Modern Mechanical Engineering, 2012, 2, 1-5 1 http://dx.doi.org/10.4236/mme.2012.21001 Published Online February 2012 (http://www.SciRP.org/journal/mme) Research on Lubrication Model of Rock Bit Bearings with * Non-Newtonian Medium Guorong Wang#, Juan Wang, Lin Zhong State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, China Email: #[email protected], [email protected], [email protected] Received December 3, 2011; revised December 28, 2011; accepted January 8, 2012 ABSTRACT Lubrication media of rock bit journal bearing is non-Newtonian fluid, and because of structure limitations, lubricant supply is only from the inside track and its rheological properties are relatively complex. In general studies, the non- Newtonianism of lubrication media of rock bit journal bearing is neglected to simplify the research process. Based on the universal Reynolds equation of non-Newton media, the mathematical model of lubrication analysis was suitable for the rock bit bearing working conditions was established. Then according to the experimental results of rheological properties of RB-type lubricating grease, the equivalent viscosity of the model was established and this model can be solved by the method used in solving the Reynolds equation of Newton fluid. Keywords: Journal Bearing; Rock Bit; Lubrication Model; RB Grease 1. Introduction bricant is presented in an infinite lubricated inclined slider bearing [4]. The apparent viscosity and the perfor- In the complicated working conditions and special envi- mance of 1D slider bearings are analyzed and discussed ronment of rock bit bearings, lubricating oil ,which is com- in a porous media model [5,6]. The performance charac- monly called Newtonian fluid ,can’t satisfy the need of teristics of bearing operating with micropolar lubricant lubrication. In order to improve the bearing lubrication are presented and compared with that of Newtonian lu- performance, people added many additives of long-chain bricant, for a wide range of non-dimensional load, capil- polymer materials in base oil. These additives can de- lary restrictor design parameter and micropolar parame- creese sliding frication coefficient, reducing heat and wear, ters [7]. According to the upper documents .It can be seen increasing oil viscosity and capacity, but it makes lubri- that non-Newton media has great influence on lubrication cant present non-Newtonian characteristic at different de- and load carrying capacity of the sliding bearings, so the gree, and makes rheological property of lubricants beco- study of Newton’s media with improvement on lubrica- me the noticeable factors of lubrication designs. RB grea- tion of rock bit bearings is of great significance. In order se used in rock bit bearings is a type of non-Newtonian to greatly accord with actual working condition, the non- fluid, because of its complicate rheological property ,the Newtonian behavior of the grease should be considered non-Newtonian behavior of rock bit lubrication medium during establishing the lubrication equation of rock bit is usually neglected in the general study. The Reynolds sliding bearings, which is convenient to deeply study the equations of non-Newtonian fluid and Newton fluid are rock bit sliding bearings. Based on common Reynolds equ- unified by two equivalent viscosities [1], a dynamically ation of non-Newtonian medium, this paper establish a loaded finite journal plastic bearing lubricated with a non- model of the lubrication analysis, which is suitable for the Newtonian fluid, which account for some non-Newtonian actual working condition of rock bit bearings, and study its characteristics through an equivalent power-law [2]. A solving method. power law model is developed for the elastohydrody- namic lubrication regime. The modified Reynolds equa- 2. The Determination on Equivalent tions with elasticity equation were formulated for the ela- Viscosity of Rock Bit Sliding Bearings stohydrodynamic journal bearings under heavy load con- with Non-Newtonian Medium ditions [3]. Non-Newtonian Powell-Eyring fluid as lu- In the lubrication model of rock bit sliding bearings with * The project was supported by the National Natural Science Foundation non-Newtonian medium, the expression of , of China (50905149). bx by # Corresponding author and bx indicates that the final integral results contain Copyright © 2012 SciRes. MME 2 G. R. WANG ET AL. clearance function h(x, y), in order to get the expression Newtonian fluid through the ratio of shear stress given h(x, y) of clearance function from ex and ex , the shear velocity and shear velocity, it is called the nominal most critical parameters—equivalent viscosity x y , viscosity (or the apparent viscosity) [9], by the parameter the equivalent viscosity under working conditions of rock estimation method, the fitting equation of nominal vis- bit sliding bearings will be confirmed in this paper. cosity changing with shear velocity, whose temperature The RB grease constitutive equation of rock bit is: ranges from 100˚C to 150˚C, is as follow: n u * xsxB m (6) z z (1) v It is known that the nominal viscosity of RB grease ysyBz and shear velocity become the relation of power expo- nent in the same temperature, is the nominal viscos- About Equation (1), Thereinto, is shear stress, B ity of RB grease, m n is the fitting parameter, the special is Bingham viscosity, is ultimate shear stress. s data is showed in Table 1. According to the lubrication problem of journal bea- Through the experiment data processing, Regulation ring, no matter it is Newtonian rheological model or non- that the nominal viscosity of RB grease vary with tem- Newtonian rheological model, theirs constitutive equa- perature is as Figure 1, Which show that viscosity of RB tion can be uniformly wrote into the following forms: grease vary with temperature at different shear rate. u Because the expression of equivalent viscosity is the f z x same as the definition of the nominal viscosity, in order (2) to making the lubrication model more suitable for the v f () y actual working condition of rock bit sliding bearing, the z nominal viscosity of RB grease is took as equivalent vis- For Newtonian fluid cosity in this paper, therefore it can be showed in the following: f (3) n * u Therefore, for unifying the flow of non-Newtonian x m z fluid and that of Newtonian fluid, the two equivalent n (7) * v viscosity x y only need to be defined: m y z f x x * x (4) Table 1. The fitting parameters of the relationship between y viscosity and shear rate. f y * y 100˚C 110˚C 120˚C 140˚C 150˚C Taking Equation (4) into Equation (2), m 5.7527 6.0565 5.7645 6.8554 6.5065 * x n –0.5960 –0.5781 –0.6038 –0.6648 –0.6739 x uz/ (5) * y y vz/ By Equation (5), equivalent viscosity is equal to the ratio of shear stress and shear velocity with non-New- tonian fluid. In view of the actual working condition of rock bit, the rheological property of RB grease about rock bit, whose temperature is between 50˚C and 150˚C is carried out the experimental research in the literature [8]. According to the experimental data ,the change rule of shear stress and shear velocity with RB grease was obtained in the above temperature range, for the Bingham law containing two rheological parameters about s and B Equation (1), in order to simplify the analysis process, applying New- Figure 1. The nominal viscosity-temperature curve of RB tonian viscous law, showing the viscosity of the non- grease in different shear rate. Copyright © 2012 SciRes. MME G. R. WANG ET AL. 3 The lubrication property under relevant working con- with lubrication mechanics was derived from reference dition can be analyzed by taking the expression of equi- [10], with the boundary conditions of rock bit working valent viscosity into the lubrication model rock bit slid- conditions’ speed: ing bearing with non-Newtonian fluid. zuUvVwW0, 00, , 0 According to the derivation of the lubrication analysis Obtaining that zhuUvVwW, , , mathematical model of rock bit sliding bearing with non- hh h Newtonian medium, during establishing the model in this zz pz ex d z uU0 d z h paper, it unifies the Reynolds equation with non-Newto- x ** nian fluid and the Reynolds equation with Newtonian 00xex x (8) fluid in the forms, and determines the expression of equi- Z dz ex UU valent viscosity by the previous experimental results, h 0 * h 0 x while in solving the lubrication analysis mathematical mo- del of rock bit sliding bearing with non-Newtonian me- When the rock bit sliding bearings are in work, its dium, it only need to solve according to the general Rey- pressure are usually less than 300 MPa [11], grease den- nolds equation with Newtonian fluid. sity changes can be neglected in this range, namely the grease is regarded as incompressible fluid in the working 3. The Lubrication Analysis Mathematical conditions of rock bit bearings, the density ρ is constant, Model of Rock Bit Sliding Bearings with fluid mass along X direction can be expressed as Non-Newtonian Medium h Qu dz (9) 3.1. Hypotheses about Non-Newtonian Medium x 0 Lubrication of Rock Bit Sliding Bearings Substituted in Equation (8), obtaining that According to rock bit of special working conditions and RB grease of complicate rheological property, there are 1 3 p Qhx bx hu bx (10) some hypotheses about non-Newtonian medium lubrica- 12 x tion of rock bit bearings: Including 1) Ignoring the physical impacts of rock bit and gre- ex ax ase. Due to structure limits of rock bit, the size of bear- bx12 ax ings system is smaller, and the drilling pressure is much ex bigger during working .Therefore the physical effects can UU U u be neglected .