ANALYSIS OF THE POWER LOSSES IN GEARED TRANSMISSIONS - MEASUREMENTS AND CFD CALCULATIONS BASED ON OPEN SOURCE CODES

Franco Concli1, Carlo Gorla

2

1) Bonfiglioli Mechatronic Research, via Fortunato Zeni 8, 38068, Rovereto (Tn)

2) Politecnico di Milano, Dipartimento di Meccanica, via Giuseppe La Masa 1,

20156, Milano (Mi)

ABSTRACT

The increasing demand of more and more efficient gearboxes implies the need of predictive models. Models provided by literature for load independent power losses are mainly based on experimentally derived relations that are not so accurate on real applications. A new trend suggests to use CFD for the calculation of the no-load losses of gear transmissions. In this sense, literature provides some works but most of them involve only one single phase. In this paper, some investigations of both the oil-windage losses and the churning losses in which the gears are immersed in an air/oil lubricant mixture are studied. Furthermore, some attempts to reduce the computational effort have been successfully tested.

Keywords: efficiency, churning, power losses, CFD, gear, multiphase

1. INTRODUCTION

Gear power transmissions have been the most common way to transmit mechanical power for decades. For this reason, thanks to years of investigations, many calculation methods are nowadays available, for example, in order to optimize the transmission against many failures like for instance tooth breakage, pitting, micropitting, wear, scuffing... Many of these failure modes are strongly related to the tribology of the system., which strongly depends on the lubricant properties which are, in turn, a function of the temperature. One challenge in the design of gear power transmissions is to find out how to accurately calculate the operating temperatures and the efficiency of the gearbox. Increasing the efficiency of power transmissions can bring a significant contribution not only in terms of pollutant emissions limitation and energy saving, but also for other aspects such as reliability, related to different (lower) thermal regimes, and allows the application of downsizing, weight reduction and system architecture simplification. For this reason, the authors maintain the importance of finding an appropriate, reliable and not excessively time consuming approach for such kind of investigations. At the present

state of the art, in fact, literature provides many analytical and empirical relations derived from experiments in order to calculate the various contributions to the power losses. With these models it is possible to accurately predict the losses generated, for example, by the gears due to sliding or by the bearings. What still need to be improved are on the other hand models for the accurate prediction the power losses generated by the interaction between the mechanical components and the lubricant that in some applications can even represent up to 50% of the total losses [1]. The majority of the works presented on this topic are in many cases results of experimental tests. And sometimes from the measurements some analytical equation are derived. The weakness of all this calculation methods is that they are accurate only as long as the real operating conditions are similar to the conditions of the experiments. Furthermore, these models often neglect some important influencing parameters. Some work on the hydraulic losses was presented by Dowson [2] regarding the windage losses or by Mauz [3] and by Seetharaman et al. [4] who have concentrated on the churning power losses. Nevertheless, the authors maintain that a deeper understanding of the physical phenomena responsible of the losses due to the interaction with the lubricant is still needed in order to improve existing models, and CFD simulation can be an effective approach for such investigation. Marchesse et al. [5], on the basis of a state-of-the art on the application of CFD to gear power losses, applied CFD models to study air-windage losses of gears, and have validated their results by means of experimental tests. A CFD based method was proposed also by the authors in previous papers and applied in order to study the churning losses generated by the planet carrier in a planetary gearbox [6, 7]. Also the squeezing [8, 9] and the windage [9, 10] power losses of ordinary gears have been investigated by the authors.

This paper presents a parametric study of both the windage and the churning power losses generated by a single rotating gear. Simulations were performed considering different operating conditions and geometrical parameters like rotational speed, tip diameter, face width and lubricant level, in order to understand their influence on the power losses. The appropriateness of CFD as prediction tool was validated with some experimental data already presented by the authors in previous publications [6, 7, 9, 10].

Furthermore, beside the classical dynamic meshing method, a new approach, MRF, based on the idea that, in order to compute the power losses, only the regime solution is needed and not the whole startup, is applied to the calculation of both windage and churning losses. For this purpose the open-source CFD tool OpenFOAM® was used. The choice of an open-source code has been made because it allows more flexibility with respect to any close-source commercial software, making it possible to customize the code with the implementation of specific tools for the analysis of the physical problem of interest. Some other computational effort reduction methods are theoretically already available, like the one presented by Pallas et All. [11]. This method allows to reduce the computational time through a reduction of the size of the mesh (by limiting the computational domain and substituting the missing part with appropriate boundary conditions), but is not applicable, for example, to mixture lubrication since it is not capable to predict the repartition of the phases. Also by considering the axial symmetry of the gear and by modelling just one section as shown by Marchesse et All. [4] or by Chaari et All. [12] it is not possible to simulate a real multiphase system which is, due to the dispersed position of the lubricant, intrinsically not symmetric. 2. COMPOSITION OF THE POWER LOSSES IN A GEARBOX

According to [13] the power losses of geared transmissions can be subdivided into load

dependent and load independent (subscript ).

[1]

According to this classification the losses, one subdivision is between losses depending

or not from the transmitted torque. The other useful subdivision can be made according

to the mechanical component responsible for the power dissipation: gears (subscript ),

bearings (subscript ), seals (subscript ) or other generic components like clutches or

synchronizers (subscript ). A sub classification of the load independent power losses of

gears separates the contribution of windage (by adding the subscript ), churning

(subscript ) and squeezing (subscript ).

Basically the windage power losses arise due to the interaction of mechanical components and a single phase fluid that can be either air or lubricant. In gearboxes this losses are significant only if the tangential velocities of the gear are important (high rotational speeds or large gears) or if the gearbox is completely filled with lubricant (fluid with a large viscosity grade and, above all, density). These losses evolve with rotational speed power three as shown by Mauz [3]. The churning losses are conceptually similar to the windage ones, but involve at least two phases and, in the practice, appear as a completely different phenomenon. These losses, in fact, evolve with a power law that is allways less than three, but, at the actual state of the art, the knowledge regarding this contribution are very limited. The main problem of calculating this source of losses is that the subdivision between the phases cannot be calculated analytically and, in addition, due to his nature, does not allows to reduce the computational domain taking advantage of the symmetry. These losses are the most important load independent power losses in geared transmissions: the majority of the gearboxes, in fact, are deep lubricated and, therefore, subjected to churning losses. The last source of load independent power losses of gear are the squeezing power losses. This dissipation is of a lower order of magnitude and arises due to the fact that the cavity between two mating teeth is reducing its volume during the engagement causing pressure gradients and additional flux circulation.

3. PROBLEM DESCRIPTION:

In a previous paper [10], some single phase simulations have been performed adopting a commercial CFD code and compared with measurements [14] showing good agreement. After the validation of the CFD approach, additional simulations using OpenFOAM have been performed and the results are here presented. The configuration of the analyzed system is simple. A single gear is mounted on a cantilevered shaft supported by two bearings inside a rectangular box (which dimensions can be modified) and connected to an electric motor. The gear can be set into rotation and the losses generated by the interaction of the gear with the air/oil lubricant mixture measured with a torque meter mounted between the motor and the gear as shown in figure 1. The contribution given by the bearings was previously measured with the same system configuration without the gear and the lubricant so to be able, at the end, to isolate the contribution from the total power loss measure. This geometry has been also used for the numerical calculations.

a) b)

Figure 1: a) schematic representation of the test rig [10, 12]; b) geometrical model for the numerical calculations

4. NUMERICAL MODEL

For this study, an open-source code (OpenFOAM®) was adopted. The choice of an open-

source tool was made because it allows more flexibility with respect to close-source

commercial software, making it possible to customize the code with the implementation

of specific models for the analysis of the physical problem of interest. The power losses

have been numerically analyzed be means of two different CFD approaches.

4.1 Sliding mesh

The first approach is based on a sliding mesh technique. The sliding mesh model is

theoretically the most accurate method for simulating rotating flows and it is able to

correctly describe the whole transient startup but it is also the most computationally

demanding. This technique, applied to the specific case, results in two cell zones that are

created separately (the first one is a cylindrical cell containing the gear, the second one

is the remaining volume). Each cell zone is bounded by an interface where it meets the

opposing cell zone. The two cell zones will slide relative to one another along the mesh

interface in discrete steps. The AMI (arbitrary mesh interface) operates by projecting

one of the patches geometry onto the other. In other words, the two sub-domains are

geometrically separated but numerically connected by the AMI that ensures that the

values of a generic field are the same on both sides of the interface. The external cell is a

simple steady partition which mesh does not move during the calculations. The internal

partition instead, the cylindrical one, is a cell zone that rotates during the simulations.

After each time step, in fact, the internal mesh is rotated of a prescribed angle and,

consequently, also the motion of the gear faces is in this manner correctly reproduced.

The advantage of this technique, is that it ensure the best accuracy avoiding, at the same

time, mesh deformation (that means an additional computational effort).

4.2 Multi reference frame (MRF)

The MRF model is a steady-state approximation in which individual cell zones move at

different rotational speeds. The flow in each moving cell zone is solved using the moving

reference frame equations in which additional terms related to the Coriolis acceleration

( ) and the centripetal acceleration ( ) are added to the momentum

conservation equation. At the interfaces between cell zones, a local reference frame

transformation is performed to enable flow variables in one zone to be used to calculate

fluxes at the boundary of the adjacent zone. It should be noted that the MRF approach

does not account for the relative motion of a moving zone with respect to the adjacent

zone; the grid remains fixed for the computation. This is analogous to freezing the

motion of the moving part in a specific position and observing the instantaneous flow

field with the rotor in that position. Although the MRF approach is clearly an

approximation, it can provide a reasonable model of the flow for many applications

considerably reducing the computational effort with respect to the sliding meshes

approach. However, the multiple reference frame model does not accurately simulate

the transient startup but gives just a regime solution of an unsteady problem. In this

specific case, the mesh is similar to the mesh adopted for the sliding mesh method with

the only difference that the two domains are not meshed separately as before, but they

are just two separate cell zones of the same mesh, resulting therefore in a conformal

mesh.

4.3 Model settings

In both cases, the adopted solver uses a PIMPLE (merged PISO-SIMPLE) algorithm. PISO

is an acronym for Pressure Implicit Splitting of Operators [15] for time dependent flows

while SIMPLE stands for Semi-Implicit Method for Pressure Linked Equations [16]

which is used for steady state problems. In the SIMPLE algorithm a pressure correction

term is used while the velocity corrections are neglected because they are unknown.

This results in rather slow convergence. The PISO algorithm also neglects the velocity

correction in the first step, but then performs one in a later stage, which leads to

additional corrections for the pressure. PIMPLE main structure is inherited from the

original PISO, but it allows equation under-relaxation, as in SIMPLE, to ensure the

convergence of all the equations at each time step. Furthermore, the presence of more

than one phases, implies the need of an additional transport equation for the volume

fraction α so to be able to calculate each mixture property in each cell as the weighted

average value between the properties of the two phases in that cell.

5. RESULTS Figure 2 shows the results in terms of windage power losses versus

circumferential velocity for different tip diameters and helix angles, the values of

which are specified in . It can be noticed that both the CFD-based approaches are in good

agreement with the measurements for the 102.5 mm tip diameter. On the other hand an

overestimation of the resistant power can be observed in the case of the 96.5 mm tip

diameter. The mismatch is mainly due to the uncertainty related to the real operating

conditions in which the experiments were carried out. As matter of fact, simulations

were performed imposing a constant oil temperature of 90°C. On the contrary, the oil

temperature during the experimental tests cannot be assumed as constant, since the

cooling/heating system (calibrated on 90°C) was not able to immediately compensate a

temperature fluctuation of the lubrication bath [7]. The oil temperature is a key

parameter for the prediction of the resistant torque, because directly influences both the

density and the viscosity of the fluid. Therefore, the uncertainty related to its value

strongly affects the results of the simulation.

a)

b) Figure 2: windage power losses versus circumferential velocity for a)

different tip diameters and b) different helix angles

The differences between the two numerical approaches depend on the fact that while the sliding meshes approach describes the whole transitory, along with the effects of non-steady instantaneous fluctuation of the velocity field, the MRF approach is a rough simplification that returns only the steady-state solution. The advantage of the latter technique, however, is the substantial reduction of computational effort with respect to the other approach. Adopting the same mesh for both the approaches (about 1.01E6 cells) and the same computational power (76GFLOPS), the computational time using the MRF approach was of approximately 30 min, about 50 times less than the time needed with the sliding meshes approach. Figure 3 shows the results of the numerical simulations in terms of churning power

losses vs. tangential velocity . The tip diameter seems to be the most influencing

parameter also on the churning power losses: an increment of the tip diameter from

96,5 mm to 102.5 mm (about 5%), in fact, induces an increment of about 100% in the

churning power loss. As expected, by halving the face width, the churning dissipation

reduces less than 50% due to the viscous contribution of the gear sides that remain

0

10

20

30

40

50

60

0 10 20 30 40

PV

G0

,W [

kW

]

vt [m/s]

b=40mm; β=0

;

Measured da=96,5mmSliding mesh da=96,5mmMRF da=96,5mmSliding mesh da=98,0mmMRF da=98,0mmMeasured da=102,5mmSliding mesh da=102,5mmMRF da=102,5mm

0

10

20

30

40

50

60

0 10 20 30 40

PV

G0

,W [

kW

]

vt [m/s]

da=102,5mm; b=40mm

Sliding Meshes Beta=0°MRF Beta=0°Measured Beta=0°Sliding Meshes Beta=20°MRF Beta=20°

unchanged. The influence of this two geometrical parameters is very similar to the

trends shown for a single phase lubrication (Figure 2). The main difference between the

two phenomena can be observed in the influence of the rotational speed: unlike the

churning losses, the windage ones evolve with rotational speed power three.

Figure 3: churning power losses versus circumferential velocity for a)

different tip diameters and b) different static lubricant levels

a) b)

c) d)

Figure 4: a) pressure distribution for 100% dip lubrication; b) pressure

distribution for 50% (L=0mm) dip lubrication; c- d) distribution of the oil-phase;

0

10

20

30

0 10 20 30 40

PP

VZ

0,C

[k

W]

n [rpm]

β=0

; L=0mm a)

da=102.5mm, b=40mmda=96.5mm, b=40mmda=96.5mm, b=40mm, MRF

0

20

40

60

0 10 20 30 40

PP

VZ

0,C

(W) [

kW

]

n [rpm]

da=102.5mm; b=40mm; β=0

b)

L=100%

L=0mm

Another interesting aspect is the influence of the internal volumes. Changing the sizes of

the box from 35x290xh185 mm to 100x600xh400 mm implies a drastic reduction of the

churning losses: the results with 50% oil filling (L=0) and the larger case are

comparable to the results obtained with the smaller case and less lubricant (L=-20mm).

This is due to the fact that, if the internal volume is small, the oil is necessary involved in

the circulation. Figure 4 shows clearly the differences between windage and churning.

The losses generated if a single phase is present, are dependent just from the geometry

and from the rotational speed (and were evenly distributed on all the teeth), while, if

two phases are present, the pressure distribution on the different teeth (and,

consequently, the losses) depends from the oil distribution: peaks of pressure, in fact,

arise where the volume fraction of the lubricant is higher due to its inertial effects.

Figure 4b shows clearly this phenomenon: the pressure peaks are in correspondence of

the zones with the highest oil volume fraction. This fact points out, once again, the

importance to be able to describe the dynamics of the fluids for the prediction of the

power losses. The losses, in fact, arise both due to pressure and shear contributions but

the pressure effects are of a higher order of magnitude as shown in Figure 5.

a) b)

c) d)

Figure 5: subdivision of the power losses into pressure and viscous contributions

for 100% dip lubrication

6. CONCLUSIONS

A CFD open-source code has been adopted in order to study the influence of some

operating and geometrical parameters on the windage and churning power losses of

98,7% 99,5% 99,7%

1,31% 0,51% 0,33%

90%

95%

100%

2000 5000 8000

n [rpm] β=0° da=102,5mm;

viscous inertial

97.0% 98,7% 99,2%

3,03% 1,32% 0,84%

90%

95%

100%

2000 5000 8000

n [rpm] β=0° da=98mm;

viscous inertial

96,2% 98,3% 98,9%

3,77% 1,70% 1,10%

90%

95%

100%

2000 5000 8000

n [rpm] β=0° da=96,5mm;

viscous inertial

86,8% 93,9% 96,0%

13,2% 6,1% 4,0%

85%

95%

2000 5000 8000

n [rpm] β=20° da=102,5mm;

viscous inertial

gears. From the results of the simulations it can be seen that the tip diameter as well as

the face with have an enormous influence on the power losses both for single phase and

for mixture lubrication. The lubricant level has also a big influence on the churning

losses: in the specific geometric configuration considered, a halving of the amount of

lubricant implies a reduction of about 60% of the losses, but if the volumes inside the

gearbox increases, the reduction can be also more significant. These considerations

points out the importance of being able to optimize the gearbox using predictive models.

Due to the complex internal fluid dynamics, the empirical models provided by literature

often give results that cannot be trusted. Nevertheless, being able to predict the power

losses avoiding or reducing the need of prototyping and testing can give big advantages

in terms of costs and developing times to gearbox manufacturers. For this reason the

approach proposed here, with the two possible ways of calculations (Sliding meshes and

MRF), can be effectively used as an useful tool in the design of gear transmissions. The

results have shown that the data obtained with the simplified MRF approach, even

though it can only represent the regime condition, are comparable with the

measurements and are computationally less expensive that the sliding mesh approach.

On the contrary, this last approach has the advantage of being capable of well describing

also the initial transient phase. The advanced of MRF in terms of computational time for

this kind of simulations is in the order of magnitude of 1:50, making it suitable also for

industrial applications where the time is an important issue.

The aim of this study is both to provide data that can be effectively used by engineers in

the design practice for a first rough estimation, but also to prove, once again, that CFD is

an effective approach for this kind of investigations that provides a lot of useful and

detailed informations regarding the power loss mechanism. These results will be the

basis for a larger research program regarding all the no-load power losses of gears in

which also the real condition concerning more than one single rotating gear (mating

gear pair) will be studied and simulated.

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