On the distribution of friction-induced heat in the dry sliding of metallic solid pairs

Pergamon

Int. Comm. Heat Mass Transfer, Vol. 24, No. 7, pp. 989-998, 1997 Copyright © 1997 Elsevier Science Ltd Printed in the USA. All rights reserved 0735-1933/97 $17.00 + .00

P H S0735-1933(97)00084-5

ON THE DISTRIBUTION OF FRICTION-INDUCED HEAT IN THE DRY SLIDING OF METALLIC SOLID PAIRS

H. A. ABDEL-AAL Department of Mechanical Engineering University of North Carofina at Charlotte Charlotte, N. C., 28223 U.S.A

(Communicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT We present an expression for the partition of friction-induced heat betwee~ two rubbing members. The expression stems from a variable conductivity solution of the heat equation and incoq3orates the inflmmce of the thermal capacity of the sliding pair on heat distribution. The cmrmt formulation yields matching predictions to those of constant conductivity expressions in the case of sliding materials with similar properties, o 1997 ElsevierScienceLtd

The calculation of th© temperature rise in the contact spot between two sliding solids requires the knowledge of the quantity of heat distributed among the rubbing members. The introduction of the concept of a heat partition function (HPF) is due to Blok [1] who considered contact situations where elementary circular square, and the partially contacting band s o ~

were sliding over the surface of a flat half-space.

Blok assmned that the frictional heat was partitioned in a manner that made the peak temperatures on the two contacting surfaces equal. However, because the peak temperatures that were equated occur at different locations of the interface, the assumption violates the requiremem that: at all points of intimate contact the surface temperatures ofthe two bodies be equal. Jaeger [2] enhanced Blok's approach by considering that the average temperature rise of the contact spot is a better representative of the temperature distribution of the contact. Consequently, he equates this average temperature on both rubbing members to derive an expression for the HPF. The method of Jaeger takes some account of the variation of temperature over the contact area. 989

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H.A. Abdel-Aal

Vol, 24, No. 7

However, it does not account for the point wise variation of the heat distribution over the contact. Ling [3] enforced the requirement of the pointwise temperature matching at all the points of intimate contact when calculating the interfacial temperatures. His procedure resulted in a singular integral equation that can be solved by a quasi-itmative num~cal technique [4]. A more exact treatment based on matching the interfacial by a nmnericM ~

has been produced by C ~

et aL [5]. The results of Cameron and Ling

showed that the HPF is not constant but is a function of position within the real area of contact and of velocity [6]. Interestingly I t has been found [7] that an HPF based on the Blok postulate gives reasonably accurate temlm-ature ixedictions as long as the velocityd~T~Ullce betw~g.~l~the sliding bodies is small. At higher sliding velocities, however, signifr,ant ~ i e s

can result. Barber 18] argues that unless high accuracy,is required

the apprommate solutions of Blok and Jaeger are to be use~ Whence he introduces a solution for the partition of hest in a single interactionbetween two sliding metals of comparable hardness. This solution was extended [9] to the case of multiple contacts between two sliding solids with different temperatures at infinity. In developing an expression for HPF it is normally assumed [ 1,2] that, the heat flow is independent of time and that the thermal properties are constant. The introduction of temperature dependant thermal properties, however, implies that the evolution of the interfacial temperature influences the distribution of fik:fionalheat. In this paper, we ira:sent an expression for the heat partition factor based on a variable thermal conductivity solution of the heat equation. It is shown that unlike the case of a constant conductivity the expression for the HPF incoqxwates a coupling between the properties of the rubbing pair.

Variable Conductivity Solution of The Heat Eouafion

When the thermal properties of the solid vary with the temperature, but remain independent of position, the differential equation of heat conduction takes the form

PC a-'~O= V.kVO + g dt

(I)

Where, p. C, and k me temp~ature dependent. Whereas, the heat generation term g is independent of temperature. As such, the heat equation governing the ¢~utuction ofheat in the material is non-linear.

since, k is a fun~on of temperature, equation (1) may be written in the form, p C a...oo= kV~o + v k.V 0 + g at

(2)

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DRY SLIDING OF METALLIC SOLID PAIRS

991

Expressing, ~ in the form, dk . VO

(3)

then, equation (2) takes the form,

pcaO at

~. k V 2 0

+ dk

~.(vO) 2 + g

(4)

To change the dependent variable in equation (4) we apply the Kirchoff transformaaon [ 10], to this effect define a new dependent variable, U, such that,

w = f t(e).toao

(s)

0

Where U-=U(O). Now by modeling the variation of the conductivity with temperature in the linear form,

k(O) = ko(1 +l~oO)

(6)

we may deduce the following relations, au

k ae

at

ko at

au v o

T6

1

dk

(7-a)

A.vo

to

2

Substituting equations (7-a) and (7-c) into equation (4) we obtain,

(7-b)

(7-c)

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H.A. AbdeI-Aal

! a__u_u,v 2 u + ~ at ko

Vol. 24, No. 7

(8)

Ifthe variation in the themud diffusivity, a', with temperature is neglected then; equation (8) becomes a linear differential equation that has the same form of solution as those of the original heat equation, except that the solutions would yield the function U. Frictional heating probkaus an: usually solved with boundary conditions of the second type. These are used when the magnitude of the heat flux along the boundary surface is prescribed. That is at the boundary

surfacej, k ( e ) ~ n j = fj(R,t)

(9)

Equation (9) represents a nonlmee~ equation as the conductivity is a function of temperature. This boundary condition may be ~ansfonned into a linear prescribed heat flux condition by means of equation (5). From equation (7-b) we have

ou

k(O) O0

Now, eliminating oW~j between equations (9) and (10), the transformed boundary conditions assume the form,

ko 7,.: OU

I,

(It)

at the boundary surfaccj. The solutions of equation (4), subject to the boundary oondition equation (11), yields the n~-tremsfonned temperat~ function U. To obtain the values ofthe true temperatures, 8, we apply the inverse transformation o = -~ ( V f + 2 ~ u ( K O - 1)

02)

Note that equation (12), is based on a iinca~ vm-iation in the thermal conductivity with temperature, As such,

the range of application of the Kirchoff transformation would depend on the behavior of the temp~ture

Vol. 24, No. 7

DRY SLIDING OF METALLIC SOLID PAIRS

993

coelficient, ft. If the conductivity ofthe matmal increases with temperaturethe conductivity-temperature curve willhave a positiveslope every where. Hence, the inversela'ansformationis applicableto the entiredomain of intcrfitcialtemperaturedevelopment. If,however, the conductivitydrops with temperature the curve willhave a negative slope every where and the application of the transformationis bounded by those values of the function U which yielda positivevalue to the square root in the bracketed quantityin equation (12).

Temnerature distribution in the slidin~ solids The contribution of an infinitesimal instantaneous point source of strength Q~ placed at a distance x" from the origin at a time / ', may be expressed as [ 11],

U(x:2)= ~3

• 3 EXP [ - ~

[(x-x/)2+y2+z2]}

(13)

4~t~ k(t-t/) 2 Where, 2~ is the inverse of the heat diffusivity given by, 22 = Cp/k. Whe~ the heat source has moved only for a finite interval of time the temperature field may not be at a steady state. In this case the basic solution of the linearized heat equation, equation (13), has to be integrated with respect to time to obtain the solution. Whence, the temperature at the surface z = 0 for a moving contact band is thus given by, E

U(x,y,0,,) =

ll~-~-~ ~ art/ f fF_~p[_2(t~_t~ {[x_xt]2.y2}]dxtdy 47t~r2 k j (t-tt) ~2 J, o

(14)

8t

Where, St'is the mrs of contact between the sliding solids. Equation (14) has an imaginary singularity

at (t - t'=O). To overcome this singularity,

introduce the following new variables:

(O = ( t - t / ) la, ~ = (X-X~

)' 2kuh ° , n = 2kvz~

(15)

theft,

dt t = -2~d6o , dx/= -2k~(od~ dy= 2kta(od~ ,

(16)

Substituting flora equations (15)and (16), into equation (14) the constant conductivity temperature, O(x, y,O,

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H.A. Abdel-Aal

Vol. 24, No. 7

0 may be writtenas,

07) 0

08) °

8t

Where, ® is the heat partition function.

The Heat oortifion factor

To obtain an expression for the heat partition factor at each point of the real contact area the actual temperature, ~, ( and not the function U) has to be continuous at all points of contact. That is at the surface Z=O,

0 l (R,t) = On (R,t)

(19)

Substituting from equation (12) into equation (19) we may write the temperature com~tion at the surface as,

Substituting from equations (17) and (18) in equation (20) the imposed condition at the surfecc assumes the form, =

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DRY SLIDINGOF METALLICSOLID PAIRS

995

Whfre,

An ~

At = 2 ~ ,

F, --

2~

(22-a)

TI

l°el"

ff

(22-b)

I -2~.k.[e.,e]} d~,~,l

(22-c)

Using the binomial theorem we may expand bracketed terms in equation (21) as,

{1 + At[3,~ FI ~ } Ir2 = 1 + A I ~ , Q FI ~ - . .......

{1 * Att~tl~F2(l

(23-a)

_~)}Jn = l * Au[~n(~F2 - A n ~uQF2~

.......

(23-b)

Substituting equations (23-e end b) in equation (21) the tempenlture continuity condition at the contact band may be expressed as, [3n + ~31~HAIC)F, ~ -" ......... -~1, = [~, * [3]~n AIIC)F2 ( 1 - ~ )

-. ......... -~,

(24)

Finallyupon substitutingfrom equation (22-a) the variableconductivityheat partition function, • takes the form, •

(25)

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H.A. Abdel-Aal

Vol. 24, No. 7

Here, k is the actual thermal conductivity, p is the mass density and C is the specific heat. Note that equation (25) reduces to that of Jaeger [2] for sliding pairs of the same material. However, for dissimilar ~

the expremion for Oincoq3oratesa conpling between the thermal capacity of the mating

wa,erials ( the product of the density and the specific heat ). This falls in line with the observation of Ling and Pu [4] that the heat capacity of one or both rubbing members has an effect on the continuity of the temperature at the interface. Note also that the temperature coefficient//doesn't appear in equation (25). This implies that the linear modeling of the conductivity may not influence the heat partition.

Results and discussion TABLE 1 Comparison Between the Values of the Heat Partition Factor Predicted by the Current Model and that Predicted by Barber [91

Slidingpair

G/Q, (eq. iS)

Q~I~ (eq. 14) 191

[.AiSl I020-AIS1302

1.948

1.7825

Bronze-AISll020

1.3018

1.7659

Bronze-AIS1302A

0.668

0.9912

1.14

1. ! 43

Sapphire-Steel 1.0% C

0.937

!.268

Duralumin-Zinc

0.8127

0.959

,

i

Zinc-Brass

Table (1) presents a mmpamon between the ratio of the heat conducted to each of the rubbing members at room temperature. The numbers in the second column are calculated by means of the present model (equation (25)). Whereas the numbers in the third column are calculated from the.relation [9],

Q,

: . .o.,ff,

c26)

It will be noticed that for materials of comparable properties (e.g Zinc-Brass pairs) both expressions yield matching results. When the thermal conductivity of both sliding materials is almost the same (Sapphire- Steel 1.0 % C ), but the density and specific heat are different, equation (26) yields a higher value. This may be atlributed to the absence of the influence of the physical p ~ t t i e s in the constant conductivity based expressions.

This is achieved here by coupling the physical properties explicitly in equation (25) and

Vol. 24, No. 7

DRY SLIDING OF METALLIC SOLID PAIRS

997

implicitly through the functions F1, and F2. 0.5

0.4 o

o3

t~

0.2

•"r'

0.1

\ °'5

01,

0',

0',

Non-.dimensional radius ( R/a )

FIG. 1 Variation of the heat partition factor, O, across a circular contact spot Steel AIM 1020- AISI 1020 sliding pair, plastic contact conditions

Figure (1) is a plot of the variation of the HPFacross a circular contact spot of radius a. The values presented here pertain to the sliding of a steel (AISI 1020) pair under plastic contact conditions for two rubbing speeds. It may be noticed that at a relatively high sliding speed (8 m/see) most of the heat transfer is concentrated at the towards the center of the contact. At low speeds, however, the heat distribution is almost uniform across the contact spot. This may be attributed to the act that: at lower sliding speeds, low Peklct number, there is considerable time for heat to be conducted radially every where on the contact spot. At high speeds, high Pddet number, the short duration of the contact doesn't permit full conduction except at the center for the stationary member and at the trailing section of the slider.

Conclusion

An ~pression for the partition of fi-ictional heat between the robbing member has been presented. The expression stems from a variable conductivity solution of the heat equation. For sliding pairs of similar physical and thermal properties the expression reduces to that based on a constant conductivity solution. The expression reflects to a c~min extent the effect of the thermal capacity of the solids on the partition of heat by coupling the thermal and physical properties of the sliding pair. The current model however is valid for materials of comparable hardness, under quasi-stationary conditions. The extension to transient contact

998

H.A. Abdel-Aal

Vol. 24, No. 7

situations should be mdeavtm~dcarefully as other factors ( such as the relative motion of the contact spot with respect the rubbing plane) are influential to the generation and the partition of heat. Part of this work was written while the author was sponsored by the Department of Mechanical Engineering at UNCC. The atal~ would like to thank Dr. S. T. Smith for many helpful discussions and valuable comments regarding this work.

lkfuy.ema 1.

H. Blok, Proc. Inst. Mech.. Eng. General Discussion on Lubrication, London, 222 (1939).

2.

J.C. Jaeger, Proc. Roy.,~c. N.S.t~., 76,203(1942).

3.

Cmneron A., Gtr&m,A. N., and, Symm, G. T.,Proc. Roy. Soc. London, A 286, 245 (1962).

4.

F.F. Ling, and S. L. Pu, WEAR,7, 23 (1964).

5.

F. F. Ling, Z Angew. Math. Phys., 10, 461 (1959).

6.

A. Floquet, D. Play and M. Godet, A~CEd. Lubr. Tech., 99, 277 (1977).

7.

G.T. Symra, QJ. Mech. Appl.Math.,20,381(1967).

8.

J.R. Bather, J. Mech. Engng Sci., 9, 351 (1967).

9.

J. R. Barher, lnt.J. Heat Mass l"ransfer, 13, 85 7 (1970).

10.

0zi~ik, M. Necati, Boundary Value Problems of Heat Conducaon, Dover Publications, New York, (1989).

1 !-

Rosenthal, D., Trans. A.S~.E., 68, 849 (1946). Received March 31, 1997