Investigation of pressure distributions for a finite elastohydrodynamic journal bearing lubricated by Ferro fluids with couple stresses

Transcription

1 Aerican Journal of Applied Mateatics 14; (4): Publised online Auust 3, 14 (ttp:// doi: /j.aja ISSN: (Print); ISSN: 33-6X (Online) Investiation of pressure distributions for a finite elastoydrodynaic journal bearin lubricated by erro fluids wit couple stresses Kiua Daniel 1, Kinyanjui Matew 1, Kiati Mark 1 Pure and Applied Mateatics Departent, Joo Kenyatta University of Ariculture and Tecnoloy, Nairobi, Kenya Pure and Applied Mateatics Departent, Tecnical University of Kenya, T. U. K, Nairobi, Kenya Eail address: Kiuadaniel36@ail.co (K. Daniel), atewkiny@ail.co (K. Matew), ekiati@ail.co (K. Mark) To cite tis article: Kiua Daniel, Kinyanjui Matew, Kiati Mark. Investiation of Pressure Distributions for a inite Elastoydrodynaic Journal Bearin Lubricated by erro luids wit Couple Stresses. Aerican Journal of Applied Mateatics. Vol., No. 4, 14, pp doi: /j.aja Abstract: Te perforance of finite elastoydrodynaic journal bearins lubricated by ferrofluids wit couple stresses as been studied. We derive a Reynolds equation tat takes into account aneto elastoydrodynaics, by usin te continuity and oentu equations. Te equation as been interated locally across te fil tickness and an equation for te pressure radient obtained as a function of fil tickness and total ass flow rate of lubricant, te odified Reynolds equation. Te odified Reynolds equation obtained is solved nuerically by te finite difference tecnique since it is ily non-linear. Te nuerical scee used is ipleented in MATLAB so as to obtain te approxiate solutions. Te pressure distributions are obtained and te solutions obtained represented in raps. It is clear fro te results tat te pressure increases wit te increase in anetis and te couple stress. Keywords: Journal Bearin, Reynolds Equation, Maneto-Elasto-ydrodynaic Lubrication, Couple Stresses 1. Introduction A bearin is a syste of acine eleents wose function is to support an applied load by reducin friction between te relatively ovin surfaces. Journal bearin is subset of bearins used to support rotatin safts tat use te principle of ydrodynaic lubrication. Te journal bearin is ade up of four ain parts as in te i 1. Tis type of bearins is one of te ost coon fors and is used in a wide variety of acines. Wen desinin suc lubricatin syste, te eat, load and flow rate required for a bearin ust be known to properly size te lubricatin oil pups, coolers aon oter bearin acines. Bearin type and size are based on rotor weit, rotor rotations per inute (RPM), and lubricant caracteristics Govindaraj etal (1). Te caracteristics of ydrodynaic journal bearin ave been te subject of any researces. Soe bein directed to te bearin eoetry desin and oters devoted to te study of te lubricant properties. Te recent solutions (Nada & Osan, 7) tat ave encouraed te study of te lubricant properties ave been used wit oter paraeters bein considered in predictin te journal bearin beavior. Tese beaviors suc as te load carryin capacity, type of lubricant flow reion (lainar or turbulent flow), type of lubricant (Newtonian or non-newtonian), inertia and acceleration effects, and anetic effect in te case of usin ferrofluid (Nada, ). iure 1. Scee of te exained bearin Altou any aspects of bearin perforance are fully solved, tere is still need for furter investiations on ow to

2 Aerican Journal of Applied Mateatics 14; (4): iprove te bearin perforance and its caracteristics. Literature Review Te journal bearin desin paraeter suc as load capacity can be deterined fro Reynolds equation bot analytically and nuerically (arock, 1994). Saynatjoki and olber (198) described a anetic fluid as not a siple fluid; it is a stable suspension of sall particles of ferroanetic aterials in a base fluid. or te purpose of ensurin te colloid stability, a surfactant of polyers (suc as oleic acid) is usually introduced into te suspension. Tis will create around eac sinle particle a coatin layer to prevent te aloeration of te particles by te anetic field effect or by te olecular attraction. Wen a anetic field is applied to te anetic fluid, eac particle experiences a force tat depends on te anetization of te anetic aterial of te particles and on te strent of te applied field. A nuber of works ave applied te icro continuu teory (Stokes, 1966) to investiate te effect of couple stresses on te perforance of different types of fluid-fil bearins. Based on te icro continuu teory, te proble of lubrication of finite ydrodynaic journal bearin lubricated by anetic fluids wit couple stresses was investiated and by includin tese couple stresses due to icro structure additives and te anetic effects due te anetization of te anetic fluid, te odified Reynolds equation was obtained to study bearin caracteristics. Te studies cited above consider te anetic fluid to beave as a Newtonian fluid but oter researces treated te anetic fluid as non- Newtonian fluid usin te power-law odel accordin to Osan (1999), Osan et al () and Osan et al (3). Bujurke and Naduvinaani (199) invested te lubrication of litly loaded cylinders in cobined rollin, slidin, and noral otion wit a couple stress fluids as lubricant under cavitation boundary conditions was done. Tey found tat te load capacity and te frictional dra increased as te squeeze velocity increased. Increasin te couple stress paraeter enanced tis increase. Presentation on te study of perforance of finite journal bearins lubricated wit a fluid wit couple stresses takin into account te elastic deforation of te liner ave been done by Mokiaer et al (1999). Tey concluded tat te influence of couple stresses on te bearin caracteristics was apparently sinificant. Naduvinaani et al () investiated te static caracteristics of rotor bearin systes lubricated wit couple stress fluids and takin into account of surface rouness. Nada and Osan (7) investiated te static perforance of finite ydrodynaic journal bearins lubricated by anetic fluids wit couple stresses and aon teir conclusion was tat te anetic lubricant ives ier load carryin capacity, ier attitude anle, lower friction coefficient, and ier side leakae. Tese effects are ore sinificant were te ydrodynaic effects are low, at te lower values of couple stress paraeter and eccentricity ratios. ydrodynaic lubrication investiations ave been te key researc studies for any researcers, owever in tis study, we focus on te velocity and pressure distributions in a aneto-elasto-ydrodynaic lubrication syste. 3. Governin Equations Te induced anetic force depends ainly on te applied anetic field and on te anetization of te particles anetic aterial. Te effects of te anetic force on te pressure of elasto-ydrodynaic journal bearin are also considered studied. Te effects of couple stresses on te pressure of elasto-ydrodynaic journal bearin are also considered Modified Reynolds Equation ro icro-continuu teory, te eneral oentu equation of an incopressible fluid wit couple stress is iven as; In x direction In y direction Dq 1 ρ = P B ( η ) q Dt ρ η u u 4 u = P u u 4 (3.1) (3.) 4 v v v v ρ v = y η 4 (3.3) In z direction 4 w w w w ρ w = z η 4 (3.4) Were:, yand are te coponents of external forces z per unit volue. w w w u u ρ u, v v ρ v and ρ are te tree coponents of acceleration of te fluid. Equations (3.) to (3.4) contain te four unknowns' u, v, w and p. te fourt equation is supplied by te continuity equation: ρu ρv ρw (3.5) Applyin te assuptions below; 1. Te lubricant is assued to be Newtonian fluid.. Te flow is lainar; consequently, neiter vortex flow nor turbulence is occurrin anywere in te flow. 3. Te lubricant is assued incopressible; i.e. its density is constant.

3 137 Kiua Daniel et al.: Investiation of Pressure Distributions for a inite Elastoydrodynaic Journal Bearin Lubricated by erro luids wit Couple Stresses 4. Te curvature of te fluid fil is nelected since te fil tickness in y-direction is very tin copared wit te span in x and z-directions. Tus, no radient of te applied anetic field across te fluid fil, no anetic force in y-direction and tus no pressure radient in tis direction 5. Te fluid flow is steady. Iplyin tat u = t t = w t 6. Te fluid inertia force is nelected copared to te viscous force and te induced anetic force 7. No slippin at te bearin surface 8. Except u and w, all oter velocity radients are considered neliible. 9. No eat conducted to or fro te fluid fil to surfaces (adiabatic case). On application of te above assuptions te equations 3., 3.3 and 3.4 reduces to; = = 4 u u η 4 (3.6) y (3.7) z 4 w w η 4 (3.8) Were and are te anetic force z coponents in circuferential and axial directions respectively. Te boundary conditions can be stated as below; At At At y u v w (3.9) y = u w, (3.1) R u = ω, v w (3.11) u w, (3.1) y = u = ω R, v w (3.13) u w, (3.14) Equations (3.9), (3.11) and (3.13) are te no-slip boundary conditions. Equations (3.1), (3.1) and (3.14) result fro te couple stress and vanis at te solid surfaces (Nada & Osan, 7). Tis is due to te resistance of te solid surfaces for te rotation otion of te additive particles. By solvin equations (3.6) and (3.8) and usin te above boundary conditions, te velocity profiles in circuferential and axial directions can be obtained (Nada & Osan, 7). Tese velocities are substituted in te followin interated continuity equation across te fluid fil tickness for an incopressible fluid flow and ten solved to ive te specific Reynolds equation. u v w dy (3.15) Te eneralized pressure equation is obtained in te for; Were; (, l ) (, l ) (, l ) (, l ) x z = 6ω R l 3 3 (, l ) = 1l 4l tan (3.16) Abdo, 9 stated tat te coponents of anetic force ( and z ) can be obtained in te x and z directions respectively as below; z X (3.17) = X (3.18) = Substitutin equations (3.17) and (3.18) in te Reynolds equation (3.16) above we obtain; (, l ) (, l ) = 6ω R (, l) (, l) X X (3.19) 3.. Non-Diensionalization Te subject of diensional analysis considers ow to deterine te required te set of scales for any iven proble. Tis is a process tat starts wit selectin a suitable scale aainst wic all diensions in a iven pysical odel are scaled. Tis process ais at ensurin tat te results obtained are applicable to oter eoetrically siilar confiurations under siilar set of flow conditions. Te non-diensional paraeters used are defined as below: x = R z = L bz e = C ε = C l = CL λ ω P L b = Rβ = o And P = C R ( ) Usin tese non-diensional paraeters, te odified Reynolds equation (3.5) becoes;

4 Aerican Journal of Applied Mateatics 14; (4): α = 1 ( ) ( ) = G, L β G, L α β α G (, L ) G (, L ) (3.) Were 3 3 ( ) and ( ) o ω λ X ( L ) b C G, L = 1L 4L tan L It is noted tat, as te value of ( L ) tend to zero, equation (3.) reduced to te Newtonian anetic lubricant case and te effect of couple stresses vanises = β β α α (3.1) 3.3. inite Difference Tecnique Introduction Te perforance of te bearin in tis study as been predicted assuin adiabatic conditions wit anetic fluid lubricant wit couple stress. Te effect of anetic force coefficient, couple stress paraeter and lent to diaeter ratio and pressure distribution as been nuerically investiated Te Reynolds Equation in inite Difference Back to te specific non-diensional Reynolds equation (3.1) 3 P 1 3 = 6 4 β z z 3 3 4β α α (3.) Considerin te bearin eoetry and te anetic field odel, te fil tickness depends totally on te cane of te anle and ence te derivatives of wit respect to is retained wile tose of wit respect to vanises, also te derivatives of wit respect to Z disappears wile tose of wit respect to Z is retained tus te equation reduces to; 3 P 1 3 P. = 6 4 β Z α (3.3) Applyin te finite difference tecnique and usin te central difference etod, te odified Reynolds equation ets discretized to; ( ) ( ) 3 3 i, j 1 i, j i 1, j i 1, j Pi 1, j Pi 1, j i, j = 3 ( i, j ) P ( ) β ( Z ) ( i j ) ( i j ) P P P P ( ) 4β ( Z ) 3 i 1, j i 1, j, i, j 1 i, j 1, 3 α ( i, j ) Z α i 1, j i 1, j 3 i, j 1 i, j 1 ( Z ) i, j 1 i, j 1 i, j i, j 3 (3.4) 4. Results and Discussion Te journal bearin used is as in te iure 1. It as two coponents, one representin te saft and te oter te bearin. Te syste is supplied by lubricant fluid trou openins wic eere in an axial roove. iure. Plot surface of pressure wit teta and z

5 139 Kiua Daniel et al.: Investiation of Pressure Distributions for a inite Elastoydrodynaic Journal Bearin Lubricated by erro luids wit Couple Stresses Te results as in iure ive te diensionless pressure distribution in circuferential and axial directions wit eccentricity ratio for different values of te couple stress paraeter. It represents a surface of pressure aainst bot circuferential and axial directions. Te results are obtained for constant eccentricity ratio e, lent to diaeter ratio β of 1. and varied couple stress paraeter L of.,. and.4 (Abdo, 9) were L. is te Newtonian lubricant case. Te results are also deterined for anetic coefficient α. wic is te non-anetic lubricant case and α =. to 8. te anetic lubricant case (Nada & Osan, 7). Te pressure distribution in te circuferential direction at te bearin id-plane (bearin centerline) for different values of couple stress paraeter and a constant value of anetic coefficient are sown in i 3 for eccentricity ratio e.6. Considerin te anetic effect usin te anetic coefficient, we vary te value of te anetic coefficient to see te effect. Tis is evident clearly as fro iure 4. Tere is a lare increase of te pressure as te value of te anetic coefficient increases. Te axiu pressure is sifted to te anle.5 π. Due to syetry fro i it is clear tat te pressure increases in anitude in te neative way after anoter.5π ark anle. Effect of alpa is also deterined at a constant value of te eccentricity ratio. At tis value of eccentricity ratio, tere are two axiu pressure points as sown in i 1and also accordin to Abdo, 9. Te first at.5π due to te anetic effect and value of te couple stress paraeter L and te oter at nearly = π due to te ydrodynaic effect, its value depends ainly on te couple stress paraeter L. iure 5. Plot surface of pressure wit teta and z iure 3. Plot of pressure aainst teta wit increasin couple stress It is clearly sown in iure 3 tat for anetic lubricant, tere is an increase of te pressure wit increasin L at a constant eccentricity ratio used. Tis increase is ore pronounced as we increase te value of te couple stress paraeter. Tus, te pressure is increasin as te value of te couple stress paraeter increases. iure 5 represents te plot of pressure aainst te lent of te bearin for different couple stress paraeter for anetic lubricant (α =.). It is clear tat as we increase te value of te anetic coefficient te pressure also increases. Te anetic contribution is ore alon te lent of te bearin and it is iest at te iddle of te bearin as evident in iure 5. Soe syetry is seen at te center of te bearin tat is at Z =.. All tis is bein analyzed considerin te sae eccentricity ratio bein eld a constant, it can also be seen tat te pressure of te bearin lubricated wit couple stress fluid increases wit increasin te couple stress paraeter. Bearins pressure wit anetic lubricant and couple stress are reater tan tat wit nonanetic lubricant witout te couple stress. 5. Conclusion iure 4. Plot surface of pressure aainst teta wit varied anetic coefficient Te results concluded tat te anetic lubrication ives ier pressure distribution. Based on te icro-continuu teory (Stokes, 1966), and by takin into account te couple stresses due to te icrostructure additives, te effects of couple stresses on te perforance of a finite elastoydrodynaic journal bearins were studied. Te results ave sown tat lubricants wit couple stresses copared wit Newtonian lubricants increase te pressure. Tus it can be concluded tat fluids wit couple stresses are

6 Aerican Journal of Applied Mateatics 14; (4): better tan Newtonian fluids. Considerin te anetis of te fluid, it is clear tat ferrofluids ives ier pressure to te bearin ence iprovin on te efficiency of te syste. Terefore, it can be concluded tat lubricants wit additives are better tan te coon lubricants used. Te results are tereafter expected to provide to te enineers useful inforation to desin acine eleents and bearin systes wit a ier life expectancy and efficiency. Noenclature C Bearin clearance, e Eccentricity of te journal center, ε Eccentricity ratio ( ε = e C ) Unit volue value of te induced anetic force, Tesla Manetic force in x direction (circuferential direction) Manetic force in z direction (axial direction) z M Manetization of te anetic aterial, Wb M Saturation value of anetization s Lubricant fil tickness, Non-diensional fil tickness ( = C ) u i luid velocity vector in x, y and z directions u, v, w Coponents of velocity vectorq 1 ( s ) x, y, z Cartesian coordinates Anular coordinate ( x R ) R Bearin or journal radius Z Non-diensional axial distance ( Z = ) ax Maxiu fil tickness, in Miniu fil tickness, Manetic field intensity, Wb - z L b Caracteristic value of anetic field intensity Non-diensional anetic field intensity P Lubricant pressure, N/ P Non-diensional pressure P( C / R ) P = α Manetic force coefficient ( o ) ρ luid density, K/ 3 luid viscosity, pa-s Non-diensional viscosity l Couple stress paraeter L α = ω ω λ X o ( L ) Non-diensional couple stress paraeter ( L= lc) b C ω Anular speed, rad/s η Material constant responsible for te couple stress property Gradient operator i j k Body forces vector in x, y and z directions References [1] Abdo,. S. M. (9). Teral effects on ydrodynaic journal bearin lubricated by anetic fluids wit couple stresses. [] Bujurke N. M. & Naduvinaani N. B., (199). Te lubrication of litly loaded cylinders in cobined rollin, slidin and noral wit couple stress fluid, Int. Journal Mec. Science. [3] Cowley,. D., & Rosenswei, r. E. (April 1967). Te interfacial stability of ferroanetic fluid. Journal of fluid ec, [4] Govidaraj, R. & Satiskuar V. M. (1). Desin of journal bearin test ri, Sweden blekine Tekniska oskola. [5] Mokiaer U. M., Crosby W. A. &El-Gaal. A., (1999). A study of a journal bearin lubricated by fluids wit couple stress considerin te elasticity of te liner wear. [6] Nada,. S. (). Static and dynaic caracteristics of anetized journal bearins lubricated wit ferrofluid. PD tesis in ecanical enineerin, faculty of enineerin. Cairo, Eypt: Cairo University. [7] Nada,. S., & Osan, t. A. (7). Static perforance of finite ydrodynaic journal bearins lubricated by anetic fluids wit couple stresses. Triboloy letters, [8] Naduvinaani N. B., ireat P. S. & Gurubasavaraj G. (). Effects of surface rouness on te static caracteristics of rotor bearin wit couple stress fluid, Coput Struct. [9] Osan T. A., (1999). Static caracteristics of ydrodynaic anetic bearins workin by Non-Newtonian errofluid, Journal of Enineerin and Applied science, Cairo University, [1] Osan T. A., Nada G. S. & Safar Z. S., (3). Different anetic odels in te desin of ydrodynaic journal bearins lubricated wit Non-Newtonian errofluid, Triboloy letters, 11-3 [11] Stokes, V. K. (1966). Couple stresses in fluids. Pys. luids, [1] Saynatjoki, M, and olber K., (198) Manetic fluids in sealin and lubrication a state of art review. inland: Tecnical Researc Center of inland.