Research Article Numerical Investigation of …downloads.hindawi.com/archive/2014/157615.pdfResearch...

Research ArticleNumerical Investigation of Pressure Profile inHydrodynamic Lubrication Thrust Bearing

Farooq Ahmad Najar and G A Harmain

Department of Mechanical Engineering National Institute of Technology Srinagar 190006 India

Correspondence should be addressed to Farooq Ahmad Najar frqengrgmailcom

Received 3 March 2014 Revised 25 June 2014 Accepted 17 July 2014 Published 29 October 2014

Academic Editor Jan Awrejcewicz

Copyright copy 2014 F A Najar and G A Harmain This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Reynolds equation is solved using finite difference method (FDM) on the surface of the tilting pad to find the pressure distributionin the lubricant oil film Different pressure profiles with grid independence are described The present work evaluates pressure atvarious locations after performing a thorough grid refinement In recent similar works this aspect has not been addressedHoweverpresent study shows that it can have significant effect on the pressure profile Results of a sector shaped pad are presented and it isshown that the maximum average value of pressure is 12 (approximately) greater than the previous results Grid independenceoccurs after 24 times 24 grids A parameter ldquo120595rdquo has been proposed to provide convenient indicator of obtaining grid independentresults 120595 = |(119875refinedgrid minus 119875Refrence-grid)119875refinedgrid| 120595 le 120576 where ldquo120576rdquo can be fixed to a convenient value and a constant minimumfilm thickness value of 75120583m is used in present study This important parameter is highlighted in the present work the location ofthe peak pressure zone in terms of (119903 120579) coordinates is getting shifted by changing the grid size which will help the designer andexperimentalist to conveniently determine the position of pressure measurement probe

1 Introduction

In this fluid lubrication two mating surfaces are separatedby a layer of lubricant In order to have a load carrying(positive) pressure the film needs to be convergent in spaceConsequent determination of pressure profile numericallyis an important issue and values so obtained need to bechecked stringently as a function of grid size It is the pressuredistribution that balances the weight of the heavy shaft andthe turbine found in hydropower generating plants with aturbine assembly The Reynolds equation which is derivedfrom the Navier-Stokes (NS) equations using thin-filmassumptions is extensively used in tribological applicationsThe Reynolds equation in polar form can be easily found inmultiple text books if any of these sources however derivethe polar Reynolds equation directly from the cylindrical NSequation [1ndash3]TheReynolds equation is a simplified from theNS equation when analyzing a thin lubricant flow Reynoldsequation is commonly used for its practical application whileNS full equations are used to find validity limits of Reynoldsequation Both methods give similar results when working

with narrow gaps however when the minimum distance ofthe channel throat is increased the pressure values obtainedbecome quite different [4] It is assumed that the fluid flowbetween pad and collar is never turbulent and the modelapplied is only valid for laminar fluids It is a commonassumption although in some cases turbulent flow existsunder certain points of operation [5] The transition fromlaminar to turbulent flow occurs at the leading edge firstwhere the fluid flow is thicker Although it is known that theresults would be more accurate using the full NS equationsthe complexity of the calculations is increased heavily so theReynolds equation has been used for the thin lubricant filmcalculations in several research papers The fluid density andthe viscosity are some of the significant parameters To obtaindesired pressure it is often easier to switch the lubricant typeinstead of modifying other parameters as the gap height orthe relative motion between surfaces which in this case isidentical to the collar velocity

The effects of pad curvatures on thrust bearing perfor-mances have been reported in [6] It has been shown by[7 8] that the film shapes have considerable influence on the

Hindawi Publishing CorporationInternational Scholarly Research NoticesVolume 2014 Article ID 157615 8 pageshttpdxdoiorg1011552014157615

2 International Scholarly Research Notices

bearing performances It has been investigated by [9] thatthe effects of continuous circumferential surface profiles canbe signified on the performance characteristics of a sector-type thrust bearing As per study of [10 11] that as comparedwith conventional taper fluid film shape new surface profile(cycloidal catenoidal exponential polynomial) are found tooffer a significant increase in the load-carrying capacity aswell as a considerable reduction in the coefficient of frictionIn recent works as well researchers have not presented theeffect of grid size on the solution of Reynolds equation forexample the work presented by [12] is cited as one of suchcases in which it is directly presented a 9 times 9 grid and has notshown effect of coarsening or refining mesh on the pressureprofile It has been studied theoretically and experimentallythe effects of surface waviness over the load carrying capacityof finite slider bearing The author recorded enhanced loadcarrying capacity in the presence of surface waviness on thestationary pad

The shape of the converging wedge influences the bearingperformance significantly [13ndash15] Investigations made by[11 16] infinitely wide rough slider bearings isothermally forexponential hyperbolic and secant film shapes using couplestress fluids The authors have reported that the increase inpressure is more for the exponential and hyperbolic sliders[16] Moreover investigators [17 18] have studied the THDbehavior of a slider bearing having a pocket and reportedthat themaximumpressure is higher for the pocketed bearingin comparison to plane slider bearing [19 20] studied theinfluence of film shape on the performance of longitudinallyrough infinitelywide slider bearing for isothermal conditionsand reported better load carrying capacity with exponentialsecant and hyperbolic film shapes in comparison to theinclined plane film shape Calculation model of the thrustbearing is built with the assumptions made in the previouschapters It is prepared in the form of a sector shaped padwhich is totally immersed in the oil and supported on asupporting structure (different systems can be applied) Loadis transferred from the rotating runner through the oil filmto the bearing pad and the support Rotational repeatabilityof the system is used so the model can be limited to asingle sector In the present work a sector shaped six padthrust bearing and its characteristic dimensions are shownin Figure 1 At normal speeds the surface of pad and runnerare separated by a thin film lubricant Bearing geometry andproperties are shown in tabulated form in Table 1

2 Reynolds Equation

The following assumptions are made in the analysis

(a) Steady-state conditions exist in the oil film

(b) The lubricant is incompressible

(c) The lubricant is Newtonian in nature

(d) Flow in the convergent wedge is laminar

(e) Pressure and shear effects on the viscosity are negligi-ble

Table 1 Thrust bearing geometry and properties

Grade QuantityDescription

Inner radius 5715mmOuter radius 1143mmNumber of pads 6Pad angle 500∘

Pivot angle 300∘

Pad thickness 2858mmOperating conditions

Axial load 52265NShaft speed 1500 rpmInlet temperature 40∘C

Oil PropertiesOil type VG46Viscosity at 40∘C 390mPasViscosity at 100∘C 54mPasDensity 8550 kgm3

Thermal conductivity 013WmK

120579RI

RO

Figure 1 Thrust bearing pads and their characteristic dimensions

The analysis of hydrodynamic thrust bearings has beenbased on the Reynoldsrsquo equation for the pressure distribu-tion With the increase in capacity of computers numericalmodels including the influences of viscosity variations alongand across the lubricating film have been developed TheReynolds equation is used to calculate the pressure field inthe oil film The variation of viscosity across the thicknessof the oil film is neglected Reynolds equation for a sectorshaped thrust bearing pad with an incompressible lubricantunder steady state condition as reported by [3] Equation(1) presents cylindrical coordinates of Reynolds equationConsider

120597

120597119903

(

119903ℎ3

120583

120597119875

120597119903

) +

1

119903

120597

120597120579

(

ℎ3

120583

120597119875

120597120579

) = 6120596119903

120597ℎ

120597120579

(1)

International Scholarly Research Notices 3

This equation can be converted into nondimensional form byputting the following substitutions

119877lowast

=

119903

1198770

120579lowast

= 120579 119867lowast

=

ℎ

1198670

120583lowast

=

120583

120583119868

119875lowast

=

1198751198672

0

12120587119873120583119868

1198772

0

(2)

When the above substitutions are made after some simplifi-cations the equation in its nondimensional form is as follows

120597

120597119877lowast

(

119867lowast

3

120583lowast

119877lowast120597119875lowast

120597119877lowast

) +

1

119877lowast

120597

120597120579lowast

(

119867lowast

3

120583lowast

120597119875lowast

120597120579lowast

) =

119877lowast

120597119867lowast

120597120579lowast

(3)

3 Equation for Film Thickness

For sector shape geometry film thickness is expressed interms of (119903-120579) coordinates The compact film thicknessexpression reported by [15] considers variation in circum-ferential and radial direction The oil film shape has beenobtained using (4)

119867 = 1198670+ 119867119904(1 minus

120579

120579119905

) (4)

Converting above equation into nondimensional form bydividing above equation by119867

0we get

119867lowast

= 1 +

119867119904

1198670

(1 minus

120579

120579119905

) (5)

4 Load Carrying Capacity (LCC)

Once the pressure distribution is determined the load capac-ity can be calculated [12] In nondimensional form the loadcapacity is given by (6)

LCC = 119882

1198701199032

119874

= int

119877119900

119877119894

int

120579119905

0

(119875119877) 119889120579 119889119877 (6)

5 Numerical Procedure

Numerical treatment of Reynolds equation (2D) using finitedifference method for discretization of the sector shapedbearing pad is performed by considering different grid sizesin terms of (119872 times 119873) nodes and various convergence ratiosldquo119870rdquo as given by (7)

119870 =

119867119904+1198670

1198670

(7)

The finite difference equation is derived by approximatingthe derivatives in the differential equation through truncatedTaylor series expansion for successive grid points Writingthe Reynolds equation in the finite difference form as in(8) results in set of linear algebraic equations which are

converted into the matrix form for the solution using Gauss-Seidel scheme for iteration along with the relevant boundaryconditions and hence the nodal pressure (dimensionless) iscomputed This will determine the nondimensional pressureat each nodeThe iteration will repeat until the oil pressure isconverged as per the algorithm is shown in Figure 2 and theconvergence criteria used for nodal pressure are given in (9)Consider

119875lowast

119894+1119895

[

3119867lowast2

119894119895

120583lowast

119894119895

119877lowast

119894119895

(

119867lowast

119894+1119895

minus 119867lowast

119894minus1119895

4Δ 120579lowast2

)

119867lowast3

119894119895

120583lowast2

119894119895

119877lowast

119894119895

(

120583lowast

119894+1119895

minus 120583lowast

119894minus1119895

4Δ120579lowast2

)]

+ 119875lowast

119894minus1119895

[

119867lowast3

119894119895

120583lowast2

119894119895

119877lowast

119894119895

(

120583lowast

119894+1119895

minus 120583lowast

119894minus1119895

4Δ120579lowast2

)

minus

3119867lowast2

119894119895

120583lowast

119894119895

119877lowast

119894119895

(

119867lowast

119894+1119895

minus 119867lowast

119894minus1119895

4Δ120579lowast2

)]

+ 119875lowast

119894119895+1

[

119867lowast3

119894119895

2Δ119877lowast

120583lowast

119894119895

+

119867lowast3

119894119895

119877lowast

119894119895

Δ119877lowast2

120583lowast

119894119895

minus

119867lowast3

119894119895

119877lowast

119894119895

120583lowast2

119894119895

(

120583lowast

119894119895+1

minus 120583lowast

119894119895minus1

4Δ119877lowast2

)]

+ 119875lowast

119894119895minus1

[

[

119867lowast3

119894119895

119877lowast

119894119895

Δ119877lowast2

120583

lowast

119894119895

minus

119867lowast3

119894119895

120583lowast

119894119895

+

119867lowast3

119894119895

119877lowast

119894119895

2Δ119877lowast

120583lowast2

119894119895

(

120583lowast

119894119895+1

minus 120583lowast

119894119895minus1

2Δ119877lowast

)]

]

+ 119875lowast

119894119895

[

[

2119867lowast3

119894119895

119877lowast

119894119895

Δ119877lowast2

120583

lowast

119894119895

+

2119867lowast3

119894119895

Δ120579lowast2

120583lowast

119894119895

119877lowast

119894119895

]

]

= 119877lowast

119894119895

[(

119867lowast

119894+1119895

minus 119867lowast

119894minus1119895

2Δ120579lowast

)]

(8)

The calculation treatment uses these pressure values alongthe numerical methods (as one-third Simpsons rule) forintegration in order to calculate the load carrying capacity(LCC) A very tight tolerance value is considered hereto ensure that the numerical derivative calculated by thealgorithm shown is precise Consider

119872minus1

sum

119868=2

119873minus1

sum

119868=2

10038161003816100381610038161003816119875new119868119869

minus 119875old119868119869

10038161003816100381610038161003816

10038161003816100381610038161003816119875new119868119869

10038161003816100381610038161003816

le 120598119903 (9)

where ldquo120598119903rdquo is the tolerance limit In this study a constant

value of minimum film thickness 75120583m has been kept underconsideration [5]The result of grid refinement study is foundmatching with the work reported [5] Number of mesh sizesis there and their corresponding results in terms of pressuredistribution and film thickness are presented in this paper


Start

Solve for film thickness

Initialization

Read input data

Output data

Convergence test

Calculate parameters[load carrying capacity friction force etc]

Solve Reynolds equation using FDM

Stop

No

Yes

Iter = K

K + 1

If H0 = (required value)

(120598r) le 0001

Figure 2 Flow chart for computation

6 Results and Discussion

To ensure numerical accuracy the pressure distribution asshown in Figure 9 satisfies the 01 convergence limit InFigure 9 it is clear that the magnitude of pressure generationin the oil film for a sector shaped pad is changing from smallgrid size to larger one In the leading edge side the pressureon the pad surface is small meanwhile large pressuregeneration occurs on the pad surface in the vicinity of thetrailing edge The maximum values of pressure are locatednearer to the trailing edge because of the peak pressure isslightly towards the trailing edge The save minimum filmthickness has been a limiting parameter used by analysisThe nondimensional oil film thickness distribution alongthe circumferential direction for center line and outer arcare shown in the Figure 8 grid independent study playsan eminent role in order to find the better solution of thenumerical model As we go on increasing the grid size from12 times 12 to 96 times 96 the significant change comes in practiceVarious 3D meshes of nondimensional pressure distributionare shown from Figures 3 4 5 6 and 7The results generatedby the researchers [17 18] are closely matching the path withgrid size of 4 times 4 8 times 8 and so forthThe results computed by[12] with the limited grid size of 9 times 9 are also showing goodagreement with the present higher order grid size In generalthe results are showing monotonic increase in accuracy andstability while shifting from course meshes to fine meshes

The maintained sustainable oil film thickness is toenhance the load bearing capacity In the present case theminimum oil film thickness is assumed to be a constant

05

1015

0

5

1015

00002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)

Figure 3 Nondimensional pressure distribution with grid size (12 times12) in radial and circumferential directions

0

10 1050

1520

2520

300

0002

0004

0006

0008

001

0012N

ondi

men

siona

l pre

ssur

e

r(dir)120579(dir)

Figure 4Nondimensional pressure distributionwith grid size (24 times24) in radial and circumferential directions

value of 75 120583m A film profile is called a global optimum iffor a given set of operating conditions and minimum filmthickness it can produce the top load carrying capacity amongall possible film profiles This is due to accommodation ofthe oil film thickness Detailed results of pressure generationfilm thickness values of Reynolds equation are reported inthe work of [12 15 17 18] From the graphical interpretationshown in Figure 10(a) it is clearly understood that the nondi-mensional pressure distribution increases gradually fromzero to a maximum value at centre of the pad almost in allcases of grid sizes along the radial direction In Figure 10(b) itis noted here that the nondimensional pressure distribution isintensifying towards the trailing edge of the pad but varies inmagnitude in varying grid sizes In Figure 10(c) it is observedthat the nondimensional pressure distribution is much smalland it abruptly reaches to the peak value at theminimumfilmthickness and thus counteracts the external load of the slidingsurface The results obtained are shown in Table 2 using ascaling factor ldquo119878rdquo for computing the pressure at differentgrid levels A relationship is developed to get the change inpercentage in terms of a parameter ldquo120595rdquo based on the reference


010

2030

0

1020

300

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)


010

2030

40

010

2030

400

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)


grid at 9 times 9 grid size used by [12] It is evident from thepresent study that there is a 12 increase in pressure valueswith the incorporation of grid independence

The location of the peak pressure zone is also changingin terms of (119903 120579) coordinates with respect to changing ofgrid sizes from courser to refine grids hence on the basisof the present investigation it is possible to find the exactlocation where the pressure probes in the matrix form canbe introduced in order to get the effective pressure values onthe surface of the pad

7 Validation

It is clear that the present study is in close agreement withsome of the works available in open literature [12 15 17 18]Since the present work is reporting the grid independenceof pressure beyond 9 times 9 grid size this feature is limitedlyreported by [12 17 18] The results for values of pressuredistribution therefore limit the authors regarding one-to-one comparison for the refined mesh size with the previousworks of [12 15 17 18]

020

4060

80100

0

50

1000

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)


02

02

04

04

06

06

08

08

10 1

15

2

25

r(dir)

120579(d

ir)

Figure 8 Nondimensional film thickness

8 Conclusion

This work analyzes the hydrodynamic performance char-acteristics of thrust bearing sector shaped pad taking fullscale pressure generation effects The governing equationsare broken using FDM expressed in their nondimensionalform which finally has been solved for pressure distributionusing appropriate boundary conditions Anumerical solutionis proposed and an algorithm has been developed alongwith a numerical code When the mesh size changes it hasbeen observed that improvements in accuracy of the resultswere significant The pressure value has been changed con-siderably with the embodiment of grid refinement analysisIt is evident from the present work that maximum averagevalue of pressure is 12 greater than the results obtainedby using coarse grid At 24 times 24 grid the analysis shows anindependent behavior of results and it does not show furthersignificant improvements although result changes (albeitinsignificantly) when the grid is further refined beyond 24 times24 An important design parameter has been coined in theform of ldquo120595rdquo during the present work This will provide


02 04 06 08 10

02

04

06

08

(a)

02 04 06 08 10

02

04

06

08

(b)

02 04 06 08 10

02

04

06

08

(c)

02 04 06 08 10

02

04

06

08

(d)

02 04 06 08 10

02

04

06

08

1

(e)

Figure 9 (a) Nondimensional pressure distribution contour on 12 times 12 grid size (b) nondimensional pressure distribution contour on 24 times24 grid size (c) nondimensional pressure distribution contour on 30 times 30 grid size (d) nondimensional pressure distribution contour on 40times 40 grid size (e) nondimensional pressure distribution contour on 96 times 96 grid size

Table 2 Effect of grid size on the results with grid refinement

Grid size (119872times119873) Pressure (ND) 119875lowast Pressure = 119878 times 119875lowastWhere ldquo119878rdquo = 1000 120595 = |(119875newgrid minus 119875Refrence-grid)119875newgrid|

12 times 12 00110 110 0054224 times 24 00119 119 0127330 times 30 001261 1261 0175140 times 40 001260 1260 0176596 times 96 001255 1255 01713Where ldquo119878rdquo is a scaling factor which makes it convenient for noticing pressure values at different grid sizes


05 06 07 08 09 10

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)

(a)

0

0002

0004

0006

0008

001

0012

0 01 02 03 04 05 06 07 08 09

Non

dim

ensio

nal p

ress

ure

120579(deg)

(b)

0

0002

0004

0006

0008

001

0012

1 15 2 25 3Film thickness

Non

dim

ensio

nal p

ress

ure

Grid (12 times 12)

Grid (24 times 24)

Grid (30 times 30)

Grid (40 times 40)

Grid (96 times 96)

(c)

Figure 10 (a) Nondimensional pressure distribution for different grid sizes along the radial direction for sector shaped oil film (b) nondi-mensional pressure distribution for different grid sizes along the circumferential direction for sector shaped oil film (c) nondimensionalPressure distribution along and across the centre line of flow direction for different grid sizes of sector shaped oil film

experimentalists a logical procedure for location of the pres-sure sensors

Nomenclature

120576119903 Convergence criteria for computation

1198670 Minimum oil film thickness (120583m)

119867119904 Amount of taper

119872 Number of grid points along radialdirection

119873 Number of grid points alongcircumferential direction

119889120579 Angular division of the grid (radian)119889119903 Radial division of the grid (m)

120579119905 Angular extent of the pad in degrees

119894 Index of node in radial direction119895 Index of node in circumferential direction119875 Hydrodynamic pressure (Nm2)119877119874 Outer radius of the pad (m)

119877119868 Inner radius of the pad (m)

120588 Density of lubricating oil (kgm3)120583 Viscosity (Pas)120596 Angular velocity of shaft (rads)119908 Load on bearing (k N)119875lowast Non dimensional Pressure119877lowast Non dimensional Radial coordinate120579lowast Non dimensional circumferential

coordinate


120583lowast Non dimensional viscosity119867lowast Non dimensional oil film thickness

120595 Grid refinement calculation Parameter120576 Percentage tolerance limit for grid size

variations119870 Convergence ratio

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B BhushanPrinciples andApplications of TribologyWiley NewYork NY USA 2013

[2] B J Hamrock B O Jacobson and S R Schmid Fundamentalsof Fluid Film Lubrication Marcel Dekker New York NY USA2004

[3] O Pinkus and B Sternlight Theory of Hydrodynamic Lubrica-tion McGraw-Hill New York NY USA 1961

[4] D J Song D K Seo and W W Schultz ldquoA comparison studybetweenNavier-Stokes equation and reynolds equation in lubri-cating flow regimerdquo KSME International Journal vol 17 no 4pp 599ndash605 2003

[5] M Tanaka ldquoRecent thermohydrodynamic analyses and designsof thick-film bearingsrdquo Journal of Engineering Tribology vol214 no 1 pp 107ndash122 2000

[6] M B Dobrica and M Fillon ldquoThermohydrodynamic behaviorof a slider pocket bearingrdquo Journal of Tribology vol 128 no 2pp 312ndash318 2006

[7] P I Andharia J L Gupta and G M Deheri ldquoOn the shapeof the lubricant film for the optimum performance of a lon-gitudinal rough slider bearingrdquo Industrial Lubrication andTribology vol 52 no 6 pp 273ndash276 2000

[8] P I Andharia J L Gupta and G M Deheri ldquoEffect of surfaceroughness on hydrodynamic lubrication of slider bearingsrdquoTribology Transactions vol 44 no 2 pp 291ndash297 2001

[9] B P Huynh ldquoNumerical study of slider bearings with limitedcorrugationrdquo Journal of Tribology vol 127 no 3 pp 582ndash5952005

[10] R K Sharma and R K Pandey ldquoExperimental studies of pres-sure distributions in finite slider bearing with single continuoussurface profiles on the padsrdquoTribology International vol 42 no7 pp 1040ndash1045 2009

[11] S B Glavatskih A Method of Temperature Monitoring in FluidFilm Bearings Lulea University of Technology Sirius Labora-tory Division of Machine Elements Lulea Sweden 2003

[12] D V Srikanth K K Chaturvedi and A C K Reddy ldquoDeter-mination of a large tilting pad thrust bearing angular stiffnessrdquoTribology International vol 47 pp 69ndash76 2012

[13] S Abramovitz ldquoTheory for a slider bearing with a convex padsurface side flow neglectedrdquo Journal of the Franklin Institutevol 259 no 3 pp 221ndash233 1955

[14] H P F PurdayAn Introduction to theMechanics of Viscous FlowConstable London UK 1949

[15] N Heinrichson and I Ferreira Santos ldquoReducing friction intilting-pad bearings by the use of enclosed recessesrdquo Journal ofTribology-transactions of The ASME vol 130 no 1 2008

[16] N C Das ldquoStudy of optimum load capacity of slider bearingslubricated with power law fluidsrdquo Tribology International vol32 no 8 pp 435ndash441 1999

[17] N M E Ashour K Athre Y Nath and S Biswas ldquoElastic dis-tortion of a large thrust pad on an elastic supportrdquo TribologyInternational vol 24 no 5 pp 299ndash309 1991

[18] C Bagci and A P Singh ldquoHydrodynamic lubrication of finiteslider bearings effect of one-dimensional experimental filmshape and their computer aided optimum designsrdquo Journal ofLubrication Technology vol 105 no 1 pp 48ndash66 1983

[19] D T Gethin ldquoLubricant inertia effects and recirculatory flowin load-capacity optimized thrust pad bearingsrdquoASLE Transac-tions vol 30 no 2 pp 254ndash260 1987

[20] A P Singh ldquoAn overall optimum design of a sector-shapedthrust bearing with continuous circumferential surface pro-filesrdquoWear vol 117 no 1 pp 49ndash77 1987

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bearing performances It has been investigated by [9] thatthe effects of continuous circumferential surface profiles canbe signified on the performance characteristics of a sector-type thrust bearing As per study of [10 11] that as comparedwith conventional taper fluid film shape new surface profile(cycloidal catenoidal exponential polynomial) are found tooffer a significant increase in the load-carrying capacity aswell as a considerable reduction in the coefficient of frictionIn recent works as well researchers have not presented theeffect of grid size on the solution of Reynolds equation forexample the work presented by [12] is cited as one of suchcases in which it is directly presented a 9 times 9 grid and has notshown effect of coarsening or refining mesh on the pressureprofile It has been studied theoretically and experimentallythe effects of surface waviness over the load carrying capacityof finite slider bearing The author recorded enhanced loadcarrying capacity in the presence of surface waviness on thestationary pad

The shape of the converging wedge influences the bearingperformance significantly [13ndash15] Investigations made by[11 16] infinitely wide rough slider bearings isothermally forexponential hyperbolic and secant film shapes using couplestress fluids The authors have reported that the increase inpressure is more for the exponential and hyperbolic sliders[16] Moreover investigators [17 18] have studied the THDbehavior of a slider bearing having a pocket and reportedthat themaximumpressure is higher for the pocketed bearingin comparison to plane slider bearing [19 20] studied theinfluence of film shape on the performance of longitudinallyrough infinitelywide slider bearing for isothermal conditionsand reported better load carrying capacity with exponentialsecant and hyperbolic film shapes in comparison to theinclined plane film shape Calculation model of the thrustbearing is built with the assumptions made in the previouschapters It is prepared in the form of a sector shaped padwhich is totally immersed in the oil and supported on asupporting structure (different systems can be applied) Loadis transferred from the rotating runner through the oil filmto the bearing pad and the support Rotational repeatabilityof the system is used so the model can be limited to asingle sector In the present work a sector shaped six padthrust bearing and its characteristic dimensions are shownin Figure 1 At normal speeds the surface of pad and runnerare separated by a thin film lubricant Bearing geometry andproperties are shown in tabulated form in Table 1

2 Reynolds Equation

The following assumptions are made in the analysis

(a) Steady-state conditions exist in the oil film

(b) The lubricant is incompressible

(c) The lubricant is Newtonian in nature

(d) Flow in the convergent wedge is laminar

(e) Pressure and shear effects on the viscosity are negligi-ble

Table 1 Thrust bearing geometry and properties

Grade QuantityDescription

Inner radius 5715mmOuter radius 1143mmNumber of pads 6Pad angle 500∘

Pivot angle 300∘

Pad thickness 2858mmOperating conditions

Axial load 52265NShaft speed 1500 rpmInlet temperature 40∘C

Oil PropertiesOil type VG46Viscosity at 40∘C 390mPasViscosity at 100∘C 54mPasDensity 8550 kgm3

Thermal conductivity 013WmK

120579RI

RO

Figure 1 Thrust bearing pads and their characteristic dimensions

The analysis of hydrodynamic thrust bearings has beenbased on the Reynoldsrsquo equation for the pressure distribu-tion With the increase in capacity of computers numericalmodels including the influences of viscosity variations alongand across the lubricating film have been developed TheReynolds equation is used to calculate the pressure field inthe oil film The variation of viscosity across the thicknessof the oil film is neglected Reynolds equation for a sectorshaped thrust bearing pad with an incompressible lubricantunder steady state condition as reported by [3] Equation(1) presents cylindrical coordinates of Reynolds equationConsider

120597

120597119903

(

119903ℎ3

120583

120597119875

120597119903

) +

1

119903

120597

120597120579

(

ℎ3

120583

120597119875

120597120579

) = 6120596119903

120597ℎ

120597120579

(1)



119877lowast

=

119903

1198770

120579lowast

= 120579 119867lowast

=

ℎ

1198670

120583lowast

=

120583

120583119868

119875lowast

=

1198751198672

0

12120587119873120583119868

1198772

0

(2)


120597

120597119877lowast

(

119867lowast

3

120583lowast


120597119877lowast

) +

1

119877lowast

120597

120597120579lowast

(

119867lowast

3

120583lowast

120597119875lowast

120597120579lowast

) =

119877lowast

120597119867lowast

120597120579lowast

(3)



119867 = 1198670+ 119867119904(1 minus

120579

120579119905

) (4)


0we get

119867lowast

= 1 +

119867119904

1198670

(1 minus

120579

120579119905

) (5)



LCC = 119882

1198701199032

119874

= int

119877119900

119877119894

int

120579119905

0

(119875119877) 119889120579 119889119877 (6)



119870 =

119867119904+1198670

1198670

(7)



119875lowast

119894+1119895

[

3119867lowast2

119894119895

120583lowast

119894119895

119877lowast

119894119895

(

119867lowast

119894+1119895

minus 119867lowast

119894minus1119895

4Δ 120579lowast2

)

119867lowast3

119894119895

120583lowast2

119894119895

119877lowast

119894119895

(

120583lowast

119894+1119895

minus 120583lowast

119894minus1119895

4Δ120579lowast2

)]

+ 119875lowast

119894minus1119895

[

119867lowast3

119894119895

120583lowast2

119894119895

119877lowast

119894119895

(

120583lowast

119894+1119895

minus 120583lowast

119894minus1119895

4Δ120579lowast2

)

minus

3119867lowast2

119894119895

120583lowast

119894119895

119877lowast

119894119895

(

119867lowast

119894+1119895

minus 119867lowast

119894minus1119895

4Δ120579lowast2

)]

+ 119875lowast

119894119895+1

[

119867lowast3

119894119895

2Δ119877lowast

120583lowast

119894119895

+

119867lowast3

119894119895

119877lowast

119894119895

Δ119877lowast2

120583lowast

119894119895

minus

119867lowast3

119894119895

119877lowast

119894119895

120583lowast2

119894119895

(

120583lowast

119894119895+1

minus 120583lowast

119894119895minus1

4Δ119877lowast2

)]

+ 119875lowast

119894119895minus1

[

[

119867lowast3

119894119895

119877lowast

119894119895

Δ119877lowast2

120583

lowast

119894119895

minus

119867lowast3

119894119895

120583lowast

119894119895

+

119867lowast3

119894119895

119877lowast

119894119895

2Δ119877lowast

120583lowast2

119894119895

(

120583lowast

119894119895+1

minus 120583lowast

119894119895minus1

2Δ119877lowast

)]

]

+ 119875lowast

119894119895

[

[

2119867lowast3

119894119895

119877lowast

119894119895

Δ119877lowast2

120583

lowast

119894119895

+

2119867lowast3

119894119895

Δ120579lowast2

120583lowast

119894119895

119877lowast

119894119895

]

]

= 119877lowast

119894119895

[(

119867lowast

119894+1119895

minus 119867lowast

119894minus1119895

2Δ120579lowast

)]

(8)


119872minus1

sum

119868=2

119873minus1

sum

119868=2

10038161003816100381610038161003816119875new119868119869

minus 119875old119868119869

10038161003816100381610038161003816

10038161003816100381610038161003816119875new119868119869

10038161003816100381610038161003816

le 120598119903 (9)




Start


Initialization

Read input data

Output data

Convergence test



Stop

No

Yes

Iter = K

K + 1


(120598r) le 0001





05

1015

0

5

1015

00002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)


0

10 1050

1520

2520

300

0002

0004

0006

0008

001

0012N

ondi

men

siona

l pre

ssur

e

r(dir)120579(dir)




010

2030

0

1020

300

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)


010

2030

40

010

2030

400

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)




7 Validation


020

4060

80100

0

50

1000

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)


02

02

04

04

06

06

08

08

10 1

15

2

25

r(dir)

120579(d

ir)


8 Conclusion



02 04 06 08 10

02

04

06

08

(a)

02 04 06 08 10

02

04

06

08

(b)

02 04 06 08 10

02

04

06

08

(c)

02 04 06 08 10

02

04

06

08

(d)

02 04 06 08 10

02

04

06

08

1

(e)






05 06 07 08 09 10

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)

(a)

0

0002

0004

0006

0008

001

0012

0 01 02 03 04 05 06 07 08 09

Non

dim

ensio

nal p

ress

ure

120579(deg)

(b)

0

0002

0004

0006

0008

001

0012


Non

dim

ensio

nal p

ress

ure

Grid (12 times 12)

Grid (24 times 24)

Grid (30 times 30)

Grid (40 times 40)

Grid (96 times 96)

(c)



Nomenclature











coordinate







References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119877lowast

=

119903

1198770

120579lowast

= 120579 119867lowast

=

ℎ

1198670

120583lowast

=

120583

120583119868

119875lowast

=

1198751198672

0

12120587119873120583119868

1198772

0

(2)


120597

120597119877lowast

(

119867lowast

3

120583lowast


120597119877lowast

) +

1

119877lowast

120597

120597120579lowast

(

119867lowast

3

120583lowast

120597119875lowast

120597120579lowast

) =

119877lowast

120597119867lowast

120597120579lowast

(3)



119867 = 1198670+ 119867119904(1 minus

120579

120579119905

) (4)


0we get

119867lowast

= 1 +

119867119904

1198670

(1 minus

120579

120579119905

) (5)



LCC = 119882

1198701199032

119874

= int

119877119900

119877119894

int

120579119905

0

(119875119877) 119889120579 119889119877 (6)



119870 =

119867119904+1198670

1198670

(7)



119875lowast

119894+1119895

[

3119867lowast2

119894119895

120583lowast

119894119895

119877lowast

119894119895

(

119867lowast

119894+1119895

minus 119867lowast

119894minus1119895

4Δ 120579lowast2

)

119867lowast3

119894119895

120583lowast2

119894119895

119877lowast

119894119895

(

120583lowast

119894+1119895

minus 120583lowast

119894minus1119895

4Δ120579lowast2

)]

+ 119875lowast

119894minus1119895

[

119867lowast3

119894119895

120583lowast2

119894119895

119877lowast

119894119895

(

120583lowast

119894+1119895

minus 120583lowast

119894minus1119895

4Δ120579lowast2

)

minus

3119867lowast2

119894119895

120583lowast

119894119895

119877lowast

119894119895

(

119867lowast

119894+1119895

minus 119867lowast

119894minus1119895

4Δ120579lowast2

)]

+ 119875lowast

119894119895+1

[

119867lowast3

119894119895

2Δ119877lowast

120583lowast

119894119895

+

119867lowast3

119894119895

119877lowast

119894119895

Δ119877lowast2

120583lowast

119894119895

minus

119867lowast3

119894119895

119877lowast

119894119895

120583lowast2

119894119895

(

120583lowast

119894119895+1

minus 120583lowast

119894119895minus1

4Δ119877lowast2

)]

+ 119875lowast

119894119895minus1

[

[

119867lowast3

119894119895

119877lowast

119894119895

Δ119877lowast2

120583

lowast

119894119895

minus

119867lowast3

119894119895

120583lowast

119894119895

+

119867lowast3

119894119895

119877lowast

119894119895

2Δ119877lowast

120583lowast2

119894119895

(

120583lowast

119894119895+1

minus 120583lowast

119894119895minus1

2Δ119877lowast

)]

]

+ 119875lowast

119894119895

[

[

2119867lowast3

119894119895

119877lowast

119894119895

Δ119877lowast2

120583

lowast

119894119895

+

2119867lowast3

119894119895

Δ120579lowast2

120583lowast

119894119895

119877lowast

119894119895

]

]

= 119877lowast

119894119895

[(

119867lowast

119894+1119895

minus 119867lowast

119894minus1119895

2Δ120579lowast

)]

(8)


119872minus1

sum

119868=2

119873minus1

sum

119868=2

10038161003816100381610038161003816119875new119868119869

minus 119875old119868119869

10038161003816100381610038161003816

10038161003816100381610038161003816119875new119868119869

10038161003816100381610038161003816

le 120598119903 (9)




Start


Initialization

Read input data

Output data

Convergence test



Stop

No

Yes

Iter = K

K + 1


(120598r) le 0001





05

1015

0

5

1015

00002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)


0

10 1050

1520

2520

300

0002

0004

0006

0008

001

0012N

ondi

men

siona

l pre

ssur

e

r(dir)120579(dir)




010

2030

0

1020

300

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)


010

2030

40

010

2030

400

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)




7 Validation


020

4060

80100

0

50

1000

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)


02

02

04

04

06

06

08

08

10 1

15

2

25

r(dir)

120579(d

ir)


8 Conclusion



02 04 06 08 10

02

04

06

08

(a)

02 04 06 08 10

02

04

06

08

(b)

02 04 06 08 10

02

04

06

08

(c)

02 04 06 08 10

02

04

06

08

(d)

02 04 06 08 10

02

04

06

08

1

(e)






05 06 07 08 09 10

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)

(a)

0

0002

0004

0006

0008

001

0012

0 01 02 03 04 05 06 07 08 09

Non

dim

ensio

nal p

ress

ure

120579(deg)

(b)

0

0002

0004

0006

0008

001

0012


Non

dim

ensio

nal p

ress

ure

Grid (12 times 12)

Grid (24 times 24)

Grid (30 times 30)

Grid (40 times 40)

Grid (96 times 96)

(c)



Nomenclature











coordinate







References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Start


Initialization

Read input data

Output data

Convergence test



Stop

No

Yes

Iter = K

K + 1


(120598r) le 0001





05

1015

0

5

1015

00002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)


0

10 1050

1520

2520

300

0002

0004

0006

0008

001

0012N

ondi

men

siona

l pre

ssur

e

r(dir)120579(dir)




010

2030

0

1020

300

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)


010

2030

40

010

2030

400

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)




7 Validation


020

4060

80100

0

50

1000

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)


02

02

04

04

06

06

08

08

10 1

15

2

25

r(dir)

120579(d

ir)


8 Conclusion



02 04 06 08 10

02

04

06

08

(a)

02 04 06 08 10

02

04

06

08

(b)

02 04 06 08 10

02

04

06

08

(c)

02 04 06 08 10

02

04

06

08

(d)

02 04 06 08 10

02

04

06

08

1

(e)






05 06 07 08 09 10

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)

(a)

0

0002

0004

0006

0008

001

0012

0 01 02 03 04 05 06 07 08 09

Non

dim

ensio

nal p

ress

ure

120579(deg)

(b)

0

0002

0004

0006

0008

001

0012


Non

dim

ensio

nal p

ress

ure

Grid (12 times 12)

Grid (24 times 24)

Grid (30 times 30)

Grid (40 times 40)

Grid (96 times 96)

(c)



Nomenclature











coordinate







References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










010

2030

0

1020

300

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)


010

2030

40

010

2030

400

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)




7 Validation


020

4060

80100

0

50

1000

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)120579(dir)


02

02

04

04

06

06

08

08

10 1

15

2

25

r(dir)

120579(d

ir)


8 Conclusion



02 04 06 08 10

02

04

06

08

(a)

02 04 06 08 10

02

04

06

08

(b)

02 04 06 08 10

02

04

06

08

(c)

02 04 06 08 10

02

04

06

08

(d)

02 04 06 08 10

02

04

06

08

1

(e)






05 06 07 08 09 10

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)

(a)

0

0002

0004

0006

0008

001

0012

0 01 02 03 04 05 06 07 08 09

Non

dim

ensio

nal p

ress

ure

120579(deg)

(b)

0

0002

0004

0006

0008

001

0012


Non

dim

ensio

nal p

ress

ure

Grid (12 times 12)

Grid (24 times 24)

Grid (30 times 30)

Grid (40 times 40)

Grid (96 times 96)

(c)



Nomenclature











coordinate







References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










02 04 06 08 10

02

04

06

08

(a)

02 04 06 08 10

02

04

06

08

(b)

02 04 06 08 10

02

04

06

08

(c)

02 04 06 08 10

02

04

06

08

(d)

02 04 06 08 10

02

04

06

08

1

(e)






05 06 07 08 09 10

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)

(a)

0

0002

0004

0006

0008

001

0012

0 01 02 03 04 05 06 07 08 09

Non

dim

ensio

nal p

ress

ure

120579(deg)

(b)

0

0002

0004

0006

0008

001

0012


Non

dim

ensio

nal p

ress

ure

Grid (12 times 12)

Grid (24 times 24)

Grid (30 times 30)

Grid (40 times 40)

Grid (96 times 96)

(c)



Nomenclature











coordinate







References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










05 06 07 08 09 10

0002

0004

0006

0008

001

0012

Non

dim

ensio

nal p

ress

ure

r(dir)

(a)

0

0002

0004

0006

0008

001

0012

0 01 02 03 04 05 06 07 08 09

Non

dim

ensio

nal p

ress

ure

120579(deg)

(b)

0

0002

0004

0006

0008

001

0012


Non

dim

ensio

nal p

ress

ure

Grid (12 times 12)

Grid (24 times 24)

Grid (30 times 30)

Grid (40 times 40)

Grid (96 times 96)

(c)



Nomenclature











coordinate







References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Numerical Investigation of …downloads.hindawi.com/archive/2014/157615.pdfResearch...

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Transcript of Research Article Numerical Investigation of …downloads.hindawi.com/archive/2014/157615.pdfResearch...