XK'Sj

Friction losses and heat transfer in high-speed electrical machines

Juha Saari

DJSTR10UTK)N OF THI6 DOCUMENT IS UNLIMITED

Teknillinen korkeakoulu Sahkotekniikan osasto Sahkomekaniikan laboratorio

Otaniemi 1996 150

Helsinki University of Technology Faculty of Electrical Engineering Laboratory of Electromechanics

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Saari J. 1996. Friction losses and heat transfer in high-speed electrical machines. Helsinki University of Technology, Laboratory of Electromechanics, Report 50,

Espoo, Finland, p. 34.

Keywords: Friction loss, heat transfer, high-speed electrical machine

Abstract

High-speed electrical machines usually rotate between 20 000 and 100 000 rpm and the peripheral speed of the rotor exceeds 150 m/s. In addition to high power density, these figures mean high friction losses, especially in the air gap. In order to design good electrical machines, one should be able to predict accurately enough the friction losses and heat-transfer rates in the air gap. This report reviews earlier investigations concerning frictional drag and heat transfer between two concentric cylinders with inner cylinder. rotating. The aim was to find out whether these studies cover the operation range of high-speed electrical machines. Friction coefficients have been measured at very high Reynolds numbers. If the air-gap surfaces are smooth, the equations developed can be applied into high-speed machines. The effect of stator slots and axial cooling flows, however, should be studied in more detail. Heat-transfer rates from rotor and stator are controlled mainly by the Taylor vortices in the air-gap flow. Their appearance is affected by the flow rate of the cooling gas. The papers published do not cover flows with both high rotation speed and flow rate. Furthermore, the effect of stator and rotor slots cannot, be predicted by the existing methods.

ISBN 951-22-3119-0 ISSN 0784-4662 TKK OFFSET

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Preface

The report gives a literature review concerning friction losses and heat transfer between concentric cylinders. This information is very essential in the design of

high-speed electrical machines.

This study is one part of a research work dealing with cooling of high-speed

electrical motors. The work belongs to a project aiming at increasing the performance of high-speed electrical motors developed in 1991-1994. The research is

financed by High Speed Tech Ltd.I would like to express my gratitudes to Dr Jaakko Laijola and Dr Antero

Arkkio for their advices concerning this work. Financial supports from Graduate School of Electrical Power Engineering, as well as foundations “Tekniikan edistamissaatio” and “Suomen Kulttuurirahasto, Hameen rahasto” are gratefully

acknowledged.

Espoo, 22.05.1996

Juba Saari

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Contents

Abstract......................................................................................................................................... 2Preface............................................................................................................................................3Contents......................................................................................................................................... 4List of symbols............................................................................................................................. 51 Introduction............................................................................................................................. 72 Fluid flow in the air gap.........................................................................................................8

2.1 Tangential and axial velocity distributions...................................................... 8

2.2 Taylor vortices.......................................................................................................103 Friction torque........................................................................................................................124 Heat transfer.......................................................................................................................... 185 Discussion............................................................................................................................... 26

5.1 Friction torque.......................................................................................................265.2 Heat transfer......................................................................................................... 27

6 Conclusions.............................................................................................................................29References................................................................................................................................... 30Appendix A: Friction coefficient........................................................................................... 33Appendix B: Nusselt number................................................................................................ 34

DISCLAIMER

Portions of this document may be illegible in electronic image products. Images are produced from the best available original document

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List of symbols

Cf Friction coefficienth1 Heat-transfer coefficient of inner cylinderh2 Heat-transfer coefficient of outer cylinderhl2 Heat-transfer coefficient between inner and outer cylindersk Thermal conductivityl Length of the inner cylinderNu-i Nusselt number of inner cylinderNu2 Nusselt number of outer cylinderNu12 Nusselt number between inner and outer cylindersPr Prandtl numberqm Mass flow rate of cooling fluidr Radiusrx Radius of inner cylinder (rotor)r2 Radius of outer cylinder (stator)rm Mean air-gap radius: r^=0.5(r^+rg)Rea Axial Reynolds number for axial flowRes Couette Reynolds number for tangential flowReT Tip Reynolds number of rotating cylinderT TorqueTa Taylor numberTam Modified Taylor numberTae Critical Taylor numberu Tangential fluid velocity between the cylinders

Peripheral speed of inner cylinder um Mean tangential fluid velocity between the cylinders v Tangential fluid velocity between the cylindersve Effective fluid velocity between the cylinders vm Mean axial fluid velocity between the cylindersy Radial location between the cylindersz Axial location between the cylinders

8 Length of the clearance between cylinders% Eddy diffusivity of momentumfi Dynamic viscosityv Kinematic viscosityp DensityT Shear stressxl Shear stress on the inner cylinderco Angular velocity

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1 Introduction

High-speed electrical machines usually rotate between 20 000 and 100 000 rpm and the peripheral speed of the rotor exceeds 150 m/s. These machines have high power

densities but also high loss densities. Especially the friction losses are very large and effective cooling methods are needed. In order to design good high-speed

machines, the designer has to know how the friction losses and heat transfer

depend on the machine geometry, rotation speed, and cooling method. The performances of high-speed electrical machines have been reported, for example, by Boglietti et al. (1992), Gilon (1994), and Saari et al. (1995a and 1995b).

The most critical friction losses for the electrical machine are located in the air gap. The rotor is cooled mainly by the air-gap gas and, therefore, the temperature of rotor is increased by the friction losses. On the other hand, one can lower the air-gap temperature by introducing an axial cooling flow through the air gap. In addition to losses, the rotor temperature depends on the heat-transfer coefficient on the rotor surface. This parameter is set by the peripheral speed of the rotor and axial velocity of the cooling gas. Naturally, the temperature of the stator depends also on the losses and heat-transfer coefficients in the air gap.

This report reviews the earlier investigations concerning friction losses and heat transfer between concentric cylinders. The aim of the study was to find out whether one can apply these calculation methods on high-speed electrical machines. Only analytical methods were considered because they are preferred in the cooling design by thermal networks. The analysis was focused on the turbulent flow which is the case of high-speed machines.

Friction losses and convection heat transfer are set by the velocity field in the

air gap. This includes the tangential and axial flows as well as Taylor vortices discussed in Chapter 2. In Chapters 3 and 4, the most important research works concerning friction generation and heat transfer between concentric cylinders are reported. The papers are summarised in chronological order and the experimental setups are listed at the end of the corresponding chapter. In Chapter 5, the applicability of analysis methods on high-speed machines is discussed. Chapter 6 gives the conclusions of this study.

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2 Fluid flow in the air gap

Friction losses and heat transfer are set by the velocity field. The velocity distribution in the air gap of electrical machines is controlled by the following flows:

1. Tangential flow due the rotor rotation.2. Axial flow of the cooling gas through the air gap.3. Taylor vortices due to centrifugal forces.

The importance of each flow depends on the peripheral speed of the rotor, flow rate of the cooling gas, gas properties and air-gap geometry. Typical peripheral speeds of rotors are 150-500 m/s and the axial velocity of the cooling gas is 20-60 m/s.

2.1 Tangential and axial velocity distributions

The nature of a fluid flow is determined by the ratio between the inertia and viscous forces, called the Reynolds number. The tangential flow is forced by the rotating rotor and the turbulence is described by the Couette Reynolds number

(1)

where p is the density and ji is the dynamic viscosity of the fluid, u1 is the peripheral

speed of the rotor and 5 is the radial air-gap length. For flows through the air gap

we have

(2)P

where »n is the mean axial fluid velocity and 25 is the hydraulic diameter.

When the Reynolds number is below 2000, all the individual fluid particles are flowing in the same direction and the flow is laminar. If we increase the fluid velocity, the Reynolds number increases and velocity fluctuations appear in the flow. Although the flow has some mean direction of velocity, the individual fluid particles have no longer the same velocity direction. This kind of a flow is turbulent and is usually the case in electrical motors.

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Laminar flow Turbulent flow

Tangentialflow

stator'S^>>SsSsduy^=constant "X. viscous layer

fully turbulent layer

X^viscous layer

rotor “l

Axial flow

statorviscous layer

ua— ► fully turbulent layer

__s viscous layer

rotor

Fig. 1. Tangential and axial velocity profiles of laminar and turbulent air-gap flows.

Fig. 1 presents the tangential and axial velocity distributions of the air-gap flow. The tangential flow is viewed from the end of the machine and the curvature of the air gap is neglected. These kinds of flows are called the Couette flows. The fluid

velocity near the rotor is the same as the peripheral speed of the rotor. Correspondingly, the velocity near the stator is zero. For the laminar flow, the velocity distribution is linear. In the turbulent case, one can separate three layers: Two viscous layers near the walls and one fully turbulent layer in the middle flow.

In the viscous layers, the generation of friction, as well as energy transfer, is determined mainly by the molecular viscosity of the fluid. The thickness of the layer decreases with an increasing Reynolds number. In the middle flow, the chaotic motion of fluid particles is independent of viscosity. The highest velocity gradients in the mean velocity are in the viscous layers.

The lower figures show sideviews of axial air-gap flows. In the laminar flow, the fluid velocity has a parabolic distribution. In the turbulent flow, one can separate the same regions as in the tangential flow.

In addition to Reynolds number, the velocity distributions are affected by the roughness of the wall surfaces. In general, roughness decreases the velocity gradients in the flow.

The calculation of fluid velocities is based on Navier-Stokes and continuity equations. These can be solved analytically in the case of the laminar flow. Turbulent flows, however, are more difficult to analyse, because the viscosity terms

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in the equations have to be replaced by terms for turbulent Motional stresses. These parameters are semi-empirical constants. If turbulent flows are analysed

analytically, the flow is usually divided into several layers and each layer is analysed separately. Turbulent flows in engineering have been discussed, for example, by Reynolds (1974).

22. Taylor vortices

Taylor vortices are circular velocity fluctuations appearing in the air gap (Fig. 2). They are due to the centrifugal force affecting of fluid particles. At low rotation speeds the flow is laminar, because the creation of vortices is damped by frictional

forces. In addition to the Couette Reynolds number, the creation depends on the radial air-gap length. These parameters are included in the Taylor number

6 rl h(3)

where a> is the angular velocity and r1 is the rotor radius. The rotor radius should be

replaced with the mean radius of rotor and stator if the air-gap length is large. Taylor vortices occur when the Taylor number exceeds 1.7xl03 (Gazley 1958).

The critical Taylor number is affected by many aspects, such as the ratio between the rotor and stator radii, temperature of the rotor and stator, as well as the

stator1500-

500-rotor

1E+02 1E+04 1E+061E+00

Fig. 2. In the left: Sideview of the Taylor vortices in the air gap. To the right: Flow regimes for superimposed tangential and axial flows. I Laminar, II Laminar+Taylor vortices, III Turbulent and IV Turbulent+Taylor vortices. The solid lines are according to Dorfman (1963) and the dashed line by Simmers (1979).

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flow rate of the cooling fluid. Especially the effect of the axial flow is significant and

four different flow regimes have been separated according to Taylor vortices and turbulence of the axial flow. These regimes are shown in Fig. 2. The figure, however, does not show the disappearance of Taylor vortices which is supposed to

happen above jRe5=5xl04 (Smith et al. 1982). A comprehensive review, dealing with

the stability of flows between concentric cylinders, has been reported by Di Prima et

al. (1981).

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3 Friction torque

The tangential force per area is described as a shear stress. If we consider a laminar flow between two concentric cylinders, the shear stress has the equation

dT = pvr— or

(4)

where v is the kinematic viscosity, r is the radius and u is the tangential fluid

velocity. Fig. 2 suggests that Eq. 4 may be valid near the air-gap surfaces when the flow is turbulent. In the fully turbulent layer, however, the shear stress is set by the chaotic motion of fluid particles, while the fluid viscosity is a minor factor. For this reason, the shear stress for turbulent flows is written into the form

T = p(v+sM) (5)

where is the eddy diffusivity of momentum increasing with the distance from the

walls. In order to solve Eq. 5, one should know the velocity distribution and eddy diffusivity in the flow. There are no complete models for turbulence and these factors, therefore, are based on measured data of different kinds of flows.

The shear stresses near the walls are the ratio between the forces and surface areas. In the axial direction, the frictional forces are balanced by pressure forces in the air gap. The force needed to blow the cooling gas through the air gap is maintained by a ventilator. The ventilator is supplied by an external power source or it is driven directly by the electrical motor. In the tangential direction, the frictional forces are balanced by the electromagnetic torque acting on the rotor. This report focuses on the friction torque, pressure forces are not discussed.

As the shear stresses in turbulent flows are difficult to solve, the frictional drag in usually defined by a dimensionless friction coefficient. It is an empirical coefficient depending on many factors, such as the nature of the flow and the surface quality. For rotating cylinders, the friction coefficient is

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Cf = (6)

where t, is the shear stress on the rotor surface. By using Eq. 6, one can write the

friction torque into the form

T = CfpmxPril . (7)

where l is the axial air-gap length.

A historical review concerning friction coefficients of rotating cylinders is given in the following pages. One paper deals with rotating cylinders in free space and the rest with concentric cylinders. The papers are discussed in chronological order. The cylinders and flow ranges used in the measurements are presented at the end of this chapter (Table 1).

One of the earliest studies dealing with the friction torque of rotating cylinders has been published by Wendt (1933). Three different cylinders were tested at Couette

Reynolds numbers up to 105. Pure water and water-glycerol mixtures were used in the experiments. His measured data fit well with the following equations

0.46Cf =•

nO.25

ri2

Re,0.5

n0.25).073&Z2bT

Cf = -l ri2

(400<Re5<104)

(104<Re5<105)

(8a)

(8b)

where r2 is the radius of the outer cylinder.Theodorsen et al. (1944) have made a detailed study of friction torque of

rotating cylinders in free space. The tests were carried out in air, oil, kerosene and water and the highest tip Reynolds number achieved was 4x10s. Both smooth and rough cylinders were used in the measurements. At a Reynolds number of 10s, the friction coefficient increased from 6.3xl0'3 to lS.SxlO"3, when the cylinder surface was roughened with grains of sand. The size of the particles was 3% of the cylinder radius. The friction-coefficient curves of rough cylinders deviated from the curve of

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the smooth cylinder at lower Reynolds numbers when the particle size was increased. The deviation occurred, when the viscosity layer was decreased below the particle size. Above the critical point the friction coefficient was independent of Reynolds number. Theodorsen et al. measured rotating disks with supersonic speeds and found that the friction coefficient does not depend on the Mach number.

Yamada (1962a and 1962b) measured the friction torque between concentric cylinders with the inner cylinder rotating. He used smooth and circumferentially grooved cylinders in his experiments. In addition to the tangential flow, he had also an axial flow through the air gap. The tests were carried out in water and spindle oil, and the highest Couette and axial Reynolds numbers obtained were 6xl04 and 2xl04, respectively. The torque was measured in the test section where the velocity field was assumed to be fully developed. By neglecting the curvature of the air-gap

surfaces and the occurrence of Taylor vortices, Yamada developed some equations for the friction coefficient

1Wi

7.54+ 11.51og10(Re5V2Q:) (9)

which can be written into a more practical form

Cf =0.0152 (800<Re5<6xl04) (10)

He assumed that the fluid velocity varies proportionally to the distance from the air- gap midpoint with the power of 1/7. When the fluid has both tangential and axial

velocity components, the friction coefficient is

Cf =0.0152Re0.24

1+112 f 0.38

4 Re-Red )

(11)

Eq. 9 and 10 gave lower friction coefficients than the measurements. The difference increased with an increasing radial air-gap length. This was due to the Taylor vortices, which appear at lower Couette Reynolds number when the air-gap length is increased. Under an axial Reynolds number of 2x10s; the friction coefficient

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decreased with an increased axial Reynolds number. Above Rea=2xl03, the friction coefficient increased with an increased axial flow. Above a Couette Reynolds number of 3x10% the friction coefficient was independent of the axial Reynolds

number. This was tested up to Rea=104. These variations were due to the stabilising effect of the axial flow (Fig. 2) . Eq. 11 does not model these effects. The friction coefficients for circumferentially grooved rotors were mainly lower than for smooth

rotors. The effect of grooves, however, decreased at high Couette Reynolds numbers

(104).

Bilgen and Boulos (1973) have measured friction torque of enclosed smooth cylinders having Couette Reynolds numbers between 2xl04 and 2x10% Based on their own measurements and the experiments by some other authors, they developed equations for the friction coefficient. In the turbulent region, the friction coefficients are

(500<Res<104) (12a)

(104<Re5) (12b)

The experimental friction coefficients deviated less than 8.35% from the results calculated by Eq. 12. Two more equations were fit with results for flows in the laminar and transition regimes.

Polkowski (1984) has applied the general solution of momentum equation (Dorfman 1963) into rotor-stator systems with axial throughflows. For pure tangential flows he compared the calculated results with those determined for Couette flows (rx and r2 —> infinity). Approximation of the flow by the Couette flow results in an underestimation of the friction coefficient. The error grew when the ratio between the radial air-gap length and rotor radius was increased. In his paper, Polkowski underlined that rather large friction torques may be associated with the entrance effects of axial flows. Polkowski suggested that the torque needed to accelerate the axial cooling gas into a tangential movement follows the equation

0.3

C{ = 0.515 v

Cf = 0.0325^/

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r = |ffp(r|-r13)umMm (13)

where um and um are the mean axial and tangential fluid velocities, respectively. Eq. 13 assumes that the cooling flow has only an axial velocity component before entering the air gap. The mean tangential velocity is usually expected to be half of the peripheral speed of the rotor. In addition, Polkowski derived the axial air-gap length in which the acceleration takes place. This length is increased by the flow rate of the cooling gas and decreased by the rotation speed.

Ueyama et al. have measured in 1990 the friction losses for different kinds of rotors. The rotors were levitated with active magnetic bearings. The separation

between iron and friction losses was made by carrying out the tests both in air and vacuum. The measured data agreed well with Eq. 10 by Yamada.

Very accurate torque and wall shear stress measurements, as well as flow visualisation of rotating cylinders, have been reported by Lathrop et al. (1992a and 1992b). They operated at Couette Reynolds numbers from 800 to 1.23x10s, and different water-glycerol mixtures were used as the working fluid. The measured

friction torques were in agreement with Eq. 8 (Wendt 1933) within ±15 %. A careful analysis, however, showed that the Reynolds-number dependency of the torque is not constant but increases from 1.23 to 1.87 at 3xl03<Reg<106. Above the transition to

a fully irregular turbulent flow (Re5>1.3xl04), Lathrop et al. were able to predict the

torque in the same way as for open-wall bounded shear flows (pipe flow, duct flow

etc.).The discussion of scaling laws for the angular momentum of a completely

turbulent Couette flow has been continued by Panton (1992). He presented a theory similar to that for channel flow, except that the angular momentum was replaced by the velocity. Thin wall layers and an inviscid core region were assumed in the analysis. The theory is applicable at high Couette Reynolds numbers (>104) when the Taylor vortices have disappeared.

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Table 1. Experimental setups used in the friction torque measurements.

Reference Fluid Rotation Cylinder Res Surfacespeed rvl, 8 Rea quality[rpm] [mm] Re,

Wendt water — 100.0-137.5 — smooth(1933) water- 580 -

glycerine 9.5-47.0 —

Theodorsen air, oil — 152.4-1219.2 — smooth(1944) kerosene 6.4-76.2 - roughness

water free cylinder 4-4xl05 (1) in rotorYamada water 90-5xl03 180 6xl03-6xl04 smooth

(1962a and b) spindle oil 28.8-31.7 0—2xl04 cir. grooves0.43-3.32 - in rotor

Bilgen water 300-1.5xl03 212.7-249.2 2xl04-6xl04 smooth(1973) water- 127 -

glycerine 0.31-1.75 -

Ueyama air 0-105 13.8 - smooth(1990)

Lathrop water- 0-900 160.0 800-1.2x10® smooth(1992) glycerine 695.0 -

60.85 -

(1) Tip Reynolds number Rej. = BBBX

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4 Heat transfer

The Nusselt number is a dimensionless measure for the convection heat-transfer coefficient. If we consider a pure tangential air-gap flow, the Nusselt number is

Nu12 =k

(14)

where h12 is the heat-transfer coefficient across the air gap and k is the thermal conductivity of the fluid. If cooling fluid is pumped through the air gap, the Nusselt number is

= M (15)

where hx is the heat-transfer coefficient between the rotor surface and air-gap flow. Eq. 15 can be written for the stator surface by replacing the subscript 1 by 2.

The calculation of Nusselt number is usually based on Reynolds analogy in which heat transfer is linked with momentum transfer in the flow (Knudsen et al 1954). The analogy assumes that the shear stress and heat distributions vary in the same way from the wall. Moreover, the Prandtl number has to be near unity (for air Pr=0.7). For example according to the Reynolds analogy, the Nusselt number for the heat transfer between rotor and stator is

Nu12=~CfRes (16)

where Ct is the friction coefficient discussed in Chapter 3.The following pages give a historical review concerning heat transfer of

rotating cylinders follows. The papers are discussed mainly in chronological order. The cylinders and flow ranges used in the measurements are presented at the end of this chapter (Table 2).

One of the earliest papers dealing with the heat transfer between concentric cylinders was published by Gazley (1958). He tested several combinations of smooth and slotted air-gap surfaces. For a pure tangential flow, Gazley analysed the heat transfer based on the momentum-transfer theory valid in the velocity boundary

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layers near the air-gap surfaces. He neglected the Taylor-vortices and curvature of the air gap in his analysis. After calculating the friction coefficient, he obtained a Nusselt number by using the analogy between heat and momentum transfer

(Reynolds analogy). For smooth air-gap surfaces, the theory gave about 30% lower values than the measurements. The highest Couette Reynolds number tested was 104. Gazley confirmed the earlier observations that an axial flow through the air gap

increases the critical Taylor number. The measured Nusselt numbers were noticed

to follow the axial Reynolds number (Eq. 2) with the mean axial fluid velocity replaced by the effective velocity

ve=-Jvm+Jul2 (17)

The calculated Nusselt numbers with equations of turbulent channel flows were

below the measured data. Gazley used the average cylindrical surface area to compare the results of smooth and slotted cylinders. In a turbulent region, the

slotted cylinders had slightly higher Nusselt numbers than smooth cylinders. The heat-transfer coefficient of smooth rotors was noticed to be independent of the stator

slots.Bjorklund et al. (1959) have reported heat-transfer measurements for

concentric cylinders. Their research covered the Taylor numbers up to 2xl03. Above a Taylor number of 90, the measured Nusselt numbers were increasing with the square root of the Taylor number. In addition to measurements, Bjorklund suggested an analogy heat-transfer solution based on the friction-coefficient data. This theory gave about 20% higher Nusselt numbers than the experiments. The authors expected that the reason was in the fully turbulent layer in the air gap which was supposed to have a constant temperature.

Tachibana et al. (1960) extended the earlier studies by testing different fluids in the air gap and increasing the Taylor number up to a value of 4.5xl04. In the Taylor-vortex region, the measured data followed the equation

Nu12 = 0.21Ta°-5Pr°-25 (41<To<4.5xl04) (18)

where Pr is the Prandtl number. This experimental equation also agreed with Gazleys (1958) measurements.

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Tachibana et al. (1964) continued their earlier work by measuring the heat transfer of rotor-stator systems with an axial flow in the air gap. First, they modified Eq. 17 to fit the data for Taylor numbers up to 6xl05.

Nu12 = 0.046To°"67Pr0'33 (2xl03<2h<6xl05) (19)

By combining the equations for the axial flow in an stationary annulus and Eq. 19, Tachibana et al. predicted the heat-transfer rate for the rotor when there was an axial flow through the air gap. The method agreed relatively well with the experiments. The heat transfer from an axially slotted and smooth rotor was the same, when average cylindrical areas were used in the calculations.

Becker et al. (1962) have studied the heat transfer with both a tangential and axial flow in the air gap. They operated at Taylor and axial Reynolds numbers up to

3.3x10s and 1.2xl04, respectively. The Nusselt numbers due to the rotor rotation were

Nu12 = 0.128Tom°'37

Nui2 = 0.40971am0-24

(1700<7bm<104)

(104<Tam<107)

(20a)

(20b)

where the modified Taylor number can be calculated with equations

(21a)

KA(21b)

1697 1-

0.00056+0.0571( 2rm - 2.304(5 ^[2

>. 2rm — 8 j(21c)

21

The modified Taylor number is about the same as the Taylor number when the radial air-gap length is very small. Eq. 20 agreed well with the measured data and also with the results by Bjorklund et al. (1959). In addition, Becker et al. studied qualitatively the effect of axial flows. As one can see from Fig. 2, the critical Taylor number increases with an increasing axial flow. This means that at axial Reynolds number below 2xl03 the Nusselt number decreases with an increasing axial flow.

Above 2xl03, the flow becomes turbulent, and the Nusselt number increases again.

Fig. 3 shows clearly the importance of Taylor vortices discussed earlier in Chapter 2.2.

Aoki et al. (1967) have reported theory and experiments dealing with the heat transfer in rotor-stator systems. Their measurements covered the modified Taylor numbers up to 2x10s and several Prandtl numbers. The experimental Nusselt numbers were found to follow the equation

Nu12 = 0.22Tam°-25Pr0-3 (5xl03<Tom<2xl05) (22)

Their theoretical analysis was based on the analogy between friction coefficient and Nusselt number.

T 1 I I I I I 1 I I I I I | t i-r i m.____ .REYNOLDS NCe__ _2Z^-5960 -

\ 592

l t » t l t J_L I I M I i'll!ffl |

T TTTTTT T TTTTTT

,10531592

t * t t J t \ t > i i t i t t ti

MODIFIED TAYLOR NUMBER,

Fig. 3. Nusselt number for mixed tangential and axial air-gap flows (Becker et al. 1962).

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Kreith (1968) has made a literature review dealing with convection heat transfer in rotating systems. In addition to cylinders in enclosures, the paper discusses rotating disks and tubes. Another review report concerning the same topic has been published by Dorfman (1963).

Gardiner et al. (1978) have measured the heat transfer with and without axial flow through the air gap. In the case of a pure tangential flow, their results agreed well with Eq. 22 by Aoki et al. (1967) at modified Taylor numbers up to 4x10s. Measurements with axial flow were carried out with Prandtl numbers of 2.5, 4.5 and 6.5. When the axial Reynolds number was 800, the Nusselt number increased with Ta0A but at higher axial flows the dependency was much lower. The axial flow improved the heat transfer in the whole range of modified Taylor numbers measured. The authors were not able to give any general equation to predict the heat transfer with mixed tangential and axial flows. Gardiner et al. tested also a slotted rotor. By using the rotor radius and the area based on the cylindrical surface, improvements from 20 to 50% in the heat transfer were observed.

Simmers et al. (1979) have presented a Reynolds analogy solution to calculate the heat transfer between the stator surface and the air-gap flow. Firstly, they divided the flow into three regions: a laminar sublayer close to the outer wall, a buffer layer and a core region of constant velocity in the central part of the air gap. Secondly, they derived the thickness of layers according to the friction coefficients. After several simplifications they were able to use the Reynolds analogy and give the Nusselt number for the stator. It has the equation

(23a)

(23b)

where the terms A and N are defined by

23

A = (24a)

(24b)

The heat-transfer coefficient for stator surface can be calculated by

(25)

where h2 is the heat-transfer coefficient between the stator and the air-gap flow, and k is the thermal conductivity of the fluid. Eq. 23-25 are capable of modelling the stabilising effect of an axial flow in the air gap. The occurrence of Taylor vortices, however, should be known beforehand. Simmers et al. tested the equation within the range 400<Rea<1.2xl03 and 2xl04<Ta<2xl06. The agreement with the experiments was relatively good with a narrow (N=0.955) air gap. With a larger gap (N=0.8), an underestimation of 40% was obtained in the worst case.

Rao (1979) has reported a theoretical and an experimental investigation of heat transfer between two cylinders. He tested five rotors with different kinds of circumferential surface roughness. As water was used as the air-gap fluid, rather high modified Taylor numbers were achieved. The measurements were carried out at regimes 105<Tcm<2xl08 and 180<Rea<550, i.e. the flow was laminar with Taylor vortices. Compared with a smooth rotor an increase of 70% in the heat-transfer was achieved with a discrete type of roughness. For evexy rotor the Nusselt number decreased with an increasing axial Reynolds number. This agree with the tests of Becker et al. (1962). The experimental results were not compared with those calculated by the finite-difference techniques.

By modelling the air-gap flow as a Couette flow in crossflow, Polkowski (1984) has derived the Nusselt number for rotor and stator surfaces:

NuW=2¥PrC{Res (26)

---- Cfpuiz+Pr9m

24

where Cf is the friction coefficient (Chapter 3), qm is the mass flow rate of cooling gas and z is the axial location in the air gap. Radius r depends on whether one calculates the Nusselt number for the rotor or the stator:

r = ri+8 for rotor surface (27a)

r = r% --j 8 for stator surface (27b)

Eq. 26 is valid in flow regimes I and III (Fig. 2) i.e. laminar and turbulent flows

without Taylor vortices.The experiments by Lee et al. (1989) were based on the analogy between heat

and mass transfer: naphthalene sublimation technique was used to determine the heat-transfer coefficients in rotor and stator surfaces. Smooth and slotted air-gap

surfaces were tested. The experimental setup had such a design that the entrance effects could be studied. Measurements were carried out with and without an axial flow through the air gap. For smooth air-gap surfaces, the Nusselt number was independent of the axial Reynolds number (130 and 1500) above a Taylor number of 2x10s. With a slotted rotor the corresponding Taylor number was below 104. Furthermore, the Nusselt number did not vary with the rotor length indicating a very short entrance length. The results with a slotted stator surface followed the

same trend as with a slotted rotor.Hayase et al. (1992) have studied numerically the heat transfer of slotted

concentric cylinders. Their three-dimensional calculations were restricted to laminar flows. They were able to model both the Taylor vortices and recirculation fluctuations in the air gap and slots, respectively. The calculations showed that rotor slots influence the flow and heat transfer more than stator slots.

25

Table 2. Experimental setups used in the heat-transfer measurements.

Reference Fluid Speed [rpm] Axial vel.

[m/s]

Cylinder rvl, 8 [mm]

Res

Ta

Surfacequality

Gazley air 540-4700 57.66-63.31 -104 smooth(1958) 0-90 - 200-104 (1) axial slots in

0.43-6.10 — surfacesBjorklund air _ 28.70 - smooth

(1959) no axial flow 463.6 no axial flow1.55-7.06 20-2xl03

Tachibana air 3-2840 29-60 Ixl04-5xl04 smooth(1960) spindle oil no axial flow 210 no axial flow

mobile oil 0.88-55 5000-4.5xl04Tachibana air 44-2700 15-60 — smooth

(1964) water 4-32 210 380-5xl04 axial slots inmethanol 4-53 0-6xl04 rotor

Becker air — 34.53 — smooth(1962) — 1422 0—1.2xl04

8.12 0-3.3xl05 (2)Aoki air, water — 35.33-37.92 0-4170 smooth(1967) alcohol no axial flow 485 no axial flow

spindle oil 2.08-4.67 0-2x10® 09Gardiner water 0-160 51.5 — smooth

(1978) - 1220 0-7xl03 axial slots in5.5 0-10® (2) rotor

Simmers air — 55.9-66.7 — smooth(1979) - 1820 400-1.2xl03

3.15-13.95 2xl04-2xl0®Rao water up to 400 68 - smooth

(1979) 522 180-550 cir. grooves11 10®—2x10® e) in rotor

Lee air 50-4000 52.45-65.85 — smooth(1989) — 64-127 52-103 axial slots in

0.85-7.95(3) 103-2xl07 surfaces

(1> Mean axial velocity is replaced by the effective velocity of Eq. 16(2) Modified Taylor number(3) The smallest clearance between the inner and outer cylinder

26

5 Discussion

From the cooling point of view, some important data concerning high-speed electrical machines are presented in Table 3. The parameters are typical for motors driving turbomachines, such as pumps or compressors.

Table 3. Typical parameters of air-cooled high-speed electrical machines

Power range [kW] 30-500Rotation speed [rpm] 20000-100000Peripheral speed of rotor [m/s] 150-500Flow rate of cooling gas [m3/min] 2-20Axial gas velocity in air gap [m/s] 20-60Couette Reynolds number 5xl03—10sAxial Reynolds number 103-104Taylor number 106-109

As one can see in Table 3, the Reynolds and Taylor numbers become very high because of the high peripheral speed of the rotor. One should also notice that the speed can exceed the speed of sound (i.e. Mach number maybe >1).

Electrical machines have stator slots. At high rotation speeds, the laminar

velocity layer becomes very thin and the surface is hydraulically rough, even if the axial grooves are filled with epoxy resin. Besides, the stator surface has always some circumferential roughness because of the laminated structure. In order to sustain peripheral speeds over 200 m/s, the rotor has to be made of solid steel. Poor electrical performance of such a rotor can be improved by using axial and circumferential grooves on the rotor surface. Even better results can be achieved

with end rings and rotor bars.

5.1 Friction torque

Earlier experiments of the friction torque cover the Couette Reynolds numbers up to 10s which is above the operation range of high-speed machines. Appendix A compares friction coefficients calculated by some of the equations developed. The method by PolkowsM (1984) was found to be veiy sensitive to turbulence constants and, therefore, the calculated results are not presented. With small air-gap lengths, equations by Wendt (1933), Yamada (1962b) and Bilgen et al. (1973) give about the

27

same results. When the relative air-gap length is increased, the Taylor vortices influence more the friction coefficient. For that reason, the ratio between the air-gap length and rotor radius should be included to the equation. Both Eq. 8 and 12 can be used to calculate the friction losses in high-speed machines. One should remember,

however, that the equations are valid only for smooth air-gap surfaces.

Both tangential and axial flows are considered in two papers only. Yamada (1962b) measured the dependency between the friction torque and axial flow through the air gap. He expected that the flow was fully developed in the test section. Polkowski (1984), however, showed that the friction torque is independent of the

axial flow after it has been fully developed. Therefore, he assumed that the axial flow was not fully developed in all the cases measured by Yamada. There is no any detailed experimental investigation dealing with the acceleration of cooling gas in the air gap.

The papers dealing with friction torque of rough rotors have been published by Theodorsen et al. (1944) and Yamada (1962b). Theodorsen used sand grains on the rotors and showed that the friction torque increases considerably with roughness at

high Reynolds numbers. Yamada noticed that circumferential grooves decrease the friction coefficient. This results from the decreased mean cylindrical area of the rotor. Laijola et al. (1991) have obtained in the testing of a high-speed generator that axial grooves in the rotor increase the friction torque by 2.5 times from the value for a smooth rotor. It is self evident from other similar cases that axial grooves increase more the friction torque than circumferential ones. In addition, the tangential velocity distribution for a smooth rotor and rough stator differs surely from that for both surfaces smooth.

Theodorsen (1944) has measured that the friction torque of a rotating disk is independent of the Mach number. This result may be valid also for rotating

cylinders.

5.2 Heat transfer

Heat transfer between rotor and stator has been investigated widely. Papers by Rao (1979) and Lee (1989) cover the Taylor number range of high-speed electrical machines. The axial Reynolds numbers used were below 1000 which is much less than in high-speed machines. Taylor vortices have a strong effect on the heat transfer and the appearance of these fluctuations is controlled by the axial flows through the air gap. This trend has been noticed by several authors and it is shown

l

28

in Pig. 3. Furthermore, Smith et al. (1982) and Lathrop et al. (1992) have reported the

disappearance of Taylor vortices at J?e5>104.

Electrical machines often have slotted air-gap surfaces and this topic has been discussed in many papers. These studies show that slotting increases heat transfer. It seems to be difficult to estimate how large this increase is. When compared with smooth rotor surfaces, the highest increase has been 70% (Rao 1979). The surface area was calculated based on the mean rotor radius in his paper.

Appendix B compares the Nusselt numbers between rotor and stator calculated with equations presented by Becker et al. (1962) and Aoki et al. (1967). The authors have tested the equations up to Taylor numbers of 107 and, therefore, they may not be valid in the whole range calculated. In addition, results based on Reynolds analogy (Eq. 16) are shown. Friction coefficients were calculated according to Eq. 12 by Bilgen et al. (1973). In Eq. 18 (Tachibana et al. 1964), the Nusselt number increases proportionally to the square root of the Taylor number. This is much more than in the other equations and, therefore, it was not tested. In Fig. Bl, the curves at

the top were calculated with a 1 mm air gap. An air-gap length of 3 mm was used in

the curves at the bottom of the page. In both cases Aoki’s equation gave the lowest Nusselt number. Results obtained by the Reynolds analogy increase more sharply than the other curves. Becker’s equation or Reynolds analogy could be suggested for the analyses of high-speed electrical machines.

Tachibana et al. (1964), Polkowski (1984) and Lee et al. (1989) have developed equations which take into account both the tangential and axial air-gap flows. The

first one covers the Taylor numbers up to 3x10s, the second one has not been tested at all. The third one is restricted to below axial Reynolds numbers of 2x10®. These studies do not cover high-speed electrical machines and further research is needed.

29

6 Conclusions

The aim of this work was to find out methods to analyse fiiction losses and convection heat transfer in high-speed electrical machines. The study was focused

on analytical methods and turbulent flows.Flows between two concentric cylinders have been studied intensively for

more than 70 years. Above a certain rotation speed, circular vortices (Taylor

vortices) occur in the air gap which affect the fiiction generation and heat transfer.

An axial fluid flow through the air gap stabilises the flow and the occurrence of vortices is delayed. At very high speeds Taylor vortices disappear, laminar velocity layers become very thin and the velocity field is similar to the flow over a fiat plate.

Due to the high peripheral speed of the rotor, high-speed machines have large friction losses. Friction torques at Couette Reynolds numbers over 10s have been measured. The equations developed can be used to predict the losses in high-speed

machines if the air-gap surfaces are smooth. Besides to the rotor rotation, the friction torque depends on the flow rate of the cooling gas in the section where the axial flow is developing. This topic needs to be investigated further. In addition, the effect of axial stator and rotor slots cannot be estimated with the existing methods. Friction torque has been noticed to be independent of Mach number.

At low rotation speeds the air-gap flow is laminar and the heat is transferred

by conduction, i.e. the heat-transfer rate is independent of the rotation speed. After a certain critical speed, Taylor vortices appear and the heat transfer increases with rotation speed. The equations for the heat-transfer between rotor and stator have been confirmed to Taylor numbers of 107. A typical high-speed machine operates above this flow range, and cooling gas is pumped through the air gap. Heat is transferred from the rotor and stator to the air-gap gas. One should notice that the heat-transfer coefficients are no longer the same as when the heat is transferred from the rotor across the air gap to the stator. With the existing equations, the effect of the axial flow can be predicted only at low axial Reynolds numbers. Roughness on the rotor surface has been measured to increase the heat transfer by up to 70% when

compared with smooth rotors.

30

References

Aoki H., Nohira H. and Aral H. 1967. Convective heat transfer in an annulus with an inner rotating cylinder. Bulletin of JSME, Vol. 10, No. 39, pp. 523-532.

Becker K.M. and Kaye J. 1962. Measurements of diabatic flow in an annulus with an inner rotating cylinder. Transactions of the ASME, Journal of Heat Transfer, Series C, Vol. 84, No. 2, pp. 97-105.

Bilgen E. and Boulos R. 1973. jFunctional dependence of torque coefficient of coaxial cylinders on gap width and Reynolds numbers. Transactions of ASME, Journal of Fluids Engineering, Series I, Vol. 95, No. 1, pp. 122-126.

Bjorklund I.S. and Kays W.M. 1959. Heat transfer between concentric rotating cylinders. Transactions of the ASME, Journal of Heat Transfer, Series C, Vol. 81, pp. 175-186.

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Dorfman L.A. 1963. Hydrodynamic resistance and the heat loss of rotating solids. Oliver & Boyd, Edinburgh and London, Great Britain, p. 244.

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31

Lathrop D.P., Fineberg J. and Swinney H.L. 1992a. Turbulent flow between concentric rotating cylinders at large Reynolds number. Physical Review Letters, Vol. 68, No. 10, pp. 1515-1518.

Lathrop D.P., Fineberg J. and Swinney H.L. 1992b. Transition to shear-driven turbulence in Couette-flow. Physical Review A, Vol. 46, No. 10, pp. 6390-6405.

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Polkowski J.W. 1984. Turbulent flow between coaxial cylinders with the inner cylinder rotating. Transactions of the ASME, Journal of Engineering for gas turbines and power. Vol. 106, No. 1, pp. 128-135.

Rao K.V.C. 1979. Heat transfer in an annulus with a rotating rough inner cylinder. Doctoral Thesis, Indian institute of Technology, Department of Mechanical Engineering, Heat Transfer and Thermal Power Laboratory, Madras, India, p. 210.

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33

Appendix A: Friction coefficient

o + :

Couette Reynolds number

2.5 xlO-3

S 1.5I8

i0.5

+ °o

X X

;°°o0• •"i*» >•}» ••••••

+ + +

x X x X X

?..?..ft. O.O.Q

+ + + + + +|

M .x x .x .x .x

■ 0*0' O'

+ + +

x-XX'

4 6Couette Reynolds number

10xl04

++ Wendt (1933) xx Yamada (1962) oo Bilgen (1973)

Fig. Al. Friction coefficient of a rotating cylinder. The radius of the rotor is 45 mm and the radial air-gap length is 1 mm (top) and 3mm (bottom). Rotation speed is between 30 000 and 120 000 rpm. The deviation between results are larger for the larger air gap. This result probably from the Taylor vortices which are difficult to model. Their effect is stronger at large air-gap lengths. The relative air-gap lenght is taken into account in the equations by Wendt and Bilgen.

Nus

selt

num

ber

Nus

selt

num

ber

34

Appendix B: Nusselt number

+ + +b

..........+.t.+

+ *

„ o °

oo0

...+..........o.°X X X X X X X £

0l--------------- :--------------- :--------------- :----------------:---------------0 0.5 1 1.5 2 2.5

Taylor number Xl07

Taylor number xl08

++ Becker (1962) xx Aoki (1967) oo Reynolds analogy

Fig. Bl. Nusselt number between rotor and stator. The radius of the rotor is 45 mm and the radial air-gap length is 1 mm (top) and 3mm (bottom). Rotation speed calculated is between 30 000 and 120 000 rpm. The friction coefficients by Bilgen et al. (1973) was used in the Reynolds analogy solution.