A solution of the thermohydrodynamic problem for laminar flow bearings

A solution of the thermohydrodynamic problem for laminar flow bearings z.s. Safar*

This paper introduces a semi-analytical method for solving the thermohydrodynamic problem in journal bearings. The momentum equation is reduced to three ordinary differential equations by separation of variables. Two of the resulting equations are integrated directly, while the third is interpreted and solved as an isoperimetric problem. The energy equation with its boundary conditions is solved by the Galerkin Kantarovich method.

A numerical solution was first obtained by Dowson and March 1 for the two dimensional thermodynamic analysis of laminar flow journal bearings. McCallion et al 2 used a simplified method which uncouples the energy and the Reynolds equations by neglecting the effect of pressure on temperature distribution. Yu and Szeri 3 eliminated the equation of heat conduction and solved the remaining equations by a numerical iteration method. This paper presents a semi-analytical solution of the equations governing the laminar thermohydrodynamic behavior of journal bearings. A comparison with numerical results is made.

Analysis The dimensionless pressure equation for the lubricant is

a

ap

D

ao-- [a3r(°) 5b I + (L-)2 =-21r [ d~ - ( 1 - ~ )

a

ap

[H3r(0) ~,~ ]

~2 (0,1) d - ~ ( H -/j,- (0,1) )]

(1)

with the boundary conditions, P(O,,~) = P(02,~) = 0 0 1 P(,k)

1 = P(O,-~-) = 0

(2)

P(02,~)/)O It was shown by McCallion et al 2 that the energy equation may be written in the following approximate dimensionless form

*Mechanical Engineering Department, Cairo University, Cairo, Egypt. Present address: University of California, Mechanical Engineering Department, Berkeley, California 94720, USA

Nomenclature C bearing radial clearance D shaft diameter H dimensionless film thickness (h/c) L bearing length Nu Nusselt number (X (Ri + b)/Ko) oil thermal conductivity go bearing thermal conductivity KB lubricant pressure P Pecklet number (pCp wC2 /Ko) Pe Pr Prandtl number (Cplli[Ko) pad inner and outer radius Ri, Re Sommerfeld number (#iNLD(R/C) 2/W*) Se T lubricant temperature inlet oil, shaft temperature Ti, Ts bearing load W* b pad thickness Cp specific heat of lubricant dimensionless lubricant pressure (P/laiN (R/C) 2) P h film thickness

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TRIBOLOGYinternational October 1978

t u X, y , g

e

0, r/, ~ /d, V

#i ~2

dimensionless lubricant temperature velocity component rectangular Cartesian coordinates pad angle journal eccentricity ratio dimensionless coordinates (X/R, y/h, 2z/L) absolute, kinematic viscosity of lubricant inlet viscosity of lubricant dimensionless viscosity Oa/lai) ratio of bearing to shaft speed (1/Nu + (Ko/KB) 9.n (1 + b[R) coefficient

(-c/R)

, (0, a) ft~ o

ft~ o

r(o)

dr/ ~(o,r/)

r/dr/ ~(o,r/)

f= [~2(O,r/) ~2(0, 1) /jt(O,r/)] dr/ o /jl (0, 1) attitude angle

)

1.0 09 08

L/D = I

L/D = I

B --I00*

B -- ICO°

Pe = 4 0 0

Pe = 3

( B = 360 ° )

Theoretical ( / ~ = 3 6 0 °)

* = 0.2 Pe = 2 0 0

Theoretical ( ~ = 180 ° )

55 Yu end Szeri Present analysis

w

Experimental

- .....

~ o.7 ~ ' , , , , x _

o o

>" 0.6 u

o

"

05

,'x

50

E oJ

I-

0.4

.

0.2

45

0.1 0

I

I

I

I

I I II

0.1

I 1.0

Sommerfeld

I

I

I

I

variable

at _ 1 a2t +~(au)2 Po art2

(3)

I 160

I 200

By assuming that the heat flow in the bush is in the radial direction, Safar e t a l 4 showed that equation 3 is subjected to the boundary conditions.

(4)

0 = 0

This system of equations is solved by a semi-analytical method which is shown in the Appendix.

Sample calculations The Sommerfeld number and the attitude angle have been calculated for 120 ° bearing when Pr = 500 andNu = 100,

Appendix To solve equation 1 subjected to the boundary conditions 2, a pressuTe distribution is assumed in the form p(O,~) = e=(O)+O ( O ) E ( ~ ) Substituting the assumed solution into equation 1, three ordinary differential equations are obtained. One equation is integrated directly to yield the long bearing solution 4 . Direct integration of the second equation gives

I 240

I 280

I 320

:360

0

Fig 2 Theoretical predictions overestimate circumferential bush temperature change

Table 1 Thermohydrodynamic case

( c)

~) nl =

1 120

Angular position from inlet ,

a7

t = S pCp ( T - ri) R2

t (0, 7) = ti (7) t(O, 1) = t s at (t+3,H(0)

I 80

I0.0

Fig I Comparison of numerical and semi-analytical results .for two values of Pecklet number

H2u aO where

I 40

0

I III

Isothermal case

e

LID

S~

¢

Se

¢

0.2

1 0.25

1.6419 10.383

68.902 85.171

1.01 8.45

58 66

0.6

1 0.25

57.7 69.846

0.102 1.12

36 40

0.19308 1.800

with two values of aspect ratios, 1 and 0.25. Table 1 compares the thermohydrodynamic case and the isothermal case 6 and Fig 1 shows a comparison with the numerical values given by Yu et al 3 for a 100 ° bearing. The influence of the Pecklet number was relatively small. It is interesting to compare the theoretical bush temperature and the experimental results Obtained by Dowson and March I (Fig 2). The theoretical prediction overestimates the circumferential temperature change measured experimentally: there is, however, a general agreement between the forms of the theoretical and experimental curves. Fig 2 also shows that bearing cooling is improved by using a partial bearing.

hL E = B cosh - ~ The third equation is transformed by changing the variable 0 t e a = [~ (0) 1/2] 0 where ~ (0) = H a ( 0 ) F (0) and then it is interpreted as an isoperimetfic problem in the calculus of variations. The approximate eigenfunctions are assumed in the form Ak

= nT, X n k

sin [n/r

:~-01)

--oT) ]

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