A Parametric Study of Oil-Jet
Lubrication in Gear Wheels
TQFT30 -Master Thesis Report
Dona Biju (donbi027)
Link¨oping University Division of Applied Thermodynamics and Fluid Mechanics
Link¨oping University Department of Management and Engineering Division of Applied Thermodynamics and Fluid Mechanics Master Thesis 2018|LIU-IEI-TEK-A–18/03169-SE
A Parametric Study of Oil-Jet
Lubrication in Gear Wheels
TQFT30- Master Thesis Report
Dona Biju (donbi027)
Academic supervisor: Roland G˚ardhagen
Industrial supervisors: Henrik Hagerman, Erik Nordlander Examiner: Hossein Nadali Najafabadi
Link¨oping University SE-581 83 Link¨oping, Sweden
Abstract
A parametric study of oil-jet lubrication in gear wheels is conducted using Computa-tional Fluid Dynamics (CFD) to study the effect of the different design parameters on the cooling performance in a gearbox. Flow in oil jet lubrication is found to be complex with the formation of oil ligaments and droplets. Various hole radii of 1.5, 2 and 2.5 mm along with five oil velocities is analyzed and it is found that at lower volumetric rates, velocity has more effect on the cooling and at higher volu-metric rates, hole size has more effect on the cooling. At higher velocities, the heat transfer is much greater than the actual heat production in the gear wheel, hence these velocity ratios are considered less suitable for jet lubrication. At low velocity ratios of below 2, the oil doesn’t fully impinge the gear bottom land and the sides leading to low cooling. Based on the cooling, impingement length and amount of oil lost to the casing surface, 2 mm hole with a velocity ratio of 2.225 is selected for a successful oil jet lubrication. Varying the inlet position in X, Y and Z directions (horizontal, vertical and lateral respectively) is found to have no improvement on the cooling. Making the oil jet hit the gear wheel surface at an angle is found to increase the cooling. Analysis with the use of a pipe to supply oil was conducted with circular and square inlet and it was found that the heat transfer decreases in both cases due to the splitting of oil jet caused by the combination of the effects of high pressure from the pipe and vorticity in the air field. A method has been developed for two gear analysis using overset meshes which can be used for further studies of jet lubrication in multi-gear systems. Single inlet is found to be better for cooling two gear wheels as it would require a reduced volumetric flow rate compared to double inlets. Oil system requirements for jet lubrication was studied and it was concluded that larger pumps have to be used to provide the high volumetric rates and highly pressurized oil required. On comparing the experimental losses from dip lubrication and the analytical losses for jet lubrication, dip lubrication is found to have lesser loses and more suitable for this case. Good quality lubrication would reduce the fuel consumption and also increase the longevity of gearboxes and hence more research into analyzing alternate lubrication systems can be carried out using the results from this thesis.
Acknowledgements
The thesis has been carried out at ‘Powertrain engineering , Drivelines and Hybrids Division’ at Volvo Group Trucks technology in G¨oteborg. I would like to thank my manager Erik Nordlander for giving me the opportunity to conduct this thesis at Volvo Group. I would like to express my deepest gratitude to my supervisor Henrik Hagerman for his undying support and enthusiastic encouragement during this re-search. His ceaseless stream of splendid ideas and kindness has been an inspiration for me throughout the thesis.
I would like to thank my academic supervisor Roland G˚ardhagen and examiner Hossein Nadali Najafabadi from Link¨oping University for their valuable inputs for this work. I would also like to convey my appreciation to my opponent Neeti Shetty for her help with the report and presentation.
I would also like to convey my thanks to Jonathan Scott and Roman Thiele from STAR Support and Nahidh Sahrif from Volvo for their help with various problems encountered during the thesis. A special thanks to all my colleagues in Volvo Driv-eline Division for their support and for making my time there memorable and fun. Last but never the least, I would like to thank my family for always being there for me in whatever I do.
Dona Biju 2018-05-25 G¨oteborg
Nomenclature
Abbreviations and Acronyms
Abbreviation Meaning
CFD Computational Fluid Dynamics LiU Link¨oping University
MRF Moving Reference Frame Oh Ohnesorge Number Pr Prandtl Number
RANS Reynolds Averaged Navier Stokes RB Rayleigh-Benard
RBM Rigid Body Motion Re Reynolds Number RKE Realizable k-
SST Shear Stress Transport VoF Volume of Fluid We Weber Number
Latin Symbols
Symbol Description Units
Cp Specific heat Jkg−1K−1
D Diameter [m]
dv Small volume m3
F Force [N ]
g Acceleration due to gravity ms−2 k Turbulent kinetic energy Jkg−1 m. Mass flow rate kgs−1
p Pressure [P a]
Q Volumetric flow rate m3s−1
t Time [s]
U Velocity ms−1
u∧ Averaged velocity ms−1
V Volume m3
y+ Dimensionless wall distance [−]
Greek Symbols
Symbol Description Units
α Averaged effective density kgm−3
Void fraction [−]
Symbol Description Units ν Kinematic eddy viscosity m−2
φ Flux kgm−2s−1
ρ Density kgm−3
σ Surface tension coefficient N m−1
τ Shear stress N m−2
Subscripts and superscripts
Abbreviation Meaning i Interfacial
j Spatial sub-script
Contents
1 Introduction 1 1.1 Background . . . 1 1.2 Aim . . . 2 1.3 Limitations . . . 2 1.4 Literature Study . . . 3 2 Theory 5 2.1 Computational Fluid Dynamics (CFD) . . . 52.2 Multiphase Flow Modeling . . . 5
2.3 Turbulence Modeling . . . 7
2.3.1 RANS Turbulence Model . . . 7
2.3.2 SST k-ω Turbulence Model . . . 8
2.3.3 Gamma Transition Model . . . 8
2.4 Motion Modeling . . . 8
2.4.1 Moving Reference Frame Method . . . 9
2.4.2 Rigid Body Motion . . . 9
2.5 Jet Flow Regimes . . . 9
3 Method 12 3.1 Model and Domain . . . 12
3.2 Mesh . . . 13
3.2.1 Mesh Sensitivity Study . . . 15
3.3 Solver Settings . . . 16
3.3.1 Initialization with MRF Model . . . 17
3.4 Analysis with Variation in Hole Radius and Jet Velocity . . . 18
3.5 Laminar Study . . . 19
3.6 Analysis with Variation in Hole Position in X and Y Directions . . . 20
3.7 Analysis with Variation in Hole Position in Z Direction . . . 20
3.8 Analysis with Variation in Jet Angles . . . 21
3.9 Analysis with Variation in Gear Speed . . . 22
3.10 Pipe Analysis . . . 22
3.11 Analysis with Two Gear Wheels . . . 23
4 Results and Discussion 26 4.1 Initialization with MRF Model . . . 26
4.2 Basic Analysis . . . 26
4.3 Analysis with Variation in Hole Radius and Jet Velocity . . . 29
4.4 Laminar Analysis . . . 32
4.5 Analysis with Variation in Inlet Position in X and Y Direction . . . 32
4.6 Analysis with Variation in Inlet Position in Z Direction . . . 33
4.7 Analysis with Variation in Inlet Angles . . . 34
4.8 Analysis with Variation in Gear Speed . . . 35
4.9 Pipe Analysis . . . 36
4.11 Oil-System Requirements . . . 40
4.12 Dip Vs Jet Lubrication System . . . 43
5 Conclusion 45 5.1 Further Work . . . 45
5.2 Perspective . . . 46
Appendices 49 A Appendix 49 A.1 Heat Transfer for Hole Radius 2 mm . . . 49
A.2 Dimensionless Numbers . . . 49
A.3 Pressure Loss in Jet Lubrication . . . 50
List of Figures
1 Jet Breakup Regimes for the Various Hole Radii and Velocities . . . 10
2 (a) AT2612F Gearbox [1], (b) The Third Gear Wheel, (c) The Cleaned Model from ANSA . . . 12
3 (a) The Computational Domain from STAR CCM+, (b) The XY Plane showing the Dimensions of the Model . . . 13
4 The XY Plane showing the Mesh, where section A shows the refine-ments in sliding region and section B shows the refinement for oil jet from inlet . . . 13
5 The y+ Contour in the Gear Wheel Surface . . . 14
6 Temperature Contour . . . 14
7 Mesh Sensitivity Study . . . 15
8 Boundary Conditions . . . 16
9 XY Plane showing Mesh for MRF Method . . . 18
10 XY plane showing the variation of inlet hole position in X direction 20 11 Computational domain showing the variation of inlet hole position in Z Direction (a) 4 mm to the left and (b) 4 mm to the right . . . 21
12 XY Plane showing the variation of inlet hole angles from the vertical (a) 13◦ to the left, (b) 11.3◦ to the left, (c) 7◦ to the right and (d) 9◦ to the right . . . 21
13 Computational domain showing the dimensions and boundary condi-tions of the pipe analysis . . . 22
14 XY Plane showing the mesh for pipe analysis . . . 23
15 (a) Model from ANSA and (b) Computational Domain from STAR CCM+ for Two Gear Analysis . . . 24
16 XY Plane showing the dimensions of the domain for Two Gear Wheel Analysis . . . 24
17 XY Plane showing the mesh (after hole cutting) for Two Gear Analy-sis, where section A shows the refinements in the background domain for the oil jet and around the gears. Section B shows the mesh in the intermeshing region of the gear wheels. . . 25
18 (a) Velocity Field (m/s) of Air from Single Phase MRF Analysis, (b) Turbulent Kinetic Energy(J/kg) contour of the MRF Analysis . . . . 26
19 Contour of Volume Fraction of Oil for the Basic Model . . . 27
20 Oil flow in the domain colored by temperature . . . 27
21 Heat Transfer at the Gear Wheel Surface (W) for the Basic Model . 28 22 (a) Same as Section A of Figure 21, (b) Contour of Volume Fraction of Oil as the oil jet falls on the bottom land of gear wheel, (c) Contour of Volume Fraction of Oil as the oil jet falls on the top land of gear wheel . . . 28
23 Heat Transfer Contour across the gear surface . . . 29
24 Velocity of Jet (m/s) Vs Heat Transfer at Gear Wheel Surface (W) for different hole radius . . . 30
25 Volume Fraction of Oil at Velocity Ratios of (a) 0.5 and (b) 2.25 for a hole radius of 2 mm . . . 31
26 Contour of Volume Fraction of Oil for variation in inlet position in X direction in (a) 75 mm to the left (b) 75 mm to the right . . . 33 27 Heat Transfer Coefficient at the Gear Wheel Surface for Inlet Hole
Positions in Z directions at (a) Center of Gear Thickness (b) 4 mm to the left (c) 4 mm to the right . . . 34 28 Contours of volume fraction of oil for the analysis with variation of
inlet hole angles from the vertical (a) 13◦ to the left, (b) 11.3◦ to the left, (c) 7◦ to the right and (d) 9◦ to the right . . . 35 29 Contour of Volume Fraction of Oil for Pipe Analysis with circular
inlet at (a) XY Plane (b) YZ Plane . . . 36 30 Contour of Velocity rendered on the oil jet that hits the gear wheel
surface. The oil is moving in downward direction. . . 36 31 Contour of Velocity of oil jet and air (vectors) at a plane where the
oil starts to be squeezed . . . 37 32 Contour of Velocity of oil jet and air (vectors) at a plane where the
jet splits . . . 37 33 Contour of Velocity of oil jet that hits the gear wheel surface for a
square inlet at (a) XY Plane (b) YZ Plane . . . 38 34 Contour of Velocity rendered on the oil jet that hits the gear wheel
surface for a square inlet. The oil is moving in downward direction. . 38 35 Contour of Volume fraction of Oil for analysis of two gears . . . 39 36 (a) Side View and (b) Front View of oil system requirements for 10
Inlets . . . 41 37 (a) Side View and (b) Front View of oil system requirements for 4 Inlets 42 38 (a) Side View and (b) Front View of oil system requirements for 5 Inlets 42 39 (a) Side View and (b) Front View of oil system requirements for 2 Inlets 43 40 Heat Transfer at the gear surface for all the five velocities for a 2 mm
radius hole . . . 49 41 Variation of Oil Properties with Temperature for Chevron Multi-gear
MTF HD 97307, (a) Density, (b) Specific Heat, (c) Thermal Conduc-tivity . . . 51
List of Tables
1 Reynolds Numbers for each Hole Radius (Refer to Table 7 in section
3.4 for corresponding velocities) . . . 7
2 Weber Numbers for each Hole Radius(Refer to Table 7 in section 3.4 for corresponding velocities) . . . 10
3 Ohnesorge Numbers for each Hole Radius . . . 10
4 Mesh Sensitivity Study . . . 15
5 Velocity Ratios chosen for each Hole Radius . . . 19
6 Volumetric Flow Rates chosen for each Hole Radius . . . 19
7 Jet Velocities chosen for each Hole Radius . . . 19
8 Heat Transfer for Laminar Study . . . 32
9 Heat Transfer for Analysis with variation in inlet position in X direc-tion . . . 32
10 Heat Transfer for Analysis with variation in inlet position in Z direc-tion . . . 33
11 Heat Transfer for Analysis with variation in inlet angles . . . 34
12 Heat Transfer for Analysis with variation in gear speed . . . 35
13 Heat Transfer for Pipe Analysis . . . 38
14 Heat Transfer for Two Gear Wheel Analysis . . . 40
1
Introduction
The main aim of this thesis is to conduct a parametric study of oil-jet lubrication in helical gear wheel using Computational Fluid Dynamics (CFD). The effect of the different design parameters on the cooling performance is analyzed using STAR CCM+. The cooling can be optimized by iteratively tuning the various parameters through simulations.
The thesis has been carried out at ‘Powertrain Engineering, Drivelines and Hybrids Division’ at Volvo Group Trucks Technology, G¨oteborg in 2018. Volvo Group is one of the world’s leading manufacturers of trucks, buses, construction equipments and marine and industrial engines.
1.1
Background
Gears form an important aspect of transmission system in vehicles to transmit power with minimal losses. They can be of spur gear, helical gear, bevel gear, worm gear etc depending on the geometry and application. When gear wheels rotate with high speed, heating takes place in the surface of gear wheels due to friction. Lubrication is necessary in transmission systems to increase the life of gear wheels by reducing the tooth wear and surface fatigue that occur due to the friction of the mating parts and also to provide cooling to the gear wheel surfaces.
In the present scenario, transmission manufacturers mostly use dip lubrication sys-tem, where the gear system is partially immersed in transmission oil. Although this method provides substantial lubrication and cooling, the losses (specifically, churn-ing losses due to the shear force and pressure that acts on the gear teeth due to its movement in the fluid) are high. This is where jet lubrication comes into play. In jet lubrication system, the oil is impinged to the gear tooth surface from a hole, thus maintaining only the required amount of oil for sufficient lubrication and cooling, which in turn would allow for reducing the losses in the gearbox. The reduction of these load independent power losses in transmissions can decrease the fuel con-sumption of vehicles. In the future, oil jet lubrication in gear wheels may reduce or eliminate the need for dip lubricated systems and thus companies are now looking into designing jet lubrication systems for gearboxes so as to improve the efficiency of components, be fuel-efficient and also to be competitive in the market.
CFD analysis of gear wheels is a quite recent and growing approach to analyzing the flow in gear systems. This is because until recently, CFD codes didn’t possess the meshing capability to model the motion of gear wheels and also the solver capability to analyze the complex multiphase flows such as those in gearboxes. However, the recent advancement in CFD softwares in these fields has made these analysis viable. Some research has been conducted in analyzing dip lubrication using CFD, but jet lubrication hasn’t been looked into in Volvo. Some of the literature reviewed have conducted CFD analysis on jet lubrication and one of them have done an
experi-mental analysis of jet lubrication too (as explained in section 1.4).
A parametric study of the oil jet lubrication would provide details about how much the different parameters like oil velocity, hole diameter and jet angle would vary so as to provide the best cooling performance for the gear wheel. An important challenge in doing this would be the limited amount of volumetric flow rates that is available for jet lubrication due to the restricted space to increase the size of the oil pump. Based on this, a jet lubrication system that best caters to the cooling performance of the gear wheel can be designed. The method developed in this thesis can also be applied to jet lubrication in gear wheels used in other applications, provided the model is changed.
1.2
Aim
A parametric study of oil-jet lubrication in gear wheels is analyzed by carrying out the following objectives:
• By implementing a parametrized model in STAR CCM+ for the analysis and thus developing a method for jet lubrication analysis using this software. • By analyzing the influence of variation of oil jet velocity and hole diameter on
the cooling performance. For each hole diameter, a set of jet velocities will be evaluated in terms of their cooling performance. For a particular volumetric rate, the combination of hole diameter and jet velocity that gives the best cooling also has to be evaluated.
• By analyzing the effect of different positions of the inlet in horizontal, vertical and lateral directions on the cooling performance.
• By analyzing the effect of variation of jet angle on the cooling performance. • By analyzing the effect of adding a supply pipe to the model on the cooling
performance.
• Based on the studies mentioned above and an in-depth post-processing of the results, a design can be proposed for a jet lubricated system with the best hole position, jet angle, jet diameter and jet velocity that satisfies the cooling requirements while requiring a minimized flow rate.
1.3
Limitations
• A single gear wheel is considered for the parametric study as this would reduce the need to handle gear contacts (in the case of two gear wheels), which is quite complex and computationally expensive.
• Oil velocity, hole diameter, inlet position, inlet angles and gear tip velocity are considered for the parametrization. Oil type and inlet nozzles are not being considered.
• Although the high velocities of the jet used would make the flow quite close to heavy breakup and atomization regime (as explained in section 2.5), the scale of jet is small enough such that the breakup effects can be neglected and Eulerian multiphase model can be used to model the flow of the oil jet. • The temperature variations (localized high temperatures) across the gear wheel
surface are neglected, thus keeping the gear wheel surface temperature con-stant. Due to this, the overall heat transfer at the gear wheel surface will be under predicted.
• Combination of jet and dip lubrication is not considered so as to isolate and study only the effect of jet lubrication.
1.4
Literature Study
A thorough literature study has been conducted to obtain an overview of the previ-ous research done in this arena so as to establish the meshing strategies and solver settings to be followed for analyzing oil-jet lubrication in gears. The literature that provided the most relevant insights are given below:
Renjith S et al. [2] analyzed the thermal performance of a jet lubricated trans-mission system consisting of two gear wheels using overset meshing technique , VoF modeling and Realizable k- (RKE) turbulence model in STAR CCM+. It was ob-served that the cooling increases with an increase in speed of gear wheel and the jet velocity. It was also inferred that the air has to stay turbulent so as to facilitate heat transfer directly from the gear wheel surface.
Fondelli T et al. [3] analyzed the torque in a single gear wheel subjected to jet lubrication using ANSYS CFX with the adaptive mesh feature, VoF modeling and k- turbulence model. It was observed that the oil falling on the gear top land doesn’t impinge on the surface and breaks up into ligaments and small droplets, which do not contribute to the lubrication. The resistant torque is mainly dependent on the pressure on the tooth flank and not on the shear force.
Akin LS et al. [4] carried out several experimental studies about spur gear oil jet lubrication where the gear velocity and oil jet velocity has been varied for an-alyzing the penetration depth of the jet and this was compared with an analytical model. The pictures of oil flow was taken by using a high speed motion camera and the lubricant was illuminated by means of a xenon lamp. It was concluded that an optimal penetration depth and subsequently lubrication was achieved when the jet was not atomized. For very small depths and velocities, the jet doesn’t impinge the gear wheel surface. Small eddies with very low velocities are seen in between gear teeth. At lower pressures (velocities), there is a considerable difference between the calculated and experimental impingement depths, caused by the viscous losses in the nozzle due to very low viscous oil used.
Olga S et al. [5] studied the thermal boundary layer thickness of turbulent con-vection by using Rayleigh-Benard (RB) concon-vection method wherein a layer of fluid
confined between two horizontal plates is heated from below and cooled from above. It was observed that for very high Prandtl numbers (Pr), the kinetic boundary layer is thicker than the thermal boundary layer and due to this, the thickness of velocity boundary layer can be used for finding the ideal distribution of nodes in these cases. Arisawa H et al. [6] did the experimental and numerical analysis of the windage and churning losses for a jet lubricated bevel gearbox. Many jet nozzles were used to lubricate the gear wheel. For the numerical analysis, VoF method was used to model the two-phase flow and porous body methods were used for gear meshing. A good agreement between the CFD results and experiments of average power losses were found.
2
Theory
A brief overview of the theory behind the concepts used in the thesis is given in this section.
2.1
Computational Fluid Dynamics (CFD)
CFD uses computer based simulations to analyze problems related to fluid flow, heat transfer, aerodynamics and other phenomena. Numerical algorithms are used to solve governing equations of these physical models. Experimental analysis of some of these cases can be quite costly (large models), complex and unsafe (safety/accident studies) and hence using CFD in these scenarios is more viable. The availability of af-fordable high performing computing hardware and user-friendly interfaces has made CFD analysis of complex physics model much more suitable and preferred in some cases over actual experimental studies. CFD analysis are quite useful in analyzing conceptual designs, detailed product development, redesign and troubleshooting of flow problems. The total effort required in experiment design and data acquisition can be reduced by the use of CFD.
The numerical algorithms used to solve the governing equations in the physical models are based on Navier-Stokes equations. Navier-Stokes equations use the con-servation of mass, Newton’s second law of motion and the first law of thermody-namics to govern the flow of a viscous, heat conducting liquid. According to the law of conservation of mass (continuity equation), the rate of increase in mass in a fluid element is equal to the net rate of flow of mass into it. Newton’s second law of motion states that the rate of change of momentum is equal to the sum of forces on the fluid particle. This is also called conservation of momentum. The first law of thermodynamics (or energy equation) states that energy can neither be created nor be destroyed, and it can only be transferred from one form to another, hence, the total energy of an isolated system is a constant.
2.2
Multiphase Flow Modeling
Liquids of different density are considered as phases in CFD. A fluid system is de-fined by a primary phase and one or more secondary phases. Multiphase flow model is used to calculate the flow behavior of oil and air phases in the computational domain. Continuity and momentum equations along with energy/or turbulence equations are used in solving the flow variables in multiphase flow regimes. Along with Navier-Stokes equations and turbulence modeling, multiphase flow regimes also use additional modeling for capturing the complex behavior of phase interaction. Multiphase analysis is a quite recent development in commercial CFD codes and is necessary in analyzing the complex multiphase flow behavior in gearboxes [7]. Multiphase flows are modeled primarily using two techniques- Eulerian-Lagrangian technique and Eulerian-Eulerian technique. In Eulerian-Lagrangian technique, the
individual particles are tracked inside a continuous medium, thus making it com-putationally high demanding compared to Eulerian-Eulerian technique, where all phases are treated as continuous. Eulerian-Eulerian modeling technique best aligns with the multiphase flow in oil-jet lubrication in gears as both phases are treated as continuous [7].
The Volume of Fluid (VoF) model is a type of Eulerian-Eulerian modeling tech-nique where the phases are considered to not inter penetrate each other and a phase indicator function is provided for tracking the phase interface. VoF method can analyze flows with several immiscible fluids. The free surface between the phases can also be modeled using VoF model and since information about the oil surface is a very important factor in oil- jet lubrication in gearboxes, VoF modeling has been used for analyzing the multiphase behavior [7] [8].
The governing equations in multiphase flows using Eulerian-Eulerian method is ex-plained below. p denotes the void fraction and it is a representation of how much
of the region of fluid is filled by phase p. p= 1 V Z V αpdv (1)
where αp is the averaged effective density of the small volume dv and V is the total
volume. This gives the average flux (φp) across the phase as:
φp = 1 pV Z V αpφdv (2)
where φ is the flux. The velocity (u∧p) is given as: u∧p = 1
pV
Z
V
upαpdv (3)
Thus, the total mass of phase p in a control volume is given byR
V
upαpdv. Since the
mass that leaves one phase must add to another phase, P m.
p = 0, which gives the
average yield of mass conservation equation as: ∂
∂t(pρp) + ∇. (pρpup) = m
.
p (4)
Based on all the equations mentioned above, the conservation of momentum equa-tions (as explained in section 2.1) for VoF based Eulerian-Eulerian modeling can be given as:
∂
∂t(pρpup) + ∇. (pρpupup) = −p∇pp+ ∇. (pµpDp) + pρpg + ∇. (pρpuu) + Fi (5) where Fi is the interfacial forces, pp is the pressure across the phase, pρpuu
corre-sponds to the Reynolds stresses (as explained in section 2.3.1) and g is the acceler-ation due to gravity.
Fi in equation 5 denotes the interfacial forces and it is given as:
Fi = kp(u − up) + g
ρp− ρ
ρ + Fother (6) where kp(u − up) is the drag force, gρpρ−ρ is the gravity buoyancy and Fother is the
other forces due to mass, pressure etc. Hence, in addition to the Reynolds stresses, multiphase modeling also models interfacial forces [9].
2.3
Turbulence Modeling
The air in the gearbox in the case of oil-jet lubrication is considered as turbulent so as to transfer heat directly from gear surface [2]. For the various velocities that are used in the analysis (as mentioned in Table 7 in section 3.4), the corresponding Re (refer Appendix A.2 for Reynolds Number) at the inlet of the pipe is as shown in Table 1.
Table 1: Reynolds Numbers for each Hole Radius (Refer to Table 7 in section 3.4 for corresponding velocities)
Hole Radius (mm) Reynolds Numbers
1.5 1721 2733 6147 12294 18441 2.0 2049 4611 9221 13831 18319 2.5 1639 3688 7376 11065 14651
Re less than 2300 would mean a laminar flow and Re greater than 4000 would mean a turbulent flow, considering pipe flows. However, the onset of turbulence doesn’t occur immediately at the inlet and a continuous turbulent flow will only be developed at a very long distance from the inlet of the pipe. For oil pipes used in the industry, the thickness of the steel sheet is around 1 mm, which gives a rather small distance for the turbulence to develop at the inlet, even at very high Re. Since distance between the inlet and the gear surface is also very short (as explained in section 3.1), the oil flow hitting the gear may still be laminar or close to transition [10]. A turbulence model that best suits this scenario has to be chosen.
2.3.1
RANS Turbulence Model
Reynolds Averaged Navier Stokes (RANS) is a type of turbulence modeling where the Navier-Stokes equations are reformulated such that all the instantaneous flow variables will have a steady mean value and fluctuating values with respect to time. This is also called as Reynolds decomposition and the additional quanti-ties (Reynolds stresses) are calculated either by using an eddy viscosity model or by Reynolds shear stress transport model. Hence, the numerical algorithms solve the governing equations in a time averaged way.
2.3.2
SST k-ω Turbulence Model
The Shear Stress transport (SST) k-ω has been developed by Menter [11] and it uses a combination of both k-ω model and k- model. It has been used for the flow analysis for jet lubrication in gear as it captures the flow physics well and has an advantage in capturing the near wall physics best compared to other turbulence models (near wall behavior is very important to analyze heat transfer in gear surface [5]). It uses a k-ω formulation in the near wall region and switches to a k- model in free stream. Thus, the sensitivity to inlet turbulence and extra damping is avoided. The turbulence kinetic energy k is given by:
∂k ∂t + Uj ∂k ∂xj = Pk− βkω + ∂ ∂xj [(ν + σkνT) ∂k ∂xj ] (7)
where νT is the kinematic eddy viscosity and Pkis the production term. This shear
stress limiter stops having excessive turbulent kinetic energy near stagnation points. Turbulence modeling in multiphase flows is a developing field of research. If the continuous phase is turbulent, either k, and kp is solved in the dispersed phase
or the values from the dispersed phase are ignored. The first method gives more accurate results, however it is computationally demanding due to the addition of new parameters [9].
Thus, the Turbulence kinetic Energy (k) equation in multiphase modeling becomes: Dk
Dt = Eqn.7 + U.Fp= Eqn.7 + τ
ρ(u(u − up)) (8) where τ is the shear stress [9].
2.3.3
Gamma Transition Model
SST Kω turbulence model with gamma transition (so that oil doesn’t become tur-bulent) has been used for modeling the flow in the gearbox. The Gamma transi-tion model is a simplified version of the Gamma ReTheta model in STAR CCM+ where the transport equation for transition momentum thickness Reynolds number is avoided (hence the momentum thickness is modeled only for laminar and tran-sition) and only solves for turbulence intermittency (irregular dissipation of kinetic energy that shows transition from laminar model). The complex correlations in the equations are simplified and free stream edge is omitted. All these simplifications makes it less computationally intensive and hence it has been used in for modeling oil jet lubrication [7]. It has been used commonly for modeling transition onset in flows and flows that are in laminar to transition regimes.
2.4
Motion Modeling
Two approaches have been used to model the rotational motion of the gear- a moving reference frame method to obtain the initial air field and sliding mesh technique (rigid body motion) for all the other analysis.
2.4.1
Moving Reference Frame Method
The moving reference frame (MRF) is a simple, steady state, robust modeling tech-nique for rotating bodies. It assumes a constant rotational speed to the assigned volume and the non- wall boundaries as surface of rotation, hence it is also called as a ‘frozen rotor approach’. A weak interaction is assumed between the MRF volume and the stationary volumes surrounding it. An interface is also created between the MRF volume and stationary parts. Since it is less computationally intensive and works well for single phase and steady state simulations, the MRF can be used for obtaining the values for the initialization of the air flow field in the gearbox [7] .
2.4.2
Rigid Body Motion
In rigid body motion (RBM), the rotating body (moving volume) is assumed to have strong interactions with the surrounding boundaries. It is a transient method and in most commercial solvers, RBM is implemented by using sliding meshing or by using overset meshing. RBM is more computationally intensive than MRF method and it also uses an ‘interface’ between the moving part and the stationary part [7].
Sliding Mesh
Sliding mesh is less computationally intensive compared to the overset meshing tech-nique and in modeling a single gear wheel rotation, a perfect interface can be created using sliding mesh and hence sliding meshing technique has been used to model the rotation of gear wheel. This meshing strategy uses two regions -a stationary one (re-gion far away from the gear wheel) and a rotating one (re(re-gion near the gear wheel) which are connected by a sliding interface. The rotating mesh moves with respect to the stationary one [7].
Overset Mesh
Overset meshing works well for multiple overlapping boundaries like modeling the rotation of two gear wheels in contact. A very small gap has to given between the gear wheels to model the contact between gear teeth. The region around the gear wheels are given as overset domain, which creates an overset interface with the background domain. Overset meshes have active, passive, acceptor and donor cells. In active cells, the discretized governing equations are solved and in passive cells, these equations are not solved. The acceptor and donor cells are present in the overset boundary and background mesh respectively and they link the overset and background meshes. When an overset region moves, overlapping zone (overset interface) undergoes changes based on the information passed between the overset and background mesh through the overlapping cells [7].
2.5
Jet Flow Regimes
Jet break up regimes are used to depict free surface dynamics such as dispersion of liquids in gases and spray technology. Break up regimes are divided into Rayleigh
breakup (droplets and jets), first wind induced regime (wave and droplets), second wind induced regime (wave and droplets) and atomization regime (atomized particle) [12]. Ohnesorge number (Oh) and Re are used for finding the break up regime of the liquid jet (refer Appendix A.2). Oh is calculated on the basis of Weber Number (We) and these values are shown in Tables 2 and 3.
Table 2: Weber Numbers for each Hole Radius(Refer to Table 7 in section 3.4 for corresponding velocities)
Hole Radius (mm) Weber Numbers (We)
1.5 3588.8 9044.3 45775 183080 411950 2.0 3814.6 19312 77239 173800 304880 2.5 1953.1 9887.5 39544 88979 156010
Table 3: Ohnesorge Numbers for each Hole Radius
Hole Radius (mm) Ohnesorge Numbers (Oh) 1.5 0.0348
2.0 0.0301 2.5 0.0269
Based on these values, the oil jets can be classified into various schemes as shown in Figure 1.
Figure 1: Jet Breakup Regimes for the Various Hole Radii and Velocities
Lagrangian particle approach would have to be used to model the oil jet in atomiza-tion regime. Although some of the oil jets are in atomizaatomiza-tion regime, the diameter of the jet is so large that liquid intact length (or jet breakup length) is much longer than the distance from the inlet to the gear wheel [13], hence, the oil is still in
the form of jet when it hits the gear wheel and so Eulerian multiphase modeling technique is still suitable for the analysis.
3
Method
3.1
Model and Domain
A helical gear wheel of 81 mm radius and 48 mm thickness is used for the analysis as shown in Figure 2b. It is the third gear wheel in the gearbox AT2612F as shown in Figure 2a. This gear wheel was chosen so as to keep the size minimal to reduce computational cost. The model is cleaned using the software ANSA. A shaft of radius 66 mm is extruded to both sides of the gear wheel, constituting a total length of 91 mm (this length is chosen arbitrarily such that the shaft is not too long). A surface mesh for the gear wheel is also obtained from ANSA as the surface meshing capabilities of ANSA is much better than that of STAR CCM+ (especially in areas such as gear teeth edges). The cleaned model from ANSA is as shown in Figure 2c.
Figure 2: (a) AT2612F Gearbox [1], (b) The Third Gear Wheel, (c) The Cleaned Model from ANSA
This model is then exported into STAR-CCM+ where the casing for the gear wheel is given as a cylinder of radius 120 mm. A cylindrical duct of 2 mm radius is given at the casing surface to introduce the lubricating jet. The distance between the gear wheel and the casing (about 9 jet diameters) is chosen such a way that there is enough distance in the radial direction for a fully formed jet flow to impinge the gear wheel surface and also considering the space limitations for casing for gear wheels in vehicles. The oil is directed from the top of the casing (vertical jet) for the initial case.
A cylinder of radius 95 mm has been given to model the sliding domain. This distance has been chosen because it is expected to capture most of the oil-air inter-action within this limit. A cylinder of 10 mm radius has also been given below the casing surface for the pressure outlet. The model from STAR CCM+ along with its dimensions is as shown in Figure 3.
Figure 3: (a) The Computational Domain from STAR CCM+, (b) The XY Plane showing the Dimensions of the Model
3.2
Mesh
The rotation of the gear wheel is modeled as rigid body motion using the sliding meshing technique. A stationary region (far away from the gear wheel) and a ro-tating region (near the gear wheel) are created which are connected by a sliding interface. The rotating mesh moves with respect to the stationary one. Figure 4 shows the mesh for the model.
Figure 4: The XY Plane showing the Mesh, where section A shows the refinements in sliding region and section B shows the refinement for oil jet from inlet
Polyhedral mesh has been used as it will reduce the cell count as compared to tetrahedral meshing, thus providing a computational advantage. Inflation has been provided around the gear wheel surface to obtain the required y plus and also to capture the thermal gradient in the boundary layer. A refined mesh has been pro-vided in the sliding region to capture the oil flow around the gear wheel and the
oil-air interface in this region as shown in section A of Figure 4. A refinement has also been provided in the region of the velocity jet to capture the flow of the oil from the velocity inlet as shown in section B of Figure 4 .
The y+ obtained around the gear wheel as shown in Figure 5 is within accept-able limits as required for the SST k-ω turbulence model. The region where the oil jet hits the gear wheel surface has the highest y+ and the y+ value decreases away from the oil jet.
Figure 5: The y+ Contour in the Gear Wheel Surface
The Pr of the oil is 163 and at such high Pr values, the kinetic boundary layer is thicker than the thermal boundary layer. Hence, for analysis of heat transfer of oil with high Pr, the inflation has to be sufficient enough to capture the non-linearity of the thinner thermal boundary layer (as explained below) and the thicker kinetic boundary layer.
Figure 6: Temperature Contour
It can be seen from section A of the temperature contour in Figure 6 that the thermal gradient in the boundary layer in the region where the oil jet hits the gear wheel surface is captured by the first layer of the inflation provided in the mesh. This thermal gradient shows the difference in temperature at the gear wheel surface and the oil which drives the heat flux at the gear wheel surface. The thermal gradient spans from 363.15 K to 393.15 K (temperature of oil and gear wheel as mentioned in 3.3). From section A of Figure 6, it can be seen that the first layer thickness of infla-tion has temperature of 369 K, which means 80% of the thermal gradient is within
the first layer. Increasing the prism layers would resolve more thermal gradient, however it would require very small time-steps and thus increase the computational cost. Refining the prism layer resolution didn’t make much difference in the heat transfer. Since both a suitable y + and a decent thermal gradient in boundary layer is captured, this mesh is deemed fit for further analysis.
3.2.1
Mesh Sensitivity Study
Figure 7: Mesh Sensitivity Study
The mesh has approximately 10 million elements and is proven to work acceptably after a mesh sensitivity analysis as depicted in Figure 7 where the heat transfer at the gear wheel surface (W) is compared against the values obtained from the simulations run by meshes with approximately 5.77 million and 18 million elements. The elements in the sliding region of the mesh were reduced by changing the element size for the 5.77 million mesh. For the 18 million mesh, the elements in the sliding region were increased uniformly by introducing the ‘volumetric control’ option in the solver. The percentage difference of the average heat transfer in gear wheel surface is given in Table 4.
Table 4: Mesh Sensitivity Study
Number of Elements Percentage Difference of Average of Heat Transfer at Gear Wheel Surface
5779048
-10038999 33.09 %
18030974 2.37 %
It is observed from Table 4 that considering the mesh with 18 million elements over the mesh with 10 million elements would increase the computational expense with no significant changes to the results. Also, considering the 5.77 million mesh would reduce the computational cost but with a considerable change in the accuracy of the results. Hence, the mesh with 10 million elements has been considered for further analysis.
3.3
Solver Settings
A single gear wheel rotating counter clockwise at 1400 rpm (this speed has been chosen as this is the speed that gives the maximum torque for this gear wheel and thus the ratio between power loss and pump flow is highest in this case) with oil supplied from a circular hole of 2 mm radius is considered for the basic model (this radius is chosen such that a higher value of radius would mean a very large hole, which is impracticable and a lower value of radius would mean a very small hole that is hard to manufacture). The oil considered is ‘Chevron Multi-gear MTF HD 97307’. The oil jet velocity for the basic model is considered as 59.4 m/s (oil jet velocity to gear wheel tip velocity ratio = 5). The boundary conditions for the model are shown in Figure 8.
Figure 8: Boundary Conditions
The flow is three dimensional with segregated multiphase temperature option en-abled to predict the heat transfer (segregated flow works best when compressibility or high Mach numbers are not a concern [7]). The gear wheel is given as no slip wall at 120◦C (the rule of thumb is that gear wheel average temperature shouldn’t exceed this value to maintain its lifespan). The shaft of the gear wheel is given as an adiabatic wall. The oil jet at 90 ◦C is impinged through a velocity inlet, provided as a circular hole at the top of the domain such that the jet impinges vertically at the gear wheel. The cylindrical surface of the casing region is given as a no-slip wall at 90 ◦C. The flat surfaces of the casing (where the air flows) is given as symmetry condition. The flat surfaces of the rotating cylinder is given as adiabatic no-slip walls and the rotational speed of the gear wheel is specified by using zone motion. A sliding interface is created between the rotating (sliding ) domain and the stationary domain. A pressure outlet for air is also given at the bottom of the domain. The surface of the cylinder for pressure outlet is given as adiabatic wall. All walls are given a contact angle of 15 degrees to take into account the effects of surface tension.
The air flow is turbulent and the oil is laminar or close to transition Re in gearboxes (as explained in section 2.3), hence SST Kω turbulence model with gamma transi-tion (so that oil doesn’t become turbulent) has been used for modeling the flow in the gearbox as mentioned in section 2.3.3. The VoF method is used to model the multiphase flow in the gearbox. Oil is given as the primary phase and air as an ideal gas is used as the secondary phase. The properties of the oil vary with temperature and hence various parameters such as density, specific heat and thermal conductivity of the oil are specified using polynomials in temperature so as to accurately predict them (these polynomials are obtained based on the empirical formulas provided by the lubricant manufacturer [14] and their relation with temperature is given in Ap-pendix A.4). The surface tension coefficient between the phase is given as 0.03 N/m. The gravity model and all y + wall treatment is also applied. The minimum and maximum allowable temperature are limited to 353 K and 403 K respectively in the domain to provide robustness and stability in the analysis. However, it doesn’t affect the results as these values are outside the temperature bounds of the oil and the gear wheel surface.
The initial volume fraction of air is kept as 1 and all the other initial conditions like intermittency, pressure, static temperature, turbulence intensity, turbulent vis-cosity ratio and velocity are obtained from the single phase simulation conducted using MRF model as described in section 3.3.1. Implicit unsteady model with a time-step of 1E-05 s along with first order temporal discretization is used. 10 inner iterations have been used per time step. Various quantities such as heat transfer at gear wheel and casing surface, mass flow on inlet, moment at gear wheel surface, surface integral of mass flux of oil at inlet, volume averages and volume integrals of velocity, enthalpy , temperature and turbulent kinetic energy were monitored and the steadying out of these quantities would determine the convergence of the simulation. This basic model is used to visualize the flow patterns in a jet lubricated gear-box and evaluate the cooling. The heat transfer at the gear wheel surface is used to calculate the cooling performance of the oil. Furthermore, the analysis using the basic model can also be considered as a method development in CFD for further studies involving jet lubrication of gear wheels.
3.3.1
Initialization with MRF Model
The air in the gearbox would take quite some time for heating and speeding up (the time for which would be much greater than the time taken for analysis of oil flow field), hence it is advantageous to have a single phase simulation run with only air and the converged result can be used as the initialization for the multiphase analysis. This would reduce considerable time for the subsequent multiphase analysis. The mesh for the MRF method is shown in Figure 9.
Figure 9: XY Plane showing Mesh for MRF Method
The MRF method is less computationally expensive than the sliding mesh method as it doesn’t need the refinements to model the oil flow, hence it is a good way to obtain the initial air flow field here. MRF works on steady state and all the boundaries of the casing are given as no-slip walls at 90◦C for this method. Other than this, most of the solver settings are similar to those in section 3.3. Once the converged result is obtained, tables of intermittency, pressure, static temperature, turbulence intensity, turbulent viscosity ratio and velocity are exported into the initial conditions of the rigid body simulation. The MRF model also provides the heat transfer at gear wheel surface for the case without any lubrication.
3.4
Analysis with Variation in Hole Radius
and Jet Velocity
The cooling performance in gear lubrication systems mainly depend on the amount of lubricant that is being used. In jet lubricated systems, the amount of lubricant is largely dependent on the size of the hole through which the jet is being impinged and the velocity of the jet. Hence one of the key parameters identified for this analysis is the jet velocity and hole radius.
Different hole radii of 1.5 mm, 2 mm and 2.5 mm are considered to obtain the combination of hole radius and jet velocity that gives the best cooling with a min-imized requirement of volumetric flow rate. These values of hole radii were chosen considering the practicality of constructing the hole in actual gearbox. The mesh is adapted to suit the geometry for variation in inlet size.
For a particular hole radius, the jet velocity that gives the best cooling can be evaluated by performing a number of simulations with different ratios of jet velocity to gear tip velocities in the range of 10 to below one as shown in Table 5. These ratios are chosen such that the volumetric flow rates corresponding to each of these values follow a certain pattern as shown in Table 6, where all the three hole radii have certain volumetric flow rates in common. The oil pumps used in industry have
a designation of 25 L/min (for high shaft speeds as is the case in this analysis), so some of these values are well within the limit. It is also important to analyze cases with high volumetric rates (which would mean using bigger pumps and/or a higher gear ratio) to know the effect it would have on jet lubrication. The analysis of 1.5 mm hole with velocity ratio of 5.99 was done using a timestep of 5E-06 s (due to its higher velocity). For the same volumetric flow rate, the combination of hole radius and jet velocity that gives the best cooling can be determined using this. From section 4.2, it can be inferred that velocity ratios greater than 5 isn’t nec-essary to evaluate as it gives higher heat transfer than necnec-essary, so the range of velocity ratios have been limited to values between 0 and 6. The jet velocity values corresponding to these velocity ratios are provided in Table 7.
Table 5: Velocity Ratios chosen for each Hole Radius
Hole Radius (mm) Velocity Ratio
1.5 0.56 0.89 2.00 3.99 5.99 2.0 0.50 1.13 2.25 3.38 4.47 2.5 0.32 0.72 1.44 2.16 2.86
Table 6: Volumetric Flow Rates chosen for each Hole Radius
Hole Radius (mm) Volumetric Flow Rates (L/min) 1.5 2.82 4.48 10.08 20.16 30.24 2.0 4.48 10.08 20.16 30.24 40.05 2.5 4.48 10.08 20.16 30.24 40.05
Table 7: Jet Velocities chosen for each Hole Radius
Hole Radius (mm) Jet Velocity (m/s)
1.5 6.65 10.56 23.77 47.53 71.29 2.0 5.94 13.37 26.73 40.10 53.12 2.5 3.80 8.56 17.11 25.67 33.98
3.5
Laminar Study
To study the effect of using a laminar model instead of a turbulent one, an analysis was conducted using laminar model on the model with 2 mm inlet hole and 26.73 m/s inlet velocity. An analysis was also conducted on the model with 1.5 mm inlet hole radius and 71.29 m/s velocity (timestep used was 5E-06 s) to check if the laminar model is suitable for all the range of analysis that has been conducted previously. The initialization using MRF was done for laminar analysis and the resulting air field was exported for the multiphase analysis.
3.6
Analysis with Variation in Hole Position
in X and Y Directions
To study the effect of the position of inlet hole on the cooling, different analysis were carried out with varying the hole position in both X and Y directions (horizontal and vertical respectively). The variation of inlet hole position in X direction is as shown in Figure 10.
Figure 10: XY plane showing the variation of inlet hole position in X direction
The inlet hole was moved 75 mm to the left and right directions from its original position. These values are chosen because these are the farthest values at which the oil jet will hit the mid-point of the teeth as per the dimensions of the model used in this study. This change in inlet position in X direction would also mean a change in Y direction of hole due to the cylindrical shape of the casing using in this research. Thus, the effect of inlet hole position in both X and Y directions can be studied using this analysis. The mesh is adapted to suit the new position of the inlet.
3.7
Analysis with Variation in Hole Position
in Z Direction
To study the effect of the Z position (lateral direction) of inlet hole on the cooling, different analysis were carried out with varying the hole position in Z direction as shown in Figure 11.
Figure 11: Computational domain showing the variation of inlet hole position in Z Direction (a) 4 mm to the left and (b) 4 mm to the right
The inlet hole has been moved 4 mm to the left and right directions in Z position as shown in section A and B of Figure 11. If a higher value was chosen as the distance to be varied in Z direction, then this would mean that the inlet jet would hit the end of the gear wheel thickness (due to the restricted space available) and much oil would be lost to the casing surface, which is not desirable. The mesh is adapted to suit the change in inlet position.
3.8
Analysis with Variation in Jet Angles
To study the effect of the position of inlet hole on the cooling, different analysis were carried out with varying the inlet jet angles as shown in Figure 12.
Figure 12: XY Plane showing the variation of inlet hole angles from the vertical (a) 13◦ to the left, (b) 11.3◦ to the left, (c) 7◦ to the right and (d) 9◦ to the right
These angles were chosen such that the oil jet would hit the gear teeth at just below the top land and just above the bottom land in both left and and right directions are shown by the dotted black lines in Figure 12. The size of the inlet hole and the space between the two adjacent gear teeth is such that the same angles cannot be used in both directions as doing so would make the jet hit at different positions along the gear teeth in both directions. The mesh is adapted to suit the modified inlet angle in each case.
3.9
Analysis with Variation in Gear Speed
To study the effect of the parameter gear speed, an analysis was conducted using an increased gear speed of 2000 rpm. This speed was chosen as this is the highest speed on which the engine operates. A lower speed was not chosen as it is not relevant from a cooling perspective (heating is less at lower speeds). The initialization using MRF method was done for this speed and the initial air field from this speed was then exported into the multiphase analysis. The oil jet velocity is kept constant (26.73 m/s) as it was identified as the best velocity for cooling (as explained in section 4.3) and also to facilitate ease of comparison with analysis having 1400 rpm.
3.10
Pipe Analysis
To study the effect of using a pipe that supplies oil to the inlet, an analysis was carried out by adding an oil supply pipe of length 165 mm and diameter 16 mm to the model. This would make the study much more comprehensive as it would be much closer to the real scenario in gearboxes. The modified model along with the supply pipe is as shown in Figure 13.
Figure 13: Computational domain showing the dimensions and boundary conditions of the pipe analysis
The diameter for the supply pipe is chosen considering the reasonable sizes of pipes used in oil systems and the length is chosen considering the distance needed to form a fully developed oil flow and also considering the computational expense. For the oil pipes actually used in the industry, the thickness of inlet hole is around 1 mm, hence the distance between the pipe and the casing surface is given as 1 mm.
Figure 14: XY Plane showing the mesh for pipe analysis
Figure 14 shows the mesh for the pipe analysis. An inflation has been given around the pipe to capture the velocity gradient. A refinement has been given around the inlet area and a single prism layer around the casing as shown in section A to capture the effects happening at the inlet region. Rest of the mesh settings are same as in the basic case.
A velocity inlet is given at one end of the pipe with a specification of 1.67 m/s (this value is chosen so as to get the same volumetric flow rate (20.16 L/min) in the supply pipe as well as the small inlet pipe). All other surfaces in the pipe are given as no slip walls with temperature specification of 90◦C. The oil is initialized inside the pipe using custom field functions. The initialization using MRF method was done for the new geometry and the initial air field obtained was then exported into the multiphase analysis. The timestep used is 2E-06 s (due to smaller cell size caused by refinements at the region where the jet enters the casing). All the other solver settings are same as the basic analysis.
In the pipe analysis, another study was also conducted using a square inlet of side 3.54 mm to study the effect of shape of inlet on the heat transfer. The size of square was selected such that it gives the same volumetric flow rate (20.16 L/min) as the circular inlet. Other shapes such as rectangle was not studied as using a rectangular inlet that would cover the axial length of the gear wheel (so as to maximize the area that the oil jet hits) would mean a physically unrealistic breadth for the inlet to achieve the same volumetric rate as the circular inlet.
3.11
Analysis with Two Gear Wheels
In order to get a more comprehensive idea about the use of jet lubrication in gear wheels, an analysis was conducted on using jet lubrication for two gear wheels with a small gap in between them. Two helical gear wheels (the third gear wheel and its countershaft gear wheel from the AT2612F Gearbox [1]) are assembled such that
there is a very small distance between them. Shafts are added to both the gear wheels in ANSA and a surface mesh is also obtained from ANSA. The model from ANSA is as shown in Figure 15 (a).
Figure 15: (a) Model from ANSA and (b) Computational Domain from STAR CCM+ for Two Gear Analysis
The model is then exported to STAR CCM+ where cylinders of radius 84 mm and 92 mm are added to both the gear wheels so as to comprise the overset domain (this dimension was chosen such that there should be atleast 4 cells in between the gear wheel surface and the overset boundary, as per the rules for overset mesh in STAR CCM+). A background domain has also been created in STAR CCM+ with dimensions as shown in Figure 16. These dimensions are chosen such that the domain will capture the effects of jet lubrication and also considering the computational expense incurred. The computational domain for the analysis in STAR CCM+ is as given in Figure 15 (b).
Figure 16: XY Plane showing the dimensions of the domain for Two Gear Wheel Analysis
The mesh for the two gear analysis is given in Figure 17. The overset meshing strategy is used to model the rotation of gear wheels (as explained in section 2.4.2). The overset meshes are given in blue and pink colors. The background domain is meshed using polyhedral cells and refinements are given in the path of the oil jet and around both the gear wheels as shown in section A. The overset domain for both the gear wheels have the same cell size as the refinements applied to the background domain around the gear wheels (this is one of the rules for overset mesh [15]). An inflation has been provided around both the gears to capture sufficient velocity and thermal gradient. The overset meshes cut holes in the background mesh on which it lays (as seen in section B in the intermeshing region) and after the hole cutting process, this mesh has 4 million cells.
Figure 17: XY Plane showing the mesh (after hole cutting) for Two Gear Analysis, where section A shows the refinements in the background domain for the oil jet and around the gears. Section B shows the mesh in the intermeshing region of the gear wheels.
The casing is given as a no-slip wall of 90◦C and the gear wheel surfaces are given as walls of 120◦C. The front and back surfaces of both the background and overset domains are given as symmetry (as the surfaces that are in both the domains should have same boundary condition). The lateral surface of the overset cylinders are given as overset mesh boundary condition. A velocity inlet of radius 2.82 mm and 26.73 m/s is given for the oil jet (this radius is chosen such that the volumetric flow is double (40 L/min) than the single gear analysis, so that enough oil hits both the gear wheels). A pressure outlet is also given for air. The countershaft gear wheel is rotating at 1256.41 rpm in the clockwise direction while the other gear wheel is rotating at 1400 rpm in the anti-clockwise direction (value for the countershaft gear wheel chosen based on the gear ratio). Most of the other solver settings are similar to the analysis for a single gear wheel (as explained in section 3.3). The timestep used is 3E-06 s (the timestep has to be small enough to travel through half the distance of the cells in the overset interface of the smaller gear wheel [15]) and mass conservation per iteration is also monitored. The initial air field was obtained from MRF analysis for a two gear wheel analysis (the gear wheels have more space in between them in the intermeshing region to create the MRF interface).
4
Results and Discussion
4.1
Initialization with MRF Model
The air flow field obtained from MRF analysis can be visualized as shown in Figure 18 (a).
Figure 18: (a) Velocity Field (m/s) of Air from Single Phase MRF Analysis, (b) Turbulent Kinetic Energy(J/kg) contour of the MRF Analysis
It can be seen from Figure 18 (b) that there isn’t much turbulence in the air, which can be due to the fact that there isn’t much pressure difference to create eddies due to the tight enclosure of the gearbox (however, the need for use of turbulence model is verified in section 4.4). Very small turbulence effects can be observed near the gear teeth. This was also observed by Akin LS et al. [4]. The heat transfer at the gear wheel surface without any lubrication is obtained as 45 W from this analysis.
4.2
Basic Analysis
The results for the basic model with a hole radius of 2 mm and jet velocity of 59.4 m/s (oil jet velocity to gear tip velocity ratio = 5 and corresponding volumetric flow rate= 44.79 Liters/min) is explained in this section. Figure 19 shows the contour of the volume fraction of oil around the gear wheel for the basic analysis.
Figure 19: Contour of Volume Fraction of Oil for the Basic Model
From section A in Figure 19, it can be seen that the oil jet fully impinges the space between the gear teeth for this velocity. The oil forms a thin film that sticks to the surface of the gear teeth and thus provides cooling. However, it can also be seen that the oil jet that hits the top land of the gear wheel does not stick to this surface, instead immediately breaks into ligaments and small droplets (this was also predicted by Fondelli T et al. [3]). Thus, the cooling will be less at the top land of the gear wheel, however, since the top land of the gear wheel is not involved in load transmission during the inter-meshing of gear wheels, the wear would also be less at this surface. The oil that is broken up forms oil droplets that stick to the casing surface as shown in Figure 20 and moves along with the air.
Figure 20: Oil flow in the domain colored by temperature
It can be seen from Figure 20 that the flow in the case of oil-jet lubrication in gear wheels is rather complex with the formation of droplets and ligaments of oil. It can also be seen that the oil that attaches to the gear wheel surface is able to cool the gear wheel for half the rotation of the gear wheel. The plot of heat transfer at the gear wheel surface for the basic model is given in Figure 21.
Figure 21: Heat Transfer at the Gear Wheel Surface (W) for the Basic Model
It can be observed from Figure 21 that the heat transfer curve follows the shape of the gear teeth, i.e, the rise and fall in the graph corresponds to the movement of oil jet through the gear teeth surface from bottom land to top land. At around 0.044 s, one rotation of the gear wheel is completed and it can be seen that the heat transfer decreases from 2500 W to 2300 W after the first rotation. This is because the oil that hits the gear wheel surface during the first rotation covers the surface and it insulates the gear wheel from fresh cold oil.
Another interesting phenomena observed from the graph is the fluctuations in the curve when the oil jet is moving through the top land of the gear wheel, as shown by section C in Figure 22 (a). At a certain point in the journey of the oil jet through the top land, the oil starts falling to both the sides of the gear teeth as shown by Figure 22 (c), thus increasing the cooling as shown by the spike in the curve. The highest heat transfer on the gear teeth is observed at the bottom land (section B in Figure 22 (a)), when the oil-jet is fully impinging on the gear bottom land as shown in Figure 22 (b) (also explained in Appendix A.1).
Figure 22: (a) Same as Section A of Figure 21, (b) Contour of Volume Fraction of Oil as the oil jet falls on the bottom land of gear wheel, (c) Contour of Volume Fraction of Oil as the oil jet falls on the top land of gear wheel
Highest cooling is observed when the jet fully impinges till the bottom land of the gear wheel as observed in Figure 23 and thus it can be concluded that full impingement of jet to bottom land of gear wheel is one of the necessary factors for a successful jet lubrication system.
Figure 23: Heat Transfer Contour across the gear surface
The actual losses during the rotation of this gear wheel is calculated to be around 1700 W. From the analysis conducted in this thesis for the basic model, the thermal losses (heat transfer) alone contribute to around 2500 W, hence, it can be concluded that such high volumetric flow rate is not necessary for jet lubrication. Based on this, lower volumetric flow rates (and correspondingly lower velocity ratios, in the range of 0 to 6) were chosen for the next set of analysis as mentioned in section 3.4. Thus, it can be concluded from this basic analysis that full impingement of jet to the bottom land of the gear wheel along with a heat transfer of around 1700 W will correspond to the best jet lubrication system for this gear wheel.
4.3
Analysis with Variation in Hole Radius
and Jet Velocity
The heat transfer at gear surface (W) for the three hole diameters for five different velocities each is shown in Figure 24.
Figure 24: Velocity of Jet (m/s) Vs Heat Transfer at Gear Wheel Surface (W) for different hole radius
In the case of no lubrication, the air will cool the gear wheel to a very small value of 45 W. As expected, the cooling increases with velocity for all the three hole sizes. All the three graphs show an initial steep rise and then a gradual increase in heat transfer. It can be seen that for lower volumetric rates (4.5 L/min), the heat transfer in the case of hole radius 1.5 mm is greatest followed by 2 mm and 2.5 mm radius hole. This is due to the fact that for the same volumetric rate, the 1.5 mm hole needs a higher velocity (10.56 m/s) than the 2 mm hole (5.94 m/s) and 2.5 mm (3.80 m/s), which is contributing to its higher heat transfer.
However, for the next higher volumetric rate (10.1 L/min), the heat transfer by 2 mm hole is greater than that by both 1.5 mm and 2.5 mm hole. It is interesting to note that both 1.5 mm and 2.5 mm hole provides similar heat transfer. With increase in volumetric rate, the effect of hole size on the heat transfer also increases and at this volumetric rate, the combined effect of both hole size and jet velocity is highest for the 2 mm hole and this why it shows the highest heat transfer for this case. For higher volumetric rates of 20.2 L/min and 30.2 L/min, it can be seen that the heat transfer is similar for both 2 mm and 1.5 mm hole. This is due to the fact that at higher volumetric rates, full impingement takes place for both the radii and thus a similar heat transfer is observed. From volumetric rates of 20.2 L/min and higher, the highest heat transfer is observed for the 2.5 mm hole and this shows that at higher volumetric rates and velocity ratios, the effect of the hole size is more prominent than the jet velocity. This phenomenon is reversed at lower volumetric rates.
It can be concluded that for lower volumetric ratios, the smallest hole gives the highest cooling (due to high velocities) and for higher volumetric rates, the cooling is highest for 2.5 mm (due to larger hole size) and cooling is similar for both 1.5 mm and 2 mm hole (due to full impingement). Using a lower radius hole means higher
velocity of jet and higher pressure of oil to be pumped, hence in cases where the heat transfer is same for a particular volumetric rate, it is advantageous to use the larger hole.
Figure 25: Volume Fraction of Oil at Velocity Ratios of (a) 0.5 and (b) 2.25 for a hole radius of 2 mm
Figure 25 shows the volume fraction of oil at two different velocities for a hole radius of 2 mm. It can be seen that for the low velocity ratio of 0.5 [Figure 25 (a)] , the oil jet doesn’t fully impinge till the gear wheel bottom land, but stops at the gear wheel top land and hops from one gear tooth to the adjacent one. The oil is only cooling right side of the gear teeth due to this.
The impingement length of the oil increases with velocity as shown in Figure 25 (b) and for a velocity ratio of 2.25, the jet fully impinges the gear wheel bottom land and oil is flowing through both the sides of the gear teeth, which leads to high cooling. As the velocity increases more oil is also broken up into small droplets from the gear wheel top land, which leads to more oil being attached at the casing surface. This phenomenon is not as pronounced in lower velocities.
For low velocity ratios of below two, the impingement length (length of penetra-tion of oil jet into the space between the gear teeth) is very low for all the hole radii (although impingement length increases with velocity ratio) and thus velocity ratios of below two are considered unsuitable for jet lubrication. As an effect of this, the heat transfer is very low for low velocities (this was also observed by Akin LS et al. [4]). This reinforces the importance of full impingement of jet to the bottom land of the gear wheel for a good jet lubrication system.
Since heat transfer of above 1700 W isn’t necessary (as concluded in section 4.2), velocity ratio of 2.25 for 2 mm radius hole and velocity ratio of 3.98 for 1.5 mm hole are considered as viable for the jet lubrication. Both of these have the same volumetric flow rate. Using the 1.5 mm hole for this case would mean a higher velocity (and higher pressure) for getting similar heat transfer as the 2 mm hole for the same volumetric flow rate and hence, the 2 mm hole can be selected among the two.
The volumetric flow rate of 20.2 L/min can also be chosen as the ideal value for jet lubrication in this case as it offers full impingement and necessary heat transfer
