- Email: [email protected]
Thermal effects in lubrication of asymmetrical rollers
Thermal effects in lubrication of asymmetrical rollers D. Prasad*, J. B. Shukla*, P. Singh*, P. Sinha* and R. P. Chhabra **'t Unlike earlier lubrication studies using power law lubricants, where symmetric conditions were explicitly or implicitly assumed, this work presents a theoretical analysis of heavily loaded rigid asymmetrical rollers under combined rolling and sliding motions, by considering cavitation. The modified Reynolds' and energy equations under isothermal boundaries are obtained and solved simultaneously for many values of a sliding parameter 0 close to unity. It is seen that the presence of sliding enhances both pressure as well as temperature, and hence the consistency variation, as compared to the pure rolling.
Keywords: lubrication, thermal effect, non-Newtonian, roller bearing
Introduction Hydrodynamically lubricated cylindrical roller bearings are widely employed in heavily loaded systems owing to their simplicity and high load carrying capacity. In such applications, severe pressure and temperature conditions are encountered which exert an appreciable influence on the physical properties (especially viscosity) of lubricants. In recent years, there has been an increasing trend towards the use of synthetically constituted lubricants including modified oils, aqueous suspensions of fine particles, etc., which invariably exhibit non-Newtonian behaviour in the range of stress and strain rates encountered in roller bearings. Within the general framework of non-Newtonian fluids, the power law model has received considerable attention in recent years ~-5 and its use has also yielded promising results in a variety of engineering applications including lubrication. Owing to the non-linearity of stress and strain rates, most previous studies of lubrication have involved varying degrees of approximations. For instance, in the case of lubrication of rolling bearings, pure rolling condition (symmetry) has been tacitly employed: that is, both surfaces are assumed to be of equal size and to rotate with the same velocity¢' s. Sinha and Singh '~ used the power law fluid model to examine the lubrication characteristics of a cylindrical plane bearing and they assumed the flow about midplane to be symmetrical. Moreover, these investigators treated the fluid properties to be constant in the flow region of interest. In a subsequent publication, Sinha and Raj ~' presented another analysis incorporating the exponential viscosity variation but still retaining the symmetry of flow about the mid-plane. Numerous other analyses making similar approximations have also been reported in the literature ~~.J2. More recently,
Prasad et a113 have examined the effect of temperature and pressure dependent power law consistency index, m, of the lubricant flowing between cylindrical rollers, although the assumption of the symmetry of flow was invoked. In fact, there have not been many studies in which the symmetry of flow has not been used. For instance, the pioneering work of Dyson and Wilson ~4 is also based on this assumption, though subsequently this restriction has been partly relaxed. Admittedly, their analysis takes into account the EHD effects but ignores the thermal effects. Savage ~5 has employed a perturbation method to seek analytical/numerical results for this flow configuration for isoviscous flow of power law lubricants. Dein and Elrod ~6 presented the Reynold's equation for power law lubricants with special reference to the case of a journal bearing and assuming that the strain rates in the fluid arise primarily from the relative velocity between the two surfaces. They too sought an approximate solution using the perturbation method for isoviscous fluids. In this study, a new analysis is presented which takes into account the rolling/sliding (antisymmetry) condition of lubricated rigid cylinders by a power law model fluid. The effect of hydrodynamic pressure and temperature on the power law consistency index (ie viscosity) has been incorporated in the treatment w-2°. The coupled Reynolds' and energy equations have been solved simultaneously to obtain extensive results on temperature fields prevailing in the flow domain.
Mathematical analysis The governing equations, with the usual assumptions of hydrodynamic lubrication, are ~
de
0 [ IOu["-lOu)
ck = oylmloy[ *Department of Mathematics, Indian Institute of Technology, Kanpur208016, India. **Department of Chemical Engineering, Indian Institute of Technology, Kanpur-208016, India. ~Author for correspondence.
TRIBOLOGY INTERNATIONAL
oy.
Ou Ov + = 0 Ox Oy
(1) (2)
where m follows the relation 7
0301-679X/91/040239-8 © 1991 Butterworth-Heinemann Ltd
239
D. Prasad et a I - - T h e r m a l effects in lubrication of asymmetrical rollers
Notation An cn
d,, h h, h~, h2 H K m mo
th n
P
-)
R T F
T~ Tu T,, Tm
Ln b/, V
n2[(2n+ 1 ) ( 3 n + 1)] [( 4n+ 2)/n]" (U/h,)"(2R/h,,) ''2 n/[( 3n+ l )2"+ ~l Lubricant film thickness Minimum film thickness Film thickness at x = -Xl and x = x: respectively 1 + X 2 (= h/ho) Lubricant thermal conductivity Lubricant consistency Consistency at ambient pressure and temperature Consistency (= 2mc,,a) etc Flow index Hydrodynamic pressure Pressure (= ap) Equivalent radius Lubricant temperature Temperature (= [3T) etc Traction force Surface temperature (constant) Ambient temperature Surface temperature (at y - 0) Mean film temperature Lubricant velocities in x, y directions
m = mo e x p [ a p - ~ ( T m - T o ) ]
t? W
UE/U2 Velocities of plane and cylindrical surface respectively Normal load
x, y x~ x2 X a, 13
Coordinate axes x~ is the point of maximum pressure (x~ > 0) Cavitation point (x2 > 0) x/(2Rho) I/2 etc Pressure and temperature coefficients respectively
U I , U2
Load (= W/[(2Rho)'"2/cx)]
U2ho6( h,,/2R ) '/2 / ( KeO 8
Loci of points at which the lubricant velocity gradients vanish 8/ho ~2 Value of ~ at X = X_~ 8" Values of ~ in the neighbourhood of x = -+ x~ * 8" / hu Subscripts 1,2 usually refer to the respective quantities in the inlet and outlet regions separated by the point X =
/J etc
XI
Bar denotes a dimensionless quantity
(3)
with U1 ~ U 2
Tm = h
)
Tdy
(4)
F r o m the point of view of mathematical analysis, the contact between two cylinders of radii R . and Rb (as shown in Fig l ( a ) ) can be described by an equivalent cylirider near a plane (shown in Fig l ( b ) ) 21.
x
--~Ol
b
Film thickness h for this g e o m e t r y is expressed as X2
h = h,, + 2R
x
(5)
Fig 1 (a) Schematic representation o f two cylindrical rollers with different dimensions and velocities. (b) Cylinder-plane contact
where 1
1
1
R = R,, +
(6)
The b o u n d a r y conditions for the governing equations (1) and (2) are as follows: u=Ui,v=0
aty=0 dh
// = U 2 , v = U 2 ~
aty = h
p = 0 atx -~;dp/dx p=0,
dp dx=0
(7)
atx=x2
(8)
= 0 a t x -- -x~ (9)
In order to solve the system of equations (1) and (2) analytically, one has to take care of the m o d u l u s signs of the velocity gradient, Ou/Oy. For pure rolling, four different regions exist, but due to s y m m e t r y only two are required 9. H o w e v e r , the a n t i s y m m e t r y necessitates consideration of all the four regions. This is easily visualized for the N e w t o n i a n case (n = 1) as follows. 240
Equations (1) and (2) can be solved for n = 1 and m to be constant or a function of x alone; one can obtain
u = Ui + ( U 2 - U 1 ) h -
2m
dp = 6 ( U , + U a ) m ( h _ h , ) / h 3 dx
( l(1) (ll)
It can be seen that u is a linear function of y at points where d p / d x = 0. At these points, the velocity gradient Ou/Oy = ( U 2 - U ~ ) / h l which can never be equal to zero, since U2 4= U~. It can also be seen from Fig 2 that for each x, Ou/Oy can vanish at one point of y ( = 5 , 0 <~ 5 ~< h) in the regions -~c < x < -x~ and -xL < x < x2. Thus these regions can be divided into four sub-regions separated by the 8-profile, having velocities u~, u2, u3 and u4 as shown in Fig 2. For any e > 0, let us and u~, be the velocities in the regions -x~ - ~ ~< x ~< -x~ + ee and x~ - e3 ~< x ~ x2 respectively, August 91 Vol 24 No 4
D. Prasad et aI--Thermal effects in lubrication of asymmetrical rollers
then according to the velocity profile, shown in Fig 2, one may observe the following:
or
Q= 0u~ >0, a
)
u2 dy +
u l dy
or
Ou~, ~<0, 0 ~ y < a Oy
I: - ~ < x < ~ - x l . e t
(12)
2n+l
\m, d x / (18)
Since the flux Q is constant through every cross section, Q may be equated to the flux through the point x = - x l [Ref 22]. This point is selected in order to enforce the continuity of flow.
0u3>0, 0~
Q= U,a+ U2(h-a)-
[a <2''+')/'' + ( h - a ) <2"+')/"]
Ou~=Ou~= 0 a t y = a Oy Oy
I1: -Xl"~E2~x~x2-~_ 3
(13a)
0/J3 ~_.0/~4 ~ 0 a t y = a
Oy
I
Thus flux Q through the point x = - x ~ (where dp/dx = 0) can be evaluated directly from Eq (1). yielding
Oy
( U1 + U2)h
O(-xl) - ~
OUs
lll: --Xl--~l<~X<~--Xt+e2 (13b)
00u;<0, O<~y<~h
IV: x2- %<~x<~x2
(13c)
It is expected that the nature of the velocity profiles for different values of n as well as variable m, will remain the same as it is true for pure rolling'~,~3. Using the sign for the velocity gradients given in (12) and integrating equation (1) twice for the region ~ ~< y ~< h, one can obtain the velocity profile u~ as
t
(19)
Finally equating the fluxes given by (18) and (19) and simplifying, one can obtain the following Reynolds' equation:
((Ut-U2)a
+ U2h-½(UL++Uz)h
#2,,7i;.,+ (h_S)<:,~,).,
l)"
-~
(20)
Similarly for the other region, the Reynolds' equation may be written as
[(h-a)<,,, ,)/,,_(y-a)<,,,,v,,] Similarly for the other regions, expressions can be obtained:
the
(14) following
½(U, + U2)hA - U2h - ( U I - U 2 ) a ~ " a<
)
+
-xt<~x<~x2
(21)
U2 = UI - (F/~ l) ( m l / d 2 ) l / n
Use of the matching conditions [a{,,~
,):,,
_ (a-y){,, ~ ,)/,,]
u3= U , + n + ] m2clx] [a{,,+')/- - ( a _ y ) { - + , ) / . ]
(15)
(16)
Ul = /'/2
I aty = a
U3 ~ ~/4
I
into (14-17) yields
(.)(_ 1.2t,.,
U4 = U2 -[- r/-it- 1
m2 dx /
[(h-a){,,, ,)/,, _ (y-a)¢,,{ ')/,,]
(17)
The velocities in the regions III and IV are calculated numerically by extending the velocities u2 and u4 respectively, where a is replaced by a*. (For more details see the Results section, under the title numerical solution.) Now the volume flux Q for the region I is obtained as
Q=
f
h u dy
TRIBOLOGY INTERNATIONAL
[(h-S)
{''+'):'' - a { " * '):"] = 0
-oo
n+l
(23) m2
<'+')''' - a ~'+')''1
% - x l <~x<~x2-%
= 0
(24)
Eliminating dpl/dx and dpz/dx from Eqs (23) and (24), using Reynolds' equations (20) and (21), one can obtain a single relation as follows: 241
D. Prasad et aI--Thermal effects in lubrication of asymmetrical rollers implies
F
cl = c~ - c (say)
i
d l - d~ - d (say) The constants c and d are evaluated using the boundary conditions (27) and are given below: c=
h
+\
h K / m l dx
[(h_a)13,, { .1.',, _ 8¢~,,, ,,,,] d ]
II1
11
[(h-a)'"'
''"-
where A,, -
Finally the mean temperature Eq (4), may be written as
a'"~ ' " " l [ ( U , - V 2 ) ~
+ U,h - ~(U,+U,_)h,] (25)
= II
OF
,if,
T,,,] = h
be 7
. Oy
~Oy/
y
of Eq gives
ml
=0 (27) (26)
twice
T"=-(2n+l)(3n+l)(mK
,
'
for
the
)(m,
region,
nS~
-(4n+l)h[(h-8)'4" (28)
t ),',, + c2y + d2
(32)
L,,._= T,,+T.~2+ &v[ ( h - a ) ~ 3 " .
l"
n-' ( K dP') (''' ')"" T12 = - ( 2 n + l ) ( 3 n + l ) dx (8_y)(3,,,
dx
Proceeding in the same way, the mean temperature for the region -x~ ~< x ~< x~ can also be written as
I
( y - a ) c~''~ ])"" + c,y + d,
(3])
,,, (m,A,,)(, d,,l)"" ....
at y = h
Integration -~
]
where
The boundary conditions for this equation are
Th
Tll dy
nS, [ ( h _ a ) ~ , , ~,~,,, + a~4,,,.~.,,, ] ( 4 n + 1)h
(26)
w.here the heat convection has been neglected.
T
;
7"/, + To] SI + 2 l ( h - a ) ~ ' " ' " " " + S'-'"'"""l 2
The energy equation for this problem is considered to
at
~ Ti2 dy +
or
Energy equation
= T.t
T,,,~, as defined in
T'"l : h 1 if'~ T d y
It must be noted here that Eq (25) cannot be used to evaluate 8 at x = -x~ and x = x2 since au/Oy 4= 0 at these points.
T
( 2 n + l ) (3n+ l)
Thus T]~ and T~2 are explicitly known functions of x and y.
[~3~2,,' '),',, + (h_.,3)~2,, , 1,,',,]
ay 2 = -
] m~ elf
n2
(2n+l \ n+l )
U 2 - Uj
K
IV
Fig 2 Qualitative velocity profiles shown by arrows; 6distribution indicated by A B C and EFG, and 6*distributions shown by broken lines _
T~,l+\
(30)
,,.,, + a~,,,,, ,.,,]
' ) " " + S '4'' ')""]
where
= m~A,,(~ _ 1 d~c2)("~ )j'' S~ (29)
where T~ and TIe are the lubricant temperatures in the regions, g ~< y ~< h and 0 ~< y ~< g respectively, c~, d] and c2, d2 are the integration constants.
(33)
K
(34)
o12
Using the non-dimensional scheme X = x/\,;2Rh, etc., p = o~p, rh = 2rnc,,o~,
Use of the temperature matching condition T~ = T]2
at
y=
H = 1 + X2, etc.
and the matching heat flux KOT~ 33, 242
=
KOTI2 3y
at
y =
7",,, -
6T.,.
g =
a/h,,. 0
U./U~
(W,
>U_~)
(351
Eqs (20), (21), (25), (31), (33) may be rewritten as follows: August 91 Vol 24 No 4
D. Prasad et aI--Thermal effects in lubrication of asymmetrical rollers
(36)
dX - rh'0Zx)"
d&_
).
(37)
dX
1-
V¢ = -
TFO = --
[ ( H - ~ ) t'+')/'' - ~t"+')/"] [ ( 0 - 1 ) ~ + H - ½(1+/--?)Ht] 8(2"+1)/'' + ( H - ~ ) ( 2''+')/''
= 0
/.
(38) :
-
/~h+ L'l 2
L,+2 L,,
+th'd"~l@x)'+'g'x + rhe d, ~(
(39)
_L.).,+, gx
(40)
(51)
%,=o dx
(52)
dx
(53)
and Vv,, = -
Dimensionless tractions are i'Fo (= ~ T v o / h o ) = -
"%=0 d X
(54)
Similarly,
where rh~ = rho exp(p~-/~m,+/],), me = rho exp(p2-/'m~+/'~)
(41)
(g2ho~)(ho) 1/2 "9 = \ Kc~- J \2R]
(42)
d,, = n / [ ( 3 n + l )2 "+~]
(43)
(/.)- 1)~ + H - ½(/]+ 1)H, fx =
X dx dX
The surface traction force T F is obtained from the integration of shear stress r over the entire length, that is,
L? ( 2 n + l 1 -\ n+l ]
/-,,,
p dX = -
(44)
~,2,,+,)/,; + (H--6~(~.,+,)/~
gx = 1 [ ( H - 6 ) ° " + ')/" + ~<3,+,)/,,1 n
[(H_~)(4,+,)/,, + ~4,,+,)/,]
(4n+a)U
(45)
Equations (36), (39), and (37), (40) have to be solved simultaneously using the following conditions: Pl = 0
at
X---~-~
d0,_dp2_0 dX dX p2=0-
0,o2 dX
at
X=-X1
at
X=Xz
(46a) (46b)
(47)
From Eq (37), it may be noted that the last condition implies
at X = X2, H = H2
(48)
which is independent of n. Now for n = 1, H~ = H 2 or X2 = Xt when d p 2 / d , x = 0, see Eq (11). Hence from Eq (48), it follows that 82 = H1/2 (see Fig 2).
TFh = --
f.2
"%= . dX
(55)
Results and discussion Numerical solutions of the Reynolds' equations (36, 37) and the energy equations (39, 40) have been obtained for antisymmetrical, viscous incompressible flow of a power law fluid through the gap between a cylinder and a plane (see Fig (1)). The results of this investigation are assessed in terms of parameters n and U (= U~/U2, U~ > U2). The sliding parameter (J is chosen to lie between 1.0 and 1.2 (ie maximum of 20% slip). The parameter (J arises due to the consideration of antisymmetry conditions and is important because the presence of sliding ( / d > l ) is likely to produce greater pressure and temperature as compared to that for pure rolling ((J = 1). The significance of (.1 along with n has been demonstrated through table and graphs. For the numerical calculation, the following representative values have been used: U, = 500cm s -1, ho = 5 x 10 4cm, et = 1.6 X 10 -9 dyne c m 2, R = 3 cm, ~c~ = 1.5, Tot = 1, ~/ = 5 where + = 0 corresponds to the isothermal solutions. It is of interest to note that the flow configuration considered herein includes several known situations as limiting cases: for instance, when U~ = Uz and m is constant, it reduces to the case examined by Sinha and Singhg; for the case of [3 = 0, ie temperatureindependent viscosity, one obtains the case considered by Sinha and Raj l°. Finally, when U~ = UE, the present analysis is equivalent to that of Prasad et al 7.
Numerical solutions Load and traction The load component W in the y direction is calculated as
The dimensionless load 1 ~ / ( = W e d ~ o ) TRIBOLOGY INTERNATIONAL
is given by
The Reynolds' and energy equations are coupled through rh and contain two unknowns ~ (the locus of points at which 3u/Oy = 0) and XI (X = - X t is the point of maximum pressure). These unknowns are also present in Eq (38). As there is no symmetry ((.t v~ 1) for ~-distribution, it is necessary to solve Eq (38) for over the complete region under consideration, along with the Reynolds' and energy equations. The actual
243
D. Prasad et aI--Thermal effects in lubrication of asymmetrical rollers
process followed for numerical computation is briefly described below. The algebraic equation (38) in 8 contains X and X~ explicitly. First of all, an initial value of X is assigned ie the point at minus infinity is replaced by a large but a finite negative value. An arbitrary value of Xj is chosen. The value of 8 at this inlet is obtained by solving the algebraic equation (38) for 8 using the bisection method with a reasonable tolerance (say 10 4). These values of X and 8 are substituted in the energy equation (39). 7",,,~at the inlet is then obtained from solving the algebraic energy equation by prescribing rh~ and pj (= 0 at X - - + - ~ ) and using the earlier numerical technique. The same X, rho and the computed 8 and T,,,, are used in the differential equation (36). Fourth order Runge-Kutta method is used to evaluate/)~ at X = X + AX(AX > 0). For this value of X ( = X + AX), Eq (38) is solved for 8. These new X, 8 and/51 are substituted in the energy equation which again yields T,,~ as a solution. These Tmj, X and 8 are used to calculate/)~ as a solution of Eq (36) at another point X = X + 2AX. This process is repeated for the other values of X until such time that Eq (38) y_ields values of 8 satisfying 0 ~< 8 <~ H (because ~ cannot exceed H). It may be noted that at X = -X~, 8 does not exist (see previous section). Further, in the neig_hbourhood of X - - X ~ , the determined values of 8 do not lie in the interval 0 ~< ~< H. Hence in the neighbourhood of X - - X , , 8 (= say 8*) has to be determined solely on the basis of physical considerations. It should be emphasised here that 8" does not refer to the locus of points at which the velocity gradient vanishes. The e-neighbourhood: ( - X I - cj ~< X ~ - X I + %) of X = - X I is to be determined as the region where there exists no 8 lying in the interval 0 ~< 8 ~< H and satisfying Eq (38). To ease the mathematical complexity, a linear profile for 8", is given below
is calculated using Eq (57) together with/)2 and T,,eIf the computed value of/)2 at X = X2 (or Xl) satisfies the condition (47) (ie/)2 = 0 at X = X2), the assumed arbitrary value of X, was correct. Otherwise, another value of X~ is assigned and the whole process is repeated so long as/)2 vanishes at X X,. Thus X~ and 6 are computed along with pressure /)~, /)2 and the mean temperatures 7",,,~ and T,,2. The complete mean temperature profiles as functions of n and U are elaborated upon in the following sections.
Temperature profile The mean temperature T,,, has been shown in Figs (3) and (4) as a function of n and fJ. It may be observed that the qualitative behaviour of T,,, versus X is very similar to the temperature profile obtained in Refs 7, 20, 24. For a fixed value of n, T,, increases with [J which indicates that the sliding temperature is higher than that of pure rolling 2°-24. For fixed U, T,, increases with n showing that the temperature for dilatant fluid is higher than that for Newtonian and pseudoplastic fluids ~3.
Load and traction The calculated values of load carrying capacity I~ and the traction force T~. are given in Table 1. It can be seen from the table that W increases with n and (J. This confirms the enhancement of load for dilatant fluid which is consistent with the previous finding 1-~-:3-25. The increase seen in fV with (J is caused simply by the increase in hydrodynamic pressure with sliding 2°,24. The traction force Tv- has been evaluated for both the surfaces at y = 0 and y = h, and they have different
U=1.2
8*=-HI
2e,
(X+XI-~)
-X~-e1<~X<~-X,
1.9
(56a) n=1.15 1.8
= ~
-
.
-Xl<~X<~-Xl+ez
(56b) 1.7
is assumed. This profile was chosen in such a manner that it not only satisfies the Reynolds' equation at X = -X1 but also makes the pressure curve continuous. Having determined 8 = ~* using Eq (56) in the neighbourhood of the point X - -X~, the procedure obtained earlier is adopted to evaluate/51 and T,,,, for the region -X~ - e l ~< X ~< - X I , and P2 and Tin2 for the region -X1 ~< X ~< -X1 + %. Since in the interval - X , + % ~< X <~ X2 - ~3, ~ (roots of E q ( 3 8 ) ) satisfies the inequality 0 ~< 8 ~< H, so 8 in the region -X~ + e2 <~ X ~< X2 - e3 is determined by the same procedure followed near the inlet along with /)2 and 7",,2. Subsequently in the interval X2 - ~ <~ X <~ X2 (where Eq (38) is not valid) the same type of linear profile, namely 8*(X) = H~ (X-X2+~3) 2%
(57)
is assumed (the complete 8 - 8" profile has been shown in Fig 2). ~* in the region X2 - e3 ~< X ~< )(2 244
1.6
1.5
. . . . .
1.0 1.4
0 0,4 1.2
1.1
-5
I
I
I
I
-4
-3
-2
-1
X
Fig 3 Mean film temperature T,,1 versus X f o r various values o f n August 91 Vol 24 No 4
D. Prasad et aI--Thermal effects in lubrication of asymmetrical rollers
1=_ .20 ~ /~ I .15 ---I .I0 -- -I. 05 --X--
I .9
,.oo--e-
~.~
_ ~_._ .__~___~ _ _ _ _ . : ~ . . ~ .
traction forces approach towards the same value. This is plausible, since once U~ ~ U2, both the surfaces will experience the same force. The load and the traction forces both have been calculated for [3 = 0 also and are also given in Table 1. It can be noted from the table that W and TF are both higher than those based on the temperaturedependent viscosity 7-j3 and these observations are in line with the findings (traction coefficients) of Ghosh and Hamrock 2°.
1.5
Cavitation point and its E-neighbourhood
~~~X~~
/
I.q
xh
"'1'I
n=ll
-5
I
I
-q
-3
I -2
-I
0
X
Fig 4 7f,,,-distribution versus X for various values of (J characteristics. The dependence of Tvo on n and /.J is qualitatively similar to those obtained for loads, that is, ~rvo increases with n and U (Ref_13). However, T~, increases with n but decreases as U increases. This is because an increase in [E(=U~/U2) is tantamount to the fact that only U] is increasing as U2 is fixed. However, it was observed that as U ~ 1, both the
Cavitation points X2 for all n and U have been calculated but have not been given here. Their trend with n is quite similar to those reported earlier" for (J = 1; however, for a fixed value of n, cavitation point shifts towards the centre line of contact, but not significantly. Similarly, the position of pressure peak ( X = - X l ) shifts towards the centre line of contact 2°. The value of • is found to be lie the interval 0 < • ~< 0.01 which is true for both at X = -X~ and X = )(2.
Conclusions With the usual assumption of hydrodynamic lubrication, an attempt has been made to study thermal lubrication of rolling/sliding contacts by an incompressible power law fluid. The thermal Reynolds' and energy equations which are functions of consistency (&), sliding parameter (U) and the location of points at which the velocity gradients vanish (8), are derived. Semi-analytical solutions for pressure p and the mean temperature T,,, are obtained for the isothermal
Table 1 Load and traction
13:~o
n/mo
13=o
0
d/
1.15/0.56
1.20 1.15 1.10 1.05 1.00
0.6840 0.6726 0.6615 0.6503 0.6390
0.3815 0.4070 0.4344 0.4616 0.4889
0.7182 0.6767 0.6362 0.5951 0.5534
1.0537 1.0187 0.9846 0.9513 0.9192
0.6539 0.6789 0.7022 0.7240 0.7444
1.1337 1.0582 0.9840 0.9110 0.8392
1.0/0.75
1.20 1.15 1.10 1.05 1.00
0.1421 0.1394 0.1367 0.1337 0.1312
0.1007 0.1039 0.1072 0.1097 0.1137
0.1583 0.1506 0.1430 0.1343 0.1270
0.1555 0.1520 0.1484 0.1449 0.1415
0.1150 0.1175 0.1200 0.1227 0.1255
0.1761 0.1672 0.1580 0.1489 0.1398
1.20 1.15 1.10 1.05
0.0383 0.0379 0.0376 0.0372 0.0368
0.0327 0.0333 0.0341 0.0349 0.0356
0.0459 0.0444 0.0429 0.0412 0.0395
0.0393 0.0389 0.0385 0.0381 0.0377
0.0340 0.0348 0.0353 0.0362 0.0370
0.0475 0.0461 0.0443 0.0426 0.0410
0.545/86
1.00
0.4/126
1.20 1.15 1.10 1.05
1.00 TRIBOLOGY INTERNATIONAL
0.008127 0.008084 0.008032 0.007975 0.007910
0.007412 0.007540 0.007678 0.007827 0.007998
0.009987 0.009735 0.009441 0.009132 0.008790
0.008188 0.008146 0.008091 0.008031 0.007952
0.007530 0.007663 0.007799 0.007945 0.008068
0.010110 0.009872 0.009570 0.009257 0.008765 245
D. Prasad et aI--Thermal effects in lubrication of asymmetrical rollers
boundaries. The following may be concluded from the results obtained here. (1)
(2)
There is a significant increase in pressure (hence the load and traction) with the flow rate index n for a fixed value of (J. A similar trend follows when /J varies (from 1 to 1.2) and n is held fixed. This is so for the traction at y - 0. However, at y = h, the traction force shows similar trend with n but an opposite trend with respect to tj. There is also a signficant change in the mean temperature with respect to n and tj.
It is worthwhile noting here that the temperature variation coupled with the pressure variation as shown schematically in Fig 2 exert significant influence on the power law consistency index (rh); hence it is not justifiable to treat it as a constant.
References 1. Griskey R.G. and Green R.G. Flow of dilatant (Shear thickening) fluids. AICHEJ, 1971, 17, 725 2. Buckholz R.H. and Hwang B. The accuracy of short bearing theory for non-Newtonian lubricants. J. Tribology, 1986, 108, 73-79 3. Elkouh A.F., Nigro N.J. and Glowacz A. A generalized squeezing flow of a power law fluid between two plane annuli. J. Tribology, 1986, 108, 80 4. Salem E.A., Khalil M.F. and Bedewi M.A. Thermal and inertia effects in externally pressurized spherical bearing lubricated with non-Newtonian fluids. Wear, 1983, 91, 15 5. Kawase Y. and Moo-Young M. Approximate solutions for power law fluid flow past particles at low Reynolds' numbers. J. NonNewtonian Fluid Mech., 1986, 21, 187 6. Sinha P., Shukla J.B., Prasad K.R. and Singh C. Non-Newtonian power law fluid lubrication of lightly loaded cylinders with normal and rolling motion. Wear, 1983, 89, 313-322
7. Prasad D., Singh P. and Sinha P. Thermal and squeezing effects in non-Newtonian fluid film lubrication of rollers. Wear, 1987, 119, 175 8. Wang S.H., Hua D.Y. and Zhang H.H. A full numerical EHD solution for line contacts under pure rolling condition with a non-Newtonian theological model. J. Tribology, 1988, 110, 583-586
246
9. Sinha P. and Singh C. Lubrication of cylinder on a plane with a non-Newtonian fluid considering cavitation. J. Lubr. Technol., 1982, 104, 168
10. Sinha P. and Raj A. Exponential viscosity variation in the nonNewtonian lubrication of rollers considering cavitation. Wear, 1983, 87, 29-38 i1. Safar Z.S. Journal bearing operating with non-Ncwtonian lubricating films. Wear, 1979. 53, 95 12. Safar Z.S. Dynamical loaded bearings operating with non Newtonian lubricating films. Wear, 1979. 55, 295 13. Prasad D., Sinha P. and Singh P. Non-uniform temperature m non-Newtonian fluid film lubrication of rollers..1. Triholo,t,,y, 1988, I10. 65,? 14, Dyson A. and Wilson A.R. Film thickness in EHD lubrication by siliconc fluids., Proc. lnstn. Mech. Engrs. Lubrication amt Wear fourth convention, 1965-66, 180, PI-3K, 97 15. Savage M.D. Variable speed coating with purely viscous nonNcwtonian fluids. J. AppI. Moth. Phys. (ZAMP), 1983, 34, 358-369 16. Dein I.K. and Elrod H.G. A generalized steady Reynolds' equation for non-Ncwtonian fluids with application to journal bearings. J. Lubr. Technol., 1983, 105. 38,5 17. Crook A.W. The lubrication of rollers. Ill. A theoretical discussion of friction and temperature in the oil lihn. Phil. Trans. R. Soc. London, 1961, 254, 237 18. Cheng H.S. and Sternlieht B. A numerical solution for pressure. temperature and film thickness between two infinitely long lubricated rolling and sliding cylinders, under heavy loads. J Basic Engng., 1965, 87, 69,5 19. Kannel J.W. and Walowit J.A. A simplified analysis for traction between rolling/sliding E H D contacts. J. Lubr. Tech., 1971, 93, 39 20. Ghosh M.K. and Hamroek B.J. Thermal clastofiydrodynamic lubrication of line contacts. ASLE, 1985. 28, 159-171 21. Dowson D. and Higginson G.R. The fundamentals of roller-gear lubrication EHD. Pergamon Press, 1966, 3 22. Shew S. and Wilson W.R.D. Viscoplastic lubrication of asperities. J. Lubr. Technol., 1982. 104, 568 23. Safar Z.S. and Shawki G. Performance of thrust bearings operating with non-Newtonian lubricating tilms. Tribology Int. 1979, 12, 31 24. Sadeghi F., Dow T.A. and Johnson R.R. Thermal effects in rolling sliding contacts--Part I:Expcrimental measurement of surface pressure and temperature. Trans. ASME, 86-Trib-37 25. Shukla J. B. and lsa M. Thermal effects in squeeze films and externally pressurized bearing with power law lubricants. Wear. 1978, 51, 237
August 91 Vol 24 No 4
Keywords: lubrication, thermal effect, non-Newtonian, roller bearing
Introduction Hydrodynamically lubricated cylindrical roller bearings are widely employed in heavily loaded systems owing to their simplicity and high load carrying capacity. In such applications, severe pressure and temperature conditions are encountered which exert an appreciable influence on the physical properties (especially viscosity) of lubricants. In recent years, there has been an increasing trend towards the use of synthetically constituted lubricants including modified oils, aqueous suspensions of fine particles, etc., which invariably exhibit non-Newtonian behaviour in the range of stress and strain rates encountered in roller bearings. Within the general framework of non-Newtonian fluids, the power law model has received considerable attention in recent years ~-5 and its use has also yielded promising results in a variety of engineering applications including lubrication. Owing to the non-linearity of stress and strain rates, most previous studies of lubrication have involved varying degrees of approximations. For instance, in the case of lubrication of rolling bearings, pure rolling condition (symmetry) has been tacitly employed: that is, both surfaces are assumed to be of equal size and to rotate with the same velocity¢' s. Sinha and Singh '~ used the power law fluid model to examine the lubrication characteristics of a cylindrical plane bearing and they assumed the flow about midplane to be symmetrical. Moreover, these investigators treated the fluid properties to be constant in the flow region of interest. In a subsequent publication, Sinha and Raj ~' presented another analysis incorporating the exponential viscosity variation but still retaining the symmetry of flow about the mid-plane. Numerous other analyses making similar approximations have also been reported in the literature ~~.J2. More recently,
Prasad et a113 have examined the effect of temperature and pressure dependent power law consistency index, m, of the lubricant flowing between cylindrical rollers, although the assumption of the symmetry of flow was invoked. In fact, there have not been many studies in which the symmetry of flow has not been used. For instance, the pioneering work of Dyson and Wilson ~4 is also based on this assumption, though subsequently this restriction has been partly relaxed. Admittedly, their analysis takes into account the EHD effects but ignores the thermal effects. Savage ~5 has employed a perturbation method to seek analytical/numerical results for this flow configuration for isoviscous flow of power law lubricants. Dein and Elrod ~6 presented the Reynold's equation for power law lubricants with special reference to the case of a journal bearing and assuming that the strain rates in the fluid arise primarily from the relative velocity between the two surfaces. They too sought an approximate solution using the perturbation method for isoviscous fluids. In this study, a new analysis is presented which takes into account the rolling/sliding (antisymmetry) condition of lubricated rigid cylinders by a power law model fluid. The effect of hydrodynamic pressure and temperature on the power law consistency index (ie viscosity) has been incorporated in the treatment w-2°. The coupled Reynolds' and energy equations have been solved simultaneously to obtain extensive results on temperature fields prevailing in the flow domain.
Mathematical analysis The governing equations, with the usual assumptions of hydrodynamic lubrication, are ~
de
0 [ IOu["-lOu)
ck = oylmloy[ *Department of Mathematics, Indian Institute of Technology, Kanpur208016, India. **Department of Chemical Engineering, Indian Institute of Technology, Kanpur-208016, India. ~Author for correspondence.
TRIBOLOGY INTERNATIONAL
oy.
Ou Ov + = 0 Ox Oy
(1) (2)
where m follows the relation 7
0301-679X/91/040239-8 © 1991 Butterworth-Heinemann Ltd
239
D. Prasad et a I - - T h e r m a l effects in lubrication of asymmetrical rollers
Notation An cn
d,, h h, h~, h2 H K m mo
th n
P
-)
R T F
T~ Tu T,, Tm
Ln b/, V
n2[(2n+ 1 ) ( 3 n + 1)] [( 4n+ 2)/n]" (U/h,)"(2R/h,,) ''2 n/[( 3n+ l )2"+ ~l Lubricant film thickness Minimum film thickness Film thickness at x = -Xl and x = x: respectively 1 + X 2 (= h/ho) Lubricant thermal conductivity Lubricant consistency Consistency at ambient pressure and temperature Consistency (= 2mc,,a) etc Flow index Hydrodynamic pressure Pressure (= ap) Equivalent radius Lubricant temperature Temperature (= [3T) etc Traction force Surface temperature (constant) Ambient temperature Surface temperature (at y - 0) Mean film temperature Lubricant velocities in x, y directions
m = mo e x p [ a p - ~ ( T m - T o ) ]
t? W
UE/U2 Velocities of plane and cylindrical surface respectively Normal load
x, y x~ x2 X a, 13
Coordinate axes x~ is the point of maximum pressure (x~ > 0) Cavitation point (x2 > 0) x/(2Rho) I/2 etc Pressure and temperature coefficients respectively
U I , U2
Load (= W/[(2Rho)'"2/cx)]
U2ho6( h,,/2R ) '/2 / ( KeO 8
Loci of points at which the lubricant velocity gradients vanish 8/ho ~2 Value of ~ at X = X_~ 8" Values of ~ in the neighbourhood of x = -+ x~ * 8" / hu Subscripts 1,2 usually refer to the respective quantities in the inlet and outlet regions separated by the point X =
/J etc
XI
Bar denotes a dimensionless quantity
(3)
with U1 ~ U 2
Tm = h
)
Tdy
(4)
F r o m the point of view of mathematical analysis, the contact between two cylinders of radii R . and Rb (as shown in Fig l ( a ) ) can be described by an equivalent cylirider near a plane (shown in Fig l ( b ) ) 21.
x
--~Ol
b
Film thickness h for this g e o m e t r y is expressed as X2
h = h,, + 2R
x
(5)
Fig 1 (a) Schematic representation o f two cylindrical rollers with different dimensions and velocities. (b) Cylinder-plane contact
where 1
1
1
R = R,, +
(6)
The b o u n d a r y conditions for the governing equations (1) and (2) are as follows: u=Ui,v=0
aty=0 dh
// = U 2 , v = U 2 ~
aty = h
p = 0 atx -~;dp/dx p=0,
dp dx=0
(7)
atx=x2
(8)
= 0 a t x -- -x~ (9)
In order to solve the system of equations (1) and (2) analytically, one has to take care of the m o d u l u s signs of the velocity gradient, Ou/Oy. For pure rolling, four different regions exist, but due to s y m m e t r y only two are required 9. H o w e v e r , the a n t i s y m m e t r y necessitates consideration of all the four regions. This is easily visualized for the N e w t o n i a n case (n = 1) as follows. 240
Equations (1) and (2) can be solved for n = 1 and m to be constant or a function of x alone; one can obtain
u = Ui + ( U 2 - U 1 ) h -
2m
dp = 6 ( U , + U a ) m ( h _ h , ) / h 3 dx
( l(1) (ll)
It can be seen that u is a linear function of y at points where d p / d x = 0. At these points, the velocity gradient Ou/Oy = ( U 2 - U ~ ) / h l which can never be equal to zero, since U2 4= U~. It can also be seen from Fig 2 that for each x, Ou/Oy can vanish at one point of y ( = 5 , 0 <~ 5 ~< h) in the regions -~c < x < -x~ and -xL < x < x2. Thus these regions can be divided into four sub-regions separated by the 8-profile, having velocities u~, u2, u3 and u4 as shown in Fig 2. For any e > 0, let us and u~, be the velocities in the regions -x~ - ~ ~< x ~< -x~ + ee and x~ - e3 ~< x ~ x2 respectively, August 91 Vol 24 No 4
D. Prasad et aI--Thermal effects in lubrication of asymmetrical rollers
then according to the velocity profile, shown in Fig 2, one may observe the following:
or
Q= 0u~ >0, a
)
u2 dy +
u l dy
or
Ou~, ~<0, 0 ~ y < a Oy
I: - ~ < x < ~ - x l . e t
(12)
2n+l
\m, d x / (18)
Since the flux Q is constant through every cross section, Q may be equated to the flux through the point x = - x l [Ref 22]. This point is selected in order to enforce the continuity of flow.
0u3>0, 0~
Q= U,a+ U2(h-a)-
[a <2''+')/'' + ( h - a ) <2"+')/"]
Ou~=Ou~= 0 a t y = a Oy Oy
I1: -Xl"~E2~x~x2-~_ 3
(13a)
0/J3 ~_.0/~4 ~ 0 a t y = a
Oy
I
Thus flux Q through the point x = - x ~ (where dp/dx = 0) can be evaluated directly from Eq (1). yielding
Oy
( U1 + U2)h
O(-xl) - ~
OUs
lll: --Xl--~l<~X<~--Xt+e2 (13b)
00u;<0, O<~y<~h
IV: x2- %<~x<~x2
(13c)
It is expected that the nature of the velocity profiles for different values of n as well as variable m, will remain the same as it is true for pure rolling'~,~3. Using the sign for the velocity gradients given in (12) and integrating equation (1) twice for the region ~ ~< y ~< h, one can obtain the velocity profile u~ as
t
(19)
Finally equating the fluxes given by (18) and (19) and simplifying, one can obtain the following Reynolds' equation:
((Ut-U2)a
+ U2h-½(UL++Uz)h
#2,,7i;.,+ (h_S)<:,~,).,
l)"
-~
(20)
Similarly for the other region, the Reynolds' equation may be written as
[(h-a)<,,, ,)/,,_(y-a)<,,,,v,,] Similarly for the other regions, expressions can be obtained:
the
(14) following
½(U, + U2)hA - U2h - ( U I - U 2 ) a ~ " a<
)
+
-xt<~x<~x2
(21)
U2 = UI - (F/~ l) ( m l / d 2 ) l / n
Use of the matching conditions [a{,,~
,):,,
_ (a-y){,, ~ ,)/,,]
u3= U , + n + ] m2clx] [a{,,+')/- - ( a _ y ) { - + , ) / . ]
(15)
(16)
Ul = /'/2
I aty = a
U3 ~ ~/4
I
into (14-17) yields
(.)(_ 1.2t,.,
U4 = U2 -[- r/-it- 1
m2 dx /
[(h-a){,,, ,)/,, _ (y-a)¢,,{ ')/,,]
(17)
The velocities in the regions III and IV are calculated numerically by extending the velocities u2 and u4 respectively, where a is replaced by a*. (For more details see the Results section, under the title numerical solution.) Now the volume flux Q for the region I is obtained as
Q=
f
h u dy
TRIBOLOGY INTERNATIONAL
[(h-S)
{''+'):'' - a { " * '):"] = 0
-oo
n+l
(23) m2
<'+')''' - a ~'+')''1
% - x l <~x<~x2-%
= 0
(24)
Eliminating dpl/dx and dpz/dx from Eqs (23) and (24), using Reynolds' equations (20) and (21), one can obtain a single relation as follows: 241
D. Prasad et aI--Thermal effects in lubrication of asymmetrical rollers implies
F
cl = c~ - c (say)
i
d l - d~ - d (say) The constants c and d are evaluated using the boundary conditions (27) and are given below: c=
h
+\
h K / m l dx
[(h_a)13,, { .1.',, _ 8¢~,,, ,,,,] d ]
II1
11
[(h-a)'"'
''"-
where A,, -
Finally the mean temperature Eq (4), may be written as
a'"~ ' " " l [ ( U , - V 2 ) ~
+ U,h - ~(U,+U,_)h,] (25)
= II
OF
,if,
T,,,] = h
be 7
. Oy
~Oy/
y
of Eq gives
ml
=0 (27) (26)
twice
T"=-(2n+l)(3n+l)(mK
,
'
for
the
)(m,
region,
nS~
-(4n+l)h[(h-8)'4" (28)
t ),',, + c2y + d2
(32)
L,,._= T,,+T.~2+ &v[ ( h - a ) ~ 3 " .
l"
n-' ( K dP') (''' ')"" T12 = - ( 2 n + l ) ( 3 n + l ) dx (8_y)(3,,,
dx
Proceeding in the same way, the mean temperature for the region -x~ ~< x ~< x~ can also be written as
I
( y - a ) c~''~ ])"" + c,y + d,
(3])
,,, (m,A,,)(, d,,l)"" ....
at y = h
Integration -~
]
where
The boundary conditions for this equation are
Th
Tll dy
nS, [ ( h _ a ) ~ , , ~,~,,, + a~4,,,.~.,,, ] ( 4 n + 1)h
(26)
w.here the heat convection has been neglected.
T
;
7"/, + To] SI + 2 l ( h - a ) ~ ' " ' " " " + S'-'"'"""l 2
The energy equation for this problem is considered to
at
~ Ti2 dy +
or
Energy equation
= T.t
T,,,~, as defined in
T'"l : h 1 if'~ T d y
It must be noted here that Eq (25) cannot be used to evaluate 8 at x = -x~ and x = x2 since au/Oy 4= 0 at these points.
T
( 2 n + l ) (3n+ l)
Thus T]~ and T~2 are explicitly known functions of x and y.
[~3~2,,' '),',, + (h_.,3)~2,, , 1,,',,]
ay 2 = -
] m~ elf
n2
(2n+l \ n+l )
U 2 - Uj
K
IV
Fig 2 Qualitative velocity profiles shown by arrows; 6distribution indicated by A B C and EFG, and 6*distributions shown by broken lines _
T~,l+\
(30)
,,.,, + a~,,,,, ,.,,]
' ) " " + S '4'' ')""]
where
= m~A,,(~ _ 1 d~c2)("~ )j'' S~ (29)
where T~ and TIe are the lubricant temperatures in the regions, g ~< y ~< h and 0 ~< y ~< g respectively, c~, d] and c2, d2 are the integration constants.
(33)
K
(34)
o12
Using the non-dimensional scheme X = x/\,;2Rh, etc., p = o~p, rh = 2rnc,,o~,
Use of the temperature matching condition T~ = T]2
at
y=
H = 1 + X2, etc.
and the matching heat flux KOT~ 33, 242
=
KOTI2 3y
at
y =
7",,, -
6T.,.
g =
a/h,,. 0
U./U~
(W,
>U_~)
(351
Eqs (20), (21), (25), (31), (33) may be rewritten as follows: August 91 Vol 24 No 4
D. Prasad et aI--Thermal effects in lubrication of asymmetrical rollers
(36)
dX - rh'0Zx)"
d&_
).
(37)
dX
1-
V¢ = -
TFO = --
[ ( H - ~ ) t'+')/'' - ~t"+')/"] [ ( 0 - 1 ) ~ + H - ½(1+/--?)Ht] 8(2"+1)/'' + ( H - ~ ) ( 2''+')/''
= 0
/.
(38) :
-
/~h+ L'l 2
L,+2 L,,
+th'd"~l@x)'+'g'x + rhe d, ~(
(39)
_L.).,+, gx
(40)
(51)
%,=o dx
(52)
dx
(53)
and Vv,, = -
Dimensionless tractions are i'Fo (= ~ T v o / h o ) = -
"%=0 d X
(54)
Similarly,
where rh~ = rho exp(p~-/~m,+/],), me = rho exp(p2-/'m~+/'~)
(41)
(g2ho~)(ho) 1/2 "9 = \ Kc~- J \2R]
(42)
d,, = n / [ ( 3 n + l )2 "+~]
(43)
(/.)- 1)~ + H - ½(/]+ 1)H, fx =
X dx dX
The surface traction force T F is obtained from the integration of shear stress r over the entire length, that is,
L? ( 2 n + l 1 -\ n+l ]
/-,,,
p dX = -
(44)
~,2,,+,)/,; + (H--6~(~.,+,)/~
gx = 1 [ ( H - 6 ) ° " + ')/" + ~<3,+,)/,,1 n
[(H_~)(4,+,)/,, + ~4,,+,)/,]
(4n+a)U
(45)
Equations (36), (39), and (37), (40) have to be solved simultaneously using the following conditions: Pl = 0
at
X---~-~
d0,_dp2_0 dX dX p2=0-
0,o2 dX
at
X=-X1
at
X=Xz
(46a) (46b)
(47)
From Eq (37), it may be noted that the last condition implies
at X = X2, H = H2
(48)
which is independent of n. Now for n = 1, H~ = H 2 or X2 = Xt when d p 2 / d , x = 0, see Eq (11). Hence from Eq (48), it follows that 82 = H1/2 (see Fig 2).
TFh = --
f.2
"%= . dX
(55)
Results and discussion Numerical solutions of the Reynolds' equations (36, 37) and the energy equations (39, 40) have been obtained for antisymmetrical, viscous incompressible flow of a power law fluid through the gap between a cylinder and a plane (see Fig (1)). The results of this investigation are assessed in terms of parameters n and U (= U~/U2, U~ > U2). The sliding parameter (J is chosen to lie between 1.0 and 1.2 (ie maximum of 20% slip). The parameter (J arises due to the consideration of antisymmetry conditions and is important because the presence of sliding ( / d > l ) is likely to produce greater pressure and temperature as compared to that for pure rolling ((J = 1). The significance of (.1 along with n has been demonstrated through table and graphs. For the numerical calculation, the following representative values have been used: U, = 500cm s -1, ho = 5 x 10 4cm, et = 1.6 X 10 -9 dyne c m 2, R = 3 cm, ~c~ = 1.5, Tot = 1, ~/ = 5 where + = 0 corresponds to the isothermal solutions. It is of interest to note that the flow configuration considered herein includes several known situations as limiting cases: for instance, when U~ = Uz and m is constant, it reduces to the case examined by Sinha and Singhg; for the case of [3 = 0, ie temperatureindependent viscosity, one obtains the case considered by Sinha and Raj l°. Finally, when U~ = UE, the present analysis is equivalent to that of Prasad et al 7.
Numerical solutions Load and traction The load component W in the y direction is calculated as
The dimensionless load 1 ~ / ( = W e d ~ o ) TRIBOLOGY INTERNATIONAL
is given by
The Reynolds' and energy equations are coupled through rh and contain two unknowns ~ (the locus of points at which 3u/Oy = 0) and XI (X = - X t is the point of maximum pressure). These unknowns are also present in Eq (38). As there is no symmetry ((.t v~ 1) for ~-distribution, it is necessary to solve Eq (38) for over the complete region under consideration, along with the Reynolds' and energy equations. The actual
243
D. Prasad et aI--Thermal effects in lubrication of asymmetrical rollers
process followed for numerical computation is briefly described below. The algebraic equation (38) in 8 contains X and X~ explicitly. First of all, an initial value of X is assigned ie the point at minus infinity is replaced by a large but a finite negative value. An arbitrary value of Xj is chosen. The value of 8 at this inlet is obtained by solving the algebraic equation (38) for 8 using the bisection method with a reasonable tolerance (say 10 4). These values of X and 8 are substituted in the energy equation (39). 7",,,~at the inlet is then obtained from solving the algebraic energy equation by prescribing rh~ and pj (= 0 at X - - + - ~ ) and using the earlier numerical technique. The same X, rho and the computed 8 and T,,,, are used in the differential equation (36). Fourth order Runge-Kutta method is used to evaluate/)~ at X = X + AX(AX > 0). For this value of X ( = X + AX), Eq (38) is solved for 8. These new X, 8 and/51 are substituted in the energy equation which again yields T,,~ as a solution. These Tmj, X and 8 are used to calculate/)~ as a solution of Eq (36) at another point X = X + 2AX. This process is repeated for the other values of X until such time that Eq (38) y_ields values of 8 satisfying 0 ~< 8 <~ H (because ~ cannot exceed H). It may be noted that at X = -X~, 8 does not exist (see previous section). Further, in the neig_hbourhood of X - - X ~ , the determined values of 8 do not lie in the interval 0 ~< ~< H. Hence in the neighbourhood of X - - X , , 8 (= say 8*) has to be determined solely on the basis of physical considerations. It should be emphasised here that 8" does not refer to the locus of points at which the velocity gradient vanishes. The e-neighbourhood: ( - X I - cj ~< X ~ - X I + %) of X = - X I is to be determined as the region where there exists no 8 lying in the interval 0 ~< 8 ~< H and satisfying Eq (38). To ease the mathematical complexity, a linear profile for 8", is given below
is calculated using Eq (57) together with/)2 and T,,eIf the computed value of/)2 at X = X2 (or Xl) satisfies the condition (47) (ie/)2 = 0 at X = X2), the assumed arbitrary value of X, was correct. Otherwise, another value of X~ is assigned and the whole process is repeated so long as/)2 vanishes at X X,. Thus X~ and 6 are computed along with pressure /)~, /)2 and the mean temperatures 7",,,~ and T,,2. The complete mean temperature profiles as functions of n and U are elaborated upon in the following sections.
Temperature profile The mean temperature T,,, has been shown in Figs (3) and (4) as a function of n and fJ. It may be observed that the qualitative behaviour of T,,, versus X is very similar to the temperature profile obtained in Refs 7, 20, 24. For a fixed value of n, T,, increases with [J which indicates that the sliding temperature is higher than that of pure rolling 2°-24. For fixed U, T,, increases with n showing that the temperature for dilatant fluid is higher than that for Newtonian and pseudoplastic fluids ~3.
Load and traction The calculated values of load carrying capacity I~ and the traction force T~. are given in Table 1. It can be seen from the table that W increases with n and (J. This confirms the enhancement of load for dilatant fluid which is consistent with the previous finding 1-~-:3-25. The increase seen in fV with (J is caused simply by the increase in hydrodynamic pressure with sliding 2°,24. The traction force Tv- has been evaluated for both the surfaces at y = 0 and y = h, and they have different
U=1.2
8*=-HI
2e,
(X+XI-~)
-X~-e1<~X<~-X,
1.9
(56a) n=1.15 1.8
= ~
-
.
-Xl<~X<~-Xl+ez
(56b) 1.7
is assumed. This profile was chosen in such a manner that it not only satisfies the Reynolds' equation at X = -X1 but also makes the pressure curve continuous. Having determined 8 = ~* using Eq (56) in the neighbourhood of the point X - -X~, the procedure obtained earlier is adopted to evaluate/51 and T,,,, for the region -X~ - e l ~< X ~< - X I , and P2 and Tin2 for the region -X1 ~< X ~< -X1 + %. Since in the interval - X , + % ~< X <~ X2 - ~3, ~ (roots of E q ( 3 8 ) ) satisfies the inequality 0 ~< 8 ~< H, so 8 in the region -X~ + e2 <~ X ~< X2 - e3 is determined by the same procedure followed near the inlet along with /)2 and 7",,2. Subsequently in the interval X2 - ~ <~ X <~ X2 (where Eq (38) is not valid) the same type of linear profile, namely 8*(X) = H~ (X-X2+~3) 2%
(57)
is assumed (the complete 8 - 8" profile has been shown in Fig 2). ~* in the region X2 - e3 ~< X ~< )(2 244
1.6
1.5
. . . . .
1.0 1.4
0 0,4 1.2
1.1
-5
I
I
I
I
-4
-3
-2
-1
X
Fig 3 Mean film temperature T,,1 versus X f o r various values o f n August 91 Vol 24 No 4
D. Prasad et aI--Thermal effects in lubrication of asymmetrical rollers
1=_ .20 ~ /~ I .15 ---I .I0 -- -I. 05 --X--
I .9
,.oo--e-
~.~
_ ~_._ .__~___~ _ _ _ _ . : ~ . . ~ .
traction forces approach towards the same value. This is plausible, since once U~ ~ U2, both the surfaces will experience the same force. The load and the traction forces both have been calculated for [3 = 0 also and are also given in Table 1. It can be noted from the table that W and TF are both higher than those based on the temperaturedependent viscosity 7-j3 and these observations are in line with the findings (traction coefficients) of Ghosh and Hamrock 2°.
1.5
Cavitation point and its E-neighbourhood
~~~X~~
/
I.q
xh
"'1'I
n=ll
-5
I
I
-q
-3
I -2
-I
0
X
Fig 4 7f,,,-distribution versus X for various values of (J characteristics. The dependence of Tvo on n and /.J is qualitatively similar to those obtained for loads, that is, ~rvo increases with n and U (Ref_13). However, T~, increases with n but decreases as U increases. This is because an increase in [E(=U~/U2) is tantamount to the fact that only U] is increasing as U2 is fixed. However, it was observed that as U ~ 1, both the
Cavitation points X2 for all n and U have been calculated but have not been given here. Their trend with n is quite similar to those reported earlier" for (J = 1; however, for a fixed value of n, cavitation point shifts towards the centre line of contact, but not significantly. Similarly, the position of pressure peak ( X = - X l ) shifts towards the centre line of contact 2°. The value of • is found to be lie the interval 0 < • ~< 0.01 which is true for both at X = -X~ and X = )(2.
Conclusions With the usual assumption of hydrodynamic lubrication, an attempt has been made to study thermal lubrication of rolling/sliding contacts by an incompressible power law fluid. The thermal Reynolds' and energy equations which are functions of consistency (&), sliding parameter (U) and the location of points at which the velocity gradients vanish (8), are derived. Semi-analytical solutions for pressure p and the mean temperature T,,, are obtained for the isothermal
Table 1 Load and traction
13:~o
n/mo
13=o
0
d/
1.15/0.56
1.20 1.15 1.10 1.05 1.00
0.6840 0.6726 0.6615 0.6503 0.6390
0.3815 0.4070 0.4344 0.4616 0.4889
0.7182 0.6767 0.6362 0.5951 0.5534
1.0537 1.0187 0.9846 0.9513 0.9192
0.6539 0.6789 0.7022 0.7240 0.7444
1.1337 1.0582 0.9840 0.9110 0.8392
1.0/0.75
1.20 1.15 1.10 1.05 1.00
0.1421 0.1394 0.1367 0.1337 0.1312
0.1007 0.1039 0.1072 0.1097 0.1137
0.1583 0.1506 0.1430 0.1343 0.1270
0.1555 0.1520 0.1484 0.1449 0.1415
0.1150 0.1175 0.1200 0.1227 0.1255
0.1761 0.1672 0.1580 0.1489 0.1398
1.20 1.15 1.10 1.05
0.0383 0.0379 0.0376 0.0372 0.0368
0.0327 0.0333 0.0341 0.0349 0.0356
0.0459 0.0444 0.0429 0.0412 0.0395
0.0393 0.0389 0.0385 0.0381 0.0377
0.0340 0.0348 0.0353 0.0362 0.0370
0.0475 0.0461 0.0443 0.0426 0.0410
0.545/86
1.00
0.4/126
1.20 1.15 1.10 1.05
1.00 TRIBOLOGY INTERNATIONAL
0.008127 0.008084 0.008032 0.007975 0.007910
0.007412 0.007540 0.007678 0.007827 0.007998
0.009987 0.009735 0.009441 0.009132 0.008790
0.008188 0.008146 0.008091 0.008031 0.007952
0.007530 0.007663 0.007799 0.007945 0.008068
0.010110 0.009872 0.009570 0.009257 0.008765 245
D. Prasad et aI--Thermal effects in lubrication of asymmetrical rollers
boundaries. The following may be concluded from the results obtained here. (1)
(2)
There is a significant increase in pressure (hence the load and traction) with the flow rate index n for a fixed value of (J. A similar trend follows when /J varies (from 1 to 1.2) and n is held fixed. This is so for the traction at y - 0. However, at y = h, the traction force shows similar trend with n but an opposite trend with respect to tj. There is also a signficant change in the mean temperature with respect to n and tj.
It is worthwhile noting here that the temperature variation coupled with the pressure variation as shown schematically in Fig 2 exert significant influence on the power law consistency index (rh); hence it is not justifiable to treat it as a constant.
References 1. Griskey R.G. and Green R.G. Flow of dilatant (Shear thickening) fluids. AICHEJ, 1971, 17, 725 2. Buckholz R.H. and Hwang B. The accuracy of short bearing theory for non-Newtonian lubricants. J. Tribology, 1986, 108, 73-79 3. Elkouh A.F., Nigro N.J. and Glowacz A. A generalized squeezing flow of a power law fluid between two plane annuli. J. Tribology, 1986, 108, 80 4. Salem E.A., Khalil M.F. and Bedewi M.A. Thermal and inertia effects in externally pressurized spherical bearing lubricated with non-Newtonian fluids. Wear, 1983, 91, 15 5. Kawase Y. and Moo-Young M. Approximate solutions for power law fluid flow past particles at low Reynolds' numbers. J. NonNewtonian Fluid Mech., 1986, 21, 187 6. Sinha P., Shukla J.B., Prasad K.R. and Singh C. Non-Newtonian power law fluid lubrication of lightly loaded cylinders with normal and rolling motion. Wear, 1983, 89, 313-322
7. Prasad D., Singh P. and Sinha P. Thermal and squeezing effects in non-Newtonian fluid film lubrication of rollers. Wear, 1987, 119, 175 8. Wang S.H., Hua D.Y. and Zhang H.H. A full numerical EHD solution for line contacts under pure rolling condition with a non-Newtonian theological model. J. Tribology, 1988, 110, 583-586
246
9. Sinha P. and Singh C. Lubrication of cylinder on a plane with a non-Newtonian fluid considering cavitation. J. Lubr. Technol., 1982, 104, 168
10. Sinha P. and Raj A. Exponential viscosity variation in the nonNewtonian lubrication of rollers considering cavitation. Wear, 1983, 87, 29-38 i1. Safar Z.S. Journal bearing operating with non-Ncwtonian lubricating films. Wear, 1979. 53, 95 12. Safar Z.S. Dynamical loaded bearings operating with non Newtonian lubricating films. Wear, 1979. 55, 295 13. Prasad D., Sinha P. and Singh P. Non-uniform temperature m non-Newtonian fluid film lubrication of rollers..1. Triholo,t,,y, 1988, I10. 65,? 14, Dyson A. and Wilson A.R. Film thickness in EHD lubrication by siliconc fluids., Proc. lnstn. Mech. Engrs. Lubrication amt Wear fourth convention, 1965-66, 180, PI-3K, 97 15. Savage M.D. Variable speed coating with purely viscous nonNcwtonian fluids. J. AppI. Moth. Phys. (ZAMP), 1983, 34, 358-369 16. Dein I.K. and Elrod H.G. A generalized steady Reynolds' equation for non-Ncwtonian fluids with application to journal bearings. J. Lubr. Technol., 1983, 105. 38,5 17. Crook A.W. The lubrication of rollers. Ill. A theoretical discussion of friction and temperature in the oil lihn. Phil. Trans. R. Soc. London, 1961, 254, 237 18. Cheng H.S. and Sternlieht B. A numerical solution for pressure. temperature and film thickness between two infinitely long lubricated rolling and sliding cylinders, under heavy loads. J Basic Engng., 1965, 87, 69,5 19. Kannel J.W. and Walowit J.A. A simplified analysis for traction between rolling/sliding E H D contacts. J. Lubr. Tech., 1971, 93, 39 20. Ghosh M.K. and Hamroek B.J. Thermal clastofiydrodynamic lubrication of line contacts. ASLE, 1985. 28, 159-171 21. Dowson D. and Higginson G.R. The fundamentals of roller-gear lubrication EHD. Pergamon Press, 1966, 3 22. Shew S. and Wilson W.R.D. Viscoplastic lubrication of asperities. J. Lubr. Technol., 1982. 104, 568 23. Safar Z.S. and Shawki G. Performance of thrust bearings operating with non-Newtonian lubricating tilms. Tribology Int. 1979, 12, 31 24. Sadeghi F., Dow T.A. and Johnson R.R. Thermal effects in rolling sliding contacts--Part I:Expcrimental measurement of surface pressure and temperature. Trans. ASME, 86-Trib-37 25. Shukla J. B. and lsa M. Thermal effects in squeeze films and externally pressurized bearing with power law lubricants. Wear. 1978, 51, 237
August 91 Vol 24 No 4