with a great specific load and heat energy, with ar-. /. Ibitrary shaped heating surface. Thcrmopcnctrators arc radiaUy symmetric and in some cases ring-shaped,.
253 Third International Conference on Inver_ Design Concepts and Optimization in Engineering (ICIDES-III). Editor: G.S. Dulikravich. Washinzton D.C.. Oztobcr 23-25. 1991.
Sc.icnces
-92-i3947
F OPTIMIZATION
OF
IN
THE
THE
HEATING
CONTACT
SURFACE
MELTING
SHAPE
PROBLEM
. Department
of Appplied
mathematics,
Kazan
Fomin
"_tY_b(.'_. _I!"
University,
K-g-fan, 420008,
A.
Sergei State
. / USSR
t'
";
and Shangmo Department
of Power
Engineering,
Huazhong
Cheng
University
/O
_.
of Science
and Technology,
}-i
China
"
Wuhan,
"3
430074,
,.. :
..... ..
.
heat
source
with
ABSTRACT The work an arbitrary tions
is devoted
shaped
to the theoretical
isothermal
are transformed
heating
analysis
surface.After
to thc convenient
are used
for numerical
prediction
constant
and for tcmpcraturc
of contact
dependent
shape
physical
by the migrating
the substantiated
for engineering
of optimal
melting
calculations
of the heating
properties
simplification
the governing
relationships.
Analytical
surface.
Problem
equa-
solutions
is invcstigctcd
for the
of the melt.
1. INTRODUCTION
Melting
of
technological to contact
solids
processes. melting
contact
Thcse
problem
hcating
surface
melting
material).
contact
melting
melting
its way through
[3], nuclear
by
and
against
This situation
technology
ling is commonly
uscd
traditional
three
facts
of excavation
source
two groups.
drilling
rotary
new method
drilling•
(rock
The
fracturing,
in an enclosure
in mining
n:o_t considerable debris
removal
material
lies on the oF the
a moving
heat
source
[21], geology
[l l, 17, 19]. Thermal
glaciers.
Boring
It has
advantage
and
22] devoted
of the weight
fields as welding
engineering.
and
[1, 16, 22] and in other
involves
[4, 6, 18, 20] and glaciers of boring
natural
[2, 4, 12-14,
the forcc
arises in such
method
numero,,,
the melting
of applications
This situation
in
works
(for instance,
solid melts
group
of rocks
place
In one group
force
an unfixcd
solid.
takes
in thc previous
external
[8]. Another
is a relatively
with
This work
arises when
in industry
surface
now as the most effective
comparison
in a single integrated
into
it by some
[9, 10] thermal
recognized
heating
are cnumeratcd
the surrounding
soil by thermopenctrators
major
processes
a
and are divided
pressed
devices
with
rocks,
some
and
advantages
of thcrmodrilling
wall stabilization)
dril-
sands
in
is that
are accomplished
operation.
is dcvotcd
with an arbitrary
to the theoretical shaped
isothermal
analysis heating
of" the contact
melting
process
by the moving
hcating
surface.
2. ANALYSIS 2.1.
The physical Obviously
particular Ibitrary L_
model and governing
every
technological
case of thcrmodrilling, shaped
heating
surface.
equations.
process
where
it is contact Thcrmopcnctrators
contact
melting
melting
with a great
arc radiaUy
occurs
has its own specific
specific
symmetric
load
and
character.
heat energy,
and in some cases
In
with ar- / _J
ring-shaped,
25% Third International Conference on Inverse Design Concepts and Optimmation in Engineering OCIDES-HB. Editor: G.S. Dulikravich. Washinmon D.C.. October 23-25.1991. For toroidal, of the contact etrating
with
into
central
the melting
solid
is separated
tween
surface
the heating
defined
two-dimensional. pendent
-2
surface
and
layer
properties
the effect
in the contact
and assumptions
in the molten
melting
enumerated
layer can be written
flow
be-
interface
is laminar
and
with a temperature
de-
that
ve-
liquid
of rocking.
F. The
the solid-liquid
[6, 15] indicated
initiation
force
the thin channcl
t_, melt
Newtonian
results
external
along
that
at temperature
Experimental
transfer
of applied flowing
_ _. It is assumed
value V soon ahcr
model
core sample [6]. A schcmatic diagram I in Fig.l. Axisymmctric heater -1 is pen-
of melt -3,
prcciscly
1
the
to be incompressibIe
density).
heat and mass
transfcr
V under a layer
intcrfacc
occurs
is assumed
to the physical
tions of heat and mass
melting
constant
of quasi-steady
According
the velocity
the solid with
(except
its quasi-steady,
assumption
with
from
_ _ and solid-Iiquid
Molten
physical
locity attains
and
a large forming extracting melting for the thermodrilIing conditions is shown
thermopenetrator
is a sharply
hole for
Sciences
the heat source
This fact justif']cs the next
problem.
above
the governing
differential
equa-
as follows:
div_ = 0 pL(E"
_')_=
CLPL(_ where
uid properties
lindrical
for further
coordinates
dimensionless
C(t=-t
--,
)
CLPL(t=
t
All the quantities
(1), the analysis drilling
terion
rock
10 -3 is the ratio
Re--
the molten After
10 -6-
layer,
Pccklet
neglecting
mass transfer
of the characteristic
number
terms
in the molten
of 0(Kh,
Ou + _
H2dP -- 8 (II--c2Ou ) ds 8r/ otI
_ are i_.ticated
method
cy-
in Fig. 1.
in a prcliminary
analysis
= Re=Vp
d/PL
k"
,
CL
=pLgd/W
Br--
Each
Re,
Br)
= 0
of the dimensionless
numbers
Kh--
force 10 _
10 -3-
number
the governing
Stc_
and
external
1-- 10; K a, K,--
nondimensiona]
of thermal
10 -_ physically
the viscous
(2) has equations
conditions
size of the heating
of the melt represents
numbers
of the governing
for the concrete
and characteristic
mass 10 -s-
(2)
the simplification
parameter
layer
10-- 100; Stefan Kg,
S and
surfacc:
'
_ional
layer will take the following
1 _ R" gs (R'Hu)
L
number Pc--
C =--, _
, K
Dimensionless
of the molten
10-4; Brinkman
CL, PL, 2L liq-
fixed to the he.ring
coordinates
c
to substantiate
of these non-dimcn
thickness
equation;
pressure
[7]:
in the Nomenclature
out.
transfer
velocity,
are standard.
, K
,t L
In order
was carried
of characteristic
Ks--
number
'
here are defined
the liquids
are
of coordinates
arc gcncrated
K a-.'=
K
of the values
in heat
tL
2 _
meaning.
terms
layer
0,taLV =_\pLWct( _--==. = )'/'
K h
)
physical
of ice and
the ratio
,
, Pc=
--
g, p,
and using the similarity
PeK aK "
obvious
boundary
form
L WP
stresses;
to use two systems
and numbers
Ste--
Br=
an exact
of internal
the rest of the symbols
analysis
parameters
(])
the dissipative
(1) to non-dimensional
Pc --Vd a
+ q)
(r, z) and local orthogonal
Transforming the main
¢ represents
in Nomcnclaturc;
It is convenient
+ divT
part of the tensor
respectively; defined
pj.Vp
• _TtL) = div().L'_]tL)
T is the deviator
and temperature
G--
represents
surface force;
dissipation
d; cri-
Reynolds of heat in
1. equations
of heat
and
form:
(3) (4)
J
255 Third International Conference on Inverse Design Concepts and Optimtzauon in Engineering (ICIDES-IIII. Editor: G.S. Dulikravich. Washington D.C.. October 23-25. 1991 PcCH(Hu,
where
r/=
.m L,
=2L/Z
,
+ u"vr/
P=p/w
,
o_/
,
u
-
F h of heating
, H=h/Khd
v pL =--, p Vp.
, u
'
Vp,
_ 6; h = h(s) -thickness
surface
Fh; v,, v, -longitude
and transverse
(5)1
c,r/
S=s/d
v pL R=r/d
i'h--t
nondimensionalizcd
m ) is determined
by
after
their
Here
t_
is initial
ring-shaped
temperature
penetrator
with
0(Kh) it is possible
to ignore
z) as the Cartesian
coordinates;
The boundary
conditions
At the heating
u Oh=(t
h-t
of
0,= (t,-t_.
layer
normal
0.
The
assumption
t
In
temperature
properties
normal
to
of liquid
CL,
tin;reference
temperature
(
condition,
the
behaviour
equation since the
form
(3)
v=0
thickness
of heat and mass
to the continuous
heating
corresponds
transfer
surface
to
of the liquid
the
film is of
and to consider
without
(r,
hole.
are following
0=0h;
(7)
); t h is the unknown
=0;
temperature
on _ h
_ _(r/= 1)
u_=
dR ---ds,
0=0
(8)
PccQ+ (
in the exit points
The function
)/Ste
]; Q=(--_n
)IE.;
of the solid material,n
(9)
is an external
rclativcly
tomoltcn
molten
is [4]
1-
2 2 R 2 -R
(10) s= s: then
heat
in the molten
condition
depends
and mass layer.
in this case s t = 0 is the critical
transfer
couses
This condition
the equality
with the defined
point
where
of external accuracy
which
is the same
u, = 0
force
F
of 0(K h) in
(1 I)
(9) is the non-dimensional
on the temperature
problem
material
s= s_ and s= s 2
2 f_2 RPdR I _,
Q in Stefan
_ _ Iffs value
layer
= 0
stresses
form
tion of the heat transfer weldpool
= P(s2)
ofthe
of quasi-stationary
of internal
non-dimensional
k
all the physical
the intcrnal
Ill
In this case
symmetric
v= 1 it is only one exit point
ds=
rounding
along
to _ _.
Since when
surface
material. hole.
v = 1 corresponds
interface
P(sl)
and the force
line
of gcncrating
(6)
/ (tin--t**), t, is a temperature
For the pressure
dP/
central
the axially
H1 0r/l_._o0 =[ where
mersurcd
layer;
.111
of Stcfans
melting
a large
=0;
)/(th--
u
L
_ h(r/= 0)
--u
At the solid-liquid
C=CL/C
)
in dimensionless
surface
laycr
CL,ZL,/Z L at
nondimensionalization
_h--tm=PeKc(t_--t
,
m ,R=R(s)-cquation t h -- tm
in the molten
values
_
t L -- t 6==
m
2L, PI. are
, tZ=pL/p
of the molten
velocities
Sciences
distribution
density in the solid
as the problem
of heat flux
to the solid from
and is obtained
of temperature
distribution
from
the soluin the sur-
[21]:
1
256 Third International Conference on Inverse Design Concepts and Optimization in Engineering (IGIDES-III). Editor: G.S. Dulikravich. Washington D.C.. October 23-25. 1991,
' +--'
-- Pe-_0 [E.
+
c_Z:
=1;
lim
R
aK
at_
0
=0;
0,IEe
Sciences
)=0;
(12)
=0_;
(R,Z)=(r,z)/d;
(13)
R2 +Zl_o_
0r solvcd 2.2.
lion
is temperature
distribution
in [4, 18] numerically Analytical
by the boundary
that
of one
variable
conditions
independent
coordinates
boundary when
on _[ m; z =-c_,O,=
0,--
eric(x)
According
1, in infinity: Pc
:.
= _
z
)du,
2
.
(12) (1 3) was
_.2dR _
the fact
and pressure
u
of equations distribution
_ R'+_ -- R;+_ (v+ 1)HDR'
=
3
l)R"
_s[
_P= r/: o is a critical
/z point
equasion
as a func-
_[ _ is parabola.
(12) has the Following
In
2) with boundary solution
l (14)
v = 0
"
1
U
v=
du
x> 0
on 3-"= is
I (15)
v= 0 that
the distancc
bctwcen
_ h and X-"_ is thc value
2
(16)
v=l v = 0
(3) and
in the molten
(R'+,
p = +____ v f '_ R'+_ - R'+'" , R'H3D
D =-of ' (r/"_ r/°) r/dr/,
F_of
(4) with invoked Laver
boundary
conditions
--
(7), (8) and
are obtained
[_r/o -- r/ _o # dr/
I (v+
curve
solution
of 0(K h)
.
{_ El(--ce ), 4-a-a_erfc(7),
/
an analytical
2
fee El(--ce ), / _" x_ncecrfc(_),
into account
"transformation
u
L
Problem
R and Z by R = z_z, Z = 0.5(a2-z
_m )'
Ei( - x) =
(15) with the accuracy
Q = Pe ° e
(10) the velocities
generating
(9) and (14) the heat flux distribution
_2 = Pcr,_ / 2 Taking
simple
the
v=
(J_'-
2
R.
(Fig.l).
:-
ltTzm),
z)/erfc
Pe e x /z: 2 m+ a
of 0(K h) we can rewrite
only when
,E.,-Pe
cxp( --
the formulae
Q=
where
melting
(12), (13) admits
x--_oo, 0,-_0,
)/
(J_e
[eric
problem
and
--It 1
After
after
method.
to lhe coordinates
t:.lt-Tz
where
element
value
a and z related
i_..,
where
_" r, formed
solutions
In [4] it was proved
parabolic
on surface
(17)
R:+')
D
_o]
(18)
ds
(19)
r/dr/ /aof'dll # r/° = f' : o-_-
dr/ which
is determined
by (19) and the boundary
coudition
P(sl) = 0
]
257 Third International Conference on Inverse Design Concepts and OpUmiT_auon in Engineering (IGIDES-III). Editor: G.S. Dulikravich. Washington D.C., October 23-25. 1991 V
R,+_= • when
/f
(20) 1
'2 as ,: R'H3D
v=litissupposcdthatR.=0
According function
I_2 Rds ._ H3D
Sciences
the assumption
Oh= const
of one independent
variable
the temperature
distribution
in thc molten
layer
is sought
rl (21)
0 = 0(r/) As follows
from
interracial
H dR
Last formulae
condition
-- H=
(9) in this case
const.
(22)
(22) and (21) simplify 1 dR
u ....
equalities
(23)
R,+_
R,+1
and heat transfer
--
I)D f'2,
(v+
equation
Integration
d0 dr/
of this equation
•
H 3R '
in the molten
Peg_0(r/) D
(18), (19)
_(.)
D ds |
P-
(24)
ds
layer
d 0d0) dr/ "dr/
(25)
with the associated
boundary
conditions 1
01,.,--0;
reduce
-d01dr/,-'=HE;
to the following
E-pQ
R_ ds
(26)
+_e
relationship 1
-puting
PcH
1 _0C
(27)
e×p(---ff-f --y dr/)dr/
in (27) r/= 0 the tcmpcraturc _
11
of thc hcating
Pert
surfacc
is dctcrmincd
_oC
0h--EHf°exp(5- I.T dr/ dr/ In order
to simplify
further
In this case heat flux distribution
computations
assume
that
(2S) F h is specified
Q is determined
by the equality
dR
1
ds P(s) introduction
_
+ 4A:(R-
(R_ - R_)(_+
One heating
I-R,
/
R(R
J 1)_D.,
important
characteristics
(16), where
v+l
-R.
)
dR
(29)
I + 4A_(R-- R. )'
to the melt. Combining
of contact
heat energy N--
with the equation
)2.
if v=l
of the most surface
Z = A(R-R.
R .)2
v+l
1=R.=0
by the equation
into (I 1) yields 1
whereR
as a
(27) we have
melting
definition 2nf R2 R R,H (-
is the heat
in non-dimensional
energy
rcmoval
from
the
form
),d0)l odR "dr/ "-
J
258 Third International Conference on Inverse Design Concepts and Optimization in Engineenng (ICIDES-III], Editor: G.S. Dulikravich, Washington D.C.. October _.- .... 1991. F
N = r_(R2 _ R:I )Ecxp(--_'Pert
The quantity necessary
of heat
to sustain
energy
the chosen
calculatcd melting
j"1c¢o 0-_-. dr/)
according
velocity
Here
N o in comparison
with N does
in the surrounding
The main eiTectivety we'll have
scope
of the contact
paper
melting
(30) excesses
-
R_)(I
not contain
thermopcnetrator
of present
the minimum
heat power
N O ,_hich is
form
+ I/Stc)
the energy
ratc
for hcating
melt
and
useless
heat
solid material.
is to elucidate
process.
(30)_
V. In non-dimensional
N o = _(R_
dissipation
Sciences
Defining
the influence efficiency
of the heating
of the heating
surface
surface
shape
upon
as a ratio/]
the
= N o/ N
I 1 + Ste exp( -- Pert / D fl = ---g--
-7 dr/) flC09 _0
Equations with the
(23), (24), (27)-(31)
accuracy
with variable When
of 0(Kh).
physical
They
properties
the physical
simulate
for prediction
of contact
kinds of rocks
and sands.
of melt are constant
6(R "+I _R;
u
=
p=
-
N = n(R_
the considerable
melting
mclting
process
problem
for materials
in the case of ice melting)
equations
simplification.
r/(l -- r/)
dR ds r/2(3-
0 ----EHexp(PeH
(for example
in contact
+')
(v + I)R'H
12 ['" v + I J,
processes
are convenient
(l 7), (23), (24), (27), (28), (30), (31). allows
=
and mass transfer
such as different
properties
u
heat
(31)
A
(32)
211)
(33)
R'+lR'+I _]-R ;°
I'
/ 2
ds
(34)
exp[-PeHr/3 (1 -- 0.5r/)]dr/
(35)
-- R2i)Eexp(- Pert. T-)
(36)
l (I +S_c )exp( ----)P2H E
fl
(37)
3. RESULTS
Numerical
prediction
rock thermodrilling --(31).
Relatively
6]. Non-linear are obtained process used. _eating
The
conditions. complete
equation automatically
solution values
surface
of u,, u,, P, H, O, fl and
description
of basalt
(27) is solved
of ice physical
When
physical
properties
of interest are based
properties
by the iteration
process
one can find of parameter
at high temperature
for example
After
As the initial
is investigated
fl It is shown
is carried on equations
procedure.
(23), (24), (28)--(31).
the ice boring
by the value
quantities
of rock melting
numerically
in a view of equations
(35) is chosen.
is estimated
All the calculations
other
formulae
out
for ice and
(23), (24), (27) is available
this other estimate
of iterative
(29), (32)--(37)
in [4, 17, 19]. Effectiveness
in previous
works
in [5,
quantities
[4, 17l that
are of the
in
corn-]
259 Third
International
(ICIDE$-III).
Conference
Editor;
Parison with the parabolic more
analysis
characterized
for
the
heat
by
slow
of contact
the
value
the material.
tions
presented
when
pcnetrator
energy So
For
(v=
corresponding and
numerically;
case
the
when
physical
results
melting
elongate
is the
interval
molten
form for
(A
