Influence of Critical Flow Rates on Characteristics of

Research ArticleInfluence of Critical Flow Rates on Characteristics of Enforcedand Shear Flows in Circular Convergent-Divergent Channels

Leonid A Savin1 Alexey V Kornaev2 and Elena P Kornaeva3

1Department of Mechatronics and International Engineering Orel State University Named after I S Turgenev Orel 302026 Russia2Mechatronics and International Engineering Research Center Orel State University Named after I S Turgenev Orel 302026 Russia3Modeling of Hydro and Mechanical Systems Research Laboratory Orel State University Named after I S TurgenevOrel 302026 Russia

Correspondence should be addressed to Elena P Kornaeva lenoks_boxmailru

Received 24 February 2017 Revised 8 May 2017 Accepted 16 May 2017 Published 12 June 2017

Academic Editor Steven Chatterton

Copyright copy 2017 Leonid A Savin et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Analysis of the reasons of critical flow rate occurrence in hydraulic tracts of cryogenic machines has been carried out Theoreticalexpressions have been derived to calculate critical velocities in a boiling multiphase medium Applied to hybrid fluid-film bearingswith throttles for lubricant supply a mathematical model has been developed to calculate pressure distribution and hydrodynamicreaction forces of a lubricant considering the influence of steam content and critical flows in throttle devices Numerical results ofphase state and load capacity calculations of a hybrid fluid-film bearing under lubricantrsquos critical flow rates condition have beenpresented

1 Introduction

One could note the tendency to widen the field of applicationof cryogenic and low-boiling liquids for example liquidO2 H2 methane and noble gases as fuel componentsand parts of technological machines and cooling systemsThese materials have particular thermophysical propertiesnamely low viscosity compressibility in liquid state andlow boiling temperatures 119879boil For instance under pressureof 1 atm 119879boil of oxygen is around 90K hydrogenmdasharound20K methanemdasharound 110 K propanemdasharound 230K andneonmdasharound 28K [1] Phase state diagram presented inFigure 1 in119875-119879 (pressure-temperature) coordinates shows thefollowing zones of states of aggregation gas liquid plasmaand solid Under ultrahigh pressures (119875 gt 10MPa) evenhydrogen can have liquid and solid metal state Triple (crit-ical) point K separates liquid and gas from some metastaticzones where no difference between phase properties can beobserved [2]

Under real conditions these materials exist in liquidstate under pressures of 1ndash50MPa and are transported inspecial tanks During throttling and dissipative and heatexchange processes in elements and hydraulic tracts some

phase transitions can occur as a result of boiling or vaporouscavitation resulting in multiphase flows

Research of multiphase phenomena in hydraulic tracts ofcryogenic machines has been covered in many papers Forinstance in [3] modeling of cavitation has been consideredtaking thermal effects into account In [4] a joint problemof rotorrsquos axial and radial oscillations has been consideredin case of presence of hydrodynamic forces in a multiphaseflow Cooling of impulse systems of high energy by meansof using cryogenic fluids and implication of micro gasturbines in aircrafts in order to generate energy have beenpresented in [5] Study of multiphase phenomena in case offluid hammer in propulsion systems with liquid nitrogen ispresented in [6] Theoretical and experimental research ofprocesses occurring in bearings of turbopumps of cryogenicrocket engines have been considered in [7ndash9] Influence ofboiling and steam-fluid content on characteristics of fluid-film bearings is considered in [10 11] Generally the effectof steam phase occurrence is marked negative for the fluidfilm of bearings due to decrease of effective viscosity andrapid increase of compressibility which results in decrease ofload capacity and damping properties and also in the increaseof rotorrsquos oscillation amplitudes and occurrence of nonlinear

HindawiInternational Journal of Rotating MachineryVolume 2017 Article ID 8761375 8 pageshttpsdoiorg10115520178761375

2 International Journal of Rotating Machinery

Solid

Liquid

Liquid

Gas

Plasma

KK

1

10

102

103

104

105

106T

(K)

1010107 108 109 1011 1012106105

P (Pa)

Gas

Figure 1 Phase state diagram of X2

biharmonic oscillations As for positive effects it could benoted that there are a slight decrease of power losses due tofriction and some (sim20) increase of load capacity due toanticipatory boiling of lubricant in the unloaded area of abearingwhich results in redistribution of pressure and changein the main reaction force vectorrsquos direction of a fluid filmIn [12ndash17] there is information on evaluation of operation offluid-film bearings and hydrodynamic and gas dynamic sealsunder multiphase conditions of lubricants In [14] the resultsof investigation of multiphase critical flows of liquid oxygenand liquid nitrogen through channels of fixed geometryare presented The experimental results demonstrate thepossibility of application of obtained results to the theory oftwo-phase flows It has also been shown that the observedrelations could be extrapolated and applied to other typesof fluids In [18] a hydrodynamic analysis is shown in anonisothermal formulation for elliptic bearings based onthe joint solution of the Navier-Stokes equation continuityequation and energy equation for multiphase flows In orderto provide the continuity condition for the area of cavitationthe steam transfer equation has been introduced along withtheRayleighndashPlesset equation to take into account the growthand collapse of cavitation bubbles This approach spares theneed to impose artificial outlet boundary conditions in theform of various cavitation algorithms that are often employedto deal with lubricant film rupture and reformation In [16]a quite critical review has been presented of the publishedstudies namely of the 10 models of critical water flowsand the application limits have been set considering theindustrial energy field In [17] a homogenousmodel of a one-dimensional multiphase flow has been presented consideringthe boundary slip in the area of gasfluid separation It hasbeen shown that the developed model matches experimentaldata better than the well-known Fauske model

Fluid-film bearings occupy a particular niche as partsof rotor machines and in some cases their applicationhas practically no alternative It is true first of all forthe heavy energy machines internal combustion enginesand high-speed machines with a turbine driver During thehydrodynamic lubrication regime when the surfaces are

0

50

100

150

200

04 06 08 102120594

alowast

(ms

)

Figure 2 Speed of sound-steam content relation

fully separated by the fluid film their lifetime is almostunlimited and is defined by the endurance of the materialsused To provide such regime a few requirements have tobe met namely absence of misalignment significant forceand thermal deformations sufficient load capacity of thefluid film and also preservation of thermophysical andrheological properties of the film in a certain operationalrange It is related largely to viscosity on which pressuredistribution is dependent and hydrodynamic reaction forcesload capacity and power losses due to friction stiffnessand damping of hydrodynamic bearings In a number ofcases heat capacity thermal conductivity compressibilityand coefficient of expansion are also of great significance information of load bearing fluid film

2 Mathematical Model

The present paper focuses on studying the influence of aphysical effect of rapid decrease of the speed of sound ina liquid medium when steam phase occurs due to boilingA good example of such effectrsquos occurrence is the fact thatthe speed of sound in water at 119879 = 20∘b and atmosphericpressure is 1450ms and with steam content 120594 = 0001ndash001the same value decreases to 3ndash5ms [12] that is the valuechanges hundreds of times Generally the dependence of thespeed of sound on the steam content can be represented by acurve with an apparent peak at 120594 lower than 01 In Figure 2the same curve is shown for Freon-113 with various values ofdryness rate

Distribution of small disturbances in a multiphasemedium is accompanied by a complex of specific physicalprocesseswhich are subjects of thermodynamics gas kineticsacoustics and gas dynamics [19 20] Presently there area number of approaches thermodynamic based on themolecular-kinetic theory of gases acoustic and gas dynam-ics For multiphase medium the thermodynamic theory isused more often and is based on single velocity continuum

International Journal of Rotating Machinery 3

The processes in the medium are assumed to be stationarythe influence of amplitude and frequency on the speed ofsound is neglected and nonadiabaticity is also neglectedof the processes of compression and expansion due to theheat supply and energy dissipation Generally the Laplaceequation is used in thermodynamic methods to determinethe value of speed of sound

119886lowast = radic119889119875119889120588 = 120592119889119875119889119879radic 119879119862120592 (1 minus 120594) + 119862120592120594 (1)

where119875 is pressure119879 is temperature 120588 is density 120592 is specificvolume 119862120592 is isochoric heat capacity of liquid and steamphase along the saturation line and 120594 is mass steam content

The following expression can be obtained after a fewtransformations

11198862 = 119889120588119889119875 = (1 minus 120601) 1198891205881015840119889119875 minus 1205881015840 (1 minus 120588101584010158401205881015840 ) 119889120601119889119875 + 12059311988912058810158401015840119889119875 (2)

where 120601 is gas content 1205881015840 is density of liquid phase and 12058810158401015840 isdensity of gas phase

Considering the liquid phase incompressible (1198891205881015840119889119875 =0) one could obtain the expression to determine the speedof sound waves of low frequency distribution in somemultiphasemediumwithout considering the curvature of thesurfaces of bubbles

119886lowast = radic 119875120601120594 (3)

For a fine multiphase medium (119889119901 lt 10minus5m) whenthe surface tension 120590 and the bubblersquos diameter 119889119887 influencepressure in the gas phase the expression takes the followingform

119886lowast = 119875 + 8120590 (3119889119901)120601120594 (4)

For gas lubrication there is also a possibility of occurrenceof critical flows in throttle channels of a bearing Duringsupersonic drops the flow rate of an ideal gas through acylindrical channel is maximal and counterpressure doesnot affect it If this process is considered isentropic and inequilibrium then the critical velocity of a gas is determinedas follows

119886lowast = 1205722 [(21205721 minus 1)11988121 + 21198940] plusmn radic12057222 [(21205721 minus 1)11988121 minus 21198940]2 minus 811989401198811 (1205722 minus 1205721) (1205721 minus 212057211205722 + 21205722)212057211198811 (21205721 minus 1) (5)

where indexes 1 and 2 correspond to the parameters of theflow before and after the drop of condensation 1198811 is flowvelocity and 120572119894 = 119894120588119875 is a function of enthalpy densityand pressure determined using the phase state diagram (orcorresponding tables) where

120572119894 = 119894 120588119875 = 119896119896 minus 1 (6)

where 119896 is isentropic indexFor the ideal gas the expression for critical velocity at the

outlet of the channel takes the following form

119886lowastid = radic21198940 (119896 minus 1)119896 + 1 (7)

For a case of a multiphase boiling medium expression (1)is transformed to the following form [11]

119886 = 120594[(12059712059210158401015840120597119875 )119879

+ 2119879119903 (12059712059210158401015840120597119879 )119875

(12059210158401015840 minus 1205921015840)minus 119879119862101584010158401198751199032 (12059210158401015840 minus 1205921015840)] + (1 minus 120594) [(1205971205921015840120597119875 )

119879

+ 2119879119903 (1205971205921015840120597119879 )119875

(12059210158401015840 minus 1205921015840) minus 11987911986210158401198751199032 (12059210158401015840 minus 1205921015840)] (8)

where1198621015840119875 is isobaric heat capacity of liquid and11986210158401015840119875 is isobaricheat capacity of gas

The derivatives in this expression are approximated asthe ratio of finite increments Δ12059210158401015840 and Δ1205921015840 and increments ofpressureΔ119875 and temperature using the tables of saturated gas[1]

In hydraulic channels with fixed geometry the chockedflow regime could occur at the outlet cross-sectional areaswhere local flow velocity is equal to the speed of sound If thecalculated value of the velocity 119881 at the outlet of the throttleis higher than the critical 119886lowast which is principally impossiblein the channels of the considered shape then the critical flowrate through the cylindrical channel is as follows

119876lowast = 12058512058711988921198674 119886lowast (9)

where 120585 = 11ndash13 is an empirical coefficient of flow rate duringthe critical regime of a multiphase medium flow and 119889119867 isdiameter of a throttle

When the lubricant is supplied under pressure by meansof a pump in case of a hydrostatic bearing (see Figure 3) thepresence of gas or steam medium (see Figure 4) can resultin occurrence of critical flow rates when the velocity of themedium in the elements of the hydraulic tract reaches thevalue 119881 = 119886lowast which can result in a radical change of thelubrication regime and all bearingrsquos characteristicsThe valuesof the critical flow rate in throttle devices of hybrid bearings


oZ

D

h

L

e

X

ℎ0

120573

120593

ℎmin

dHlH

o1

120572(x)

Figure 3 Calculation scheme of a hybrid bearing

Multiphase area

Fluid

Rotation direction

Row of cameras

Figure 4 Phase diagram of a lubricant film

are most directly linked to the pressure in feeding chambers119875119867119894 MPa that are determined based on the flow rate balanceequation

119899sum119894=1

119876119867119894 = sum(119876119909 + 119876119911)= 119899 119899sum119894=1

1205951205880 (1 + 120588119867119894) 12058711988921198674 radic1198750 (1 minus 119875119867119894)1205880 (1 minus 120588119867119894) (10)

where sum119899119894=1 119876119867119894 is mass flow rate through the throttlingdevices in a bearing 119899 is number of throttle channels119876119909119876119911Kgs is flow rate from the chamber along the circumferentialand axial coordinate and 120595 is loss coefficient at the inlet ofthe channel

When the flow velocity in the channels of a throttlingdevice reaches the speed of sound the chocking flow regime

can be observed that is attainment of maximal possible flowrate of a lubricant with the given parameters

119899sum119894=1

119876119867119894 = 119899 sdot 119876lowast119867 (11)

The values 119875119867119894 are internal boundary conditions whendetermining the pressure distribution119875(119909 119910) in the lubricantfilm The values 119875119867119894 and 119875(119909 119911) are obtained by means ofjoint solution of (10) and the modified Reynolds equation(12) using Seidel iterationmethod For a steady-state flow thecharacteristics of a fluid film are determined given the fixedposition of a rotor in the bearing (119890 = const 120593 = 0) and theReynolds equation is as follows

120597120597119909 (120588ℎ3120583119896119909 120597119875120597119909) + 120597120597119911 (120588ℎ3120583119896119911 120597119875120597119911)= 6 120597120597119909 (119880119909120588ℎ) minus 12120588119880119910

(12)


where 120588 is density kgm3 ℎ = ℎ(119909) is a radial gap functionm 119880119909 and 119880119910 are accordingly the tangential and normalcomponents of the rotorrsquos surface velocity ms 120583 is dynamicviscosity Pasdots 119896119909 119896119911 are turbulence coefficients and 119875 ispressure Pa

Calculation of pressure distribution in the fluid film119875(119909 119910) under possible phase transition conditions unavoid-ably implies an isothermal formulation of the problemassuming there is a change in temperature in this caseenthalpy that depends on dissipative and heat exchangeprocesses [10] The energy equation for an adiabatic processtaking thermal conductivity across the fluid film and dissipa-tion into account is as follows

120588(119881119909 120597119894120597119909 + 119881119911 120597119894120597119911) = 120597120597119909 (120582120597119879120597119909) + 120597120597119911 (120582120597119879120597119911 )+ 120583[(120597119881119909120597119910 )2 + (120597119881119911120597119910 )2] (13)

where 119894 is lubricantrsquos enthalpy m2s2 120582 is heat conductioncoefficient W(msdotK) 119881119909 119881119911 are velocities in circumferentialand axial direction accordingly ms and 120583 is dynamicviscosity Pasdots

In the set of (12) and (13) the following parameters areto be determined pressure temperature (enthalpy) densityviscosity heat capacity and turbulence coefficients To closethe system a number of additional expressions are used gapfunction ℎ(119909) = ℎ0 minus 119890 sdot cos(120572(119909)) approximate relationsbetween thermophysical parameters and coefficients of tur-bulence 119896119909 and 119896119911 [21] as follows

119896119909 = 1 + 0044 sdot (119896lowast2 sdot Re)0725 119896119911 = 1 + 00247 sdot (119896lowast2 sdot Re)065 (14)

Total load capacity of a bearing can be determined asfollows

119882 = radic1198772119894 + 1198772119895 (15)

where 119877119894 119877119895 are reaction forces of a lubricant film in acorresponding direction obtained by means of integratingthe pressure distribution over the bearingrsquos area

119877119894 = 1198750119863119871int10119889119911int21205870

119875 (119909 119911) cos (120572 (119909) minus 120593) 119889120593119877119895 = 1198750119863119871int1

0119889119911int21205870

119875 (119909 119911) sin (120572 (119909) minus 120593) 119889120593 (16)

where119863119871 are diameter and length of the bearing accordinglyand 120593 is position angle of the center line

Scheme to calculation of rotor trajectories is presented inFigure 5

Rotor is represented by a pointmass on a bearingMotionis studied in the perpendicular plane relative to the bearingrsquosaxis Rotor trajectories are sets of points that describe thelocation of a center of a shaft in the fluid film at some fixed

BearingcentreGap

boundary

I

J

Y

X

m

120593

Fi Fj

Ri

Rj

120596

Figure 5 Rotor trajectory calculation scheme

points in time and are results of numerical integration of rotormotion equations which in polar coordinates are as follows

119898( 119890 minus 1198902) = 119865119894 minus 119877119894119898 (119890 minus 2 119890) = 119865119895 minus 119877119895 (17)

where 119898 119902 are rotor mass and imbalance accordingly 119890 and120593 are eccentricity and rotation angle of the center lines and119877119894 119877119895 are reaction forces of the lubricant film

119865119894 = 119898119892 cos120593 + 1199021205962 cos (120596119905 minus 120593 + 1205930) 119865119895 = 119898119892 sin120593 + 1199021205962 sin (120596119905 minus 120593 + 1205930) (18)

where 119902 is rotorrsquos imbalance and 119892 is free fall acceleration

3 Test Rig

In order to verify the obtained theoretical results and to checkthe working capacity of the bearing under the conditionsof boiling and multiphase state of the lubricant a series ofexperimental studies have been carried out using a speciallydesigned test rig which includes a rotor machine (see Fig-ure 6(a)) rotational driver the data acquisition system andthe Freon contour (see Figure 6(b)) The test rig consists oftwo bearings (3 and 4) a shaft (16) a loading device (13) anda number of sensors

Freon-113 is used as a lubricant and allows provision ofmultiphase boiling in the elements of the hydraulic tract inan acceptable range of pressuresThe supply system is used tostore and refine Freon to feed it to the tested bearings undercertain pressure and temperature to condensate the vaporand to cool it down to the storage temperature The systemconsists of filters pumps and so on

4 Results and Discussion

In Figure 7 the numerical results are presented of therelative rotor position (similar to the load capacity in vertical


1 2

3 4 56 7

8 910 11 12

13

15 16

(a)

Vacuum

Heater

Air drainage

Bearing

Freon discharge

Heat exchanger

Freo

n re

ctifi

catio

n

1

2

34

5

6

7

~U

(b)

Figure 6 Rotor machine test rig (a) with the scheme of the Freon contour (b)

1 2 3

4 5

6

Fluid Gas

e y

TK TC

0

+02

+04

+06

+08

+10

T0 (K)380360340320300

⟨Multiphase area⟩

Figure 7 Influence of steam phase concentration on the position ofrotor in the bearing

direction) in a hybrid bearing with the following parametersrotorrsquos mass 119898119877 = 12Kg imbalance 119902 = 42 sdot 10minus5 Kgsdotmangular velocity 120596 = 0ndash2000 rads diameter and length ofthe bearing 119863 = 0054m 119871 = 007m diameter and lengthof a throttle 119889119867 = 8 sdot 10minus4m 119897119867 = 3 sdot 10minus3m number ofthrottles 119899 = 8 (1 row) radial gap ℎ0 = 75 sdot 10minus5m pressureand temperature of supplied medium 1198750 = 05ndash06MPa 1198790 =340ndash350K lubricantmdashFreon-113 thermodynamic propertiesof which under atmospheric pressure are 120588 = 1540Kgm3120583 = 13 sdot 10minus4 Pasdotsm2 and 119879boil = 320K

The presented temperature range along the horizontalaxis which partially occupies the areas of liquid gas andmultiphase flow is arbitrarily divided into a few characterareas the borders of which are marked with correspondingnumbers With the increase of temperature on the inlet of ahybrid bearing in the area of liquid flow (1-2) a slight decreaseof bearing capacity is observed due to decrease of dynamicviscosity and density of the lubricant Here enthalpy of asingle-phase flow is insignificantly increased due to energydissipation as a result of friction which also results in thedecrease of load capacity119882

It shall be noted that it has been experimentally proventhat load capacity of a hybrid bearing increased in the begin-ning of the multiphase area (2-3) which can be explainedwith boiling in the unloaded area of the film The loadcapacity is increased due to redistribution of pressures inthe lubricantrsquos film in the direction of pressuresrsquo drop in thehemisphere with a big radial gap which provides additionallifting force The width of temperature interval of increasingload capacity is 8ndash12∘b and the increment of load capacity119882 given certain supply pressure eccentricity and rotationalspeed could be 5ndash20 With the increase of temperature ofthe supplied lubricant (given that 1198750 = const) the boilingprocess spreads across the whole radial gap and resultsin rapid decrease of load capacity (3-4) due to a generaldecrease of pressure in the film This phenomenon occursdue to decreasing density and viscosity of the steam-fluidmedium with increasing steam content Occurrence of gasphase in the feeding chambers of the unloaded part resultsin the flow rate limitation through them and violation ofthe flow rate balance equation which results in decreasinghydrostatic pressure and provides the displacement of a rotor(4-5) The shift of the process of critical flows in pressurecompensators towards the loaded part evens the pressuredistribution in circumferential direction and decreases thecombined reaction force (5-6) Here the critical velocity of theflow through the throttles begins to increase which increasesthe volumetric flow rate through them and the hydrodynamicand gas dynamic pressure in this area of the lubricantrsquos filmIn the surroundings of the point (6) load capacity is formedto a high extent only due to the gas dynamic effect

It is assumed that operation in this regime is unwantedsince it is quite possible for a bearing to seize functioning dueto low stiffness of the film Increase of critical velocity withincreasing steam content in the throttles provides increasingload capacity (6) and operation in a more stable gas dynamicand gas static regime where with the increase of 1198790 someslight increase of 119882 is observed due to increased viscosity ofthe gas It should also be noted that critical flow can occurin the throttles at 1198750 gt 119875lowast in case of a gas flow The relative


T0 (K)

Qtimes10

6(m

3s

)

300 330 36040

50

60

P0 = 05MPa120596 = 270 rads

Figure 8 Volumetric flow rate of the lubricant and the concentra-tion of steam content relation

curvature in the areas (4 5 6 and 7) depends mainly on 1198750119890 119866 119902 and 120596 and is 5ndash15 in terms of load capacity 119882 and10ndash20K in terms of temperature

The numerical results (solid line) of flow rate relative toincreasing temperature 1198790 eccentricity and angular velocitymatch the experimental results (slash-dotted line) quite well(see Figure 8) There is some increase of volumetric flow rateto be observed with increasing temperature and decreasing of119876 due to chocked flow through the throttle as a result of rapiddecrease of the speed of sound in the multiphase medium

Mass flow with increasing temperature of the liquidinsignificantly decreases due to a decrease in density Whenthe speed of sound is reached a sharp decrease in thelubricant flow in the throttle devices in the unloaded areais observed With an increase in temperature of the suppliedlubricant process of critical flows proceeds to other throttleswhich reduces the total flow rate through the bearing

5 Conclusion

Based on the developed mathematical and simulation mod-els a complex of numerical experiments has been carriedout in order to evaluate the influence of critical flows andflow rates of lubricants on the characteristics of pressure-shear flows In particular a hybrid bearing with throttles andpoint feeding chambers was considered It has been identifiedthat load capacity decreases and amplitudes of oscillationsincrease once the critical flows occur in all throttling devicesThis happens due to rapid pressure drop in the throttlingdevices and consequently in the fluid film So decrease offlow rate of the lubricant due to rapid drop in speed of soundin a multiphase medium shall not be considered positiveDuring critical regimes of flow almost all characteristics of abearing change and in case of a low value of critical flow ratea regime of ldquooil starvationrdquo could occurThis could lead to the

boundary friction regime and increased wear rate of surfacesof the journal and the sleeve The obtained theoretical resultshave been substantiated using the developed test rig with aFreon contour

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the Russian Science Founda-tion under Project no 16-19-00186 and under Grant no14Z56171643-MKof the President of the Russian FederationThe authors gratefully acknowledge this support

References

[1] N B Vargaftik Handbook on thermophysical properties of gasesand liquids Eneroatomizdat Press Moscow Russia 1990

[2] V A Kirillin V V Sychev and A E Sheyndlin EngineeringThermodynamics Energia Press Moscow Russia 1968

[3] S Shi GWang and C Hu ldquoA Rayleigh-Plesset based transportmodel for cryogenic fluid cavitating flow computationsrdquo ScienceChina PhysicsMechanics andAstronomy vol 57 no 4 pp 764ndash773 2014

[4] C A Evrensel H E Kimmel and D M Cullen ldquoAxial rotoroscillations in cryogenic fluid machineryrdquo in 3rd ASMEJSMEJoint Fluids Engineering Conference 1999 ASME Fluids Engineer-ing Division Summer Meeting San Francisco Calif USA July18ndash23 1999

[5] S R Nuzum R A Roberts and M Wolff ldquoVarious integratedaerospace systems utilizing a cryogenic fluidrdquo in Proceedingsof the 13th International Energy Conversion Engineering Confer-ence (IECEC) July 2015

[6] J-B Gouriet J-M Buchlin M Vetrano and J Steelant ldquoMul-tiphase fluid hammer with cryogenic fluidsrdquo in SP-636 EnvisatSymposium 2016

[7] J M Reddecliff and J H Vohr ldquoHydrostatic bearings forcryogenic rocket engine turbopumpsrdquo Journal of LubricationTechnology vol 91 no 3 p 557 1969

[8] J Xu X Yuan C Zhang and XMiao ldquoDynamic characteristicsof high-T119888 superconductor and hydrodynamic fluid-film com-pound bearing for rocket enginerdquo IEEE Transactions on AppliedSuperconductivity vol 26 no 3 2016

[9] A E Jahromi and F K Miller ldquoDevelopment of a proofof concept low temperature 4He superfluid magnetic pumprdquoCryogenics vol 82 pp 68ndash82 2017

[10] L Savin and O Solomin ldquoApplied theory of steam-liquidlubricationrdquo in Proceedings of the Sixth International Conferenceon Rotor Dynamics (IFToMM) vol II pp 637ndash645 2002

[11] L A Savin and O V Solomin Modeling of rotor systemswith fluid-film bearings monography Mashinostroenie-1 PressMoscow Russia 2006

[12] Y N Vasiliev ldquoThe theory of the two-phase gas-liquid ejec-tor with cylindrical mixing chamberrdquo in Coll Articles BladeMachines And Inkjet Devices vol 5 Mechanical EngineeringPress Moscow Russia 1971


[13] R C Hendricks and R J Simoneau ldquoTwo-phase chokedflow in tubes with very large LDrdquo in Advances in CryogenicEngineering vol 23 pp 265ndash275 Springer US Boston MAUSA 1978

[14] R C Hendricks R J Simoneau and R F Barrows ldquoCriticalflow and pressure ratio data for LOX flowing through nozzlesrdquoin Proceedings of the 14th International Congress of the Interna-tional Institute of Refrigeration 1975

[15] M R Hajmohammadi S Salman Nourazar and A CampoldquoAnalytical solution for two-phase flow between two rotatingcylinders filled with power law liquid and a micro layer of gasrdquoJournal of Mechanical Science and Technology vol 28 no 5 pp1849ndash1854 2014

[16] E Elias and G S Lellouche ldquoTwo-phase critical flowrdquo Interna-tional Journal of Multiphase Flow vol 20 no 1 pp 91ndash168 1994

[17] M M Petrovic and V D Stevanovic ldquoTwo-component two-phase critical flowrdquoFMETransactions vol 44 no 2 pp 109ndash1142016

[18] H Shahmohamadi R Rahmani H Rahnejat C P Garner andDDowson ldquoBig end bearing losseswith thermal cavitation flowunder cylinder deactivationrdquo Tribology Letters vol 57 no 2article no 2 2015

[19] E M Deitch and G FilippovGas dynamics of two-phase mediaMoscow Energia Publiching Energia Publiching Moscow Rus-sia 1968

[20] G Wallis The one-dimensional two-phase flow McGraw-HillNew York NY USA 1969

[21] V N Constantinescu Gas lubrication ASME United EngCenter NY USA 1965

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of


International Journal of

RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Solid

Liquid

Liquid

Gas

Plasma

KK

1

10

102

103

104

105

106T

(K)

1010107 108 109 1011 1012106105

P (Pa)

Gas

Figure 1 Phase state diagram of X2

biharmonic oscillations As for positive effects it could benoted that there are a slight decrease of power losses due tofriction and some (sim20) increase of load capacity due toanticipatory boiling of lubricant in the unloaded area of abearingwhich results in redistribution of pressure and changein the main reaction force vectorrsquos direction of a fluid filmIn [12ndash17] there is information on evaluation of operation offluid-film bearings and hydrodynamic and gas dynamic sealsunder multiphase conditions of lubricants In [14] the resultsof investigation of multiphase critical flows of liquid oxygenand liquid nitrogen through channels of fixed geometryare presented The experimental results demonstrate thepossibility of application of obtained results to the theory oftwo-phase flows It has also been shown that the observedrelations could be extrapolated and applied to other typesof fluids In [18] a hydrodynamic analysis is shown in anonisothermal formulation for elliptic bearings based onthe joint solution of the Navier-Stokes equation continuityequation and energy equation for multiphase flows In orderto provide the continuity condition for the area of cavitationthe steam transfer equation has been introduced along withtheRayleighndashPlesset equation to take into account the growthand collapse of cavitation bubbles This approach spares theneed to impose artificial outlet boundary conditions in theform of various cavitation algorithms that are often employedto deal with lubricant film rupture and reformation In [16]a quite critical review has been presented of the publishedstudies namely of the 10 models of critical water flowsand the application limits have been set considering theindustrial energy field In [17] a homogenousmodel of a one-dimensional multiphase flow has been presented consideringthe boundary slip in the area of gasfluid separation It hasbeen shown that the developed model matches experimentaldata better than the well-known Fauske model

Fluid-film bearings occupy a particular niche as partsof rotor machines and in some cases their applicationhas practically no alternative It is true first of all forthe heavy energy machines internal combustion enginesand high-speed machines with a turbine driver During thehydrodynamic lubrication regime when the surfaces are

0

50

100

150

200

04 06 08 102120594

alowast

(ms

)

Figure 2 Speed of sound-steam content relation

fully separated by the fluid film their lifetime is almostunlimited and is defined by the endurance of the materialsused To provide such regime a few requirements have tobe met namely absence of misalignment significant forceand thermal deformations sufficient load capacity of thefluid film and also preservation of thermophysical andrheological properties of the film in a certain operationalrange It is related largely to viscosity on which pressuredistribution is dependent and hydrodynamic reaction forcesload capacity and power losses due to friction stiffnessand damping of hydrodynamic bearings In a number ofcases heat capacity thermal conductivity compressibilityand coefficient of expansion are also of great significance information of load bearing fluid film

2 Mathematical Model

The present paper focuses on studying the influence of aphysical effect of rapid decrease of the speed of sound ina liquid medium when steam phase occurs due to boilingA good example of such effectrsquos occurrence is the fact thatthe speed of sound in water at 119879 = 20∘b and atmosphericpressure is 1450ms and with steam content 120594 = 0001ndash001the same value decreases to 3ndash5ms [12] that is the valuechanges hundreds of times Generally the dependence of thespeed of sound on the steam content can be represented by acurve with an apparent peak at 120594 lower than 01 In Figure 2the same curve is shown for Freon-113 with various values ofdryness rate

Distribution of small disturbances in a multiphasemedium is accompanied by a complex of specific physicalprocesseswhich are subjects of thermodynamics gas kineticsacoustics and gas dynamics [19 20] Presently there area number of approaches thermodynamic based on themolecular-kinetic theory of gases acoustic and gas dynam-ics For multiphase medium the thermodynamic theory isused more often and is based on single velocity continuum









119886lowast = radic 119875120601120594 (3)


119886lowast = 119875 + 8120590 (3119889119901)120601120594 (4)




120572119894 = 119894 120588119875 = 119896119896 minus 1 (6)





119886 = 120594[(12059712059210158401015840120597119875 )119879

+ 2119879119903 (12059712059210158401015840120597119879 )119875


119879

+ 2119879119903 (1205971205921015840120597119879 )119875





119876lowast = 12058512058711988921198674 119886lowast (9)




oZ

D

h

L

e

X

ℎ0

120573

120593

ℎmin

dHlH

o1

120572(x)


Multiphase area

Fluid

Rotation direction

Row of cameras



119899sum119894=1

119876119867119894 = sum(119876119909 + 119876119911)= 119899 119899sum119894=1





119899sum119894=1

119876119867119894 = 119899 sdot 119876lowast119867 (11)


120597120597119909 (120588ℎ3120583119896119909 120597119875120597119909) + 120597120597119911 (120588ℎ3120583119896119911 120597119875120597119911)= 6 120597120597119909 (119880119909120588ℎ) minus 12120588119880119910

(12)




120588(119881119909 120597119894120597119909 + 119881119911 120597119894120597119911) = 120597120597119909 (120582120597119879120597119909) + 120597120597119911 (120582120597119879120597119911 )+ 120583[(120597119881119909120597119910 )2 + (120597119881119911120597119910 )2] (13)





119882 = radic1198772119894 + 1198772119895 (15)


119877119894 = 1198750119863119871int10119889119911int21205870

119875 (119909 119911) cos (120572 (119909) minus 120593) 119889120593119877119895 = 1198750119863119871int1

0119889119911int21205870

119875 (119909 119911) sin (120572 (119909) minus 120593) 119889120593 (16)




BearingcentreGap

boundary

I

J

Y

X

m

120593

Fi Fj

Ri

Rj

120596







3 Test Rig






1 2

3 4 56 7

8 910 11 12

13

15 16

(a)

Vacuum

Heater

Air drainage

Bearing

Freon discharge

Heat exchanger

Freo

n re

ctifi

catio

n

1

2

34

5

6

7

~U

(b)


1 2 3

4 5

6

Fluid Gas

e y

TK TC

0

+02

+04

+06

+08

+10

T0 (K)380360340320300








T0 (K)

Qtimes10

6(m

3s

)

300 330 36040

50

60

P0 = 05MPa120596 = 270 rads





5 Conclusion





Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















119886lowast = radic 119875120601120594 (3)


119886lowast = 119875 + 8120590 (3119889119901)120601120594 (4)




120572119894 = 119894 120588119875 = 119896119896 minus 1 (6)





119886 = 120594[(12059712059210158401015840120597119875 )119879

+ 2119879119903 (12059712059210158401015840120597119879 )119875


119879

+ 2119879119903 (1205971205921015840120597119879 )119875





119876lowast = 12058512058711988921198674 119886lowast (9)




oZ

D

h

L

e

X

ℎ0

120573

120593

ℎmin

dHlH

o1

120572(x)


Multiphase area

Fluid

Rotation direction

Row of cameras



119899sum119894=1

119876119867119894 = sum(119876119909 + 119876119911)= 119899 119899sum119894=1





119899sum119894=1

119876119867119894 = 119899 sdot 119876lowast119867 (11)


120597120597119909 (120588ℎ3120583119896119909 120597119875120597119909) + 120597120597119911 (120588ℎ3120583119896119911 120597119875120597119911)= 6 120597120597119909 (119880119909120588ℎ) minus 12120588119880119910

(12)




120588(119881119909 120597119894120597119909 + 119881119911 120597119894120597119911) = 120597120597119909 (120582120597119879120597119909) + 120597120597119911 (120582120597119879120597119911 )+ 120583[(120597119881119909120597119910 )2 + (120597119881119911120597119910 )2] (13)





119882 = radic1198772119894 + 1198772119895 (15)


119877119894 = 1198750119863119871int10119889119911int21205870

119875 (119909 119911) cos (120572 (119909) minus 120593) 119889120593119877119895 = 1198750119863119871int1

0119889119911int21205870

119875 (119909 119911) sin (120572 (119909) minus 120593) 119889120593 (16)




BearingcentreGap

boundary

I

J

Y

X

m

120593

Fi Fj

Ri

Rj

120596







3 Test Rig






1 2

3 4 56 7

8 910 11 12

13

15 16

(a)

Vacuum

Heater

Air drainage

Bearing

Freon discharge

Heat exchanger

Freo

n re

ctifi

catio

n

1

2

34

5

6

7

~U

(b)


1 2 3

4 5

6

Fluid Gas

e y

TK TC

0

+02

+04

+06

+08

+10

T0 (K)380360340320300








T0 (K)

Qtimes10

6(m

3s

)

300 330 36040

50

60

P0 = 05MPa120596 = 270 rads





5 Conclusion





Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










oZ

D

h

L

e

X

ℎ0

120573

120593

ℎmin

dHlH

o1

120572(x)


Multiphase area

Fluid

Rotation direction

Row of cameras



119899sum119894=1

119876119867119894 = sum(119876119909 + 119876119911)= 119899 119899sum119894=1





119899sum119894=1

119876119867119894 = 119899 sdot 119876lowast119867 (11)


120597120597119909 (120588ℎ3120583119896119909 120597119875120597119909) + 120597120597119911 (120588ℎ3120583119896119911 120597119875120597119911)= 6 120597120597119909 (119880119909120588ℎ) minus 12120588119880119910

(12)




120588(119881119909 120597119894120597119909 + 119881119911 120597119894120597119911) = 120597120597119909 (120582120597119879120597119909) + 120597120597119911 (120582120597119879120597119911 )+ 120583[(120597119881119909120597119910 )2 + (120597119881119911120597119910 )2] (13)





119882 = radic1198772119894 + 1198772119895 (15)


119877119894 = 1198750119863119871int10119889119911int21205870

119875 (119909 119911) cos (120572 (119909) minus 120593) 119889120593119877119895 = 1198750119863119871int1

0119889119911int21205870

119875 (119909 119911) sin (120572 (119909) minus 120593) 119889120593 (16)




BearingcentreGap

boundary

I

J

Y

X

m

120593

Fi Fj

Ri

Rj

120596







3 Test Rig






1 2

3 4 56 7

8 910 11 12

13

15 16

(a)

Vacuum

Heater

Air drainage

Bearing

Freon discharge

Heat exchanger

Freo

n re

ctifi

catio

n

1

2

34

5

6

7

~U

(b)


1 2 3

4 5

6

Fluid Gas

e y

TK TC

0

+02

+04

+06

+08

+10

T0 (K)380360340320300








T0 (K)

Qtimes10

6(m

3s

)

300 330 36040

50

60

P0 = 05MPa120596 = 270 rads





5 Conclusion





Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












120588(119881119909 120597119894120597119909 + 119881119911 120597119894120597119911) = 120597120597119909 (120582120597119879120597119909) + 120597120597119911 (120582120597119879120597119911 )+ 120583[(120597119881119909120597119910 )2 + (120597119881119911120597119910 )2] (13)





119882 = radic1198772119894 + 1198772119895 (15)


119877119894 = 1198750119863119871int10119889119911int21205870

119875 (119909 119911) cos (120572 (119909) minus 120593) 119889120593119877119895 = 1198750119863119871int1

0119889119911int21205870

119875 (119909 119911) sin (120572 (119909) minus 120593) 119889120593 (16)




BearingcentreGap

boundary

I

J

Y

X

m

120593

Fi Fj

Ri

Rj

120596







3 Test Rig






1 2

3 4 56 7

8 910 11 12

13

15 16

(a)

Vacuum

Heater

Air drainage

Bearing

Freon discharge

Heat exchanger

Freo

n re

ctifi

catio

n

1

2

34

5

6

7

~U

(b)


1 2 3

4 5

6

Fluid Gas

e y

TK TC

0

+02

+04

+06

+08

+10

T0 (K)380360340320300








T0 (K)

Qtimes10

6(m

3s

)

300 330 36040

50

60

P0 = 05MPa120596 = 270 rads





5 Conclusion





Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2

3 4 56 7

8 910 11 12

13

15 16

(a)

Vacuum

Heater

Air drainage

Bearing

Freon discharge

Heat exchanger

Freo

n re

ctifi

catio

n

1

2

34

5

6

7

~U

(b)


1 2 3

4 5

6

Fluid Gas

e y

TK TC

0

+02

+04

+06

+08

+10

T0 (K)380360340320300








T0 (K)

Qtimes10

6(m

3s

)

300 330 36040

50

60

P0 = 05MPa120596 = 270 rads





5 Conclusion





Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










T0 (K)

Qtimes10

6(m

3s

)

300 330 36040

50

60

P0 = 05MPa120596 = 270 rads





5 Conclusion





Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Influence of Critical Flow Rates on Characteristics of