Numerical Study for Transfer in Speed Rotating Components · 2019. 8. 1. · systems. Asexpected, the ... Keywords: Internal heat transfer, Flow in rotating disks with orifices, Numerical

International Journal of Rotating Machinery1998, Vol. 4, No. 3, pp. 151-161

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Numerical Study for Optimizing Heat Transfer in HighSpeed Rotating Components

S. WITTIG, S. KIM, TH. SCHERER and I. WEISSERT*

Lehrstuhl und Institut fiir Thermische Str6mungsmaschinen, Universitdit Karlsruhe, 76128 Karlsruhe, Germany

(Received in finalform 24 March 1997)

Cooling of high speed rotating components is a typical situation found in turbomachinery aswell as in automobile engines. Accurate knowledge of discharge coefficients and heattransfer of related components is essential for the high performance of the whole engine.This can be achieved by minimized cooling air flows and avoidance of hot spots. In highspeed rotating clutches for example aerodynamic investigations improving heat transferhave not been considered in the past. Advanced concepts of modern plate design try toreduce thermal loads by convective cooling methods. Therefore, secondary cooling air flowshave to be enhanced by an appropriate design of the rotor stator system with orifices. CFDmodelling is used to improve the basic understanding of the flow field in typical geometriesused in these systems.The computational results are obtained by a 3-D-finite-volume-code based on body fitted

structured grids. The Navier Stokes equations are solved by a pressure-correction methodcombined with the standard k-e-turbulence model. Considering the rotation of orifices indisks or shafts, the frame of reference has to be changed to the rotating system. The flowthrough orifices in high speed rotating disks can be calculated with a high level of accuracyin comparison with experiments as shown in Wittig et al. [1994].

Numerical results of the flow in a high speed rotating system are presented with emphasison geometrical variations. Calculations are carried out in order to find an optimum design interms of position and size of the orifices in the housing. These variations induce differentphysical phenomena. Special consideration is directed towards the basic problems of theflow through orifices in high speed rotating disks and shafts and the flow inside rotor-statorsystems. As expected, the very complex flow fields are dominated by rotational effects. Inaddition it is shown that differences occur between the configuration of optimized mass flowrate and the geometry with a maximum of total heat transfer. Obviously, optimizationprocedures are dependent on the knowledge of the local flow field and cannot be performedwithout advanced CFD-methods. It is demonstrated that the approach presented here issuitable for these tasks.

Keywords: Internal heat transfer, Flow in rotating disks with orifices, Numerical simulation,Rotating frame of reference, Secondary air system, Clutch

Corresponding author. Tel.: / 721 608 3634. Fax: + 721 699222.

151

152 S. WITTIG et al.

INTRODUCTION

Cooling of thermal highly loaded components is atypical problem found in turbomachinery as well asin automotive components. As it is well knownfrom aeroengine secondary air systems, an opti-mization of such cooling systems requires theknowledge of the coolant dischargb and the heattransfer characteristics of various components likelabyrinth seals, discs or annular gaps. A genericproblem within these systems is the flow in rotatingorifices. In recent years, numerous investigationshave been published concerning the dischargebehavior of orifices (Lichtarowicz et al. [1965],McGreehan and Schotsch [19871, Hay and Spencer[1991]). However, most of them do not take intoaccount the effects of rotation. At the Institut ffirThermische Str6mungsmaschinen (ITS), Universityof Karlsruhe, several investigations predicting tur-bulent heat and mass transfer processes have beenperformed for complex geometries consideringrotational effects (e.g. Waschka et al. [1992] andScherer [1994]) and major interest has been focusedto the discharge characteristics of rotating orifices(Wittig et al. [1994], [1995]).As it has been pointed out in this paper, the

fundamental strategy of considering rotationaleffects is given by the transformation of the basicequations from the stationary into a local system,i.e. a rotating frame of reference. As another appli-cation of this approach, the present paper investi-gates the flow and the heat transfer in a high speedrotating clutch. High thermal loads of standardclutch design can only be reduced by inner heatconduction. However, the clutch fails if the torquewill not be transmitted to the gear box by a slippingoverheated clutch disc located between the pressureplates. Therefore, advanced design concepts tryto reduce thermal loads by convective coolingmethods. It will be shown in some detail, thatnumerical methods as the one described in thispaper, is a valuable tool for the assessment of dif-ferent design concepts. However, prior to a param-eter variation which has been performed in order to

get some indications for an effective and reliableclutch design, the numerical procedure will bepresented and analyzed.

NUMERICAL PROCEDURE

Governing Equations

The air flow through a complex three dimensionalgeometry is described by the Reynolds averagedNavier Stokes equations in combination with theconservation equations ofmass and energy. The setof partial differential equations can be written inthe form

where represents the three velocity components,the enthalpy and the turbulence quantities. Theturbulent stresses are modeled by the Boussinesqapproach (e.g. Jischa [1982]) in combination withthe standard k-e turbulence model (Launder andSpalding [1974]) which has been used successfullyin previous studies concerning the flow throughstationary and rotating orifices (e.g. Wittig et al.[1994], [1995]). For the determination of the nearwall velocities, the logarithmic wall function hasbeen used. In order to deal with complex geome-tries, the transport equations are transformed intoa general curvilinear coordinate system and discre-tized by the finite volume method. The advectionterms are represented by a second order discretiza-tion scheme, namely the Linear Profile Scheme(LPS), combined with Physical Advection Correc-tion terms (PAC; Raithby [1976]). Further detailsof the numerical code are described in Raw et al.[1989].

Rotating Frame of Reference

From a stationary point of view, the flow throughrotating orifices is periodic and unsteady.

NUMERICAL STUDY OPTIMIZING HEAT TRANSFER 153

Switching to a co-rotating frame of reference, thesystem is transfered to steady state conditions.However, solving the Navier Stokes equations,additional source terms including coriolis andcentripetal forces have to be considered. The sub-stantial time derivative of the rotating system (withthe velocity vector ) can be obtained from thestationary system (with the velocity vector 6) e.g.Spurk [1989]

Rotatingsystem

Stationary Coriolis Centripetalsystem acceleration acceleration

dfx--a.

Unsteady Acceleration

(2)

y E

FIGURE Finite volume in cylindrical coordinates.

B

Eq. (2) can be simplified by assuming the rotat-

ing system as consisting of steady state conditionssuperimposed by a constant angular speed f.Finally, the additional source terms arising fromrotation are given by

S2--2pfixff-pfix(fixF). (3)

In the following discussion it will be shown thatdepending on the discretization scheme additionalterms of the rotating frame of reference occur.

Regarding flow problems in rotating machineryit is preferable to use cylindrical coordinates

(Fig. 1) in case of rotation around the z-axis. Thevelocity vector in the stationary system is given by

C

Wz(4)

whereas in the rotating system it is given by

W-- PzWz p,r,z"

The discretization of a finite volume (Fig. 1) byusing the Gauss divergence theorem for advection

and diffusion terms is given by

0(puirb)dV J’o/ (pui)dA

peueAeee pwuwAww

+ pnunAnn psusAss

-Jr- ptutAtt pbubAbb (6)

and (F&-

lttAe ,,,k4 #tAw /k4w + #t,An/k4nre/k99 rwAgw Ar,

#,.,AA + #t,At A #tAb A. (7)Ar Ar Ar

It is well known that the coefficients (I) and Ain Eq. (6) and (7) must be replaced by q taken fromthe control volume center. In order to access theimpact of the discretization scheme on the trans-

formation, two representatives for different dis-cretization orders have been compared. As shownin Fig. 2, using UPWIND differencing for the


flow direction

W w P e E W w P e E

UPWIND Central Differencing

FIGURE 2 UPWIND and Central Differencing discretiza-tion scheme.

advection terms an estimation for Be is given by

Be-BP. (8)

The determination of AB using the CentralDifferencing Scheme follows from

Z(I) (BE Bp). (9)

Inserting Eq. (8) and (9) into Eq. (2) and sub-stituting the local velocity vector (Eq. (5)) into theleft hand side and the stationary velocity vector(Eq. (4)) into the right hand side, equivalent expres-sions for both sides of Eq. (2) should be given.Contrary to this, additional terms, e.g. for thetangential velocity (B Uz) like

S, k. co. Ar (10)

Using the Central Differencing Scheme for theadvection terms an advanced estimation for Be isfound instead of Eq. (8) (Fig. 2):

Be Bp --- (Be. (11)

The correction term 6Be can be obtained from

with

6Be -- X (12)

O )eZXX

Therefore, the use of the Central DifferencingScheme results in a correction proportional (BE--Bp). In comparison with Eq. (9) error terms can bedetermined, too.A second method for reducing numerical diffu-

sion is the adaptation of a finer mesh in order tominimize the error terms. Considering this strategyduring the mesh generation, a compromise betweenhigh spatial resolution and efficiency, e.g. CPUtime and memory requirement must be found.

APPLICATION OF THE ROTATING FRAMEOF REFERENCE

are obtained. Thus, the coefficient Ar depends onthe direction of the radial velocity component.Summarizing the derivation above, it is shown thatswitching to a rotating frame of reference and usingthe UPWlND-discretization a numerical error offirst order is found. This discretization error issimilar to the well known problem of numericaldiffusion (Patankar [1980], Noll [1993]). Thus, it iscalled an additional numerical diffusion. Anappropriate method for its reduction, is the appli-cation of a higher order discretization scheme. Thefollowing example will show, that a dramaticaldecrease can be achieved, but the numerical diffu-sion caused by the rotation of the system cannot beavoided.

The objective of an inner cooling system of anautomotive high speed rotating clutch is a reduc-tion of thermal loads at the pressure plates and thedisc as shown in Fig. 3. In order to generate aninternal flow, several orifices have been drilled intothe clutch housing. It has to be pointed out that theflow pattern shown in Fig. 3 expresses the desiredflow path. Any detailed information is not avail-able and, therefore, one task of the flow simulationis given by an initial assessment of the potential foran internal flow generation as it has been envisagedby the design of the setup shown in Fig. 3.Fundamental investigations are required aboutthe flow structure which is influenced by the flowthrough the orifices. Because of this, a research


program has been performed investigating theparameters which affect the flow pattern. As abasis of the numerical study, the geometry of theDamped Flywheel Clutch (DFC) has been taken(Fig. 3).Analogous to numerical studies performed by

Wittig et al. [1995] the clutch geometry has beenabstracted by a simple model. As shown in Fig. 4the analysis of fundamental flow and heat transferphenomena has been performed in a numerical gridconsisting of three main components.

transmission side engine side

desired flow path

orifice (05) desired flow path

disc

pressure plate

orifice (03)orifice 4 (04) orifice (O1)

rotating clutchhousing

FIGURE 3 Geometry of the clutch (LuK [1994]).

Rotating clutch disc: It consists of the pressureplates and the disc (Fig. 3). If the clutch isreleased, both components must be cooled byheat conduction or an internal air flow.Rotating clutch housing: It encloses the clutchdisc and has been supplied with several orificesdrilled into the housing. Influences oflocation andsize ofthe orifices will be investigated with respectto improvements of the cooling air flow.Stationary housing: As it connects the gear boxto the engine, the different sides of the clutchassembly will be referred to as transmission andengine side.

The hot section of the clutch disc’ (Fig. 4b)remains the same for all configurations and is setto a temperature of Tw,h--200C. All other partsof the geometry were set to a temperature of

Tw, 100C. Since the clutch geometry is periodicin circumferential direction, the computationalarea can be reduced to 1/8 of the whole geometry.The circumferential areas are linked together usingperiodic boundary conditions. In order to avoiddiscretization errors a high order discretizationscheme as described above has been used in combi-nation with a mesh of 40 x 19 x 45 points.

transmissi cold part of theside \ computationall clutch disc (Tw c )

domain

hot part of thesc (T W,h )

engine side __.._-

Ensemble model b) Clutch disc

FIGURE 4 Numerical model.


05

transmissions de /

z

hot part of/the clutch disc

engineside

\03

BASIS-DFC’Orifice rounded, in clutch discOrifice 3: rounded, in the rotating clutch

housing at the engine sideOrifice 4: gap between rotating clutch

housing and shaft at the gearbox side

Orifice 5: squared, radial in rotating clutchhousing

DF:yri-O3 I)FC,Fin,! DFC-Fjn-2with radial fins

radial position orifice with radial finsradial position orifice

FIGURE 5 Geometry definition.

TABLE Geometry variations

Configuration Orifice (O1) Orifice 3 (03) Orifice 4 (04) Orifice 5 (05)

BASIS-DFC d 20 d 28 25 Az 15

ro 40 ro 80 ra 45 Ax 18

DFC-vari-O3 d 20 d 28 ri 25 Az 15

ro=40 ro= 100 ra =45 Ax 18"

DFC-Fin-1 d= 20 d= 28 ri-- 25 Az 15

ro 40 ro 80 ra 45 Ax 18

DFC-Fin-2 d 20 d 28 ri 25 Az 15

ro=40 ro= 100 ra=45 Ax= 18

The configurations investigated in the presentstudy are shown in Fig. 5 and Table I summarizesthe appropriate dimensions. The entire numericalstudy includes numerous variations of orificepositions and sizes. In addition, the geometry ofthe radial orifice (05) inside the housing has beenchanged from a rounded to squared shape whichallows an increase of orifice area. As it will be seenin the discussion of numerical results, only thevariation of the axial orifice in the housing (03)gives a suitable effect in cooling the clutch disc. Inaddition, two configurations with radial fins on thetransmission side of the rotating clutch housinghave been studied. The main objective of this partof the investigation is the analysis of tangentialacceleration effects on the flow passing through theclutch and, therefore, on the heat transfer. The finsare simulated by a wall boundary condition intangential direction.

RESULTS

The discussion of numerical results comprises an

examination of the internal clutch flow which is

presented first in terms of the geometry variationdefined above and, secondly, a discussion of heattransfer phenomena which are highly affected bythe flow structure.

Flow Structure

As a first insight into internal clutch flows, Fig. 6illustrates the flow between clutch housing andstationary outer housing of the basic configurationBASIS-DFC defined in Fig. 5. The flow pattern onthe engine side (Fig. 6d) is similar to the flowbetween stationary and rotating discs discussed byOwen [1984]. In the small gap between rotor andstator the boundary layer at the stationary and at


a) Flow between rotor andstator (transmission side)

,.:-’i’,........

transmissionside

b) Flow in theradial gap

Vector scale

lOm/s

rotating wall

stationary wall

engineside

c) Flow field atorifice 3

FIGURE 6 Basic flow structure.

d) Flow between rotorand stator (engine side)

the rotating wall have different characteristics.Fluid moves radially outward near the rotatingclutch housing and radially inward at the station-ary outer wall. The flow pattern at 03, which isdrilled in the rotating clutch housing at the engineside, shows an axial in- and outflow caused by acircumferential acceleration of the air (Fig. 6c). Asshown in Fig. 6b the flow structure in the radial gapis similar to a Taylor vortex flow and is very sensi-tive to geometric and aerodynamic variations.Therefore, the strong velocity gradients between ro-

tating clutch housing and stationary outer housingcause a highly mixed, turbulent flow pattern in thisregion. In comparison with the engine side the samekind of flow structure is found at the transmission

side (Fig. 6a). However, a small difference occursdue to the inclined stationary outer wall, whichcauses an additional deceleration of the radialinflow.As it may have a significant influence on the heat

transfer on the clutch disc, the discharge character-istics of the orifices have to be investigated. Theintegrated mass flux is shown in Fig. 7a for eachorifice varying the geometrical parameters as

defined above (Fig. 5). As it is well known, theintegrated mass flux did not include the effects ofrecirculation zones inside orifices, i.e. amounts oflocal velocity caused by simultaneously in- andoutflow at the orifice are much higher than themass averaged mean velocity (Fig. 6c). Only a very


k BASIS-DFC IiW/A DFC-vari-03t DFC-Fin-1

0.4 DFC-Fin-2

0.2

0.0

-0.4O1 03 04 05

0.8

0.6

0.4

0.2

0.0BASIS-DFC DFC-Fin-1

DFC-vari-03 DFC-Fin-2

Mass flow through single orifices b) Main mass flow

FIGURE 7 Discharge characteristics of the clutch.

small mass flow entering the clutch is obtainedfor configurations without fins. Most of the airsupplied by 04 passes the clutch in radial directionat the transmission side. The integrated mass flowsthrough O and 03 show values next to zero, whichindicates that a weak radial secondary flow existsat the engine side. Different locations of 03 (DFC-vari-O3) have a small effect on inner mass flow.However, an increase of the turbulence level causedby the in- and outflow through 03 is obtained,which affects the heat transfer from the hot part ofthe clutch disc to the cooling air flow.

Radial fins fixed as shown in Fig. 9d cause astrong increase of convective flow at the transmis-sion side of the clutch disc, which increases massflow through all orifices. This is shown by an

increase up to a factor of three for 05 and a factorof about two for 04. The air is accelerated intangential direction where a radial pressure gradientin the rotating clutch housing is caused. The flowpattern at the engine side of the clutch disc haschanged to a radial inflow, which causes an increaseof the integrated mass flows through O1 and 03(DFC-Fin-1). A decrease of the radial inflow at theengine side will be obtained, if the location of 03 ischanged. For this case an additional radial pressuregradient in the gap between disc and rotating clutchhousing reduces the air flow.

Because it gives an information about the totalconvection inside the clutch, all air flows passingthrough the orifices into the rotating clutch housingare summarized to a mean mass flow shown inFig. 7b. This figure illustrates the strong increase ofmass flows through the clutch for finned configura-tions. It is assumed that in comparison with theBASIS-DFC the heat transfer of this configura-tions will be increased, too.

Heat Transfer

The objective of the present clutch design study isto improve the heat transfer from the hot part ofthe clutch disc to the cooling air. In order todescribe influences of geometrical variations, localas well as mean heat transfer coefficients have beendetermined.

local A T’ (14)

fA OeAidA (15)Omean fA dA

Figure 8 shows the mean heat transfer coeffi-cients (Eq. (15)) of the configurations investigatedin this study. As generally, the heat transfer at theengine side of the unfinned configurations is higher

NUMERICAL STUDY OPTIMIZING HEAT TRANSFER

10

0BASIS-DFC DFC-Fin-1

DFC-vari-O3 DFC-Fin-2

FIGURE 8 Averaged heat transfer coefficient.

159

vector scale

a) BASIS-DFC b) BASIS-DFC c) DFC-vari-O3 d) DFC-Fin-1transmission side engine side engine side transmission side

FIGURE 9 Velocity vectors at the hot section of the clutch disc.

than the one at the transmission side. With fins, a

significant increase of heat transfer coefficients atthe transmission side is observed as well as at theengine side. The parameters influencing the heattransfer may be examined by the use of local heattransfer coefficients as well as local flow conditionsas shown in Fig. 9 and Fig. 10.

Figure 9a shows the tangential flow pattern atthe transmission side of the clutch disc. Only a

small radial outflow is observed, which provides alow convective transport of the heated air in radialdirection through 05. Therefore, heat transfercoefficients decrease in radial direction (Fig. 10a).At the engine side the tangential main flow direc-tion is disturbed by the influence of 03 (Fig. 9b).The flow pattern shown in Fig. 6c reveals an in-and outflow which causes a high level of turbu-lence. Therefore, an increase of averaged heat


90 90 90 90

" 60 60 6060

: 300 300 300 30

0

a) BASIS-DFC b) BASIS-DFC c) DFC-vari-O3 d) DFC-Fin-1transmission side engine side engine side transmission side

FIGURE 10 Local heat transfer coefficients at the hot section of the clutch disc.

transfer is observed (Fig. 8). Comparing the flowpattern and the local heat transfer coefficients ofFig. 9b with Fig. 9c (Fig. 10b with Fig. 10c) it isdemonstrated that the variation of the 03 positionlead to an increase of heat flux perpendicular to thedisc. This can be seen by a peak of the local heattransfer distribution shown in Fig. 10c. The flowpattern will be changed totally with a finned radialgap (Fig. 9d and 10d). The fins prevent a flowpattern in tangential direction as shown above. Theradial pressure gradient induces a higher massthrough flow, which leads to a strong increase ofthe heat transfer at the transmission side. Further-more, a secondary flow between pressure and suc-tion side of the finned sections causes an additionalincrease of local c-numbers.

Summarizing these effects at the transmissionand the engine side, the DFC-Fin-2 configuration(Table I) can be discussed exemplary, because itgives a maximum reduction of thermal loads of theclutch disc. As the dominating parameter at thetransmission side a finned clutch disc which causesa high radial mass flow is favored. At the engineside the increase of turbulence caused by a hightangential velocity of 03 has a significant influenceon the heat transfer. Discussing mean heat transfercoefficients, the decrease of Omean at the transmis-sion side is prevailed by the increase at the engineside. The total mass flow through the core of theclutch decreases in comparison with the configura-tion DFC-Fin- 1.

CONCLUSIONS

A detailed investigation of effects arising from thetransformation of a fixed coordinate system to a

co-rotating frame of reference is presented. As a

typical application, numerical results of the flow ina high speed rotating clutch are presented withemphasis on geometrical variations. Calculationsare carried out in order to find an optimum designin terms of location and size of the orifices in thehousing supplied for a reduction of high thermalloads in the core region of the clutch. Thesevariations induce different physical phenomena.Special consideration is directed towards the basicproblems of the flow through orifices in high speedrotating discs and shafts and the flow inside rotor-stator systems. As expected, the very complex flowfields are dominated by rotational effects. In addi-tion, it is shown that differences occur between theconfiguration of optimized mass flow rate and thegeometry with a maximum total heat transfer.Obviously, optimization procedures depend on theknowledge of local flow fields and cannot be per-formed without advanced CFD-methods. It isdemonstrated that the approach presented here issuitable for these tasks.

Acknowledgements

This work was supported in part by LuK GmbH &Co Germany. Thanks are due to Dr. A. Albers,


K. Kasper and M. Lipps of LuK GmbH for theircontinuous support and helpful discussions.

NOMENCLATURE

Symbols

A m2

m/sd rnk m2/s2kuzrh kg/sn 1/mingl W/m2p, r, z (C), m, rn

SeT Ku, v, w m/s

Uz, Vz, Wz m/s

wx, y, z

#t

P

m/s

W/(m2. K)

m2/skg/(m, s)

kg/m3

1/s1/s

area

velocity vectororifice diameterturbulent kinetic energyfactor of the error termmass flowrotational speedspecific heat fluxcylindrical coordinatessource termtemperaturevelocity components incartesian coordinates

velocity components incylindrical coordinates

local velocity vectorcartesian coordinatesheat transfer coefficientdiffusion coefficientdissipation rateturbulent dynamicviscosity

densityflow parameterangular velocityangular velocity vector

Subscripts

c

e, w, s, n, t, bE,W,S,N,T,B,Pho

W1,3...

coldsurface of a finite volumecenter of a finite volumehotorificewallorifice number

References

Hay, N. and Spencer, A., 1991. Discharge coefficients of coolingholes with radiused and chamfered inlets, ASME91-GT-269.

Jischa, M., 1982. Konvektiver Impuls-, W/irme- und Stoffaus-tausch, F. Vieweg & Sohn, Braunschweig/Wiesbaden.

Launder, B.E. and Spalding, D.E., 1974. The numericalcomputation of turbulent flows, Computer Methods inApplied Mechanics and Engineering, Vol. 3, pp. 269-289.

Lichtarowicz, A., Duggins, R.K. and Markland, E., 1965.Discharge coefficients for incompressible non-cavitating flowthrough long orifices, Journal Mechanical EngineeringScience, Vol. 7, No. 2.

LuK GmbH & Co., 1994. Comfort and economy, LuK drivetrain systems, 5. LuK Kolloquium, Biihl/Baden, pp. 32.

McGreehan, W.F. and Schotsch, M.J., 1987. Flow character-istics of long orifices with rotation and corner radiusing,ASME 87-GT-162.

Noll, B., 1993. Numerische Str6mungsmechanik, SpringerVerlag, Berlin Heidelberg New York.

Owen, J.M., 1984. Fluid flow and heat transfer in rotating discsystems, Heat and Mass Transfer in Rotating Machinery,Hemisphere Publishing, New York.

Patankar, S., 1980. Numerical heat transfer and fluid flow,Hemisphere Publishing, New York.

Raithby, G.D., 1976. Skew upstream differencing schemes forproblems involving fluid flow, Computer Methods in AppliedMechanics and Engineering, Vol. 9, pp. 153-164.

Raw, M.J., Galpin, P.F. and Hutchinson, B.R., 1989. AColocated Finite Volume Method for Solving the Navier-Stokes Equations for Incompressible and CompressibleFlows in Turbomachinery: Results and Applications FirstCanadian Symposium on Aerospace Populsion, Ottawa,Ontario, Canada.

Scherer, T., 1994. Grundlagen und Vorausetzungen der numer-ischen Beschreibung von Durchflul3 und Wfirmetibergang inrotierenden Labyrinthdichtungen, Dissertation, Universitg’t

Karlsruhe, Lehrstuhl und Institut fiir Thermische StrO’mungs-maschinen.

Spurk, J.H., 1989. Str6mungslehre, Springer Verlag, BerlinHeidelberg New York.

Waschka, W., Scherer, T., Kim, S. and Wittig, S., 1992. Study ofheat transfer and leakage in high rotating stepped labyrinthseals, ISROMAC-4.

Wittig, S., Kim, S., Jakoby, R. and Weissert, I., 1994.Experimental and numerical study of orifice dischargecoefficients in high speed rotating disks, ASME 94-GT-142.

Wittig, S., Kim, S., Scherer, T., Jakoby, R. and Weissert, I.,1995. Durchflul3 an rotierenden Wellen- und Scheibenboh-rungen und Wirmeiibergang an rotierenden Wellen,Abschluflbericht, Forschungsvereinigung Verbrennungskraft-maschinen (FVV), Vorhaben Nr. 465 und 536, Heft R 574.

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Numerical Study for Transfer in Speed Rotating Components · 2019. 8. 1. · systems. Asexpected, the ... Keywords: Internal heat transfer, Flow in rotating disks with orifices, Numerical