In developing electric motors in general and induction motors in particular temperature limit is a key factor affecting the efficiency of the overall design. Since conventional loading of induction motors is often expensive, the estimation of temperature rise by tools of mathematical modeling becomes increasingly important. Excepting for providing a more accurate representation of the problem, the proposed model can also reduce computing costs. The paper develops a three-dimensional transient thermal model in polar co-ordinates using finite element formulation and arch shaped elements. A temperature-time method is employed to evaluate the distribution of loss in various parts of the 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 the temperature 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 enclosed structure.

1. Computational Analysis of 3-Dimensional Transient
Heat Conduction in the Stator of an Induction Motor
during Reactor Starting using Finite Element Method
A. K. Naskar1
and D. Sarkar2
1
Seacom Engineering College, Howrah, India
Email: naskar73@gmail.com
2
Bengal Engineering & Science University, Howrah, India
Email: debasissrkr@yahoo.co.in
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. Since
conventional loading of induction motors is often expensive, the estimation of temperature
rise by tools of mathematical modeling becomes increasingly important. Excepting for
providing a more accurate representation of the problem, the proposed model can also
reduce computing costs. The paper develops a three-dimensional transient thermal model in
polar co-ordinates using finite element formulation and arch shaped elements. A
temperature-time method is employed to evaluate the distribution of loss in various parts of
the 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 the
temperature 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 enclosed
structure.
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 converter
represents a continuous challenge for the design engineers, any achievements in this area meaning important
energy 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. on Advances in Power Electronics and Instrumentation Engineering, PEIE

2. 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 of
large 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] also
described 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 temperatures
obtained 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 iron
and winding has been chosen for modeling the problem and the geometry is bounded by planes passing
through 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 the
heat 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 gas
from 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 δδ / , the temperature gradient normal to the surface.
Axial centre of package 0=
pn
T
δ
δ (1)
Mid-slot surface 0=
Sn
T
δ
δ (2)
Mid-tooth surface 0=
tn
T
δ
δ
(3)
Air-gap surface ( )
GA
rAG
n
T
VTTh
δ
δ
−=− (4)
Annular end surface of core ( )
D
zD
n
T
VTTh
δ
δ
−=− (5)
Back-of-core surface, ( )
BC
rBC
n
T
VTTh
δ
δ
−=− (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
T
CPQ
Z
T
V
T
r
V
r
T
rV
rr
mmzr
δ
δ
δ
δ
θδ
δ
δ
δ
δ
δ θ
(7)
To find the solution of (7) together with the boundary and initial conditions by Galerkin’s weighted residual
approach, we first express the approximate behaviour of the nodal temperature within each element
according 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 the
integral 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 nodal
values and some weighting functions N1, N2… such that

3. 36
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
)(
=⎥
⎦
⎤
⎢
⎣
⎡
−+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∫∫ dvol
t
T
CPQ
z
T
V
z
T
r
V
r
T
V
r
N
e
mm
e
z
ee
ri
vol
δ
δ
δ
δ
δ
δ
θδ
δ
θδ
δ
δ
δ
δ
δ θ
(9)
Integrating all the terms through integration by parts, the equation takes the form
rdzddr
r
N
r
T
V
dzdrN
r
T
Vdrdzdr
r
T
V
r
N
i
e
r
D
i
e
r
S
e
ri
D
e
ee
θ
δ
δ
δ
δ
θ
δ
δ
θ
δ
δ
δ
δ
)(
)()(
)(
)(
2
)(
∫∫
∫∫∫
−
=⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
rdzddr
r
N
r
T
VdNn
r
T
V i
e
r
D
ir
e
r
S ee
θ
δ
δ
δ
δ
δ
δ
.
)()(
)()(
2
∫∫∑∫ −=
(10)

4. 37
B. Arch-Element Shape Functions
Consider the arch-shaped prism element of Fig. 3, formed by circle arcs radii a, b, radii inclined at an angle
2∝, and prism faces at positions z = -c and z = c.
Fig.3: Three-dimensional arch-shaped prism element suitable for discretisation of induction motor rotors
The shape functions can now be defined in terms of a set of non-dimensional co-ordinates by non-
dimensionalising the cylindrical polar co-ordinates r, θ and z using
2; ;
r z
a c
π
θ
ρ ν τ
α
−
= = = (11)
The arch element with non-dimensional co-ordinates is shown in Fig. 4.
Fig.4: The non-dimensional arch element
The temperature at any point within the element be given in terms of its nodal temperatures, by
HHBBAA NTNTNTT +⋅⋅⋅⋅⋅⋅⋅++= (12)
Where the N’s are shape functions chosen as follows:

5. 38
( )( )( )
( )
( )( )( )
( )ab
ab
N
ab
ab
N
B
A
/14
11/
/14
11/
−
++−
=
−−
+−−
=
τνρ
τνρ ( )( )( )
( )
( )( )( )
( )ab
ab
N
ab
ab
N
F
E
/14
11/
/14
11/
−−
−+−
=
−
−−−
=
τνρ
τνρ
( ) ( )( )
( )
( ) ( )( )
( )ab
N
ab
N
D
C
/14
111
/14
111
−
+−−
=
−−
++−
=
τνρ
τνρ ( )( )( )
( )
( ) ( )( )
( )ab
N
ab
N
H
G
/14
111
/14
111
−−
−−−
=
−
−+−
=
τνρ
τνρ
(13)
It is seen that the shape functions satisfy the following conditions:
(a) That at any given vertex ‘A’ the corresponding shape function NA has a value of unity, and the other
shape 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 on
the faces and within the element.
C. Approximate Numeric Form
According to Galerkin’s weighted residual approach, the weighting functions are chosen to be the same as
the shape functions. Substituting (12) and (13) into (10), gives
{ }[ ] +−
Δ
+−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎥
⎦
⎤
⎢
⎣
⎡
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
∫
∫
V
ii
emm
e
i
e
z
e
i
ee
i
e
r
V
dVNTNTN
t
CP
NQ
z
T
Tz
T
V
dV
T
T
T
r
V
r
T
Tr
T
V
0
)(
)()(
)()(
2
)()(
2][2
2
0
δ
δ
δ
δ
δ
δ
θδ
δ
δ
δ
θδ
δ
δ
δ
δ
δ
δ
δ θ
⎣ ⎦ { }( ) ∑∫ ∞− )()(
)(
2
e
ii
e
S
dNThNTNh
e
for i = A, B,…., H (14)
These equations, when evaluated lead to the matrix equation
[ ] [ ] [ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ]CTHTZR SRTSTSSSSS ++=++++ 0θ
(15)
III. DISCRETIZED MODEL FOR FEM APPLICATION
The stator of an induction motor, under transient conditions is designed to maintain all temperatures below
class A insulation limits of 105o
C 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 to
provide 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 period
and 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.

6. 39
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 determined
from 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/mm3
.
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 voltage
till 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

7. 40
x1=8.15Ω ; Ic=0.176 Amp; r1=2.04Ω; Im=2.41Amp
0425.0;415;39.2 1
/
2 ==Ω= 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 input
current of the reactor is=38.75/2 =19.38A and 50% of total impedance of the reactor will be
=415/(√3×19.38×2)= 6.18 Ω
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= (‫ݎ‬ଵ+‫ݎ‬ଶ
ᇱ
/s) = (2.04+2.39/1) =4.43Ω
Reactance in the circuit (x1) =8.15Ω
Impedance of the circuit (z1) = √(4.43)2
+ (8.15)2
=9.3Ω
As the stator is delta connected and 50% of full voltage is applied across the stator winding, the stator current
at s=1 will be I1=415/ (9.3+6.18)=415/15.48 =26.85A
To 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=0.2
Resistance of the circuit= r1+r’
2/s = (2.04+2.39/0.2)= 13.99Ω
When full voltage is provided across the stator winding by short-circuiting the reactors, the stator current at s
= 0.2 will be I1=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.
TABLE – I. 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 curved
surface 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=5.25
w/m2 o
C.

8. 41
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 / m2 o
C.
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 PROPERTIES FOR 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 temperature
variation with time in each node of copper is taken as an index to understand the temperature profile during
the 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 temperatures
persisting 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. 7 to investigate the magnitude of the temperature variation with time at different nodal points along
the stator copper winding.
TABLE III. SOLUTION FOR THREE DIMENSIONAL STRUCTURE
Magnetic Steel Wedge Copper &Insulation
Vr 33.070 2.007
Vθ 0.8260 1.062
VZ 2.874 358.267
Pm 7.86120 8.9684
Cm 523.589 385.361
NodeNos.
InitialTemperature
Temperatures exceeding 44 0
C in 1st
time step with convection in three dimensional structure of totally enclosed machine
for different stator current during Reactor starting.
Q=0.003257
I=26.85Amp
Startingtime=4.29secs
Q=0.0031967
I=26.6Amp
Startingtime=3.88secs
Q=0.003125
I=26.3Amp
Startingtime=3.48secs
Q=0.00303065
I=25.9Amp
Startingtime=3.09secs
Q=0.0029148
I=25.4Amp
Startingtime=2.702secs
Q=0.0027563
I=24.7Amp
Startingtime=2.33secs
Q=0.0025078
I=23.56Amp
Startingtime=1.99secs
Q=0.00213725
I=21.75Amp
Startingtime=1.7secs
Q=0.00296779
I=25.63Amp
Startingtime=1.1975
secs
Q=0.00105249
I=15.26Amp
Startingtime=1.222secs
31 400
C 44.563
0
C
47.695
0
C
49.906
0
C
51.488
0
C
52.610
0
C
53.368
0
C
53.799
0
C
53.925
0
C
54.427
0
C
53.982
0
C
67 400
C 44.567
0
C
47.707
0
C
49.933
0
C
51.531
0
C
52.670
0
C
53.443
0
C
53.887
0
C
54.023
0
C
54.533
0
C
54.095
0
C
103 400
C 44.587
0
C
47.781
0
C
50.073
0
C
51.737
0
C
52.935
0
C
53.758
0
C
54.244
0
C
54.415
0
C
54.946
0
C
54.531
0
C
137 400
C 44.061
0
C
47.099
0
C
49.466
0
C
51.320
0
C
52.753
0
C
53.819
0
C
54.544
0
C
54.950
0
C
55.584
0
C
55.403
0
C
138 400
C 44.014
0
C
47.112
0
C
49.550
0
C
51.458
0
C
52.924
0
C
54.007
0
C
54.739
0
C
55.148
0
C
55.761
0
C
55.597
0
C
139 400
C 44.741
0
C
48.092
0
C
50.532
0
C
52.322
0
C
53.622
0
C
54.524
0
C
55.067
0
C
55.276
0
C
55.842
0
C
55.437
0
C
175 400
C 44.408
0
C
47.668
0
C
50.105
0
C
51.925
0
C
53.265
0
C
54.210
0
C
54.802
0
C
55.068
0
C
55.621
0
C
55.312
0
C

9. 42
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 is
carried 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 i
e
S e
)(
)(
2
∫
( )dsNNTNNTNTh HAHBABAA
S e
+⋅⋅⋅⋅⋅⋅⋅⋅++= ∫
2
)(
2
(16)
Now performing the integration in non-dimensional notation
dsNh A
2
∫
( ) ( ) )(
16
11
22
1
1
1
1
dvcda
v
h ταρ
τ −+
= ∫∫ −−

10. 43
( ) ( ) dvvd
cha
∫ ∫− −
−+=
1
1
1
1
22
11
16
ττ
αρ
αρcha
9
4
= (17)
dsNNh BA∫
( ) ( ) ( ) ( ) )(
4
11
4
111
1
1
1
dvcda
vv
h ταρ
ττ ++
×
−
−+
= ∫∫ −−
( ) ( ) ττ
αρ
ddvv
cha
∫ ∫− −
+−
−
=
1
1
1
1
22
11
16
αρcha
9
2
= (18)
Evaluating the other terms, we obtain
[ ]
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
0
00
00
00
00000
000000
0000
0000
9
4
9
2
9
4
9
2
9
1
9
4
9
1
9
2
9
2
9
4
αρcahSH (19)
B. On annular end surface
By performing the integration over ( ρ ,v) space,
dsNh A
S e
2
)(
2
∫
( ) ( )
( )
)(
/14
1/
2
22
1
1
1
1
dvcda
ab
vab
h ταρ
ρ
−
−−
= ∫∫ −−
( ) ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+−−
−
= 2
2
4
4
2
2
23
2
124
1
/13
2
a
b
a
b
a
b
ab
ha α
(20)
dsNNh BA∫
( ) ( )
( )
)(
/14
1/
2
22
1
/
1
1
dvcda
ab
vab
h
ab
ταρ
ρ
−−
−−
= ∫∫−
( )
⎟
⎠
⎞
⎜
⎝
⎛
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+−−
−−
=
3
4
23
2
124
1
/14
2
2
4
4
2
2
a
b
a
b
a
b
ab
ha α
( ) ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+−−
−
= 2
2
4
4
2
2
23
2
124
1
/13 a
b
a
b
a
b
ab
ha α
(21)

11. 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
)
66
1212
1
(
/13
2
)
66
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
S H
α
αα
ααα
αααα
(22)
C. Heat Convection Vector On cylindrical curved surface
From (13), the first term of the heat convection vector
( )( ) dvcda
v
ThdsNTh
ee
s
A
S
τρα
τ
∫∫ −
−+
= ∞∞
)(
2
)(
2
4
11
2
( )( ) dvdv
caTh
ττ
αρ
11
4
1
1
−+
−
= ∫−
∞
( ) ( )dvvd
caTh
11
4
1
1
1
1
−+
−
= ∫ ∫− −
∞
ττ
αρ
αρcaTh ∞= (23)
Evaluating the other terms, we obtain
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
= ∞
0
0
1
1
0
0
1
1
][ αρcaThsC
(24)

12. 45
D. On annular end surface
The first term of the heat convection on ( ρ ,v) space
( )( )
( )
dvcda
ab
vab
Th
dsNTh
ab
A
S e
ρρα
ρ
∫ ∫
∫
−
∞
∞
−−
−−
=
1
/
1
1
2
/12
1/
)(
2
( )
( )∫∫ −
∞
−⎟
⎠
⎞
⎜
⎝
⎛
−
−−
=
1
1
1
/
2
2
1
/12
dvvd
a
b
ab
aTh
ab
ρρρ
α
( ) ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+−
−
= ∞
3
32
623
1
/1 a
b
a
b
ab
aTh α
(25)
Evaluating the other terms, we obtain
( )
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−+
−+
+−
+−
−
= ∞
0
0
0
0
236
1
236
1
623
1
623
1
/1
][
2
2
3
3
2
2
3
3
3
3
3
3
2
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
ab
aTh
s C
α
(26)
VIII. LIST OF SYMBOLS
Vr, Vθ and 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 o
C
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 Σ
TD Core-end gas temperature.
dΣ Differential arc length along the boundary
[SR], [Sθ ] 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.

13. 46
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