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M. Groper I. Etsion Fellow ASME Dept. of Mechanical Engineering, Technion, Haifa 32000, Israel

The Effect of Shear Flow and Dissolved Gas Diffusion on the Cavitation in a Submerged Journal Bearing Two possible, long standing speculated mechanisms are theoretically investigated in an attempt to understand previous experimental observations of pressure build up in the cavitation zone of a submerged journal bearing. These mechanisms are (1) the shear of the cavity gas bubble by a thin lubricant film dragged through the cavitation zone by the rotating shaft and (2) the mass transfer mechanism which dictates the rate of diffusion of dissolved gas out of and back into the lubricant. A comparison with available experimental results reveals that while the cavitation shape is fairly well predicted by the ‘‘shear’’ mechanism, this mechanism is incapable of generating the level of the experimentally measured pressures, particularly towards the end of the cavitation zone. The ‘‘mass transport’’ mechanism is found inadequate to explain the experimental observations. The effect of this mechanism on the pressure build up in the cavitation zone can be completely ignored. 关DOI: 10.1115/1.1308026兴

Introduction Cavitation in bearings and seals was the subject of a great deal of research work as well as controversy until about the mid 1980s. Issues like cavitation content, shape, pressure distribution, and boundary conditions inspired many researchers and generated a large volume of publications. Several review papers, e.g., Dowson and Taylor 关1兴, Brewe 关2兴, and Heshmat 关3兴, give an excellent summary of the knowledge on cavitation. In spite of all this, the subject is not completely closed yet and several questions remained open. One of these questions concerns the mechanism of pressure build up inside an enclosed cavitation, as in submerged journal bearings, pressure which was measured and reported by Etsion and Ludwig 关4兴 and by Braun and Hendricks 关5兴. Surprisingly, the more recent theoretical works on cavitation, e.g., Yu and Keith 关6兴, Mistry et al. 关7兴 and Ramesh et al. 关8兴 overlooked the experimental findings of the pressure distribution in the cavitation region and adhered to the common assumption of a uniform cavitation pressure. In their theoretical work Coyne and Elrod 关9,10兴 analytically solved the shape of the fluid-gas interface, which occurs when a liquid film separates from a fixed surface and is swept away by the opposing moving surface. The physical situation is shown in Fig. 1. The experimental works of Etsion and Ludwig 关4兴, Heshmat 关3兴, and others clearly revealed the existence of a thin lubricant layer, which adheres to the rotating shaft and completely covers it. The remaining space between the rotating shaft and the stationary surface is occupied by a gas cavity, divided by narrow lubricant streamers. An interface separates between the liquid phase 共the lubricant兲 and the gas phase 共the cavity兲. The flow of the lubricant layer swept by the rotating shaft shears this interface. The shear in the gas bubble may be a possible source of pressure generation but so far has not been investigated. The content of the cavitation bubble has been a matter of argument for a long time. Experimental works by Etsion and Ludwig 关4兴 and Braun and Hendricks 关5兴 revealed that in the case of their submerged hydrodynamic journal bearing under a constant load

the content of the cavity was air. The dissolved air in the lubricant 共up to 15 percent in mineral oils兲 may be liberated when the pressure in the film falls below the saturation pressure of the solution. It was postulated by Etsion and Ludwig 关4兴 that the mass transfer mechanism which dictates the rate of diffusion of the air out of the solution and the rate of absorption of air back in the solution may control the pressure variation observed in the cavitation zone. An attempt is made in the present paper to investigate theoretically the shear of the air bubble and the continuing process of air coming out and going back into solution as possible mechanisms of pressure build up in the cavitation zone of a submerged journal bearing.

Theory The Shear of the Cavity Bubble by the Coating Flow. Figure 2 illustrates a cross section of the submerged journal bearing in the cavitation region. The gas cavity adheres to the stationary surface while a thin layer of lubricant having a thickness of h ⬁ is swept away by the rotating shaft. It is assumed that all the lubricant flow, including that in the streamers, is contained within the layer of thickness h ⬁ . Based on the indication in Ref. 关3兴 that the streamers are ‘‘weakly bonded’’ to the stationary wall, an estimation of the flow rate through the streamers observed in the pictures of Ref. 关4兴 was performed. It was found that the flow rate through

Contributed by the Tribology Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for presentation at the STLE/ASME Tribology Conference, Seattle, WA, October 1–4. Manuscript received by the Tribology Division February 25, 2000; revised manuscript received June 30, 2000. Paper No. 2000-TRIB-27. Associate Editor: J. Freˆne.

494 Õ Vol. 123, JULY 2001

Fig. 1 Description of lubricant separation and the creation of a liquidÕgas interface †9‡

Copyright © 2001 by ASME

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d 2u g d P cav ⫽␮g Rd ␪ dy 2

(1)

d P cav d 2u l ⫽␮l 2 Rd ␪ dy

(2)

and

Fig. 2 A schematic cross section of the cavitation region in a submerged journal bearing

the streamers accounts for less than 20 percent of the total lubricant flow rate through the cavity. For this reason and for the interest of simplicity the illustration of Fig. 2 is the basis of the developed models. The usual assumptions of the hydrodynamic theory are applied to the gas and the liquid phases, i.e., negligible inertia, film thickness small compared with the other dimensions, etc. Mass transfer between the gas and liquid phase is ignored and thus it is assumed that the total rate of transport of gas in the x direction is zero. Experimental works performed by Etsion and Ludwig 关4兴 and Braun and Hendricks 关5兴 revealed that the axial pressure variations within the cavitation region are negligible. Thus, the lubricant layer cannot be redistributed in the direction transverse to the surface motion. A schematic description of the cavity boundary is shown in Fig. 3. The thickness, h ⬁ , of the swept lubricant sublayer is obtained from the Coyne and Elrod 关9兴 theory assuming that the angular distance between the film rupture start ( ␪ start) and the film rupture end ( ␪ t ) is small. Thus, the clearance variation between these two locations is negligible. Surface tension effects are neglected 共the Laplace pressure for the lubricant meniscus is about two orders of magnitude smaller than the pressure rise observed in 关4兴兲. Finally, based on the Coyne and Elrod 关9兴 analysis, it is assumed that the transition region from the full film h rupture to the constant thickness lubricant layer h ⬁ 共see Fig. 1兲 is short compared to the overall angular extent of the cavity. Based on these assumptions the momentum equations for the gas in the cavity and for the lubricant layer swept by the rotating shaft are, respectively,

As shown in Fig. 2 liquid of viscosity ␮ l occupies a thickness h ⬁ of the total clearance, the remaining space, h⫺h ⬁ , being filled by a gas of viscosity ␮ g . P cav is the cavity pressure. The usual assumptions of no slip at the solid boundaries and at the interface between liquid and gas are given by u l 兩 y⫽0 ⫽ ␻ R,

u g 兩 y⫽h ⫽0,

u l 兩 y⫽h ⬁ ⫽u g 兩 y⫽h ⬁

(3)

The continuity of the shear stress at the interface is given by

␮l

冏

du l dy

⫽␮g y⫽h ⬁

冏

du g dy

(4) y⫽h ⬁

The condition of no net gas transport in the angular direction gives the following:

␳g

冕

h

u g dy⫽0

(5)

h⬁

The velocities u g and u l can be obtained from the integration of Eqs. 共1兲 and 共2兲 and the conditions given by Eqs. 共3兲 and 共4兲. In addition, making use of the condition given by Eq. 共5兲 allows the calculation of the cavity pressure gradient d P cav /d ␪ . The cavity pressure gradient is d P cav 6 ␻ R 2␮ g ⫽ d␪ 共 h⫺h ⬁ 兲 2 ⫹ ␮ g / ␮ l • 共 4hh ⬁ ⫺h ⬁2 兲

(6)

The ratio ␮ g / ␮ l , which is usually a very small number, allows neglecting expressions multiplied by this ratio. In this case the cavity pressure gradient is given by d P cav 6 ␻ R 2 ␮ g 6 ␻ R 2 ␮ g ⫽ ⫽ d␪ 共 h⫺h ⬁ 兲 2 h 2g

(7)

The velocity gradient du l /dy is of order of ␮ g / ␮ l . Under the assumption that ␮ g / ␮ l is usually a very small number, for the purpose of flow rate calculation, the velocity gradient can be neglected. In this case the velocity of the lubricant swept by the rotating journal is almost uniform and equal to the journal speed, ␻ R. Normalizing Eq. 共7兲 and using the ‘‘Bearing number’’ ⌳⫽

冉冊

6 ␮ l␻ R P sat C

2

(8)

The dimensionless form of the cavity pressure gradient is d ¯P cav d␪

⫽

␮ ¯ ¯h 2g

(9)

where ␮ ¯ is the viscosity ratio ␮ ¯ ⫽␮g /␮l . At the angular location ␪ ⫽ ␪ start the lubricant pressure falls to the saturation pressure of the air/lubricant solution and the film ruptures. Thus, the boundary condition for Eq. 共9兲 is ¯P cav共 ␪ 兲 ⫽ ¯P sat⫽0 start

Fig. 3 Boundary of the cavity—schematic description

Journal of Tribology

(10)

At this stage the cavity pressure field cannot be explicitly calculated because the cavity boundary as well as the cavity start and end angular locations ␪ start and ␪ end are unknown. The dimensionless Reynolds equation for the lubricant outside the cavity can be written as JULY 2001, Vol. 123 Õ 495

rate is defined by ¯⫽ Q

Q ␻ RCL/2

(17)

Making use of this definition and of the dimensionless expressions in 共12兲 the dimensionless form of the equations for the flow rates are ¯ 1⫽h ¯ ⬁ 共¯z cav共 ␪ 兲 ⫺z Q ¯ cav共 ␪ ⫹⌬ ␪ 兲 兲 3

¯h ␪ 兲 1 共 ¯ 2⫽ ¯h 共 ␪ 兲 共 1⫺z Q ¯ cav共 ␪ 兲 兲 ⫺ 2 2

冕

3

¯h ␪ ⫹⌬ ␪ 兲 1 共 ¯ 3⫽ ¯h 共 ␪ ⫹⌬ ␪ 兲 共 1⫺z Q ¯ cav共 ␪ ⫹⌬ ␪ 兲 兲 ⫺ 2 2 ¯ 4⫽ Q

Fig. 4 Flow rate balance over the control volume located between the bearing edge z Ä L Õ2 and the cavitation boundary

冉 冊

冉 冊

¯ ⳵ ⳵ ¯P ⳵ ¯P 1 ⳵ dh ¯h 3 ¯h 3 ⫹ 2 ⫽ ⳵␪ ⳵␪ ␭ ⳵¯z ⳵¯z d␪ 2z , L

␭⫽

L , 2R

¯h ⫽

h , C

¯P ⫽

P⫺ P sat P sat⌳

(11)

(12)

For the submerged journal bearing, in the angular direction, periodical conditions are used, namely,

⳵ ¯P ⳵␪

冏

⫽ 0,z¯

⳵ ¯P ⳵␪

冏

, 2 ␲ ,z¯

¯P 兩 0,z¯ ⫽ ¯P 兩 2 ␲ ,z¯

(13)

¯⫽1) the pressure In the axial direction, at the bearing edge (z equals the ambient pressure, thus ¯P 兩 ␪ ,1⫽ ¯P sup

(14)

The second boundary condition in the axial direction depends on the angular position at which the calculation is performed. In the region where cavitation is not present, the boundary condition is obtained by making use of the axial symmetry, namely,

⳵ ¯P ⳵¯z

冏

⫽0

(15)

␪ ,0

In the region where cavitation is present the Reynolds equation is solved only in the ‘‘side bands,’’ between the cavity boundary and the bearing edge, where full film exists. In this case, neglecting surface tension effects, the second boundary condition in the axial direction is given by ¯P 兩 ␪ ,z¯ ⫽ ¯P cav共 ␪ 兲 cav

(16)

Hypothetically, in the case the cavity boundary was given, the cavity pressure field could be calculated from the solution of the differential equation 共9兲 with the boundary condition 共10兲. Then, the pressure field through the bearing could be calculated by solving the Reynolds equation 共11兲 subjected to the boundary conditions 共13兲 to 共16兲. Unfortunately the cavity boundary is unknown a priori, thus an additional condition is needed. The basis for this additional condition is flow conservation throughout the control volume of Fig. 4. A dimensionless flow 496 Õ Vol. 123, JULY 2001

冕

冏 冊

⳵ ¯P ¯z cav共 ␪ ⫹⌬ ␪ 兲 ⳵ ␪ 1

冏

(18) dz ¯ ␪ ⫹⌬ ␪

⌬␪

¯z ⫽1

Because of the very small value of du l /dy inside the cavitation zone 共see the comment following Eq. 共7兲兲 in the expression for ¯ 1 the contribution of the pressure gradient was neglected. InQ deed, as will be shown later, this does not cause a significant error in the prediction of the cavity boundary. The condition for flow conservation is given by ¯ 1⫹Q ¯ 2⫺Q ¯ 3⫹Q ¯ 4⫽0 Q

where the dimensionless parameters are given by z ¯⫽

冉

1 ¯h ⬘ 3 ⳵ ¯P ⬘ ␭ 2 2 ⳵¯z

冏

⳵ ¯P dz ¯ ¯z cav共 ␪ 兲 ⳵ ␪ ␪ 1

(19)

Equation 共19兲 is the additional condition required for the determination of the unknown cavity boundary. Following Nau’s 关11兴 work the procedure for the calculation of the pressure field throughout the bearing and of the cavity boundary uses the finite differences method. The Reynolds equation is discretized using central differences along axial lines and backward differences circumferentially. Thus it is possible to solve the difference equations implicitly for each successive angular mesh line. The calculation progresses around the bearing until an angular position is reached where the calculated pressure is lower than the air/lubricant solution saturation pressure. This location is denoted as ␪ start for this iteration and the following procedure is introduced: The cavity pressure at the next angular location is calculated using the solution of the differential equation 共9兲 subjected to the boundary condition 共10兲. Then, the position of the cavity boundary at the next angular location is given a trial value and the Reynolds equation is solved between the bearing edge and the cavity boundary to obtain the axial pressure distribution. The previously calculated cavity pressure serves as a boundary condition for the solution of the Reynolds equation. Then, making use of the calculated pressures, the flow rates through the control volume can be calculated by Eq. 共18兲 and flow continuity can be checked by Eq. 共19兲. Depending on the sign of the error the cavity boundary is corrected until flow continuity is satisfied. The calculation then proceeds to the next angular location until the calculated cavity boundary intersects the ¯⫽0 z axis and the cavity is terminated. The Mass Transport Mechanism. Figure 5 illustrates a cross section of the submerged journal bearing in the cavitation region. A gas/liquid interface exists through which a diffusive flux of gas is possible. It is postulated that the main source of gas penetrating into the cavity is in the film rupture region where ‘‘fresh’’ lubricant is rapidly depressurized below the saturation pressure. The lubricant ruptures releasing the excess gas. In a steady state condition all the gas released at the film rupture must be eventually reabsorbed back into the lubricant. The process of gas absorption into the lubricant layer through the gas/liquid interface is controlled by a mass transfer mechanism. It is assumed that the gas concentration in the lubricant along the gas/liquid interface equals the saturation concentration at the specific local Transactions of the ASME

¯ ⫽0) is impermeable to the diffusive flux, thus The journal wall (y

冏

⳵¯c g ⳵¯y

⫽0

(28)

␪ ,0

Finally, based on the initial assumptions the gas concentration at the interface equals the saturation concentration at the local lubricant pressure. Following Sun and Brewe 关12兴, Henry’s law gives the solubility of gas in liquid: cg Mg cg ␳l ⫹ Mg Ml Fig. 5 Model for diffusion effect

pressure. It is also assumed that the gas concentration in the solution is a function of the location only, there is no chemical reaction and the mass transfer process does not affect the physical properties of the lubricant (D, ␳ l , ␮ l ). Equation 共20兲 governs the gas concentration in the lubricant layer where c g denotes the concentration, v the velocity vector, and D is the diffusion coefficient of the gas in the lubricant,

⳵cg ⫹v•ⵜc g ⫽Dⵜ 2 c g ⳵t

冉

⳵cg ⳵cg ⳵cg ⳵ 2c g ⳵ 2c g ⳵ 2c g ⫹ ⫹ 2 ⫹v ⫹w ⫽D ⳵x ⳵y ⳵z ⳵x2 ⳵y2 ⳵z

冊

冉冉 冊

1 ⳵ 2c g ⳵ 2c g ⳵ 2c g ⫹ ⫹ 2 R2 ⳵␪2 ⳵y2 ⳵z

or in a dimensionless form: D ⳵¯c g ⫽ ⳵␪ ␻C2

冉冉 冊 C R

2

冊

冉 冊 冊

⳵ 2¯c g ⳵ 2¯c g 2C ⫹ ⫹ ⳵␪2 ⳵¯y 2 L

2

⳵ 2¯c g ⳵¯z 2

cg c sat共 ␪ start兲

冉冊

␻R2 C D R

(22)

(23)

2

(25)

Making use of this definition and neglecting expressions multiplied by the ratio (C/R) 2 and (2C/L) 2 in Eq. 共23兲, the following equation is obtained: (26)

At the film rupture, ␪ ⫽ ␪ start and the gas concentration in the lubricant equals the initial saturation pressure of the solution. Thus, ¯c g 共 ␪ start ,y¯ 兲 ⫽1 Journal of Tribology

(30)

Thus, Eq. 共29兲 may be written approximately as

c g 共 ␪ ,h ⬁ 兲 ⫽

(31)

␳l M g P H M l cav共 ␪ 兲

(32)

Using the expressions for the dimensionless pressure, the dimensionless concentration 共24兲, and the connection obtained from Henry’s law 共32兲 given by c sat ␳ l M g ⫽ P sat H M l

(33)

the dimensionless boundary condition at the liquid/gas interface is obtained:

(24)

⳵¯c g 1 ⳵ 2¯c g ⫽ ⳵ ␪ PeD⬘ ⳵¯y 2

cg ␳l Ⰶ Mg Ml

Hence, the boundary condition for the gas concentration at the liquid/gas interface is

The ‘‘corrected’’ Pe´clet number expresses the ratio between the convective diffusion and the molecular diffusion and is defined here by

⬘⫽ PeD

which states that the mole fraction of the gas in the gas/liquid solution is proportional to the pressure of the gas outside the solution. The coefficient H appearing in this relation is Henry’s constant and its value depends on the solution property. Henry’s law applies to weak solutions, i.e.,

(21)

The dimensionless concentration is given by ¯c g ⫽

(29)

(20)

where u, v , w are the velocity components in the x, y, z directions, respectively. Due to the fact that in the lubricant layer there is no velocity in the axial direction, z, and under the assumption that the diffusive flux velocity is negligible in comparison with the sweeping velocity of the lubricant layer, the velocity components v and w can be neglected. In this case, and following the substitution x⫽R ␪ and u⬵U⫽ ␻ R, Eq. 共22兲 is obtained:

⳵cg ␻ ⫽D ⳵␪

P cav H

cg M g P cav ⫽ ␳l H Ml

In a Cartesian coordinates system equation 共20兲 is given by u

⫽

(27)

¯c g 共 ␪ ,h¯ ⬁ 兲 ⫽1⫹⌳ ¯P cav共 ␪ 兲

(34)

The partial differential equation 共26兲 with the boundary conditions 共27兲, 共28兲, and 共34兲 defines the gas concentration in the lubricant layer. The solution of this PDE depends on the cavity pressure field, ¯P cav共␪兲 , that is unknown at this stage. Figure 6 illustrates a control volume 共marked C.V.兲 starting at the angular location ␪ start and ending at the angular location ␪. Two cases are analyzed: 共a兲 ␪ ⬍ ␪ t and 共b兲 ␪ ⬎ ␪ t . For case 共a兲 the mass flux penetrating the cavity is a function of the angular location ␪ and is given by m ˙ inz cav共␪兲 . For case 共b兲 the mass flux penetrating the cavity is independent of the angular location ␪ and is given by m ˙ inz cav( ␪ t ) . m ˙ in is the penetrating mass flux per unit length at the film rupture. This flux describes the gas flow into the cavity following the lubricant depressurization and rupture. Flow conservation throughout the control volume in Fig. 6 in case 共a兲 is m ˙ ␪ z cav共 ␪ 兲 ⫽m ˙ inz cav共 ␪ 兲 ⫺R

冕

␪

m ˙ d z cav共 ␪ 兲 d ␪

(35)

␪ start

where m ˙ ␪ is the convective mass flux per unit width of the cavity. This flux is driven by a combined Couette and Poiseuille mechanism and is given by JULY 2001, Vol. 123 Õ 497

Substituting the expressions for the mass flux given in Eqs. 共36兲 and 共37兲 into the flow conservation equations of cases 共a兲 and 共b兲 共Eqs. 共41兲 and 共42兲兲, assuming that the gas in the cavity obeys the ideal gas law: P cav⫽ ␳ g RT

(43)

and using the dimensionless expressions 共8兲, 共12兲, 共24兲, and the relation obtained from Henry’s law 共33兲, one can find the dimensionless form of the pressure gradients. For case 共a兲, where ␪ ⬍ ␪ t : d ¯P cav d␪

⫽␮ ¯

再 冉

1 ¯h 2g

⫻

⫺

⫺

2⌽

⬘ 共 ¯P cav⌳⫹1 兲¯h 3g¯z cav PeD

¯z cav共 ␪ 兲 ¯z cav共 ␪ t 兲

冕

␪

冕

␪ end ⳵¯ c

冏

⳵¯c g ⳵¯y

␪ start

冏

g

⳵¯y

␪ start

¯z cavd ␪ ¯ ¯y ⫽h ⬁

¯z cavd ␪ ¯ ¯y ⫽h ⬁

冊冎

(44)

and for case 共b兲, where ␪ ⬎ ␪ t : d ¯P cav Fig. 6 Diffusion effect—flow balance across the control volume

d␪

⫽␮ ¯

再 冉冕 冏 1

¯h 2g

冉

冊

h 3g d P cav ␻ Rh g ⫺ ␳ 2 12␮ g Rd ␪ g

(36)

m ˙ d is the diffusive mass flux per unit area of the liquid/gas interface. Fick’s law gives this mass flux which is absorbed by the swept lubricant layer m ˙ d ⫽D

冏

⳵cg ⳵y

(37) y⫽h ⬁

Flow conservation throughout the control volume in Fig. 6 in case 共b兲 is

冕

m ˙ ␪ z cav共 ␪ 兲 ⫽m ˙ in z cav共 ␪ t 兲 ⫺R

␪

m ˙ d z cav共 ␪ 兲 d ␪

m ˙ in z cav共 ␪ t 兲 ⫽R

冕

␪ start

m ˙ d z cav共 ␪ 兲 d ␪

(39)

or m ˙ in ⫽

R z cav共 ␪ t 兲

冕

␪ end

␪ start

m ˙ d z cav共 ␪ 兲 d ␪

(40)

Using this relation in the flow conservation Eqs. 共35兲 and 共38兲 one finds, for case 共a兲 m ˙ ␪ z cav共 ␪ 兲 ⫽R

z cav共 ␪ 兲 z cav共 ␪ t 兲

冕

␪ end

␪ start

m ˙ d z cav共 ␪ 兲 d ␪ ⫺R

冕

␪

m ˙ d z cav共 ␪ 兲 d ␪

␪ start

(41) and for case 共b兲: m ˙ ␪ z cav共 ␪ 兲 ⫽R

冕

498 Õ Vol. 123, JULY 2001

␪ end

␪

⌽⫽

m ˙ d z cav共 ␪ 兲 d ␪

(42)

¯z cavd ␪ ¯ ¯y ⫽h ⬁

冊冎

(45)

RT ␳ l M g H Ml

(46)

Basically, the right hand side of Eqs. 共44兲 and 共45兲 contains two expressions; the first expression describes the contribution of the convective mass flux 共by the shear of the interface兲 to the pressure gradient d ¯P cav /d ␪ in the cavity. This expression is identical to the expression in Eq. 共9兲 as would be expected. The second expression describes the contribution of the diffusive mass flux to the pressure gradient in the cavity. The boundary condition for the solution of the differential equation 共44兲 is

(38)

␪ end

g

⳵¯y

where ⌽ is given by

¯P cav共 ␪ 兲 ⫽0 start

␪ start

At steady state all the gas penetrating the cavity at the film rupture is reabsorbed back into the lubricant layer, thus

␪

2⌽

⬘ 共 ¯P cav⌳⫹1 兲¯h 3g¯z cav PeD

␪ end⳵¯ c

⫻ m ˙ ␪⫽

⫺

(47)

By solving Eq. 共44兲 the pressure ¯P cav( ␪ t ) is obtained. This pressure is used as the boundary condition for the solution of the differential equation 共45兲.

Results and Discussion The submerged hydrodynamic bearing tested by Etsion and Ludwig 关4兴 was selected to evaluate the proposed mechanisms in the present analysis for pressure build up in the cavitation region. The various geometrical and operational parameters are summarized in Table 1. The two suggested models, i.e., the ‘‘shear’’ Table 1 Geometrical data, operation conditions, and lubricant properties L D C ␧ n P sup ␮l ␳l

3.81 cm 5.08 cm 0.0114 cm 0.4 1840 rpm 154.4 kPa 0.026 N sec/m2 828.5 Kg/m3

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Table 2 Properties of the oilÕair solution

␮g R Mg Ml D T P sat H

2•10⫺5 N sec/m2 287.04 J/Kg°K 28.97 g/mole 422.8 g/mole 0.495•10⫺9 m2/sec 303 K 100 kPa 13.3 MPa

experimental cavity pressure distribution along the cavity center at z⫽0. Contrary to the cavity shape results the correlation of the theoretical and experimental cavity pressure field is rather poor mainly towards the cavity end. The model under estimates the higher experimental cavity pressures by about 20 percent at the cavity end. Based on the above findings it seems that the algorithm for the calculation of the cavity boundary may be correct. However, the suggested ‘‘shear’’ mechanism is insufficient to generate the cavity pressure field that was observed in the experiments.

Fig. 7 Predicted versus measured †4‡ cavity boundary

model and the ‘‘mass transport’’ model were analyzed and the results were compared with the experimental ones described in Etsion and Ludwig 关4兴. Shear of the Cavity Bubble by the Coating Flow. Figure 7 presents a comparison between the cavity boundary calculated by the present ‘‘shear’’ model 共using the algorithm described in the Theory section兲 and the experimental boundary obtained by Etsion and Ludwig 关4兴. As can be seen the theoretical and experimental results for the cavity start angle, ␪ start , are very close. The rupture portions of the cavity boundaries in the two cases deviate by up to about 50 degrees at ␪ t but the maximum width of the cavity is almost identical. The reformation portions of the cavity boundaries as well as the results for the cavity end angle ␪ end show reasonable correlation. Figure 8 presents a comparison between the calculated and the

The Mass Transport Mechanism. The differential equations 共44兲 and 共45兲 with the boundary conditions 共47兲 define the pressure field throughout the cavity. The solution of these ordinary differential equations depends on the cavity boundary, which is unknown a priori as was the case in the ‘‘shear’’ model. To overcome this obstacle an algorithm based on flow equilibrium similar to the shear model was used here for the ‘‘mass transport’’ model. Solution of the cavity pressure field based on the mass transport model requires some additional physical properties of the lubricant and the gas. These properties are summarized in Table 2. Based on these properties and using Eqs. 共25兲 and 共46兲, the nu⬘ , and for merical values for the ‘‘corrected’’ Pe´clet number, PeD ⬘ ⫽5058 and ⌽⫽0.3712. Based on these values the con⌽ are PeD tribution of the diffusive mass flux to the pressure gradient, d ¯P cav /d ␪ , in Eqs. 共44兲 and 共45兲 is only about one-thousandth that of the convective mass flux. As it turns out the diffusion process is too slow for the dissolved gas to liberate itself and create a significant mass flux. Thus, the diffusion effect on the cavity pressure field at steady state as well as on the cavitation boundaries is extremely small and can be easily neglected. Similar results were obtained for the other operation conditions of Ref. 关4兴, namely, good prediction of the cavity boundary, insufficient pressure build up by the ‘‘shear’’ mechanism, and negligible effect of the gas diffusion mechanism. Hence, the discus-

Fig. 8 Predicted versus measured †4‡ cavity pressure field „at the bearing center…

Journal of Tribology

JULY 2001, Vol. 123 Õ 499

sion above is not limited to the results presented in Figs. 7 and 8 but can be considered as representative of the general behavior of submerged journal bearings.

Conclusion Two possible mechanisms were theoretically investigated in an attempt to understand previous experimental observations of pressure build up in the cavitation zone of a submerged journal bearing. These mechanisms are 共1兲 Shear of the cavity gas bubble by the coating lubricant flow dragged through the cavitation zone by the rotating shaft and 共2兲 the mass transfer mechanism that dictates the rate of diffusion of dissolved gas out of and back into the lubricant. Appropriate algorithms were constructed that allow calculation of the cavitation shape and the pressure distribution inside the cavitation zone of a submerged bearing. The theoretical results were compared with an available experimental one. It was found that the cavitation shape could be fairly well predicted by the ‘‘shear’’ mechanism. This mechanism, however, was found incapable of generating the level of pressures that were measured in the experiments particularly towards the end of the cavitation zone. The ‘‘mass transport’’ mechanism was found completely inadequate to explain the experimental observations. The effect of this mechanism on the pressure build up in the cavitation zone is three orders of magnitude less than that of the ‘‘shear’’ mechanism and in fact can be completely ignored. Hopefully, the results of the present work will bring to an end the speculation in some previous publications regarding the possible effect of dissolved gas diffusion on the pressure build up inside the cavitation. It seems, however, that the actual mechanism, which generates the pressures inside the cavitation zone, is yet to be found.

Nomenclature C Ca cg ¯c g D D e h ¯h h⬁ H L m ˙␪ m ˙d Mg Ml P ¯P ⬘ PeD Q ¯ Q

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

radial clearance capillary number, ␮ l U/ ␴ mass concentration of gas dimensionless mass concentration of gas, c g /c sat( ␪ start) bearing diameter coefficient of binary diffusion bearing eccentricity lubricant film thickness, C(1⫹␧ cos ␪) dimensionless film thickness, h/C asymptotic thickness of the swept lubricant layer Henry’s constant bearing width convective mass flux per unit width diffusive mass flux per unit area molecular weight of the gas molecular weight of the liquid pressure dimensionless pressure, ( P⫺ P sat)/ P sat⌳ modified Pe´clet number, ␻ R 2 /D•(C/R) 2 volumetric flowrate dimensionless volumetric flowrate, Q/( ␻ RCL/2)

500 Õ Vol. 123, JULY 2001

R R T u U v v w x, y, z ¯y ¯z ␧ ␭ ␮ ␮ ¯ ␪ ␪t ␪ start ␪ end ␳ ␻ ⌳

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

bearing radius gas constant temperature fluid velocity in circumferential direction journal tangential velocity, ␻ •R fluid velocity vector fluid velocity in axial direction fluid velocity in radial direction coordinates system dimensionless coordinate, y/C dimensionless coordinate, 2z/L dimensionless eccentricity, e/C bearing width over diameter ratio, L/2R viscosity viscosity ratio, ␮ g / ␮ l angular coordinate angular location of the film rupture end angular location of the cavity start angular location of the cavity end density angular speed of journal bearing number, 6 ␮ l ␻ / P sat•(R/C) 2

Subscripts cav g l sat sup

⫽ ⫽ ⫽ ⫽ ⫽

cavity gas liquid saturation supply 共pressure兲

References 关1兴 Dowson, D., and Taylor, C. M., 1979, ‘‘Cavitation in Bearings,’’ Annu. Rev. Fluid Mech., 11, pp. 35–66. 关2兴 Brewe, D. E., 1988, Current Research in Cavitating Fluid Films 共NASA Technical Memorandum 103184兲. 关3兴 Heshmat, H., 1991, ‘‘The Mechanism of Cavitation in Hydrodynamic Lubrication,’’ Tribol. Trans., 34, No. 2, pp. 177–186. 关4兴 Etsion, I., and Ludwig, L. P., 1982, ‘‘Observation of Pressure Variation in the Cavitation Region of Submerged Journal Bearings,’’ ASME J. Lubr. Technol., 104, pp. 157–163. 关5兴 Braun, M. J., and Hendricks, R. C., 1984, ‘‘An Experimental Investigation of the Vaporous/Gaseous Cavity Characteristics of an Eccentric Journal Bearing,’’ Tribol. Trans., 27, No. 1, pp. 1–14. 关6兴 Yu, Q., and Keith, G., 1994, ‘‘A Boundary Element Cavitation Algorithm,’’ Tribol. Trans., 37, No. 2, pp. 217–226. 关7兴 Mistry, K., Biswas, S., and Athre, K., 1997, ‘‘A New Theoretical Model for Analysis of the Fluid Film in the Cavitation Zone of a Journal Bearing,’’ ASME J. Tribol., 119, pp. 741–746. 关8兴 Ramesh, J., Majumdar, B. C., and Rao, N. S., 1997, ‘‘Thermohydrodynamic Analysis of Submerged Oil Journal Bearings Considering Surface Roughness Effects,’’ ASME J. Tribol., 119, pp. 100–106. 关9兴 Coyne, J. C., and Elrod, H. G., 1970, ‘‘Conditions for the Rupture of a Lubricating Film, Part 1: Theoretical Model,’’ ASME J. Lubr. Technol., 92, pp. 451–456. 关10兴 Coyne, J. C., and Elrod, H. G., 1971, ‘‘Conditions for the Rupture of a Lubricating Film, Part 2: New Boundary Conditions for Reynolds’ Equation,’’ ASME J. Lubr. Technol., 93, pp. 156–167. 关11兴 Nau, B. S., 1980, ‘‘Observation and Analysis of Mechanical Seal Film Characteristics,’’ ASME J. Lubr. Technol., 102, pp. 341–349. 关12兴 Sun, D. C., and Brewe, D. E., 1992, ‘‘Two Reference Time Scales for Studying the Dynamic Cavitation of Liquid Films,’’ ASME J. Tribol., 114, pp. 612–615.

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