CHINESE JOURNAL OF MECHANICAL ENGINEERING
Vol. 28,aNo. 3,a2015 ·541·
DOI: 10.3901/CJME.2015.0403.037,available online at www.springerlink.com; www.cjmenet.com; www.cjme.com.cn
Influence of Particles on the Loading Capacity and the Temperature Rise
of Water Film in Ultra-high Speed Hybrid Bearing
ZHU Aibin
*
, LI Pei,ZHANG Yefan, CHEN Wei, and YUAN XiaoyangKey Laboratory of Education Ministry for Modern Design and Rotor-Bearing System,Xi’an Jiaotong University, Xi’an 710049, China
Received September 12, 2014; revised January 7, 2015; accepted April 3, 2015
Abstract: Ultra-high speed machining technology enables high efficiency, high precision and high integrity of machined surface. Previous researches of hybrid bearing rarely consider influences of solid particles in lubricant and ultra-high speed of hybrid bearing, which cannot be ignored under the high speed and micro-space conditions of ultra-high speed water-lubricated hybrid bearing. Considering the impact of solid particles in lubricant, turbulence and temperature viscosity effects of lubricant, the influences of particles on pressure distribution, loading capacity and the temperature rise of the lubricant film with four-step-cavity ultra-high speed water-lubricated hybrid bearing are presented in the paper. The results show that loading capacity of the hybrid bearing can be affected by changing the viscosity of the lubricant, and large particles can improve the bearing loading capacity higher. The impact of water film temperature rise produced by solid particles in lubricant is related with particle diameter and minimum film thickness. Compared with the soft particles, hard particles cause the more increasing of water film temperature rise and loading capacity. When the speed of hybrid bearing increases, the impact of solid particles on hybrid bearing becomes increasingly apparent, especially for ultra-high speed water-lubricated hybrid bearing. This research presents influences of solid particles on the loading capacity and the temperature rise of water film in ultra-high speed hybrid bearings, the research conclusions provide a new method to evaluate the influence of solid particles in lubricant of ultra-high speed water-lubricated hybrid bearing, which is important to performance calculation of ultra-high speed hybrid bearings, design of filtration system, and safe operation of ultra-high speed hybrid bearings.
Keywords: hybrid bearing, liquid-solid flow, solid particles, loading capacity, temperature rise
1 Introduction
As a new machining technology, ultra-high speed machining technology enables high efficiency, high precision and high integrity of machined surface, etc. As one of key techniques of ultra-high speed machine, the hybrid bearing is commonly used in main shaft supporting element due to its advantages. To ensure the precision of the main shaft, the bearing radius clearance is only ten microns, where the conventional lubricants are not easy to form film for its high viscosity. With advantages of low viscosity and high specific heat, water can be used as lubricant to form complete oil film and decrease the temperature rise in ultra-high speed hybrid bearing.
During the operation of bearing, the lubricant is inevitably contaminated with solid particles because of the influence of the outside or its own attrition, and then form liquid-solid two-phase flow. Particles are significantly
* Corresponding author. E-mail: abzhu@mail.xjtu.edu.cn
Supported by National Natural Science Foundation of China (Grant No. 51275395), and Major National Basic Research Program of China (973 Program, Grant Nos. 2009CB724304-2, 2009CB724404)
© Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2015
responsible for abrasive wear and thus failure of bearing operation[1], especially under the ultra-high speed and
micro-space conditions. The particles affect not only the characteristics of the lubricant but also the lubricant properties, which have been investigated by many researchers with different mathematical models and sophisticated experiments. Based on the hypothesis of rounded grain and perfectly smooth surface, KORDILLA, et al[2], used a three-dimensional multiphase smoothed
particle hydrodynamics (SPH) model to simulate surface tension dominated flow on smooth fracture surfaces. BINU, et al[3], theoretically evaluated the pressure distribution and
load carrying capacity by using a modified Reynolds equation for various TiO2 nanoparticle concentrations and
aggregate sizes. GRUPP, et al[4], evaluated the impact of a
biphase anterior-posterior(AP) and internal-external(IE) motion restraint system on the wear behavior, tibio-femoral kinematics and particle release of a mobile bearing posterior stabilized knee design in comparison to the widely used linear restraint. MENG, et al[5], concluded the
influence of particles on the piston ring and the cylinder sleeve, and solved the lubricant film force on the particles and the influence of particles on the stress suffered by the piston ring and the cylinder sleeve. Aiming at the soft grit
ZHU Aibin, et al: Influence of Particles on the Loading Capacity and the Temperature Rise of Water Film in Ultra-high Speed Hybrid Bearing
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in the lubricant, NIKAS[6] analyzed the lubricant
heat-transfer model of the particles rolling on the lubricating surface and concluded that high temperature is inevitability produced when particles contact with friction surface is greater than the chance. KHONSARI[7] dealt with
the law of the film thickness under condition of liquid-solid two-phase flow, and a problem with extreme dimension of particles from the perspective of temperature. ZHOU, et al[8], reported that the reduction of friction and wear must
come from the addition of Fe3O4 magnetic nanoparticles
(MNPs) in the base oil which can offer protection between the rubbing surfaces. CHOU, et al[9], used statistical
regression method to investigate the thermal behavior and experimentally demonstrated the influence of the friction and the abrasion of the particles on the capability of the lubricant. DAI, et al[10], presented a study on the
liquid-solid two-phase flow lubrication and the contact performance of the particles based on the hypothesis that the surface is smooth. ZHAO, et al[11], further studied
tribological properties of particles as lubricant additives. LENGIEWICZ, et al[12], carried out an efficient wear model
modeling evolution of wear, contact pressure and contact zone in sliding contact. KANG, et al[13], dealt with the
solid-particle effect on lubrication in the state of hydrodynamic lubrication with considering the liquid-solid two-phase flow as elastic fluid. It is obvious that most of the previous research works were focused on analyzing the liquid-solid two-phase flow with a simple wedge ramp model concerning particle impact during the operation of bearings lubricated with oil. As for performance calculation of hybrid bearing, previous researches rarely consider influence of solid particles in lubricant and ultra-high speed of hybrid bearing, which cannot be ignored under the high speed and micro-space conditions of ultra-high speed water-lubricated hybrid bearing. Considering the impact of solid particles in lubricant, turbulence and temperature viscosity effects of lubricant, the influences of particles on pressure distribution, loading capacity and the temperature rise of the lubricant film with four-step-cavity ultra-high speed water-lubricated hybrid bearing are presented in the paper. The research conclusions provide a new method to evaluate the influence of solid particles, which is important to performance calculation of ultra-high speed hybrid bearings, design of filtration system, and safe operation of ultra-high speed hybrid bearings.
2 Analysis for Four-step-cavity Hybrid
Bearings
Hybrid bearing combines advantages of hydrodynamic bearing and hydrostatic bearing. In the operation of hybrid bearing, external static pressure and dynamic pressure produced by high speed rotating ensure that the bearing can be formed within the full range of high-pressure oil film, which is important to the normal operation of the bearing. A four-step-cavity hybrid bearing shown in Fig. 1, is
chosen as the research object.
Fig. 1. Schematic of four-step-cavity hybrid bearing 2.1 Reynolds equation
The modified Reynolds equation governing the turbulent flow in the clearance space of a journal and bearing in non-dimensional form is given as: [14]
2 3 3 , x z H P D H P H k L k æ ö æ ö æ ö ¶ çççç ¶ ÷÷÷÷+ççè øç ÷÷÷ ¶ çççç ¶ ÷÷÷÷= ¶ ¶ è ¶ ø ¶ è ¶ ø ¶ (1)
Introducing the following non-dimensional parameters: , x R = / =2z L/ , P= /p ps, H= /h h0, = / 0, 0 , h R = / U=R, 2 0 6 (ps ). = /
Where and h0 are respectively the viscosity and the
radius clearance under the liquid-solid two-phase flow; U is the speed of the linear velocity of the rolling neck journal;
P is the water film pressure; is the attitude angel of the bearing; kx and kz are the coefficient of turbulent lubrication, and their expressions are as follows: [15]
0.9 12 0.013 6 , x k = + Re (2) 0.98 12 0.004 3 , z k = + Re (3) where Re is the Reynolds number in the state of bearing operating.
2.2 Fluid film thickness
The fluid film thickness between journal and bearing is as follows:
0 0
In the bearing lands: cos( ),
In the bearing recesses: r cos( ),
h h h h h ì = + -ïï íï = + + -ïî (4)
where h and hr are respectively nominal fluid-film thickness and recess depth.
2.3 Restrictor flow equation
In a compensated journal-bearing system, the continuity of flow between restrictor and bearing is to be maintained. Therefore, flow through the restrictor is taken as a constraint in the solution domain. The flow of the lubricant through the orifice restrictor is defined as Eq. (5)[16]:
2 0 π 2( ) . 4 s r r d p p q -= (5)
CHINESE JOURNAL OF MECHANICAL ENGINEERING ·543· 2.4 Boundary conditions
Hybrid bearings, like the general bearings, have periodic boundary condition: P L1=P L4, side vent boundary
condition: P L2=0, Reynolds boundary condition:
3 0, ( ) 3 0,
L L
P = ¶ ¶P = and so on (Fig. 2). Besides, hybrid bearings have its typical inlet-port boundary condition: P L0=Pr. The pressure boundary Pr at the inlet should be calculated according to the flow continuity equation and the restrictor flow equation.
Fig. 2. Schematic of boundary conditions 2.5 Choice of viscosity
During the operation of ultra-high speed hybrid bearings, we should consider the temperature-viscosity effect of the lubricant. The viscosity is mainly influenced by the particle concentration and the lubricant temperature[17], and its
computing formula is as follows:
0 0 2.5 1 1 1 exp , 1 (1/ 1) 273 273 L s Tm T æ ö é ù ç æç ö÷÷ ê ú ç ÷÷ = ê + ú ç çç - ÷÷÷÷ ç ÷ ç - - è è + + øø ë û (6) where ,s are respectively the solid phase density and L liquid density. λ is solid volume fraction of liquid-solid two-phase flow. μ0 is dynamic viscosity at T0. is the
coefficient of viscosity. Tm is the temperature under operating conditions.
2.6 Numerical calculation
With semi-differential eight-node format, Reynolds equation is transformed into difference schema. At the discontinuities of the water-film thickness, the differential format should be solved by using continuous flow equation. Thus we can transform the differential form of the Reynolds equation to equations set.
The transformed equations set has been integrally calculated with the over-relaxation iterative method. And with corresponding convergence criterion, the pressure distribution can be obtained. And then, the bearing characteristics would be worked out, such as fluid loading capacity, friction, and so on, based on the pressure distribution of the lubricant film. The overall solution is shown in Fig. 3.
3 Theoretical Calculation of Liquid-solid
Two-phase Flow
Owing to liquid-solid two-phase flow lubricant in
ultra-high speed hybird bearing, corresponding film capacity would be produced. Meanwhile, it produces heat because of the friction and increases the temperature of the lubricant. In terms of the two-phase flow theory, the loading capacity and the friction in process of the two-phase flow movement should be divided into three parts: loading capacity and friction of fluid, loading capacity and friction of solid particles, and loading capacity and friction of asperities on the bearing surface. These should be solved respectively.
Fig. 3. Flow chart of overall solution
3.1 Loading capacity and friction of fluid
After working out the pressure distribution by solving the Reynolds equation, integrating the pressure field structure on the loading surface, the fluid loading capacity of the bearing would be calculated. And the fluid friction is made up with two parts: the pressure-flow friction force and the shear-flow friction force of the oil film.
3.2 Loading capacity and friction of solid particles
Particle-motion theory is used to represent the movement of the solid particles which is considered as particle swarm moving forward at the solid-phase velocity of Vs within the lubricant. However, because of the difference between particles and lubricant density, there exists speed difference between the liquid and the solid phase. And the kinematic velocity of the solidoid is lower than that of the liquidoid. The flow model[18] and formula to calculate the kinematic
ZHU Aibin, et al: Influence of Particles on the Loading Capacity and the Temperature Rise of Water Film in Ultra-high Speed Hybrid Bearing
·544· (1 ) , (1 ) , L s Q V B c Q V B c ì -ïï = ïï -ïí ïï = ïï ïî (7)
where Q is the mixed volume flow rate of the two-phase flow; is the volume flow rate fraction of the solidoid occupied in the two-phase flow; is volume fraction of the solidoid occupied in the two-phase flow; B is the bearing width; c is the bearing perimeter.
The speed difference between the solidoid and liquidoid indicates that the particle swarm in the lubricant would increase the flow resistance loss of the liquidoid which can be considered as the loss of the two-phase slip friction in the liquid-solid phase. The method in the hydromechanics of calculating the subsidence resistance of small balls is used, and the friction force of the single particle suffered in the movement of the liquidoid is[19]:
2
( ) 2,
D s L s
D=C A V -V / (8) where CD is the resistance coefficient of particles; A is the projection area of particles.
After calculating the friction resistance the single particle suffered in the movement of the liquidoid, the product of the resistance each particle suffered with the total number of the particles is the total resistance loss of the total particle swarm in the movement of the liquidoid, which is the loss the two-phase slip friction in the liquid-solid phase.
When particles flow through minimum film thickness region, particles may contact with the bearing wall and it would cause frictional wear. The deformation and pressure distribution of the single pressed particle should be analyzed. Also the total number of the particles involved and the total friction loss should be calculated. The pressure distribution model of particle compression studied by Hammer is used: 2 ( ) , k k R r P u h -= + (9)
where k is the shear yield stress; u is the fluid friction coefficient; h is the minimum film thickness. The specification of the formula is referenced in Ref. [20].
By integrating the expression on its contact surface and multiplying it by number of the particles involved in carrying, the total loading capacity can be calculated. Finally, the friction force can be obtained by multiplying the total loading capacity of the solid particles and coefficient of friction (COF) between the particles and the bearing surface. 0 2π R d , p W =N
ò
Pr r (10) , p p F = fW (11)where N is number of particles, f is coefficient of friction between journal and bearing.
3.3 Loading capacity and friction of asperities
The loading capacity and the friction of asperities can be calculated according to the asperities bearing model studied by GREENWOOD, et al[21]: 2 5/2 4.378( ) ( ), a c W E AF H = (12) 2 2 0π ( ) 2( ) , A c a F = A F H +bW (13) ( ) !exp( ), n F H =n -H (14) where and are contact density and radius of c curvature of the asperities respectively; is the synthetically roughness; E is the synthetically elastic modulus; A is the nominal area of contact; H is the film thickness; is the shear strength of boundary film; b is 0
the COF of the boundary.
With adding the above three parts together, the total loading capacity of the film and the total friction loss in process of the hybrid bearing operating are obtained. At last, the average temperature rise of the lubricant can be calculated by choosing the reduced thermal equilibrium condition of the integral bearing. It is reckoned that the friction heat (contains the friction heat of lubricant film, solid particles and asperities) generated within the bearing in unit time is approximately equal to the heat taken away by the flowing oil at the same time period.
After solving the above model with the method of numerical calculation, the temperature rise of the oil film can be worked out. With substituting the result in the viscosity formula and calculating the pressure of the lubricant film, the loading capacity and friction force of each part, the temperature of the lubricant and the loading capacity and the friction force of each part in the steady state are worked out. Different results can be obtained with different conditions.
4 Results Analysis
The object of the calculation is water-lubrication four-step-cavity ultra-high speed hybrid bearings. The related parameters are as follows: the bearing width is 60 mm; the bearing radius clearance is 0.015 mm; the depth of oil chamber is 0.09 mm; the type of throttleer is small-hole throttleer and the diameter of the spatialfilter is 1.5 mm; the oil supply pressure is 6 MPa; water density is 1000 kg/m3;
the density is 0.001 003 Pa • s. With numerical simulation, bearing characteristics are worked out, such as the lubricant film pressure distribution shown in the Fig. 4, lubricant
CHINESE JOURNAL OF MECHANICAL ENGINEERING ·545·
film loading capacity and the temperature, and so on.
Fig. 4. Pressure distribution of the bearing 4.1 Calculations under the assumption of liquid-solid
two-phase flow
With Fig. 5 and Fig. 6, it can be observed that particles in lubrication obviously influence the performance of hybrid bearing. The loading capacity and temperature rise of pure lubrication are lower than that of lubrication with considering particles. Therefore, the calculation error might be produced when the solid particles in lubrication are ignored.
Fig. 5. Curve for the loading capacity on eccentricity of bearing
Fig. 6. Curve for the each temperature rise on eccentricity of bearing
The influences of the particle concentration and diameter on the hybrid bearing performance are analyzed in the
following.
4.2 Influences of particles concentration on loading capacity and temperature rise
Fig. 7 shows the influence of particles concentration on loading capacity of bearing. It is observed that the value of loading capacity increases along with the increase of the paricle concentration. The similar trend is observed when particle diameters are respectively 2, 4, 6 μm. It may due to two aspects: on one hand the lubricant viscosity gets higher under the assumption of liquid-solid two-phase flow, on the other hand the loading capacity of solid particles is taken into account. As we know, the bearing capacity of the liquid phase increases along with the increase of the lubrication viscosity. Simultaneously, a higher solid concentration lead to the increment of the loading capacity of solid phase, liquid phase and asperities. When the solid particle concentration increases, the loading capacity of liquid phases, solid phases and asperities increase and the lubrication film loading capacity increases accordingly.
Fig. 7. Influence of particles concentration on loading capacity of bearing
Fig. 8 shows that temperature has little increment for particle diameter at 2 μm, but it obviously rises for diameters at 4 μm and 6 μm. This is due to the fact that friction loss caused by a tiny, single particle is very low and can be negligible. The particle with 2 μm diameter hardly affect the bearing surface. When the particle diameter is getting larger, a single particle can generate a certain amount of friction loss when contacting with the bearings surface. Along with the increase of the particle concentration, the number of particles which contacting with the bearings surface per unit time also increases. Finally a corresponding increase in friction losses caused by the total particles will lead to a increment of temperature rise of water film.
4.3 Influences of particles diameter on loading capacity and temperature rise
Fig. 9 and Fig. 10 are curves of hybrid bearing water film temperature rises and loading capacities with various particle diameters. The particle diameter is variable and the particle concentrations are 0%, 1%, 5%, 10%, respectively.
ZHU Aibin, et al: Influence of Particles on the Loading Capacity and the Temperature Rise of Water Film in Ultra-high Speed Hybrid Bearing
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Fig. 8. Influence of particles concentration on temperature rise of bearing
Fig. 9. Influence of particles diameter on loading capacity of bearing
Fig. 10. Influence of particles diameter on temperature rise of bearing
Fig. 9 shows that particles of the bigger diameter lead to a higher loading capacity. While the diameter is below 5 μm, loading capacity stabilize at some certain value instead of varying with diameter and concentration. However, when the diameter is close to the minimum film thickness, there is a significant change in loading capacity. And it is more obviously while diameter is larger than the minimum film thickness. Such an influence is due to loading capacity of particles. When diameter changes, particles have little effects on loading capacity of liquid and asperities. So significant increment of loading capacity can be achieved for particles of larger diameter.
The similar trend for the value of temperature rise was observed in Fig. 10. But the value of temperature rise is at almost the same value, while the diameter of particles is below 5 μm, which is different from loading capacity. Such
change is due to friction of solid particles, which mainly affects temperature. At this moment, the influence from liquid and asperities can be ignored. Therefore, the variation laws of loading capacity and temperature rise are similar.
4.4 Influences of particles yield strength on loading capacity and temperature rise
It may be noticed from Fig. 11 and Fig. 12 that influences of hard particles and soft particles on bearing performance are quite different. Good solid additives mean that they have suitable yield strength of particles. When the particles material is soft, the loading capacity of solid particles depends on shear yield strength. When the particles material is hard, the loading capacity of solid particles depends on tensile yield strength. As the above figures show, loading capacity and temperature rise are higher when the bearing is operating with water contained hard particles as compared to the bearing operating with water contained soft particles. With the increment of shear yield strength, the loading capacity and temperature rise of particles increase. The tensile yield strength becomes the main factor affecting particles when the shear yield strength is bigger than tensile yield strength. Meanwhile, there are increments in the loading capacity and temperature rise of particles. However, hard particles probably scratch the bearing surface, which greatly increases the temperature of water film. Therefore, the increment of loading capacity with hard solid particles doesn’t make sense. The situation should be prevented.
Fig. 11. Influence of particles yield strength on loading capacity of bearing
Fig. 12. Influence of particles yield strength on temperature rise of bearing
CHINESE JOURNAL OF MECHANICAL ENGINEERING ·547· 4.5 Influences of particles on loading capacity and
temperature rise under different speeds
The results are presented in Fig. 13 and Fig. 14, for the specific diameter of 10 μm.
Fig. 13. Influence of particles on loading capacity at different speeds
Fig. 14. Influence of particles on temperature rise at different speeds
With Fig. 13 and Fig. 14, it is observed that the loading capacity and temperature rise are affected slightly when bearing is operating slowly. However, there is a significant change in the case of high speed. The trend of loading capacity and temperature rise are similar. Thus, it is of great significance to improve the ability of filtration system when bearing is operating with high speed.
5 Conclusions
Considering the impact of solid particles in lubricant, turbulence and temperature viscosity effects of lubricant, the influences of particles on pressure distribution, loading capacity and the temperature rise of the lubricant film with four-step-cavity ultra-high speed hybrid bearing are presented in the paper.
(1) Solid particles in lubricant have a significant effect on loading capacity of the hybrid bearing. No matter the size of solid particles, loading capacity of the hybrid bearing can be affected by changing the viscosity of the lubricant, and large particles can improve the bearing loading capacity higher. However, large particles may directly collide with the bearing surface, which may cause severe wear of the bearing surface.
(2) The impact of water film temperature rise caused by solid particles in lubricant is related with particle diameter and minimum film thickness. When the particle diameter is much smaller than the minimum film thickness, solid particles have little influence on temperature rise. However, when the particle diameter is larger than the minimum film thickness, its impact on the oil film temperature rise will be more obvious. When the particle diameter is larger than the minimum film thickness, solid particles come into contact with the bearing surface friction, solid particles have a significant effect on the water film temperature rise, which may also cause damage to bearing surface. Occurrence of such a situation must be minimized to ensure the normal operation of the hybid bearing.
(3) The yield stress of solid particles has obvious influence on temperature rise and loading capacity of hybid bearing. Compared with the soft particles, hard particles cause the more increasing of water film temperature rise and loading capacity.
(4) When the speed of hybid bearing increases, the impact of solid particles on hybid bearing becomes increasingly apparent. Therefore, under such extreme condition of ultra-high speed of hybid bearing, filtration system of bearing must be sure to be well designed to prevent unexpected accidents from the effects of solid particles.
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Biographical notes
ZHU Aibin, born in 1975, is currently an associate professor at
School of Mechanical Engineering, Xi’an Jiaotong University, China. He received his PhD degree from Xi’an Jiaotong University, China, in 2007. His research interests include
machinery science, tribology, rotor-bearing dynamics and product design.
Tel: +86-29-82669162; E-mail: abzhu@mail.xjtu.edu.cn.
LI Pei, born in 1990, is currently a master candidate at School of
Mechanical Engineering, Xi’an Jiaotong University, China.
E-mail: overcanopy@foxmail.com
ZHANG Yefan, born in 1990, is currently a master candidate at
School of Mechanical Engineering, Xi’an Jiaotong University, China.
E-mail: 709674846@qq.com
CHEN Wei, born in 1957, is currently the head of Theory of
Lubrication and Bearing Institute, and a professor at School of Mechanical Engineering, Xi’an Jiaotong University, China. His research interests include machinery science, rotor
dynamics, tribology.
Tel: +86-29-82668552; E-mail: chenw@mail.xjtu.edu.cn
YUAN Xiaoyang, born in 1963, is currently a professor at Theory
of Lubrication and Bearing Institute, School of Mechanical Engineering, Xi’an Jiaotong University, China. His research
interests include lubrication theory, rotor dynamics. Tel: +86-29-82669152; E-mail: xyyuan@mail.xjtu.edu.cn
