Rochester Institute of Technology
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Theses
Thesis/Dissertation Collections
1011983
Radiation heat transfer analysis of a Czochralski
furnace with a radiation shield
Frederick A. Merz
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Recommended Citation
RADIATION HEAT TRANSFER ANALYSIS
OF A CZOCHRALSKI FURNACE
WITH A RADIATION SHIELD
by
Frederick A. Merz
A Thesis Submitted
in
Partial Fulfillment
of the
Requirements for the Degree of
MASTER OF SCIENCE
in
M
e cha nica l Engineering
Approved by:
P
ro .
f
Dr. Bhalchandra
V.
Karlekar
(
The s I
's
Ad
vi s or )
Requirements for the Degree of
MASTER OF SCIENCE
in
M
e cha nica l Engineering
Approved by:
P
ro .
f
Dr. Bhalchandra V.
Karlekar
(
The s I
's
Ad
vi s or )
Robert A.
Elison
Prof.    
_
Signature not legible
ACKNOWLEDGEMENTS
It
is
apleasure
for
the
authorto
expresshis
sinceregraditude
to
Dr.
Bhalchandra
V.
Karlekar
for
his
able supervisionand
patient
encouragement
throughout
this
work.The
authoris
also
thankful
to
Dr.
W.
E.
Langlois
for
his
useful
_{help}
and suggestions.Appreciation
also goesto
Peggie
TABLE
OF
CONTENTS
Page
Abstract
Introduction
Nomenclature
The
Crystal
Growth in
aCzochralski
Furnace
1
History
ofVarious
Works
3
Statement
ofthe
Problem
5
Table
1
6a
Mathematical
Formulation
7
Method
ofSolution
16
Svanberg
Vorticity
Parameter
18
Rotational Equation
24
Energy
Equation
24
Radiant
_{Energy}
Balance
atthe
Melt
Surface
27
Melt
Surface
Equilibrium
27
Radiation
Network
Equation
29
View
Factors
Computer
Program
and_{Stability}
Fluid,
Temperature,
andRadiation
Profiles
47
Results
51
Discussions
andConclusions
54
References
>>Appendix
56
NaiverStokes
Equation
in Polar Coordinates
Rotational
Velocity
in
Production
Svanberg Vorticity
Equation
Factors
34
TABLE
OF
FIGURES
Figure
Title
Page
1
Layout
ofCzochralski
Furnace
6
2
aAxial
Grid
Cell
8
b
Off
Axial Grid Cell
8
3
r+ andzGrid Locations
21
4
Network
ofSystem
withoutShield
31
5
Network
ofSystem
withShield
32
6
aSurfacetoshield
Viewfactor
35
Shield
Point
Locations
b
Surfacetoshield
Viewfactor
36
Furnace
Views
7
_{Instability}
Grid
Locations
42
8
_{Instability}
Grid
Values
43
9
Flow
Streamlines
45
10
Temperature
Isotherms
(with Shield)
49
ABSTRACT
This
paperreports
onthe
radiationheat
transfer
from
the
melt
in
aCzochralski
crystal_{growing}
furnace
in
the
presence of aradiation
shield
The
change
of radiationheat
transfer
effectsthree
basic
_{areas,}
the
_{cooling}
rate ofthe
_{crystal,}
the
bulk
flow
and
the
bulk
temperature
ofthe
melt.The
shieldis
installed
above
the
meltto
inhibit
the
heat
transfer
from
the
meltto
the
cooling
crystal,
allowing
the
crystalto
grow40%
faster.
Radiation
is
the
_{primary}
mode ofheat
transfer
in
the
furnace
wherethe
temperature
to
grow silicon crystalsis
around1773K.
The
governing
equations arethe
threedimensional
Navier
Stokes
equations
in
cylindrical coordinatesfor
laminar
_{flow,}
withthe
_{buoyancy}
term
present,
andthe
_{energy}
equation.The
_{boundary}
conditions at
the
free
surface ofthe
melt are nonlineardue
to
the
presense
ofthe
radiation.Instead
of_{introducing}
the
streamfunction
to
facilitate
the
solution,
the
_{Svanberg}
_{vorticity}
parameter,
w/r,
wasintroduced.
The
_{resulting}
equations
weresolved
_{by}
aniterative
processto
yieldthe
_{velocity}
distribution
in
the
melt.Then
these
equations were usedto
solvethe
_{unsteady}
state
_{energy}
equation.The
radiation viewfactors were calculatedby
using
a program calledCONFACII.
The
temperature
valuesthus
obtained were used
for
the
_{buoyancy}
terms
in
the
Navier
Stokes
INTRODUCTION
In
the
_{manufacturing}
of solid state electronicdevices
andin
solar
cells,
siliconwafers
are_{primarly}
used asthe
base
material.The
wafers areproduced
by
slicing
a solid silicon crystal grownin
a
Czochralski
furnace.
The
crystals are grownin
alow
pressure,
(20
torr)
high
temperature
(1773K)
environment,
so radiationis
the
_{primary}
method ofheat
transfer
in
the
furnace.
The
objectiveof
this
workis
to
showthe
effect of_{introducing}
a radiationshield
to
enhance crystal growth.This
paper willbegin
with aNOMENCLATURE
Cv
specific
heat
_{(J/kgK)}
g
gravitational
acceleration
(cm/sec^)
H
melt
depth
_{(cm)}
p
pressure
_{(N/m2)}
r
radial
coordinate
_{(cm)}
Rcru
crucible
radius_{(cm)}
Rcrys
crystal
radius
(cm)
Riw
spacialthermal
resistance
(1/cm2)
Ri
surface
thermal
resistance
(1/cm
)
T
absolutetemperature
(K)
1cru absolute
temperature
ofthe
crucible(K)
^crys
absolutetemperature
of crystal_{(K)}
u outward radial
_{velocity}
_{(cm/sec)}
v azimuthal
_{velocity}
_{(cm/sec)}
w upward axial
_{velocity}
(cm/
sec)
z axial
coordinate,
measuredfrom
the
cruciblebottom
(cm)
S
_{Svanberg}
vorticity
1/
(cmsec)
a volumetric expansion coefficient
(1/K)
e
_{emissivity}
v
kinematic
viscosity
(m^/sec)
P
density
(gm/cm"5)
o
StefanBoltzmann
constantW/(K^m2)
4*
Stokes
streamf unction(cmJ/sec)
co
vorticity,
8w/8r
8u/3z
(1/sec)
THE
CRYSTAL
GROWTH
IN A
CZOCHRALSKI
FURNACE
This
_{study}
was
done
to
determine
the
effects of a radiationshield
onthe
temperature
and_{velocity}
profiles ofthe
meltin
acrucible
andthe
effect
it
has
onthe
radiationheat
transfer
in
the
furnace.
This
radiation shieldis
located
abovethe
moltensurface
in
afurnace
usedto
grow silicon crystalsfor
integrated
circuits
or solar cells.The
crystalgrowing
process starts_{by}
melting
pure siliconin
a pure quartz cruciblein
an oxygen_{free,}
low
pressure
surroundings._{Then,}
a perfect silicon crystal(either
1:0:0
or1:1:1
_{orientation),}
called aseed,
is
lowered
into
the
melt.
The
temperature
ofthe
meltis
then
lowered
untilthe
solidseed
maintains equilibrium.The
crucible and seed are_{continuously}
rotated
in
opposite_{directions,}
andthe
seedis
then
raised at aspeed of
2.5
to
10
cm perhour
for
a7.5
cmdiameter
crystal.The
seed
and crystaldiameter
is
controlled_{by}
the
temperature
and pullspeed;
asthe
temperature
is
lowered
orthe
pull speedis
reducedthe
crystal will growlarger
and_{conversely,}
the
oppositeconditions will reduce
the
diameter.
The
crystal_{growing}
process consists offour
_{steps,}
1)
meltthe
solidpolysilicon,
2)
growthe
_{seed,}
_{3)}
crystal_{growth,}
and_{4)}
cool
down
to
prevent cracking.The
time
required
to
produce a5
_{kg}
crystal
is
about12.5
hours.
The
breakdown
ofthis
time
periodThe
crystal_{growing}
_{time}
_{for}
_{a}_{10cm}
_{diameter}
crystal_{being}
pulled
at9
cm/hour
is
72%
ofthe
total
time.
The
goal of_{any}
crystal
_{growing}
process
is
to
produce as much material per unittime
as possible.To
achieve
this
_{goal,}
the
_{growing}
time
shouldbe
reduced.
To
reduce
this
_{time,}
the
growth ratehas
to
be
increased.
The
growth
rate
ofthe
crystalis
determined
_{by}
the
rate atwhich
the
silicon
solidifies.
The
heat
offusion
given off_{during}
this
solidificationis
transported
through
the
crystal andinto
the
supercooled
meltjust
underthe
crystal surface.To
increase
the
pull
_{speed,}
this
heat
has
to
be
dissipated
faster.
This
canbe
done
in
two
ways,
1)
the
melttemperature
canbe
lowered
or,
2)
the
crystal
_{surrounding}
temperature
canbe
reduced.The
method ofreducing
the
melttemperature
is
to
lower
the
heater
output.This
cools
the
crucible wall andthen
the
melt.There
is
a point whenthe
meltgets
so
coldthat
it
will startto
freeze
at nucleationsites on
the
crucible wall.This
is
referredto
as wallfreeze.
The
wallfreeze
then
startsto
growtowards
the
center andif
left
to
grow_{long}
enough,
it
will_{actually}
reachthe
crystal.To
savethe
_{crystal,}
it
has
to
be
pulledout,
thus
_{producing}
smallercrystals.
The
other methodfor
_{increasing}
the
pullspeed
is
to
allow
the
crystal
to
lose
moreheat
to
the
surroundings.In
atypical
introduced
between
the
meltsurface
andthe
crystal,
the
heat
transfer
is
reduced_{allowing}
the
crystal
to
coolfaster.
The
goal
ofthis
thesis
is
to
showthat
the
melt surface willlose
heat
at
aslower
rate
andthe
crystal willlose
moreheat
_{by}
radiation
withthe
installation
of aradiation
shield.This
willthen
allowhigher
pullspeeds
andbetter
yields ofthe
crystallized material.The
shield
alsoresults
in
anotherbeneficial
byproduct,
lower
_{energy}
consumption.This
is
achieved_{by}
insulating
the
meltsurface
andthus
_{requiring}
lower
heater
temperature
to
maintainthe
same melttemperature.
HISTORY
OF
VARIOUS WORKS
The
first
work of numericalsolutions
to
flow
andtemperature
profiles
in
aCzochralski
furnace
wasdone
_{by}
Kobayski
andArizumi
in
_{19751 1]}
During
the
late
1970's,
Langlois,
[2
,3,4,5]in
his
four
papers also covered
bulk
flow
parameter of meltin
the
crucible.These
papers present severalimprovements.
The
verticalgrid
sizein
the
numerical scheme was reduced atthe
_{top}
andbottom
to
show moredetail
andto
revealthe
reverseflow
patternin
the
meltjust
under
the
crystal.A
direct
methodto
solvethe
NaiverStokes
equation_{by}
matrix manipulation was presentedto
reducethe
computing
time
overthe
iterative
method.The
last
improvement
wasfactor
that
was notconsidered
_{in any}
paper sofar
is
the
influence
of
the
heat
offusion
from
the
crystalin
the
bulk
flow
patterns.STATEMENT
OF
THE
PROBLEM
The
crystal
_{growing}
system
consists
of six_{elements,}
_{crystal,}
crucible,
melt,
furnace
_{wall,}
_{top}
cover and shield as shownin
Figure
1.
The
furnace
(walls
and_{top}
_{cover)}
is
_{typically}
watercooled and
is
sealedto
prevent
airleakage.
An
oxygenfree
environment
is
provided
_{by}
_{continuously}
_{purging}
argon atlow
pressure
(20
torr).
This
reduces
the
build
up
of silicon monoxidethat
can
cause
problems
asthe
crystal
grows.The
crucibleis
madefrom
ultra purefused
quartz.The
crystalis
started_{by}
a seed ofeither
1:0:0
or1:1:1
orientation.
The
meltis
pure polysilicondoped
withboron
to
achievethe
proper_{resistivity}
for
applicationsin
integrated
circuits or solar cells.The
radiation
shield,
asfar
asthe
author_{knows,}
has
_{only}
been
used on an experimentalbasis.
The
materialfor
the
shield usedwas
puregraphite
that
wasprebaked under vacuum
to
remove contaminations.The
physicalparameters
used arelisted
in Table
1.
We
wishto
determine
the
effects of a radiation shield onthe
cooling
rate ofthe
crystal andthe
temperature
ofthe
moltensilicon.
This
is
done
_{by}
_{comparing}
the
heat
loss
from
the
_{crystal,}
FURNACE
TOP PLATE
CRYSTAL
RADIATION
SHIELD
MELT
w
u
FURNACE
WALL
i
=_{0}
j
=_{0}
f
CRUCIBLE
J
=_{H}
H
Ocru
h
R
i
=_{R}
cru
cru
FIGURE
1
TABLE
1
Melt
density:
2.33
gm/cm3Specific
heat:
9.75
x 104J/kg~K
Kinematic
viscosity:
_{233M2/sec.}
Thermal
diffusivity:
1.4
cm2/sec.
Volumetric
expansion
coeff:
1.41
x10~6/K
Emissivities:
Surface:
0.318
Shield:
0.15
Crystal:
0.2
Wall:
0.40
Furnace
_{Top}
Place:
0.40
Shield
angle:30
Crucible
radius:11.55
cmCrystal
radius:
3.75
cmCrystal
height:
20
cmMelt
depth:
10.91
cmCrucible
rotation rate:1.47
rad/sec.
Crystal
rotation rate: 2.31rad/sec.
(counterrotation)
Crucible
temperature:
1773.0
K
Crystal
temperature:
1685.0
K
Wall
temperature:
900K
MATHEMATICAL
FORMULATION
The
equations
usedto
solve
this
probleminvolved
first
casting
them
in
differential
form
and_{then,}
in
finite
difference
form.
The
finite
difference
form
will consist of aniterative
solution over a
grid
pattern_{containing}
two
types
of_{cells,}
axisand
offaxis
_{elements,}
_{fig}
_{2a,}
b.
The
four
_{governing}
equationsdescribing
this
problem
are;
1)
continuity
equation,
2)
three
dimensional
Navier
Stokes
_{equation,}
_{3)}
conservation of_{energy}
equation with
the
free
melt surface_{energy}
_{boundary}
conditiondescribed
_{by}
radiation network_{equation,}
_{4)}
_{Svanberg}
_{Vorticity}
equation.
The
_{following}
assumptions were made:_{(i)}
a_{steady}
statesystem;
(ii)
noslip
velocity
condition;
(iii)
specifiedtemperatures
on all solid surfaces exceptthe
radiationshield;
(iv)
noheat
generationdue
to
the
heat
offusion
or_{by}
fluid
friction
and(v)
temperature
independent
physical properties.In
the
equationsthat
_{follow,}
the
independent
variables are:T
the
temperature,
u,
w,
and vrespectively,
the
radial,
axial,
androtational
fluid
velocities,
p
the
_{pressure,}
m_{vorticity}
parameter,
and
V
the
Stream
function.
The
constant parameters used_{are,}
kthe
thermal
_{diffusivity,}
P
the
_{density,}
vthe
kinematic
viscosity,
g
the
gravitational acceleration and a
the
volume expansion_{coefficient,}
FIGURE
2a
AXIAL GRID
CELL
(0,Zi)
z3
Rr
The
sevendependent
variables_{(u,}
_{w,}
_{v,}
_{T,}
_{p,w}
,
y)
arefunctions
ofr,
z,
andtime.
_{They}
areindependent
with9
the
angulardirection.
The
rotational
_{velocity}
is
nonzero
andthe
pressureterm
will notbe
solved
for.
Presented
in
functional
_{form,}
the
dependent
variables
are:u =
u(r,
z,
t)
w =
w(r,
z,
t)
v =
v(r,
z,
t)
^
0
T
=T(r,
z,
t)
w =
(r,
z,
t)
=
(r,
z,
t)
The
_{first governing}
equationis
the
_{continuity}
_{equation,}
Id
_{dw}
The
conservation of momentumfor
the
fluid
is
described
_{by}
the
wellknown
NaiverStokes
equation.
The
NaiverStokes
equationin
cylindrical coordinates are simplified
_{using}
the
_{Boussinesq}
approximation.
The
_{Boussinesq}
approximation assumesthat
_{compressibility}
may
be
neglectedbut
the
thermal
expansion_{may}
not andthis
_{only}
_9_
30
modifies
the
gravitationforce.
With
 =0,
the
Naiver
Stokes
equations
become
2
,v 1
_{9p}
, ,2 u
9u
,9u
,9u
TT +
U+
W9w
9w
3w
1_{3P}
o3t
+
U9r+Wi7
=
t+W
w+go,
(T_,
3z~~Iz
+
vVw+
g^(TT
_{)}
3
9v
9v
.9v
, uv2
v9T
+
U7
+
w^
+
=
_{w}
vv^
4
where,
V1 = a'/ar2
+
(l/r)3/3r
+
a2/az2 .The energy
equation
is
f
+^^(ruT)
+
^f
(WT)
=KV2T
5
The
five
dependent
_{variables,}
_{T,}
_{u,}
_{v,}
_{w,}
and_{p,}
are coupledin
the
five
equations.
The
fluid
properties
are assumedto
be
constant
overthe
range
ofthe
dependent
variables.To
solve
equations
2
and_{3,}
the
radialand
axialvelocities
are expressed
in
terms
ofthe
Stokes
stream_{function,}
Y
,u =
(1/r)
3^/3
2, w =
(l/r)
3
^/3r
This
definition
satisfiesthe
_{continuity}
equation,
equation1.
Introducing
the
vorticity
parameter co _{,}defined
_{by,}
co =
_{3w/3r}
3u/9Z
The
streamfunction
and
_{vorticity}
are related_{by,}
a/
1
3^
\
1
_{92^}
)r\r
9r
_{f}
r9z 2
*
~u ,
The
pressure
term
in
the
NaiverStokes
equation was eliminated_{by}
cross
differentiation
ofthe
radial and axial components_{yielding}
(See
appendixI
for
_{derivation)}
&
+
jf
(.)
?
^f
()
+
*2
.I
+
**,,
*j
The
evaluation of coin
equation was notdone
directly.
A
newvariable called
the
_{Svanberg Vorticity}
_{term,}
_{S,}
wasintroduced
asproposed
_{by}
Langious[4].
It
is
S
= _{co/r}10
This
variableis
constantfor
anincompressible
fluid
whererotational
symmetry
is
present.When
the
substitutionfor
10is
made
in
equation(9)
and simplifiedit
becomes
3S.
1
9
_{(rUS)}
_{._!_}(wS)
3
/
tf\
It
r T"^
az
\
r'4/
11
r
9r
r 3r[
r3r
^
S>J
where
the
angular momentumis
defined
_{by}
Q
= _{rv}From
this
point
_{onwards,}
we will usethe
set ofequations
_{4,}
5,
6,
7, 8, 9,
10,
and11
instead
of
equations
_{1,}
_{2, 3,}
4
and5.
In
orderto
formulate
the
_{boundary}
conditionsthe
_{following}
assumptions
were used.1.
The
temperatures
ofthe
_{crystal,}
the
crucible andthe
walland
the
_{top}
cover ofthe
furnace
are uniform.2.
There
is
noheat
transfer
from
the
crucibleto
the
_{melt,}
so
the
temperature
ofthe
melt_{touching}
the
crucible andthe
crucible arethe
same.3.
The
melt canbe
treated
as an opaque material.4.
At
the
melt_{surface,}
radiationis
the
_{only}
mode ofheat
transfer.
The
_{boundary}
conditionsin
differential
form
are asfollows
[1]:
AXIS,
r =_{0}
12a
V
=0,
co =0,
v =_{0,}
9T/3r
=_{0}
CRUCIBLE
_{WALL,}
r =_{R}
cru0,
w^,
V=R
_{n}
,T=T
dr
_{cru}
cru' Llcru
12c
CRYSTAL
_{FACE,}
z =H,
R
Rcry
4
0,
v  _{v}=R
fi
,T =T
_{12d}
dr
crys
crys'crys
FREE MELT
_{SURFACE,}
z =H,
_{Rcry}
<
r<
_{Rcru}
3t
a9T
3T
9T
T=0,
w=0,
=0,
=12e
melt radThe
problem as expressedin
the
differential
form
cannotbe
solveddirectly.
The
common approachto
a solutionis
to
express eachequation
in
atime
dependent
difference
form.
These
equations canthen
be
iterated
_{by}
_{marching}
along
using
smalltime
steps.Since
the
equations
arecoupled,
each equation mustbe
processed at eachtime
_{step}
and usedfor
approximate valuesfor
the
next step.Equations
_{4,}
_{5,}
_{6,}
_{7,}
_{8,}
and11
will nowbe
recastin
finite
difference
form.
The
rotational_{velocity}
equation4
becomes
Tt
ui,J
rrS1.J
'z.
r.
i,.)
r.jj2
13
The
energy
equation5
becomes
The
radial and axialvelocities
in
equation6
are.
AV.
. .AY.
,
Ui,j
"
(r.}
Az
'Wi,j
"(r
}
Ar.
1
_{J}
i iThe
_{vorticity}
_{parameter,}
as
defined
_{by}
equation7
becomes
Aw.
.Au.
co =
X?J
_LaJ
i,j
Ar.
Az.
1
_{J}
.
i
+ v
=^i>J
+^L_
Af
1_
_A_ ,2
."1r.
Ar.
r.
Ar.
I
r.
Ar.
i
i.j
J
l l l l  l l 'd
Az2
16
The
vorticity
parameter and streamfunction
are_{related,}
andequation
8
becomes
a i
AT.
.2
AfA^j^L)
.^.AA_
17
Ar.
lr.
Ar.
'
+
r.
2
_{_a)i}
i
17
ii
i
i
Az
'JEquation
11
for
the
_{Svanberg}
_{vorticity}
parameterbecomes
AS,
._{Q}
2
^2i+
7^(r u
T
_{)}
+A(w
S
_{)}
+ AiJ
)
At
rAr
vri
_{i,j}
i,j;Az
lvi,j
bijj
+_{Az.}
l
_{k}
11
_{J}
jr.
Before
the
finite
difference
equations canbe
solved,
a gridpattern
had
to
be
developed.
There
aretwo
different
types
of gridcells
that
modelthe
fluid
in
the
_{melt,}
onaxis and offaxis cells.This
wasrequired
to
handle
terms
of1/r
that
wouldbe
undefined atthe
axis
in
the
_{Svanberg}
equation.
_{So,}
an_{entirely}
separate set ofintegral
equations
had
to
be
developed
asdescribed
in
appendix2.
The
cells
_{bounding}
the
_{top}
andbottom
surface are_{only}
onehalf
in
height.
This
putsthe
grid
coordinate onthe
boundary.
The
interior
pointsin
the
_{energy}
and rotational equations were solvedusing
centraldifference
method and another set of equations wererequired
to
evaluatethe
_{boundary}
points_{using}
eitherforward
orbackward
difference
method.As
canbe
seen several sets ofequations were required
to
solvethis
problem.These
methods wereemployed
for
eachterm
in
equations_{13,}
_{14,}
_{15,}
_{16,}
17
and18.
These
equations are coupled with eachother,
so_{they}
had
to
be
solved as a set of equations rather
than
separate equations.The
finite
difference
form
ofthe
_{boundary}
conditionsis
presentedMETHOD OF
SOLUTION
A
single
solution
can nowbe
found
_{by}
_{solving}
the
difference
form
of
the
equations
for
each particular set of conditions.The
basic
approach wasto
assume
initial
valuesfor
the
independent
variables,
t,
u,
v,
w,
co, and
Y
.Then
adelta
that
represents achange
in
these
variables wouldbe
calculatedto
improve
the
initial
values.Now
_{using}
the
improved
values,
a newdelta
is
calculated and
this
process wasrepeated
untilthe
delta
changebecame
_{very}
small and couldbe
neglected.There
were six sectionsto
the
solution as outlinedbelow.
1.
Dimension
andinitialize
the
parameters(or
use valuesfrom
last
run)
2.
Calculate
new radial and axial_{velocity}
componentsCalculate:
a.
A S
b.
S
from
A S
c. co
from A S
d.
4*_{from A co,}
boundary
conditionse. u,cofromA
Y
3.
Calculate
newtemperature
distribution
3.
_{(cont'd)}
b.
AT
_{conduction,}
Surface
c.
_{AT radiation,}
_{Surface}
1.
Viewfactor
a.
grid
locations
_{(surface,}
wall,
shield)
b.
Confac
II
d
_{Tnew}
=Told
+
_{AT/}
At
4.
Calculate
newrotational
_{velocity}
a.
_{Boundary}
condition(rotation
of crystal+
b.
_{Velocity}
profile(axial
&
_{radial)}
c
_{vnew}
"vold
+
Av/At
5.
Check
for
the
_{vorticity}
parameter_{(feedback)}
a.
Calculated
from
velocities and averaged with_{vorticity}
calculated
from
S.
6.
Check
for
steadystateto
_{stop}
calculation.Now
each ofthese
steps
will
be
discussed
in
more_{detail,}
starting
with
the
initialization
ofthe
independent
variables.This
was
done
in
two
methods
_{1)}
initial
guess or_{2)}
usethe
valuesfrom
the
previous
run.In
the
first
case,
S.Y
, co,u,
and w wereset
to
zeroand
estimated
values
for
T
and w were calculated.Estimated
valuesfor
T
and wwere
obtained_{by}
a proportionalaverage of
the
fixed
_{boundary}
conditions.
In
the
second_{case,}
the
values of all
the
independent
variables
were stored and usedto
initiate
the
next run.Svanberg
Vorticity
Equation
The
next_{step}
is
to
calculatethe
new values of axial andradial velocities via
S
andV
.The
new value ofS
is
obtained_{by}
calculating
eachterm
of equation18
andsumming,
then
_{multiplying}
by
At.
Equation
18
canbe
symbolicly
writtenin
the
_{following}
form.
AS.
.(AS.
.)(AS.
.)(AS
)
1J
=1>1
ra i,jza V i,j;
C
19
At
At
At
At
(AS
,)b
AS.
.)(AS.
.)^
b
+
hLL
rv_{+}
LJ_
_{zv}The
physical_{meaning}
of
each
ofthe
terms
in
equation19
canbe
related_{by}
the
subscripts
asfollows:
radial advection(ra),
axial advection
_{(za),}
_{coupling}
term
_{(c), bouyancy}
term
_{(b),}
radialviscious
_{(rv),}
axial
viscious
(zv).
Each
term
in
equation
(19)
is
integrated
over each cellto
obtain
S
_{by}
the
_{following}
scheme.One
ofthe
terms
willbe
expanded(for
moredetail
seeappendix
2).
V
i^ii
=f
f
_A_ __(ruS)
dv
At
"71
I f
r
3r
20
V
ttA
z (r2r2)
=_{2ir}
rA
rAz
21
(AS.
_{J}
2+
ri"
k^*
[
_{H}
J
7V
^T2lT
/
M
dZ
22
ri+
(A
_{S.}
.)ijir
<ri
<u>v <s>v
"
r
_{<u>i+}
<s>
The
evaluation ofthe
weighted
average over each cell(Figure
2)
designated
_{by}
_{<f>+}
where
f
is
ageneric
variable andis
defined
as;
<f>
dz
j
J
z<0
z1
^
/*rf
_{(r,z+)}
<uf>
R
_{<v>}
_{>}
"V
<v2+><f2+>
where,
r+ = _{r}
_{+}
_{A}
_{r/2} z+ = _{z}_{+}
_{A}
_{z/2}For
example<f>R+
"2A7
(fr+,z
~
fr+,z)
Az
To
calculate<f>r+,z+
=fr
+
A
r/2 z+
A
z/2(Point
A in Figure
3)
=
(fi.j
+
fi+l,j
+
fi,j+l,
+
fi",j+l)/4
24
"
7te+
_{I}
rf
<r'z+>dr
25
ri26
27
<f>r+z
=fr
+
A
r/2, z
A
z/2(Point
B
Figure
_{3)}
(fi,j
+
fl+l,J
+
_{fi,jl}
+
_{fil}
_{ji)/4}A
_{r+,}
z+C
r+,
z(i,j)
B
_{r+,}
zFIGURE
3
r+ and
zGrid
Locations
Then
_{returning}
to
equation_{(27),}
equationfor
pointC
in Figure
(3)
becomes
<f>r+
=(fj.j+1, fi,jl,
fi+l,j+l
fi+l,jl)M
or
the
average ofthe
four
exterior points ratherthan
_{simply}
the
average
between
the
_{(i,j)}
and(i+l,j)
terms.
Due
to
the
fact
that
someterms
of equation18
have
a1/r
term,
two
types
of grid cells were used as seenin
Figure
2
to
evaluate values at r =
_{0}
Once
has
been
_{calculated,}
_{the}
new
value ofS
is
_{simply}
arrived
at_{by}
equation
Snew
=Sold
+/"a_s\
a
t
W
Using
the
new
value
of_{S,}
cois
then
calculated
_{by}
equation10.
Equation
8
wasthen
usedto
be
solved
for
the
Y
valuesfor
each
point
in
the
interior.
This
is
done
_{directly}
presented
in
_{[3]}
rather
than
iterative
method.
This
method
uses matrixmanipulation,
in
the
_{following}
_{simplified}
_{sequence.}
1.
Form
a matrix ofgrid
locations.
2.
Calculate
eigenvalues
for
the
above matrix_{using}
the
subroutine
TQL2
ref.
[9].
3.
Transform
axesto
getdiagonal
matrix_{using}
the
QL
algorithm.
4.
Solve
transformed
equation.5.
Retransform
solutionfor
actual values.A
brief
_{summary}
ofthe
iterative
method willbe
given
here
[2].
This
solution ofthe
streamfunction
entails
_{evaluating}
the
equation
(7).
This
is
done
_{by}
_{applying}
successive
overrelaxation
to
obtainthe
relationbetween
successiveiterates1*'
and
where
/3
is
arelaxation
factor.
As
presentin
[3],
the
direct
method
is
about5
times
faster
than
the
iterative
method andthere
ROTATIONAL
EQUATION
Equation
13
was usedto
calculate
the
rotational
velocity
(v).
An
initial
estimate
ofthe
rotational
_{velocity}
was neededbefore
the
other velocities couldbe
calculated.
These
estimates wereobtained
_{by}
_{iterating}
the
static
_{term,}
[V]^term
of equation3.
To
speed
up
convergence,
aproportional
average
for
the
interior
points was calculated
from
the
_{boundary}
values.It
turned
outthat
a
_{relatively}
good
estimate ofthe
rotational_{velocity}
was requiredto
reduce_{instability}
in
the
main program.These
estimated valueswere
fed
into
the
main programto
obtainthe
radial and axialvelocities.
There
wassome
_{difficulty}
in
_{evaluating}
the
dynamic
terms
_{(velocity}
coupledterms)
nextto
the
boundaries.
To
overcomethis
problemthe
dynamic
terms
nextto
the
boundaries
were setto
zero so
the
rest ofthe
problem couldbe
solved._{Updating}
the
rotation velocities required
two
arrays,
an old and a new.The
newvalue
was
obtained_{by}
_{multiplying}
Av/At
_{by}
A
t
andthen
_{adding}
to
the
old value.ENERGY
EQUATION
The
energy
equation,
equation14
wasthe
third
major equationto
be
solved.It
is
I+iA(rumfz(wn=K72r'
it
rbriZ
where
<is
the
thermal
_{diffusivity.}
This
equationwas
handled
in
the
same
manner asthe
rotationalequation
in
terms
of
_{establishing}
an estimate ofinitial
profileand
_{updating}
the
new
profile.This
equation was processedin
the
same
manner asthe
rotational
equation
was.The
initial
estimatewas
a
proportional
average
and
the
_{updating}
ofthe
new profile wasdone
in
the
same
way.The
_{velocity}
coupledterms
were not setto
zero as
in
the
case
ofthe
rotational
equations.The
_{boundary}
conditions
were more_{complicated,}
sincethe
convection andradiation
components
wereto
be
combined.Once
the
interior
values ofV
are_{obtained,}
the
_{boundary}
conditions can
be
calculated._{A summary}
ofthe
_{boundary}
conditionsadapted
for
the
finite
difference
schemeis
givenbelow
[2].
AXIS,
r =_{0}
*
=0,
co =_{0,}
_{v} =0,
AT/Ar
=_{0}
3Aa
CRUCIBLE
BOTTOM,
z =_{0}
_2
(
Az
)
*
T vx'""''"cru
"cru
^
=0,
co =r^JZ
*
(r,Az),
v_{=Rcruncru>}
T
=
T^
34b
CRUCIBLE
_{WALL,}
r =Rcru
*
=_{'}
w =r
U7T7cru
y
I
R
Ar_{,} z)
, v =I
cru/
2 ,.,
/
\
" _{=}_{R}
Q
,T
=T
34C
CRYSTAL
_{FACE,}
z =H,
R
<
_{Rcry}
*
=0,
ur
2
TTT?
*
(r'
_{"A*),}
v_{E^a^T}
=_{T}
_{34d}
:rys
crys'crys
FREE MELT
_{SURFACE,}
z =H,
_{Rcry}
<
r
<
_{Rcru}
*
=_{}
_{}o,
Av/Az
o,
AT/At
.ATcon
v.
_{ATred.}
34e
RADIANT ENERGY
BALANCE
AT THE MELT SURFACE
There
arethree
majorconsiderations
in
the
radiation_{boundary}
conditions at
the
melt
_{surface,}
the
netheat
loss
from
the
surface,
the
heat
transfer
from
the
_{cooling}
_{crystal,}
andthe
temperature
profile of
the
melt surface.These
weredetermined
_{by}
considering
a systems with and without a radiation shield.
To
facilitate
this
evaluation,
the
problem wasbroken
into
three
_{areas,}
surfaceequilibrium,
radiation viewfactors andthe
radiation networkequations.
Melt
Surface Equilibrium
The
melt surface elements are atthermal
equilibrium whentheir
changein
internal
energy
is
zero.The
changein
internal
energy, A
uin
the
finite
difference
form is
AU
mocV
_{AT}
r
v
l.J.
35
A*
_{At}
where
p
is
the
_{density,}
_{cv}
is
the
specific_{heat,}
V
is
the
volume.For
equilibrium,
ATij/
A
t
mustbe
equalto
zero.This
change
in
temperature ofthe
surface elementis
zero whenthe
convective
heat
transfer
Qconv
from
the
meltequals
the
The
quantity
_{Qconv}
in
the
absence of other mechanisms ofheat
transfer
from
agiven
surface
element causes atemperature
change of
the
element.The
rate ofthis
temperature
changeis
designated
_{by}
(AT/At)conv.
In
the
presence of radiativeheat
transfer
from
the
surface
ofthe
_{element,}
the
temperature
ofthe
element
changes
at a rate(AT/At)rad.
The
difference
between
these
two
ratesis
responsible
for
the
net changein
the
internal
energy
ofthe
element.Thus
_{rewritting}
equation35
AU
_
Q
q
At
~
conv
_{rad}
36
and
then
_{solving}
for
AT.
. ,X>1
= A(q
_Q
37
At
pcV
conv radThe
convectionterm
of equation36
canbe
calculated_{by}
the
energy
equation,
soto
solvethis
_{equation,}
an expressionfor
the
Radiation Network
Equations
To
calculatethe
heat
loss
due
to
_{radiation,}
an electricalequivalent
circuit ofthe
radiationheat
flow
was constructed(see
Figures
4
and5).
The
wall,
top
plate,
andthe
crystal are atknown
temperatures
andthe
N
surface elementtemperatures
variedfrom
iteration
to
iteration.
The
iterations
were neededto
solvethe
_{governing}
momentum equations untilthe
heat
transfer
_{by}
radiation equaled
the
heat
loss
due
to
conduction.
The
generalnodal equation
has
the
form
V?
+
Vfi
+
fsft
,*i
Riv
Ris
where,
E,
=a
_{T\}
RL
=(lC^/CAj
Vw
=lMlFln
o
_{4}
_{2}
a
=_{5.67}
x10
W/
(K
m)
.In
the
aboveequation,
E^
is
the
_{blackbody}
emissirepower,
J
is
the
radiosity,
_{Ri}
is
the
surfaceresistance,
Rj_x
is
the
spacial
resistance,
E
is
the
emissivity,
Fi_n
is
the
radiation
viewfactor,
A
is
the
surface area and ais
the
Ste
fanBo
ltzman
constant.
Equation
38
is
then
putin
matrixform
andsolved.
This
This
equationis
broken
into
_{3 matrixes,}
A
the
resistanceelements,
J
the
_{radiosity}
unknowns andB
the
emissive power ofthe
elements.The
areas
usedfor
_{calculating}
the
resistance values were ofthe
complete
_{element,}
for
_{example,}
a_{ring}
for
the
surfaceelements,
orthe
total
surface
area ofthe
crystal.The
radiation viewfactor's
will
be
discussed
in
the
next section.The
system of equationsin
matrix
form
was;
HW=M
39
J
=Jl
J2
J
nJ
wcry
ftp
sh
out
sh. in
B
=Hi
R2
\nl
nK
I
R
DW W
^cry 
cry
\
0
I
Ice
to
I
Oh
I
Oh Eh
I
>
I Oh Eh
I PL, Eh
Eh
I
J3
Eh
H Ph
H
O, b,
Eh 1
&h ~ _{>>}
1 1 (h
>. o
Jh it
3
I
Ebftp
= .084,084
sec/
in
Ji
~(5.058
5.660)
(.0486
E
= .084Rftp
(.0016)
Jftp
=.715,
)crys=
5.207
Rcry
(.017)
UNITS
1
cm'
"crysftp
(.0061)
Rftpshout
(007)
Jcry
=_{1.94}
"crysh
(.0141)
Jshout
2.07
Rsh
=(.0109)
Tsh
=1334.
K
Jshin
=6.51
Rsh
(.0109)
Rshw
(.0333!
Jw
=2.78>
(.0038)
sh
(.0268
.061)
fjl
Ebw
 084Ji
=(7.82
7.7)
JE
cal
sec
in
/Y
18
W
\
V
cm
/
Rj.w
(.1335
.0726),
Ri
=(.0486
.1155)
Z
Ebj
=The
resistance matrix was17
x17
for
the
case with shield.This
set
of simultaneous equations werethen
solvedfor
the
_{radiosity}
values
_{by}
_{using}
a subroutine calledLEQ2S
from
the
Statistical
andMathematics
library.
This
routine
allowedthe
inverse
matrix of(A)
to
be
saved
for future
calculations.The
next
_{step}
wasto
calculatethe
radiantheat
loss
_{Qracj}
from
the
surface elements.Heat
loss
in
the
presence of radiationis
:(EbsVA
<>rd
* ~_{40}
[(1ej/e,)]
.Using
equation37
_{(Qconv}
=)
and equation40
(AT.
.)Q
,(E,

_{J}
)
i^L_rad=^l
^
1
_{41}
At
PCvV[(le.)/E.)
(PC
%]
11 v i.
Equation
37
canbe
rewrittenin
the
_{following}
form.
V
At
_{/ronv}
Vtc
AT,.
/AT
.\
/ATI
42
il
lil
lAtlrad
At
/_{conv}
The
newtemperature
atthe
surfacebecomes
AT.
T
=_{T}
+_{(}
^?J
_{)}
At
43
new old
At
.A
_{steady}
state was assumedto
have
been
reached when rate ofchange
oftemperature
wasless
than
,05K/sec.This
rate of changeproduced
a changein
the
final
temperature
values ofless
than
1K
when
allowed
to
run200
iterations.
The
resistance
valuesthat
make
_{up}
the
network
matrix
mustbe
calculated
and
this
requires
the
evaluation
of
the
radiation
viewfactors
between
the
_{radiating}
surfaces.
View Factors
The
radiation
viewfactors
between
the
free
surface
ofthe
melt,
crystal,
wall,
shield,
andfurnace
_{top had}
to
be
obtained.A
computer
_{program,}
Confac
II
_{[7]}
was usedto
evaluatethese
viewfactors.
Consider
_{any}
two
elemental areasA
andB.
In
generalarea
A
_{may}
see allof
areaB
or part of area_{B,}
or none ofit.
Which
one ofthese
three
possibilitesis
applicableto
a given pairof
elemental
_{surfaces,}
depends
upon whetherthe
rays_{connecting}
the
edges of surfaces
A
andB
areblocked
_{by}
another surface such asthe
crystal.Radiant
energy
received_{by}
surface#2
Radiation
viewfactor_{?\2}
=T.
Radiant
_{energy}
_{leaving}
surface#1
The
surfacetoshield viewfactors were calculated_{by}
determining
the
area"seen"