Computational Analysis of 3-Dimensional Transient Heat Conduction in the Stator of an Induction Motor during Reactor Starting using Finite Element Method

8/12/2019 Computational Analysis of 3-Dimensional Transient Heat Conduction in the Stator of an Induction Motor during Re

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Computational Analysis of 3-Dimensional Transient

Heat Conduction in the Stator of an Induction Motor

during Reactor Starting using Finite Element Method

A. K. Naskar1and D. Sarkar

2

1Seacom Engineering College, Howrah, IndiaEmail: [email protected]

2

Bengal Engineering & Science University, Howrah, IndiaEmail: [email protected]

Abstract In developing electric motors in general and induction motors in particular

temperature limit is a key factor affecting the efficiency of the overall design. Sinceconventional loading of induction motors is often expensive, the estimation of temperaturerise by tools of mathematical modeling becomes increasingly important. Excepting for

providing a more accurate representation of the problem, the proposed model can alsoreduce computing costs. The paper develops a three-dimensional transient thermal model in

polar co-ordinates using finite element formulation and arch shaped elements. Atemperature-time method is employed to evaluate the distribution of loss in various parts ofthe machine. Using these loss distributions as an input for finite element analysis, more

accurate temperature distributions can be obtained. The model is applied to predict thetemperature rise in the stator of a squirrel cage 7.5 kW totally enclosed fan-cooled induction

motor. The temperature distribution has been determined considering convection from the

back of core surface, outer air gap surface and annular end surface of a totally enclosedstructure.

Index Terms FEM, Induction Motor, Thermal Analysis, Design Performance, Transients

I.INTRODUCTION

Considering the extended use of squirrel cage induction machine in industrial or domestic applications both

as motor and generator, the improvement of the energy efficiency of this electromechanical energy converterrepresents a continuous challenge for the design engineers, any achievements in this area meaning importantenergy savings for the world economy. Thus to design a reliable and economical motor, accurate prediction

of temperature distribution within the motor and effective use of the coolant for carrying away the heat

generated in the iron and copper are important to designers[18].

Traditionally, thermal studies of electrical machines have been carried out by analytical techniques, or by

thermal network method [1], [8]. These techniques are useful when approximations to thermal circuit

parameters and geometry are accepted. Numerical techniques based on finite element methods [5],[7] and[10]-[20] are more suitable for analysis of complex system. Rajagopal, M.S, Kulkarni, D.B, Seetharamu,K.N,and Ashwathnarayana P. A [4], [6] have carried out two-dimensional steady state and transient thermal

DOI: 02.PEIE.2014.5.14 Association of Computer Electronics and Electrical Engineers, 2014

Proc. of Int. Conf. onAdvances in Power Electronics and Instrumentation Engineering, PEIE


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35

analysis of TEFC machines using FEM. Compared to the finite difference method, the finite element method

can easily handle complicated boundary configurations and discontinuities in material properties.

The finite element method was first introduced for the steady state thermal analysis of the stator cores oflarge turbine-generators by Armor and Chari [2]. However, their works are restricted to core packages far

from the ends and they do not consider the influence of the stator coil heat. Sarkar and Bhattacharya [9] alsodescribed a method based on arch-shaped finite elements with explicitly derived solution matrices for

determining the thermal field of induction motors.In this paper, the finite element method is used for predicting the temperature distribution in the stator of an

induction motor using arch-shaped finite elements with explicitly derived solution matrices. A 100-element

three-dimensional slice of armature iron, together with copper winding bounded by planes at mid-slot, mid-tooth and mid-package, are used for solution to a transient stator heating problem, and this defines the scope

of this technique. The model is applied to one squirrel cage TEFC machine of 7.5 KW and the temperaturesobtained are found to be within the permissible limit in terms of overall temperature rise computed from the

resulting loss density distribution.

II.POLYPHASE INDUCTION MOTOR MODEL AND BOUNDARY CONDITIONS

The details of the induction motor are shown in Fig. 1. In this analysis, the 3-dimensional slice of core ironand winding has been chosen for modeling the problem and the geometry is bounded by planes passingthrough the mid-tooth, the mid-slot and the package centre. This is shown in Fig.2, taken from the shaded

region A of Fig.1. The temperature distribution is assumed symmetrical across these three planes, with theheat flux normal to the three surfaces being zero. From the other three boundary surfaces, heat is transferred

by convection to the surrounding gas. It is convected to the air-gap gas from the teeth, to the back of core gasfrom the yoke iron, and to the core end gas from the annular end surface of core. The boundary conditions

may be written in terms of nT / , the temperature gradient normal to the surface.

Axial centre of package 0=pn

T

(1)

Mid-slot surface 0=Sn

T

(2)

Mid-tooth surface 0=tn

T

(3)

Air-gap surface ( )GA

rAG

n

TVTTh

= (4)

Annular end surface of core ( )D

zDn

TVTTh

= (5)

Back-of-core surface, ( )BC

rBCn

TVTTh

= (6)

A. Finite Element Formulation

The governing differential equation for transient heat conduction is expressed in the general form as

0~1

2

2

2

2

2 =+++

t

TCPQ

Z

TV

T

r

V

r

TrV

rrmmzr

(7)

To find the solution of (7) together with the boundary and initial conditions by Galerkins weighted residualapproach, we first express the approximate behaviour of the nodal temperature within each elementaccording to equation (8). Since substitution of this approximation into the original differential equation and

boundary conditions results in some error called a residual, the method of weighted residual requires that theintegral of the projection of the residual on a set of weighting functions is zero over the solution region.

The approximate behaviour of the potential function within each element is prescribed in terms of their nodalvalues and some weighting functions N1, N2 such that


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Fig.1. Half sectional end & sectional elevation of a 7.5 kW squirrel cage induction motor

Fig. 2. Slice of core iron & winding bounded by planes at mid-slot, mid-tooth and mid-package

ii

mi

TNT =

=....2,1

(8)

The required equation governing the behaviour of an element is given by the expression:

0~

)()()(

2

)(

=

+

+

+

dvolt

TCPQ

z

TV

z

T

r

V

r

TV

rN

e

mm

e

z

ee

ri

vol

(9)

Integrating all the terms through integration by parts, the equation takes the form

rdzddrr

N

r

TV

dzdrNr

TVdrdzdr

r

TV

rN

ie

r

D

i

e

r

S

e

ri

D

e

ee

)(

)()(

)(

)(2

)(

=

rdzddrr

N

r

TVdNn

r

TV i

e

r

D

ir

e

r

See

.

)()(

)()(2

= (10)


4/13


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( ) ( )( )( )

( ) ( )( )( )abab

N

ab

abN

B

A

/14

11/

/14

11/

++

=

+

=

( ) ( )( )( )

( ) ( )( )( )abab

N

ab

abN

F

E

/14

11/

/14

11/

+

=

=

( ) ( )( )( )

( ) ( )( )( )ab

N

abN

D

C

/14

111

/14111

+=

++=

( ) ( )( )( )

( ) ( )( )( )ab

N

abN

H

G

/14

111

/14111

=

+=

(13)

It is seen that the shape functions satisfy the following conditions:

(a) That at any given vertex A the corresponding shape function NAhas a value of unity, and the othershape functions NB, NC, ,have a zero value at this vertex. Thus at node j, Nj = 1 but Ni= 0 , i

j .(b) The value of the potential varies linearly between any two adjacent nodes on the element edges.

(c) The value of the potential function in each element is determined by the order of the finite element.The order of the element is the order of polynomial of the spatial co-ordinates that describes the

potential within the element. The potential varies as a cubic function of the spatial co-ordinates onthe faces and within the element.

C. Approximate Numeric Form

According to Galerkins weighted residual approach, the weighting functions are chosen to be the same as

the shape functions. Substituting (12) and (13) into (10), gives

{ }[ ] +

+

+

+

=

V

ii

emm

e

i

e

z

e

i

ee

i

e

r

V

dVNTNTNt

CPNQ

z

T

Tz

TV

dVT

T

T

r

V

r

T

Tr

TV

0

)()()(

)()(

2

)()(

2][22

0

{ }

)()(

)(2

e

ii

e

S

dNThNTNhe

for i = A, B,., H (14)These equations, when evaluated lead to the matrix equation

[ ] [ ] [ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ]CTHTZR SRTSTSSSSS ++=++++ 0 (15)

III.DISCRETIZED MODEL FOR FEM APPLICATION

The stator of an induction motor, under transient conditions is designed to maintain all temperatures belowclass A insulation limits of 105

oC hot spot. The hottest spot is generally in the copper coils. Thermal

conductivity of copper and insulation in the slot are taken together for simplification of calculation [2].

In the case of transient stator heating caused by reactor starting, the transient analysis procedure is able toprovide an estimate of the temperatures throughout the volume of the stator at an interval of time required to

bring the motor from rest to rated speed by providing reduced voltage and current during the starting periodand as the motor has reached a sufficiently high speed near to the operating speed, rated voltage and current

are provided by short circuiting the reactors during the starting action of the induction motor.Assuming that the machine is at rest with its stator winding at normal ambient temperature, respective

voltage and current are injected to the stator winding of the machines. The temperatures within the volume of

the stator are calculated at all nodal points for a period of time required for the reactor starting action.


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In this analysis, because of symmetry, the 3-dimensional slice of core iron and winding, chosen for modeling

the problem are divided into arch shaped finite elements as shown in fig 5.

Fig.5. Slice of core iron & winding bounded by planes of mid-slot & mid-tooth divided into arch shaped Finite Elements

IV.CALCULATION OF HEAT LOSSES

Heat losses in the tooth and yoke of the core are based on calculated magnetic flux densities (0.97 wb / m2

and 1.293 wb/m2 respectively) in these regions. Tooth flux lines are predominantly radial and yoke flux lines

are predominantly circumferential. The grain orientation of the core punching differs in these two directions

and therefore influences the heating for a given flux density. Copper losses in the winding are determinedfrom the length as well as the area required for the conductors in the slot.

Iron loss of stator core per unit volume = 0.0000388 W/mm3.Iron loss of stator teeth per unit volume= 0.0000392 W/mm

3.

A. Stator copper loss

In reactor starting of the induction motor, the equivalent circuit of which is shown in Fig.6, we are interested

to calculate the temperature distribution in the stator during the starting period. For the purpose of starting we

will take the starting voltage at 50% of full voltage to start with and calculations will be done on that voltagetill the reactor acts as impedance in the motor circuit. Finally, the temperature distribution within the stator

due to reduced voltage reactor starting are calculated by splitting the entire slip range (i.e. from s=1 to full

load slip s=0.04) into small intervals.

Fig.6. Equivalent Circuit of Induction Motor


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x1=8.15;Ic=0.176 Amp; r1=2.04;Im=2.41Amp

0425.0;415;39.2 1/2

=== sVVr

To calculate winding impedance of the reactor

Voltage across the reactor is=415/3 V

The stator winding current per phase during starting = )15.843.4(

4155.0

j+

= 22.37A.

Line current = 22.37 3 =38.75A, which is the output current of the reactor. From VA balancing the inputcurrent of the reactor is=38.75/2 =19.38A and 50% of total impedance of the reactor will be

=415/(319.382)= 6.18 To calculate stator current at starting when reactor is connected in the circuit from s=1.0 to s=0.2,

At start s=1.0

Resistance of the circuit= (+

/s) = (2.04+2.39/1) =4.43

Reactance in the circuit (x1) =8.15

Impedance of the circuit (z1) = (4.43)2+ (8.15)

2=9.3

As the stator is delta connected and 50% of full voltage is applied across the stator winding, the stator current

at s=1 will beI1=415/ (9.3+6.18)=415/15.48 =26.85ATo calculate stator current at different slips when motor is directly connected to the supply from slip s=0.2 to

full load slip s=0.0425,

At slip s=0.2Resistance of the circuit= r1+r

2/s = (2.04+2.39/0.2)= 13.99

When full voltage is provided across the stator winding by short-circuiting the reactors, the stator current at s= 0.2 will beI1=415/ (13.99+j8.15) =25.63A

The stator currents, stator copper losses and the time required for starting action at different slips are

calculated and tabulated as shown in Table I.

TABLEI.THE DIFFERENT VALUES OF STATOR CURRENT,STATOR COPPER LOSS /SLOT /UNIT VOLUME AND TIME REQUIRED FOR

STARTING ACTION AT DIFFERENT SLIPS IN REACTOR STARTING

B. Convective heat transfer co-efficient [2,8]

Three separate values of convection heat transfer co-efficient have been taken for the cylindrical curvedsurface over the stator frame and the cylindrical air gap surface and the annular end surfaces. The natural

convection heat transfer co-efficient on cylindrical curved surface over the stator frame is taken as h=5.25w/m

2 oC.


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The heat transfer co-efficient on forced convection for turbulent flow in cylindrical air gap surface is taken as

h=60.16 w/m2 o

C. The heat transfer co-efficient on forced convection for turbulent flow in annular end

surface is taken as,h= 34.67W / m2oC.

C. Thermal constants [3,9]

For a transient problem in three-dimensions, the following properties are taken for each different element

material as shown in Table II.

TABLE II. TYPICAL SET OF MATERIAL PROPERTIESFOR INDUCTION MOTOR STATOR

V.RESULTS AND DISCUSSIONS

Since the hottest spots are found to be in the stator copper as envisaged from the calculated temperatures for

the three-dimensional structure during the reactor starting period as shown in Table III, the temperaturevariation with time in each node of copper is taken as an index to understand the temperature profile duringthe transient. It is to be noted that the temperature is found to be maximum at the nodes pertaining to copper

in the axis of symmetry. The temperature rise is steady at different stator currents under the reactor starting

region at different slips from s = 1 to s = 0.2. It is also to be noted that under DOL run the motor has reached

a steady speed under full load condition and as such there is a slight decrease of hot spot temperaturespersisting across the axis of symmetry after the reactors are shorted.As a consequence, the temperature variation with time at hottest spots has been depicted in graphs as shown

in Fig. 7to investigate the magnitude of the temperature variation with time at different nodal points alongthe stator copper winding.

TABLE III.SOLUTION FOR THREE DIMENSIONAL STRUCTURE

Magnetic Steel Wedge Copper &Insulation

Vr 33.070 2.007

V 0.8260 1.062

VZ 2.874 358.267

Pm 7.86120 8.9684

Cm 523.589 385.361

NodeNos.

InitialTemperature

Temperatures exceeding 440

C in 1st

time step with convection in three dimensional structure of totally enclosed machinefor different stator current during Reactor starting.

Q=0.0

03257

I=26.8

5Amp

Startingtime=4.2

9secs

Q=0.0

031967

I=26.6

Amp

Startingtime=3.8

8secs

Q=0.0

03125

I=26.3

Amp

Startingtime=3.4

8secs

Q=0.0

0303065

I=25.9

Amp

Startingtime=3.0

9secs

Q=0.0

029148

I=25.4

Amp

Startingtime=2.7

02secs

Q=0.0

027563

I=24.7

Amp

Startingtime=2.3

3secs

Q=0.0

025078

I=23.5

6Amp

Startingtime=1.9

9secs

Q=0.0

0213725

I=21.7

5Amp

Startingtime=1.7secs

Q=0.0

0296779

I=25.6

3Amp

Startingtime=1.1

975

Q=0.0

0105249

I=15.2

6Amp

Startingtime=1.2

22secs

31 400C 44.563

0C

47.6950C

49.9060C

51.4880C

52.6100C

53.3680C

53.7990C

53.9250C

54.4270C

53.9820C

67 400C 44.567

0C

47.7070C

49.9330C

51.5310C

52.6700C

53.4430C

53.8870C

54.0230C

54.5330C

54.0950C

103 400C 44.587

0

C

47.7810

C

50.0730

C

51.7370

C

52.9350

C

53.7580

C

54.2440

C

54.4150

C

54.9460

C

54.5310

C137 400C 44.061

0C

47.0990C

49.4660C

51.3200C

52.7530C

53.8190C

54.5440C

54.9500C

55.5840C

55.4030C

138 400C 44.014

0C

47.1120C

49.5500C

51.4580C

52.9240C

54.0070C

54.7390C

55.1480C

55.7610C

55.5970C

139 400C 44.7410C

48.0920C

50.5320C

52.3220C

53.6220C

54.5240C

55.0670C

55.2760C

55.8420C

55.4370C

175 400C 44.408

0C

47.6680C

50.1050C

51.9250C

53.2650C

54.2100C

54.8020C

55.0680C

55.6210C

55.3120C


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Fig.7. Corresponding Temperatures vs Time

VI.CONCLUSIONS

The numerical models using finite element method developed in this paper referring to a 7.5 kW induction

motor proved to be accurate offering useful results for engineers involved in the design of electric machines.The results based on the proposed method show close agreement for totally enclosed machines with

calculated heat transfer coefficients at the back of core surface, air-gap surface and annular end surface of the

stator winding. This analysis gives the designers a better idea of where the hot spot is and how the heat iscarried away at the outer surfaces with the help of both conduction and convection modes of heat transfer, the

magnitude of which is determined in terms of coefficients usually derived from machine parameters.

VII.APPENDIX

A.Formulation Of The Heat Convection Matrix On Cylindrical Curved Surface

The heat convection term in (13) for i = A is

[ ]{ } dsNTNh ie

S e

)(

)(2

( ) dsNNTNNTNTh HAHBABAAS

e+++=

2

)(2

(16)

Now performing the integration in non-dimensional notation

dsNh A2

( ) ( ))(

16

1122

1

1

1

1dvcda

vh

+=


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43

( ) ( ) dvvdcha

+=1

1

1

1

2211

16

cha9

4= (17)

dsNNh BA ( ) ( ) ( ) ( ) )(

4

11

4

111

1

1

1dvcda

vvh

++

+=

( ) ( ) ddvvcha +=1

1

1

1

22 1116

cha9

2= (18)

Evaluating the other terms, we obtain

[ ]

=

0

00

00

00

00000

000000

0000

0000

94

92

94

929194

91

92

92

94

cahSH (19)

B.On annular end surface

By performing the integration over (

,v) space,

dsNh AS

e

2

)(2

( ) ( )( )

)(/14

1/2

221

1

1

1dvcda

ab

vabh

=

( )

+

=

2

2

4

4

2

2

23

2

124

1

/13

2

a

b

a

b

a

b

ab

ha (20)

dsNNh BA ( ) ( )

( ))(

/14

1/2

221

/

1

1dvcda

ab

vabh

ab

=

( )

+

=

3

4

23

2

124

1

/142

2

4

4

2

2

a

b

a

b

a

b

ab

ha

( )

+

=

2

2

4

4

2

2

23

2

124

1

/13 a

b

a

b

a

b

ab

ha (21)


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44

[ ]

[( )

]

[( )

]

[( )

]

[( )

]

[( )

]

[( )

]

[( )

]

[( )

]

[( )

]

[( )

]

+

+

+

+

+

+

+

+

+

+

=

0

00

000

0000

0000

)23

2

412

1(

/13

2

0000

)23

2

412

1(

/13

)23

2

412

1(

/13

2

0000

)66

1212

1(

/13

2

)66

1212

1(

/13

2

)23

2

124

1(

/13

2

0000

)6

6

1212

1(

/13

2

)6

6

1212

1(

/13

)

23

2

124

1(

/13

)

23

2

124

1(

/13

2

2

2

3

3

4

4

2

2

2

2

3

3

4

4

2

2

2

2

3

3

4

4

2

2

3

3

4

4

2

2

3

3

4

4

2

2

2

2

4

4

2

2

3

3

4

4

2

2

3

3

4

4

2

2

2

2

4

4

2

2

2

2

4

4

2

2

SYM

a

b

a

b

a

b

ab

ha

a

b

a

b

a

b

ab

ha

a

b

a

b

a

b

ab

ha

a

b

a

b

a

b

ab

ha

a

b

a

b

a

b

ab

ha

a

b

a

b

a

b

ab

ha

a

b

a

b

a

b

ab

ha

a

b

a

b

a

b

ab

ha

a

b

a

b

a

b

ab

ha

a

b

a

b

a

b

ab

ha

SH

(22)

C.Heat Convection Vector On cylindrical curved surface

From (13), the first term of the heat convection vector

( )( ) dvcdavThdsNThee

s

A

S

+=

)(2

)(2

4

112

( )( ) dvdvcaTh

114

1

1+

=

( ) ( )dvvdcaTh

114

1

1

1

1+

=

caTh = (23)Evaluating the other terms, we obtain

=

0

0

1

1

0

0

1

1

][ caThsC

(24)


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45

D.On annular end surface

The first term of the heat convection on ( ,v) space

( )( )( ) dvcdaab vabTh

dsNTh

ab

A

S e

=1

/

1

1

2

/121/

)(2

( ) ( )

=1

1

1

/

22

1/12

dvvda

b

ab

aTh

ab

( )

+

=

3

32

623

1

/1 a

b

a

b

ab

aTh (25)

Evaluating the other terms, we obtain

( )

+

+

+

+

=

0

0

0

0236

1236

1623

1623

1

/1][

2

2

3

3

2

2

3

3

3

3

3

3

2

a

b

a

ba

b

a

ba

b

a

ba

b

a

b

ab

aThs C

(26)

VIII.LIST OF SYMBOLS

Vr, Vand Vz Thermal conductivities in the radial,Circumferential and axial directions

Pm Material density,

Cm Material specific heat

T Potential function (Temperature)V Medium permeability (Thermal conductivity) watt /m

oC

q Flux (heat flux) watt / mm2.

Q Forcing function (Heat source)

T Surface temperature

TAG Air-gap gas temperature

TBC Back of core gas temperature

n Outward normal vector to the bounding curve TD Core-end gas temperature.

d Differential arc length along the boundary

[SR], [S] and [SZ] Symmetric co-efficient matrices (thermal stiffness matrices)[SH] Heat convection matrix.[T] Column vector of unknown temperatures.

[R] Forcing function (heat source) vector.[ST] Column vector of heat convection.

[SC] Column vector heat convection.

[T0] Column vector of unknown (previous point in time) temperatures.


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46

REFERENCES

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[2] Armor, A.F., and Chari, M. V. K., Heat flow in the stator core of large turbine generators by the method of three-dimensional finite elements, Part-I: Analysis by Scalar potential formulation: Part - II: Temperature distribution inthe stator iron, IEEE Trans,Vol. PAS-95, No. 5, pp.1648-1668, September 1976.

[3] Armor, A. F., Transient Three Dimensional Finite Element Analysis of Heat Flow in Turbine-Generator Rotors,IEEE Transactions on Power Apparatus and Systems, Vol. PAS 99, No.3 May/June. 1980.

[4] Rajagopal,M.S.,Kulkarni,D.B.,Seetharamu,K.N.,and shwathnarayana P.A.,Axi-symmetric steady state thermalanalysis of totally enclosed fan cooled induction motors using FEM, 2 ndNat Conf.on CAD/CAM,19-20 Aug,1994.

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Documents

Computational Analysis of 3-Dimensional Transient Heat Conduction in the Stator of an Induction Motor during Reactor Starting using Finite Element Method