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Parallel surfaces lubricated by a bubbly oil
Wear, 35 (1975) 23 - 34 @ Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
PARALLEL
SURFACES
LUBRICATED
23
BY A BUBBLY
OIL
K. TQNDER Znstitutt for Maskindeler, Norwegian Institute of Technology, 7034 Trondheim (Norway)
University of Trondheim,
(Received January 22, 1975)
Summary A theory of the effect of gas bubbles upon parallel-surfaced hydrodynamic bearings is presented. The analysis applies to “infinitely” wide isothermal bearings, but conclusions are also drawn on the behaviour of finite bearings. It is shown that no self-sustained pressure is established, but that a pressure profile generated otherwise can be strongly modified to yield a near doubling of the load capacity and unaltered friction; and for a fixed load, lowered friction and increased mass flow through the bearing. Notation integration constant integration constant pressure function integral J-9J’1.4 film height H integration constant K bearing length in direction of motion L pressure; pressure in kp/cm2 p, PK flow intensity Q u relative speed of bearing surfaces coordinate in direction of motion X Y coordinate across direction of motion load parameter, 6~ UL/H2, in kp/cm2 ; non-dimensional film height non-dimensional pressure function; p at inlet PI PO non-dimensional flow intensities Q*Qm, Qv non-dimensional load W non-dimensional X-coordinate X non-dimensional Y-coordinate Y pressure drop over bearing, in kp/cm2 APQ Aqm,Aqu relative flow functions Aw relative load gain function
A B
24
volume ratio, gas/pure lubricant at atmospheric exponent in pressure, density relation viscosity non-dimensional density density of gas/lubricant suspension density of pure lubricant; gas, respectively arbitrary reference density shear stress
pressure
1. Introduction Although air or gas bubbles are often present in lubricating oils, little seems to have been published on the effect of such bubbles on the performance of hydrodynamic bearings. A prevailing assumption in this field seems to be that the presence of gas bubbles is undesirable. A recent theory of the effect of bubbly lubricants on hydrodynamic bearings [l] predicted an improvement of the performance for many types of common bearing designs. It was shown that the effect could be explained on the basis of density wedges near the leading and trailing edges of the bearings due to the pressure-dependency of the density of bubbly lubricants. It was further indicated that preliminary investigations showed that the presence of bubbles in an oil would not create a self-sustained load capacity in parallel-surfaced bearings. However, being based on numerical treatment of the problem the theory [l] did not give conclusive answers to this problem; the inability of the method to produce positive pressure profiles could be due to numerical shortcomings of the algorithms employed. In the case of parallel surfaces the Reynolds’ equation becomes particularly simple lending itself to reasonably easy mathematical treatment. The present paper is an account of such an analytical approach. 2. Theoretical 2.1. Hydrodynamic bearings Reynolds’ equation is assumed to hold. It reads, in Cartesian coordinates, in the steady state,
(1) P is the pressure,
pd the density, 77the viscosity and H the thickness of the oil film at any point (X, Y) and U the relative speed of the bearing surfaces. The bubbles are assumed to be (1) only slowly soluble, (2) evenly dispersed throughout the lubricant, and (3) small compared with the film height. It was shown [l] that the effect of viscosity changes due to bubble inclusion was very small. Hence, ignoring temperature changes, and since H is assumed constant, eqn. (1) may be transformed into
25
Further, putting P = cp + 1, X = L x, Y = L y and c = 6q KG/H2 the equation will read
(2) withp
(boundary) = 0. With c in kpfcm2 this clearly corresponds to atmospheric pressure at the bearing edges. Also it was shown [l] that if the mass ratio of gas to oil is constant and gross temperature changes as well as gas solubility in the lubricant are ignored, the density of the lubricant is given by 1 Pd
=Pr
1+/3&l’?
pr is constant, & is the bearing pressure in kp/cm2, y is the exponent in the pressure-density relationship and /I is the volume ratio of gas to pure lubricant at atmospheric pressure. Hence one may write 1 1 +p(cp + 1)-l’? 1
a
=-
ax:
1
[1
+p(cp + 1)-l’?
+_!__ ap. ay [ ay
l 1 + P(cp + 1)-l’?
1
1
Though containing no other terms than p and easily solved analytically in both x and y. The considering only “infinitely wide” bearings is This is equivalent to ignoring the y-term,
constants, this equation is not further simplification of therefore introduced. and eqn. (2) is reduced to
1 + p(1 + cpj-I’? Integrating ap 3X
Inserting ap
once yields =
1 - 5, where A is an integration
for p and re~~~ng~
constant.
one obtains
_ fl+~p)~‘~(l--A)--A/3
ax-
(3
(1 + cp)“Y P
K+
P
s
(1 + cp)l’Y (1 + c~)‘/~(l-
where K is a new integration
A) -A@ constant,
dp=jdx 0 whence
26
A/3
K+P+---1 -A
‘* s
~(1 -A)2
(l+‘y-
dcp A/3/(1 -A)
=X
or equivalently x=K+-
P
AP
~-A+c(~-AA)~
F(cP)
Note that the integral F is a function of p and constants only. It will be observed that to a given value of p corresponds a single value of x. This is clearly the case whatever the value of y since F does not contain x, the x-dependency of eqn. (4) thus being strictly linear. It therefore follows that infinitely wide hydrodynamic bearings of parallel surfaces cannot carry a load when lubricated by a bubbly oil in the absence of gross temperature changes. This negative result applies to infinitely wide bearings but will be assumed to hold for finite bearings also. Thus the presence of (uniformly dispersed) bubbles in an oil does not generate a load carrying capacity, it can only modify one produced otherwise. 2.2. Externally pressurized bearings Just as gas bubbles in a lubricant can increase the load capacity of normal hydrodynamic bearings, it may inflate that of externally pressurized bearings of parallel sliding surfaces. Assume therefore that at the leading edge the pressure function takes the value po. The pressure distribution is then evaluated by solving eqn. (4) subject to the following boundary conditions, p = p. at x = 0, and p = 0 at x = 1. This will be brought about by a proper choice of the integration constants A and K. The exponent y should be 1 or 1.4 for purely isothermal or adiabatic conditions respectively. The latter value may seem more likely under the present circumstances. If on the other hand the value 1 is chosen for y, the function F is readily evaluated and is given by F = In [l -A(1
+ 0) + cp(1 -A)]
(5)
Because of this fact the following results will be based upon y = 1. In principle, however, other values of y may be dealt with by evaluating F numerically. This appears to be somewhat unnecessary as the primary object of the present investigation is to determine the general features of the effect of bubbles upon parallel-surfaced bearings. Finally, the p,x relationship will be determined for various combinations of the constants c and 0. 2.2.1. Solving the pressure-equation Normally one would prefer to express p as a function of x. However, in the present case it is quite evident that takingp’as the “free” variable is much more practical, as the former mode of presentation would involve either the inherent inaccuracies in an interpolation, or the time-consuming work involved in solving eqn. (4) for a whole series of x-values. The integration constants A and K are determined as follows. K is found by inserting p = 0 and x = 1 in eqn. (4). If y = 1, this produces
(c(lAPA)2)ln[1--A(1+P)1
K=land
x=l+P
1 -A
+ -.._pA_ ~(1 -A)2
ln
1 + PC (1 -A) 1 -A(1 +fl)
(6)
1
If y = 1.4, F is to be evaluated numerically, and F may be defined to be 0 at x = 1, from which one obtains K = 1, and P
xI:l+-
PA
Fra @P)
~-A+c(~-A)~
(7)
where the index 1.4 refers to the value of the exponent y. Finding A consists of solving the equations PA
1+po+ 1-A or
ln c
~$1 - A)2 AP
1+po+ 1-A
~(1 -A)2
_I.+POCU -A)
FL4000
l-A(l+@)
1
=0
= 0
which was performed on a trial-and-error basis. The non-Dimensions load capacity, W, is found by in~~atingp
over x
Hence w is found by integrating x over p. This is done numerically by means of Simpson’s Rule. Further, the flow intensity, Q, may be found by inserting A in eqn. (3), thus Q$!!-gz
for volume flow, whence the mass flow intensity P~Q may be obtained. More conveniently non-dimensional quantities qv and qm may be employed, Q = UH qv.
( ) 1 - 2
Qv =
12 for volume flow.
Inserting eqn. (3) one obtains A q!J
=2
1 I+ (1
+&J
=:(I
+p&)
Similarly
for mass flow where pI is the density of the pure lubricant. Here pr is given
28 PO
-.
-.
P
‘.
.
P
c
=
R
= “.I
c
200
I
x
0
0
=
500
=
0.1
x
Fig. 1. Pressure profile. Fig. 2. Pressure profile.
;
x
0
Fig. 3. Pressure profile.
Fig. 4. Load gain as function of pressure drop. Fig. 5. Load gain against load factor.
by p1[ 1 + (p,/pl)/3] at atmospheric pressure which is very nearly pz, the term (p,/p I)p vanishing. (It has already been neglected in the expression for p .) Hence one obtains, after inserting for qu, qm = A/2. Thus one may put, A Qm =pp”
=qm
( > 1+p
PZ’
,q=(l+p,)/2
(9)
Here q is the non-bubbly flow intensity, applying to mass and volume flow alike.
29
3. Results The results of a series of computations along the lines indicated above are shown in Figs. 1 to 3, depictingp uersus X. The characteristic linear pressure profile is seen to have been drastically modified for reasonable values of the constants. Figure 4 shows the relative load capacity gain, A w, as a function of the pressure drop APe for a given value of 0 and three values of c. A w, defined as the load capacity difference divided by the net non-bubbly load capacity, is seen to diminish with increasing leading edge pressure. Similarly A w is plotted against c for three values of APO in Fig. 5. The relative load gain increases with c, rapidly for !ow values; less so for higher values. For moderate initial pressures the load gain has obtained a considerable value before this flattening occurs. Figure 6 represe’nts the percentual load gain as a function of the gas content, expressed in terms of the volume ratio at atmospheric pressure, 0. A w is seen to grow with an increasing P-value. This curve, being reasonably steep for low /3’s, also flattens out as /I increases.
”
0.1
0.2
0.3
Fig. 6. Load gain against gas content.
Similarly the effect on the flow functions is represented by the percentual deviations Aq of the bubbly flow-functions relative to the nonbubbly one, i.e. Aq, = (q,,, - q)/q, Aqv = (q, - q)/q. Aqu refers to the trailing edge of the bearing, in which case q, = qm (1 + 0). Figure 7 depicts Aq as a function of c. After an initial steep rise the Aq-curve is seen to change very little with increasing load-factor c. The general shape is similar for all three values of AP,. Figure 8 shows the curves of Aq uersus AP,-, for the c-values of 200, 1,000 and 5,000. Again the curves are rather steep near the origin and become gradually less so as APe increases. Finally, Fig. 9 depicts Aq as a function of the bubble-content, for c = 1,000 and AP, = 5. The curves are seen to be almost linear over the presented range.
Fig. 7. Flow
gain against load factor.
Fig. 8. Flow
gain against pressure
-1,1
drop.
I.
I,
0
Fig. 9. Flow
I
“.i
0.3
gain against gas content.
4. Discussion 4.1. Assumptions The above analysis contains several assumptions. Firstly Reynolds’ equation is taken to hold. However, the validity of that equation is so well established that this point will receive no further comment. The analysis also assumes that the gas bubbles are evenly distributed throughout the oil, i.e. the mass content of gas in a small volume does not vary, and the bubbles are small compared with the film height. If the buboles are very small, their tendency to collect at specific regions is indeed very small, so the assumption is primarily one of bubble size. As for the question of solubility, it appears probable that the short transit time of the bubbly liquid through the pressure zone of a normal bearing prevents any important solubility effects. On the other hand the expansion and contraction of the bubbles due to pressure variations should not be seriously affected by the times involved. Gross temperature changes have been ignored. This is clearly not in
31
accordance with the normal situation in real bearings. However, it should be possible to heat or cool the bearing externally to remove such changes. At any rate the assumption should be looked upon as a mathematical simplification. Further the simplification of ignoring the viscosity increase due to the “particle” inclusion represented by the bubbles, was shown [l] to be reasonable. The types of bearings described and that treated in the present paper are basically similar so the same order of magnitude of such corrections (order of percent) is to be expected. The choice of value of y was made primarily because of its inherent mathematical advantages. The whole range of values between 1 and 1.4 may find support: it is felt, however, that this choice will primarily affect the magnitude of the results rather than the general pattern itself. This assumption was supported by results obtained [ 11. Finally, one of the conclusions reached in this work is the inability of parallel-surfaced bubbly bearings to create a self-sustained positive pressure. This was, in the general case, based upon an assumption, its validity having been demonstrated for “infinitely” wide bearings only. Physically this is a reasonable proposal. It is difficult to envisage that a pressure build-up can take place in the case of a finite bearing in which the pressure will drop more readily than in the “infinite” bearing, the fluid being allowed to flow over the sides. It is also supported by findings [1] ; since under the present circumstances it seems reasonable that the algorithm used in that investigation did produce correct results. 4.2. Findings The results given above can be explained as follows. The basis of the load capacity gain is the effective wedge being created in the bearing in which the pressure drops from 1 + AP,-, to 1, causing the density to decrease correspondingly. This will produce an extra lift and, because of the increased pressure level, a steeper negative pressure gradient at the outlet, which in turn means a greater volume flow. On the other hand, an inflated pressure level must be due, physically, to the tendency of a given bearing configuration to reduce the mass flow. Thus one may conclude, bubbles mean positive AqU and Aw and a negative Aqm. However, Aq, and Aq, are connected through q, = qm (1 + fl), i.e. qv - q, = q,fl and qu -q (qm - q) = qmP whence Aqu - Aq, = (q, /a)/.3 = p since qm /q is of the order of unity. Thus one may put
Aa, - Aq, x/3, Aq,
and
q,
>--‘-
1+P
(10)
The q,-curve of Fig. 9 is seen to be very nearly linear over the range shown. Since the flow quantities do not change much by the inclusion of bubbles, A is reasonably constant, hence, considering eqn. (4) in which x and p are linearly related, the above relationship is explained. The behaviour of qm follows immediately from relation (10). An inspection of eqn. (3) indicates that p. and c play reasonably
32
similar roles in the generation of the pressure gradient, and hence of the volume flow. This explains why the flows increase with c, p. and hence APO, while the asymptotic character stems from relation (10). Since the pressure must necessarily drop from the inlet to the outlet, the limit of the load capacity gain is 100%. Hence the flattening of the A wcurves of Figs. 4 and 6 is explained. The pressure inflation is related to the diminishing of the mass flow, or A. Equation (8) shows that c and p and thereby APO have rather opposite functions in the determination of A. This explains why the A w-curve shows a somewhat opposite behaviour on Figs. 4 and 5. Physically speaking one may think of c as the “dynamic” part of the load gain. The static case was investigated by writing P = cop + 1, where co is arbitrary and definesp. Further a/ax (p ap/ax) = 0, whence aptax = B/p where B is an integration constant. Developing this along the lines of the dynamic case, one obtains cop-/3ln
(1+%)
=(x-l)
[/3ln(1+p)--cop01
This equation produces positive but very small load gains. Hence the load capacity gain is primarily a dynamically generated quantity and will thus increase with c, and hence A w must diminish with APO, 4.3. Consequences The consequences of the above are the following. The load capacity, for a given c-value i.e. for given values of q, U and H of a bearing, is increased by the inclusion of gas bubbles. For finite bearings this has not been demonstrated to be the case in this investigation. However, by comparison with results [l] such a gain must always take place. This can be stated since ref. 1 deals with several bearings having flat trailing lands, under which circumstances the inlet part of the bearing may be looked upon as a generator of the pressure at the leading edge of the flat land. Secondly the friction is unaltered. This again may be stated [l] , or shown directly by integrating the shear stress r= Uq/H + (H/2) aP/aX. The frictional force per unit width, will be given by L
s 0
1 7dX=
=T
dx 2 3 .Iap 0 -ax [1?3[dp]
=T
dx I
(1+3Po)
This is general and applies to the bubbly and non-bubbly cases alike. But p. is not a function of the bubble content, so the friction is completely unaltered. Thirdly the volume flow is increased but the more important mass flow is somewhat diminished, again for a given geometry. However, the percent-
33
ual gain in load capacity is much greater than the corresponding lowering of the mass flow. Therefore the bearing will run with a thicker lubricant film for given values of initial pressure, sliding speed and viscosity, for a given load. This results in a gain in lubricant mass flow and hence better cooling; a reduction of the friction and friction coefficient and hence heat generation; and finally reduced chances of metallic contact in the case of imperfect surfaces. 5. Conclusions The effect of bubbles on the performance of parallel-surfaced isothermal bearings has been analysed. The following has been found. (1) The presence of gas bubbles will not generate positive pressures and thereby a load carrying ability, but will modify and inflate the pressure profile produced by other means. This has been proved to hold for “infinitely” wide bearings while strong arguments for its general validity have been produced. (2) Externally pressurized bearings will, for a given geometry, have their load capacity increased when bubbles are entrapped in the lubricant. The load gain increases with bubble content if the latter is reasonable. It also increases with the load factor, c, but decreases with the pressure drop. It follows that the greatest load gain is obtained for moderate load levels, and may attain values corresponding to a near doubling for realistic parameter values in the case of wide bearings. (3) Bearing friction is unaltered, compared with the non-bubbly case, and again for a given geometry. Accordingly the coefficient of friction is greatly reduced. (4) The lubricant volume flow is increased while the mass flow is reduced. (5) For a given load, the presence of bubbles will make the bearing run on a thicker oil film, resulting in a reduced frictional force and coefficient of friction and in an increased mass flow and therefore improved cooling. Considering the above findings the use of flat bubbly bearings appears to be recommendable. Inherent advantages are the reduced chances of metallic contact under running conditions, and low intermetallic pressure when the bearing is out of operation. The disadvantage is the necessity of producing an initial pressure. This may be obtained hydrodynamically by a leading step or wedge, though it may not be possible to produce optimal conditions in this manner. Extemal pressurization of the inlet may readily ensure such conditions. Finally, compared with a purely hydrostatic operation the described bubbly bearing offers at least the advantages of moderate specific loads during periods of shut-down.
34
References 1 K. Tdnder, Effect on bearing performance of a bubbly JSLE - ASLE Int. Lubric. Conf., Tokyo, June 1975.
lubricant,
to be published,
PARALLEL
SURFACES
LUBRICATED
23
BY A BUBBLY
OIL
K. TQNDER Znstitutt for Maskindeler, Norwegian Institute of Technology, 7034 Trondheim (Norway)
University of Trondheim,
(Received January 22, 1975)
Summary A theory of the effect of gas bubbles upon parallel-surfaced hydrodynamic bearings is presented. The analysis applies to “infinitely” wide isothermal bearings, but conclusions are also drawn on the behaviour of finite bearings. It is shown that no self-sustained pressure is established, but that a pressure profile generated otherwise can be strongly modified to yield a near doubling of the load capacity and unaltered friction; and for a fixed load, lowered friction and increased mass flow through the bearing. Notation integration constant integration constant pressure function integral J-9J’1.4 film height H integration constant K bearing length in direction of motion L pressure; pressure in kp/cm2 p, PK flow intensity Q u relative speed of bearing surfaces coordinate in direction of motion X Y coordinate across direction of motion load parameter, 6~ UL/H2, in kp/cm2 ; non-dimensional film height non-dimensional pressure function; p at inlet PI PO non-dimensional flow intensities Q*Qm, Qv non-dimensional load W non-dimensional X-coordinate X non-dimensional Y-coordinate Y pressure drop over bearing, in kp/cm2 APQ Aqm,Aqu relative flow functions Aw relative load gain function
A B
24
volume ratio, gas/pure lubricant at atmospheric exponent in pressure, density relation viscosity non-dimensional density density of gas/lubricant suspension density of pure lubricant; gas, respectively arbitrary reference density shear stress
pressure
1. Introduction Although air or gas bubbles are often present in lubricating oils, little seems to have been published on the effect of such bubbles on the performance of hydrodynamic bearings. A prevailing assumption in this field seems to be that the presence of gas bubbles is undesirable. A recent theory of the effect of bubbly lubricants on hydrodynamic bearings [l] predicted an improvement of the performance for many types of common bearing designs. It was shown that the effect could be explained on the basis of density wedges near the leading and trailing edges of the bearings due to the pressure-dependency of the density of bubbly lubricants. It was further indicated that preliminary investigations showed that the presence of bubbles in an oil would not create a self-sustained load capacity in parallel-surfaced bearings. However, being based on numerical treatment of the problem the theory [l] did not give conclusive answers to this problem; the inability of the method to produce positive pressure profiles could be due to numerical shortcomings of the algorithms employed. In the case of parallel surfaces the Reynolds’ equation becomes particularly simple lending itself to reasonably easy mathematical treatment. The present paper is an account of such an analytical approach. 2. Theoretical 2.1. Hydrodynamic bearings Reynolds’ equation is assumed to hold. It reads, in Cartesian coordinates, in the steady state,
(1) P is the pressure,
pd the density, 77the viscosity and H the thickness of the oil film at any point (X, Y) and U the relative speed of the bearing surfaces. The bubbles are assumed to be (1) only slowly soluble, (2) evenly dispersed throughout the lubricant, and (3) small compared with the film height. It was shown [l] that the effect of viscosity changes due to bubble inclusion was very small. Hence, ignoring temperature changes, and since H is assumed constant, eqn. (1) may be transformed into
25
Further, putting P = cp + 1, X = L x, Y = L y and c = 6q KG/H2 the equation will read
(2) withp
(boundary) = 0. With c in kpfcm2 this clearly corresponds to atmospheric pressure at the bearing edges. Also it was shown [l] that if the mass ratio of gas to oil is constant and gross temperature changes as well as gas solubility in the lubricant are ignored, the density of the lubricant is given by 1 Pd
=Pr
1+/3&l’?
pr is constant, & is the bearing pressure in kp/cm2, y is the exponent in the pressure-density relationship and /I is the volume ratio of gas to pure lubricant at atmospheric pressure. Hence one may write 1 1 +p(cp + 1)-l’? 1
a
=-
ax:
1
[1
+p(cp + 1)-l’?
+_!__ ap. ay [ ay
l 1 + P(cp + 1)-l’?
1
1
Though containing no other terms than p and easily solved analytically in both x and y. The considering only “infinitely wide” bearings is This is equivalent to ignoring the y-term,
constants, this equation is not further simplification of therefore introduced. and eqn. (2) is reduced to
1 + p(1 + cpj-I’? Integrating ap 3X
Inserting ap
once yields =
1 - 5, where A is an integration
for p and re~~~ng~
constant.
one obtains
_ fl+~p)~‘~(l--A)--A/3
ax-
(3
(1 + cp)“Y P
K+
P
s
(1 + cp)l’Y (1 + c~)‘/~(l-
where K is a new integration
A) -A@ constant,
dp=jdx 0 whence
26
A/3
K+P+---1 -A
‘* s
~(1 -A)2
(l+‘y-
dcp A/3/(1 -A)
=X
or equivalently x=K+-
P
AP
~-A+c(~-AA)~
F(cP)
Note that the integral F is a function of p and constants only. It will be observed that to a given value of p corresponds a single value of x. This is clearly the case whatever the value of y since F does not contain x, the x-dependency of eqn. (4) thus being strictly linear. It therefore follows that infinitely wide hydrodynamic bearings of parallel surfaces cannot carry a load when lubricated by a bubbly oil in the absence of gross temperature changes. This negative result applies to infinitely wide bearings but will be assumed to hold for finite bearings also. Thus the presence of (uniformly dispersed) bubbles in an oil does not generate a load carrying capacity, it can only modify one produced otherwise. 2.2. Externally pressurized bearings Just as gas bubbles in a lubricant can increase the load capacity of normal hydrodynamic bearings, it may inflate that of externally pressurized bearings of parallel sliding surfaces. Assume therefore that at the leading edge the pressure function takes the value po. The pressure distribution is then evaluated by solving eqn. (4) subject to the following boundary conditions, p = p. at x = 0, and p = 0 at x = 1. This will be brought about by a proper choice of the integration constants A and K. The exponent y should be 1 or 1.4 for purely isothermal or adiabatic conditions respectively. The latter value may seem more likely under the present circumstances. If on the other hand the value 1 is chosen for y, the function F is readily evaluated and is given by F = In [l -A(1
+ 0) + cp(1 -A)]
(5)
Because of this fact the following results will be based upon y = 1. In principle, however, other values of y may be dealt with by evaluating F numerically. This appears to be somewhat unnecessary as the primary object of the present investigation is to determine the general features of the effect of bubbles upon parallel-surfaced bearings. Finally, the p,x relationship will be determined for various combinations of the constants c and 0. 2.2.1. Solving the pressure-equation Normally one would prefer to express p as a function of x. However, in the present case it is quite evident that takingp’as the “free” variable is much more practical, as the former mode of presentation would involve either the inherent inaccuracies in an interpolation, or the time-consuming work involved in solving eqn. (4) for a whole series of x-values. The integration constants A and K are determined as follows. K is found by inserting p = 0 and x = 1 in eqn. (4). If y = 1, this produces
(c(lAPA)2)ln[1--A(1+P)1
K=land
x=l+P
1 -A
+ -.._pA_ ~(1 -A)2
ln
1 + PC (1 -A) 1 -A(1 +fl)
(6)
1
If y = 1.4, F is to be evaluated numerically, and F may be defined to be 0 at x = 1, from which one obtains K = 1, and P
xI:l+-
PA
Fra @P)
~-A+c(~-A)~
(7)
where the index 1.4 refers to the value of the exponent y. Finding A consists of solving the equations PA
1+po+ 1-A or
ln c
~$1 - A)2 AP
1+po+ 1-A
~(1 -A)2
_I.+POCU -A)
FL4000
l-A(l+@)
1
=0
= 0
which was performed on a trial-and-error basis. The non-Dimensions load capacity, W, is found by in~~atingp
over x
Hence w is found by integrating x over p. This is done numerically by means of Simpson’s Rule. Further, the flow intensity, Q, may be found by inserting A in eqn. (3), thus Q$!!-gz
for volume flow, whence the mass flow intensity P~Q may be obtained. More conveniently non-dimensional quantities qv and qm may be employed, Q = UH qv.
( ) 1 - 2
Qv =
12 for volume flow.
Inserting eqn. (3) one obtains A q!J
=2
1 I+ (1
+&J
=:(I
+p&)
Similarly
for mass flow where pI is the density of the pure lubricant. Here pr is given
28 PO
-.
-.
P
‘.
.
P
c
=
R
= “.I
c
200
I
x
0
0
=
500
=
0.1
x
Fig. 1. Pressure profile. Fig. 2. Pressure profile.
;
x
0
Fig. 3. Pressure profile.
Fig. 4. Load gain as function of pressure drop. Fig. 5. Load gain against load factor.
by p1[ 1 + (p,/pl)/3] at atmospheric pressure which is very nearly pz, the term (p,/p I)p vanishing. (It has already been neglected in the expression for p .) Hence one obtains, after inserting for qu, qm = A/2. Thus one may put, A Qm =pp”
=qm
( > 1+p
PZ’
,q=(l+p,)/2
(9)
Here q is the non-bubbly flow intensity, applying to mass and volume flow alike.
29
3. Results The results of a series of computations along the lines indicated above are shown in Figs. 1 to 3, depictingp uersus X. The characteristic linear pressure profile is seen to have been drastically modified for reasonable values of the constants. Figure 4 shows the relative load capacity gain, A w, as a function of the pressure drop APe for a given value of 0 and three values of c. A w, defined as the load capacity difference divided by the net non-bubbly load capacity, is seen to diminish with increasing leading edge pressure. Similarly A w is plotted against c for three values of APO in Fig. 5. The relative load gain increases with c, rapidly for !ow values; less so for higher values. For moderate initial pressures the load gain has obtained a considerable value before this flattening occurs. Figure 6 represe’nts the percentual load gain as a function of the gas content, expressed in terms of the volume ratio at atmospheric pressure, 0. A w is seen to grow with an increasing P-value. This curve, being reasonably steep for low /3’s, also flattens out as /I increases.
”
0.1
0.2
0.3
Fig. 6. Load gain against gas content.
Similarly the effect on the flow functions is represented by the percentual deviations Aq of the bubbly flow-functions relative to the nonbubbly one, i.e. Aq, = (q,,, - q)/q, Aqv = (q, - q)/q. Aqu refers to the trailing edge of the bearing, in which case q, = qm (1 + 0). Figure 7 depicts Aq as a function of c. After an initial steep rise the Aq-curve is seen to change very little with increasing load-factor c. The general shape is similar for all three values of AP,. Figure 8 shows the curves of Aq uersus AP,-, for the c-values of 200, 1,000 and 5,000. Again the curves are rather steep near the origin and become gradually less so as APe increases. Finally, Fig. 9 depicts Aq as a function of the bubble-content, for c = 1,000 and AP, = 5. The curves are seen to be almost linear over the presented range.
Fig. 7. Flow
gain against load factor.
Fig. 8. Flow
gain against pressure
-1,1
drop.
I.
I,
0
Fig. 9. Flow
I
“.i
0.3
gain against gas content.
4. Discussion 4.1. Assumptions The above analysis contains several assumptions. Firstly Reynolds’ equation is taken to hold. However, the validity of that equation is so well established that this point will receive no further comment. The analysis also assumes that the gas bubbles are evenly distributed throughout the oil, i.e. the mass content of gas in a small volume does not vary, and the bubbles are small compared with the film height. If the buboles are very small, their tendency to collect at specific regions is indeed very small, so the assumption is primarily one of bubble size. As for the question of solubility, it appears probable that the short transit time of the bubbly liquid through the pressure zone of a normal bearing prevents any important solubility effects. On the other hand the expansion and contraction of the bubbles due to pressure variations should not be seriously affected by the times involved. Gross temperature changes have been ignored. This is clearly not in
31
accordance with the normal situation in real bearings. However, it should be possible to heat or cool the bearing externally to remove such changes. At any rate the assumption should be looked upon as a mathematical simplification. Further the simplification of ignoring the viscosity increase due to the “particle” inclusion represented by the bubbles, was shown [l] to be reasonable. The types of bearings described and that treated in the present paper are basically similar so the same order of magnitude of such corrections (order of percent) is to be expected. The choice of value of y was made primarily because of its inherent mathematical advantages. The whole range of values between 1 and 1.4 may find support: it is felt, however, that this choice will primarily affect the magnitude of the results rather than the general pattern itself. This assumption was supported by results obtained [ 11. Finally, one of the conclusions reached in this work is the inability of parallel-surfaced bubbly bearings to create a self-sustained positive pressure. This was, in the general case, based upon an assumption, its validity having been demonstrated for “infinitely” wide bearings only. Physically this is a reasonable proposal. It is difficult to envisage that a pressure build-up can take place in the case of a finite bearing in which the pressure will drop more readily than in the “infinite” bearing, the fluid being allowed to flow over the sides. It is also supported by findings [1] ; since under the present circumstances it seems reasonable that the algorithm used in that investigation did produce correct results. 4.2. Findings The results given above can be explained as follows. The basis of the load capacity gain is the effective wedge being created in the bearing in which the pressure drops from 1 + AP,-, to 1, causing the density to decrease correspondingly. This will produce an extra lift and, because of the increased pressure level, a steeper negative pressure gradient at the outlet, which in turn means a greater volume flow. On the other hand, an inflated pressure level must be due, physically, to the tendency of a given bearing configuration to reduce the mass flow. Thus one may conclude, bubbles mean positive AqU and Aw and a negative Aqm. However, Aq, and Aq, are connected through q, = qm (1 + fl), i.e. qv - q, = q,fl and qu -q (qm - q) = qmP whence Aqu - Aq, = (q, /a)/.3 = p since qm /q is of the order of unity. Thus one may put
Aa, - Aq, x/3, Aq,
and
q,
>--‘-
1+P
(10)
The q,-curve of Fig. 9 is seen to be very nearly linear over the range shown. Since the flow quantities do not change much by the inclusion of bubbles, A is reasonably constant, hence, considering eqn. (4) in which x and p are linearly related, the above relationship is explained. The behaviour of qm follows immediately from relation (10). An inspection of eqn. (3) indicates that p. and c play reasonably
32
similar roles in the generation of the pressure gradient, and hence of the volume flow. This explains why the flows increase with c, p. and hence APO, while the asymptotic character stems from relation (10). Since the pressure must necessarily drop from the inlet to the outlet, the limit of the load capacity gain is 100%. Hence the flattening of the A wcurves of Figs. 4 and 6 is explained. The pressure inflation is related to the diminishing of the mass flow, or A. Equation (8) shows that c and p and thereby APO have rather opposite functions in the determination of A. This explains why the A w-curve shows a somewhat opposite behaviour on Figs. 4 and 5. Physically speaking one may think of c as the “dynamic” part of the load gain. The static case was investigated by writing P = cop + 1, where co is arbitrary and definesp. Further a/ax (p ap/ax) = 0, whence aptax = B/p where B is an integration constant. Developing this along the lines of the dynamic case, one obtains cop-/3ln
(1+%)
=(x-l)
[/3ln(1+p)--cop01
This equation produces positive but very small load gains. Hence the load capacity gain is primarily a dynamically generated quantity and will thus increase with c, and hence A w must diminish with APO, 4.3. Consequences The consequences of the above are the following. The load capacity, for a given c-value i.e. for given values of q, U and H of a bearing, is increased by the inclusion of gas bubbles. For finite bearings this has not been demonstrated to be the case in this investigation. However, by comparison with results [l] such a gain must always take place. This can be stated since ref. 1 deals with several bearings having flat trailing lands, under which circumstances the inlet part of the bearing may be looked upon as a generator of the pressure at the leading edge of the flat land. Secondly the friction is unaltered. This again may be stated [l] , or shown directly by integrating the shear stress r= Uq/H + (H/2) aP/aX. The frictional force per unit width, will be given by L
s 0
1 7dX=
=T
dx 2 3 .Iap 0 -ax [1?3[dp]
=T
dx I
(1+3Po)
This is general and applies to the bubbly and non-bubbly cases alike. But p. is not a function of the bubble content, so the friction is completely unaltered. Thirdly the volume flow is increased but the more important mass flow is somewhat diminished, again for a given geometry. However, the percent-
33
ual gain in load capacity is much greater than the corresponding lowering of the mass flow. Therefore the bearing will run with a thicker lubricant film for given values of initial pressure, sliding speed and viscosity, for a given load. This results in a gain in lubricant mass flow and hence better cooling; a reduction of the friction and friction coefficient and hence heat generation; and finally reduced chances of metallic contact in the case of imperfect surfaces. 5. Conclusions The effect of bubbles on the performance of parallel-surfaced isothermal bearings has been analysed. The following has been found. (1) The presence of gas bubbles will not generate positive pressures and thereby a load carrying ability, but will modify and inflate the pressure profile produced by other means. This has been proved to hold for “infinitely” wide bearings while strong arguments for its general validity have been produced. (2) Externally pressurized bearings will, for a given geometry, have their load capacity increased when bubbles are entrapped in the lubricant. The load gain increases with bubble content if the latter is reasonable. It also increases with the load factor, c, but decreases with the pressure drop. It follows that the greatest load gain is obtained for moderate load levels, and may attain values corresponding to a near doubling for realistic parameter values in the case of wide bearings. (3) Bearing friction is unaltered, compared with the non-bubbly case, and again for a given geometry. Accordingly the coefficient of friction is greatly reduced. (4) The lubricant volume flow is increased while the mass flow is reduced. (5) For a given load, the presence of bubbles will make the bearing run on a thicker oil film, resulting in a reduced frictional force and coefficient of friction and in an increased mass flow and therefore improved cooling. Considering the above findings the use of flat bubbly bearings appears to be recommendable. Inherent advantages are the reduced chances of metallic contact under running conditions, and low intermetallic pressure when the bearing is out of operation. The disadvantage is the necessity of producing an initial pressure. This may be obtained hydrodynamically by a leading step or wedge, though it may not be possible to produce optimal conditions in this manner. Extemal pressurization of the inlet may readily ensure such conditions. Finally, compared with a purely hydrostatic operation the described bubbly bearing offers at least the advantages of moderate specific loads during periods of shut-down.
34
References 1 K. Tdnder, Effect on bearing performance of a bubbly JSLE - ASLE Int. Lubric. Conf., Tokyo, June 1975.
lubricant,
to be published,