An analysis for estimating the variation of the temperature profile through a bearing clearance

Wear,27(1974)189-194 $; Elsevier Sequoia S.A., Lausanne

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in The Netherlands

AN ANALYSIS FOR ESTIMATING THE VARIATION OF THE TEMPERATURE PROFILE THROUGH A BEARING CLEARANCE

D. DEWAR Sperry Gyroscope

Dioision, Sperry Rand Ltd., Downshire

(Received

21, 1973)

August

Way, Bracknell,

Berks (Ct. Britain)

SUMMARY

This analysis estimates the variation in the temperature profile of a fluid as it moves through a bearing clearance using a simple formula; it also enables a physical interpretation of the commonly used Peclet number. The main use envisaged is for planning the numerical analysis of a bearing. A through flow spiral grooved thrust bearing is used as an example.

INTRODUCTION

There are various methods for approximately determining the ratio of conducted to convected heat dissipation of a bearing’. These methods can be adapted to include the Peclet Number (C; T/. p’ h)/k. However there is not readily available a method for simply estimating the temperature profile of the lubricant as it progresses along the bearing film. The need for this arose when a numerical model was being developed for a spiral grooved oil lubricated bearing2, where it was desirable to know at the outset the approximate attenuation the temperature profile experiences from one grid-point to the next. If for example the attenuation of the inlet temperature profile is significant then a great simplification of the analysis is possible. The object of the analysis was to provide a simple formula that without too many drastic assumptions would give a reasonable estimate of the attenuation involved. From this information such parameters as grid spacing could be optimised, also the possible restrictions of the numerical analysis could be assessed from the outset. NOMENCLATURE

Ai

CP C” Cl>

h

J(4) K

c2

Airy integral of a function (c) Specific heat at constant pressure Specific heat at constant volume Constants of integration Film thickness Bessel function (5) Thermal conductivity of lubricant

D. DEWAK

Peclet number Temperature profile as a function of z Velocity profile as a function of z Circumferential velocity Radial velocity Co-ordinate along the bearing film Co-ordinate across the bearing film Circumferential grid spacing Radial grid spacing Lubricant density Lubricant viscosity ANALYSIS

The general outline of the analysis is to examine the thermodynamic energy equation, and by making various assumptions to separate the homogeneous and particular solutions. These are compared to give a simple expression which can be used to approximate the upper bound effect of the homogeneous solution.

dsttibutim

Fig. 1. Bearing

equation

model.

Consider a two dimensional for an oil will be

model,

Fig.

1, then3.4

the general

energy

For this analysis consider a linear velocity distribution, the effect of this assumption is discussed later. Then 1/ = M.z . Hence eqn. (1) becomes

Consider first the particular solution of eqn. (2). This contributes the temperature rise due to the shear heat generation in the film. The solution of this for T, = 0 at z = 0 and z = h is well known, i.e., T,=rj Considering

-$ {hz-z2}. the homogeneous

solution

BEARING

TEMPERATURE

VARIATION

191

ANALYSIS

(4 As shown in Fig. 1 define the inlet temperature profile as some arbitrary function of z &(O, z) = T(z). The solution will be of the form i=m %(x,.2)-

+Ci.~(z)-e-a”x.

1

(5)

i=l

Substituting this into eqn. (4) -panic;

i=cc C

i=co C

.Ci.Z.Pi.e-B”X.~((z)=K.

i=l

+Ci.r(~)*e-fli’X.

i=l

Non dim~nsionalising with respect to bearing clearance then T’L+K~.Z*q = 0 where xi=p.a.jIi. C;h3/K. At this point it is noted that if the velocity profile was a more complex function of gap, say a pressure induced flow, the homogeneous solution becomes insoluble by classical means and the final solution is no longer simple and explicit. Now let Z= -4. Then Ty- xi*q’ q=O. This is a Bessel equation of fractional order and the solution is the Airy integral q= A,($.q). Transforming into 5 co-ordinates ?;:= A,( -&z)

= 4(2.x~)t(J~(~)+J-,(~)}

(6)

and

Using the conditions cited earlier i.e. q=O at Z=O and i= 1 then as g-0 J_+(<)-+co. Hence for T=O at Z= 1 then eqn. (6) simplifies to 0 = +(#{J&“)+}.

then (7)

Equation (7) yields eigen values of pi, hence giving the series solution shown in eqn. (5). Further the higher eigen values will be negligible after only a relatively small number of film thicknesses (x/h). Hence extracting the first solution of x from eqn. (7) using ref. 5 x = 18.9, since J+(r) = 0 at (=2.9. Hence for the solution i.e. eqn. 5 we can state the inequality that

i.e. using the Peclet number then T,(x, z)
18.9/Pe,(x/h)].

Note that we do not know the constants Ci but we can easily estimate an upper bound for the value of the homogeneous solution. Ideally we want to approach zero so that we have local heat balance in the film. DISCUSSION

OF RESULTS

Consider the grid configuration

for a bearing analysis shown in Fig. 2, then

192

D. DEWAR Grid nodes

Inm-

radius

Outer radius

Fig. 2. Grid configuration

for bearing

analysis

depending upon the prevailing conditions any temperature profile that exists at point 1 that is not the normal parabolic profile (i.e. the particular solution) may be carried on to any of the neighbouring points. This would be likely to be the lubricant inlet temperature profile or perhaps the temperature profile of the lubricant from a feed hole etc. The two assumptions that could well affect accuracy of this method firstly are that the velocity profile is assumed to be linear and secondly there is no significant change in viscosity from one point to the next. It has been shown2 that the velocity approximation introduces only some 10-2Oy;, error in the overall picture. Suppose the following example is considered; this is taken from ref. 2 a stiffness stiffness/power optimised spiral grooved through flow thrust bearing with the following parameters. Outer diameter of the bearing Inner diameter Clearance Specific heat of the oil Effective thermal conductivity of the oil Oil density Oil viscosity

153 mm 61 mm 0.125 mm 2.0 I kJ/‘kg‘ C 0.127 J-s/m’ C 885.0 kgjm” 120 c. St (lO_” m2i;s) at 20°C

At a rotational speed of 3,000 r.p.m. the radia1 and circumferential velocities at the periphery2, are V, = 28.8 mjs, V,= 250 mm/s. Thus suppose it is envisaged that 36 circumferential nodes and 10 radial nodes are needed. Then AC= 13.1 mm. 4 = 5.2 mm. Then the Peclet numbers for the two nodes are (i) Circumferential (ii) Radiai

Pe= 5.06 x lo* PC)= 440.0.

Hence suppose points 1, 4, 7 etc. are fed with oil at a constant temperature, what effect does this profile have on points 2, 5, 8 (Fig. 2)? The temperature profile by the time it reaches point 5 from point 4 will be

BEARING

TEMPERATURE

T,
[-

VARIATION

g*

ANALYSIS

193

&&,)I

-c T(z)exp(-1.79) < T(z) x 0.85 . Hence the inlet temperature profile in this case has been attenuated to only SSo/:,of its original value by the time it reaches point 5 and so any thermodynamic consideration of the point 5 must take account of the temperature profile existing at point 4. Now citing an example under a different set of conditions. Consider the clearance being reduced to h=0.04 mm and keeping the speed still at 3000 r.p.m.2, V,= 28 m/s, t: = 230 mm/s, giving Peclet numbers of ( 1) Circumferential (2) Radial

Pe = 15,700 Pe= 129.

Again considering the effect of the inlet temperature point 5.

progressing from point 4 to

< T(z)exp(-19.2) < T(z)O.5 x 1o-8 i.e. completely negligible thus the inlet temperature wili play no effect in this second case. As can be seen this analysis could rapidly give either an estimation as to the appropriate grid spacing for given operating conditions; or given the grid spacing the kind of thermodynamic model that has to be applied to any given point. It seems feasible to apply this approach to any temperature bearing investigation that involves a numerical analysis. As can be seen it is simple and easy to apply and subject to the approximation of the velocity in the bearing profile. CONCLUSIONS

A simple formula for giving a quantitative estimate of the change of temperature profile as the fluid moves through the bearing film is presented. The application envisaged is in the field of numerical analysis of bearings. At the outset of a numerical analysis.it is necessary to know whether the temperature profile existing at one grid point is going to significantly affect its neighbour; from this formula it is simple to assess the most stringent limit of a particular numerical approach. Two examples have. been shown which illustrate two conditions, one where little attenuation of the inlet temperature profile occurs and the other, resulting from a change in bearing clearance shows a complete suppression of the inlet temperature profile from one grid point to its neighbour.

194

D. DEWAR

ACKNOWLEDGEMENTS

This work was carried out at the Mechanical Engineering Department of Southampton University under the joint sponsorship of the Science Research Council and the Sperry Gyroscope Division, Sperry Rand Ltd. The author gratefully acknowledges the help of Professor H. G. Elrod of Columbia University, New York, who inspired this piece of work.

REFERENCES 1 A. Cameron, Principles of Lubrication, Longmans, Green and Co., London, 1966. 2 D. M. Dewar. The Development of a Design Procedure ,for Grease and Oil Lubricuted Groooed Bearings, Mechanical Eng. Dept., Southampton University, June, 1972. 3 H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, London, 1967. 4 H. P. Skelland, Non-Newtonian Flow and Heat Transfer, Wiley, London, 1969. 5 Janke-Ende-Losche, Tables of Higher Function.s, McGraw-Hill. Stuttgart. 1966.

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