Development and Application of a Two-Dimensional,
Transient Numerical Model for Temperature Distribution
within a Cylinder Subjected to Electromagnetically Induced
Heat Generation
Conduction Heat Transfer
MEAE 6630
Submitted by
Michael J. Guidos
April 17, 2000
1
Table of Contents
Section
Title
Page
1.0
2.0
3.0
4.0
Abstract
Introduction
Nomenclature
Formulation
Self- and Mutual Inductance Model
Heat Conduction Model
Application of Models: Discussion of Results
Conclusion
References
3
3
4
5
5
9
13
15
16
(i)
(ii)
5.0
6.0
7.0
List of Appendices
Section
Title
Page
A
B
Figures of Reference
Results from “Currentsolver” Model: Determination of
Volumetric Heat Generation
Results from “Tempdist” Model: Determination of
Temperature Distribution within Cylinder
Inputs to “Currentsolver” model and their Descriptions
“Currentsolver” Visual Basic Code
“Tempdist” Visual Basic Code
17
C
D
E
F
2
21
30
38
46
56
1.0
Abstract
A numerical model is presented for calculating the two-dimensional, transient
temperature distribution within a metallic cylinder subjected to non-uniform volumetric
heat generation caused by electromagnetically induced eddy currents. Based on the heat
equations and boundary conditions used, the model is valid for low temperatures only,
when radiation heat exchange is insignificant. Based on the model results, for all times,
the maximum temperature was consistently at the outermost cylinder radius at each axial
location. However, due to the imposed boundary conditions and neglect of heat loss from
the cylinder, temperature growth was uniform at all locations within the cylinder for all
times except for the first few seconds of eddy current application.
2.0
Introduction
Induction heating is the application of electrical current to a metallic material in order to
generate volumetric heating within the material. Induction heating also occurs when a
metallic material is subjected to the magnetic field produced by another conductor. Eddy
currents arise within the non-load carrying material, opposite in direction to the current in
the load-carrying conductor, and therefore, simultaneously cause volumetric heat
generation within the non-load carrying conductor.
When alternating current (or similarly eddy currents) is passed through a mass of
material, the current flow and manner of heating of the material are highly dependent on
the electrical conductivity of the metal and the frequency of the heating. If the
conductivity is high and the frequency is high, the current tends to concentrate at the
outside of the mass. If the conductivity and the frequency are low, the current
distribution tends to be more distributed, heating the mass uniformly. The depth of this
maximum concentration of electrical current is known as the skin depth or skin thickness.
Therefore, for a high conductivity material such as a metal, the current density and
similarly the volumetric heat generation are typically non-uniform.
In practice, typical values for frequency of heating roughly range from 60 Hz to 1 Mhz as
given by (4). Practical applications of induction heating include the casehardening of
wear-intensive parts, such as gears, shafts, or rotors, and the purification of
semiconductor metals, known as zone refining, to melt and remove impurities.
The effect of induced eddy currents and similarly their concentration will be analyzed to
determine quantitatively the volumetric heat generation that is produced when a metallic
cylinder is subjected to the magnetic field created from a surrounding alternating current
coil. The basic setup for this type of problem is shown in Figure 1 of Appendix A, in
which a radio frequency (low frequency) generator is applied to the primary coil, which
in turn induces currents in the load material, which acts as short-circuited secondary.
Once the volumetric heat generation is obtained, a heat conduction model will then be
used to determine the two-dimensional, transient temperature distribution throughout the
cylinder through the use of symmetry.
3
3.0
Nomenclature
Symbol
Description
Units
0
r

free space magnetic permeability
relative magnetic permeability
electrical conductivity
frequency
angular frequency
skin depth
self-inductance
mutual inductance
current
voltage
resistance
complex number
radial dimension of discretized circuit (Fig. 3 or 4a)
radial dimension of discretized circuit (Fig. 3)
cross-sectional area of discretized circuit
volume of discretized circuit
length of discretized circuit
one-half discretized circuit thickness (Fig. 4b)
height of discretized circuit
cylinder radius
cylinder height
radial dimension
axial dimension
temperature
thermal conductivity
thermal diffusivity
volumetric heat generation
index for radial position
index for axial position
index for time
H/m
(-m)-1
Hz
rad/s
m
H
H
A
V

m
m
m2
m3
m
m
m
m
m
m
m
C
W/mC
m2/s
W/m3
-
f

s
L
M
I
V
R
j
a
b
A
V
l
a
h
b
c
r
z
T
k

g
i
j
n
4
4.0
Formulation
The formulation for solution of the temperature distribution within the cylinder may be
divided into two sections,
(i) The development of the self- and mutual-inductance model used to calculate the
induced eddy currents in the cylinder and thereby arrive at the heat generation at each
node, and
(ii) The development of the heat conduction governing equations and boundary
conditions using an explicit finite control volume method and combining with the output
volumetric heat generation from the inductance model at each node to obtain the
temperature distribution within the cylinder.
(i) Self- and Mutual-Inductance Model
To determine the volumetric heat generation within the cylinder, the cylinder and coil
must be considered as a coupling of circuits. The model used in this work was based on a
model developed and applied to a cylindrical cross-section in (2). In (2), the crosssections of both the cylinder and the coil are discretized into several smaller-area circuits
to arrive at the current density at the center of each of these circuits.
From electromagnetic theory, mutual-inductance is defined as the induced voltage in one
circuit due to current flowing in another circuit. Alternatively, self-inductance is the
induced voltage within a circuit due to changes in its own current. Both inductance
effects are reliant upon the geometry of the coupled circuits. Therefore, for the model in
consideration, for each of the discretized cylinder and coil circuits, the inductance of each
circuit must be properly addressed to accurately determine the heat generation within the
cylinder.
The basic problem setup, given from (4), is represented in Figure 2 of Appendix A, which
displays an equivalent circuit representing a coil of inductance L1 and resistance R1,
placed across the terminals of an emf generator developing a voltage V1. The metallic
cylinder is represented as the shorted secondary turn with resistance R2 and inductance
L2. The circuit relations for this simplified model are given by
V1  R1  jL1 I 1  jM 12 I 2
0   jM 21 I 1  R 2  jL2 I 2
( 4 1)
(42)
where  = 2f and the coefficients R, L, and M are determined from the geometry of the
respective components. From this simplified model, the more complex model in (2) is
based and is now approached for solution of the volumetric heat generation for the
present problem.
5
Following from (2), the radial cross-sections of both the cylinder and coil are first
discretized into arrays of smaller cross-sectional areas. For the arrays of discretized
circuits of the cylinder and coil, equations similar to (4-1) and (4-2) are developed, where
there is one equation for every circuit, generally set up as
 N 1

VC  RC  jLC I C  j 
M Cn I n 


 n 1,n  C



 N 1

0   j 
M Pn I n   R P  jL P I P


 n 1,n  P


( 43)
(44)
where
C = a discretized circuit of the coil
N = total number of circuits
P = a discretized circuit of the cylinder
For proper use of these equations, the following basic guidelines must be followed:
1) The voltage across every discretized circuit of the coil is non-zero and is given by the
setup of the problem, and conversely, the voltage across every cylinder circuit is zero
2) A self-inductance term, L, exists for each circuit, and a mutual-inductance term, M,
exists between every pair of circuits.
3) If the equation has voltage that is non-zero, each term is preceded by a “plus” sign (for
coil circuits), and conversely, each term in an equation where voltage is zero is preceded
by a “minus” sign (for cylinder circuits).
Since eddy current concentration is a significant factor in the heat generation and thermal
distribution for this type of problem, where the skin depth is defined from (3) as
s
2
 r 
(45)
where  0  4 x 10 7 H/m, from (9)
the selected cylinder size and the mesh used to analyze it are based on this value.
Therefore, since the outer radius of the cylinder experiences the densest concentration of
current (and similarly volumetric heat generation), a varying mesh is used to allow more
concentration of nodes at the cylinder outer radius.
The mesh used for the solution of this problem will be presented in Section 5.0 along
with the results and discussion of all inputs in Appendix D. However, the mesh is
mentioned now because the discretization of circuit sizes is necessary by procedure
before proceeding with the determination of all of the coefficients for (4-3) and (4-4).
6
Figures 3 and 4 of Appendix A depict the critical geometric dimensions necessary for the
calculation of the mutual- and self-inductance terms. In Figure 3, a pair of discretized
circuits from the cylinder-coil system may be viewed as two independent, parallel,
coaxial conducting loops. Based on this simple configuration, the relative physical
dimensions between the loops as noted in Figure 3 and the physical dimensions for a
single loop as given in Figure 4a are then used to calculate the mutual inductance terms
for every given pair of circuits in (4-3) and (4-4). As given from (5), the mutual
inductance terms are then
 2


M   ab 
 k 2  K k 2   E k 2 

 k 2

(46)
where
k 22 
4ab
d  a  b 2
2
Similarly, for calculation of the self-inductance terms from (4-3) and (4-4), a single
discretized circuit may be viewed as a conducting loop as shown in Figure 4b, where the
geometry of interest is noted. Therefore, for every single discretized cylinder and coil
circuit and its known geometry, the following relations as given from (5) may then be
used to calculate the total self-inductance term for a specified circuit as,
for the external inductance,


k2 
L0   2r  a 1  1  K k1   E k1 


2 


where
k12 
(47)
4r r  a 
2r  a 2
for the internal inductance,
Li 

l
8
(4  8)
where l is the length of the discretized circuit based on the radius to the centerline of the
discretized circuit, r, calculated as
l  2r
(49)
and therefore, the internal inductance becomes
Li 
r
( 4  10 )
4
The total self-inductance for a given circuit as given from (4-7) and (4-10) is then
 k 2
L  L0  Li   2r  a 1  1
2



 K k1   E k1   r

 4

7
( 4  11 )
For both (4-6) and (4-11), E(k) and K(k) are respectively the complete elliptic integrals of
the first and second kind, given by
E( k ) 
/2

1  k 2 sin 2 d
( 4  12 )
0
K( k ) 
d
/2

0
1  k 2 sin 2
( 4  13 )
For the resistance terms of the discretized cylinder and coil circuit relations given in (4-3)
and (4-4), the following relation is used, as given for a homogeneous metallic conductor
from (3),
l
R
( 4  14 )
A
where l is the length of the conductor given from (4-9), and A is the cross-section of the
discretized circuit. The cross-sectional area of a discretized circuit, as referring to the
circuit radial thickness geometry from Figure 4b, is determined as
A = (2a) h
(4-15)
where h is the height of the circuit as determined from the selected geometry of
discretization.
Once all L, M, and R coefficients are known for the set of equations generalized by (4-3)
and (4-4), the set of equations can be solved simultaneously for the current in each
discretized circuit of the coil and the cylinder.
After the value of the current at the center of each discretized circuit in the cylinder is
known, denoted as node (i,j), the volumetric heat generation term, g(i,j) is now calculated
as,
g( i , j ) 
I 2R
(4-16)

where  is the volume of the discretized circuit containing node (i,j), calculated using the
definitions from (4-9) and (4-15),
  lA
(4-17)
Application of the inductance model formulation using the Visual Basic program
“Currentsolver”, given in Appendix C and its results are discussed in Section 5.0.
8
(ii)
Heat Conduction Model
The heat conduction model is based on a two-dimensional cylinder, with transient
temperature variation in the r- and z-directions only. The cylinder has radius, r = b, and
height, z = c.
For simplification, the cylinder is assumed to be insulated at all surfaces, and radiation
heat transfer is neglected at all times. Based on these assumptions, the model is only
valid while the temperature remains relatively low. Therefore, the governing exact heat
equation and boundary conditions are,
1   T   2T 1
1 T
 g r , z  
for 0  r  b, 0  c  z , t  0
r

r r  r  z 2 k
 t
dT
 0 at r  b , 0  z  c , t  0
dr
(4-19)
dT
 0 at z  0, 0  r  b , t  0
dz
(4-20)
dT
 0 at z  c , 0  r  b , t  0
dz
(4-21)
(4-18)
and from the symmetry of a cylinder,
dT
 0 at r  0, 0  z  c , t  0
dr
( 4  22 )
The initial condition of the cylinder is assumed as
T  Ti throughout cylinder, at t  0
(4-23)
Due to the non-uniform volumetric heating condition, the thermal model must be solved
numerically. A control volume scheme, explicit in time will be used. Details of the exact
mesh employed as well as all input parameters will be discussed in Section 5.0.
Prior to discretizing (4-18), due to the singularity at r = 0 when applying numerical
techniques for solution, L’Hospital’s rule is first applied to the r derivative term, as given
from (7), results as
2
  T   2 T 1
1 T
at r  0, 0  z  c , t  0

  2  g( r , z ) 
r  r  z
k
 t
( 4  24 )
The numerical approximations for the exact heat equations can now be performed. In
order to coordinate between the obtained volumetric heat generation data and the
9
temperature distribution, the nodes (i,j) at the centers of the discretized circuits from the
inductance model must also be the nodes (i,j) at which the temperature is calculated.
From this point on, all references to the nodes shall be designated as (i,j) instead of (r,z).
Using the explicit finite volume method, the r-dependent term of (4.18) is represented
similarly for either a constant mesh between nodes i – 1, i, and i + 1 or for a varying
mesh size between the nodes i – 1, i, and i + 1. About node i with respect to the nth step
in time, the r-dependent term is discretized as,
1   T 
r

r r  r 
n
n
  T  n
 T 
r
r


1   r  ri 1 / 2  r  ri 1 / 2
 
1
ri 
ri 1  ri 1 

2








(4-25)
where
n
Ti n1, j  Ti n, j
 T 
r

r


i 1 / 2
ri 1
 r  ri 1 / 2
and
n
 T 
 ri 1 / 2
r

 r  ri 1 / 2
where
1
ri 1
2
1
 ri  ri 1
2
ri 1 / 2  ri 
Ti n, j  Ti n1, j
ri 1
ri 1 / 2
ri 1  the interval to the forward side of the node i,j in the i direction , between nodes i and i  1
ri 1  the interval to the backward side of the node i,j in the i direction , between nodes i  1 and i
As can be seen from (4-25) and the definitions of its terms, if the mesh size is constant
about node i,j in the i direction then the familiar central difference numerical
approximation of the second derivative r term from (4-18) will result, in terms of
constant r only.
Similarly for the z-dependent term in both (4-18) and (4-24), a representation is necessary
for both a constant z and a changing z (for varying size mesh) about a specified node.
The finite difference representation for the z-dependent term for either a constant or
changing interval size about node j with respect to the nth time step is given from (8) as
 2T
z 2
where

  Ti n, j 1  Ti n, j Ti n, j  Ti n, j 1 
2




 z j 1  z j 1   z j 1
z j 1 



( 4  26 )
z j 1  the interval to the forward side of the node i,j in the j direction , between nodes j and j  1
r j 1  the interval to the backward side of the node i,j in the j direction , between nodes j  1 and j
10
Similar to the numerical approximation for the r derivative term, if mesh size is constant
about the node along the z-axis, the familiar central difference formula in terms of only
constant z will result from (4-26).
For r = 0 (i = 1), the r-dependent term from (4-24) must be discretized accordingly with
respect to the nth time step using the symmetry boundary condition at r = 0, (4-22), as
given from (6) as
2
n
Ti n1, j  Ti n, j
T2n, j  T1n, j
  T 
4
4


r  r  r 0 ( i 1 )
ri 1
ri 1
4  27 
For the t-dependent term in both (4-18) and (4-24), an explicit, backward difference
formula is used, yielding
n 1
n
T Ti , j  Ti , j

t
t
4  28 
For selection of a proper time step t, as defined in (6) for a two dimensional explicit
finite difference model, the following stability criterion must be adhered to,
1
t 
2
 1
1 



2
z 2 
 r 
1
4  29 
where r and z are the minimum interval sizes for the given applied mesh.
Assembling the numerical approximations for the r- and z-dependent terms, the
governing discretized heat conduction equations used for the model are
1) for i = 1, 1 < j < N + 2, n > 1 (where N = total number of control volumes in j
direction), combining (4-27), (4-26), (4-28), and the discretized volumetric heat
generation term as calculated from the inductance model,
4
T2n, j  T1n, j
ri 1
n 1
n

  Ti n, j 1  Ti n, j Ti n, j  Ti n, j 1  1
2
1 Ti , j  Ti , j

  g( i , j ) 


 z j 1  z j 1   z j 1
z j 1  k

t



( 4  30 )
2) for 2 < i < M + 1, 1 < j < N + 2, n > 1 (where M = total number of control volumes in i
direction)
11
n

 T n  Ti n, j  
 n

   r  1 r  Ti , j  Ti 1, j  
  ri  1 ri 1  i 1, j
i
i

1
 

2
2
 ri 1
 ri 1
1 




1
ri 

ri 1  ri 1 


2




n
n
n
n
n 1
n

  Ti , j 1  Ti , j Ti , j  Ti , j 1  1
2
1 Ti , j  Ti , j


  g( i , j ) 

 z j 1  z j 1   z j 1
z j 1  k

t



( 4  31 )
where for the model setup, the nodes and time steps are based on: r = 0 when i = 1 and r
= b when i = M + 2; z = 0 when j = 1 and z = c when j = N + 2, and t = 0 when n = 1.
Discretization of the boundary conditions using backward and forward difference
equations as appropriate are now performed for (4-19), (4-20), and (4-21) respectively as
1) for i = M + 2, 1 < j < N + 2, n > 1,
TMn  2 , j  TMn 1, j
ri 1
 0  TMn  2 , j  TMn 1, j
( 4  32 )
2) for 1 < i < M + 2 , j = 1, n > 1,
Ti n,2  Ti n,1
z j 1
 0  Ti n,1  Ti n,2
( 4  33 )
3) for 1 < i < M + 2 , j = N + 2, n > 1,
Ti n,N  2  Ti n,N 1
z j 1
 0  Ti n,N  2  Ti n,N 1
12
( 4  34 )
5.0
Application of Models: Discussion of Results
For application of the models, a setup very similar to that used in (2) was utilized with the
exception of using a larger radius cylinder. Based on this setup, constant properties with
temperature are assumed for both the inductance model and the temperature model.
All inputs for the inductance model are given in Appendix D, Tables 1 and 2. The radial
cross-section of the circuit mesh used is given in Figure 5 of Appendix A, which shows
detail of the size and relative positions of each coil and cylinder circuit. Referring to this
mesh, the current is solved at the center of each discretized cylinder circuit where the
temperature mesh nodes are located. This method allowed for direct input of the
volumetric heat generation output from the electrical model directly into the temperature
model. Dimensions between all temperature nodes are also denoted in Figure 5.
The system of circuits is composed of a two-winding copper coil composed of eight
equal-sized discretized circuits with 1 V across each and an aluminum cylinder composed
of 250 discretized circuits of varying size with 0 V across each, as given by requirements
of Section 4.0. The cylinder circuits are sized smaller towards the outer radius of the
cylinder, r = b, based on a skin depth calculated from (4-5) to be .0197 m for an input
frequency of 50 Hz. The finer mesh toward r = b was employed to allow for more
accurate determination of the larger gradient of the current/volumetric heating that should
exist there due to higher current concentration as compared to the center of the cylinder
as based on the skin depth phenomena.
A Visual Basic routine was developed, based from a program provided in (10), for
calculation of all coefficients given from (4-6), (4-11), and (4-14) to construct the 258
simultaneous electrical current equations derived from (4-3) and (4-4). The Visual Basic
routine “Currentsolver” is provided in Appendix E and follows the method of the
formulation using numerical techniques for the simultaneous calculation of the currents
and similarly the volumetric heat generation from (4-16) at the center of each discretized
cylinder circuit. The output of volumetric heat generation term, current, resistance, and
its (r, z) location within the cylinder are presented in Table 1 of Appendix B for a
frequency of 50 Hz.
Based on the employed mesh, Figure 1 of Appendix B shows the comparative volumetric
heating for frequencies of 50, 150, 300, and 1500 Hz across the radius of the cylinder at
z = .006875 m from the bottom surface of the cylinder. This trend in heat generation over
the cylinder radius was similar at every axial level of the cylinder. From Figure 1, the
skin depth is apparent in that the concentration of heat generation increases outer radius
as the frequency increases from 50 Hz to 1500 Hz as compared to the heat generation
towards the cylinder center. The heat generation increases at each radial increment from
50 Hz to 300 Hz. However, at 1500 Hz, Figure 1 shows a sharp decrease in heating,
which strongly indicates that the mesh used was not fine enough at the cylinder outer
surface to capture the concentration between r = .0295 m and the outer cylinder surface, r
= .030 m.
13
Figure 2 of Appendix B shows the comparative volumetric heating for the same
frequencies across the axial length of the cylinder at r = .0295 m. The trend in heat
generation term over the cylinder axial length was similar at every radial mesh increment
of the cylinder, where the volumetric heat generation is consistently highest at the
outermost radius of the cylinder. Figure 2 shows that for frequencies of 50 to 300 Hz,
maximum heat generation occurs at the mid-axial length of the cylinder. At 1500 Hz, the
minimum occurs at the mid-axial length. However, as shown by Figure 3 of Appendix B,
at smaller radial fixed position of r = .0215 m, g at 1500 Hz is a maximum at the midaxial length. 2-D plots would have been better suited for this problem but were
unavailable with the utilized software.
The output volumetric heat generation at each circuit center was then applied to the
temperature model as developed from the formulation in Section 4.0. To meet the
requirements of (4-30), the volumetric heat generation at the center of the cylinder, g (1,
j), 0 < j < 12 was approximated by setting equal to g (2, j), 0 < j < 12. Since the g values
were very low as approaching r = 0 for each level of z within the cylinder and it was
impossible to directly calculate g at this point using the inductance model, this
approximation is justified and should introduce very little error into the solution.
Temperature distribution was determined only for a frequency of 50 Hz because the
higher frequencies caused high temperature growth in very short times. This effect was
due to the boundary conditions used, which impose restrictions of being a low
temperature model (since there is no radiation heat transfer consideration in the heat
equations).
The Visual Basic routine “Tempdist” was developed to use the discretized heat equations
(4-30) and (4-31) and boundary conditions (4-32) through (4-34). The routine is
provided in Appendix F. Inputs to the code are from the “Currentsolver” outputs in
Appendix C and the positions of the each temperature node, which are included in the
“Tempdist” code as taken from the mesh layout shown in Figure 5 of Appendix A. Also,
the thermal properties of pure aluminum are used as given from (7), as k = 204 W/mC
and  = 8.418 x 10-5 m2/s.
As given by (4-29), for stability of the explicit temperature model, where the minimum
node increments in each direction were z = .000625 and r = .0005 m as given from
Figure 5 of Appendix A, the time steps were selected as t = .0009 s. Output data is
given in Tables 1 to 4. This data was used to respectively construct the plots in Figures 1
to 4 of Appendix C.
Figure 1 of Appendix C shows that for various radial locations at z = .00685 m, the
temperature growth is linear except for the first few seconds. This effect is most likely
due to the non-uniform volumetric heat generation, which at first has a very noticeable
effect on the temperatures of the nodes where it is applied. However, with increasing
time, the prescribed boundary conditions do not allow for any heat losses of the cylinder,
thereby causing a uniform growth in heat generation. Although g is much greater at the
outer portion of the cylinder radius than it is as r = 0 is approached, the heat has no where
14
else to go but toward r = 0, hence having the effect of causing a uniform growth in
temperature across the radius of the cylinder. This trend in temperature growth was
similar for all radial dimensions for all fixed axial locations.
Figure 2 of Appendix C shows temperature profile snapshots across the cylinder radius at
various times at a constant axial location z = .00685 m. Due to the uniform growth in
temperature caused by the boundary conditions, the differences in temperature across the
radius are small (< 4 C as given by Table 2 of Appendix C), with the maximum
temperature consistently occurring at maximum r for all times. This trend was expected
although had convection or radiation been accounted for in the boundary conditions, the
temperature difference across the radius would be much greater due to heat leaving the
cylinder rather than going to the center of the cylinder. The trend in temperature
distribution from Figure 2 was similar for all radial dimensions for all fixed axial
locations
Figure 3 of Appendix C shows the temperature profile across the radius for various axial
locations at a frozen frame in time, at t = 27 s. As shown, temperature distribution over
the radius at each axial location is practically the same due to the uniform growth in
temperature at all cylinder locations as caused by the boundary conditions. At a much
less time, specifically within the first couple of seconds, small differences in temperature
should be more noticeable due to the initial application of the non-uniform volumetric
heat generation as shown in Figure 1.
6.0
Conclusion
The results from this coupled model seem correct and grounded in real world phenomena.
It was at first expected that even with insulated boundary conditions, due to the nonuniform volumetric heat generation, which is much greater at the outermost radius of the
cylinder, large differences in temperature across the cylinder radius would be calculated
with increasing time. However, as shown by the results, for all times except for the first
few seconds of eddy current application, the temperature growth was uniform throughout
the cylinder.
Since there was no mechanism for heat loss, the heat could only redistribute itself inward
towards the center of the cylinder where the volumetric heating was much lower than that
at the outer radius. This effect then gave the cylinder the appearance of uniform heating
over time for all radial and axial locations.
An improvement for this thermal model would be to incorporate radiation heat losses
from the cylinder for all temperatures and times or, for a low temperature model only,
incorporate only convection heat losses with a surrounding medium. In either of these
cases, the cylinder would exhibit less uniform temperature growth throughout the
cylinder with increasing time and experience much larger differences in temperature
distribution across the cylinder radius. In both of these cases, more heat concentrated at
the outer cylinder radius would be lost to the surroundings and therefore provide less heat
15
to the cylinder interior. Much lower temperatures towards the center of the cylinder
would then result.
The other limitation of the model was the very small time steps required for stability due
to the 2-D mesh used. A coarser-mesh was not attempted so that accuracy was not lost in
the representation of the volumetric heat generation field. Greater times could be
observed more readily if an implicit temperature model was used but this would add more
complexity to solving the equations.
References
1.
J.P. Shields, ABC’s of Radio Frequency Heating, Howard W. Sams & Co., 1969.
2.
E. Kolbe and W. Reiss, “Eine Methode zur numerischen Bestimmung der
Stromdicteverteilung in induktiv erwarmten Korpern unterschiedlicher
gemetrischer Form”, Wissenschaftliche Zeitschrift der Hochschule fur
Elektrotechnik Ilmonau, Jg. 9, Heft 3, 1963.
3.
D.H. Tamboulian, Electric and Magnetic Fields, Harcourt, Brace & World, Inc.,
1965.
4.
G.H. Brown, C.N. Hoyler, and R.A. Bierwirth, Theory and Application of Radio
Frequency Heating, D.Van Nostrand Co., Inc., 1947.
5.
S. Ramo, J.R. Whinnery, and T. Van Duzer, Fields and Waves in Communication
Electronics, John Wiley & Sons, Inc., 1984.
6.
J.D. Jackson, Classical Electrodynamics, John Wiley & Sons, Inc., 1962.
7.
M.N. Ozisik, Heat Conduction, John Wiley & Sons, Inc., 1993.
8.
C. Hirsch, Numerical Computation of Internal and External Flows, Volume 1,
Fundamentals of Numerical Discretization, John Wiley & Sons, 1988.
9.
D. Halliday and R. Resnick, Fundamentals of Physics, John Wiley & Sons, Inc.,
1988.
10.
E. Gutierrez-Miravete, FORTRAN code “mim.f”, RPI-Hartford, 2000.
16
APPENDIX A:
FIGURES OF REFERENCE
17
FIGURE 1: BASIC SETUP OF INDUCTION HEATING PROBLEM, FROM (1)
FIGURE 2: SIMPLE MODEL OF ALTERNATING CURRENT COIL CIRCUIT
COUPLED TO A CONDUCTING MATERIAL CIRCUIT, FROM (4)
18
FIGURE 3: GEOMETRY OF A PAIR OF COAXIAL, PARALLEL CIRCUITS
FOR MUTUAL-INDUCTANCE CALCULATION, FROM (5)
(a)
(b)
FIGURE 4: (a) GEOMETRY OF SINGULAR CIRCUIT LOOP FOR MUTUAL
INDUCTANCE CALCULATION, (b) GEOMETRY OF CIRCUIT LOOP FOR
SELF-INDUCTANCE CALCULATION
19
r
z
FIGURE 5: DISCRETIZED MESH OF RADIAL CROSS-SECTIONS OF CYLINDER AND AC COIL FOR
CALCULATION OF VOLUMETRIC HEAT GENERATION –SECTIONS AND TEMPERATURE AT THE NODES AT
THE CENTERS OF EACH DISCRETIZED CIRCUIT CROSS SECTIONAL AREA
**(NOTE:ALL DIMENSIONS SHOWN IN MILLIMETERS)
20
APPENDIX B:
RESULTS FROM “CURRENTSOLVER” MODEL:
DETERMINATION OF VOLUMETRIC HEAT GENERATION
21
Table 1: Heat generation vs. radial dimension, r, within cylinder for various frequencies at a
fixed axial location, z=.006875 m
6.00E+07
5.00E+07
g (W/cu. m)
4.00E+07
f=50 Hz
f=150 Hz
3.00E+07
f=300 Hz
f=1500 Hz
2.00E+07
1.00E+07
0.00E+00
0.000
0.005
0.010
0.015
0.020
r (m)
22
0.025
0.030
0.035
Figure 2: Volumetric heat generation, g, vs. axial dimension, z, for constant r = .0295 m, for
various frequencies
60000000
50000000
g (W/cu. m)
40000000
f=50 Hz
f=150 Hz
f=300 Hz
f=1500 Hz
30000000
20000000
10000000
0
0
0.002
0.004
0.006
0.008
z (m)
23
0.01
0.012
0.014
Figure 3: Volumetric heat generation, g, vs. axial location, z, for constant radial position,
r = .0215 m, for various frequencies
18000000
16000000
14000000
g (W/cu. m)
12000000
f=50 Hz
f=150 Hz
f=300 Hz
f=1500 Hz
10000000
8000000
6000000
4000000
2000000
0
0
0.002
0.004
0.006
0.008
z (m)
24
0.01
0.012
0.014
Table 1: Output Current, Resistance, and Volumetric Heat Generation for Each
Discretized Cylinder Circuit for a frequency of 50 Hz
(g(i,j) are inputs to “Tempdist”)
CIRCUIT
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Re{I}
-0.5267
-1.57955
-2.63115
-3.67981
-4.724
-2.75374
-3.01279
-3.27051
-3.5268
-3.78146
-4.03426
-4.28491
-4.53311
-4.7785
-5.02068
-5.25919
-5.4935
-5.72305
-5.94718
-6.16516
-6.37617
-6.57929
-6.77347
-6.95736
-7.1286
-0.54523
-1.63544
-2.72481
-3.81198
-4.89569
-2.85433
-3.12379
-3.39215
-3.65928
-3.92499
-4.18908
-4.45127
-4.71126
-4.96868
-5.22313
-5.47412
-5.7211
-5.96341
-6.2003
-6.43089
-6.65412
-6.86874
-7.07316
-7.26527
-7.44198
Im{I}
-0.60197
-1.8125
-3.04256
-4.30628
-5.61813
-3.31968
-3.67252
-4.03615
-4.41168
-4.80035
-5.20351
-5.62258
-6.05912
-6.51478
-6.99139
-7.49092
-8.0155
-8.56753
-9.14961
-9.76465
-10.4159
-11.1072
-11.8429
-12.6284
-13.4706
-0.59915
-1.80425
-3.02996
-4.29118
-5.60332
-3.31396
-3.66843
-4.0344
-4.41312
-4.80597
-5.21443
-5.64013
-6.08482
-6.55043
-7.03908
-7.5531
-8.0951
-8.66798
-9.27501
-9.91994
-10.607
-11.3414
-12.129
-12.9775
-13.8967
ABS(I)
0.799863
2.40419
4.022453
5.664362
7.340269
4.313156
4.75019
5.194878
5.64812
6.110882
6.584206
7.069224
7.567168
8.079389
8.60737
9.15275
9.71735
10.3032
10.91257
11.54806
12.21259
12.9096
13.64313
14.41807
15.24057
0.810096
2.435157
4.074956
5.739808
7.440764
4.373735
4.818246
5.270965
5.732882
6.20507
6.688697
7.185044
7.695518
8.221678
8.765259
9.328204
9.912698
10.52122
11.15659
11.82207
12.52145
13.25921
14.04074
14.87275
15.76389
I^2
6.3978E-01
5.7801E+00
1.6180E+01
3.2085E+01
5.3880E+01
1.8603E+01
2.2564E+01
2.6987E+01
3.1901E+01
3.7343E+01
4.3352E+01
4.9974E+01
5.7262E+01
6.5277E+01
7.4087E+01
8.3773E+01
9.4427E+01
1.0616E+02
1.1908E+02
1.3336E+02
1.4915E+02
1.6666E+02
1.8613E+02
2.0788E+02
2.3228E+02
6.5626E-01
5.9300E+00
1.6605E+01
3.2945E+01
5.5365E+01
1.9130E+01
2.3215E+01
2.7783E+01
3.2866E+01
3.8503E+01
4.4739E+01
5.1625E+01
5.9221E+01
6.7596E+01
7.6830E+01
8.7015E+01
9.8262E+01
1.1070E+02
1.2447E+02
1.3976E+02
1.5679E+02
1.7581E+02
1.9714E+02
2.2120E+02
2.4850E+02
25
ABS(R)
0.000193
0.00058
0.000967
0.001353
0.00174
0.00406
0.004447
0.004833
0.00522
0.005607
0.005993
0.00638
0.006767
0.007153
0.00754
0.007926
0.008313
0.0087
0.009086
0.009473
0.00986
0.010246
0.010633
0.01102
0.011406
0.000193
0.00058
0.000967
0.001353
0.00174
0.00406
0.004447
0.004833
0.00522
0.005607
0.005993
0.00638
0.006767
0.007153
0.00754
0.007926
0.008313
0.0087
0.009086
0.009473
0.00986
0.010246
0.010633
0.01102
0.011406
g(r,z)
7.874233E+03
7.114009E+04
1.991400E+05
3.948922E+05
6.631328E+05
9.158555E+05
1.110858E+06
1.328579E+06
1.570523E+06
1.838418E+06
2.134241E+06
2.460255E+06
2.819054E+06
3.213614E+06
3.647351E+06
4.124201E+06
4.648709E+06
5.226136E+06
5.862610E+06
6.565297E+06
7.342644E+06
8.204696E+06
9.163567E+06
1.023412E+07
1.143508E+07
8.076999E+03
7.298449E+04
2.043725E+05
4.054818E+05
6.814150E+05
9.417629E+05
1.142917E+06
1.367782E+06
1.618015E+06
1.895527E+06
2.202519E+06
2.541532E+06
2.915495E+06
3.327803E+06
3.782389E+06
4.283835E+06
4.837493E+06
5.449651E+06
6.127731E+06
6.880563E+06
7.718736E+06
8.655098E+06
9.705475E+06
1.088978E+07
1.223385E+07
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
-0.56092
-1.68269
-2.80403
-3.92388
-5.04123
-2.94006
-3.21844
-3.49596
-3.77249
-4.04785
-4.32184
-4.5942
-4.86464
-5.13281
-5.39828
-5.66056
-5.91908
-6.17313
-6.42189
-6.66437
-6.89936
-7.12539
-7.34063
-7.54278
-7.72898
-0.57023
-1.71071
-2.85099
-3.99014
-5.12727
-2.99071
-3.27431
-3.55717
-3.83916
-4.1201
-4.3998
-4.678
-4.95441
-5.22866
-5.50034
-5.76893
-6.03381
-6.29426
-6.54938
-6.79809
-7.03907
-7.2707
-7.491
-7.69755
-7.88736
-0.59363
-1.78791
-3.00361
-4.2562
-5.56188
-3.29178
-3.64583
-4.0119
-4.39136
-4.78569
-5.19652
-5.6256
-6.07487
-6.54646
-7.04272
-7.5663
-8.12015
-8.70761
-9.33245
-9.99904
-10.7124
-11.4784
-12.3041
-13.1978
-14.1701
-0.59221
-1.78383
-2.99742
-4.24892
-5.55502
-3.28916
-3.64415
-4.01155
-4.39278
-4.78941
-5.20316
-5.63588
-6.08962
-6.56668
-7.06957
-7.60115
-8.16462
-8.76359
-9.40222
-10.0852
-10.8181
-11.6072
-12.4597
-13.3842
-14.3904
0.816716
2.455212
4.109042
5.788968
7.506567
4.413588
4.863171
5.321379
5.789274
6.26801
6.758854
7.2632
7.782594
8.318761
8.873632
9.449387
10.0485
10.6738
11.32852
12.01644
12.74193
13.51019
14.32741
15.20114
16.14087
0.822117
2.471552
4.136744
5.828767
7.559574
4.445549
4.899078
5.361526
5.834006
6.317731
6.814036
7.324397
7.850457
8.394056
8.957266
9.542435
10.15223
10.78973
11.45845
12.16249
12.9066
13.69633
14.5382
15.43984
16.41017
6.670255E-01
6.028065E+00
1.688423E+01
3.351215E+01
5.634855E+01
1.947976E+01
2.365044E+01
2.831708E+01
3.351569E+01
3.928795E+01
4.568210E+01
5.275407E+01
6.056877E+01
6.920178E+01
7.874134E+01
8.929091E+01
1.009724E+02
1.139299E+02
1.283354E+02
1.443947E+02
1.623568E+02
1.825252E+02
2.052746E+02
2.310747E+02
2.605275E+02
6.758768E-01
6.108571E+00
1.711265E+01
3.397453E+01
5.714716E+01
1.976291E+01
2.400096E+01
2.874597E+01
3.403563E+01
3.991373E+01
4.643108E+01
5.364679E+01
6.162967E+01
7.046018E+01
8.023262E+01
9.105806E+01
1.030679E+02
1.164183E+02
1.312961E+02
1.479261E+02
1.665802E+02
1.875895E+02
2.113594E+02
2.383887E+02
2.692936E+02
26
0.000193
0.00058
0.000967
0.001353
0.00174
0.00406
0.004447
0.004833
0.00522
0.005607
0.005993
0.00638
0.006767
0.007153
0.00754
0.007926
0.008313
0.0087
0.009086
0.009473
0.00986
0.010246
0.010633
0.01102
0.011406
0.000193
0.00058
0.000967
0.001353
0.00174
0.00406
0.004447
0.004833
0.00522
0.005607
0.005993
0.00638
0.006767
0.007153
0.00754
0.007926
0.008313
0.0087
0.009086
0.009473
0.00986
0.010246
0.010633
0.01102
0.011406
8.209544E+03
7.419157E+04
2.078059E+05
4.124572E+05
6.935206E+05
9.590037E+05
1.164329E+06
1.394072E+06
1.650003E+06
1.934176E+06
2.248965E+06
2.597124E+06
2.981847E+06
3.406857E+06
3.876497E+06
4.395861E+06
4.970947E+06
5.608858E+06
6.318050E+06
7.108663E+06
7.992950E+06
8.985854E+06
1.010582E+07
1.137599E+07
1.282597E+07
8.318484E+03
7.518241E+04
2.106172E+05
4.181480E+05
7.033498E+05
9.729431E+05
1.181586E+06
1.415186E+06
1.675600E+06
1.964984E+06
2.285838E+06
2.641073E+06
3.034076E+06
3.468809E+06
3.949914E+06
4.482858E+06
5.074111E+06
5.731362E+06
6.463810E+06
7.282518E+06
8.200874E+06
9.235176E+06
1.040538E+07
1.173606E+07
1.325753E+07
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
-0.5749
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2.479831
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7.58643
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4.91727
5.381869
5.856673
6.342928
6.842003
7.355417
7.884862
8.432238
8.999688
9.589644
10.20488
10.84859
11.52443
12.23665
12.99017
13.79069
14.64477
15.55983
16.544
0.824854
2.479831
4.150779
5.848931
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6.342928
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16.544
6.803839E-01
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1.687446E+02
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27
0.000193
0.00058
0.000967
0.001353
0.00174
0.00406
0.004447
0.004833
0.00522
0.005607
0.005993
0.00638
0.006767
0.007153
0.00754
0.007926
0.008313
0.0087
0.009086
0.009473
0.00986
0.010246
0.010633
0.01102
0.011406
0.000193
0.00058
0.000967
0.001353
0.00174
0.00406
0.004447
0.004833
0.00522
0.005607
0.005993
0.00638
0.006767
0.007153
0.00754
0.007926
0.008313
0.0087
0.009086
0.009473
0.00986
0.010246
0.010633
0.01102
0.011406
8.373956E+03
7.568695E+04
2.120488E+05
4.210462E+05
7.083560E+05
9.800441E+05
1.190378E+06
1.425945E+06
1.688646E+06
1.980689E+06
2.304641E+06
2.663491E+06
3.060729E+06
3.500438E+06
3.987416E+06
4.527325E+06
5.126874E+06
5.794060E+06
6.538459E+06
7.371599E+06
8.307427E+06
9.362868E+06
1.055849E+07
1.191917E+07
1.347465E+07
8.373956E+03
7.568695E+04
2.120488E+05
4.210462E+05
7.083560E+05
9.800441E+05
1.190378E+06
1.425945E+06
1.688646E+06
1.980689E+06
2.304641E+06
2.663491E+06
3.060729E+06
3.500438E+06
3.987416E+06
4.527325E+06
5.126874E+06
5.794060E+06
6.538459E+06
7.371599E+06
8.307427E+06
9.362868E+06
1.055849E+07
1.191917E+07
1.347465E+07
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
-0.57023
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0.822117
2.471552
4.136744
5.828767
7.559574
4.445549
4.899078
5.361526
5.834006
6.317731
6.814036
7.324397
7.850457
8.394056
8.957266
9.542435
10.15223
10.78973
11.45845
12.16249
12.9066
13.69633
14.5382
15.43984
16.41017
0.816716
2.455212
4.109042
5.788968
7.506567
4.413588
4.863171
5.321379
5.789274
6.26801
6.758854
7.2632
7.782594
8.318761
8.873632
9.449387
10.0485
10.6738
11.32852
12.01644
12.74193
13.51019
14.32741
15.20114
16.14087
6.758768E-01
6.108571E+00
1.711265E+01
3.397453E+01
5.714716E+01
1.976291E+01
2.400096E+01
2.874597E+01
3.403563E+01
3.991373E+01
4.643108E+01
5.364679E+01
6.162967E+01
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1.030679E+02
1.164183E+02
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1.665802E+02
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2.113594E+02
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6.670255E-01
6.028065E+00
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1.009724E+02
1.139299E+02
1.283354E+02
1.443947E+02
1.623568E+02
1.825252E+02
2.052746E+02
2.310747E+02
2.605275E+02
28
0.000193
0.00058
0.000967
0.001353
0.00174
0.00406
0.004447
0.004833
0.00522
0.005607
0.005993
0.00638
0.006767
0.007153
0.00754
0.007926
0.008313
0.0087
0.009086
0.009473
0.00986
0.010246
0.010633
0.01102
0.011406
0.000193
0.00058
0.000967
0.001353
0.00174
0.00406
0.004447
0.004833
0.00522
0.005607
0.005993
0.00638
0.006767
0.007153
0.00754
0.007926
0.008313
0.0087
0.009086
0.009473
0.00986
0.010246
0.010633
0.01102
0.011406
8.318484E+03
7.518241E+04
2.106172E+05
4.181480E+05
7.033498E+05
9.729431E+05
1.181586E+06
1.415186E+06
1.675600E+06
1.964984E+06
2.285838E+06
2.641073E+06
3.034076E+06
3.468809E+06
3.949914E+06
4.482858E+06
5.074111E+06
5.731362E+06
6.463810E+06
7.282518E+06
8.200874E+06
9.235176E+06
1.040538E+07
1.173606E+07
1.325753E+07
8.209544E+03
7.419157E+04
2.078059E+05
4.124572E+05
6.935206E+05
9.590037E+05
1.164329E+06
1.394072E+06
1.650003E+06
1.934176E+06
2.248965E+06
2.597124E+06
2.981847E+06
3.406857E+06
3.876497E+06
4.395861E+06
4.970947E+06
5.608858E+06
6.318050E+06
7.108663E+06
7.992950E+06
8.985854E+06
1.010582E+07
1.137599E+07
1.282597E+07
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
-0.54699
-1.64076
-2.73377
-3.82474
-4.9125
-2.86434
-3.13493
-3.40446
-3.67283
-3.93985
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-4.46895
-4.73048
-4.98955
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-4.30279
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-4.26745
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-4.7809
-5.18754
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-0.59867
-1.80261
-3.02605
-4.2831
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-4.38901
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-5.17732
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-8.52846
-9.10861
-9.7217
-10.371
-11.0603
-11.794
-12.5773
-13.4175
0.808797
2.43125
4.068418
5.730598
7.428826
4.366718
4.810516
5.262508
5.723683
6.195114
6.677966
7.173516
7.683171
8.208486
8.751196
9.313237
9.896793
10.50434
11.13869
11.80311
12.50136
13.23794
14.01821
14.84889
15.7386
0.79858
2.400333
4.015999
5.655273
7.328491
4.306236
4.742568
5.186543
5.639058
6.101077
6.573642
7.057881
7.555027
8.066426
8.59356
9.138065
9.701759
10.28667
10.89507
11.52953
12.193
12.88889
13.62124
14.39494
15.21613
6.541519E-01
5.910976E+00
1.655203E+01
3.283976E+01
5.518745E+01
1.906822E+01
2.314106E+01
2.769399E+01
3.276055E+01
3.837944E+01
4.459523E+01
5.145933E+01
5.903111E+01
6.737925E+01
7.658343E+01
8.673638E+01
9.794652E+01
1.103411E+02
1.240704E+02
1.393133E+02
1.562841E+02
1.752430E+02
1.965103E+02
2.204894E+02
2.477034E+02
6.377301E-01
5.761599E+00
1.612825E+01
3.198212E+01
5.370678E+01
1.854367E+01
2.249195E+01
2.690023E+01
3.179897E+01
3.722314E+01
4.321277E+01
4.981369E+01
5.707843E+01
6.506723E+01
7.384927E+01
8.350424E+01
9.412413E+01
1.058155E+02
1.187024E+02
1.329300E+02
1.486693E+02
1.661236E+02
1.855382E+02
2.072142E+02
2.315305E+02
29
0.000193
0.00058
0.000967
0.001353
0.00174
0.00406
0.004447
0.004833
0.00522
0.005607
0.005993
0.00638
0.006767
0.007153
0.00754
0.007926
0.008313
0.0087
0.009086
0.009473
0.00986
0.010246
0.010633
0.01102
0.011406
0.000193
0.00058
0.000967
0.001353
0.00174
0.00406
0.004447
0.004833
0.00522
0.005607
0.005993
0.00638
0.006767
0.007153
0.00754
0.007926
0.008313
0.0087
0.009086
0.009473
0.00986
0.010246
0.010633
0.01102
0.011406
8.051101E+03
7.275048E+04
2.037173E+05
4.041816E+05
6.792301E+05
9.387433E+05
1.139252E+06
1.363396E+06
1.612827E+06
1.889449E+06
2.195457E+06
2.533382E+06
2.906147E+06
3.317132E+06
3.770261E+06
4.270099E+06
4.821983E+06
5.432177E+06
6.108083E+06
6.858501E+06
7.693986E+06
8.627346E+06
9.674355E+06
1.085486E+07
1.219463E+07
7.848986E+03
7.091198E+04
1.985015E+05
3.936260E+05
6.610066E+05
9.129189E+05
1.107296E+06
1.324319E+06
1.565488E+06
1.832524E+06
2.127398E+06
2.452366E+06
2.810015E+06
3.203310E+06
3.635656E+06
4.110978E+06
4.633803E+06
5.209380E+06
5.843813E+06
6.544248E+06
7.319102E+06
8.178391E+06
9.134188E+06
1.020131E+07
1.139843E+07
APPENDIX C:
RESULTS FROM “TEMPDIST” MODEL:
DETERMINATION OF TEMPERATURE DISTRIBUTION
WITHIN CYLINDER
30
Figure 1: Temperature vs. time for various radial locations, r, for fixed axial location,
z=.006875 m, at f=50 Hz
120
Temperature (deg. C)
100
80
r=0 m
r=.003 m
r=.0105 m
r=.0155 m
r=.0205 m
r=.0255 m
r=.0285 m
r=.030 m
60
40
20
0
0
5
10
15
20
25
t (sec)
31
30
35
40
Figure 2: Temperature vs. radial position, r, for fixed axial location, z = .006875, for
various times ("snapshots") at 50 Hz
120
Temperature (deg. C)
100
80
t=9 s
t=18 s
t=27 s
t=36 s
60
40
20
0
0
0.005
0.01
0.015
0.02
r (m)
32
0.025
0.03
0.035
Figure 3: Temperature vs. radial position, r, at various axial locations, z, for fixed time of
t = 27 sec. at 50 Hz
83
82.5
Temperature (deg. C)
82
81.5
z=0 m
z=.003125 m
81
z=.006875 m
z=.010625 m
z=.0125 m
80.5
80
79.5
79
0
0.005
0.01
0.015
0.02
r (m)
33
0.025
0.03
0.035
Table 1: Temperature as a function of time, t, for various radial locations, r, for a
constant axial dimension z = .006875 m
(Data used for Figure 1 of Appendix C)
t
0
0.36
0.72
1.08
1.44
1.8
2.16
2.52
2.88
3.24
3.6
3.96
4.32
4.68
5.04
5.4
5.76
6.12
6.48
6.84
7.2
7.56
7.92
8.28
8.64
9
9.36
9.72
10.1
10.4
10.8
11.2
11.5
11.9
12.2
12.6
13
13.3
13.7
14
14.4
14.8
15.1
15.5
15.8
16.2
16.6
16.9
17.3
17.6
18
18.4
18.7
19.1
19.4
19.8
20.2
20.5
20.9
21.2
21.6
22
22.3
22.7
T at r=0
25.00
25.08
25.35
25.78
26.34
26.98
27.66
28.37
29.10
29.84
30.58
31.33
32.08
32.84
33.59
34.34
35.10
35.85
36.61
37.36
38.12
38.87
39.63
40.38
41.14
41.89
42.64
43.40
44.15
44.91
45.66
46.42
47.17
47.93
48.68
49.44
50.19
50.95
51.70
52.46
53.21
53.97
54.72
55.48
56.23
56.99
57.74
58.50
59.25
60.00
60.76
61.51
62.27
63.02
63.78
64.53
65.29
66.04
66.80
67.55
68.31
69.06
69.82
70.57
t
0
0.36
0.72
1.08
1.44
1.8
2.16
2.52
2.88
3.24
3.6
3.96
4.32
4.68
5.04
5.4
5.76
6.12
6.48
6.84
7.2
7.56
7.92
8.28
8.64
9
9.36
9.72
10.08
10.44
10.8
11.16
11.52
11.88
12.24
12.6
12.96
13.32
13.68
14.04
14.4
14.76
15.12
15.48
15.84
16.2
16.56
16.92
17.28
17.64
18
18.36
18.72
19.08
19.44
19.8
20.16
20.52
20.88
21.24
21.6
21.96
22.32
22.68
T at r=.003
25.00
25.09
25.37
25.81
26.38
27.02
27.70
28.42
29.14
29.88
30.63
31.38
32.13
32.88
33.64
34.39
35.14
35.90
36.65
37.41
38.16
38.92
39.67
40.43
41.18
41.94
42.69
43.45
44.20
44.96
45.71
46.47
47.22
47.97
48.73
49.48
50.24
50.99
51.75
52.50
53.26
54.01
54.77
55.52
56.28
57.03
57.79
58.54
59.30
60.05
60.81
61.56
62.32
63.07
63.83
64.58
65.33
66.09
66.84
67.60
68.35
69.11
69.86
70.62
t T at r=.0105
0
25.00
0.36
25.24
0.72
25.67
1.08
26.22
1.44
26.86
1.8
27.54
2.16
28.25
2.52
28.98
2.88
29.72
3.24
30.47
3.6
31.22
3.96
31.97
4.32
32.72
4.68
33.47
5.04
34.23
5.4
34.98
5.76
35.74
6.12
36.49
6.48
37.24
6.84
38.00
7.2
38.75
7.56
39.51
7.92
40.26
8.28
41.02
8.64
41.77
9
42.53
9.36
43.28
9.72
44.04
10.08
44.79
10.44
45.55
10.8
46.30
11.16
47.06
11.52
47.81
11.88
48.57
12.24
49.32
12.6
50.08
12.96
50.83
13.32
51.59
13.68
52.34
14.04
53.09
14.4
53.85
14.76
54.60
15.12
55.36
15.48
56.11
15.84
56.87
16.2
57.62
16.56
58.38
16.92
59.13
17.28
59.89
17.64
60.64
18
61.40
18.36
62.15
18.72
62.91
19.08
63.66
19.44
64.42
19.8
65.17
20.16
65.93
20.52
66.68
20.88
67.44
21.24
68.19
21.6
68.95
21.96
69.70
22.32
70.45
22.68
71.21
t Tat r=.0155
0
25.00
0.36
25.45
0.72
26.06
1.08
26.73
1.44
27.44
1.8
28.17
2.16
28.90
2.52
29.65
2.88
30.40
3.24
31.15
3.6
31.90
3.96
32.65
4.32
33.41
4.68
34.16
5.04
34.92
5.4
35.67
5.76
36.43
6.12
37.18
6.48
37.94
6.84
38.69
7.2
39.45
7.56
40.20
7.92
40.95
8.28
41.71
8.64
42.46
9
43.22
9.36
43.97
9.72
44.73
10.08
45.48
10.44
46.24
10.8
46.99
11.16
47.75
11.52
48.50
11.88
49.26
12.24
50.01
12.6
50.77
12.96
51.52
13.32
52.28
13.68
53.03
14.04
53.79
14.4
54.54
14.76
55.30
15.12
56.05
15.48
56.81
15.84
57.56
16.2
58.31
16.56
59.07
16.92
59.82
17.28
60.58
17.64
61.33
18
62.09
18.36
62.84
18.72
63.60
19.08
64.35
19.44
65.11
19.8
65.86
20.16
66.62
20.52
67.37
20.88
68.13
21.24
68.88
21.6
69.64
21.96
70.39
22.32
71.15
22.68
71.90
34
t T at r=.0205
0 25.00
0.36 25.76
0.72 26.56
1.08 27.35
1.44 28.13
1.8 28.90
2.16 29.66
2.52 30.42
2.88 31.18
3.24 31.93
3.6 32.69
3.96 33.45
4.32 34.20
4.68 34.96
5.04 35.71
5.4 36.47
5.76 37.22
6.12 37.98
6.48 38.73
6.84 39.48
7.2 40.24
7.56 40.99
7.92 41.75
8.28 42.50
8.64 43.26
9 44.01
9.36 44.77
9.72 45.52
10.08 46.28
10.44 47.03
10.8 47.79
11.16 48.54
11.52 49.30
11.88 50.05
12.24 50.81
12.6 51.56
12.96 52.32
13.32 53.07
13.68 53.83
14.04 54.58
14.4 55.34
14.76 56.09
15.12 56.84
15.48 57.60
15.84 58.35
16.2 59.11
16.56 59.86
16.92 60.62
17.28 61.37
17.64 62.13
18 62.88
18.36 63.64
18.72 64.39
19.08 65.15
19.44 65.90
19.8 66.66
20.16 67.41
20.52 68.17
20.88 68.92
21.24 69.68
21.6 70.43
21.96 71.19
22.32 71.94
22.68 72.70
t T at r=.0255
0 25.00
0.36 26.11
0.72 27.07
1.08 27.94
1.44 28.76
1.8 29.55
2.16 30.33
2.52 31.10
2.88 31.87
3.24 32.63
3.6 33.39
3.96 34.14
4.32 34.90
4.68 35.65
5.04 36.41
5.4 37.16
5.76 37.92
6.12 38.67
6.48 39.43
6.84 40.18
7.2 40.94
7.56 41.69
7.92 42.45
8.28 43.20
8.64 43.96
9 44.71
9.36 45.47
9.72 46.22
10.08 46.98
10.44 47.73
10.8 48.49
11.16 49.24
11.52 50.00
11.88 50.75
12.24 51.51
12.6 52.26
12.96 53.02
13.32 53.77
13.68 54.53
14.04 55.28
14.4 56.03
14.76 56.79
15.12 57.54
15.48 58.30
15.84 59.05
16.2 59.81
16.56 60.56
16.92 61.32
17.28 62.07
17.64 62.83
18 63.58
18.36 64.34
18.72 65.09
19.08 65.85
19.44 66.60
19.8 67.36
20.16 68.11
20.52 68.87
20.88 69.62
21.24 70.38
21.6 71.13
21.96 71.89
22.32 72.64
22.68 73.39
t T at r=.0285
0
25.00
0.36
26.27
0.72
27.26
1.08
28.15
1.44
28.98
1.8
29.79
2.16
30.57
2.52
31.34
2.88
32.11
3.24
32.87
3.6
33.63
3.96
34.38
4.32
35.14
4.68
35.90
5.04
36.65
5.4
37.41
5.76
38.16
6.12
38.92
6.48
39.67
6.84
40.43
7.2
41.18
7.56
41.94
7.92
42.69
8.28
43.44
8.64
44.20
9
44.95
9.36
45.71
9.72
46.46
10.08
47.22
10.44
47.97
10.8
48.73
11.16
49.48
11.52
50.24
11.88
50.99
12.24
51.75
12.6
52.50
12.96
53.26
13.32
54.01
13.68
54.77
14.04
55.52
14.4
56.28
14.76
57.03
15.12
57.79
15.48
58.54
15.84
59.30
16.2
60.05
16.56
60.80
16.92
61.56
17.28
62.31
17.64
63.07
18
63.82
18.36
64.58
18.72
65.33
19.08
66.09
19.44
66.84
19.8
67.60
20.16
68.35
20.52
69.11
20.88
69.86
21.24
70.62
21.6
71.37
21.96
72.13
22.32
72.88
22.68
73.64
t
0
0.36
0.72
1.08
1.44
1.8
2.16
2.52
2.88
3.24
3.6
3.96
4.32
4.68
5.04
5.4
5.76
6.12
6.48
6.84
7.2
7.56
7.92
8.28
8.64
9
9.36
9.72
10.08
10.44
10.8
11.16
11.52
11.88
12.24
12.6
12.96
13.32
13.68
14.04
14.4
14.76
15.12
15.48
15.84
16.2
16.56
16.92
17.28
17.64
18
18.36
18.72
19.08
19.44
19.8
20.16
20.52
20.88
21.24
21.6
21.96
22.32
22.68
T at r=.03
25.00
26.29
27.28
28.17
29.01
29.81
30.60
31.37
32.13
32.90
33.65
34.41
35.17
35.92
36.68
37.43
38.19
38.94
39.70
40.45
41.21
41.96
42.72
43.47
44.23
44.98
45.74
46.49
47.25
48.00
48.76
49.51
50.27
51.02
51.77
52.53
53.28
54.04
54.79
55.55
56.30
57.06
57.81
58.57
59.32
60.08
60.83
61.59
62.34
63.10
63.85
64.61
65.36
66.12
66.87
67.62
68.38
69.13
69.89
70.64
71.40
72.15
72.91
73.66
23
23.4
23.8
24.1
24.5
24.8
25.2
25.6
25.9
26.3
26.6
27
27.4
27.7
28.1
28.4
28.8
29.2
29.5
29.9
30.2
30.6
31
31.3
31.7
32
32.4
32.8
33.1
33.5
33.8
34.2
34.6
34.9
35.3
35.6
36
71.33
72.08
72.84
73.59
74.35
75.10
75.86
76.61
77.36
78.12
78.87
79.63
80.38
81.14
81.89
82.65
83.40
84.16
84.91
85.67
86.42
87.18
87.93
88.69
89.44
90.20
90.95
91.71
92.46
93.21
93.97
94.72
95.48
96.23
96.99
97.74
98.50
23.04
23.4
23.76
24.12
24.48
24.84
25.2
25.56
25.92
26.28
26.64
27
27.36
27.72
28.08
28.44
28.8
29.16
29.52
29.88
30.24
30.6
30.96
31.32
31.68
32.04
32.4
32.76
33.12
33.48
33.84
34.2
34.56
34.92
35.28
35.64
36
71.37
72.13
72.88
73.64
74.39
75.15
75.90
76.66
77.41
78.17
78.92
79.68
80.43
81.19
81.94
82.69
83.45
84.20
84.96
85.71
86.47
87.22
87.98
88.73
89.49
90.24
91.00
91.75
92.51
93.26
94.02
94.77
95.53
96.28
97.04
97.79
98.55
23.04
23.4
23.76
24.12
24.48
24.84
25.2
25.56
25.92
26.28
26.64
27
27.36
27.72
28.08
28.44
28.8
29.16
29.52
29.88
30.24
30.6
30.96
31.32
31.68
32.04
32.4
32.76
33.12
33.48
33.84
34.2
34.56
34.92
35.28
35.64
36
71.96
72.72
73.47
74.23
74.98
75.74
76.49
77.25
78.00
78.76
79.51
80.27
81.02
81.78
82.53
83.29
84.04
84.80
85.55
86.31
87.06
87.81
88.57
89.32
90.08
90.83
91.59
92.34
93.10
93.85
94.61
95.36
96.12
96.87
97.63
98.38
99.14
23.04
23.4
23.76
24.12
24.48
24.84
25.2
25.56
25.92
26.28
26.64
27
27.36
27.72
28.08
28.44
28.8
29.16
29.52
29.88
30.24
30.6
30.96
31.32
31.68
32.04
32.4
32.76
33.12
33.48
33.84
34.2
34.56
34.92
35.28
35.64
36
72.66
73.41
74.16
74.92
75.67
76.43
77.18
77.94
78.69
79.45
80.20
80.96
81.71
82.47
83.22
83.98
84.73
85.49
86.24
87.00
87.75
88.51
89.26
90.02
90.77
91.52
92.28
93.03
93.79
94.54
95.30
96.05
96.81
97.56
98.32
99.07
99.83
35
23.04 73.45
23.4 74.20
23.76 74.96
24.12 75.71
24.48 76.47
24.84 77.22
25.2 77.98
25.56 78.73
25.92 79.49
26.28 80.24
26.64 81.00
27 81.75
27.36 82.51
27.72 83.26
28.08 84.02
28.44 84.77
28.8 85.53
29.16 86.28
29.52 87.04
29.88 87.79
30.24 88.55
30.6 89.30
30.96 90.06
31.32 90.81
31.68 91.56
32.04 92.32
32.4 93.07
32.76 93.83
33.12 94.58
33.48 95.34
33.84 96.09
34.2 96.85
34.56 97.60
34.92 98.36
35.28 99.11
35.64 99.87
36 100.62
23.04 74.15
23.4 74.90
23.76 75.66
24.12 76.41
24.48 77.17
24.84 77.92
25.2 78.68
25.56 79.43
25.92 80.19
26.28 80.94
26.64 81.70
27 82.45
27.36 83.21
27.72 83.96
28.08 84.72
28.44 85.47
28.8 86.23
29.16 86.98
29.52 87.74
29.88 88.49
30.24 89.24
30.6 90.00
30.96 90.75
31.32 91.51
31.68 92.26
32.04 93.02
32.4 93.77
32.76 94.53
33.12 95.28
33.48 96.04
33.84 96.79
34.2 97.55
34.56 98.30
34.92 99.06
35.28 99.81
35.64 100.57
36 101.32
23.04
23.4
23.76
24.12
24.48
24.84
25.2
25.56
25.92
26.28
26.64
27
27.36
27.72
28.08
28.44
28.8
29.16
29.52
29.88
30.24
30.6
30.96
31.32
31.68
32.04
32.4
32.76
33.12
33.48
33.84
34.2
34.56
34.92
35.28
35.64
36
74.39
75.15
75.90
76.65
77.41
78.16
78.92
79.67
80.43
81.18
81.94
82.69
83.45
84.20
84.96
85.71
86.47
87.22
87.98
88.73
89.49
90.24
91.00
91.75
92.51
93.26
94.01
94.77
95.52
96.28
97.03
97.79
98.54
99.30
100.05
100.81
101.56
23.04
23.4
23.76
24.12
24.48
24.84
25.2
25.56
25.92
26.28
26.64
27
27.36
27.72
28.08
28.44
28.8
29.16
29.52
29.88
30.24
30.6
30.96
31.32
31.68
32.04
32.4
32.76
33.12
33.48
33.84
34.2
34.56
34.92
35.28
35.64
36
74.42
75.17
75.93
76.68
77.44
78.19
78.95
79.70
80.46
81.21
81.97
82.72
83.48
84.23
84.98
85.74
86.49
87.25
88.00
88.76
89.51
90.27
91.02
91.78
92.53
93.29
94.04
94.80
95.55
96.31
97.06
97.82
98.57
99.33
100.08
100.84
101.59
Table 2: Temperature as a function of radius, r, for various times, t (snapshots of
temperature distribution), for a constant axial dimension z = .006875 m
(Data used for Figure 2 of Appendix C)
t=9 s
r
0
0.001
0.003
0.005
0.007
0.009
0.0105
0.0115
0.0125
0.0135
0.0145
0.0155
0.0165
0.0175
0.0185
0.0195
0.0205
0.0215
0.0225
0.0235
0.0245
0.0255
0.0265
0.0275
0.0285
0.0295
0.03
T
41.89
41.90
41.94
42.03
42.17
42.36
42.53
42.65
42.78
42.92
43.07
43.22
43.37
43.53
43.69
43.85
44.01
44.17
44.32
44.46
44.59
44.71
44.82
44.90
44.95
44.98
44.98
t=18 s
r
0
0.001
0.003
0.005
0.007
0.009
0.0105
0.0115
0.0125
0.0135
0.0145
0.0155
0.0165
0.0175
0.0185
0.0195
0.0205
0.0215
0.0225
0.0235
0.0245
0.0255
0.0265
0.0275
0.0285
0.0295
0.03
T
60.76
60.77
60.81
60.90
61.04
61.23
61.40
61.52
61.65
61.79
61.94
62.09
62.24
62.40
62.56
62.72
62.88
63.04
63.19
63.33
63.46
63.58
63.68
63.77
63.82
63.85
63.85
36
t=27 s
r
0
0.001
0.003
0.005
0.007
0.009
0.0105
0.0115
0.0125
0.0135
0.0145
0.0155
0.0165
0.0175
0.0185
0.0195
0.0205
0.0215
0.0225
0.0235
0.0245
0.0255
0.0265
0.0275
0.0285
0.0295
0.03
T
79.63
79.64
79.68
79.77
79.91
80.10
80.27
80.39
80.52
80.66
80.81
80.96
81.11
81.27
81.43
81.59
81.75
81.91
82.06
82.20
82.33
82.45
82.55
82.64
82.69
82.72
82.72
t=36 s
r
0
0.001
0.003
0.005
0.007
0.009
0.0105
0.0115
0.0125
0.0135
0.0145
0.0155
0.0165
0.0175
0.0185
0.0195
0.0205
0.0215
0.0225
0.0235
0.0245
0.0255
0.0265
0.0275
0.0285
0.0295
0.03
T
98.50
98.50
98.55
98.64
98.78
98.97
99.14
99.26
99.39
99.53
99.68
99.83
99.98
100.14
100.30
100.46
100.62
100.78
100.93
101.07
101.20
101.32
101.42
101.51
101.56
101.59
101.59
Table 3: Temperature as a function of radius, r, for various axial locations, z, for a
fixed time of t = 27 s
(Data used for Figure 3 of Appendix C)
t=27 s, z = 0
t=27 s, z = .003125 t=27 s, z = .006875 t=27 s, z = .010625 t=27 s, z = .0125
r
T
r
T
r
T
r
T
r
T
0 79.62878
0 79.6288
0 79.6289
0 79.62904
0 79.62909
0.001 79.63499
0.001 79.63502
0.001 79.63512
0.001 79.63526
0.001 79.63531
0.003 79.67539
0.003 79.67543
0.003 79.67558
0.003 79.67573
0.003 79.67577
0.005 79.76923
0.005 79.76931
0.005 79.76955
0.005 79.76973
0.005 79.76976
0.007 79.91146
0.007 79.91161
0.007 79.91201
0.007 79.91224
0.007 79.91224
0.009 80.09837
0.009 80.09862
0.009 80.09924
0.009 80.09953
0.009 80.09951
0.0105 80.26568
0.0105 80.26603
0.0105 80.26687
0.0105 80.26724
0.0105 80.26718
0.0115 80.38878
0.0115 80.3892
0.0115 80.39023
0.0115 80.39065
0.0115 80.39057
0.0125 80.52015
0.0125 80.52066
0.0125 80.52189
0.0125 80.52239
0.0125 80.52227
0.0135 80.65894
0.0135 80.65955
0.0135 80.66104
0.0135 80.66161
0.0135 80.66147
0.0145 80.80423
0.0145 80.80496
0.0145 80.80674
0.0145 80.8074
0.0145 80.80722
0.0155 80.95497
0.0155 80.95585
0.0155 80.95796
0.0155 80.95873
0.0155 80.9585
0.0165 81.11002
0.0165 81.11106
0.0165 81.11356
0.0165 81.11447
0.0165 81.11418
0.0175 81.2681
0.0175 81.26934
0.0175 81.27229
0.0175 81.27334
0.0175 81.27299
0.0185 81.4278
0.0185 81.42926
0.0185 81.43276
0.0185 81.43397
0.0185 81.43353
0.0195 81.58756
0.0195 81.58929
0.0195 81.59342
0.0195 81.59481
0.0195 81.59426
0.0205 81.74565
0.0205 81.74771
0.0205 81.75257
0.0205 81.75415
0.0205 81.75347
0.0215 81.90017
0.0215 81.9026
0.0215 81.90833
0.0215 81.91011
0.0215 81.90925
0.0225
82.049
0.0225 82.05187
0.0225 82.05861
0.0225 82.06058
0.0225 82.05949
0.0235 82.18979
0.0235 82.19318
0.0235 82.20109
0.0235 82.20321
0.0235 82.20181
0.0245 82.31991
0.0245 82.32393
0.0245 82.33319
0.0245 82.33538
0.0245 82.33358
0.0255 82.43647
0.0255 82.44121
0.0255 82.45198
0.0255 82.45415
0.0255 82.45182
0.0265 82.53622
0.0265 82.54179
0.0265 82.55419
0.0265 82.55615
0.0265 82.55317
0.0275 82.61552
0.0275 82.62201
0.0275 82.63605
0.0275 82.63761
0.0275 82.63387
0.0285 82.67031
0.0285 82.67772
0.0285 82.6932
0.0285 82.69419
0.0285 82.68969
0.0295 82.69607
0.0295 82.70415
0.0295 82.72047
0.0295 82.72096
0.0295 82.71592
0.03 82.69607
0.03 82.70415
0.03 82.72047
0.03 82.72096
0.03 82.71592
37
APPENDIX D:
INPUTS TO “CURRENTSOLVER” MODEL AND THEIR
DESCRIPTIONS
38
Table 1: Constants and radial thickness and height for each discretized circuit
Cylinder
Coil
Conductivities
1.30E+07 (Mho/m) (for aluminum, as given from (2))
5.20E+07
(for copper, as given from (2))
Cylinder
Coil
free space
Magnetic Permeability
0.999999
1.00001960
1.25664E-06 (H/m)
(for diamagnetic materials, as given from (6))
(for paramagnetic materials, as given from (6))
(from (9))
1/2 RADIAL THICKNESS OFEACH RADIAL ELEMENTS PER EACH LEVEL OF Z, BEGINNING FROM R = 0:
RELEM1
0.001 (m)
cylinderelems. 1 thru 5 from r = 0 (see Figure 5 of Appendix A)
RELEM2
0.0005
cylinder elems. 6 thru 25 from r = 0
RELEMC
0.00125
all coil elems.
OVER ALL HEIGHT OF EACH RADIAL ELEMENT:(see Figure 5 of Appendix A)
HELEM
0.00125 (m)
all cylinder elems.
HELEMC
0.0025
all coil elems.
Input
Skin Depth
Frequency (as calculated from (4-5))
50 Hz
0.019740751
150 Hz
0.011397328
300 Hz
0.008059128
1500 Hz
0.003604152
39
Table 2: Voltage and Positional Inputs for Each Discretized Circuit
(R(I) and Z(I) denote the distance from z=0 and r=0 to the center of the
specified circuit, and Re(V) and Im(V) are the real and imaginary voltage
components across the specified discretized circuit.)
CIRCUIT
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
R(I)
0.0010000
0.0030000
0.0050000
0.0070000
0.0090000
0.0105000
0.0115000
0.0125000
0.0135000
0.0145000
0.0155000
0.0165000
0.0175000
0.0185000
0.0195000
0.0205000
0.0215000
0.0225000
0.0235000
0.0245000
0.0255000
0.0265000
0.0275000
0.0285000
0.0295000
0.0010000
0.0030000
0.0050000
0.0070000
0.0090000
0.0105000
0.0115000
0.0125000
0.0135000
0.0145000
0.0155000
0.0165000
0.0175000
0.0185000
0.0195000
0.0205000
0.0215000
0.0225000
0.0235000
0.0245000
0.0255000
0.0265000
0.0275000
0.0285000
0.0295000
Z(I)
Re{V(I)}
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0006250
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
0.0018750
0
Im{V(I)}
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
40
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
0.0010000
0.0030000
0.0050000
0.0070000
0.0090000
0.0105000
0.0115000
0.0125000
0.0135000
0.0145000
0.0155000
0.0165000
0.0175000
0.0185000
0.0195000
0.0205000
0.0215000
0.0225000
0.0235000
0.0245000
0.0255000
0.0265000
0.0275000
0.0285000
0.0295000
0.0010000
0.0030000
0.0050000
0.0070000
0.0090000
0.0105000
0.0115000
0.0125000
0.0135000
0.0145000
0.0155000
0.0165000
0.0175000
0.0185000
0.0195000
0.0205000
0.0215000
0.0225000
0.0235000
0.0245000
0.0255000
0.0265000
0.0275000
0.0285000
0.0295000
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0031250
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0.0043750
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
41
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
0.0010000
0.0030000
0.0050000
0.0070000
0.0090000
0.0105000
0.0115000
0.0125000
0.0135000
0.0145000
0.0155000
0.0165000
0.0175000
0.0185000
0.0195000
0.0205000
0.0215000
0.0225000
0.0235000
0.0245000
0.0255000
0.0265000
0.0275000
0.0285000
0.0295000
0.0010000
0.0030000
0.0050000
0.0070000
0.0090000
0.0105000
0.0115000
0.0125000
0.0135000
0.0145000
0.0155000
0.0165000
0.0175000
0.0185000
0.0195000
0.0205000
0.0215000
0.0225000
0.0235000
0.0245000
0.0255000
0.0265000
0.0275000
0.0285000
0.0295000
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0056250
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0.0068750
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
42
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
0.0010000
0.0030000
0.0050000
0.0070000
0.0090000
0.0105000
0.0115000
0.0125000
0.0135000
0.0145000
0.0155000
0.0165000
0.0175000
0.0185000
0.0195000
0.0205000
0.0215000
0.0225000
0.0235000
0.0245000
0.0255000
0.0265000
0.0275000
0.0285000
0.0295000
0.0010000
0.0030000
0.0050000
0.0070000
0.0090000
0.0105000
0.0115000
0.0125000
0.0135000
0.0145000
0.0155000
0.0165000
0.0175000
0.0185000
0.0195000
0.0205000
0.0215000
0.0225000
0.0235000
0.0245000
0.0255000
0.0265000
0.0275000
0.0285000
0.0295000
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0081250
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0.0093750
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
43
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
0.0010000
0.0030000
0.0050000
0.0070000
0.0090000
0.0105000
0.0115000
0.0125000
0.0135000
0.0145000
0.0155000
0.0165000
0.0175000
0.0185000
0.0195000
0.0205000
0.0215000
0.0225000
0.0235000
0.0245000
0.0255000
0.0265000
0.0275000
0.0285000
0.0295000
0.0010000
0.0030000
0.0050000
0.0070000
0.0090000
0.0105000
0.0115000
0.0125000
0.0135000
0.0145000
0.0155000
0.0165000
0.0175000
0.0185000
0.0195000
0.0205000
0.0215000
0.0225000
0.0235000
0.0245000
0.0255000
0.0265000
0.0275000
0.0285000
0.0295000
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0106250
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0.0118750
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
44
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
251
252
253
254
255
256
257
258
0.0362500
0.0387500
0.0362500
0.0387500
0.0362500
0.0387500
0.0362500
0.0387500
0.0012500
0.0012500
0.0037500
0.0037500
0.0087500
0.0087500
0.0112500
0.0112500
1
1
1
1
1
1
1
1
45
0
0
0
0
0
0
0
0
APPENDIX E:
“CURRENTSOLVER” VISUAL BASIC CODE
46
Sub currentsolver()
'Declare variables and array sizes
Dim nl, nc, n, np1, nt2, nt2p1, ii, jj, k, ip1, jm1, _
im1, L, irow, nn, ee, dd, cc As Integer
'nl and n are used for physical discretization of the
'cylinder (load) and coil
nl = 250 '231
nc = 8 '4
n = 258 '235
'np1, nt2, and nt2p1 are used for matrix manipulations
np1 = 259 '236
nt2 = 516 '470
nt2p1 = 517 '471
Dim r(300), z(300), eemmee(700, 700), aiabs(300), v_r(300), _
v_c(300), resist(300), eme_r(700, 700), eme_c(700, 700), _
aai(300), reai(300), aimi(300), rc(300), areac(300), volume(300), _
isqrd(300), heatgenrate(300) As Single
Dim aaa(700, 700), aaii(700), coef As Double
'Read in constants, coordinates, and parameters of cylinder
'sigl1,sigc = conductivities of load and coil (1/ohm x m)
'emul1,emuc = relative magnetic permeabilities of load and coil (-)
'emu0 = magnetic permeability of free space
'relem1 = 1/2 radial thickness of the load elems. 1 thru 5
'relem2 = 1/2 radial thickness of the load elems. 6 thru 25
'relemc = 1/2 radial thickness of the coil elems.
'helem = overall thickness of all load circuit elements
'helemc = overall thickness of all coil circuit elements.
'omega = frequency (Hz)
'r(i) = radial coordinates of circuits
'z(i) = axial coordinates of circuits
'v_r(i) = voltage applied on circuits, real term
'v_c(i) = voltage applied on circuits, complex term
sigl1 = Sheets("sheet1").Range("b2")
sigc = Sheets("sheet1").Range("b3")
emul1 = Sheets("sheet1").Range("b6")
emuc = Sheets("sheet1").Range("b7")
emu0 = Sheets("sheet1").Range("b8")
relem1 = Sheets("sheet1").Range("b11")
relem2 = Sheets("sheet1").Range("b12")
47
relemc = Sheets("sheet1").Range("b13")
helem = Sheets("sheet1").Range("b19")
helemc = Sheets("sheet1").Range("b20")
omega = Sheets("sheet1").Range("b21")
For i = 1 To n
r(i) = Sheets("sheet2").Cells(i + 1, 2)
z(i) = Sheets("sheet2").Cells(i + 1, 3)
v_r(i) = Sheets("sheet2").Cells(i + 1, 4)
v_c(i) = Sheets("sheet2").Cells(i + 1, 5)
'Sheets("sheet9").Cells(i + 1, 2) = r(i)
'Sheets("sheet9").Cells(i + 1, 3) = z(i)
'Sheets("sheet9").Cells(i + 1, 4) = v_r(i)
'Sheets("sheet9").Cells(i + 1, 5) = v_c(i)
Next
'Compute preliminary parameters
Pi = 3.141592654
efe = omega
omega = efe * 2 * Pi
Sheets("sheet3").Range("b2") = omega
'Begin calculation
'First calculate skin depth of load cylinder to estimate effective radii for
' current flow inside conductors and for temperature distribution calculation
deltal1 = Sqr(2 / (omega * emul1 * emu0 * sigl1))
Sheets("sheet1").Range("c25") = deltal1
' Calculate discretized circuit radial thickness, cross-sectional areas, and
' volumes for load cylinder
For cc = 0 To 9
dd = cc * 25
'elem 1 from r=0 for each z level, helem, relem1
' at circuits nearest r = 0, must approximate using .99 * 1/2 radial thickness
' (relem1) to avoid singularity error in calculating coefficients L and M
i=1
ee = i + dd
rc(ee) = 0.99 * relem1
areac(ee) = (2 * relem1) * helem
'volume(ee) = Pi * helem * ((r(ee) + relem1) ^ 2 - (r(ee) - relem1) ^ 2)
volume(ee) = 2 * Pi * r(ee) * areac(ee)
'elems 2 thru 5 from r=0 for each z level, helem, relem2
For i = 2 To 5
48
ee = i + dd
rc(ee) = relem1
areac(ee) = (2 * relem1) * helem
'volume(ee) = Pi * helem * ((r(ee) + relem1) ^ 2 - (r(ee) - relem1) ^ 2)
volume(ee) = 2 * Pi * r(ee) * areac(ee)
Next
'elems. 6 thru 25 from r=0 for each z level, helem, relem3
For i = 6 To 25
ee = i + dd
rc(ee) = relem2
areac(ee) = (2 * relem2) * helem
'volume(ee) = Pi * helem * ((r(ee) + relem2) ^ 2 - (r(ee) - relem2) ^ 2)
volume(ee) = 2 * Pi * r(ee) * areac(ee)
Next
Next
'Lastly assign discretized circuit radial thickness and cross-sectional areas
' of coil, relemc, helemc
For i = 251 To 258
rc(i) = relemc
areac(i) = (2 * relemc) * helemc
'volume(i) = Pi * helemc * ((r(i) + relemc) ^ 2 - (r(i) - relemc) ^ 2)
volume(i) = 2 * Pi * r(i) * areac(i)
Next
'***********************************************************************
*
'***********************************************************************
*
'Calculate the coefficients of the equations
For i = 1 To n
For j = 1 To n
'Calculate the geometric constant eka for every pair of
' circuits
If j = i Then
eka = 2 * Sqr((r(j) * (r(j) - rc(j)) / _
((2 * r(j) - rc(j)) ^ 2)))
End If
If j <> i Then
anum = r(i) * r(j)
aden = (Abs(z(i) - z(j))) ^ 2 + (r(i) + r(j)) ^ 2
eka = 2 * Sqr(anum / aden)
End If
49
'Sheets("sheet1").Cells(i + 1, 8) = eka
'Sheets("sheet1").Cells(i + 1, 6) = i
'Sheets("sheet1").Cells(i + 1, 7) = j
'Using functions elline and ellink, calculate the values of
' the complete elliptic integrals of first and second kind.
' Then calculate the mutual and self inductances eemmee.
emu = emul1
If j > nl Then emu = emuc
If j = i Then 'Calculation of L's (eemmee)
coef = emu0 * (2 * r(j) - rc(j))
gke = (1 - ((eka ^ 2) / 2)) * ellink(eka) - elline(eka)
sii = emu * emu0 * r(j) / 4
eemmee(i, j) = coef * gke + sii
'Sheets("sheet3").Cells(i + 1, j + 5) = eemmee(i, j)
End If
If j <> i Then 'Calculation of M's (eemmee)
coef = emu0 * Sqr(r(i) * r(j))
gke = ((2 / eka) - eka) * ellink(eka) - (2 / eka) _
* elline(eka)
eemmee(i, j) = coef * gke
'Sheets("sheet3").Cells(i + 1, j + 5) = eemmee(i, j)
End If
'Sheets("sheet9").Cells(j + 1, 7) = ellink(eka)
'Sheets("sheet9").Cells(j + 1, 8) = elline(eka)
Next
Next
'Now calculate the resistances of individual circuits.
For i = 1 To n
If i > nl Then res = (2 * Pi * r(i)) / (sigc * areac(i))
If i <= nl Then res = (2 * Pi * r(i)) / (sigl1 * areac(i))
'Store resistance terms and assign proper direction per
' radial location
resist(i) = res
If i <= nl Then resist(i) = -resist(i)
'Sheets("sheet3").Cells(i + 1, 4) = resist(i)
Next
'Calculate coefficients of the matrix
For i = 1 To n
For j = 1 To n
If i <= nl Then eemmee(i, j) = -eemmee(i, j)
50
'Separate and organize real and imaginary coefficients of current
' equation coefficients
eme_c(i, j) = omega * eemmee(i, j) ' imaginary term (for both L's and M's)
If j <> i Then
eme_r(i, j) = 0 ' real term (for M's)
End If
If j = i Then
eme_r(i, j) = resist(j) 'real term (for L's)
End If
Next
Next
'********************************************************
'Prepare arrays for incorporation into matrix for solution of
' simultaneous linear algebraic equations to obtain the
' current in every circuit
'********************************************************
For i = 1 To n
For j = 1 To n
aaa(i, j) = eme_r(i, j)
'Sheets("sheet5").Cells(i, j) = aaa(i, j)
Next
Next
For i = 1 To n
For j = np1 To nt2
jj = j - n
aaa(i, j) = -eme_c(i, jj)
'Sheets("sheet5").Cells(i, j) = aaa(i, j)
Next
Next
For i = np1 To nt2
For j = 1 To n
ii = i - n
aaa(i, j) = eme_c(ii, j)
'Sheets("sheet5").Cells(i, j) = aaa(i, j)
Next
Next
For i = np1 To nt2
For j = np1 To nt2
ii = i - n
jj = j - n
aaa(i, j) = eme_r(ii, jj)
'Sheets("sheet5").Cells(i, j) = aaa(i, j)
Next
Next
For i = 1 To n
51
aaa(i, nt2p1) = v_r(i)
'Sheets("sheet5").Cells(i, nt2p1) = aaa(i, j)
Next
For i = np1 To nt2
ii = i - n
aaa(i, nt2p1) = v_c(ii)
'Sheets("sheet5").Cells(i, nt2p1) = aaa(i, j)
Next
'***Perform matrix manipulations here (based on original program
'which called for subroutine MATSOL) using LU Decomposition (Cholesky)
'Input values of interest here are aaa, nt2, and nt2p1
'Output values of interest for this part of the code are aaii,
'reai, and aimi (real and imaginary current cells of aaii)
irow = 1
big = Abs(aaa(1, 1))
For i = 2 To nt2
ab = Abs(aaa(i, 1))
If big < ab Then
big = ab
irow = i
End If
Next
If irow <> 1 Then
For j = 1 To nt2p1
temp = aaa(irow, j)
aaa(irow, j) = aaa(1, j)
aaa(1, j) = temp
Next
End If
For j = 2 To nt2p1
aaa(1, j) = aaa(1, j) / aaa(1, 1)
Next
For i = 2 To nt2
j=i
For ii = j To nt2
summ = 0
jm1 = j - 1
For k = 1 To jm1
summ = summ + aaa(ii, k) * aaa(k, j)
Next
aaa(ii, j) = aaa(ii, j) - summ
Next
If i <> nt2 Then
irow = i
52
big = Abs(aaa(i, i))
ip1 = i + 1
For ii = ip1 To nt2
ab = Abs(aaa(ii, i))
If big < ab Then
big = ab
irow = ii
End If
Next
If irow <> i Then
For j = 1 To nt2p1
temp = aaa(irow, j)
aaa(irow, j) = aaa(i, j)
aaa(i, j) = temp
Next
End If
End If
ip1 = i + 1
For j = ip1 To nt2p1
summ = 0
im1 = i - 1
For k = 1 To im1
summ = summ + aaa(i, k) * aaa(k, j)
Next
aaa(i, j) = (aaa(i, j) - summ) / aaa(i, i)
Next
Next
aaii(nt2) = aaa(nt2, nt2p1)
L = nt2 - 1
For nn = 1 To L
summ = 0
i = nt2 - nn
ip1 = i + 1
For j = ip1 To nt2
summ = summ + aaa(i, j) * aaii(j)
Next
aaii(i) = aaa(i, nt2p1) - summ
Next
'***End of matrix manipulations (formerly subroutine "MATSOL")
For i = 1 To n
ipn = i + n
reai(i) = aaii(i) '/ areac(i)
aimi(i) = aaii(ipn) '/ areac(i)
53
'convert units from per sqr. m to per sq. cm
'(used only when validating to model from tech. paper
' for current densities per discretized circuit cross' sectional area)
'reai(i) = reai(i) / 10000
'aimi(i) = aimi(i) / 10000
aiabs(i) = Sqr((reai(i)) ^ 2 + (aimi(i)) ^ 2)
isqrd(i) = (reai(i)) ^ 2 + (aimi(i)) ^ 2
heatgenrate(i) = (isqrd(i) * Abs(resist(i))) / volume(i)
'Ouput data for currents in cylinder circuits and coil
Sheets("sheet8").Cells(i + 1, 1) = i
'for validation, reai(i) and aimi(i) output units are per sq. cm
'for project use, reai(i) and aimi(i) output units are per sq. m
Sheets("sheet8").Cells(i + 1, 6) = reai(i)
Sheets("sheet8").Cells(i + 1, 7) = aimi(i)
Sheets("sheet8").Cells(i + 1, 8) = aiabs(i)
Sheets("sheet8").Cells(i + 1, 9) = isqrd(i)
Sheets("sheet8").Cells(i + 1, 10) = Abs(resist(i))
'Heat genrate units are per cu. m
Sheets("sheet8").Cells(i + 1, 11) = heatgenrate(i)
Next
End Sub
Function ellink(ak) 'numerical integration for (4-13) in Section 4.0
eme1 = 1 - ak ^ 2
eme12 = eme1 ^ 2
eme13 = eme12 * eme1
eme14 = eme13 * eme1
a0k = 1.38629436112
a1k = 0.09666344259
a2k = 0.03590092383
a3k = 0.03742563713
a4k = 0.01451196212
b0k = 0.5
b1k = 0.12498593597
b2k = 0.06880248576
b3k = 0.03328355346
b4k = 0.00441787012
fpk = a0k + a1k * eme1 + a2k * eme12 + a3k * eme13 + a4k * eme14
spk = b0k + b1k * eme1 + b2k * eme12 + b3k * eme13 + b4k * eme14
ellink = fpk + spk * Log(1 / eme1)
End Function
Function elline(ak) 'numerical integration for (4-12) in Section 4.0
eme1 = 1 - ak ^ 2
eme12 = eme1 ^ 2
eme13 = eme12 * eme1
54
eme14 = eme13 * eme1
a0e = 1#
a1e = 0.44325141463
a2e = 0.0626060122
a3e = 0.04757383546
a4e = 0.01736506451
b0e = 0
b1e = 0.2499836831
b2e = 0.09200180037
b3e = 0.04069697526
b4e = 0.00526449639
fpe = a0e + a1e * eme1 + a2e * eme12 + a3e * eme13 + a4e * eme14
spe = b0e + b1e * eme1 + b2e * eme12 + b3e * eme13 + b4e * eme14
elline = fpe + spe * Log(1 / eme1)
End Function
55
APPENDIX F:
“TEMPDIST” VISUAL BASIC CODE
56
Sub tempdist()
Dim T(27, 12, 40001), g(27, 12), r(27), rphalf(27), rmhalf(27), rstepf(27), _
rstepb(27), rfterm, rbterm, rterm, zterm, gterm, z(12) As Single
Dim i, j, aa, bb As Integer
'Parameters for thermal problem
alpha = 0.00008418 'diffusivity of pure aluminum at room temperature, units are m2/s
timestep = 0.0009 'units are s
k = 204 'thermal conductivity of pure aluminum at room temperature, units are W/m/C
Open "o:\ceno\public\guidos\temp50_n1.txt" For Output As #1
Open "o:\ceno\public\guidos\temp50_n2.txt" For Output As #2
Open "o:\ceno\public\guidos\temp50_n3.txt" For Output As #3
Open "o:\ceno\public\guidos\temp50_n4.txt" For Output As #4
Open "o:\ceno\public\guidos\temp50_n5.txt" For Output As #5
Open "o:\ceno\public\guidos\temp50_n6.txt" For Output As #6
Open "o:\ceno\public\guidos\temp50_n7.txt" For Output As #7
Open "o:\ceno\public\guidos\temp50_n8.txt" For Output As #8
Open "o:\ceno\public\guidos\temp50_n9.txt" For Output As #9
Open "o:\ceno\public\guidos\temp50_n10.txt" For Output As #10
Open "o:\ceno\public\guidos\temp50_n11.txt" For Output As #11
Open "o:\ceno\public\guidos\temp50_n12.txt" For Output As #12
'Total amount of node levels: 27 in r, 12 in z
'In r, 25 internal node levels numbered i=2 to 26
'In z, 10 internal node levels numbered j=2 to 11
'At r=0, node is i=1, but since cannot use the Mutual Inductance
' model to solve for heat generated at this edge (since symmetry
' was used to create Mutual Inductance model and currents are
' calculated at the centers of the discretized circuit elements),
' since heat generated at node i=2 is small, heat gen. at i=1
' must be smaller. From other approximations, 1st order errors already incurred.
' Assuming heat gen. at r=0 equal to heat gen. at i =2 should be on the order of
' a first order error. Also node at i=1 is very close to i=2.
' Therefore, approximate heat gen. at i=1 to be equal to heat gen.
' at i=2.
'Assign generated heat/vol. per each internal temperature node from spreadsheet
' output from program Currentsolver() for each internal node for each z level
For j = 2 To 11
For i = 2 To 26
aa = i + (j - 2) * 25
g(i, j) = Sheets("sheet8").Cells(aa, 11)
Next
Next
'Assign generated heat per each temperature node at r = 0 for each z level
57
For j = 2 To 11
aa = 1 + (j - 2) * 25
g(1, j) = Sheets("sheet8").Cells(aa + 1, 11)
'g(1, j) = 0
Next
'Apply initial temperature conditions to each temp. node.
For i = 1 To 27
For j = 1 To 12
n = 1 'this is at time, t=0 s
T(i, j, n) = 25 'initially cylinder at room temperature
Next
Next
'Set up r values for each temperature calculation by creating arrays
'based on dimensions from mesh. Refer to Figure of Mesh in Appendix
'A for dimensional information
z(1) = 0
z(2) = 0.00125 / 2
For j = 3 To 11
z(j) = z(2) + (j - 2) * (0.00125)
Next
z(12) = 0.0125
r(1) = 0
r(27) = 0.03
'for i=1
rstepf(1) = 0.001
'for i=2
r(2) = 0.001
rphalf(2) = 0.002
rmhalf(2) = 0.0005
rstepf(2) = 0.002
rstepb(2) = 0.001
'for i = 3 to i =5
For i = 3 To 5
aa = i - 3
r(i) = 0.003 + aa * 0.002
rphalf(i) = 0.004 + aa * 0.002
rmhalf(i) = 0.002 + aa * 0.002
rstepf(i) = 0.002
rstepb(i) = 0.002
Next
58
'for i=6
r(6) = 0.009
rphalf(6) = 0.00975
rmhalf(6) = 0.008
rstepf(6) = 0.0015
rstepb(6) = 0.002
'for i=7
r(7) = 0.0105
rphalf(7) = 0.011
rmhalf(7) = 0.00975
rstepf(7) = 0.001
rstepb(7) = 0.0015
'for i=8 to i=25
For i = 8 To 25
aa = i - 8
r(i) = 0.0115 + aa * 0.001
rphalf(i) = 0.012 + aa * 0.001
rmhalf(i) = 0.011 + aa * 0.001
rstepf(i) = 0.001
rstepb(i) = 0.001
Next
'for i=26
r(26) = 0.0295
rphalf(26) = 0.02975
rmhalf(26) = 0.029
rstepf(26) = 0.0005
rstepb(26) = 0.001
'Evaluate temperatures throughout mesh, for time greater than zero
'reminder: at t=0s, n=1
'This loop calculates for nodes along r=0, interior nodes, and applies
' boundary conditions for EACH timestep
For n = 2 To 40001
p = n - 1 'dummy variable for previous time step values
For i = 1 To 26
For j = 2 To 11
gterm = g(i, j) / k
If j = 2 Then
zstepf = 0.00125
zstepb = 0.000625
59
epsilon = zstepf / zstepb
End If
If j = 11 Then
zstepf = 0.000625
zstepb = 0.00125
epsilon = zstepf / zstepb
End If
If j <> 2 Or j <> 11 Then
zstepf = 0.00125
End If
'Nodes at r=0
If i = 1 Then
'Changing or constant z, constant r
rterm = 4 * (T(i + 1, j, p) - T(i, j, p)) / ((rstepf(i)) ^ 2)
zterm = (2 / (zstepf + zstepb)) * (((T(i, j + 1, p) - T(i, j, p)) / _
zstepf) - ((T(i, j, p) - T(i, j - 1, p)) / zstepb))
End If
'Interior nodes
If i <> 1 Then
'Changing or constant r, changing or constant z
rfterm = rphalf(i) * (T(i + 1, j, p) - T(i, j, p)) / rstepf(i)
rbterm = rmhalf(i) * (T(i, j, p) - T(i - 1, j, p)) / rstepb(i)
rterm = (1 / r(i)) * (rfterm - rbterm) / ((1 / 2) * _
(rstepb(i) + rstepf(i)))
zterm = (2 / (zstepf + zstepb)) * (((T(i, j + 1, p) - T(i, j, p)) / _
zstepf) - ((T(i, j, p) - T(i, j - 1, p)) / zstepb))
End If
T(i, j, n) = alpha * timestep * (rterm + zterm _
+ gterm) + T(i, j, p)
Next 'j
Next 'i
'Apply conditions at the boundaries for current time step
' (insulated at all outer surfaces (in vacuum) with temperatures low so
' that radiation heat transfer may be neglected
'Along r boundary surface
For j = 2 To 11
'At outer cylinder surface, r=b
i = 27
60
T(i, j, n) = T(i - 1, j, n)
Next
'Along z boundary surfaces
For i = 1 To 27
'At lower surface boundary, z=0
j=1
T(i, j, n) = T(i, j + 1, n)
'At upper surface boundary, z=c
j = 12
T(i, j, n) = T(i, j - 1, n)
Next
Next 'n
'Output temperature of areas of interest to table
For n = 1 To 40001 Step 400
For i = 1 To 27
Write #1, (n - 1) * timestep, r(i), z(1), T(i, 1, n)
Write #2, (n - 1) * timestep, r(i), z(2), T(i, 2, n)
Write #3, (n - 1) * timestep, r(i), z(3), T(i, 3, n)
Write #4, (n - 1) * timestep, r(i), z(4), T(i, 4, n)
Write #5, (n - 1) * timestep, r(i), z(5), T(i, 5, n)
Write #6, (n - 1) * timestep, r(i), z(6), T(i, 6, n)
Write #7, (n - 1) * timestep, r(i), z(7), T(i, 7, n)
Write #8, (n - 1) * timestep, r(i), z(8), T(i, 8, n)
Write #9, (n - 1) * timestep, r(i), z(9), T(i, 9, n)
Write #10, (n - 1) * timestep, r(i), z(10), T(i, 10, n)
Write #11, (n - 1) * timestep, r(i), z(11), T(i, 11, n)
Write #12, (n - 1) * timestep, r(i), z(12), T(i, 12, n)
Next
Next
Close #1
Close #2
Close #3
Close #4
Close #5
Close #6
Close #7
Close #8
Close #9
Close #10
Close #11
Close #12
End Sub
61