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Determination of oil distribution and churning power loss of gearboxes by finite volume CFD method
Author’s Accepted Manuscript Determination of Oil Distribution and Churning Power Loss of Gearboxes by Finite Volume CFD Method Hua Liu, Thomas Jurkschat, Thomas Lohner, Karsten Stahl www.elsevier.com/locate/jtri
PII: DOI: Reference:
S0301-679X(16)30527-8 http://dx.doi.org/10.1016/j.triboint.2016.12.042 JTRI4526
To appear in: Tribiology International Received date: 31 October 2016 Revised date: 22 December 2016 Accepted date: 24 December 2016 Cite this article as: Hua Liu, Thomas Jurkschat, Thomas Lohner and Karsten Stahl, Determination of Oil Distribution and Churning Power Loss of Gearboxes by Finite Volume CFD Method, Tribiology International, http://dx.doi.org/10.1016/j.triboint.2016.12.042 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Determination of Oil Distribution and Churning Power Loss of Gearboxes by Finite Volume CFD Method Hua Liu*, Thomas Jurkschat, Thomas Lohner, Karsten Stahl Gear Research Centre (FZG), Technical University of Munich, Boltzmannstraße 15, D-85748 Garching b. München, Germany * Corresponding author: [email protected]
Abstract Sufficient oil supply of all machine elements in gearboxes is usually required to avoid damage and to reduce friction during operation. However, dip lubrication and injection lubrication always result in hydraulic gear power losses. Especially in high-speed and dip-lubricated gearboxes with high oil levels, churning losses can represent a significant portion of the total power loss. Current analytical and empirical approaches for examining the churning losses are often limited to certain constraints and operating conditions. Furthermore, they do not provide any information about the oil distribution. CFD (Computational Fluid Dynamics) methods offer a very flexible way to investigate the oil distribution and the churning power losses, with almost no restrictions on the housing shape and operating conditions. In this paper, a CFD model based on the Finite Volume Method (FVM) is built to investigate the oil distribution and the churning losses inside a single-stage gearbox of the FZG back-to-back efficiency gear test rig. Thereby, a three-dimensional finite volume simulation model considering two-phase flow is applied. The results are compared with measurements of the churning losses and with high-speed camera recordings. The comparisons show very good agreement and high potential for predicting the oil distribution and the churning losses in modern transmission systems. Keywords: Gear efficiency, churning losses, dip lubrication, CFD
Nomenclature a
mm Center distance
αwt
° Working pressure angle
b
mm Face width
da
mm Tip diameter
ϑoil
°C Oil sump temperature J Turbulence kinetic energy Pas Turbulence viscosity - Eddy viscosity
mn
mm Normal module mm²/s Kinematic viscosity
n
rpm Rotational speed
N/m² Pressure PVD
W Seal loss
PVLP
W Load-dependent bearing loss
PVL0
W No-load bearing loss
PVX
W Auxilary losses
PVZ0
W No-load gear loss
PVZP
W Load-dependent gear loss kg/m³ Density m/s Velocity - Reynolds-stress tensor
z
x
- Addendum modification coefficient, Cartesian coordinate axis
y
- Cartesian coordinate axis
- Number of teeth, Cartesian coordinate axis
Indices 1
Pinion
2
Wheel
oil
Oil
1. Introduction Efficiency has become a major driving factor in the development of transmission systems. Churning losses can contribute a large amount to the total loss and have particularly a significant influence in high-speed gearboxes and dip-lubricated transmission systems with high immersion depths (Mauz [1]). It is possible to derive measures for reducing these losses, which allows a significant increase in efficiency and a reduction of pollutant emissions to be achieved. In general, losses occur whenever mechanical components move relative to each other. In lubricated systems, moving components result in molecular interactions in the lubricant and hence in losses. The components that cause losses in gearboxes are the gears, bearings and seals. Auxiliary losses are caused e.g. by clutches and planet carriers. Generally, the losses can be divided into load-dependent and no-load losses. The following equation gives an overview of the losses inside a gearbox (Niemann and Winter [2]): gears
bearings
seals auxiliaries (1) no-load losses load-dependent losses
Concli [3] exemplifies the proportional contributions of several kinds of power loss in a single stage geared transmission. Despite of the strong dependency of the relative portions of noload losses on different kinds of machine elements and operating conditions, the no-load
gear loss PVZ0 often represents the largest contributor to no-load losses, as can be seen in Figure 1.
sessol rewoP Figure 1: Power Power losses of single-stage gear transmission system (reproduced from Concli [3] losses with permission of the author) PZVP
PVZP
0ZVP
PVZ0
PLVP
PVLP
0LVP
PVL0
DVP
PVD
XVP
PVX
PVZP
PVZ0
PVLP
PVL0
PVD
PVX
Different approaches are available to calculate the churning losses in transmission systems. These are: -
Measurements on test rigs
-
Calculation with empirical equations
-
Simulation with Computational Fluid Dynamics (CFD)
Various studies on the churning power losses inside gearboxes have been conducted. Terekhov [4] and Boness [5] were one of the first to investigate the churning losses in diplubricated geared transmissions for a wide range of operating conditions. They introduced several empirical equations based on different Reynolds and Froud numbers. The findings show that the tip diameter, the gear width, the immersion depth, the angular velocity, the volume of oil and the density are the main influence factors on the churning loss of gears. Changenet et al. [6, 7] investigated the windage and churning losses of spur gears. According to his findings, the tip diameter and the gear width have the biggest influence on both losses. Höhn et al. [8, 9] investigated the no-load power losses of a dip-lubricated FZG efficiency gear test rig. They identified the oil level, the tip diameter and the circumferential speed as the biggest influencing factors. They ascertained that the oil distribution itself has a remarkable impact on the immersion depth of the gears, as well as on the life time and the efficiency of transmission systems. Maurer [10] and Walter [11] conducted experimental investigations on the different kinds of hydraulic losses in a gearbox. They identified a number of different influencing factors for different kinds of hydraulic losses. Empirical equations that are nowadays widely used in drive technology were subsequently derived from these investigations. However, Mauz [1] and Luke [12] pointed out that there can be considerable uncertainties when applying these empirical equations. Continuous increases in computational capacity have brought CFD methods into the spotlight as a new way of investigating churning power losses (Concli [3]). These methods have several advantages. Firstly, CFD simulations can additionally provide a very detailed view of the oil distribution in a gearbox, which is not possible with other methods. Secondly, CFD is highly flexible with respect to the gear geometry and the gear housing, which allows a wide range of operating conditions and geometric influences to be investigated. This leads to the third advantage: empirical equations are often limited to certain constraints and operating conditions, which in turn limits the applicability of these equations. This can be overcome by CFD methods. Marchesse et al. [13, 14] were among the first to investigate the windage losses of single rotating, dry-lubricated spur gears with the finite volume CFD method. The results agree well with the experimental data with a maximum deviation of 8%. Kvist [15] investigated the oil
distribution and the churning losses for a single rotating helical gear with dip lubrication using the finite volume CFD method. Both the oil distribution and the churning losses are in good agreement with experimental measurements. The oil distribution was compared to video recordings of a transparent gearbox. The recordings showed that the main features of the oil flow can be captured well by the CFD simulations. Concli et al. [16] analyzed the churning losses and the oil distribution of a planetary gearbox with dip lubrication. As the squeezing losses caused by the gear meshing were not the main focus of this study, they used a simplified model without gear meshing. The simulation results are in very good agreement with experimental results for the investigated operating conditions. A review of the current state of research shows that various studies have already indicated the effectiveness of CFD in determining the churning losses of gear drives. However, CFD approaches are still subject to three main limitations (Concli and Gorla [17]): -
The complexity of setting up a transient mesh model The computational effort The selection of a suitable simulation model
In order to improve our understanding of the effects involved in churning losses, a CFD simulation model was set up to investigate the oil distribution and the churning losses of a singlestage gearbox in the FZG efficiency gear test rig. Within the framework of this article, the comparison between numerical results and measurements shows that CFD offers a very promising way to predict the power losses and the oil distribution for transmission systems. 2. Object of Investigation and Operating Conditions The object of investigation of this study is the FZG efficiency gear test rig. In order to compare CFD simulation results with experimental data, the churning loss torque will be measured for all considered operating conditions. Furthermore, a high-speed camera system is installed in front of a transparent test gearbox to capture high-resolution recordings of the oil distribution during a start-up case of the gear pair. The FZG efficiency gear test rig (Figure 2) is a modified FZG back-to-back gear test rig, and is based on the principle of power circulation. The main components are the electric engine, the loss torque meter, the slave gearbox, the shaft with the load clutch, the driving shaft with the load torque meter and the test gearbox. The transmitted torque is applied by a load clutch and measured by a load torque meter. The electric engine only supplies the total power loss, which is measured by a loss torque meter between the electric engine and the power circle. In no-load operating conditions, only negligible torque is applied to the load clutch, so that the no-load loss can be evaluated. The measured total no-load power loss consists of the no-load losses caused by the gears, bearings and seals. The test gearbox consists of two gears, four bearings and two seals, whereas the slave gearbox consists of two gears, four bearings and three seals. In order to separate the gear no-load losses from the total no-load loss, the no-load losses of the bearings and seals are calculated by means of analytical approaches (SKF [19], ISO 14179-1 [20]), and subtracted from the total no-load loss. Since the test and slave gearbox have the same gear pairs, the resulting value is divided by two to obtain the churning loss of one gearbox. The experimental setup allows the churning losses to be evaluated at different speeds, oil tem-
peratures and lubrication conditions. During operation, the oil temperature is maintained at the desired value. Figure 2: Mechanical layout of the FZG efficiency gear test rig acc. to Lohner et al. [18] The considered gears are of type C-PT (Schedl [21]). The geometry data for both the pinion and the wheel are listed in Table 1. Table 1: Geometry of the type C-PT gears a in mm Pinion
91.5
Wheel
z
mn in mm
16
4.5
24
4.5
α in ° 20.0
x1, x2 in mm
b in mm
da in mm
0.182
14
82.46
0.171
14
118.36
Different operating conditions in terms of the oil sump temperature and the speed of the gear pair are investigated (Table 2, Table 3). The different oil sump temperatures result in different lubricant viscosities and slightly different oil fill levels for the gearboxes. The oil fill level is chosen so that at least one tooth of both the pinion and the wheel dips into the oil sump. Table 2: Operating conditions for comparison of the churning loss torque Pressure in bar
Oil sump temperature ϑoil in °C
Oil fill level
Rotational speed of the wheel n2 in rpm
Circumferential speed vt at pitch circle in m/s
1
60
21.8 mm below the middle axis
348
2.0
1444
8.3
3474
20.0
348
2.0
1444
8.3
3474
20.0
90
20.0 mm below the middle axis
Table 3: Operating conditions for comparison of the oil distribution based on high-speed camera recordings for the start-up case Pressure in bar
Oil sump temperature ϑoil in °C
Oil fill level
Rotational speed of the wheel n2 in rpm
Circumferential speed vt at pitch circle in m/s
Acceleration time for the gears
1
40
36.7 mm below the middle axis
407
2.3
~1.5s
A mineral-based oil FVA3A with 4% additive Anglamol A99 (Laukotka [22]) is considered. Investigations show that the churning losses are primarily influenced by the oil’s viscosity and density properties (Mauz [1]). The following table gives an overview of the main properties of FVA3A, as measured. Table 4: Oil properties of the considered mineral oil FVA3A (Laukotka [22]) Oil
Kinematic viscosity at 40°C in mm²/s
Kinematic viscosity at 100°C in mm²/s
Density at 15°C kg/m³
in
FVA3A
90.0
10.4
884.1
Based on the Ubbelohde-Walter viscosity-temperature relationship (DIN 51563 [23]), the data in Table 4 results in viscosity values of = 90 mm²/s, = 44 mm²/s and =15 mm²/s. 3. Numerical Model The numerical model is implemented and solved in the commercial CFD software Ansys Fluent 16.0. 3.1 Geometry and mesh For finite volume CFD methods, the conservation equations are solved iteratively on the volumes. Consequently, a mesh representing the test gearbox of the FZG efficiency test rig has to be set up first. The mesh of the single-stage gearbox essentially consists of four domains: The pinion domain, the wheel domain, the gearbox domain and the domain of the remeshing zone. In contrast to structure analysis, the entire model represents a negative model of the gearbox. The mesh of each single domain is connected with the other domains as shown in Figure 3. The mesh of the pinion and wheel domain is discretized with extruded hexahedral elements and does not undergo any mesh deformation. For a given domain size, this mesh topology can be created with far fewer cells than the equivalent mesh consisting of tetrahedral elements. The gearbox domain is meshed with extruded tetragonal elements and does also not undergo any mesh deformation. According to the Stokes adhesive condition, the velocities of fluid particles adhering to the walls must be zero and the velocities of fluid particles on the pinion and wheel must be equal to the corresponding circumferential speed. As a matter of fact, high velocity and pressure gradients arise on the contour of the rotating bodies. Therefore, the resolution on the contour of the pinion and wheel domain has to be very fine in order to obtain reasonable results for the loss torque. For this reason, a few layers consisting of long hexahedral elements are adopted near the contour lines. During the simulation, the meshes of the pinion and wheel domain rotate inside the gearbox domain at the predefined rotational speeds. The domain of the remeshing zone fills the cavity between the gearbox domain and the pinion and wheel domain. During operation, the meshing zone of pinion and wheel is a transient area that changes at every meshing position. Therefore, the so-called remeshing zone is adopted, which consists of a deformable meshing structure that changes with every time step of the rotating pinion and the wheel domain. This domain of the remeshing zone is discretized with deformable extruded tetrahedral elements that follow every rigid movement on the planar area. For numerical analysis, the element size has in general remarkable influence on the results. It is important to find a compromise between the quality of the results and the calculation time. Thus, the average element size in the remeshing zone has been varied from 0.65 mm to 1.00 mm during test studies. No significant improvement of the simulated no-load losses has been observed for an element size smaller than 0.85 mm. Hence, the average element size in the remeshing zone has been set to 0.85 mm for the simulation analyses. In order to avoid numerical singularities due to the very small gap between the tooth flanks i.e. backlash, both the pinion and the wheel are scaled to 99% of their actual size. It is not expected that the results with respect to oil distribution and loss torque will be significantly influenced by this simplification. According to Mauz [1], the squeezing losses caused by compressing the oil particles between tooth flanks can be neglected for the considered direction of rotation in Figure 3. Figure 3: Entire mesh model (top left); outer domain of the gearbox (top right); domain of remeshing zone (bottom left); pinion and wheel domain (bottom right)
During the numerical calculation process, the conservation equations are solved iteratively on the grid points. Generally, the numerical calculation can be divided into three steps: -
Integration of the conservation equations in the control volume Discretization by substitution of all gradients with the finite difference method, so that the differential equations are turned into algebraic equations Solution of the approximated equations by an iterative method
3.2 Navier-Stokes equations The Navier-Stokes equations consist of five different conservation equations: the mass equation, the conservation of momentum equation in the x-, y- and z-directions, and the conservation of energy equation (Gersten [24]). The conservation of mass or continuity equation states that mass is constant in all directions in a Cartesian coordinate system. (2) The conservation of momentum equations can be understood as Newton’s Third Law for compressible fluid particles in a continuum. With Einstein’s summation notation, the equation reads as follows: (
(
)
)
(3)
The energy conservation equation is not considered, as it is assumed that thermal influences can be neglected in no-load operating conditions (Concli [25]). The computation time of turbulent flows can become extremely high if every single whirl is dissolved. This is because an extremely fine mesh with a very high number of elements is necessary. To overcome this, Reynolds [26] derived the Reynolds-averaged Navier-Stokes equations (RANS). In these equations, the fluid flow quantities in the conservation equations are separated into a time-weighted average term and an oscillating term. The oscillating term is known as the Reynolds stress tensor: (4) The Reynolds stress tensor is unknown. Hence, further turbulent models are required. These models do not dissolve every single whirl, so a coarser mesh can be used, which requires much less computation time. 3.3 Turbulence models Boussinesq attempted to find a closed form for Equation (3) by introducing the concept of eddy viscosity. He postulated that the Reynolds stress is caused in the same way as the molecular viscosity [26]. (
(
)
)
, mit
(5)
The term is called the eddy viscosity. It describes the increase in the viscosity due to the turbulent oscillating movement. It is proportional to both the turbulent length and the turbulent velocity scale . It follows that the RANS equations can be written in closed form as long as the eddy viscosity is determined. Over the years, different eddy viscosity models have been developed. The quality of the eddy viscosity model greatly depends on the number of turbulence variables and the number of equations. There are a number of different turbulence models, ranging from simple algebraic equations to 2nd order differential equations. In this study, a k-ε model consisting of two coupled transport equations was used. The model describes the development of the kinetic energy and the isotropic dissipation rate ε with two transport equations. The first equation solves the turbulence kinetic energy k. The second equation describes the dissipation rate ε, which characterizes the transformation of turbulent kinetic energy into thermal internal energy. ( (
(
)
(
)
)
(
)
(6)
)
(
)
(7)
(
(
)
)
(8)
For test cases, different turbulence models, e.g. the k- SST-model (two equations) and the Transition SST model (two equations) have been applied to the simulation model. Marchesse et al. [13] have shown that there are no remarkable differences with respect to the windage losses between the k- SST and the k-ε model. In general, the k-ε model represents a good compromise between computation time, result quality and stability. Also the resolution of the interphase between oil and air and the loss torque can be captured well by this turbulence model. 3.4 Discretization In order to solve the conservation equations on the volumes, the differential equations need to be transformed into finite difference equations. This process is called discretization. Generally, a distinction is made between spatial and temporal discretization.
The analytical derivatives in the conservation equations are replaced and approximated by partial derivatives of a certain order. The accuracy of the results increases with the order of the partial derivatives and the fineness of the grid. In this study, a second-order formulation is used for most instances of spatial discretization, whereas a first-order transient formulation is selected for temporal discretization. 3.5 Remeshing method The geometry of the meshing zone between the pinion and the wheel changes at each time step. Hence a transient mesh is required, for which the remeshing method is applied in this study. To do this, the finite elements are bounded by grid points. Just like a solid body, the grid has a certain stiffness, which can be stretched and contracted at every location according to the grid movement. Due to the deformation of the elements, the mesh quality deteriorates relative to the initial mesh. In this study, the quality of the mesh is checked at each time step. The mesh quality is evaluated based primarily on the following mesh parameters: -
Minimal length of the element Maximal length of the elements Maximal skewness of the element face Maximal skewness of the element volume
Certain initial values for all the mesh parameters are defined based on the initial mesh. If one of the following mesh parameters falls below the predefined value, the mesh is reconstructed. At the same time, all data for fluid variables are converted into the grid points of the new elements. As the mesh quality is preserved, the local convergence criteria of 1∙10-5 with respect to the mass and momentum is fulfilled for each time step. 4. Results The CFD simulations in this study focus on the oil distribution and the churning losses. Thereby, CFD simulation results are compared with measured gear no-load torques and high-resolution camera recordings from the FZG efficiency gear test rig. Depending on the operating condition, the calculation time for the transient three-dimensional simulation model considering two-phase flow lasts about eight to ten hours per revolution. A quasi-stationary condition for the flow field is typically reached after three to four revolutions. 4.1 Velocity distribution First, the plausibility of the simulation results is discussed by velocity distribution plots. According to the Stokes adhesive condition, a Newtonian fluid particle on the pinion and wheel must be equal to the corresponding circumferential speed. Figure 4 shows an example of the velocity distributions of the fluid particles between the tooth flanks of the wheel at different circumferential velocities at 9 o´clock position according to the circumferential speeds specified in Table 2 with ϑoil=60°C. It can clearly be seen that the velocity of the fluid particles on the tip diameter is equal to the velocity at the tip diameter of the gear pair. The velocity of particles outside of the tip diameter gradually decreases as the distance from the center of the wheel increases. The fluid particles between the tooth flanks are dragged by the teeth and are expected to have similar velocity, whereas the velocity outside the tip circle drastically drops to lower values. The large difference between the velocities in this transition areas leads to turbulence, which brings about higher velocity in this area (see Figure 4). Figure 4: Simulated velocity distribution between wheel flanks at different rotational speeds for ϑoil=60°C at 9 o´clock position of the wheel
3.0
15.0
25.0
1.5
7.5
12.5
0 m/s
0 m/s
0 m/s
vt=2.0 m/s
vt=8.3 m/s
vt=20.0 m/s
4.2 Oil distribution Figure 5 shows the CFD simulation results of the oil distribution inside the test gearbox at different rotational speeds for ϑoil=60°C. The results show that a large amount of oil under the meshing zone of the pinion and wheel is replaced at the right and left sides of the gearbox. This observation correlates very well to the findings made by Concli [17]. At the same time, oil particles are propelled towards the top of the gearbox. Despite the fact that the pinion has a higher rotational speed, this spin effect is larger in the wheel, i.e. the amount of oil swept towards the left wall of the gearbox is larger than the oil swept towards the right wall by the pinion. Hence, for this particular case the spin effect is more strongly influenced by the tip diameter than by the rotational speed. As expected, the spin effect increases with the rotational speed of the gears, which can be clearly seen by comparing the three different rotational speeds shown in Figure 5. Due to the centrifugal forces of the pinion and wheel, oil droplets are fragmented into a great number of small particles, forming an oil fog in the meshing zone of the gearbox. It can be clearly observed that the amount of air fraction inside the lubricant increases with the rotational speed of the gears. This can be qualitatively observed experimentally as the color of the oil turns form dark yellow before test to very light yellow after test especially when the circumferential speed of the gears is larger than vt>5 m/s. A similar observation has been made by Otto [27]. The obtained air fractions also correlate well to results from online monitoring of lubricant aeration by Leprince et al. [28].
Figure 5: Simulated oil distribution at vt=2.0 m/s, vt=8.3 m/s and vt=20.0 m/s for ϑoil=60°C
vt=2.0 m/s
Oil fraction 1
vt=2.0 m/s
0.5
0 vt=8.3 m/s
Oil fraction 1
vt=8.3 m/s
0.5
0 vt=20.0 m/s
Oil fraction 1
vt=20.0 m/s
0.5
0
In order to validate the oil distributions obtained by CFD simulation, the oil distribution of a start-up case of the gear pair is investigated and compared to the recordings obtained from a high-speed camera system at two different times. In order to adjust the simulation to the experiment, the kinematic of the gears are described by a time dependent angular velocity with constant acceleration during start-up of the electric motor. Figure 6 shows the comparison for a start-up after 50° and after 100° of the pinion under the operating conditions listed in Table 3. From a qualitative viewpoint, the comparisons show very good agreement with the recordings. It can be clearly seen that the oil particles adhere to the tooth flanks at the very first moment (e.g. after 50° rotation). After 100° rotation of the pinion, the oil particles start to spin off from the gear flanks by forming oil threads starting at the tooth tip. The CFD results have shown to be physically plausible, and show very good agreement with recordings from a high-speed camera. In the next step, the churning losses are investigated.
vt=2.3 m/s after 50°
vt=2.3 m/s after 100°
Figure 6: Oil distribution by CFD simulations (left) and recordings by high-speed camera (right) during a start-up of the gear pair after 50° and 100° rotation of the pinion (operating conditions: ϑoil=40°C and vt=2.3 m/s) 4.3 Churning losses Figure 7 shows the measurements and CFD simulation results for the churning loss torque under the operating conditions listed in Table 2. Again, the CFD simulation results are in very good agreement with the measurement results.
C-PT FVA 3A oil=90°C oil fill level: -20 mm below middle axis
1.25
Churning loss torque
1 0.75
CFD simulation Experiment
0.25 0 5 10 15 Circumferential velocity vt in m/s
1.5 1.25 1
0.5
0
in Nm
1.5
20
Churning loss torque
in Nm
Figure 7: Comparison of gear churning loss torque TVZ0,C between CFD simulation and measurement for ϑoil=90°C (left) and ϑoil=60°C (right)
0.75
C-PT FVA 3A oil=60°C oil fill level: -21.6 mm below middle axis
CFD simulation Experiment
0.5 0.25 0 0
5 10 15 Circumferential velocity vt in m/s
20
As shown experimentally by Otto [9], increasing the rotational speed has the greatest influence on the churning losses. The turbulence in the oil sump increases with the rotational speed, the empty space of the gearbox is gradually filled with oil, which in turn leads to higher churning loss torque. Decreasing the temperature from 90°C to 60°C triples the viscosity and induces a lower oil flow velocity. Mauz [1] showed that with increasing oil viscosity and circumferential speed (vt > 5 m/s) , the replaced oil particles do not have enough time to flow back to the area between the tooth flanks before they are swept back to the oil sump by the tips of the teeth. Consequently, the gears can create an empty air space with almost no oil, which in turn leads to lower resistant torque and hence churning loss torque. This explains the lower churning loss torque for ϑoil=60°C compared to ϑoil=90°C at vt = 20 m/s (see Figure 7). The arithmetic mean value and the scatter band in Figure 7 are based on 20 measurements for each operating point. It should be noted that the experimental measurements always involve measuring uncertainties, e.g. due to imbalance in the shafts, the relative error of the loss torque meter, and limited data processing accuracy. Furthermore, the fact that the churning loss torque is determined by subtracting the seal and bearing losses (calculated analytically) from the total no-load loss (measured) introduces another source of uncertainty. Further investigations will therefore include separate measurement of bearing and sealing losses. 5. Conclusion In this study, a CFD model was built up and used to investigate the oil distribution and the churning losses inside a single-stage gearbox of the FZG back-to-back efficiency gear test rig. The investigations show that the finite volume CFD method is a valid tool for predicting the churning loss and oil distribution inside a gearbox. The results for both the oil distribution and the churning loss torque are in very good agreement with the high-speed camera recordings and experimental measurements of the loss torque. The investigations have shown that the rotational speed has the greatest influence on the churning losses, whereas the spin effect on the oil particles is mostly influenced by the tip diameter. The geometrical influence of the pinion and wheel, the impact of different immersion depths, and the influence of different oil types on the churning losses will be investigated in further studies. In the future, new power transmission systems can be developed and optimized for efficiency by means of CFD models.
References [1] Mauz, W., Hydraulische Verluste von Stirnradgetrieben bei Umfangsgeschwindigkeiten bis 60 m/s. Dissertation, Universität Stuttgart,1987, [2] Niemann, G. and Winter, H., Maschinenelemente Band 2: Getriebe allgemein Zahnradgetriebe- Grundlagen, Stirnradgetriebe. 2, Springer-Verlag, Berlin Heidelberg, 2003, [3] Concli, F., Efficiency of gear transmissions and CFD analysis of the load independent power losses. Dissertation, Politecnico di Milano,2012, [4] Terekhov, A. B., Hydraulic losses in gearboxes with oil immersion. Russian Engineering Journal, 5, 1975, [5] Boness, R. J., Churning losses of discs and gears running partially submerged in oil. Chicago, Illinois, 1989,
[6] Changenet, C., Leprince, G., Ville, F., and Velex, P., A Note on Flow Regimes and Churning Loss Modelling. 11th International Power Transmission and Gearing Conference, DOI=10.1115/DETC2011-47227, 2012, 549–555. [7] Changenet, C. and Velex, P., A Model for the Prediction of Churning Losses in Geared Transmissions—Preliminary Results. J. Mech. Des., 129, 1, 2007, 128. [8] Höhn, B.-R., Michaelis, K., and Völlmer, T., Thermal rating of gear drives: balance between power loss and heat dissipation. 1996, [9] Otto, H.-P., Michaelis, K., and Höhn, B.-R., Eds.Influences on no-load gear losses. 3rd European Conference on Tribology, Wien, 2011, [10] Maurer, J., Lastunabhängige Verzahnungsverluste schnelllaufender Stirnradgetriebe. Dissertation, Universität Stuttgart,1994, [11] Walter, P., Langenbeck, K., and Dittrich, O., FVA-Nr. 44/I. Heft 118 Anwendungsgrenzen für die Tauchschmierung von Zahnradgetrieben, Plansch- und Quetschverluste bei Tauchschmierung. Frankfurt/Main, 1982, [12] Luke P. , Olver A., A study of churning losses in dip-lubricated spur gears. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, Vol. 213, 1999, [13] Marchesse, Y., Changenet, C., Ville, F., and Velex, P., Eds.Investigations on CFD Simulation for Predicting Windage Power Losses in Spur Gears 3rd International Conference on Integrity, Reliability and Failure, Porto/Portugal. 2009, [14] Pallas, S., Marchesse, Y., Changenet, C., Ville, F., and Velex, P., Application and validation of a simplified numerical approach for the estimation of windage power losses in spur gears. Computers & Fluids, 84, 2013, 39–45. [15] Kvist, T., Oil splashing simulation using CFD. Trollhättan, Schweden, 2012, [16] Concli, F., Conrado, E., and Gorla, C., Analysis of power losses in an industrial planetary speed reducer -Measurements and computational fluid dynamics calculations. Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology, 228, 1, 2013, 11–21. [17] Concli, F. and Gorla, C., Numerical modeling of the power losses in geared transmissions: Windage, churing and cavitation simulations with a new integrated approach that drastically reduces the computational effort. Tribology International, 103, 2016, 58–68. [18] Lohner, T., Mayer, J., Michaelis, K., Höhn, B.-R., and Stahl, K., On the Running-In Behavior of Lubricated Line Contacts. Journal of Engineering Tribology. Advance online publication, 2015, [19] SKF-Gruppe, SKF Hauptkatalog. Media-Print INformationstechnologiePaderborn, 1994, [20] ISO 14179-1, Gears – Thermal capacity – Part 1: Rating gear drives with thermal equilibrium at 95 °C sump temperature. 14179-1, 2001, [21] Schedl, U., Einfluss des Schmierstoffs auf die Grübschenlebensdauer einsatzgehärteter Zahnräder. Dissertation, TU München,1998, [22] Laukotka, E. M., FVA-Heft Nr. 660 Referenzöle – Datensammlung. Frankfurt/Main, 2003, [23] DIN 51563, Prüfung von Mineralölen und verwandten Stoffen - Bestimmung des Viskosität-Temperatur-Verhaltens - Richtungskonstante m. 51563, 2011, [24] Gersten, K., Einführung in die Strömungsmechanik. Vieweg, Braunschweig, 1992, [25] Gorla, C., Concli, F., Stahl, K., Höhn, B.-R., Michaelis, K., Schultheiß, H., and Stemplinger, J.-P., CFD Simulations of Splash Losses of a Gearbox. Advances in Tribology. Advance online publication, 2012, [26] Kistner, B., Modellierung und numerische Simulaion der Nachlaufstruktu von Turbomachinen am Beispiel einer Axialturbinenstufe. Dissertation, TU Darmstadt,1999, [27] Otto, H.-P., Flank Load Carrying Capacity and Power Loss Reduction by Minimised Lubrication. Dissertation, FZG, TU München,2009, [28] Leprince G., Changenet C., Ville F., Velex Ph., Jarnias F., Influence of oil aeration on churning losses. In JSME International Conference on Motion, 2009,
Highlights Finite Volume CFD method was applied to a dip lubricated gear box
Oil distribution and churning losses were studied for various operating conditions
A 3D transient two-phase flow model based on the remeshing technique was built
Simulated oil distributions correlate very well to high-speed camera records
Churning losses correlate very well to measurements at the FZG efficiency test rig
PII: DOI: Reference:
S0301-679X(16)30527-8 http://dx.doi.org/10.1016/j.triboint.2016.12.042 JTRI4526
To appear in: Tribiology International Received date: 31 October 2016 Revised date: 22 December 2016 Accepted date: 24 December 2016 Cite this article as: Hua Liu, Thomas Jurkschat, Thomas Lohner and Karsten Stahl, Determination of Oil Distribution and Churning Power Loss of Gearboxes by Finite Volume CFD Method, Tribiology International, http://dx.doi.org/10.1016/j.triboint.2016.12.042 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Determination of Oil Distribution and Churning Power Loss of Gearboxes by Finite Volume CFD Method Hua Liu*, Thomas Jurkschat, Thomas Lohner, Karsten Stahl Gear Research Centre (FZG), Technical University of Munich, Boltzmannstraße 15, D-85748 Garching b. München, Germany * Corresponding author: [email protected]
Abstract Sufficient oil supply of all machine elements in gearboxes is usually required to avoid damage and to reduce friction during operation. However, dip lubrication and injection lubrication always result in hydraulic gear power losses. Especially in high-speed and dip-lubricated gearboxes with high oil levels, churning losses can represent a significant portion of the total power loss. Current analytical and empirical approaches for examining the churning losses are often limited to certain constraints and operating conditions. Furthermore, they do not provide any information about the oil distribution. CFD (Computational Fluid Dynamics) methods offer a very flexible way to investigate the oil distribution and the churning power losses, with almost no restrictions on the housing shape and operating conditions. In this paper, a CFD model based on the Finite Volume Method (FVM) is built to investigate the oil distribution and the churning losses inside a single-stage gearbox of the FZG back-to-back efficiency gear test rig. Thereby, a three-dimensional finite volume simulation model considering two-phase flow is applied. The results are compared with measurements of the churning losses and with high-speed camera recordings. The comparisons show very good agreement and high potential for predicting the oil distribution and the churning losses in modern transmission systems. Keywords: Gear efficiency, churning losses, dip lubrication, CFD
Nomenclature a
mm Center distance
αwt
° Working pressure angle
b
mm Face width
da
mm Tip diameter
ϑoil
°C Oil sump temperature J Turbulence kinetic energy Pas Turbulence viscosity - Eddy viscosity
mn
mm Normal module mm²/s Kinematic viscosity
n
rpm Rotational speed
N/m² Pressure PVD
W Seal loss
PVLP
W Load-dependent bearing loss
PVL0
W No-load bearing loss
PVX
W Auxilary losses
PVZ0
W No-load gear loss
PVZP
W Load-dependent gear loss kg/m³ Density m/s Velocity - Reynolds-stress tensor
z
x
- Addendum modification coefficient, Cartesian coordinate axis
y
- Cartesian coordinate axis
- Number of teeth, Cartesian coordinate axis
Indices 1
Pinion
2
Wheel
oil
Oil
1. Introduction Efficiency has become a major driving factor in the development of transmission systems. Churning losses can contribute a large amount to the total loss and have particularly a significant influence in high-speed gearboxes and dip-lubricated transmission systems with high immersion depths (Mauz [1]). It is possible to derive measures for reducing these losses, which allows a significant increase in efficiency and a reduction of pollutant emissions to be achieved. In general, losses occur whenever mechanical components move relative to each other. In lubricated systems, moving components result in molecular interactions in the lubricant and hence in losses. The components that cause losses in gearboxes are the gears, bearings and seals. Auxiliary losses are caused e.g. by clutches and planet carriers. Generally, the losses can be divided into load-dependent and no-load losses. The following equation gives an overview of the losses inside a gearbox (Niemann and Winter [2]): gears
bearings
seals auxiliaries (1) no-load losses load-dependent losses
Concli [3] exemplifies the proportional contributions of several kinds of power loss in a single stage geared transmission. Despite of the strong dependency of the relative portions of noload losses on different kinds of machine elements and operating conditions, the no-load
gear loss PVZ0 often represents the largest contributor to no-load losses, as can be seen in Figure 1.
sessol rewoP Figure 1: Power Power losses of single-stage gear transmission system (reproduced from Concli [3] losses with permission of the author) PZVP
PVZP
0ZVP
PVZ0
PLVP
PVLP
0LVP
PVL0
DVP
PVD
XVP
PVX
PVZP
PVZ0
PVLP
PVL0
PVD
PVX
Different approaches are available to calculate the churning losses in transmission systems. These are: -
Measurements on test rigs
-
Calculation with empirical equations
-
Simulation with Computational Fluid Dynamics (CFD)
Various studies on the churning power losses inside gearboxes have been conducted. Terekhov [4] and Boness [5] were one of the first to investigate the churning losses in diplubricated geared transmissions for a wide range of operating conditions. They introduced several empirical equations based on different Reynolds and Froud numbers. The findings show that the tip diameter, the gear width, the immersion depth, the angular velocity, the volume of oil and the density are the main influence factors on the churning loss of gears. Changenet et al. [6, 7] investigated the windage and churning losses of spur gears. According to his findings, the tip diameter and the gear width have the biggest influence on both losses. Höhn et al. [8, 9] investigated the no-load power losses of a dip-lubricated FZG efficiency gear test rig. They identified the oil level, the tip diameter and the circumferential speed as the biggest influencing factors. They ascertained that the oil distribution itself has a remarkable impact on the immersion depth of the gears, as well as on the life time and the efficiency of transmission systems. Maurer [10] and Walter [11] conducted experimental investigations on the different kinds of hydraulic losses in a gearbox. They identified a number of different influencing factors for different kinds of hydraulic losses. Empirical equations that are nowadays widely used in drive technology were subsequently derived from these investigations. However, Mauz [1] and Luke [12] pointed out that there can be considerable uncertainties when applying these empirical equations. Continuous increases in computational capacity have brought CFD methods into the spotlight as a new way of investigating churning power losses (Concli [3]). These methods have several advantages. Firstly, CFD simulations can additionally provide a very detailed view of the oil distribution in a gearbox, which is not possible with other methods. Secondly, CFD is highly flexible with respect to the gear geometry and the gear housing, which allows a wide range of operating conditions and geometric influences to be investigated. This leads to the third advantage: empirical equations are often limited to certain constraints and operating conditions, which in turn limits the applicability of these equations. This can be overcome by CFD methods. Marchesse et al. [13, 14] were among the first to investigate the windage losses of single rotating, dry-lubricated spur gears with the finite volume CFD method. The results agree well with the experimental data with a maximum deviation of 8%. Kvist [15] investigated the oil
distribution and the churning losses for a single rotating helical gear with dip lubrication using the finite volume CFD method. Both the oil distribution and the churning losses are in good agreement with experimental measurements. The oil distribution was compared to video recordings of a transparent gearbox. The recordings showed that the main features of the oil flow can be captured well by the CFD simulations. Concli et al. [16] analyzed the churning losses and the oil distribution of a planetary gearbox with dip lubrication. As the squeezing losses caused by the gear meshing were not the main focus of this study, they used a simplified model without gear meshing. The simulation results are in very good agreement with experimental results for the investigated operating conditions. A review of the current state of research shows that various studies have already indicated the effectiveness of CFD in determining the churning losses of gear drives. However, CFD approaches are still subject to three main limitations (Concli and Gorla [17]): -
The complexity of setting up a transient mesh model The computational effort The selection of a suitable simulation model
In order to improve our understanding of the effects involved in churning losses, a CFD simulation model was set up to investigate the oil distribution and the churning losses of a singlestage gearbox in the FZG efficiency gear test rig. Within the framework of this article, the comparison between numerical results and measurements shows that CFD offers a very promising way to predict the power losses and the oil distribution for transmission systems. 2. Object of Investigation and Operating Conditions The object of investigation of this study is the FZG efficiency gear test rig. In order to compare CFD simulation results with experimental data, the churning loss torque will be measured for all considered operating conditions. Furthermore, a high-speed camera system is installed in front of a transparent test gearbox to capture high-resolution recordings of the oil distribution during a start-up case of the gear pair. The FZG efficiency gear test rig (Figure 2) is a modified FZG back-to-back gear test rig, and is based on the principle of power circulation. The main components are the electric engine, the loss torque meter, the slave gearbox, the shaft with the load clutch, the driving shaft with the load torque meter and the test gearbox. The transmitted torque is applied by a load clutch and measured by a load torque meter. The electric engine only supplies the total power loss, which is measured by a loss torque meter between the electric engine and the power circle. In no-load operating conditions, only negligible torque is applied to the load clutch, so that the no-load loss can be evaluated. The measured total no-load power loss consists of the no-load losses caused by the gears, bearings and seals. The test gearbox consists of two gears, four bearings and two seals, whereas the slave gearbox consists of two gears, four bearings and three seals. In order to separate the gear no-load losses from the total no-load loss, the no-load losses of the bearings and seals are calculated by means of analytical approaches (SKF [19], ISO 14179-1 [20]), and subtracted from the total no-load loss. Since the test and slave gearbox have the same gear pairs, the resulting value is divided by two to obtain the churning loss of one gearbox. The experimental setup allows the churning losses to be evaluated at different speeds, oil tem-
peratures and lubrication conditions. During operation, the oil temperature is maintained at the desired value. Figure 2: Mechanical layout of the FZG efficiency gear test rig acc. to Lohner et al. [18] The considered gears are of type C-PT (Schedl [21]). The geometry data for both the pinion and the wheel are listed in Table 1. Table 1: Geometry of the type C-PT gears a in mm Pinion
91.5
Wheel
z
mn in mm
16
4.5
24
4.5
α in ° 20.0
x1, x2 in mm
b in mm
da in mm
0.182
14
82.46
0.171
14
118.36
Different operating conditions in terms of the oil sump temperature and the speed of the gear pair are investigated (Table 2, Table 3). The different oil sump temperatures result in different lubricant viscosities and slightly different oil fill levels for the gearboxes. The oil fill level is chosen so that at least one tooth of both the pinion and the wheel dips into the oil sump. Table 2: Operating conditions for comparison of the churning loss torque Pressure in bar
Oil sump temperature ϑoil in °C
Oil fill level
Rotational speed of the wheel n2 in rpm
Circumferential speed vt at pitch circle in m/s
1
60
21.8 mm below the middle axis
348
2.0
1444
8.3
3474
20.0
348
2.0
1444
8.3
3474
20.0
90
20.0 mm below the middle axis
Table 3: Operating conditions for comparison of the oil distribution based on high-speed camera recordings for the start-up case Pressure in bar
Oil sump temperature ϑoil in °C
Oil fill level
Rotational speed of the wheel n2 in rpm
Circumferential speed vt at pitch circle in m/s
Acceleration time for the gears
1
40
36.7 mm below the middle axis
407
2.3
~1.5s
A mineral-based oil FVA3A with 4% additive Anglamol A99 (Laukotka [22]) is considered. Investigations show that the churning losses are primarily influenced by the oil’s viscosity and density properties (Mauz [1]). The following table gives an overview of the main properties of FVA3A, as measured. Table 4: Oil properties of the considered mineral oil FVA3A (Laukotka [22]) Oil
Kinematic viscosity at 40°C in mm²/s
Kinematic viscosity at 100°C in mm²/s
Density at 15°C kg/m³
in
FVA3A
90.0
10.4
884.1
Based on the Ubbelohde-Walter viscosity-temperature relationship (DIN 51563 [23]), the data in Table 4 results in viscosity values of = 90 mm²/s, = 44 mm²/s and =15 mm²/s. 3. Numerical Model The numerical model is implemented and solved in the commercial CFD software Ansys Fluent 16.0. 3.1 Geometry and mesh For finite volume CFD methods, the conservation equations are solved iteratively on the volumes. Consequently, a mesh representing the test gearbox of the FZG efficiency test rig has to be set up first. The mesh of the single-stage gearbox essentially consists of four domains: The pinion domain, the wheel domain, the gearbox domain and the domain of the remeshing zone. In contrast to structure analysis, the entire model represents a negative model of the gearbox. The mesh of each single domain is connected with the other domains as shown in Figure 3. The mesh of the pinion and wheel domain is discretized with extruded hexahedral elements and does not undergo any mesh deformation. For a given domain size, this mesh topology can be created with far fewer cells than the equivalent mesh consisting of tetrahedral elements. The gearbox domain is meshed with extruded tetragonal elements and does also not undergo any mesh deformation. According to the Stokes adhesive condition, the velocities of fluid particles adhering to the walls must be zero and the velocities of fluid particles on the pinion and wheel must be equal to the corresponding circumferential speed. As a matter of fact, high velocity and pressure gradients arise on the contour of the rotating bodies. Therefore, the resolution on the contour of the pinion and wheel domain has to be very fine in order to obtain reasonable results for the loss torque. For this reason, a few layers consisting of long hexahedral elements are adopted near the contour lines. During the simulation, the meshes of the pinion and wheel domain rotate inside the gearbox domain at the predefined rotational speeds. The domain of the remeshing zone fills the cavity between the gearbox domain and the pinion and wheel domain. During operation, the meshing zone of pinion and wheel is a transient area that changes at every meshing position. Therefore, the so-called remeshing zone is adopted, which consists of a deformable meshing structure that changes with every time step of the rotating pinion and the wheel domain. This domain of the remeshing zone is discretized with deformable extruded tetrahedral elements that follow every rigid movement on the planar area. For numerical analysis, the element size has in general remarkable influence on the results. It is important to find a compromise between the quality of the results and the calculation time. Thus, the average element size in the remeshing zone has been varied from 0.65 mm to 1.00 mm during test studies. No significant improvement of the simulated no-load losses has been observed for an element size smaller than 0.85 mm. Hence, the average element size in the remeshing zone has been set to 0.85 mm for the simulation analyses. In order to avoid numerical singularities due to the very small gap between the tooth flanks i.e. backlash, both the pinion and the wheel are scaled to 99% of their actual size. It is not expected that the results with respect to oil distribution and loss torque will be significantly influenced by this simplification. According to Mauz [1], the squeezing losses caused by compressing the oil particles between tooth flanks can be neglected for the considered direction of rotation in Figure 3. Figure 3: Entire mesh model (top left); outer domain of the gearbox (top right); domain of remeshing zone (bottom left); pinion and wheel domain (bottom right)
During the numerical calculation process, the conservation equations are solved iteratively on the grid points. Generally, the numerical calculation can be divided into three steps: -
Integration of the conservation equations in the control volume Discretization by substitution of all gradients with the finite difference method, so that the differential equations are turned into algebraic equations Solution of the approximated equations by an iterative method
3.2 Navier-Stokes equations The Navier-Stokes equations consist of five different conservation equations: the mass equation, the conservation of momentum equation in the x-, y- and z-directions, and the conservation of energy equation (Gersten [24]). The conservation of mass or continuity equation states that mass is constant in all directions in a Cartesian coordinate system. (2) The conservation of momentum equations can be understood as Newton’s Third Law for compressible fluid particles in a continuum. With Einstein’s summation notation, the equation reads as follows: (
(
)
)
(3)
The energy conservation equation is not considered, as it is assumed that thermal influences can be neglected in no-load operating conditions (Concli [25]). The computation time of turbulent flows can become extremely high if every single whirl is dissolved. This is because an extremely fine mesh with a very high number of elements is necessary. To overcome this, Reynolds [26] derived the Reynolds-averaged Navier-Stokes equations (RANS). In these equations, the fluid flow quantities in the conservation equations are separated into a time-weighted average term and an oscillating term. The oscillating term is known as the Reynolds stress tensor: (4) The Reynolds stress tensor is unknown. Hence, further turbulent models are required. These models do not dissolve every single whirl, so a coarser mesh can be used, which requires much less computation time. 3.3 Turbulence models Boussinesq attempted to find a closed form for Equation (3) by introducing the concept of eddy viscosity. He postulated that the Reynolds stress is caused in the same way as the molecular viscosity [26]. (
(
)
)
, mit
(5)
The term is called the eddy viscosity. It describes the increase in the viscosity due to the turbulent oscillating movement. It is proportional to both the turbulent length and the turbulent velocity scale . It follows that the RANS equations can be written in closed form as long as the eddy viscosity is determined. Over the years, different eddy viscosity models have been developed. The quality of the eddy viscosity model greatly depends on the number of turbulence variables and the number of equations. There are a number of different turbulence models, ranging from simple algebraic equations to 2nd order differential equations. In this study, a k-ε model consisting of two coupled transport equations was used. The model describes the development of the kinetic energy and the isotropic dissipation rate ε with two transport equations. The first equation solves the turbulence kinetic energy k. The second equation describes the dissipation rate ε, which characterizes the transformation of turbulent kinetic energy into thermal internal energy. ( (
(
)
(
)
)
(
)
(6)
)
(
)
(7)
(
(
)
)
(8)
For test cases, different turbulence models, e.g. the k- SST-model (two equations) and the Transition SST model (two equations) have been applied to the simulation model. Marchesse et al. [13] have shown that there are no remarkable differences with respect to the windage losses between the k- SST and the k-ε model. In general, the k-ε model represents a good compromise between computation time, result quality and stability. Also the resolution of the interphase between oil and air and the loss torque can be captured well by this turbulence model. 3.4 Discretization In order to solve the conservation equations on the volumes, the differential equations need to be transformed into finite difference equations. This process is called discretization. Generally, a distinction is made between spatial and temporal discretization.
The analytical derivatives in the conservation equations are replaced and approximated by partial derivatives of a certain order. The accuracy of the results increases with the order of the partial derivatives and the fineness of the grid. In this study, a second-order formulation is used for most instances of spatial discretization, whereas a first-order transient formulation is selected for temporal discretization. 3.5 Remeshing method The geometry of the meshing zone between the pinion and the wheel changes at each time step. Hence a transient mesh is required, for which the remeshing method is applied in this study. To do this, the finite elements are bounded by grid points. Just like a solid body, the grid has a certain stiffness, which can be stretched and contracted at every location according to the grid movement. Due to the deformation of the elements, the mesh quality deteriorates relative to the initial mesh. In this study, the quality of the mesh is checked at each time step. The mesh quality is evaluated based primarily on the following mesh parameters: -
Minimal length of the element Maximal length of the elements Maximal skewness of the element face Maximal skewness of the element volume
Certain initial values for all the mesh parameters are defined based on the initial mesh. If one of the following mesh parameters falls below the predefined value, the mesh is reconstructed. At the same time, all data for fluid variables are converted into the grid points of the new elements. As the mesh quality is preserved, the local convergence criteria of 1∙10-5 with respect to the mass and momentum is fulfilled for each time step. 4. Results The CFD simulations in this study focus on the oil distribution and the churning losses. Thereby, CFD simulation results are compared with measured gear no-load torques and high-resolution camera recordings from the FZG efficiency gear test rig. Depending on the operating condition, the calculation time for the transient three-dimensional simulation model considering two-phase flow lasts about eight to ten hours per revolution. A quasi-stationary condition for the flow field is typically reached after three to four revolutions. 4.1 Velocity distribution First, the plausibility of the simulation results is discussed by velocity distribution plots. According to the Stokes adhesive condition, a Newtonian fluid particle on the pinion and wheel must be equal to the corresponding circumferential speed. Figure 4 shows an example of the velocity distributions of the fluid particles between the tooth flanks of the wheel at different circumferential velocities at 9 o´clock position according to the circumferential speeds specified in Table 2 with ϑoil=60°C. It can clearly be seen that the velocity of the fluid particles on the tip diameter is equal to the velocity at the tip diameter of the gear pair. The velocity of particles outside of the tip diameter gradually decreases as the distance from the center of the wheel increases. The fluid particles between the tooth flanks are dragged by the teeth and are expected to have similar velocity, whereas the velocity outside the tip circle drastically drops to lower values. The large difference between the velocities in this transition areas leads to turbulence, which brings about higher velocity in this area (see Figure 4). Figure 4: Simulated velocity distribution between wheel flanks at different rotational speeds for ϑoil=60°C at 9 o´clock position of the wheel
3.0
15.0
25.0
1.5
7.5
12.5
0 m/s
0 m/s
0 m/s
vt=2.0 m/s
vt=8.3 m/s
vt=20.0 m/s
4.2 Oil distribution Figure 5 shows the CFD simulation results of the oil distribution inside the test gearbox at different rotational speeds for ϑoil=60°C. The results show that a large amount of oil under the meshing zone of the pinion and wheel is replaced at the right and left sides of the gearbox. This observation correlates very well to the findings made by Concli [17]. At the same time, oil particles are propelled towards the top of the gearbox. Despite the fact that the pinion has a higher rotational speed, this spin effect is larger in the wheel, i.e. the amount of oil swept towards the left wall of the gearbox is larger than the oil swept towards the right wall by the pinion. Hence, for this particular case the spin effect is more strongly influenced by the tip diameter than by the rotational speed. As expected, the spin effect increases with the rotational speed of the gears, which can be clearly seen by comparing the three different rotational speeds shown in Figure 5. Due to the centrifugal forces of the pinion and wheel, oil droplets are fragmented into a great number of small particles, forming an oil fog in the meshing zone of the gearbox. It can be clearly observed that the amount of air fraction inside the lubricant increases with the rotational speed of the gears. This can be qualitatively observed experimentally as the color of the oil turns form dark yellow before test to very light yellow after test especially when the circumferential speed of the gears is larger than vt>5 m/s. A similar observation has been made by Otto [27]. The obtained air fractions also correlate well to results from online monitoring of lubricant aeration by Leprince et al. [28].
Figure 5: Simulated oil distribution at vt=2.0 m/s, vt=8.3 m/s and vt=20.0 m/s for ϑoil=60°C
vt=2.0 m/s
Oil fraction 1
vt=2.0 m/s
0.5
0 vt=8.3 m/s
Oil fraction 1
vt=8.3 m/s
0.5
0 vt=20.0 m/s
Oil fraction 1
vt=20.0 m/s
0.5
0
In order to validate the oil distributions obtained by CFD simulation, the oil distribution of a start-up case of the gear pair is investigated and compared to the recordings obtained from a high-speed camera system at two different times. In order to adjust the simulation to the experiment, the kinematic of the gears are described by a time dependent angular velocity with constant acceleration during start-up of the electric motor. Figure 6 shows the comparison for a start-up after 50° and after 100° of the pinion under the operating conditions listed in Table 3. From a qualitative viewpoint, the comparisons show very good agreement with the recordings. It can be clearly seen that the oil particles adhere to the tooth flanks at the very first moment (e.g. after 50° rotation). After 100° rotation of the pinion, the oil particles start to spin off from the gear flanks by forming oil threads starting at the tooth tip. The CFD results have shown to be physically plausible, and show very good agreement with recordings from a high-speed camera. In the next step, the churning losses are investigated.
vt=2.3 m/s after 50°
vt=2.3 m/s after 100°
Figure 6: Oil distribution by CFD simulations (left) and recordings by high-speed camera (right) during a start-up of the gear pair after 50° and 100° rotation of the pinion (operating conditions: ϑoil=40°C and vt=2.3 m/s) 4.3 Churning losses Figure 7 shows the measurements and CFD simulation results for the churning loss torque under the operating conditions listed in Table 2. Again, the CFD simulation results are in very good agreement with the measurement results.
C-PT FVA 3A oil=90°C oil fill level: -20 mm below middle axis
1.25
Churning loss torque
1 0.75
CFD simulation Experiment
0.25 0 5 10 15 Circumferential velocity vt in m/s
1.5 1.25 1
0.5
0
in Nm
1.5
20
Churning loss torque
in Nm
Figure 7: Comparison of gear churning loss torque TVZ0,C between CFD simulation and measurement for ϑoil=90°C (left) and ϑoil=60°C (right)
0.75
C-PT FVA 3A oil=60°C oil fill level: -21.6 mm below middle axis
CFD simulation Experiment
0.5 0.25 0 0
5 10 15 Circumferential velocity vt in m/s
20
As shown experimentally by Otto [9], increasing the rotational speed has the greatest influence on the churning losses. The turbulence in the oil sump increases with the rotational speed, the empty space of the gearbox is gradually filled with oil, which in turn leads to higher churning loss torque. Decreasing the temperature from 90°C to 60°C triples the viscosity and induces a lower oil flow velocity. Mauz [1] showed that with increasing oil viscosity and circumferential speed (vt > 5 m/s) , the replaced oil particles do not have enough time to flow back to the area between the tooth flanks before they are swept back to the oil sump by the tips of the teeth. Consequently, the gears can create an empty air space with almost no oil, which in turn leads to lower resistant torque and hence churning loss torque. This explains the lower churning loss torque for ϑoil=60°C compared to ϑoil=90°C at vt = 20 m/s (see Figure 7). The arithmetic mean value and the scatter band in Figure 7 are based on 20 measurements for each operating point. It should be noted that the experimental measurements always involve measuring uncertainties, e.g. due to imbalance in the shafts, the relative error of the loss torque meter, and limited data processing accuracy. Furthermore, the fact that the churning loss torque is determined by subtracting the seal and bearing losses (calculated analytically) from the total no-load loss (measured) introduces another source of uncertainty. Further investigations will therefore include separate measurement of bearing and sealing losses. 5. Conclusion In this study, a CFD model was built up and used to investigate the oil distribution and the churning losses inside a single-stage gearbox of the FZG back-to-back efficiency gear test rig. The investigations show that the finite volume CFD method is a valid tool for predicting the churning loss and oil distribution inside a gearbox. The results for both the oil distribution and the churning loss torque are in very good agreement with the high-speed camera recordings and experimental measurements of the loss torque. The investigations have shown that the rotational speed has the greatest influence on the churning losses, whereas the spin effect on the oil particles is mostly influenced by the tip diameter. The geometrical influence of the pinion and wheel, the impact of different immersion depths, and the influence of different oil types on the churning losses will be investigated in further studies. In the future, new power transmission systems can be developed and optimized for efficiency by means of CFD models.
References [1] Mauz, W., Hydraulische Verluste von Stirnradgetrieben bei Umfangsgeschwindigkeiten bis 60 m/s. Dissertation, Universität Stuttgart,1987, [2] Niemann, G. and Winter, H., Maschinenelemente Band 2: Getriebe allgemein Zahnradgetriebe- Grundlagen, Stirnradgetriebe. 2, Springer-Verlag, Berlin Heidelberg, 2003, [3] Concli, F., Efficiency of gear transmissions and CFD analysis of the load independent power losses. Dissertation, Politecnico di Milano,2012, [4] Terekhov, A. B., Hydraulic losses in gearboxes with oil immersion. Russian Engineering Journal, 5, 1975, [5] Boness, R. J., Churning losses of discs and gears running partially submerged in oil. Chicago, Illinois, 1989,
[6] Changenet, C., Leprince, G., Ville, F., and Velex, P., A Note on Flow Regimes and Churning Loss Modelling. 11th International Power Transmission and Gearing Conference, DOI=10.1115/DETC2011-47227, 2012, 549–555. [7] Changenet, C. and Velex, P., A Model for the Prediction of Churning Losses in Geared Transmissions—Preliminary Results. J. Mech. Des., 129, 1, 2007, 128. [8] Höhn, B.-R., Michaelis, K., and Völlmer, T., Thermal rating of gear drives: balance between power loss and heat dissipation. 1996, [9] Otto, H.-P., Michaelis, K., and Höhn, B.-R., Eds.Influences on no-load gear losses. 3rd European Conference on Tribology, Wien, 2011, [10] Maurer, J., Lastunabhängige Verzahnungsverluste schnelllaufender Stirnradgetriebe. Dissertation, Universität Stuttgart,1994, [11] Walter, P., Langenbeck, K., and Dittrich, O., FVA-Nr. 44/I. Heft 118 Anwendungsgrenzen für die Tauchschmierung von Zahnradgetrieben, Plansch- und Quetschverluste bei Tauchschmierung. Frankfurt/Main, 1982, [12] Luke P. , Olver A., A study of churning losses in dip-lubricated spur gears. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, Vol. 213, 1999, [13] Marchesse, Y., Changenet, C., Ville, F., and Velex, P., Eds.Investigations on CFD Simulation for Predicting Windage Power Losses in Spur Gears 3rd International Conference on Integrity, Reliability and Failure, Porto/Portugal. 2009, [14] Pallas, S., Marchesse, Y., Changenet, C., Ville, F., and Velex, P., Application and validation of a simplified numerical approach for the estimation of windage power losses in spur gears. Computers & Fluids, 84, 2013, 39–45. [15] Kvist, T., Oil splashing simulation using CFD. Trollhättan, Schweden, 2012, [16] Concli, F., Conrado, E., and Gorla, C., Analysis of power losses in an industrial planetary speed reducer -Measurements and computational fluid dynamics calculations. Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology, 228, 1, 2013, 11–21. [17] Concli, F. and Gorla, C., Numerical modeling of the power losses in geared transmissions: Windage, churing and cavitation simulations with a new integrated approach that drastically reduces the computational effort. Tribology International, 103, 2016, 58–68. [18] Lohner, T., Mayer, J., Michaelis, K., Höhn, B.-R., and Stahl, K., On the Running-In Behavior of Lubricated Line Contacts. Journal of Engineering Tribology. Advance online publication, 2015, [19] SKF-Gruppe, SKF Hauptkatalog. Media-Print INformationstechnologiePaderborn, 1994, [20] ISO 14179-1, Gears – Thermal capacity – Part 1: Rating gear drives with thermal equilibrium at 95 °C sump temperature. 14179-1, 2001, [21] Schedl, U., Einfluss des Schmierstoffs auf die Grübschenlebensdauer einsatzgehärteter Zahnräder. Dissertation, TU München,1998, [22] Laukotka, E. M., FVA-Heft Nr. 660 Referenzöle – Datensammlung. Frankfurt/Main, 2003, [23] DIN 51563, Prüfung von Mineralölen und verwandten Stoffen - Bestimmung des Viskosität-Temperatur-Verhaltens - Richtungskonstante m. 51563, 2011, [24] Gersten, K., Einführung in die Strömungsmechanik. Vieweg, Braunschweig, 1992, [25] Gorla, C., Concli, F., Stahl, K., Höhn, B.-R., Michaelis, K., Schultheiß, H., and Stemplinger, J.-P., CFD Simulations of Splash Losses of a Gearbox. Advances in Tribology. Advance online publication, 2012, [26] Kistner, B., Modellierung und numerische Simulaion der Nachlaufstruktu von Turbomachinen am Beispiel einer Axialturbinenstufe. Dissertation, TU Darmstadt,1999, [27] Otto, H.-P., Flank Load Carrying Capacity and Power Loss Reduction by Minimised Lubrication. Dissertation, FZG, TU München,2009, [28] Leprince G., Changenet C., Ville F., Velex Ph., Jarnias F., Influence of oil aeration on churning losses. In JSME International Conference on Motion, 2009,
Highlights Finite Volume CFD method was applied to a dip lubricated gear box
Oil distribution and churning losses were studied for various operating conditions
A 3D transient two-phase flow model based on the remeshing technique was built
Simulated oil distributions correlate very well to high-speed camera records
Churning losses correlate very well to measurements at the FZG efficiency test rig