 | 26 | 20 | 78.2 |

Wheel (2) | 36 | 114.1 |

**Table 2.**Measured oil and component temperatures for different speeds at no-load [21].

v_{t} in m/s | T_{oil} in K | T_{M1} in K | T_{M2} in K |
---|---|---|---|

2.0 | 303.25 | 304.75 | 304.85 |

5.0 | 306.35 | 309.05 | 308.85 |

10.0 | 313.55 | 318.15 | 317.55 |

**Table 3.**Properties of the considered mesh models Mesh

_{gear-C}and Mesh

_{gear-LL}(prism.: prismatic; hex.: hexahedral; tet.: tetrahedral).

Mesh Model | Zones | Element Size in mm | Element Number in M | Density in Elements per mm³ |
---|---|---|---|---|

Mesh_{gear-C} | (1) Gear zones | 0.75 (prism./hex.) | 222,373 | 1.47 |

(2) Cylindrical zones | 1.00 (prism./hex.) | 784,795 | 2.56 | |

(3) Transition zones | 1.00 (hex.) | 79,016 | 1.52 | |

(4) Outer zone | 1.00 (hex.) | 810,488 | 1.01 | |

Overall Model | 0.92 | 1.89 | 1.33 | |

Mesh_{gear-LL} | (1) Gear zones | 0.75 (tet.) | 1,078,204 | 6.17 |

(2) Cylindrical zones | 1.00 (prism./hex.) | 2,291,204 | 2.42 | |

(3) Transition zones | 1.00 (hex.) | 102,866 | 1.52 | |

(4) Outer zone | 1.00 (hex.) | 1,744,630 | 1.01 | |

Overall Model | 0.92 | 5.21 | 2.45 |

**Table 4.**Specifications of the prototypical conservation equation [33].

Equation | $\mathit{\varphi}$ | ${\mathit{D}}_{\mathit{\varphi}}$ | ${\mathit{Q}}_{\mathit{\varphi}}$ |
---|---|---|---|

Mass | 1 | 0 | 0 |

Impulse | $\stackrel{\rightharpoonup}{v}$ | $\nabla \xb7\tau $ | $-\nabla p+\rho \stackrel{\rightharpoonup}{g}$ |

Energy | h | $-\nabla \xb7{\stackrel{\rightharpoonup}{q}}^{\u2033}$ | $\frac{\partial p}{\partial t}+\nabla \xb7(\tau \xb7\stackrel{\rightharpoonup}{v})$ |

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## Share and Cite

**MDPI and ACS Style**

Hildebrand, L.; Dangl, F.; Paschold, C.; Lohner, T.; Stahl, K.
CFD Analysis on the Heat Dissipation of a Dry-Lubricated Gear Stage. *Appl. Sci.* **2022**, *12*, 10386.
https://doi.org/10.3390/app122010386

**AMA Style**

Hildebrand L, Dangl F, Paschold C, Lohner T, Stahl K.
CFD Analysis on the Heat Dissipation of a Dry-Lubricated Gear Stage. *Applied Sciences*. 2022; 12(20):10386.
https://doi.org/10.3390/app122010386

**Chicago/Turabian Style**

Hildebrand, Lucas, Florian Dangl, Constantin Paschold, Thomas Lohner, and Karsten Stahl.
2022. "CFD Analysis on the Heat Dissipation of a Dry-Lubricated Gear Stage" *Applied Sciences* 12, no. 20: 10386.
https://doi.org/10.3390/app122010386