Subscribe Now Subscribe Today
Research Article
 

Analysis of Two-layered Porous Journal Bearing using the Brinkman Model



T.V.V.L.N. Rao, A.M.A. Rani, T. Nagarajan and F.M. Hashim
 
Facebook Twitter Digg Reddit Linkedin StumbleUpon E-mail
ABSTRACT

The present study evaluates the effects of two-layered long porous journal bearing configuration on improvement in load capacity and reduction in coefficient of friction. The Brinkman model is utilized to model the flow in the porous region. A modified form of Reynolds equation is derived considering two-layered porous region adjacent to the bearing surface. The non-dimensional pressure and shear stress expressions are obtained using the Reynolds boundary conditions. Results of non-dimensional load capacity and coefficient of friction are presented as a function of permeability and thickness of porous layers. Based on the results presented in the study, a low permeability porous layer that adheres to high permeability porous layer on bearing surface could significantly enhance load capacity and reduce coefficient of friction.

Services
Related Articles in ASCI
Similar Articles in this Journal
Search in Google Scholar
View Citation
Report Citation

 
  How to cite this article:

T.V.V.L.N. Rao, A.M.A. Rani, T. Nagarajan and F.M. Hashim, 2012. Analysis of Two-layered Porous Journal Bearing using the Brinkman Model. Journal of Applied Sciences, 12: 2610-2615.

DOI: 10.3923/jas.2012.2610.2615

URL: https://scialert.net/abstract/?doi=jas.2012.2610.2615
 
Received: September 02, 2012; Accepted: November 04, 2012; Published: January 10, 2013



INTRODUCTION

The performance of hydrodynamic lubricated contacts would be improved by additives which form a thin porous layer adhered to bearing surfaces due to lubricant microstructure (Oliver, 1988). Tichy (1995) developed models applicable for fluid flow through porous medium considering the effects of lubricant additives. Li (1999) derived a modified form of Reynolds equation using Brinkman-extended Darcy model which takes into account the viscous shear and viscous damping effects. Lina et al. (1996) applied the Brinkman model to predict the load capacity and friction parameter for flexible long porous journal bearings. Their results showed that Brinkman model which includes viscous shear effects, predicts an increase in load capacity and reduction in coefficient of friction. Li and Chu (2004) and Elsharkawy (2005) utilized porous media model and the couple stress model to study the effects of lubricant additives on the performance of hydrodynamic contacts. By modeling the microstructure of lubricating surfaces as thin porous film press fitted on bearing surfaces, the theoretical approach on the steady state performance of hydrodynamic contacts is presented to investigate the effects of lubricant additives.

Naduvinamani (1997) presented a theoretical study of double-layered porous Rayleigh-step bearings using Darcy’s model and Beavers-Joseph velocity slip at the porous media/fluid film interface. Saha and Majumdar (2004) investigated steady state and stability characteristics of hydrostatic two-layered porous oil journal bearings. Two-layered porous bearing using highly permeable structural support topped by a thin layer with restriction to fluid flow gives better stability. The characteristics of flow through three layered porous media are investigated by Allan et al. (2009). Attia (2007) analyzed steady flow between two parallel plates in a porous medium. Amiri (2001) presented the analysis of flow through porous medium using a capillary model.

This study presents one-dimensional analysis of two-layered long porous journal bearing using Newtonian fluid as lubricant. A modified Reynolds equation is derived using Brinkman model. The nondimensional pressure and shear stress expressions are obtained. Reynolds boundary conditions are used to solve the pressure distribution. In this work, the effects of dimensionless permeability parameter and porous layer thickness on the steady state performance characteristics such as load capacity and coefficient of friction are analyzed.

MATHEMATICAL FORMULATION

The schematic of a two-layered porous journal bearing is shown in Fig. 1. The porous layer 1 of thickness δ1 is adjacent to the stationery bearing surface while the porous layer 2 of thickness δ2 is adjacent to the porous layer 1. In the present analysis, the variation of pressure across the porous layer and fluid film is assumed to be negligible.

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
Fig. 1: Geometry of two-layered porous journal bearing

The fluid motion in the two-layered porous regions (region I: 0≤y≤δ and region II: δ1≤y≤δ12) is governed by Brinkman equation considering viscous shear and viscous damping effects as:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(1)

The fluid motion in the film region obeys the conventional governing Newtonian flow equations:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(2)

The boundary conditions are: (1) no-slip boundary conditions at the bearing and journal surface and (2) velocities and viscous shear stresses match at the porous layer 1/porous layer 2 and porous layer 2/fluid film interface:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(3)

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(4)

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(5)

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(6)

Integrating the Eq. 1-2 using the boundary conditions in Eq. 3-6, the dimensionless velocity distribution is expressed as:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(7)

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(8)

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(9)

Where:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(10)

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(11)

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(12)

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(13)

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(14)

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(15)

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(16)

The equation of continuity across the film is:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(17)

Simplifying the equation of continuity across the film, yields the non-dimensional pressure gradient term for a two-layered porous journal bearing as:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(18)

Where:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(19)

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(20)

For Δ2 = 0, G1 and G2 in Eq. 19-20 reduce to:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(21)

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(22)

Where:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(23)

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(24)

The Reynolds boundary conditions are:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(25)

Integrating the Eq. 18 and substituting the first boundary condition given in Eq. 25, yields the nondimensional pressure profile as:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(26)

Substitution of the Reynolds boundary conditions for nondimensional pressure at film rupture in Eq. 26 and simplifying results in Q as:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(27)

Substituting the pressure gradient boundary condition given in Eq. 23 in the expression for nondimensional pressure gradient in Eq. 18, results in:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(28)

The Newton-Raphson iterative procedure is used to solve simultaneously both θr and Q using Eq. 27 and 28.

The radial and tangential nondimensional load capacity obtained by integration of nondimensional pressure along and perpendicular to line of centers are expressed as:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(29)

The nondimensional load capacity is expressed as:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(30)

The nondimensional shear stress in the journal bearing at Y = H is obtained as:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(31)

The nondimensional friction force on the journal surface is obtained by integrating the shear stress along the journal surface as:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
(32)

The nondimensional friction coefficient is calculated as:

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model

NUMERICAL STUDIES AND DISCUSSION

A two-layered porous journal bearing is considered in the analysis. The influence of permeability and thickness of porous layers on the nondimensional load capacity and coefficient of friction are presented.

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
Fig. 2(a-b): Nondimensional load capacity, (a) ε = 0.5, Δ1 = 0.2 and (b) ε = 0.5, Δ2 = 0.2

The parameters used in the analysis are: eccentricity ratio (ε) = 0.5; nondimensional permeability of porous layers 1 and 2, respectively (K1, K2) = 10-2, 10-3 and nondimensional thickness of porous layers 1 and 2, respectively (Δ1, Δ2)= 0.05, 0.1, 0.15, 0.2.

Figure 2a and b show the non-dimensional load capacity (W) of two-layered porous journal bearing with nondimensional thickness of porous layers 1 and 2, respectively (Δ1, Δ2) for different values of nondimensional permeability of porous layers 1 and 2 (K1, K2). The non-dimensional load capacity (W) increases with (1) decrease in the nondimensional permeability of porous layer and (2) increase in the nondimensional thickness of porous layer. For all the parameters of nondimensional thickness of porous layers considered in the study, the higher non-dimensional load capacity (W) is obtained with reduction in the nondimensional permeability of porous layer 2 (K1 = 10-3, K2 = 10-2-10-4) compared to reduction in the nondimensional permeability of porous layer 1 (K1 = 10-3, K2 = 10-2-10-4).

Figure 3a and b show the coefficient of friction (Cf) of two-layered porous journal bearing as a function of nondimensional thickness of porous layers 1 and 2, respectively (Δ1, Δ2) for different values of nondimensional permeability of porous layers 1 and 2 (K1, K2).

Image for - Analysis of Two-layered Porous Journal Bearing using the Brinkman Model
Fig. 3(a-b): Coefficient of friction, (a) ε = 0.5, Δ1 = 0.2 and (b) ε = 0.5, Δ2 = 0.2

The coefficient of friction (Cf) decreases with (1) decrease in the nondimensional permeability of porous layer and (2) increase in the nondimensional thickness of porous layer. Lower coefficient of friction (Cf) is obtained with reduction in the nondimensional permeability of porous layer 2 (K1 = 10-3, K2 = 10-2-10-4) compared to reduction in the nondimensional permeability of porous layer 1 (K1 = 10-3, K2 = 10-2-10-4).

CONCLUSION

The present study evaluates the influence of permeability and thickness of porous layers on improvement in load capacity and reduction in coefficient of friction for a two-layered long porous journal bearing. A modified Reynolds equation is derived considering permeability and thickness of two-layered porous layers using Brinkman model. The conclusions based on the analysis are:

Higher non-dimensional load capacity (W) and lower coefficient of friction (Cf) are obtained with (1) decrease in the permeability and (2) increase in the thickness of porous layer
For a given thickness of porous layers 1 and 2 (Δ1, Δ2 in the range 0.05-0.2), a low permeability porous layer 2 with high permeability porous layer 1 (K1 = 10-3, K2 = 10-4; K1 = 10-2, K2 = 10-3) would result in higher non-dimensional load capacity (W) and lower coefficient of friction (Cf) compared to a low permeability porous layer 1 with high permeability porous layer 2 (K1 = 10-4, K2 = 10-3; K1 = 10-3, K2 = 10-2), respectively

ACKNOWLEDGMENTS

This research is funded by Exploratory Research Grant Scheme of Ministry of Higher Education (ERGS-MOHE) Malaysia under grant ERGS/1/2011/TK/ UTP/02/40. The authors greatly appreciate the support provided by Universiti Teknologi PETRONAS for this research.

NOMENCLATURE

C = Radial clearance (m)
f = Friction force (N) F = fC/μuj RL
h, H = Film thickness (m) H = h/C
ki, i = 1, 2 = Permeability of porous layer 1 and porous layer 2, respectively (m2) Ki = ki/C2
L = Length of the journal bearing (m)
p = Pressure distribution (N m-2); P = pC2/μujR
q = Volume flow rate per unit length along film thickness (m2 sec-1) Q = q/ujC
R = Journal radius (m)
u = Velocity component along θ direction (m sec-1; U = u/uj)
ui, i = 1, 2, 3 = Local mean velocity components along θ direction in porous layer 1 and 2 and velocity component along θ direction in fluid film, respectively (m sec-1)
u12, u23 = Velocity component along θ direction at the interface of porous layer 1-porous layer 2 and porous layer 2-fluid film, respectively (m sec-1)
uj = Journal velocity along θ direction (m sec-1)
w = Static load, N (W = wC2/μuj R2 L)
Wε, Wφ = Nondimensional radial and tangential static load for journal bearing
x = Coordinate along circumferential (x) direction (m) θ = x/R
y = Coordinate along radial (y) direction (m) Y = y/C
δi, i = 1, 2 = Thickness of porous layer 1 and porous layer 2, respectively (m) Δi = δi/C
ε = Journal bearing eccentricity ratio
μ = Fluid viscosity (Nsec m-2)
θ = Angular coordinate measured from the position of maximum film thickness in journal bearing
θr = Angular extent of film rupture for journal bearing
τ = Shear stress component (N m-2; N m-2) Π = τC/μuj
ω = Angular velocity of journal bearing (rad sec-1)

Subscripts:
r = rExtent of outlet film in journal bearing measured
ε = long the radial direction
φ = Along the tangential direction
REFERENCES
1:  Allan, F.M., M.A. Hajji and M.N. Anwar, 2009. The characteristics of fluid flow through multilayer porous media. ASME J. Applied Mechanics, Vol. 76. 10.1115/1.2998483

2:  Amiri, M.C., 2001. Modified darcy's law to predict low reynolds flow through porous media. J. Applied Sci., 1: 8-10.
CrossRef  |  Direct Link  |  

3:  Attia, H.A., 2007. On the effectiveness of variation in the physical variables on the generalized couette flow with heat transfer in a porous medium. Res. J. Phys., 1: 1-9.
CrossRef  |  Direct Link  |  

4:  Elsharkawy, A.A., 2005. Effects of lubricant additives on the performance of hydrodynamically lubricated journal bearings. Tribol. Lett., 18: 63-73.
CrossRef  |  Direct Link  |  

5:  Li, W.L., 1999. Derivation of modified reynolds equation-a porous media model. ASME J. Tribol., 121: 823-828.
Direct Link  |  

6:  Li, W.L. and H.M. Chu, 2004. Modified reynolds equation for couple stress fluids-a porous media model. Acta Mech., 171: 189-202.
CrossRef  |  Direct Link  |  

7:  Lina, J.R., C.C. Hwang and R.F. Yang, 1996. Hydrodynamic lubrication of long, flexible, porous journal bearings using the brinkman model. Wear, 198: 156-164.
CrossRef  |  Direct Link  |  

8:  Naduvinamani, N.B., 1997. Non-newtonian effects of second order fluids on double-layered porous rayleigh-step bearings. Fluid Dynamics Res., 21: 495-507.
CrossRef  |  

9:  Oliver, D.R., 1988. Load enhancement due to polymer thickening in a short model journal bearing. J. Non-Newtonian Fluid Mech., 30: 185-196.
CrossRef  |  Direct Link  |  

10:  Saha, N. and B.C. Majumdar, 2004. Steady state and stability characteristics of hydrostatic two-layered porous oil journal bearings. Proc. IMechE J. Eng. Tribol., 218: 99-108.
CrossRef  |  

11:  Tichy, J.A., 1995. A porous media model for thin film lubrication. ASME J. Tribol., 117: 16-21.
CrossRef  |  

©  2021 Science Alert. All Rights Reserved