Thermohydrodynamic Analysis of Plain Journal Bearing With Modified Viscosity Temperature Equation

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING AND ISSN 0976 – 6359(Online), Volume 5, Issue 11, November (2014), pp. 31-46 © IAEME TECHNOLOGY (IJMET)

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THERMOHYDRODYNAMIC ANALYSIS OF PLAIN JOURNAL BEARING WITH MODIFIED VISCOSITY TEMPERATURE EQUATION Kanifnath Kadam,

S.S. Banwait,

S.C. Laroiya

National Institute of Technical Teachers Training & Research, Sector 26, Chandigarh ABSTRACT

The purpose of this paper is to predict the temperature distribution in fluid-film, fl uid-film, bush housing and journal along with pressure in fluid-film using a non-dimensional viscosity-temperature equation. There are two main governing equations as, the Reynolds equation for the pressure distribution and the energy equation for the temperature distribution. These governing equations are coupled with each other through t hrough the viscosity. The viscosity decreases as temperature increases. The hydrodynamic pressure field was obtained through the solution of the Generalized Reynolds equation. This equation was solved numerically by using finite element method. Finite difference method has been used for three dimensional energy equations for predicting temperature distribution in fluid film. For finding the temperature distribution in the bush, the Fourier heat conduction equation in the non- dimensional cylindrical coordinate has been adopted. The temperature distribution of the journal was found out using a steady-state unidirectional heat conduction equation. Keywords: Journal Bearings. Reynolds Equation, Thermohydrodynamic Analysis, ViscosityTemperature Equation. 1. INTRODUCTION

A Journal bearing is a machine element whose function is to provide smooth relative motion between bush and journal. In order to keep a machine workable for long periods, friction and wear of mating parts must be kept low. The plain journal bearings are used for high speed rotating machinery. This high speed rotating machinery machinery fails due to failure of bearings. Due to the heavy load and high speed, the temperature increases in the bearing. For prediction of temperature and pressure distribution in bearing, accurate data analysis is necessary. An accurate thermo hydrodynamic analysis is required to find the thermal response response of the lubricating fluid and bush. Therefore, a need has been felt to carry out further investigation on the t hermal effects in journal bearings. 31

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 11, November (2014), pp. 31-46 © IAEME

By considering thermal effects B. C. Majumdar [1] obtained a theoretical solution for pressure and temperature of a finite full journal bearing. D. Dowson and J. N. Ashton [2] computed a solution of Reynolds equation for plain pl ain journal bearing configuration. Operating characteristics were evaluated from the computed solutions and results were presented graphically. The optimum design objective was stated explicitly in terms of the operating characteristics and was minimized within both design and operative constraints. J. Ferron et al. [3] solved three dimensional energy, three dimensional heat conduction equation. They computed mixing temperature by performing a simple energy balance of recirculating and supply oil at the inlet. H. Heshmat and O. Pinkus [4] recommended recommended that the mixing mi xing occurs in the thin lubricant l ubricant layer attached on the surface of the journal. This implies that no mixing occurs inside the grooves. An excellent brief review of thermo hydrodynamic analysis analysis was presented by M. M. Khonsari [5] for journal bearings. H. N. Chandrawat . and R. Sinhasan [6] simultaneously solved the generalized Reynolds equation along with the energy and heat conduction equations. They studied the effect of viscosity variation due to rise in temperature of the fluid film. Also they compared Gauss- Siedel iterative scheme and the linear complementarity approach. M. M. Khonsari and and J. J. Beaman Beaman [7] presented thermohydrodynamic thermohydrodynamic effects in journal bearing operating with axial groove under steady-state loading. In this analysis, the recirculating fluid and and the supply oil was considered. S. S. Banwait and H. N. Chandrawat Chandrawat [8] proposed a non-uniform inlet temperature profiles and for correct simulation. They considered the heat transfer from the outlet edge of of the bush to fluid in the supply groove. groove. L. Costa et et al. [9] presented extensive experimental results of the thermohydrodynamic behavior of a single groove journal bearing. And developed the influence of groove location and supply pressure on some bearing performance characteristics. M. Tanaka [10] had shown a theoretical analysis of oil film formation and the hydrodynamic performance of a full circular journal bearing under starved lubrication condition. Sang Myung Chun and Dae-Hong Ha [11] examined the effect on bearing performance by the mixing between re-circulating and and inlet oil. M. Tanaka and K. Hatakenaka Hatakenaka [12] developed a three-dimensional turbulent thermohydrodynamic lubrication model was presented on the basis of the isothermal turbulent lubrication model by Aoki and Harada, this model was different from both the Taniguchi model and the Mikami model. P. B. Kosasih and A. K. Tieu [13] considered considered the flow field inside the supply region of different configurations and thermal mixing around the mixing zone above the supply region for different supply conditions. Flows in the thermal mixing zone of a journal bearing were investigated using the computational fluid dynamics. The complexity and inertial effect of the flows inside the supply region of different configurations were considered. M. Fillon and J. Bouyer [14] presented the thermohydrodynamic analysis of plain journal bearing and the influence of wear defect. They analyzed the influence of a wear defect ranging from 10% to 50% of the bearing radial clearance on the characteristics of the bearing such as the temperature, the pressure, the eccentricity ratio, the attitude angle or the minimum thickness of the lubricating film. L. Jeddi et al. [15] outlined a new numerical analysis which was based on the coupling of the continuity. This model allows to determine the effects of the feeding pressure and the runner velocity on the thermohydrodynamic behavior behavior of the lubricant l ubricant in the groove of hydrodynamic h ydrodynamic journal bearing and to emphasize the dominant phenomena in the feeding process. S. S. Banwait [16] presented a comparative critical analysis of static performance characteristics along with the stability parameters and temperature profiles of a misaligned non-circular of two and three lobe journal bearings operating under thermohydrodynamic lubrication condition. U. Singh et al. [17] theoretically performed a steady-state thermohydrodynamic analysis of an axial groove journal bearing in which oil was supplied at constant pressure. L. Roy [18] theoretically obtained steady state thermohydrodynamic analysis and its comparison at five different feeding locations of an axially grooved oil journal bearing. Reynolds equation solved simultaneously along with the energy equation and heat conduction equation in bush and shaft. B. Maneshian and S. A. Gandjalikhan Nassab [19] presented presented the computational fluid dynamic dynamic techniques. techniques. They obtained obtained the lubricant 32


velocity, pressure and temperature distributions in the circumferential and cross film directions without considering any approximations. B. Maneshian and S. A. Gandjalikhan Nassab [20] determined thermohydrodynamic characteristics of journal bearings with turbulent flow using computational fluid dynamic techniques. The bearing had infinite length and operates under incompressible and steady conditions. The numerical solution of two-dimensional Navier–Stokes equation, with the equations governing the kinetic energy of turbulence and the dissipation rate, coupled with then energy equation in the lubricant flow and the heat conduction equation in the bearing was carried out. N. P. Mehata et al. [21] derived a generalized Reynolds equation for carrying out the stability analysis of a two lobe hydrodynamic bearing operating with couple stress fluids that has been solved using using the finite element method. N. P. Arab Solghar et al. [22] carried out experimental assessment of the influence of angle between the groove axis and the load line on the thermohydrodynamic behavior of twin groove hydrodynamic journal bearings. Mukesh Sahu et al. [23] used computational fluid dynamic technique for predicting the performance characteristics of a plain journal bearing. Three dimensional studies have been done to predict pressure distribution along journal surface circumferentially as well as axially. E. Sujith Prasad et al. [24] modified average Reynolds equation that includes the Patir and Cheng’s flow factors, cross-film viscosity integrals, average fluid-film thickness and inertia term. This was used to study the combined influence of surface roughness, thermal and fluid-inertia on bearing performance. Abdessamed Nessil et al. [25] presented the journal bearings lubrication aspect analysis using non-Newtonian fluids which were described by a power law formula and thermohydrodynamic aspect. aspect. The influence of the various values of the non- Newtonian power-law index, , on the lubricant film and also analyzed the journal bearing properties using the Reynolds Reynolds equation in its generalized generalized form. The aim of this work is to predict the pressure and temperature distribution in plain journal bearing. Thermohydrodynamic analysis of a plain journal bearing has been presented with an improved viscosity-temperature equation. The equation has been modified by authors to predict the proper relation between viscosity and temperature for forecasting the correct temperature in plain journal bearing. The pressure and temperature distribution in the journal bearing which was almost equal to the temperature obtained by experimental results of Ferron J. et al. [3]. The results have been validated by comparison with experimental results of Ferron J. et al. [3]. and show good agreement. agreement. 2. GOVERNING EQUATIONS

In this present work three dimensional energy equation, heat conduction and Reynolds equation were considered for analysis of thermohydrodynamic analysis of a plain journal bearing. This bearing having a groove of 18° extent at the load line. The geometric details of the journal bearing system are are illustrated in Fig 1. Single axial groove has has been used for supplying supplying fluid to the bearing under, negligible pressure. The model based on the simultaneous numerical solution of the generalized Reynolds and three dimensional energy equations within the fluid-film and the heat transfer within the bush body.

Fig. 1: Bearing geometry 33


2.1

Generalized Reynolds Equation Navier derived the equations of fluid motion for a viscous fluid. Stokes also derived the governing equations of motion for a viscous fluid, and the basic equations are known as NavierStokes equations of motion. The Reynolds equation is a simplified version of Navier-Stokes equation. A partial differential equation governing the pressure distribution in fluid film lubrication is known as as the Reynolds equation. This equation was was first derived by Osborne Reynolds. Reynolds. The hydrodynamic pressure pressure and the velocity field within fluid flow were accurately described through the solution of the complete Navier-Stokes equations. This has provided a strong foundation and basis for the design of hydrodynamic lubricated bearings. This paper is to deal with the finite element analysis of Reynolds’ equation. It will show how the finite element technique is used to form an approximate solution of the basic Reynolds’ equation. The analysis has been incorporated in a computer programme and results from it were presented. A Reynolds equation in the following dimensionless form governs the flow of incompressible isoviscous fluid in the clearance space of a journal bearing system. This equation in the Cartesian coordinate system is written as, (1) ∂  3 ∂  3 ∂  ∂h ∂ p  ∂ p  F  ∂α

 

h F2

∂ α 

+

∂β 

h F2



∂β

 

=

∂α

1 h−h   F 0  

+

∂ t

where the non-dimensional functions of viscosity 1

F0 =

dz

∫ µ 0

1

; F1 =

z

∫µ

1

dz ; and F2 =

0

z 

F 1 

∫ µ  z − F  dz 0

F0 , F1 and F 2

are defined by,

(2)

0

The non-dimensional functions of viscosity F0 , F1 and F 2 report for the effect of variation in fluid viscosity across the film thickness. And non dimensional minimum film thickness is given by, h = 1 − X j cosα − Z j sin α

(3)

The above equation (1) was solved to satisfy the following boundary and complementarity conditions: i. On the bearing side boundaries, ( β = ± λ ), p = 0

ii.

(4)

On the supply groove boundaries, p = ps

(5)

iii. In the positive pressure region, Positive pressures will be generated only when the fil thickness is thin, Q = 0, p > 0

iv.

(6)

In the cavitated region, Q < 0, p = 0,

∂ p = 0 ∂α

(7)

34


Solution of Eq. (1) with above boundary and complementary conditions gives pressure at each node. 2.2

Viscosity-Temperature Equation for predicting temperature distribution in bearings The viscosity of fluid film was extremely sensitive to the operating temperature. With increasing temperature the viscosity of oils falls rapidly. In some cases the viscosity of oil can fall by about 80% with a temperature increase of 25°C. From the engineering viewpoint it is important to know the viscosity value at the operating temperature since it determines the lubricant fil m thickness separating two surfaces. The fluid viscosity at a specific temperature can be either calculated from the viscosity-temperature equation or obtained from the t he viscosity-temperature ASTM chart.

2.2.1

Viscosity-Temperature Equations There were several viscosity-temperature equations available; some of them were purely empirical whereas others were derived from theoretical models. The Vogel equation was most accurate. In order to keep a machine workable for long l ong periods, friction and wear of its parts must be kept low. For effective lubrication, fluid must be viscous enough to maintain a fluid film under operating conditions. Viscosity is the most most important important property property of the fluid, which utilized in hydrodynamic lubrication. The coefficient of viscosity of fluid and density changes with temperature. If a large amount of heat is generated in the fluid film, the thickness of fluid film changes with respect to temperature and viscosity. The viscosity of oil decreases with increasing temperature. Hence, the change in viscosity cannot be ignored. Due to viscous shearing of fluid layer, heat is generated; as significance, high temperatures may be anticipated. Under this condition the fluid can experience a variation in temperature, so that it is necessary to predict the bearing temperature and pressure. Therefore, a need has been felt to carry out further investigations on analysis of the thermal effects in journal bearings, so the viscosity-temperature relation given by Ferron J. et al. [3] has been modified. The viscosity µ is a function of temperature and it was assumed to be dependent on temperature. The viscosity of the lubricant was assumed to be variable across the fil m and around the circumference. The variation of viscosity with the temperature in the non-dimensional two degree equation was described by Ferron J. et al. [3]; this equation was expressed as, µ =

2 µ = k0 − k1 T f + k 2 T f µ 0

(8)

The authors authors modified and developed developed a two degree viscosity-temperature relation in to three degree polynomial viscosity-temperature relation. This modified equation as illustrated below, µ =

µ = k0 − k1 T f + k 2 T f µ 0

2

3

− k3 T f

(9)

J. Ferron et al. [3] used the viscosity coefficients, k 0 = 3.287, k 1 = 3.064 , k 2 = 0.777 while while the authors considered the following modified viscosity coefficients, k 0 = 3.1286, k 1 = 2.4817 , k 2 = 1.1605 and k 3=0.3266 . The polynomial equation was found out for getting improved results. Results obtained from viscosity-temperature equation which was developed by authors’ gives good results when compared with experimental results of J. Ferron et al. [3]. This temperature distribution in plain journal bearing shows very slight variation between temperature obtained by authors and temperature obtained by J. Ferron et al. [3]. At different load the computed maximum bush temperature and pressure are nearly equal for 1500, 2000, 3000 and 4000 rpm. The authors have found during their investigation that the developed viscosity-temperature equation gives very close values of the maximum bush temperature when compared with the experimental results of J. Ferron et al. [3] at all above speeds. To verify the validity of the above equations and the computer code, the 35


results from the above analysis was compared with experimental values of J. Ferron et al. [3] bearing. 2.3

Three dimensional energy equation for temperature distribution in bearing The solution of energy equation needs the pressure field established from solution of Reynolds equation. It is very important to carry out a three-dimensional analysis analysis to accurately predict the temperature distribution in bearings. Accurate prediction of various bearing characteristics, like temperature distribution, is very important in the design of a bearing. The heat flows inside the solid parts, such as the bearing and the shaft, and finally dissipates in the air. The total amount of heat that flows out by convection and conduction is equal to the total amount of heat generated. Temperature distribution in fluid-film is given by three-dimensional energy equation. Fluid temperature has been obtained by solving the following three-dimensional energy equation which has been modified using thin-film approximation and changing the shape of the fluid film into a rectangular field, 2





2

2

∂T f ∂T ∂T     + ν f + 1 ( w − z u ∂h ) f  =De µ   ∂u  +  ∂v  h u  ∂α β α ∂ ∂ ∂ ∂z ∂z z h      

∂2T

f  +P e 2   ∂ z

(10)

The non-dimensional effective inverse Peclect number (

Pe

) and Dissipation number ( D ) are e

as follows, k f

Pe =

(C

p

ρ ω j c

2

)

, De =

µ ω j

(C

p

ρ T r c

2

(11)

)

Values of the non-dimensional velocity components in circumferential and axial direction are as follow, u=h

2

v=h

 z ∂ p  z z 1 z d z d z − F1 ∫ d z  + ∫ ∫  ∂α µ F 0 0 µ  F 0 0 µ  0

2

 ∂ p  z z F1 z d z  d z − ∫ ∂ β  ∫ µ F 0 0 µ   0

(12)

(13)

The continuity equation is partially differentiated with respect to z to determine the nondimensional radial component of velocity ( w ) as, ∂2 w 2

+h

∂ z

∂  ∂u ∂v  ∂ +  − ∂ z  ∂α ∂β  ∂ z

 ∂u ∂ h  z  =0  ∂ z ∂α 

(14)

Integrate the above equation with finite difference method considering the following boundary conditions, w = 0 at z = 0 and w =

∂h at z = 1 ∂α

(15)

The three dimensional energy equations have been solved with the following boundary conditions, (i)

On the fluid–journal interface ( z = 1) 36


T f = T j

(16)

On the fluid–bush interface ( z = 0) ,

(ii)

T f = T b

(17)

2.4

Thermal analysis of heat conduction equation for Bush-Housing Heat conduction analysis was performed to determine the bush temperatures. The Fourier heat conduction equation in the form of non-dimensional cylindrical coordinate form has been solved for the temperature distribution in the bush and is given below, ∂ 2Tb ∂r

2

+

1 ∂Tb ∂ 2Tb 1 ∂ 2Tb + + =0 2 2 2 r ∂ r ∂ β ∂α r

(18)

Using following boundary conditions, heat conduction equation was solved. i.

On the interface of fluid–bush

( z = 0, r = R1 ) ,

Continuity of heat flux gives,  ∂ T  k b  = − f k b   ∂ r c h   | r = R1

ii.

   

∂ T f  ∂ z  | z = 0

(19)

On the outer part of the bush housing

( r = R2 ) ,

The free convection and radiation

hypothesis gives,  ∂T  h R  b = − ab  Tb |r = R − T a   ∂ r  k b  2   |r = R2

iii.

(20)

On the lateral faces of the bearing ( β = ± λ ) ,

 ∂T  h R  b = − ab  Tb = ± − T a   ∂ β  k b  |β λ    β | =± λ

(21)

iv. At the outlet edge of bearing pad, free convection of heat flow from bush to fluid in the supply groove gives,  ∂T   b  ∂α   |α =α

e

α e =

=−

h fb R (Tb − T s ) k b

(22)

Circumferential coordinate of the outlet edge of bearing.

v. At the inlet edge of the bearing Tb |

r =R 2

= T s

(α = α i ) and

at the fluid supply point on the outer surface,

(23)

37


In addition, a free convection of heat between fluid and housing has been assuming,  ∂T   b  ∂α   | α =α i

=

− h fb R k b

(Tb − T s )

(24)

Where α i = circumferential coordinate of the inlet edge of bearing. 2.5 Heat conduction equation for Journal For finding the temperature distribution in journal, the following assumptions were made,

i. Conduction of heat in the axial direction. ii. Journal temperature does not vary in radial or circumferential direction at any section. iii. Heat flows out of the journal from its axial ends. Hence the following steady state unidirectional heat conduction equation was used for a journal,  2 ∂ T j k  j  2  ∂ y

   ∆y A j  

+ ∆q = 0

(25)

Where ∆ q = the heat input to the element (q ∆y ) ; reduces to the following non-dimensional form,  2 ∂ T j   + q =0 2  ∂ β   

q=−

k f c k j

q

the length of element. Above equation

(26)

π 

where

∆ y =

is the non-dimensional non-dimensional heat input to journal journal per unit length,

 2π   1  ∂T f    d α  ∫ h  ∂ z       0

(27)

The above equations have been solved with the following fo llowing boundary condition, At the axial ends, i.e. β = ± λ ,  ∂T  j  ∂ β 

  =   β | = ± λ

−

haj R k j

  T − T a   j |β = ± λ  

(28)

2.6 Thermal mixing of fluid in a groove It was not possible for the experimenters to maintain the inlet fluid temperature at a constant value. Because of of low supply pressures pressures and high fluid viscosities, viscosities, the inlet fluid temperature would would rise. Thermal mixing analysis of hot recirculating and incoming cold fluid from supply groove was used to calculate the fluid temperature at the inlet of the groove. Energy balance equation is used to estimate the mean temperature of the fluid in a groove. In this work, the overall energy balance equation is expressed in terms t erms of mean temperature, T m, m, Q Tm = Qre Tre + Qs Ts

(29)

38

Intern Inte rnat ation ional al Jo Jour urna nall of of Mec Mecha hani ni al En Engi gine neer erin ing g and and Te Tech chno nolo logy gy (I (IJM JMET ET), ), I SN 0976 – 6340(Print), ISSN 0976 – 6359(O 6359(Online nline), ), Volum Volum 5, Issue 11, November (2014), pp. 31-46 © IAEME

Where T re - recirculating hot flui , For the unit length of bearing, 1

(

(30)

)

Q = ∫ h u d z 0

(31)

Q s = Q − Q re 1

(

(32)

)

Qre = ∫ C L h u d z 0

1

(

Tre Qre = ∫ C L h u T f 0

(33)

) d z

Mean temperature

T m

re related to the assumed temperature distributio ,

film at at the inlet of the the bearing bearing pa as below, 1

T m = ∫ T f ( z ) d z

(34)

0

Fig. 2: Solution Scheme

39

T f ( z ) across

the fluid


3. SOLUTION PROCEDURE

The overall solution scheme for thermohydrodynamic analysis of plain journal bearing is depicted in Fig 2. The non dimensional coefficient of viscosity has been found out. Reynolds equation solved by finite element method for obtaining pressure distribution in the fluid-film by iterative technique. The negative pressure nodes were set to zero and attitude angle was modified till convergence convergence was achieved. Pressure and temperature fields for the initial eccentricity ratio have been recognized. The load capacity of the journal bearing was calculated by iterative method. Values of the fluid film velocity components were calculated in circumferential, axial and radial directions. Coefficient of contraction of fluid-film was determined. Coefficient of contraction was assumed as unity in positive pressure zone. The mean temperature of the fluid was calculated. By using finite difference method three dimensional energy equation was solved for temperature distribution in fluid-film. Heat conduction equation was solved for determination of temperature distribution in bush housing. The above procedure was repeated till convergence was achieved. One dimensional heat conduction equation equation was used for temperature temperature distribution in journal. The journal temperature was revised after obtaining the converged temperature for fluid and bush. The energy and Fourier conduction equations were simultaneously solved with revised journal temperature. All the above steps were repeated until the convergence was achieved. Using modified non dimensional viscositytemperature relation the non dimensional viscosity was found out and modified until convergence was achieved. achieved. After convergence achieved achieved the temperature of of fluid, bush and journal was found. found. For the next value of the eccentricity ratio once the thermohydrodynamic pressure and temperature have been established. The data used for computation of pressure and temperature in fluid, bush and journal were depicted depicted in Table 1. Table 1: Bearing dimensions, operating conditions and lubricant properties Node No. of nodes in one element 4 R2 Outer radius bush 0.1 m Radius of journal 0.05 m R L Length of bush 0.08 m L/D Length to diameter ratio (Aspect ratio) 0.8 Attitude angle 56° Φ Radial clearance (c) 0.0029 c

Thermal conductivity of fluid Thermal conductivity of bush housing Thermal conductivity of journal Convective heat transfer coefficient of bush Convective heat transfer transfer coefficient of journal Convective heat transfer coefficient of bush housing from solid to fluid Specific heat of lubricant Density of lubricant Viscosity of lubricant at 40°C Journal Speed

k f k b k j hab haj h fb

0.13 50 50 50 50 1500

C p ρ µ N

Reference temperature of lubricant Ambient temperature of lubricant Supply temperature of lubricant

T r r T a T s

2000 J/kg °C 860 860 kg/ kg/m m 0.0277 0.0277 N-s/m N-s/m 1500, 2000, 3000 and 4000 rpm 40 °C 40 °C 40 °C

40

W/m W/m W/m W/m W/m W/m W/m W/m

°C °C °C °C °C °C


4. RESULTS AND AND DISCUSSION

Numerical calculations were performed by writing a computer program in C. The nondimensional governing equations were discretized for numerical solution. The global iterative scheme was used for solving these equations. A mesh discretization for fluid film and bush with 68 nodes in the circumferential direction, 16 nodes in the axial direction and 16 nodes across the film thickness and 16 nodes across the radius of bush thickness. For thermohydrodynamic analysis of plain journal bearing the input parameters has been taken from Table 1. The present data was assumed for aligned plain journal bearing. It was assumed that temperature of fluid equal to temperature of bush at the fluid–bush interface. Journal temperature is also equal to temperature fluid at the fluid–journal interface. The condition of mixing the recirculating fluid with the supply fluid was also considered. Fig. 3 and Fig. 4 depicts the distribution of the maximum bush temperature obtained with different eccentricity ratio for different speeds of plain journal bearing. The experimental results of J. Ferron et al. [3] were nearly equal to theoretical results of authors as per the modified viscosity-temperature equation. Fig. 5 and Fig. 6 predict the circumferential temperature distribution in the mid-plane of fluid-bush interface. Theoretical predictions and experimental results of J. Ferron et al. [3] exhibit a similar pattern, the predicted maximum temperature value and their locations are reasonably very close to the measured values of J. Ferron et al. [3]. Pressure variation in mid plane of plain journal bearing for various speeds and loading conditions were shown in the Fig. 7 and Fig. 8. In the authors developed model pressure distribution was very close to the experimental values given by J. Ferron et al. [ 3]. The mean journal temperature has been computed along axial direction. Fig. 9 depicts load versus mean journal temperature at 2000 and 4000 rpm for two different loads as 4000N 4000N and 6000N respectively. respectively. The radial temperature was negligible in the present work. Journal temperature along axial direction of the journal varies by about one degree for 2000 rpm and very close to the 4000 rpm at 4000 N and 6000 N loads respectively. A theoretical result predicted by authors’ modified viscosity-temperature equation gives good agreement agreement as compared compared with published published experimental experimental results of J. Ferron et al. [3].

Fig. 3: Comparison of Experimental values of Ferron J. et al. [3] with theoretical values of predicted model for Maximum bush temperature and eccentricity ratio at different speeds

41

Fig. 4: Comparison of Theoretical values of Ferron J. et al. [3] with theoretical values of predicted model for Maximum bush temperature and eccentricity ratio at different speeds


Fig, 5: Temperature distribution in mid-plane at 4000 rpm under 6000 N Load

Fig. 6: Temperature distribution in mid-plane at 2000 rpm under 4000 N load

Fig. 7: Pressure variation in mid plane at 4000 rpm under 6000 N Load

Fig. 8: Pressure variation in mid plane at 2000 rpm under 4000 N Load

Fig. 9: Variation in mean journal temperature at different loads and speeds

42


5. CONCLUSIONS

On the basis of results and discussions presented in the earlier sections, the following major conclusions are drawn:

• • • • • • • •

The developed viscosity-temperature viscosity-temperature equation for this work is more appropriate. The maximum pressure is noted at minimum film thickness of fluid. The temperature of fluid-film increases with increase in load and speed of shaft. Due to thermal effects the eccentricity ratio, attitude angle and side flow also changes. The effect of mixing of recirculating and supply temperatures of lubricant in the groove is quite important. Heat transfer from the outlet edge of the bush to fluid in the supply groove must be considered for correctly simulating the actual conditions. At higher speed and heavy load, developed model of viscosity-temperature predicts accurate values for temperature in fluid, bush and journal. The authors have found during their investigation that the developed equation gives very close values of the maximum fluid and bush temperature when compared with the experimental results of J. Ferron et al. [3] at different speeds and loads respectively.

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Nomenclature A j

Cross-sectional area of the journal ( π R 2 )

c

Radial clearance, (m); c = c / R

C L

Coefficient of contraction , C L is unity in positive pressure region region 1

(

C L = ∫ u h 0

)|t d z e

1

∫ ( u h )|α d z 0

C p

Specific heat heat of fluid, (J/kg °C)

D

Diameter of Journal, (m) 44


De

Dissipation number

e

= e / c Journal Eccentricity, (m); ε = Non dimensional Integration functions of Viscosity

F0 , F 1, F 2

h hab haj h fb

k0 , k 1 k2 , k 3 k , k f b k j

Thickness of fluid-film,(m); h = h / c Convective heat transfer coefficient 2 0 bush, (W/ m C) Convective heat transfer coefficient 2 0 of journal, (W/m C) Convective heat transfer coefficient from bush to fluid in groove, 20 (W/m C) Coefficient of Viscosity Thermal conductivity of fluid, bush and journal, (W/m °C)

L

Length of bearing, (m)

p

Pressure , p = p ps (N/ m )

ps

Supply pressure, (N/ m )

Pe q

Peclet number, Heat input per unit length

Q

Fluid-flow, (m /s)

r R

Radial coordinate; r = r / R Radius of journal, (m) Inner and outer radius of bush, m

R1 , R2

R1 = R1 / R, R 2 = R2 / R

T r

T a

Reference temperature, (°C) Ambient temperature, (°C); T a = Ta / T r

T b

Bush temperature, (°C); T b = Tb / T r

T f

Fluid film temperature, (°C);

T j

Journal temperature,(°C); T j = T j / T r

T s

Supply temperature , (°C); T s = Ts / T r

t

Time ; t = t / ω j

u,v,w

2

2

3

4 Q s = Q (ω j c R )

T f = T / T f r

Fluid velocity components, in circumferential, axial and radial u=

directions respectively (m/s)

u v w , v= , w= (ω j / R ) (ω j /R ) ( ω j / R)

Cartesian Coordinate in circumferential, axial and radial direction,

x, y, z X j , Z j

z = z / h Coordinates of journal centre, (m);

45


α β ε λ φ µ

µ 0 ρ ω j

X j = ε sin φ , Z j = − ε cos φ Circumferential cylindrical coordinate; x coordinate; x R

Axial cylindrical coordinate; y R Eccentricity ratio; Aspect ratio; L D Attitude angle (degrees) Viscosity Viscosity of of fluid, fluid, (N.s/m (N.s/m ); 2

Reference viscosity of fluid,(N-s/m ) Mass density density of fluid, fluid, (kg/m (kg/m ) Angular speed of the journal, (rad/s)

46

Thermohydrodynamic Analysis of Plain Journal Bearing With Modified Viscosity Temperature Equation

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