Improved Thermal Models for Predicting End Winding Heat Transfer1079866/FULLTEXT01.pdf · 2017. 3....

IN DEGREE PROJECT ELECTRICAL ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2017

Improved Thermal Models for Predicting End Winding Heat Transfer

GABRIELE LUCA BASSO

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ELECTRICAL ENGINEERING

Kungliga Tekniska Högskolan (Royal Institute of Technology)

Improved Thermal Models for Predicting End

Winding Heat Transfer

Gabriele Luca Basso

Master of Science Thesis

EJ210X : Degree Project in Electrical Machines and Drives

Stockholm, Sweden, March 2017

Supervisors: Yew Chuan Chong

Dave Staton

Examiner: Oskar Wallmark

To my family,

who has always been with me wherever I was,

who has always backed me whatever I did

and always pushed me forward whenever I needed.

i

Abstract (English)

The thermal analysis is of primary importance for the design of any kind of electrical machine.

The cooling has a significant impact on the electromagnetic performances and on the durability

and reliability of the machine. One of the main thermal issues for AC machines is the cooling of

the end windings, which usually appear to be the most important hotspots.

The thermal analysis of the end space region of an electrical machine is mainly linked to the

characterization of the convective heat transfers between the solid surfaces and the cooling fluid,

normally air. This results in a very complex analysis, mainly because of the unpredictable

turbulence which is generated inside the end space by the rotating surfaces, such as the rotor, the

shaft and the wafters. Different relations have been proposed in the literature by several authors

but the calculation of the heat transfer coefficients (HTC) is still mainly related to empirical

correction factors, based on the experience.

The aim of this work is to investigate the convection inside the end space region with the double

objective of studying how the geometry influences the HTCs and of developing a new improved

correlation for the computation of the HTCs.

The study is conducted with the support of an advanced CFD software, ANSYS Fluent CFD. The

Nissan Leaf motor is modelled and the volume of air inside the end-space region is extracted and

simulated in different conditions. These includes the study of the channels/ducts underneath the

end winding toroids, the variation in the wafters length, the evaluation of the impact of the housing

and endcap distance from the end-windings and the study of the HTCs response to a variation in

the rotor mechanical speed.

The investigation leads, as a main result, to a dimensionless equation that links the Nusselt number

to the rotational Reynolds number in the end-space region. This equation is supported by a table

of coefficients to be selected depending on the wafters length and the considered surface of the

end-space. The new relation still has to be validated with the support of experimental data.

However, it is suitable of a direct application in commercial software that uses a lumped-parameter

thermal network (LPTN) for the thermal analysis of electrical machines, such as Motor-CAD.

ii

Abstract (Italian)

Lo studio termico è di primaria importanza in fase di progettazione per qualunque tipologia di

macchina elettrica. Il raffreddamento ha infatti un impatto significativo sia sulle prestazioni

elettromagnetiche che sulla durata di vita utile della macchina. Uno dei principali problemi per le

macchine a corrente alternata è il raffreddamento delle testate (collegamenti frontali e posteriori

degli avvolgimenti), le quali risultano essere dei punti caldi critici.

L’analisi termica delle testate è principalmente legata alla caratterizzazione dello scambio termico

convettivo tra le superfici solide, inclusi gli stessi collegamenti di fine avvolgimento, e il fluido di

raffreddamento, generalmente aria. L’analisi risulta essere di particolare complessità soprattutto a

causa dei fenomeni di turbolenza che si generano per effetto delle superfici rotanti, quali il rotore,

l’asse e le eventuali alette di raffreddamento sulla superficie rotorica. Il calcolo del coefficiente di

trasmissione termica (HTC) dovuto a fenomeni convettivi è stato affrontato da diversi autori e le

correlazioni attualmente disponibili sono basate su coefficienti di natura empirica.

Lo scopo del lavoro presentato in questa tesi è quello di investigare i fenomeni convettivi che si

sviluppano nella zona compresa tra l’estremità assiale del pacco laminato, i coperchi di chiusura e

l’alloggiamento (end-space region). Lo studio è condotto con il duplice obiettivo di analizzare le

variazioni degli HTCs a seguito di modifiche apportate nella geometria e sviluppare una nuova e

più accurata equazione per il calcolo degli HTCs.

Il progetto è condotto con il supporto di un software CFD avanzato, ANSYS Fluent CFD, ed il

modello implementato è costruito sulla base del motore utilizzato nella Nissan Leaf. Il volume di

aria interno alla end-space region è estratto e simulato in diverse condizioni. Le simulazioni

comprendono lo studio dei condotti tra i collegamenti di fine avvolgimento ed il pacco laminato,

la variazione della lunghezza delle alette, la variazione della dimensione dell’alloggiamento e dei

coperchi e lo studio degli HTCs a diverse velocità di rotazione della macchina.

L’analisi conduce, tra i suoi risultati principali, alla definizione di un’equazione adimensionale che

correla il numero di Nusselt con il numero di Reynolds rotazionale. L’equazione è supportata da

una tabella di coefficienti che permettono il calcolo dell’HTC per tutte le superfici nell’end space

region e per diverse lunghezze delle alette. Tale nuova correlazione necessita ancora di una

convalida sperimentale ma sarebbe già applicabile in software commerciali che implementino una

rete termica a parametri concentrati (LPTN), quale Motor-CAD.

iii

Abstract (Swedish)

Då elektriska maskiner designas är termisk analys av yttersta vikt då en rätt avpassad kylning bade

påverkar elektromagnetisk prestanda samt maskinens livslängd. En viktig aspekt vid termisk

analys av elmaskiner är kylning av maskinens ändlindningar i vilka ofta den högsta temperaturen

uppkommer.

Termisk analys av ändregionen på en elmaskin är huvudsakligen länkad till karakteriseringen av

den konvektiva värmetransporten mellan ändlindningens och omkringliggande luft. Detta

resulterar i en komplex analys, huvudsakligen på grund av turbulensen som genereras i

ändregionen av rotorn, rotorns axel samt eventuella kylflänsar. Ett antal yttryck för

värmeöverföringskoefficienten för denna värmetransport har föreslagits i litteraturen. Dessa

yttryck är dock beroende av empiriska korrektionsfaktorer vars värden baseras på erfarenhet.

Målet med detta arbete är att undersöka hur konvektionen i ändregionen påverkas av ändringar i

geometrin samt, från de numeriska studier som genomförts, ta fram förbättrade uttryck för

värmeöverföringskoefficienten.

Studien som presenteras i arbetet utförs med hjälp av den CFD-baserade mjukvaran Ansys Fluent

CFD. Elmaskinen i elbilen Nissan Leaf har modellerats och luftvolymen i ändregionen har

simulerats i ett antal fall. Dessa fall inkluderar en studie av det radiella luftflödet vid statortänderna,

variation av kulflänsarnas dimensioner, samt det axiella avståndet till ändskölden.

Som huvudresultat har ett parametriserat, dimensionslöst uttryck som relaterar Nusselt och

Reynoldstalen i ändregionen tagits fram. Parametrarna i uttrycket beror på kylflänsarnas axiella

längd samt ändregionens yttyp. Det framtagna uttrycket, som måste valideras experimentellt i

vidare studier, kan nyttjas i mjukvara som använder termiska nätverk av lumped-parameter-typ för

termisk analys av elmaskiner (t.ex. Motor-CAD).

v

Acknowledgements

The entire work reported in this thesis was carried out at Motor Design Ltd, Ellesmere, UK.

Therefore, I would first like to thank every single member of the company for letting me feel like

home for all the six months that I was there. Motor Design Ltd has been for me like a second big

family in the foreign country of England.

The project was partially funded by The Engblom Foundation and The Malme Foundation, both

announced in the April 2016 at KTH, in Stockholm. My thanks go to the foundation, who gave me

the economic support I needed to do this beautiful experience in the United Kingdom.

I express my gratefulness to my main supervisor at Motor Design, Dr. Yew Chuan Chong, Eddie

for the friends, for guiding me through the entire project, being always patient in teaching me all

those thermal concepts that were not part of my academic background. At the same time, I would

like to thank the president, Dave Staton, for giving me the opportunity to be part of his team and

my examiner at KTH, Oskar Wallmark, for being my landmark in Stockholm, during this Master

Thesis.

Special thanks also go to Unai San Andres, senior researcher in Motor Design, for always helping

me whenever I felt lost in a CFD simulation. Thanks go to all my home mates in Ellesmere, Richard

Marsden, Giuseppe Volpe, Rebecca Line and especially Wayne Brookfield, who were true friends

and shared with me some of those moments that I will always keep in my heart.

I would then like to thank all my colleagues in these past two years at university, especially

Antonio Giannetto, Luca Roggio, Matteo Rapisarda and Giulio Siniscalchi in Turin, Florian Baron,

Alexej Schmidt, Enrico Lucca and Marc Helmer in Stockholm.

My most sincere gratitude to my girlfriend, with the hope that the effort I put on my master course

will be an important brick in the foundation of our future together.

Finally and most importantly, this thesis, my work and my studies are dedicated to my parents,

Daniela and Nuccio, and my sisters, Simona and Laura. I thank them for never letting me feel

lonely or lost and always being very close to me, even when we were at the opposite ends of

Europe.

vii

Table of Contents

Abstract (English) ............................................................................................................................ i

Abstract (Italian) ............................................................................................................................. ii

Abstract (Swedish) ......................................................................................................................... iii

Acknowledgements ......................................................................................................................... v

Table of Contents .......................................................................................................................... vii

List of Figures ................................................................................................................................. x

List of Tables ............................................................................................................................... xiii

Nomenclature ................................................................................................................................ xv

1 Introduction ............................................................................................................................. 1

1.1 Losses in electrical machines ........................................................................................... 1

1.2 Cooling of electrical machines ......................................................................................... 2

1.3 Machine enclosure............................................................................................................ 3

1.4 Convective heat transfer ................................................................................................... 4

1.5 End-space cooling ............................................................................................................ 6

1.6 Focus and aims of the M.Sc. project ................................................................................ 8

1.7 Outline of the Thesis ........................................................................................................ 9

2 Literature review ................................................................................................................... 10

2.1 Lumped Circuit Thermal Network ................................................................................. 10

2.2 Computational Fluid Dynamics (CFD) .......................................................................... 14

2.2.1 Governing equations of fluid flows ........................................................................ 14

2.2.2 Turbulent flow ........................................................................................................ 19

2.2.3 Wall treatment in CFD ............................................................................................ 22

2.2.4 Modelling of rotating regions ................................................................................. 25

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2.3 Chapter summary ........................................................................................................... 26

3 End-space region modelling ................................................................................................. 28

3.1 Nissan Leaf model in Motor-CAD ................................................................................. 28

3.1.1 Motor-CAD implementation ................................................................................... 29

3.1.2 Exportation of the geometry from Motor-CAD ...................................................... 34

3.2 ANSYS Model ............................................................................................................... 34

3.2.1 ANSYS Workbench ................................................................................................ 34

3.2.2 ANSYS Design Modeler......................................................................................... 35

3.2.3 ANSYS Meshing .................................................................................................... 45

3.2.4 ANSYS Fluent CFD ............................................................................................... 49

3.2.5 CFD-Post................................................................................................................. 59

3.3 Chapter summary ........................................................................................................... 61

4 Simulations ........................................................................................................................... 62

4.1 Propaedeutic simulations................................................................................................ 63

4.1.1 Number of inflation layers ...................................................................................... 64

4.1.2 Boundary conditions type ....................................................................................... 66

4.1.3 Viscous model ......................................................................................................... 68

4.1.4 MRF vs Sliding Mesh ............................................................................................. 68

4.2 Main Simulations ........................................................................................................... 72

4.2.1 Ducts/channels underneath the end-winding .......................................................... 73

4.2.2 Wafters/Blades on the rotor surface ........................................................................ 74

4.2.3 Variation of the geometry dimension ..................................................................... 76

4.2.4 Simulation of a range of speeds .............................................................................. 78

4.3 Chapter Summary ........................................................................................................... 78

5 Analysis and Results ............................................................................................................. 80

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5.1 Preliminary considerations ............................................................................................. 80

5.2 Effects of the ducts/channels .......................................................................................... 82

5.3 Effects of the wafters length........................................................................................... 85

5.4 Effects of variations in the geometry dimensions .......................................................... 89

5.5 New HTC relation .......................................................................................................... 93

5.6 Chapter summary ........................................................................................................... 99

6 Conclusions and further work ............................................................................................. 100

6.1 Conclusions .................................................................................................................. 100

6.2 Further work ................................................................................................................. 102

References ................................................................................................................................... 104

APPENDIX A – Results of the variation in the Housing and Endcap dimensions .................... 107

APPENDIX B – Results from the simulations of a range of rotor speeds ................................. 112

x

List of Figures

Figure 1.1. Published End-Space convection correlations [10] [12] .............................................. 7

Figure 2.1 - Equivalent lumped circuit for the general cylindrical component, Mellor 1991 [16]

....................................................................................................................................................... 11

Figure 2.2 - Equivalent lumped circuit thermal network of an induction machine, proposed by

Mellor in 1991 [16] ....................................................................................................................... 11

Figure 2.3 - Three dimensional network representation for the general cuboid, proposed by

Wrobel, 2010 [17] ......................................................................................................................... 12

Figure 2.4 - LPTN of the end-space region [18] ........................................................................... 13

Figure 2.5 - Graphical representation of the rheological law for a Newtonian fluid .................... 14

Figure 2.6 - Definition of different flow regimes depending on the Mach number ..................... 17

Figure 2.7 - Subdivision of the Near-Wall region [23] ................................................................. 23

Figure 2.8 - Wall treatment approaches [23] ................................................................................ 24

Figure 3.1 - Last version of the Nissan Leaf [24] ......................................................................... 28

Figure 3.2 - Motor Design Limited logo ...................................................................................... 29

Figure 3.3 - Lamped Circuit of the Nissan Leaf as it appears in Motor-CAD ............................. 30

Figure 3.4 - Detail view of the Lamped Circuit in Motor-CAD ................................................... 31

Figure 3.5 - Efficiency map of the Nissan Leaf motor ................................................................. 33

Figure 3.6 - Detail of the end-windings model in Motor-CAD .................................................... 33

Figure 3.7 - ANSYS Workbench .................................................................................................. 35

Figure 3.8 - Imported geometry in ANSYS Design Modeler ....................................................... 36

Figure 3.9 - End-space region in ANSYS Design Modeler .......................................................... 37

Figure 3.10 - Air-volume (obtained as internal volume) in ANSYS Design Modeler ................. 38

Figure 3.11 - Final Geometry in ANSYS Design Modeler .......................................................... 40

Figure 3.12 - Modified geometry without channels underneath the end-winding ........................ 41

Figure 3.13 - Sketch to be used for the generation of the wafters ................................................ 43

Figure 3.14 - Modified geometry with wafters/blades mounted on the rotor surface .................. 44

Figure 3.15 - Meshed geomerty in ANSYS Meshing ................................................................... 48

Figure 3.16 - ANSYS Fluent CFD user interface ......................................................................... 50

Figure 3.17 - Interface settings sub-window................................................................................. 53

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Figure 3.18 - Wall settings sub-window ....................................................................................... 55

Figure 3.19 - Thermal tab of the Wall settings sub-window ........................................................ 56

Figure 3.20 - Residual monitors settings sub-window.................................................................. 57

Figure 3.21 - Surface Monitor settings sub-window .................................................................... 58

Figure 3.22 – ANSYS CFD-Post user interface ........................................................................... 60

Figure 4.1 - Summary scheme of all the simulations run. ............................................................ 62

Figure 4.2 - Workbench project schematic for the propaedeutic simulations .............................. 64

Figure 4.3 – Y-plus with good inflation when the rotor speed is equal to 10.000 rpm ................ 65

Figure 4.4 - Y-plus with good inflation when the rotor speed is equal to 2.000 rpm ................... 65

Figure 4.5 - Walls temperature at rotational speed equal to 10.000 rpm ...................................... 67

Figure 4.6 - Residuals for a test simulation with MRF method (a) and Sliding Mesh method (b)

....................................................................................................................................................... 70

Figure 4.7 - Example of convergence of the Heat Transfer for one of the boundaries (End Winding

Lateral Internal) when the Moving Reference Frame approach is used ....................................... 70

Figure 4.8 - Channels underneath the end-winding. The red arrows indicate that the air is able to

pass through the channels to reach the external part of the end-space region. ............................. 73

Figure 4.9 - Detail of the end-winding in real machines [25] ....................................................... 74

Figure 4.10 - Example of induction motors with wafters mounted on the end-ring [26] ............. 74

Figure 4.11 - Four wafters dimensions modelled and simulated with the ANSYS software ....... 75

Figure 4.12 – Variations simulated in the housing dimensions .................................................... 76

Figure 4.13 - Variations simulated in the endcap dimensions ...................................................... 77

Figure 5.1 - Air velocity streamline for the model with channels underneath the end windings . 84

Figure 5.2 - Air Local Speed as function of the Wafters Length .................................................. 87

Figure 5.3 - Heat Transfer as function of the Wafters Length ...................................................... 87

Figure 5.4 - Heat Transfer Coefficients as function of the Wafters Length ................................. 87

Figure 5.5 – Heat Transfer and Air Velocity at the EW Outer and EW End when the housing

geometry is varied ......................................................................................................................... 90

Figure 5.6 - Heat Transfer at all the control volume boundaries when the Endcap axial dimension

is varied ......................................................................................................................................... 92

Figure 5.7 - Local speed of air for those surfaces where the velocity results to increase as a

consequence of the endcap enlarging ........................................................................................... 92

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Figure 5.8 - Results for the rotor equipped with medium wafters at different rotational speeds . 94

Figure 5.9 - Graphical analysis of the Nusselt number for the Housing surface .......................... 97

Figure A.1 - Heat Transfer and Air Velocity at the Housing when the housing geometry is varied

..................................................................................................................................................... 109

Figure A.2 - Heat Transfer and Air Velocity at the EW Bore when the housing geometry is varied

..................................................................................................................................................... 109

Figure A.3 - Heat Transfer and Air Velocity at the Bearing when the housing geometry is varied

..................................................................................................................................................... 110

Figure A.4 - Heat Transfer Coefficient at all the control volume boundaries when the Endcap axial

dimension is varied ..................................................................................................................... 111

Figure A.5 – Local Speed of Air at all the control volume boundaries when the Endcap axial

dimension is varied ..................................................................................................................... 111

Figure B.1 - Results for the smooth rotor at different rotational speeds..................................... 117

Figure B.2 - Results for the rotor equipped with good wafters at different rotational speeds .... 118

Figure B.3 - Graphical analysis of the Nusselt number for the Endcap surface ......................... 120

Figure B.4 - Graphical analysis of the Nusselt number for the Bearing surface ........................ 120

Figure B.5 - Graphical analysis of the Nusselt number for the Shaft surface ............................ 121

Figure B.6 - Graphical analysis of the Nusselt number for the Rotor surface ............................ 121

Figure B.7 - Graphical analysis of the Nusselt number for the EW_Bore surface ..................... 122

Figure B.8 - Graphical analysis of the Nusselt number for the EW_Outer surface .................... 122

Figure B.9 - Graphical analysis of the Nusselt number for the EW_End surface ...................... 123

xiii

List of Tables

Table 1.1 - IEC/NEMA standards for the thermal insulation ......................................................... 3

Table 1.2 - Coefficients proposed by different authors for the calculation of the HTC in the End-

Space region [12] ............................................................................................................................ 7

Table 2.1 - Explanation of the symbols used in the LPTN of the end-space region .................... 13

Table 3.1 - Temperatures for the different components of the end-space region, obtained from

Motor-CAD ................................................................................................................................... 32

Table 3.2 - Specified dimensions for the two sketches used for the generations of the wafters .. 43

Table 3.3 - Inflation details for the meshing ................................................................................. 47

Table 3.4 - Suggested values for the "Mesh Sizing" .................................................................... 47

Table 3.5 - Mesh quality parameters............................................................................................. 49

Table 4.1 - Simulation settings for the MRF approach (a) and the Sliding Mesh approach (b) ... 69

Table 4.2 - Comparison between MRF and Sliding Mesh............................................................ 72

Table 4.3 - Wafters dimensions as implemented in ANSYS ........................................................ 75

Table 4.4 - Housing radial dimensions, as implemented in ANSYS ............................................ 76

Table 4.5 – Endcap axial dimensions, as implemented in ANSYS .............................................. 78

Table 5.1 - Details of the locations used to calculate the local speed of air in CFD-Post ............ 81

Table 5.2 - Correspondence between the groups of surfaces in ANSYS and the Motor-CAD name

....................................................................................................................................................... 82

Table 5.3 - Results from two simulations at 10.000 rpm for the two models with and without

channels underneath the end-winding ........................................................................................... 84

Table 5.4 - Results for the air local speed for different wafters conditions .................................. 86

Table 5.5 - Results for the heat transfers (HT) and the heat transfer coefficients (HTC) for different

wafters conditions ......................................................................................................................... 86

Table 5.6 - Dragging torque and power at 10.000 rpm and for different wafters layout .............. 89

Table 5.7 - Air properties at 100°C ............................................................................................... 96

Table 5.8 - New coefficients proposed for equation (5.4) ............................................................ 97

Table A.1 – Heat Transfers, Heat Transfer Coefficients and local speed of air when smooth rotor

is simulated and the Housing radial dimension is modified ....................................................... 107

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Table A.2 - Heat Transfers, Heat Transfer Coefficients and local speed of air when rotor equipped

with medium wafters is simulated and the Housing radial dimension is modified .................... 108


with good wafters is simulated and the Housing radial dimension is modified ......................... 108


with good wafters is simulated and the Endcap axial dimension is modified ............................ 110

Table B.1 – Example of Excel analysis to get the heat transfer average upon 200 iterations .... 113

Table B.2 - Example of Excel analysis to get the HTC for a single simulation ......................... 114

Table B.3 – Data summary for smooth rotor .............................................................................. 115

Table B.4 - Summary table for good wafters .............................................................................. 116

Table B.5 - Summary table for medium wafters......................................................................... 116

Table B.7 - Summary of the HTCs for all the simulations done ................................................ 119

Table B.6 - Summary of the Nusselt numbers ............................................................................ 119

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Nomenclature

Mechanical symbols

A Cross sectional area [m2]

C Local velocity of the sound [m/s]

cp Specific heat capacity at constant pressure [J/(kg∙K)]

ΔT Temperature difference [K]

g Gravity acceleration [m/s2]

h Convective heat transfer coefficient [W/(m2K)]

k General constant [ - ]

L Characteristic length [m]

Ma Mach number [-]

Nu Nusselt number [-]

Pr Prandtl number [-]

r Radius [m]

R Ideal gas constant [J/(K∙mol)]

𝑅𝑡ℎ Thermal resistance [K/W]

Re Reynolds number [-]

T Torque [N∙m]

u x-component of velocity [m/s]

u Velocity vector [m/s]

v Velocity / y-component of velocity [m/s]

V Volume [m3]

w z-component of velocity [m/s]

y Normal distance from the wall [m]

Mechanical subscripts

ij jth component acting on the faces of the fluid element perpendicular to the i-axis

m Mechanical

t Turbulent/Friction

turb Turbulence

xvi

w Winding/Wall

Greek alphabet

α Thermal diffusivity [m2/s]

λ Thermal conductivity [W/(m∙K)]

δ Air gap [m]

𝛿𝑖𝑗 Kronecker delta function [-]

η Efficiency [-]

μ Dynamic viscosity [Pa∙s]

ν Kinematic viscosity [m2/s]

ρ Density [kg/m3]

τ Shear stress [Pa]

Φ Viscous dissipation function [-]

ω Angular speed [rad/s]

Electromagnetic symbols

A Linear current density [A/m]

B Magnetic flux density [T]

cosϕ Power factor [rad]

f Frequency [Hz]

I Current [A]

k General constant [ - ]

P Electric Power [W]

R Resistance [Ω]

Electromagnetic subscripts

Cu Copper

e Eddy currents losses

ex Excess losses

h Hysteresis losses

xvii

max Maximum value

rms Root Mean Square

Abbreviations

AC Alternate Current

CFD Computational Fluid Dynamics

DC Direct Current

DM Design Modeler

DNS Direct Numerical Simulation

EW End Windings

HTC Heat Transfer Coefficient

IEC International Electrotechnical Commission

LES Large Eddy Simulation

LPTN Lumped Circuit Thermal Network

MRF Moving Reference Frame

NEMA National Electrical Manufactures Association

ODP Open Drip Proof

PMSM Permanent Magnet Synchronous Machine

RANS Reynolds-Averaged Navier-Stokes

RMS Root Mean Square

RSM Reynolds Stress Transport Model

SM Sliding Mesh

SST Shear Stress Transport

TEFC Total Enclosed Fan Cooled

Introduction Losses in electrical machines

1

1 Introduction

This first chapter is meant to be an introduction to those basic concepts that justify and have led to

the work produced in this thesis. A brief description of the losses, the cooling and the enclosure

systems in electrical machines is provided. Moreover, the convective heat transfer and the

equations used in literature to compute the HTC in the end-space region are described, as starting

point of the work. The chapter is concluded with a summary of the thesis objectives and the thesis

outline.

1.1 Losses in electrical machines

An electrical machine is a device which converts electrical energy into mechanical energy (motor)

and vice versa (generator). The main classification is between DC and AC machines, whereas this

second group is further divided into synchronous and asynchronous (induction). The energy

conversion generates an unwanted percentage of thermal losses which lead to an increase of the

local temperature inside the machine. The losses can be divided, for an AC machine, into three

different components: copper losses, iron losses and mechanical losses [1].

The copper losses are related to the Joule heating of the conductors, both in the stator and the rotor.

It may be noticed that it is not always true that this kind of losses are located in copper material,

e.g. for an Aluminum squirrel cage rotor. Therefore, it is probably more accurate to speak about

resistive losses. They are given by the well-known Joule’s first law:

𝑃𝐶𝑢 = 𝐼𝑟𝑚𝑠2 ∙ 𝑅 (1.1 )

where R is a general electric resistance and the current is considered to be sinusoidal and the root

of main squared value is used.

The iron losses includes eddy currents losses, hysteresis losses and excess losses; they are given

by Equation (1.2) [2].

𝑃𝑖𝑟𝑜𝑛 = 𝑘ℎ𝑓𝐵𝑛 + 𝑘𝑒𝑓

2𝐵2 + 𝑘𝑒𝑥𝑓1.5𝐵1.5 (1.2)

The subscript ‘h’ refers to the hysteresis losses, the subscript ‘e’ to the eddy currents losses and

the subscript ‘ex’ to the excess losses. The hysteresis losses are proportional to the frequency,

whereas the eddy currents losses are related to the square of both the frequency and the magnetic

Introduction Cooling of electrical machines

2

flux density. ‘n’ is the Steinmetz constant and is equal to 1.6. The excess losses have been

introduced in order to take into account additional measured losses with respect to the expected

value of the eddy current losses. They are related to the material microstructure, the conductivity

and the cross-sectional area of the lamination.

The electromagnetic losses of an induction motor are increased by the presence of both space and

time harmonics, especially when the motor is driven by an inverter. The presence of current

harmonics increments the value of the RMS current, increasing the copper losses. Moreover, time

harmonic fields induce harmonic losses also in the stator and rotor cores [3].

The mechanical losses include friction losses at the bearings and windage losses, caused by the

rotor rotation.

1.2 Cooling of electrical machines

The maximum power of an electric machine is limited by the maximum admissible magnetic flux

density (𝐵𝑚𝑎𝑥) in the iron and by the maximum admissible linear current density (𝐴𝑚𝑎𝑥) in the

conductors. Given the angular speed of the rotor, the maximum mechanical torque is limited by

the same quantities as for the maximum output power. The torque density, per unit volume, can be

expressed by the following equation [4]:

𝑇𝑚𝑉𝛿

=4

√2∙ 𝐵𝑚𝑎𝑥 ∙ 𝐴𝑚𝑎𝑥 ∙ 𝑐𝑜𝑠𝜙 ∙ 𝜂 ∙ 𝑘𝑤

(1.3)

where ‘𝑇𝑚’ is the mechanical torque, ‘𝑉𝛿’ the machine volume at the air gap, ‘𝑐𝑜𝑠𝜙’ the power

factor, ‘𝜂’ the machine efficiency and ‘𝑘𝑤’ the fundamental winding factor.

The maximum flux density depends on the magnetic properties of the materials. However, the

maximum permissible current is mainly limited by the maximum acceptable temperature for the

insulation. Therefore, the cooling system of the machine becomes of primary importance in order

to reduce the insulation temperature or, from another point of view, keep the same temperature

and allow an increasing in the maximum current. Moreover, some of the power losses are

temperature dependent, e.g. the copper losses, being the electrical resistance an increasing function

of the conductors’ temperature.

Introduction Machine enclosure

3

The maximum allowed temperature for a given electrical machine is defined by its insulation class,

which is expressed by a letter or a number, according to the IEC/NEMA standards1. For a given

insulation class, the operating temperature of the machine should be always kept under the

maximum allowed temperature. Exceeding the maximum temperature limit permanently reduces

the life time of the machine. The value of the maximum admissible temperature is reported in

Table 1.1, depending on the insulation class (IEC 60085).

Table 1.1 - IEC/NEMA standards for the thermal insulation

IEC 60085

Thermal Class

Old IEC 60085

Thermal Class

NEMA

Class

NEMA

Letter Class

Maximum hot spot

temperature [°C]

90 Y - - 90

105 A 105 A 105

120 E - - 120

130 B 130 B 130

155 F 155 F 155

180 H 180 H 180

200 - - N 200

220 - 220 R 220

250 - - - 250

1.3 Machine enclosure

The electrical machines can be classified depending on the environmental protection and the

methods of cooling. The enclosure of the motor protects the windings and all the internal

mechanical parts from possible dangerous fluids or particles in the external environment.

According to the IEC/NEMA standards, the electrical machines can be divided into three main

groups: open machines, totally enclosed machines and machines with encapsulated or sealed

windings [5]. A sub-classification can then be made depending on the cooling fluid, e.g. air, water,

or oil. Among all the different possible solutions, the Open Drip Proof (ODP) and the Totally

1 The International Electrotechnical Commission (IEC) is a non-profit, non-governmental standard organization, world

leader for all electrical, electronic and related technologies standards. The National Electrical Manufactures

Association (NEMA) is the standard organization in USA.

Introduction Convective heat transfer

4

Enclosed Fan Cooled (TEFC) are two of the most common motor enclosure types for industrial

applications [6].

The ODP motor is an ideal choice for indoor applications, where the external environment is

controlled to be dry and clear. The motor housing has openings to allow air circulation, improving

the cooling of the machine, but prevents any liquid to entering into the motor through dripping.

This type of enclosure is generally cheaper than others and it is often used in normal applications.

Its main drawback is the lack of protection against dust particles or other airborne contaminants.

The TEFC motor is ideal for an external environment which is exposed to moisture or water, since

the external housing and endcaps are completely sealed. A fan is used to cool the housing outer

surfaces but the exchange of air between the inside and the outside is prevented. Compared to the

ODP motors, the TEFC is more expensive and it’s generally to be used for special applications.

1.4 Convective heat transfer

The heat transfer is the exchange of thermal energy between two systems. The three fundamental

mechanisms by which the energy moves from a system to another are conduction, convection and

radiation. Among them, convection is the main way to remove the heat from an electrical machine

[7]. The convection heat transfer can be expressed by the following equation:

𝑄 = ℎ ∙ 𝐴 ∙ Δ𝑇 (1.4)

where ‘h’ is the heat transfer coefficient (HTC), ‘A’ is the surface area and ‘ΔT’ is the temperature

difference between the surface and the fluid. Equation (1.4) can be modified in order to be similar

to Ohm’s law for the electrical circuit:

𝑄 =Δ𝑇

𝑅𝑡ℎ

(1.5)

where ‘𝑅𝑡ℎ’ is the thermal resistance between the surface and the fluid. The resistance is related to

the reciprocal of the HTC, meaning that the higher the HTC, the lower the thermal resistance.

For a given geometry, the HTC can be obtained from the dimensionless Nusselt number. The

Nusselt number [Nu] represents the ratio between the convective and conductive heat transfer

across the given boundary/surface. It is related to the HTC according to equation (1.6) [8].

Introduction Convective heat transfer

5

𝑁𝑢 =h ∙ L

𝜆

(1.6)

where ‘h’ is the convective heat transfer coefficient, ‘L’ a characteristic length and ‘λ’ the thermal

conductivity. The Nusselt number, as the HTC, is a complex function of the fluid thermo-physical

characteristics, the fluid regime, the surface geometry, the surface roughness and other related

characteristics, as shown in equation (1.7) [9].

𝑁𝑢 = 𝑓(𝐿, 𝑅𝑒, Pr) (1.7)

where ‘Re’ is the Reynolds number and ‘Pr’ is the Prandtl number.

The Reynolds number is defined as the ratio between inertial forces and viscous forces. It is used

to characterize the flow regime, differentiating between laminar and turbulent flows. The Prandtl

number is defined as the ratio between the moment diffusivity (kinematic viscosity) and the

thermal diffusivity2. The Reynolds and Prandtl numbers are given by equations (1.8) and (1.9)

respectively [8].

𝑅𝑒 =𝜌 ∙ 𝑣 ∙ 𝐿

𝜇=

𝑣 ∙ 𝐿

𝜈

(1.8)

𝑃𝑟 =𝜈

𝛼=

𝑐𝑝 ∙ 𝜇

𝜆

(1.9)

where ‘𝜌’ is the fluid density, ‘𝑣’ the fluid velocity, ‘𝐿’ the characteristic length, ‘𝜇’ the dynamic

viscosity, ‘𝜈’ the kinematic viscosity, ‘𝛼’ the thermal diffusivity, ‘𝑐𝑝’ the specific heat capacity

at constant pressure and ‘𝜆’ the thermal conductivity.

In the case of radial flux machines, the fluid flow regime in the rotor-stator annular gap can be

better characterized using the Taylor number. Taylor studied, from a theoretically point of view,

the stability of the fluid flow between concentric cylinders, finding the generation of complex

toroidal vortices when the inner cylinder speed exceeds a critical value. Among all the different

available definitions of the Taylor number, the correlation reported in equation (1.10) is the one

provided by [7].

2 The thermal diffusivity α is given by the thermal conductivity λ divided by the density ρ and the specific heat

capacity at constant pressure cp [8]:

𝛼 =𝜆

𝜌 ∙ 𝑐𝑝

Introduction End-space cooling

6

𝑇𝑎 =𝜌 ∙ 𝜔𝑚 ∙ �̅�

0.5 ∙ 𝛿1.5

𝜇

(1.10)

where �̅� is the mean annulus radius between the two cylindrical surfaces, 𝜔𝑚 the angular

mechanical speed and δ is here the rotor-stator gap size.

1.5 End-space cooling

The end-windings are typically the hottest part of an induction motor. Moreover, the convective

cooling from the surfaces inside the endcaps of an electric motor is known to be difficult to be

predicted accurately. The fluid in the end space region of an electric motor is commonly air, which

is usually much more complex than the fluid/air flow over outer surfaces. This is because the fluid

flow depends on many factors, including the shape and length of the end-winding, added fanning

effects due to simple fans and end-ring wafters (fan/bladed), the surface finish of the end sections

of the rotor and the turbulence. Several authors have studied the cooling of internal surfaces in the

vicinity of the end-winding. In the majority of cases they propose the use of a formulation such as

that given below, i.e. the convection heat transfer coefficient is affected by the local fluid velocity

for surfaces in contact with the end space fluid:

ℎ = 𝑘1 ∙ (1 + 𝑘2 ∙ 𝑣𝑘3) (1.11)

where ‘k1’, ‘k2’ and ‘k3’ are curve fit coefficients and ‘v’ is the local surface fluid velocity [10].

The 𝑘1 ∙ 1 term accounts for natural convection when the reference velocity is zero and the 𝑘1 ∙

𝑘2 ∙ 𝑣𝑘3 term accounts for the added forced convection due to rotation. Plots of some of the

published correlations with totally enclosed air cooling are shown in Figure 1.1. The values of the

coefficients proposed by the different authors are reported in Table 1.2.

The main difficulty in computing the end-space region heat transfer coefficients using equation

(1.11) is the selection of a representative value of the inner fluid velocity. The air velocity varies

with the local position because of the shielding effect of the end-windings with respect to the rotor-

shaft angular velocity. Therefore, the air speed coincides with the rotational speed of the rotor for

the rotating surfaces and goes down to a very small fraction of the rotor velocity farther away from

the rotor, e.g. in the region between the end winding and inner housing [11].

Introduction End-space cooling

7

Figure 1.1. Published End-Space convection correlations [10] [12]

Table 1.2 - Coefficients proposed by different authors for the calculation of the HTC in the End-Space region [13]

Author k1 k2 k3 Notes

Mellor 15.5 0.39 1

IM with vel

Introduction Focus and aims of the M.Sc. project

8

especially concerning the winding-insulation manufacturing. Therefore, a better insulation thermal

conductivity may be expected for modern motors, leading to an increased heat transfer coefficient

with respect to the published correlations [12].

1.6 Focus and aims of the M.Sc. project

This thesis is the final report to the M.Sc. project carried out in collaboration with Motor Design

Ltd, Ellesmere, UK. The software developed by the company, Motor-CAD, is a unique software

package dedicated to the electromagnetic performance of motors and generators and the

optimization of their cooling. Motor-CAD provides the ability to quickly and easily perform

electromagnetic and thermal performance tests on prototype designs. Accurate electromagnetic

and thermal calculations can be done in seconds [14]. In Motor-CAD, the past experience is used

to estimate the local velocity term of equation (1.11). An End Space Velocity Multiplier term is

used to give an estimate of the general End Space Reference Velocity. The End Space Reference

Velocity is calculated from the rotor peripheral velocity using the formulation:

End Space Reference Velocity =

= End Space Velocity Multiplier ∙ RPM / 60 ∙ π ∙ Rotor Diameter

A drop down box provides a choice of End Space Velocity Multiplier experience values for

different levels of rotor end condition. The smoother the rotor end, the lower the End Space

Velocity Multiplier is set. Additionally, some of the surfaces in the end space are shielded from

the main End Space Reference Velocity. For these surfaces, additional experience factors are used

to estimate the local surface velocity, i.e. the outer surface of the end winding is set to be around

20% of the End Space Reference Velocity.

The main objective of the project is to use ANSYS Fluent CFD software to study the air flow and

convection around the end windings.

The specific objectives of the project are listed below:

1. To model the chosen electric machine, using Motor-CAD.

2. To export the selected electric machine in ANSYS DesignModeler and to model the end-

space region. Moreover, to generate the mesh for the selected geometry using ANSYS

Meshing software.

Introduction Outline of the Thesis

9

3. To investigate the improved cooling due to the additional fanning effect of the rotor fan

bladed/wafters, using ANSYS Fluent CFD.

4. To investigate how local convective cooling varies with the spacing between the end

winding and housing/endcaps, using ANSYS Fluent CFD.

5. To develop an improved correlation that better accounts for the fanning and end winding

spacing factors effects.

In order to fulfill all the goals listed above, a model of the Nissan Leaf motor will be used, having

been the machine already studied in a previous available work [15].

1.7 Outline of the Thesis

The thesis consists of six chapters, including this introductive one.

Chapter 2 describes the theoretical background in the field of thermal analysis of electrical

machines. Both the analytical and the numerical approach are presented, together with the most

significant methods and equations for the modelling of the turbulence.

Chapter 3 reports a detailed description of how the selected machine is implemented in Motor-

CAD and in ANSYS. The chapter actually includes a sort of tutorial for the ANSYS package,

including ANSYS Design Modeler, ANSYS Mesh and ANSYS CFD Fluent.

Chapter 4 provides a detailed explanation of the simulations run during the project, including their

settings and the main goals of each group of simulations.

Chapter 5 analyzes all the results obtained from the simulations presented in Chapter 4. These

include a sensitivity analysis upon the impact of specific geometrical modifications and the

development of a new improved relation for the computation of the heat transfer coefficients inside

the end-space region.

Chapter 6 summarizes the project, presenting the conclusions of this thesis and introducing some

possible improvement and analysis to be included in further work.

Literature review Lumped Circuit Thermal Network

10

2 Literature review

The present chapter is intended to be a summary of the theoretical knowledge acquired during this

M.Sc. project as propaedeutic background required to fully understand and analyze the problem

of the end-space region cooling in the electrical machines.

As already introduced in the previous chapter, the cooling system is crucial to increase the

electromagnetic performance and the lifetime of the electrical machines. High temperatures inside

the machine may lead to failures in the insulation materials and risk of permanent magnet

demagnetization in the case of permanent magnet synchronous machines. Therefore, to understand

the thermal issues, it is of primary importance for all electrical machines manufacturers to model

the heat transfers and improve the cooling of the machine.

The chapter gives an overview of the thermal analysis of electrical machines, from a theoretical

point of view. Both the analytical (Lumped Circuit) and the numerical (CFD) methods are

presented and their main advantages and drawbacks are reported.

2.1 Lumped Circuit Thermal Network

The lumped circuit thermal network (LPTN) is an analytical method for the thermal analysis of

electrical machines. The LPTN consists of an equivalent circuit for the representation of the

thermal properties of the machine, based on the so-called lumped element modelling technique.

The machine is geometrically divided into a number of lumped components where each component

has a given bulk thermal storage and heat generation and is connected to the other components via

a mesh of thermal impedances.

In the first work by Mellor in 1991 [16], the solid components of the machine were based on a

general cylindrical component and the following assumptions were made:

a. Independency of the heat flow in the radial and axial directions.

b. Single mean temperature of the element of the definition of the heat flux in both the radial

and axial directions.

c. Absence of circumferential heat flow.

d. Thermal capacity and heat generation uniformly distributed.


11

The machine was subdivided into ten components and a symmetry was assumed about the shaft

and a radial plane through the centre of the machine. The equivalent lumped circuit for each

component and for the overall machine is reported in Figure 2.1 and Figure 2.2 respectively.

Figure 2.1 - Equivalent lumped circuit for the general cylindrical component, Mellor 1991 [16]

In that case, the end-winding was modelled to be a homogenous toroid surface and the model was

weighted to estimate the peak hot-spot temperature, accepting a correction factor of 1.5 for the

computation of resistances and capacitance with respect to the mean value parameters. Convection

in the end-space region was taken into account adopting constant heat transfer coefficients

obtained by previous experiments and assumed to be proportional to the rotational speed of the

machine (the equation used was equal to equation (1.11) but with k1=1).

The lumped circuit proposed by Mellor, appeared to be suitable for both steady state and transient

analyses.

Figure 2.2 - Equivalent lumped circuit thermal network of an induction machine, proposed by Mellor in 1991 [16]


12

However, the resistive element commonly used in thermal equivalent networks is derived from a

one-dimensional solution to the steady-state heat diffusion equation for zero internal heat

generation. To include the internal heat generation a combination of solutions for two heat

diffusion equations with zero internal heat generation and zero surfaced temperatures was

proposed by Wrobel in 2010 [17]. This technique also allows the implementation of material

anisotropy, in contrast with the commonly used two resistors network.

To define the general cuboidal element, Wrobel made the assumption of independent heat transfer

within axes of the Cartesian coordinate system, in which the cuboidal element is modelled. The

equivalent three-dimensional network representation of the general cuboidal element is reported

in Figure 2.3. The central node is characterized by the average temperature for the cuboid and the

thermal capacitance and heat generation are attached to this point.

Figure 2.3 - Three dimensional network representation for the general cuboid, proposed by Wrobel, 2010 [17]

The cuboidal technique allows fast three-dimensional temperature prediction using the LPTN

method. Hence, it is applicable for fast transient thermal analyses.

Compared to a full numerical method, the LPTN approach introduces approximations, such as the

temperature independency of thermal conductivity. However, the cuboidal element model

introduce the mean temperature as a separate node of the network, making the model more accurate

for the thermal transient response. The LPTN method remains a valid approach, thanks to its much

less computational time, compared to the numerical methods.

A detailed LPTN for the end-space region of a TEFC machine can be found in [12] and has been

further applied and verified in [18]. The equivalent circuit is reported in Figure 2.4 whereas Table

2.1 reports a description of the symbols used in the picture.


13

Figure 2.4 - LPTN of the end-space region [18]

Table 2.1 - Explanation of the symbols used in the LPTN of the end-space region [18]

Symbol Meaning

PS Stator winding joule losses (active conductors in the slots)

PEW Stator winding losses (end windings)

R0 Thermal resistance between motor frame and ambient

RS-MF Thermal resistance between copper in the slot and rotor frame

RNC Thermal resistance between endwindings and motor frame due to natural convection

RRAD Thermal resistance between endwindings and motor frame due to radiation

REW-IA Thermal resistance between endwindings and the inner air

RIA-MF Thermal resistance between inner air and the motor frame

Convection is the major heat transfer mechanism in the end-space region of a TEFC electrical

machine. In the analytical thermal model, the convective HTC is based on empirical correlations

such as the one proposed in chapter 1.5. As already discussed, these correlations require the value

for the local fluid speed as an input. In most of the cases, these values are suggested from previous

experience or selected to match the experimental data in the best possible way.

Literature review Computational Fluid Dynamics (CFD)

14

2.2 Computational Fluid Dynamics (CFD)

The Computational Fluid Dynamics (CFD) is a branch of fluid mechanics and it uses numerical

analysis and numerical algorithms to solve problems which involve fluid flows. Thanks to the

improvement in computer power, the CFD method is growing in popularity among industries and

universities and it is now widely used for the thermal analysis of electrical machines.

In order to model a fluid flow using a CFD approach, it is of primary importance to understand the

physical phenomena behind fluid motion and how these phenomena can be modelled by using

numerical algorithms. Moreover, pros and cons in the different available turbulence models have

to be investigated in order to choose the most suitable for the modelling of the end-space region

air flow.

2.2.1 Governing equations of fluid flows

The cooling fluids used in machines, e.g. air, water or oil, are all characterized by a Newtonian

behavior. This means that they follow the rheological law first proposed by Newton (Newton law)

and reported below [19]:

𝜏 = 𝜇 ∙𝑑𝒖

𝑑𝑦

(2.1)

where ‘y’ is the direction orthogonal to the wall considered. Equation (2.1) represents a straight

line in the share rate - share stress plane, as shown in Figure 2.5, where the slope is equal to the

constant dynamic viscosity μ of the fluid.

Figure 2.5 - Graphical representation of the rheological law for a Newtonian fluid


15

The law can be represented by a straight line passing through the origin and with angular

coefficient equal to the fluid viscosity 𝜇. It can be rewritten in an arbitrary coordinate system of

axes i and j as [7]:

𝜏𝑖𝑗 = 𝜇 ∙ (𝜕𝑢𝑖𝜕𝑥𝑗

+𝜕𝑢𝑗

𝜕𝑥𝑖)

(2.2)

where 𝜏𝑖𝑗 is the jth component of the stress acting on the faces of the fluid element perpendicular

to the i-axis.

The fluid viscosity, which is an index of the internal friction of the fluid, can then be described as

the internal friction of the fluid and the proportional coefficient between the shear stress and the

variation of the streamwise speed with respect to the distance from the solid boundary.

The problems of fluid cinematic can be all described using two different approaches:

a. Lagrangian method: the particle of fluid is described through its coordinates and the

variation of these coordinates in time. Therefore, the trajectory of the fluid particle is

tracked.

b. Eulerian method: a control volume is define and the fluid within the control volume is

described trough “fields”, e.g. velocity, pressure and temperature. In this description of the

flow, the position of every single particle is not of any importance for the analysis of the

entire flow.

The Eulerian method is generally to be preferred to the Lagrangian method of flow when the fluid

mechanics has to be studied. However, the fundamental laws which governed the flow behavior

apply directly to the particles of the Lagrangian description. The link between Lagrangian and

Eulerian analyses is obtained starting from the following expressions:

𝑑𝑥 = 𝑢 ∙ 𝑑𝑡 (2.3)

𝑑𝑦 = 𝑣 ∙ 𝑑𝑡 (2.4)

𝑑𝑧 = 𝑤 ∙ 𝑑𝑡 (2.5)

The main tool to move the fundamental laws from the Lagrangian approach to the Eulerian

coordinates system is the Euler derivative (or total derivative or substantial derivative), defined by

the following operator:


16

𝐷

𝐷𝑡=

𝜕

𝜕𝑡+

𝜕

𝜕𝑥∙

𝑑𝑥

𝑑𝑡+

𝜕

𝜕𝑦∙

𝑑𝑦

𝑑𝑡+

𝜕

𝜕𝑧∙

𝑑𝑧

𝑑𝑡=

𝜕

𝜕𝑡+ 𝑢 ∙

𝜕

𝜕𝑥+ 𝑣 ∙

𝜕

𝜕𝑦+ 𝑤 ∙

𝜕

𝜕𝑧

(2.6)

The local term 𝜕

𝜕𝑡 can be distinguished by the three advection terms which take into account the

accelerations related to the spatial variation of the field considered (e.g. the velocity field) [20].

The three fundamental equations which govern the fluid flow can then be defined to be the

following:

a. Continuity equation, i.e. the conservation of mass

b. Conservation of momentum, i.e. the Navier-Stokes equation

c. Conservation of energy, i.e. the first law of thermodynamics

For thermal fluid analysis of electrical machines, the governing equations are coupled. Therefore,

the continuity equation is the transport equation3 for density and the energy equation is the

transport equation for temperature.

Additionally, for an ideal gas, the well know equation of state can be used to calculate the density,

known the pressure and temperature of the fluid:

𝜌 =𝑝

𝑅𝑇

(2.7)

where R is here the ideal gas constant. It should be noticed that air can be considered to be an ideal

gas at standard conditions.

Moreover, to distinguish between compressible and incompressible flow the Mach number can be

used. The Mach number is defined to be the ratio between flow velocity and the local speed of

sound, i.e. the speed of sound in the given fluid:

𝑀𝑎 =𝑢

𝑐

(2.8)

Depending on the Mach number, different flow regimes can be identified, as shown in Figure 2.6.

The flow is accepted to be incompressible for M


17

sufficiently higher than the range of velocities that can be found in an electrical machine.

Therefore, the density of air inside the end-space region of an electric machine can be assumed to

be constant.

Figure 2.6 - Definition of different flow regimes depending on the Mach number

2.2.1.1 Conservation of mass

Considering an infinitesimal volume of fluid, the net mass outgoing from the volume can be written

as sum of three terms, for the three directions in the space. The total outgoing mass has then to be

equal to the variation in the air density inside the volume, in the same amount of time:

(𝜕(𝜌𝑢)

𝜕𝑥+

𝜕(𝜌𝑣)

𝜕𝑦+

𝜕(𝜌𝑤)

𝜕𝑧) 𝑑𝑥 ∙ 𝑑𝑦 ∙ 𝑑𝑧 ∙ 𝑑𝑡 = −

𝜕𝜌

𝜕𝑡𝑑𝑥 ∙ 𝑑𝑦 ∙ 𝑑𝑧 ∙ 𝑑𝑡 (2.9)

The equation can be condensed in the following form [19]:

𝐷𝜌

𝐷𝑡+ 𝜌 ∙ 𝑑𝑖𝑣 𝒖 = 0 (2.10)

Assuming constant density, as previously discussed in 2.2.1, the equation is simplified into:

𝜌 ∙ 𝑑𝑖𝑣 𝒖 = 0 (2.11)

2.2.1.2 Navier-Stokes equations

This balance equation arises from applying the Newton’s second law to the motion of fluid and

hence represents the conservation of momentum for the fluid flow. The Navier-Stokes equations

can be written, in a compact form, as [20]:


18

𝐷(𝜌 ∙ 𝒖)

𝐷𝑡= 𝜌𝒈 − 𝑔𝑟𝑎𝑑 𝑝 + 𝜇 ∙ div τ +

1

3𝜇 ∙ 𝑔𝑟𝑎𝑑 (𝑑𝑖𝑣 𝒖) (2.12)

The terms 𝜇 ∙ div τ +1

3𝜇 ∙ 𝑔𝑟𝑎𝑑 (𝑑𝑖𝑣 𝒖) takes into account the viscous behavior of the fluid.

Therefore, for an “ideal” fluid, i.e. a fluid with constant density and viscosity coefficient equal to

zero, the equation is simplified into the Euler equation:

𝜌 ∙𝐷𝒖

𝐷𝑡= 𝜌𝒈 − 𝑔𝑟𝑎𝑑 𝑝 (2.13)

For a real, incompressible fluid the divergence of the velocity vector is equal to zero. Moreover,

the term 𝜌𝒈, which takes into account for the gravity effect, is negligible in the case of rotating

flows. If the fluid is Newtonian, i.e. respect equation 2.1, the Navier-Stokes equations then

become4:

𝜌 ∙𝐷𝒖

𝐷𝑡= −𝑔𝑟𝑎𝑑 𝑝 + 𝜇∇2𝒖 (2.14)

The Navier-Stokes equations, combined with the continuity equation, create a system of equations

with five unknowns: the density ρ, the three components of the speed u,v,w, and the pressure p.

However, for incompressible flow the density is assumed to be constant; therefore, the equation of

state is not any longer required to solve the system.

2.2.1.3 Energy equation

The first principle of the thermodynamics ensures that the variation of the internal energy of a

close system is equal to the difference between the amount of heat supplied to the system and the

amount of work done by the system on the external ambient.

Considering that the heat transfer occurs at constant pressure, the following equation can then be

written as energy equation for the fluid flow [7]:

𝐷(𝜌 ∙ 𝑐𝑝 ∙ 𝑇)

𝐷𝑡= 𝑑𝑖𝑣(𝜆 ∙ 𝑔𝑟𝑎𝑑 𝑇) + Φ (2.15)

4 ∇2 is the “Laplace” operator, which is defined as:

∇2=∂2

∂x2+

∂2

∂y2+

∂2

∂z2


19

The conversion of mechanical energy into heat (viscous dissipation function) is considered by the

term Φ which can in this case be neglected, since it is important only in the case of very high fluid

viscosity and very large velocity gradient of the fluid motion.

Assuming an incompressible flow, i.e. the density to be constant, and a constant value for the

specific heat at constant pressure 𝑐𝑝 and the conductivity 𝜆, the energy equation can then be

rewritten as:

𝐷𝑇

𝐷𝑡= 𝛼 ∙ ∇2𝑇 (2.16)

where 𝐷𝑇

𝐷𝑡=

𝜕𝑇

𝜕𝑡+ 𝒖 ∙ 𝑔𝑟𝑎𝑑 𝑇 (Euler derivative) and 𝛼 =

𝜆

𝜌∙𝑐𝑝 is the thermal diffusivity.

2.2.2 Turbulent flow

The Reynolds number, which represents the ratio between inertial and viscous forces, is used to

distinguish between laminar and turbulent flow, as already discussed in 1.4. A high Reynolds

number indicates that the inertia forces are sufficiently high to trigger the turbulence vortices in

the flow [7].

The CFD simulation of turbulent flows is much more complicated than the simulation of laminar

flows, according to the governing equations showed in 2.2.1. because of the unsteady, three-

dimensional-random, chaotic swirling in the flow. Different numerical approaches can be used in

order to solve the turbulence:

a. Direct Numerical Simulation (DNS): it solves both large-scale and small-scale eddies in

the flow field. This leads to the most accurate possible solution with numerical methods.

However, the computational time is extremely high.

b. Large Eddy Simulation (LES): it solves only the large-scale eddies in the flow field.

Compared to the DNS, the LES makes the assumption that the small-scale eddies are

isotropic and behaves in a predictable way. This drawback is largely rewarded by the

reduction in the computational time.

c. Reynolds-Averaged Navier-Stokes (RANS) equations: only the effect of the turbulence

on the mean flow is considered. This is the most computational economic approach and it

includes a group of 2-transport-equations models such as the k-ε and the k-ω.


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2.2.2.1 RANS equations

In the RANS equations only the mean values of all variables are considered. Therefore, the velocity

and all other variables have to be split into a time-averaged component and a fluctuating

component. Given a general variable ‘u’, the decomposition is made as follows:

𝑢 = �̅� + 𝑢′ (2.17)

with obvious meaning of the symbols.

Using the RANS approach generates an additional term in the momentum equation, which is called

Reynolds stress tensor [7]:

𝜏𝑡𝑢𝑟𝑏 = −𝜌𝑢𝑖′𝑢𝑗′̅̅ ̅̅ ̅̅ = −𝜌 [𝑢′𝑢′̅̅ ̅̅ ̅̅ 𝑢′𝑣′̅̅ ̅̅ ̅̅ 𝑢′𝑤′̅̅ ̅̅ ̅̅

𝑢′𝑣′̅̅ ̅̅ ̅̅ 𝑣′𝑣′̅̅ ̅̅ ̅̅ 𝑣′𝑤′̅̅ ̅̅ ̅̅

𝑢′𝑤′̅̅ ̅̅ ̅̅ 𝑣′𝑤′̅̅ ̅̅ ̅̅ 𝑤′𝑤′̅̅ ̅̅ ̅̅ ̅] (2.18)

Therefore, equation 2.12 becomes:

𝜌𝐷𝒖

𝐷𝑡= −𝑔𝑟𝑎𝑑 𝑝 + div τ𝑖𝑗 (2.19)

where:

τ𝑖𝑗 = 𝜇 ∙ (𝜕𝑢𝑖𝜕𝑥𝑗

+𝜕𝑢𝑗

𝜕𝑥𝑖) − 𝜌𝑢𝑖′𝑢𝑗′̅̅ ̅̅ ̅̅ (2.20)

To close the system of equations and solve the RANS model, the Boussinesq5 approximation is

used and the Reynolds stress tensor can then be rewritten as [7]:

−𝜌𝑢𝑖′𝑢𝑗′̅̅ ̅̅ ̅̅ = 2𝜇𝑡𝑆𝑖𝑗 −2

3𝜌 ∙ 𝑘 ∙ 𝛿𝑖𝑗 (2.21)

where 𝜇𝑡 is the turbulent viscosity of eddy viscosity which characterizes the transport and

dissipation of energy in the smaller-scale flow (whereas the large-scale turbulence is solved by the

model). 𝑆𝑖𝑗 is the mean strain-rate tensor defined by [21]:

𝑆𝑖𝑗 =1

2(

𝜕𝑢𝑖𝜕𝑥𝑗

+𝜕𝑢𝑗

𝜕𝑥𝑖) (2.22)

5 Joseph Valentin Boussinesq, in 1887, proposed to relate the turbulence stress to the mean flow in order to close the

system of equations. The application of the Boussinesq makes the new turbulence viscosity term appear in the RANS

equations.


21

In equation 2.21 ‘k’ is the turbulent kinetic energy, which is equal to [7] [21]:

𝑘 =1

2𝑢𝑖′𝑢𝑖′̅̅ ̅̅ ̅̅ =

1

2(𝑢′2̅̅ ̅̅ + 𝑣′2̅̅ ̅̅ + 𝑤′2̅̅ ̅̅ ̅) (2.23)

In an analog way, the turbulent viscosity can be taken into account in the energy equation by

adding at the standard thermal diffusivity an additional turbulent thermal diffusivity. The

governing equations in RANS model become:

𝜌 ∙ 𝑑𝑖𝑣 �̅� = 0 (2.24)

𝜌 ∙𝐷�̅�

𝐷𝑡= −𝑔𝑟𝑎𝑑 𝑝 + (𝜇 + 𝜇𝑡)∇

2�̅� (2.25)

𝜕�̅�

𝜕𝑡+ �̅� ∙ 𝑔𝑟𝑎𝑑 �̅� = (𝛼 + 𝛼𝑡) ∙ ∇

2�̅� (2.26)

where all the variables which appear in the model are time-averaged.

The RANS approach is used in several turbulence models. These can be divided into algebraic,

one-equation, two-equations and Reynolds stress models. It should always be kept in mind that the

turbulence model is an approximation of the real problem and, for this reason, the CFD calculation

should be always supported by experimental data for its calibration.

2.2.2.2 k-ε model

The k-ε model for turbulence is mainly valid for fully turbulent flows but it appears to be very

poor for complex flows with severe pressure gradients and strong streamline curvature [7] [9].

Compared to the standard version of the model, the two-layer k-ε model and the low Reynolds k-

ε models improve the solution for the near-wall regions. Moreover, the normalized group (RNG)

version of k-ε model includes a modification that better takes into account for the effects of rapid

strain rate and streamline curvature.

The kinetic eddy viscosity in the k-ε model is given by [9]:

𝜇𝑡 = 𝐾𝜇 ∙𝜌 ∙ 𝑘2

𝜀 (2.27)

where 𝐾𝜇 is an empiric constant and ε is the dissipation of the turbulent kinetic energy.


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2.2.2.3 k-ω model

The k-ω model is a valid alternative to the k-ε model in the cases of boundary-layer flows in the

near-wall region treatment and of streamwise adverse pressure gradient. Generally, its application

is more indicated for low Reynolds numbers, whereas the application of the standard k-ε model is

limited to high Reynolds numbers.

The kinetic eddy viscosity in the k-ω model is given by [21]:

𝜇𝑡 = 𝜌 ∙𝑘

𝜔 (2.28)

The shear stress transport (SST) k-ω model overcomes the free stream sensitivity problem of the

standard k-ω model, enlarging its field of application. This is because it combines the k-ω standard

model in the near-wall region with the k-ε model for the far field, i.e. outside of the turbulence.

The combination obtained in SST k-ω model makes it suitable for a wide range of Reynolds

numbers [22].

2.2.2.4 Reynolds Stress Transport Model (RSM)

The Reynolds stress model (RSM) is the most elaborate type of RANS turbulence model. The

Boussinesq’s isotropic eddy-viscosity hypothesis that is used in the k-ε and k-ω models is

abandoned, in order to better predict turbulent flows with complex strain fields. The RSM closes

the RANS equations by solving six transport equations, one per each Reynolds stress, together

with an equation for the dissipation rate of the turbulent kinetic energy ε [23].

The Reynolds stress model solves a system of seven equations against the only two-equations

system solved by the k-ε and k-ω models. Therefore, it results to be more computationally

expensive and not suitable for a large number of simulations in a limited amount of time.

2.2.3 Wall treatment in CFD

Turbulent flow are significantly affected by the interaction with walls. Even if the mean velocity

field is affected by the no-slip condition at the walls, the turbulence is influenced by the presence

of the wall in a complex way. In the closest region to the wall, the viscous damping reduces the

tangential velocity fluctuations and the kinematic blocking reduces the normal fluctuations.

However, in the outer part of the near-wall region, the turbulence dominates again due to large

gradients in the mean velocity [23]. For a good result of the CFD simulation, the near-wall


23

modelling is of primary importance, especially in those cases where the walls are the main source

of mean vortices and turbulence, as it happens inside an electrical machine.

The near-wall region can be described as shown in Figure 2.7, where semi-log coordinates are

used. The closest part to the wall, called “viscous sub-layer”, is characterized by an almost laminar

flow, because of the dominant role of the viscosity in momentum and heat transfer. On the other

hand, the outer part of the near-wall region, called “fully-turbulent layer” is characterized by the

fully turbulent flow, because of the dominant role of the turbulence. The intermediate region,

called “buffer layer” is a transition region between the viscous sub-layer and fully-turbulent layer.

Figure 2.7 - Subdivision of the Near-Wall region [23]

In Figure 2.7, the flow velocity and the wall distance are shown in a dimensionless form as [23]:

𝑢+ =�̅�

𝑢𝑡 (2.29)

𝑦+ = 𝜌𝑦 ∙ 𝑢𝑡

𝜇 (2.30)

where 𝑢𝑡 is the friction velocity and is equal to:


24

𝑢𝑡 = √𝜏𝑤𝜌

(2.31)

The three regions of the near-wall boundary layer are identified by the following values of the

dimensionless distance from the wall 𝑦+:

a. 𝑦+ < 5: viscous sub-layer. The relation between 𝑢+ and 𝑦+ is linear:

𝑢+ = 𝑦+ (2.32)

b. 𝑦+ > 30: fully-developed turbulent layer. For a smooth wall:

𝑢+ = 2.5 ∙ 𝑙𝑛𝑦+ + 5.45 (2.33)

c. 5 < 𝑦+ < 30: buffer layer. This layer can be solved by setting the condition 𝑦+ = 11 as

boundary to distinguish between the laminar and the turbulent profile.

Traditionally, two different approaches are available for the modelling of the near-wall boundary

layer (Figure 2.8).

The first approach, named “Wall function approach”, does not solve the boundary layer and,

instead, it uses semi-empirical equations, called “wall functions”, as a bridge between the no-slip

condition of the wall surface and the fully-turbulent region.

Figure 2.8 - Wall treatment approaches [23]


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The second approach, named “Near-Wall model approach”, solves the boundary-layer all the way

down to the wall. To do that, high mesh resolution is needed in the viscous-dominated region and

the approach results to be more computational expensive. A value of 𝑦+ ≈ 1 is required for the

closest layer of the mesh in order to be sure that the viscous sub-layer will be solved. The

“Inflation” can be added to achieve the required 𝑦+, as it will be better described in chapter 4.1.1.

It has to be noticed that a hybrid approach is possible. With the so called “All 𝑦+ wall treatment”

a different wall-treatment approach is applied, depending on the local value of the 𝑦+, i.e. the

viscous-sub layer is solved for low values of 𝑦+ and the wall functions are applied in the case of

high 𝑦+ [7]. This hybrid approach is embedded in the SST k-ω model of ANSYS CFD Fluent.

2.2.4 Modelling of rotating regions

The modelling and analysis of electrical machines include the presence of rotating part, which

strongly influence the flow field in the considered volume. Therefore, it is crucial to embed the

rotating effects into the CFD methods, described in the previous sections. There are two

fundamental approaches for the analysis of rotating parts in a CFD software:

a. Moving Reference Frame (MRF)

b. Sliding Mesh

2.2.4.1 Moving Reference Frame (MRF)

The governing equations of the flow are generally solved in the stationary reference frame.

However, in the cases of rotating parts such as for the electrical machines, it is useful to solve the

system of equations in a moving, non-inertial, reference frame. Moving parts make the problem

unsteady when viewed from a stationary frame. However, in a moving reference frame

synchronous with the rotating parts, the flow can be modelled as a steady-state problem with

respect to the moving frame [23]. The equations of motion are in this case modified to include

additional acceleration terms that appear as a consequence of the transformation from stationary

to moving reference frame.

In solving some problems it is possible to incorporate the entire computational domain into a single

moving reference frame. However, in the case of electrical machines, it is better to split the domain

into two regions, one rotating synchronously with the rotor and one fixed with the stator. Therefore,

the problem has to be break up into multiple cell zones, with a defined interface between them, in

order to apply multiple reference frame approach.

Literature review Chapter summary

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The solution given by the MRF approach always represents the time-averaged behavior of the

flow. The “rotating” walls do not physically rotate in the CFD representation of the problem but

the rotation is taken into account by means of the Coriolis and the centrifugal forces.

The MRF approach is suitable for steady state analysis of rotating flows for all the cases where

transient effects such as vortex shedding and wake coming off rotating components can be

neglected. Compared to the sliding mesh method it results to be much less computational

expensive.

2.2.4.2 Sliding Mesh

The sliding mesh model can be considered to be a special case of the more generic dynamic mesh

model. In the case of sliding mesh, the nodes move rigidly in a certain dynamic mesh zone and

multiple cells zones are connected together through a non-conformal interface. Therefore, a non-

conformal interface between the rotating and the non-rotating regions must be created during the

meshing process, as it will be described in chapter 3.2.3.

In the sliding mesh approach, the rotating parts are physically put in rotation via a small increment

of the shifting angle between rotating and non-rotating region, at each time step. The interfaces

between the two regions slide relative to each other and the physical interaction between rotor and

stator is modelled exactly. The setting of

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