An adsorbent layer model for thin film lubrication

Wear 221 Ž1998. 9–14

An adsorbent layer model for thin film lubrication Qu Qingwen a b

a,)

, Hu Yahong b, Zhu Jun

b

Department of Mechanical Design, Shandong Institute of Engineering, Shandong 255012, China Theory of Lubrication and Bearing Institute, Xi’an Jiaotong UniÕersity, Shaanxi 710049, China Received 23 June 1997; accepted 19 June 1998

Abstract Because of the adsorption properties of bearing surfaces, the viscosity of the lubricant varies with distance from the surface to the interior of the fluid. The oil film may be divided into three layers. The layer near surfaces is known as the variable viscosity layer, the middle as conventional constant viscosity layer. The general Reynolds equation can be derived for calculating thin film lubrication. It is provided as the calculating basis for the analysis of thin film lubrication. q 1998 Elsevier Science S.A. All rights reserved. Keywords: Adsorbent layer; Viscosity; Viscosity ratio; Clearance ratio

1. Introduction There are not many reports on the practical model of theoretical calculation for thin film lubrication. The polar molecule model w1,2x was adapted to calculate polar-lubricant. The solution could be gained only when polar interaction between bearing material and lubricant is known. Because an additional bend moment is added to the stress equation, the calculation becomes more difficult. Tichy w3x introduced a surface layer model. Reynolds equation of change viscosity is derived when viscosity is considered as a function of the adsorbent effect. Qu and Zhu w4x proposed the model of average viscosity correcting equation; viscosity correcting equation was simulated by experiment results reported in Refs. w5–7x. The adsorbent layer model is proposed from lubrication state of practical bearing. Based on viscous adsorption theory w8x, the distribution of fluid density is not the same across the thickness of the film w9,10x, that is, density varies with the distance from surface to calculating point. When these changes are connected with the change of viscosity on the surface layer, the average viscosity of the surface layer is produced w11,12x. To obtain the viscosity model, the calculating method is adopted to bearing property analysis for low speed, heavy-load and micro-machines in study. The parameters of the study, adsorbent thickness d 1 , d 2 , and viscosity of surface layer m 1 , m 2 , will determine the lubricant’s characteristics.

2. Fundamental model As shown in Fig. 1, three parameters are introduced. They are inseparable adsorption layer viscosity m 1 and m 2 , thickness of adsorption layer d 1 and d 2 , and conventional viscosity m 0 . Because the thickness of adsorption layer is very

)

Corresponding author.

0043-1648r98r$ - see front matter q 1998 Elsevier Science S.A. All rights reserved. PII: S 0 0 4 3 - 1 6 4 8 Ž 9 8 . 0 0 2 5 4 - 3

Q. Qingwen et al.r Wear 221 (1998) 9–14

10

Fig. 1. Sketch of construction.

small, and the surface effect acts only at small region apart from surface, therefore, a uniform calculating viscosity is assumed in every range. Then, viscosity function becomes:

°0 F z F d : : ¢h y d F z F h:

m s m1 m s m0 m s m2

1

~d - z - h y d

2

2

Ž 1.

3. Velocity equation By momentum equation, the velocities at all kinds of regions are represented as follows:

°0 F z F d :

us

~d

u2 s

1

1 F z F h y d2 :

¢

h y d 2 F z F h:

Ep

1

2 m1 E x

z 2 q C1 z q C 2

Ep

1

2 m0 E x

m3 s

1

Ep

2 m2 E x

z 2 q C 3 z q C4

Ž 2.

z 2 q C5 z q C6

By boundary conditions

°z s 0: ~

u1 s U

z s d1 : z s h y d2 :

¢z s h:

m1

u1 s u 2

m0

u2 s u3

E u1 Ez E u2 Ez

s m0 s m2

E u2 Ez E u3

Ž 3.

Ez

u3 s 0

Because thickness of adsorption layers are very smaller than the one of oil film, adsorption property between metal and oil may be considered as the same. Then, let m 1 s m 2 s m , d 1 s d 2 s d , velocity is as follows:

°u s 1

1 Ep 2m E x

~u s 1 E p 2

2 Ex

1 Ep

z Ž z y h. y z Ž z y h.

m0

q

z

D

UqU

d Ž d y h. m U

¢u s 2 m E x z Ž z y h . y D Ž z y h . 3

where D s Ž h m .rŽ m 0 . q 2 d Ž1 y  mrm 0 4.

ž

m 1y

m0

/

mU y

m0 D

d zqUy

D

ž

m 1y

m0

/

U

Ž 4.

Q. Qingwen et al.r Wear 221 (1998) 9–14

11

4. Reynolds equation Based on the continuity of flow, one can gain the following relationships for the model shown in Fig. 1, at stability, Newtonian fluid and infinite: 1 h3 1 1 qx s y q 2 d Ž 3h 2 y 6 h d q 4d 2 . y 12 m 0 m m0

ž

=

E qx Ex

Ep Ex

mU q 2 m0 D 1

sy

E

12 E x

½

Ž h 2 q 4d 2 y 4 d h . q

h3

m0

/

U

D

Ž 2 d 2 y d h. q UŽ h y d .

q 2 d Ž 3h2 y 6 h d q 4d 2 .

ž

1

m

1 y

m0

Ep

/ 5 Ex

E q

mU

E x 2 m0 D

Ž h 2 q 4 d 2 y 4d h .

U

Ž 2 d 2 y d h. q UŽ h y d . s 0 D Let m ) s mrm 0 . Reynolds equation is derived as follows: E 1 Ep h 3 q 2 d Ž 3h 2 y 6 h d q 4d 2 . y1 ) Ex m Ex E m) 2 2 s 6Um 0 Ž h q 4d 2 y 4d h . q Ž 2 d 2 y d h . q 2 Ž h y d . Ex D D q

½

/ 5

ž

Ž 5.

5. Results and discussion 5.1. Pressure characteristics We introduce the following dimensionless parameters 6 m 0 UL ) ps p h s h b h) d s d ) h b h 2b x s x ) L m s m )m 0 Ž 6. then the dimensionless Reynolds equation is gained E 1 E p) )3 ) ) )2 h 1 q 2 d 3 y 6 d q 4 d y 1 Ž . E x) m) E x) E m) ) 2 2 s ) h q 4 d ) 2 y 4d ) h ) . q ) Ž 2 d ) 2 y d ) h ) . q 2 Ž h ) y d ) . Ž 7. ) Ž Ex D D The relationships between dimensionless pressure p ) and d , m ) are shown in Fig. 2. Dimensionless pressure p ) increases with viscosity, while thickness of adsorbent layer increases. It is believed that the change in p ) is a result of adsorption action between the surface of metal and fluid. And this is not to increase viscosity for fluid.

½

ž

/ 5

Fig. 2. Dimensionless pressure p ) .

Q. Qingwen et al.r Wear 221 (1998) 9–14

12

Fig. 3. Dimensionless load-capacity.

5.2. Load-capacity Dimensionless load-capacity is defined as: Ws

6 m 0 UL2 h2b

W)

Ž 8.

Dimensionless load-capacity per unit width is represented as W )s

1 )

H0 p

d x)

Ž 9.

The relationship between load-capacity and other parameters is shown in Fig. 3. In Fig. 3Ža., when clearance ratio h a) is uniform, and m ) is very small, W ) variation with d ) is not large. This is because the thickness of adsorption layer may be ignored when compared with the thickness of the minimum film. From Fig. 3Žb., when d ) is very small, W ) variation with m ) is not large. The reason is the same as the above. These characteristics are affected by the properties of lubricant and contact surface.

Fig. 4. Dimensionless friction force.

Q. Qingwen et al.r Wear 221 (1998) 9–14

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Fig. 5. The ratio of friction coefficient.

5.3. Friction properties By viscous law, dimensionless friction force becomes F)s

1 )

H0 t

d x)

Ž 10.

Shear stress is m0U 1 h Ep 3m0 U ) ts q s t ) ) h 1 q 2 d Ž 1rm y 1 . 2 Ex hb

t )s

1

1

3 h ) 1 q 2 d ) Ž 1rm ) y 1 .

q h)

Dimensionless friction force is defined as 3 m 0 UL ) Fs F hb

E p) E x)

Ž 11. Ž 12.

Ž 13.

Fig. 4 shows the relative curves between F ) and other parameters. The following conclusions can be gained: when d ) and m ) are small, F ) grows quickly. And when m ) is large and d ) is still small, F ) is almost constant. But, when d ) approaches 0.5, the change is large. When d ) G 0.5, thin film consists entirely of invariable viscosity layers. From Fig. 4, F ) increases as d ) and m ) increase. The friction coefficient may be represented as: F hb F ) hb ) fs s s f Ž 14. ) W 2L W 2L Fig. 5 shows the profiles of the friction coefficient ratio. f 0) is the friction coefficient at d ) s 0, or friction coefficient of conventional calculation. The effects of adsorption layer may be clearly known from Fig. 5. Fig. 5Ža. states that friction coefficient decreases as m ) increases when d ) is constant. The less m ) is, the more the change in friction coefficient with m ) , when m ) is large, friction coefficient is essentially unchanged. As shown in Fig. 5Žb., when m ) equals to constant, friction coefficient decreases first as d ) increases, then increases. If the film is made of the adsorption layer completely, friction coefficient will approach constant. As the thickness of the adsorption layer increases and the thickness of easy deformation liquid layer decreases, shear rate increases, and shear thinning is produced. Therefore, there exists a small friction coefficient under thin film lubrication. After adsorption layer is introduced, friction coefficient is always less than that at d ) s 0, because the increasing amplitude of load-capacity is greater than that of friction force when the viscosity is considered as a variation. 6. Conclusions As adsorption action of surface is considered, the following conclusions can be obtained. Ž1. The pressure increases with the thickness of adsorption layer while viscosity ratio increases. When d ) is very small, the effect of adsorption may be ignored.

Q. Qingwen et al.r Wear 221 (1998) 9–14

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Ž2. With increasing thickness of adsorption layer and the viscosity ratio, the increasing rate of load-capacity is greater than that of friction force. Therefore, the ratio of friction coefficient is less than 1, that is, the friction coefficient will decrease if the effects of adsorption layer could not be ignored. Ž3. Reynolds equation derived adapt to the closed-region. That is, the equations that are derived in this paper are still applied to calculate thick film lubrication.

7. Nomenclature f F ha hb f0 h h a) L p u W d , d 1, d 2 m1, m 2 t qx U m0 m) Ž) .

friction coefficient friction force film thickness at inlet film thickness at outlet friction coefficient at d ) s 0 film thickness clearance ratio bearing length pressure velocity load capacity adsorbent layer thickness adsorbent layer viscosity shear stress flow rate Surface velocity Conventional viscosity Viscosity ratio Dimensionless symbol

References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x

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