INCORPORATING THERMAL EXPANSION ACROSS THE FILM Viscosity is generally considered to be the single most important property of lubricants, therefore, it represents the central parameter
Trang 1line XX at F1 A horizontal line FIGl is then drawn to meet the line
P* = 500 at G 1 The point H I vertically above G1 gives the first approxima-
tion for S* to be 0.27
This value of S* is then entered as point I in the second nomogram (Fig
6.46b) Draw a vertical line IJ to meet the curve LID = 1 Then the hor- izontal line JK meets RCL/k = 2.88 line at K Point L is vertically below point K and on the line M / ( T + t9)* = 2.15 The horizontal line LMN is such that point N is vertically below point I and point M is on the eve axis Point M provides the new approximation for P* = 495 psi and point N gives
AT* 2 110"F(for LID = 1)
With AT = 1 lO"F, the new entry point A2 in the first nomogram (Fig
6.46a) is determined and the procedure is continued
After six iterations, S* = 0.16 can be accepted as the solution for the
considered example
INCORPORATING THERMAL EXPANSION ACROSS THE FILM
Viscosity is generally considered to be the single most important property of lubricants, therefore, it represents the central parameter in all lubricant analysis By far the easiest approach to the question of viscosity variation within a fluid film bearing is to adopt a representative or mean value visc- osity Examples of studies which have provided many suggestions for cal- culations of the effective viscosity in a bearing analysis are presented By Cameron [34] and Szeri [35] When the temperature rise of the lubricant across the bearing is small, bearing performance calculations are customa- rily based on the classical, isoviscous theory In other cases, where the temperature rise across the bearing is significant, the classical theory loses its usefulness for performance prediction One of the early applications of the energy equation to hydrodynamic lubrication was made by Cope [36] in
1948 His model was based on the assumptions of negligible temperature variation across the film and negligible heat conduction within the lubrica- tion film as well as into the adjacent solids The consequence of the second assumption is that both the bearing and the shaft are isothermal compo-
nents, and thus, all the generated heat is carried out by the lubricant As
indicated in a review paper by Szeri [37]: the belief, that the classical theory
on one hand, and Cope's adiabatic model on the other hand, bracket bear- ing performance in lubrication analysis, was widely accepted for a while
Trang 2Design of Fluid Film Bearings 23 I
In 1987, Pinkus [38] in his historical overview of the theory of hydro-
dynamic lubrication pointed out that one of the least understood and urgent areas of research is that of thermohydrodynamics In the discussion of parallel surfaces and mixed lubrication, he indicated that the successful operation of centrally pivoted thrust bearings cannot be explained by the hydrodynamic theory He backs his assertion by reviewing several failed attempts to explain the pressure developed for bearings with constant film thickness
Braun et al [39-41] investigated different aspects of the thermal effects in
the lubricant film and the thennohydrodynamic phenomena in a variety of
film situations and bearing configurations Dowson et al [42,43] adopted the
cavitation algorithm proposed by Elord and Adams and studied the lubri- cant film rupture and reformation effects on grooved bearing performance for a wide range of operating parameters The comparison of their theore- tical results and those presented in a design document revealed that a con- sideration of more realistic flow conditions will not normally influence the value of predicted load capacity significantly However, the prediction of the
side leakage flow rate will be greatly affected if film reformation is not
included in the analysis Braun et al [44] also experimentally investigated the cavitation effects on bearing performance in an eccentric journal bearing
Lebeck [45, 461 summarized several well-documen ted experiments of
parallel sliding or parallel surfaces The experiments clearly show that as speed is increased, the bearing surfaces are lifted such that asperity contact and friction are reduced He also suggested that the thermal density wedge, viscosity wedge, microasperity and cavitation lubrication, asperity colli- sions, and squeeze effects do not provide sufficient fluid pressure to be considered a primary source of beneficial lubrication in parallel sliders
Rohde and Oh [47] reported that the effect of elastic distortion of the bear- ing surface due to temperature and pressure variations on the bearing per- formance is small when compared with thermal effects on viscosity Most of analytical studies dealing with thermohydrodynamic lubrica-
tion utilize an explicit marching technique to solve the energy equation [47,
481 Such explicit schemes may in some situations cause numerical instabil- ity Also, the effect of variation of viscosity across the fluid film on the bearing performance was acknowledged to be an important factor in bear-
ing analysis [49-531 The effect of oil film thermal expansion across the film
on the bearing load-carrying capacity is not adequately treated in all the published work
The extensive experimental tests reported by Seireg et al [25-281 for the steady-state and transient performance of fluid film bearings strongly sug- gest the need for a reliable methodology for their analysis This is also a
major concern for many workers in the field (Pinkus [38] and Szeri [37])
Trang 3Several theoretical studies have been undertaken to address this problem (Dowson and Hudson [54 551 and Ezzat and Rohde [48, 561) but were not totally successful in predicting the experimentally observed relationship between speed and pressure for fixed geometry films
Wang [57] developed a thermohydrodynamic computational procedure
for evaluating the pressure, temperature, and velocity distributions in fluid films with fixed geometry between the stationary and moving bearing sur- faces The velocity variations and the heat generation are assumed to occur
in a central zone with the same length and width as the bearing but with a significantly smaller thickness than the fluid film thickness The thickness of the heat generation (shear) zone is developed empirically for the best fit with experimentally determined peak pressures for a journal bearing with a fixed film geometry operating in the laminar regime A transient thermodynamic computation model with a transformed rectangular computational domain
is utilized The analysis can be readily applied to any given film geometry The existence of a thin shear zone, with high velocity gradients, has been
reported by several investigators Batchelor [58] suggested that for two disks
rotating at constant but different speeds, boundary layers would develop on each disk at high Reynolds numbers and the core of the fluid would rotate at
a constant speed
Szeri et al [59] carried out a detailed experimental investigation of the flow between finite rotating disks using a laser doppler velocimeter Their measurements show the existence of a velocity field as suggeste by Batchelor
More recently [60], the experimental investigation of the flow between rotat-
ing parallel disks separated by a fixed distance of 1.27 cm in a 0.029% and
0.053 % solutions of polyacrylamide showed the existence of an exceedingly thin shear layer where the velocity gradients iire exceedingly high This effect was found to be most pronounced at higher revolution rates
Another experimental study by Joseph et al I611 demonstrates the exis-
tence of thin shear layers betweer! two immiscible liquids that have unequal viscosi t ies
6.4.1 Empirical Evaluation of Shear Zone
The computer program described by Wang [67] is used to compute the shear
zone ratio, h,/h, which gives the value for the maximum fiim pressure cor- responding to the experimental data A flow chart of the program is given in
Fig 6.48 The geometric parameters of the bearing (UW-1, UW-2 and UW- 3) investigated in this empirical evaluation are listed in Table 6.7 The non-
dimensional viscosity-temperature relation used in the computer program is
lists the viscosity coefficients and reference viscosity at 37.8 and 93.3"C for
F( F ) = y K f ( T - l j , where S is the temperature viscosity coefficient Table 6.8
Trang 4Design of Fluid Film Bearings 233
Set initial pressure and temperature
distribution in the fluid film
Solve the simplified generalized Reynolds Eq whthermal
expansion to get the pressure distribution of the film
Calculate the velocity profile in the shear zone
Solve the time dependent energy equation to obtain
the temperature distribution in the shear zone
f
Solve the heat transfer eq and energy eq to obtain the temp
distributions in the stationary and the moving fluid films
Do the pressure and temperature
distributions reach steady state ?
Yes Output the pressure and the temperature distributions,
and calculate the bearing performance
Figure 6.48
thermal expansion)
Program flow chart of developed analysis (THD with across film
Table 6.7 Geometric Parameters of Investigated Bearings
Trang 5various grades of oil In all the cases considered, the properties of the lubricants were taken as follows:
The following nomenclature is used in the analysis:
a = lubricant thermal expansion coefficient
S = temperature viscosity coefficient
F = eccentricity ratio
p = lubricant density
p = lubricant viscosity
ji = dimensionless viscosity = eSTtN('-I)
h = thickness of the film
h, = thickness of the shear layer
k = thermal conductivity of the lubricant
T = fluid film temperature
= lubricant bulk modulus
(3
= dimensionless temperature =
T,,, = oil inlet temperature
The UW-1 bearing was first selected for evaluating the shear zone due to the fact that extensive test data for various conditions are available for it The experiment covered eccentricity ratios from 0.6 to 0.9, speeds from 500 to
2400 rpm, lubricating oils from SAE 10 to 50, and inlet temperatures from
Table 6.8 Lubricant Viscosity-Temperature Table, @( F ) = e'Tn(f-')
Oil p at 37.8"C [N/(sec-m2)] p at 93.3" [N/(sec-m2)] s
Trang 6Design of Fluid Film Bearings 235
25 to 72°C The results were used to iteratively evaluate the shear zone ratio,
h,/h, which best fits the maximum values of the experimental pressure The
relationship was then applied in the computational program to check the experimental pressure data for the different test conditions of the journal bearing UW-2, as well as the slider bearing UW-3 (see Table 6.7)
6.4.2 Empirical Formula for Predicting the Shear Zone
Traditionally, the behavior of an isoviscous fluid film wedge can be char- acterized based on Newtonian fluid dynamics by a dimensionless number -
the Sommerfeld number (S) When the transient heat transfer in the fluid wedge for thermohydrodynamic analysis is considered, it stands to reason that another dimensionless number - the Peclet number ( P p ) should also
play an equally significant part Consequently, the calculated values for
h,/h based on all the considered experimental data (Table 6.9) were plotted
versus the product of Sommerfeld and Peclet numbers as shown in Fig 6.49
It can be seen that a highly correlated curve was obtained The fitted curve
can be defined by the following equations to a high degree of accuracy:
Table 6.9 Summarized Results for Empirical Evaluation of Shear Zone Ratio for Bearings (UW-1)
Speed Eccentricity (Pmax)exp Shear zone Case no (rpm) ratio, E Oil Oil T,, ("C) (106 N/m2) ratio, h,/h
0.87
0.60
0.60 0.60
SAE 10 SAE 50 SAE 10 SAE 50 SAE 10 SAE 50
SAE 30
SAE 30 SAE 30
53.3 71.1 53.3 71.1 53.3 71.1 41.7 41.7 41.7
0.644 0.700
0.93 1
0.994 1.274 1.414 0.238 0.336 0.48 3
Trang 7Bearing Characteristic No (SxP, x 106)
Figure 0.49 Fluid film shear zone ratio versus the product of Sommerfeld and Peclet numbers 0 , calculated; -, curve-fitted equation
where the Sommerfeld and Peclet numbers are defined as follows:
The following nomenclature is used in the analysis:
Trang 8Design of Fluid Film Bearings 237
The characteristic length of the journal bearing used in the Peclet number is
nR, as this is the length of significance in the transient heat transfer problem
6.4.3 Geometric Analogy Between the Lubricating Film for Journal and Slider Bearings
The film geometry of a journal bearing is considered analogus to that of a generalized slider bearing The unwrapped journal bearing, assuming the radius of curvature of the bearing is large compared with the film thickness,
is basically a slider with a convergent-divergent shape The divergent por- tion of the journal bearing cannot be expected to contribute to the load- carrying capacity and consequently the characteristic length of the journal
bearing is considered equal to nR By using a transformation based on the
following geometric analogy:
B = nR = length of the bearing in the direction of sliding
a characteristic number for a slider bearing, similar to that of a journal bearing, can be obtained The generally adopted characteristic number (Sommerfeld number, S) for journal bearings, which is based on the iso- viscous theory, and the derived characteristic number of slider bearings based on the considered analogy can be written as follows:
Trang 9The characteristic time constant for thermal expansion, At, for both bear-
ings is defined as follows:
At = 2 n ( 3 (journal bearing)
A f = 2 n ( F ) (slider bearings)
and their nondimensional expressions are:
(journal tearing) (slider bearing)
at= 2.(%)(3 = 2 n - ha,,
B
The equivalent journal bearing length for a fresh film during one rotation is
considered to be equal to 2nR, where
L = length of bearing perpendicular to sliding
To determine the minimum film thickness, based on the isoviscous theory,
a transformation is needed between S2 for slider bearings (which is devel-
oped based on the journal bearing analogy) and the commonly used char-
acteristic number Sl [62, 631, which is equal to ( l/m2)([pU/(FIB)]) In the formula for S1, m is the slope of the wedge and p is the average pressure of
the film Figure 6.50 shows the relation between S and S2 for several LIB
ratios This relationship is obtained by performing an isoviscous calcula- tion to determine the average pressure for slider bearings with different wedge geometries and consequently evaluating the corresponding charac- teristic number S1
6.4.4 Pressure Distribution Using the Conventional Assumption of Full
Film Shear
The pressure distribution in the fluid film is probably the most important behavioral characteristic in the analysis of bearing performance The other bearing characteristics, e.g., heat distribution, frictional resistance, can be calculated from the pressure field However, the pressure distribution is strongly influenced by the thermal effects in the fluid film which can not
be accurately predicted by previous analytical methods, i.e., isoviscous the-
ory and commonly adopted thermohydrodynamic (THD) analysis [64, 65,
661
Trang 10Design of Fluid Film Bearings 239
Slider Bearing Characteristic No., S1
Figure 6.50 Conversion chart for bearing dimensionless characteristic numbers based on the isoviscous theory
The isoviscous theory is understandably inadequate in the performance evaluation, due to the lack of consideration of the temperature and viscosity
variations in the model The THD analysis mentioned above are based on
the conventional assumption of full film shear and require the simultaneous solution of a coupled system of equations (the energy, momentum, and continuity equations) in the full fluid film For most cases, however, the solutions predicted from the THD model deviate considerably from experi- mental results Figure 6.51 shows a comparison between the solutions
obtained from these two theories and the experimental pressure [25] The
latter was obtained by slowly changing the speed of a variable speed motor and a continuous plot of the pressure versus speed is automatically recorded using an x-y plotter All the tests on the UW-1 bearing with different lubricants and eccentricity ratios exhibited a square root relationship between the pressure and speed, as reported by Seireg and Ezzat [25] (Table 6.10) The THD solutions in this figure are obtained by solving the Reynolds equation coupled with the energy equation for the full film with- out considering the thermal expansion The boundary of the stationary component is assumed to be thermally insulated, while the moving part surface has the same temperature as the inlet oil It can be seen that the results based on these theories do not appropriately predict the pressure distribution
Trang 11.7 g
0
Fig ire 6.51 Comparison between the solutions obtained fr m commonly adopted theories with experimental pressure The THD solution is obtained by sol- ving the Reynolds equation coupled with energy equation for the full film; no ther- mal expansion is considered (From Ref 25.)
6.4.5 Pressure Distribution Using the Proposed Model
Table 6.1 1 gives a summary of the calculated results for all three bearings (Table 6.10) using the empirical relationship for the shear zone and the computational approach described by Wang and Seireg [67] The experi- mental results for the maximum pressure are also given for comparison It can be seen that the calculated maximum pressures are in excellent correla- tion with experimental results for all cases
Figures 6.52a and 6.52b show typical normalized pressure distribution
in the film for the UW-1 bearing obtained from the proposed model (which
considers thin central shear zone and thermal expansion) and the pressure obtained experimentally [25] It can be seen from the figures that the calcu- lated pressure distribution correlates well with the experimental pressure It
is interesting to note that the calculated normalized pressure distributions in both the experimental data and to those predicted by the isoviscous theory
The same correlation is found in the case of the UW-2 bearing, as well as the
slider bearing UW-3 (Figs 6.53 and 6.54) for the test conditions given in Table 6.10
Trang 12Design of Fluid Film Bearings 24 I
Table 6.10 Test Conditions for the Bearings UW-2 and UW-3
Journal bearing (UW-2)
Eccentricity Case no Speed (rpm) ratio Oil Oil T,, (“C)
10
11
Slider bearing (UW-3)
Case no Speed (m/s) hmin (mm) Slope Oil Oil T,, (‘C)
12 0.2286 0.0 1524 0.0009 SAE 5 25
13 0.4572 0.0 1524 0.0009 SAE 5 25
Figures 6.55-6.57 show examples of the pressure-speed characteristics
of the bearings investigated under various test conditions The continuous curves shown in the figures are the best-fit square root relationship between the experimental pressure and speed starting from the origin The maximum pressure predicted by the isoviscous theory based on the inlet oil tempera- ture is also plotted in each figure for comparison The calculated maximum
Table 6.1 1 Numerical Results Based on the Empirical Formula
0.1 17 0.234
0.0933 0.1534 0.1867 0.3069 0.3734 0.6137 0.779 1.557
3.115
0.089 0.177 0.03 1 0.062
0.0038 0.0084 0.01 12 0.0202 0.0245 0.0368 0.0430 0.0576 0.063 1
0.0036 0.0 104
0.0007
0.0020
0.644 0.700 0.93 1
0.994 1.274 1.414 0.238 0.336 0.483 0.455 0.686 0.196 0.322
0.63 0.72 1 0.966 0.959 1.253 1.365 0.252 0.357 0.470 0.42 0.679 0.245 0.343
The test conditions for the bearings are given in Tables 6.9 and 6.10 with corresponding case numbers