Download Modeling of High-Current Arcs with Emphasis on Free Surface

Modeling of High-Current Arcs
with Emphasis on Free Surface Phenomena
in the Weld Pool
Calculations suggest that flow patterns in deformed weld
pools may be more complex than previously thought
BY R. T. C. C H O O , J. SZEKELY A N D R. C. WESTHOFF
ABSTRACT. An extensive set of calculations is presented to describe the effect of
the role played by the arc weld pool
interface in affecting both the arc and the
weld pool behavior. The computed results may be classified into the following
three main groups:
1) The behavior of the welding arc for
both flat and deformed weld pool surfaces. Here, the computed results were
found to be in excellent agreement with
measurements reported for flat weld pool
surfaces or flat anodes in general. However, it was shown that when the free
surface of the weld pool is significantly
deformed, the resultant heat and current
flux falling on the anode may be markedly
modified. It follows that the usually postulated Gaussian heat flux distribution may
not be generally applicable for significantly deformed weld pools.
2) Computed results reported on heat
flow and melt circulation involving deformed weld pools have shown that under these conditions the melt circulation
may be markedly affected by the shape of
the free surface. Indeed, it has been found
that deformed free surfaces may result in
very complex weld pool circulation patterns. In the present case, the actual free
surface shape was deduced from previous experimental observations. Since the
heat flux and current distribution employed at the weld pool surface were obtained from the previously described arc
calculations, the linkage between arc and
weld pool behavior has been accomplished. A further refinement, which is
being currently pursued, would allow the
calculation of the deformed weld pool
surface.
3) Finally, calculations have been presented describing the transient collapse of
R. T. C. CHOO, j SZEKELY and R. C WESTHOFF are with the Department of Materials
Science and Engineering, Massachusetts Institute of Technology, Cambridge, Mass.
346-s I SEPTEMBER 1990
a weld pool assuming isothermal conditions as an approximate representation.
These preliminary results are able to capture some essential features of experimentally observed weld pool collapse, including the entrapment of gases and the
expulsion of liquid droplets.
Introduction
The role of convection in arc welding
operations and its effects on the structure
and properties of the resultant weld is well
documented (Refs. 1-4). Convection in
weld pools is driven by a combination of
forces, which include surface tension gradients, buoyancy, electromagnetic forces
and the shear stress exerted by the gas on
the molten pool surface. Indirectly, all
these forces are derived from the arc,
which is struck between the electrode and
the workpiece —Fig. 1.
The spatial distribution of the energy
flux falling on the weld pool surface will
have a marked effect on the weld pool
shape and on the subsequent solidification
process, which in turn affects the structure and properties of the weldment produced. It follows that the interface between the welding arc and the free surface of the weld pool represents a critical
component of the welding operation.
Up to the present time, most investiga-
KEY W O R D S
High-Current Arcs
Free Surface Phenomenon
Weld Pool
Arc Weld Pool Interface
Arc Behavior
Weld Pool Behavior
Deformed Weld Pool
Weld Pool Collapse
Modeling
Welding Arc Modeling
tors have concentrated on representing
weld pool behavior on the one hand and
the modeling of welding arcs on the other,
with relatively little attention being paid to
the interfacial regions. Indeed, in virtually
all previously published papers concerning weld pool behavior, this interfacial region was introduced in terms of the
boundary conditions, with a Gaussiantype current and heat flux distribution being specified. By the same token, in previous studies of welding arc phenomena,
the weld pool was represented as a flat
surface having a constant temperature
and electric potential.
There is persuasive qualitative evidence
that these "standard" postulates may represent a gross oversimplification in many
instances. Thus, in the case of deformed
weld pools, which are observed for operation at high current levels, there may
be an important two-way interaction between the welding arc and the molten regions in that the nature of the arc may be
affected by the pool shape and viceversa; and so heat transfer to the pool
may be modified by changes in the arc.
The marked nonlinearity and hysteresis
behavior in the relationship between surface depression and arc current (Fig. 2)
reported by Lin and Eagar (Ref. 5) provide
clear support for this contention. Yet
other, more complex forms of interaction
may involve ripple or wave formation at
the free weld pool surface, the existence
of which has been reported, but not analyzed.
It is suggested that a comprehensive
representation of these free surface phenomena could play an important role in
providing greatly improved insight into
weld pool behavior and could represent
an interesting new frontier in the modeling of arc welding systems. At the same
time, these phenomena could be relevant
also to a range of other joining processes,
including electron beam and laser beam
welding operations.
To provide a realistic perspective, let us
Electrode (Cathode)
5
-
1
-
c
o
Plasma Arc
Base Metal
(Anode)
Molten Pool
240
260
280
300
Current (A)
Fig. 2— Variation of surface depression depth with current. The arrow
indicates the direction in which the current was increased and decreased
(Ref. 5).
Fig. 1 — Schematic of arc welding operation.
briefly review recent relevant work concerned with the modeling of welding operations. In the last five years, there have
been major advances made in the numerical simulation of the weld pool (Refs. 6 11). Equally important, parallel developments regarding arc behavior have allowed the numerical representations of
current density, temperature and velocity
in the welding arc (Refs. 12, 13). At the
present time, the representation of fully
three-dimensional weld pool circulation
has become an accepted fact (Refs. 14,
15); furthermore, a very elegant comprehensive modeling of continuous welding
operations in three dimensions has also
been accomplished (Refs. 16, 17). As a result of these works, predictions may now
be made with confidence regarding the
transient temperature and velocity profiles in weld pools, provided the boundary
conditions may be defined at the weld
pool-plasma arc interface. As a practical
matter, all these models are being run by
specifying the free surface shape and the
current and heat flux distributions at the
free surface. Notable exceptions are the
recent work of Zacharia, ef al. (Refs. 16,
17), and that of Paul and DebRoy (for laser beam welding) (Ref. 18), which allow
free surface deformation, but again, for a
set of thermal boundary conditions that
are specified a priori.
The main purpose of the present article
is to address a relatively unexplored but
potentially important problem area; that
of the free surface behavior of welding
arc and by implication, free surface problems in welding operations in general.
The ultimate objective of the work is to
develop an understanding and quantitative representation of the dynamic, twoway coupling between the transport phenomena in the welding arc and in the weld
pool. The results to be presented here
represent an intermediate stage in this direction but a further development of
ideas first suggested in 1986 (Ref. 19).
More specifically, we shall calculate arc
parameters for certain specified weld pool
shapes taken from experimental data, and
then use the computed heat and current
fluxes falling on the free weld pool surface
to evaluate the temperature and the velocity profiles in the weld pool. The important consequence of this approach is
that allowance can be made for nonGaussian heat and current fluxes falling on
the free surface of the weld pool.
Free Surface Problems in Welding
The schematic sketch given in Fig. 1 was
an oversimplification because the upper
surface of the weld pool was represented
as an undistorted plane. While the postulate of a flat weld pool surface may be
appropriate for certain types of operations, it is known that significant weld pool
deformation may take place in arc welding at high current levels, in submerged arc
welding, in laser and EB welding and in
Oscillating Weldpool
U)
GMA operations.
The free surface deformation of weld
pools may have several important practical consequences, as sketched in Fig. 3
(Refs. 20, 21):
1) At significant deformation levels the
welding arc itself may be affected by the
weld pool shape, and thus the heat and
current flux may be markedly modified by
the weld pool behavior.
2) The interaction of the plasma gas
stream with the weld pool may cause surface ripples, oscillating surfaces or instabilities.
3) The collapse of a significantly deformed weld pool may produce numerous weld defects including keyholes, gas
bubble entrapment and the like.
The issues addressed in this paper are
the modeling of the welding arc with a
predeformed anode surface, the modeling of the fluid flow and heat transfer in
the depressed weld pool, and the modeling of the collapse of a keyhole depression
in the weld pool. The effect of instabilities
will be discussed in a separate publication
(Ref. 22).
R i p p l i n g Effects
D e p r e s s e d Surface
(b)
Fig. 3 — Effects of free surface on weld pool. A —CTAW can produce oscillating weld pool surface
(Ref. 20); B — rippling effects can be seen in moving welds (Ref. 21); C — free surface is depressed
at high arc currents (Ref. 5).
WELDING RESEARCH SUPPLEMENT 1347-s
Modeling of the Welding Arc with
a D e f o r m e d Anode Surface
Figure 4 shows the computational domain used to model the arc. Excellent reviews of welding arc behavior are available in the literature (Refs. 23, 24). In the
present case, w e shall briefly summarize
the key features of arc behavior. When an
electric potential difference is set up between the anode (metal workpiece) and
the cathode (usually a tungsten rod), and
a spark or some initial heating ionizes the
gas between the two, an arc may form.
The ionized gas, or plasma, has a much
higher electrical conductivity than the gas
in a normal state, which allows current to
flow. This flow of current causes joule
heating in the gas, allowing it to maintain
its ionized state so that it remains electrically conductive; and the arc becomes
self-sustaining. As the electrical conductivity of the plasma is highly temperature
dependent, the arc is constrained to a
narrow path between the cathode and
anode, and the current density within the
arc column may assume a Gaussianshaped radial distribution.
At the cathode, electrons are thermally
emitted from a small, hot region known as
the cathode spot. From the cathode spot
the electrons diverge radially and move
axially toward the anode. The strong
divergence of the current at the cathode
gives rise to significant electromagnetic or
Lorentz forces which drive the cathode
jet, a high-speed flow of gas, toward the
anode. The combination of the cathode
jet and joule heating contribute to the
convective heat flux and pressure at the
anode surface. At the anode, the electrons are condensed, giving up energy
proportional to the work function of the
anode material. This electron component
of the anode heat flux is usually about 70%
of the total anode heat flux.
The following assumptions will be made
in this analysis:
1) The arc is assumed to be radially
symmetrical.
2) Steady-state conditions are assumed
throughout the arc.
3) The arc is assumed to be in local
thermodynamic equilibrium (LTE), which is
taken to mean that the electron and heavy
particle temperatures are not significantly
different. The studies by Hsu and Pfender
(Refs. 12, 25) show this assumption is accurate through most of the arc, except in
the fringes and very near the cathode and
anode surfaces.
4) The arc plasma is assumed to consist
of pure argon at atmospheric pressure.
The effect of other gases that may be entrained is neglected as is the effect of
metal vapors from the electrode and
workpiece.
5) The flow is assumed to be laminar.
This assumption was justified by McKelliget and Szekely (Ref. 13) on the basis of
348-s I SEPTEMBER 1990
laminar-turbulent transition for a free jet.
6) The plasma is assumed to be optically thin so that radiation may be accounted for using an optically thin radiation loss per unit volume.
7) The heating effects of viscous dissipation, and buoyancy forces due to gravity are neglected.
Governing Transport Equations
Using the above assumptions, the governing equations for the arc may be written as follows:
Conservation of mass:
7 Jr
(prUr) +
Tz ^,)
=0
d)
Conservation of radial momentum:
2
r
iKf)]-'
2u r
(2)
az
Conservation of axial momentum:
±|(pruruz) +
JjP
'dz
oz
(PUZ 2 ) =
+
1_5
rdr
f)]+
dz
2 -dzf
(?)
+ )rl
Conservation of thermal energy:
1 1 (pru h)
r
r or
+
(puzh) =
/d<_ db\
d_ ( k d h \
r or \ C p d r j + dz \ C p d z /
p
5 kb
SR
2 e
1 A
I
J_ dh
5h
C p 5z +
)r
Cp
fir
+
(4)
The source terms in braces in the energy
equation represent joule heating, radiation losses and transport of enthalpy due
to electron drift, respectively.
Current continuity in terms of electric potential:
1 d ( dA.df
dA
7 a? ^ r a F J + az {° Tz) =
For the purpose of modeling, the welding arc shown in Fig. 1 is transformed to
the domain shown in Fig. 4 and the corresponding boundary conditions are given
in Table 1. These specify zero velocities at
solid boundaries, zero fluxes at the axis of
symmetry, constant temperature of 1000
K at inflow boundaries, a zero radial temperature gradient at the outflow boundary, zero currents at all boundaries except
at the cathode spot and at the anode,
where the current density and the potential are specified, respectively; a constant
electric potential is specified at the anode.
Most of these boundary conditions are
self-evident but a few require some discussion. Hsu, ef al. (Ref. 12), investigated
the effect of the temperature boundary
conditions at the inflow region DE and reported that the arc characteristics remained substantially unchanged whether
a 1000 K or a 2000 K isotherm was used.
This observation is probably due to the
fact that the specific heat of the argon gas
in this temperature range does not vary
significantly. Secondly, the choice of zero
radial temperature gradient at the outflow
region FG assumes that this region is far
away from the plasma column such that
the radial temperature gradient approaches zero.
The anode and cathode surfaces require special treatment since deviations
from LTE occur in these regions and these
are given below.
Cathode Region
0{5)
Since the current distribution is axisymmetrical, the self-induced magnetic field is
given by the following relation from Ampere's law:
=7 >dr
(7)
Boundary Conditions
dP
(pUrUz) = - -Jjj +
h^+k
-oV</>
The physical properties, namely density, viscosity, thermal conductivity, heat
capacity and electrical conductivity, are
treated as temperature dependent in the
model. Compressibility effects have been
neglected, which is admissible if the plasma
velocity is significantly lower than the
speed of sound. This assumption was
tested by including the compressibility
terms of the momentum equations, and
little or no effect was noticed in the flow
field. The temperature-dependent plasma
properties were taken from the tabulated
data of Liu (Ref. 26), and the radiation loss
is adapted from the measurements of
Evans and Tankin (Ref. 27).
(6)
Current density is calculated from the
definition of electric potential:
Between the arc column and the cathode surface there is a thin transition layer
in which steep gradients occur; it supports
several of the physical processes that sustain the arc. Positive ions are accelerated
toward the cathode surface and provide
energy for the thermal emission of electrons. The emitted electrons either combine with the positive ions or are accelerated away from the cathode. Additionally,
the plasma is ionized in this layer, causing
a substantial potential drop called the
Inflow
cathode fall. This cathode boundary layer
has been investigated by Hsu and Pfender
(Ref. 25) who showed that as the cathode
surface is approached, the heavy particle
temperature approaches the cathode surface temperature while the electron temperature remains much higher (~17,000
K). This state of thermal nonequilibrium
cannot be represented using an LTE model.
In addition, the thickness of the cathode
boundary layer is of the order 0.1 mm,
which presents some practical problems
in being resolved using a 2-D finite-difference approximation.
The contribution of the cathode fall has
been approximately accounted for using a
free-fall type of expression for the cathode fall voltage, V c , as was done by McKelliget and Szekely (Ref. 13):
. , _ 5 kbTe|ec
V c
~2
e '
Qioniz = | J c | V c
(8)
Here, Qbniz is a positive source to the
plasma column at the cathode boundary
which approximates the energy used in
the cathode boundary layer to ionize the
plasma. Te|ec is approximated as the maximum plasma temperature in the column
adjacent to the cathode ( ~ 20,000 K), so
V c ~ 4.3 V.
The boundary condition for the electric
potential is approximated assuming that
the cathode current density, Jc, emitted
from the cathode normal to the surface is
constant inside the cathode spot radius,
Rc, and is zero outside:
(9)
0
Inflow - ^
r>Rc
The unknown parameter Jc must be specified by giving the cathode spot radius.
McKelliget and Szekely (Ref. 13) showed
that a single value of Jc (6.5 X 107 A/m 2 )
could be used over a range of conditions.
Anode Region
Four principal modes of heat transfer
contribute to the anode heat flux in welding: convection from the plasma, electron
flow due to the current, radiation from the
plasma, and vaporization of the anode.
The boundary layer which exists between the anode and the arc column is of
special interest as the processes which
occur there govern the current density
and heat flux to the anode. Dinulescu and
Pfender (Ref. 28) presented a model of the
anode boundary layer which showed that
the heavy particle temperature approaches the anode temperature near the
wall, while the electron temperature remains high ( ~ 10,000 K), thus maintaining
a conducting path to the anode. This
boundary layer region is approximately
0.1-mm (0.004-in.) thick and presents the
same problems to an LTE model as the
cathode boundary layer. W e have thus
chosen to take the same approach used
Outflow
Fig. 4 —Domain of
integration for the
welding arc.
10,000 K, the heat flux due to electron
flow may be written as shown:
by McKelliget and Szekely (Ref. 13).
The anode fall is assumed to be negative as seen by the modeling (Ref. 28) and
experiments (Ref. 29) of Pfender and coworkers so that the electron contribution
to the anode heat flux may be written as
follows:
Kb I elec
Q,
+ LV»
Q e = | a (2.76 + V w )
(10)
4>a is the thermal diffusion coefficient of
the electrons. The quantity in parentheses
is assumed to be 3.203, corresponding to
an argon plasma with electron temperatures of approximately 10,000 K. If the
electron temperature is assumed to be
(11)
where V w is the work function of the
anode material.
The approach taken here is to assume
that the electron temperature is equal to
the film temperature, Tf = 0.5(TW + Te),
where T e is the temperature at the edge
of the boundary layer and T w is the temperature of the wall. This is an approximate way to allow for the fact that the
electron temperature will be lower as the
radius increases near the anode. This
assumption gives an electron temperature
Table 1 - Boundary Conditions for the Integration Domain Shown in Fig. 4
BC
CD
DE
T = 3000 K
0
0
0
0
T = 3000 K
0
0
T = 3000 K
0
EF
(inflow)
dpru r
FG
(outflow)
dprur
CH
h
u2
Ur
AB
~~d~r~
T = 1000 K
az
0
0
^ =0
ar
ar
0
Q-IJcNc
T = 1000 K
ar
0
T = 1000 K
Equation 8
<p
,c =
^7
8* - •
az
ar
ar
0 = constant
HA
0
^
= 0
ar
ar
WELDING RESEARCH SUPPLEMENT | 349-S
and w r i t t e n as f o l l o w s :
=
0515
Pr w
(
a.
O
Table 2—Variable Mesh Used for the Four
Cases Studied for Welding Arc Modeling
/MePeV"
VMWPW/
S
*Y n,
IPwPw— I
(he -
u i
(13)
hw)
_i
UJ
>
x
o
The correlation as given in the literature
is a p p a r e n t l y only valid at t h e stagnation
point (i.e., at r = 0). It has b e e n m o d i f i e d
f o r use o v e r the entire a n o d e b y replacing
ce
<
t h e radial v e l o c i t y gradient —r— by —. This
UJ
a
ui
v>
ce
is d o n e because the radial velocity at the
a n o d e goes t h r o u g h a m a x i m u m at s o m e
. . ,
,.
dure
critical radius, so that —r— goes t o z e r o
5
cu
w h i l e t h e c o n v e c t i v e heat flux remains f i -
_J
UJ
nite. T h e t e r m — is selected because it a p -
Ul
a
proaches
Ul
o
>
x
o
o:
<
ui
05
Ul
o:
£
a.
o
-J
Ui
>
Ul
a
Fig. 5 - Configuration of radiation view factors.
o f a p p r o x i m a t e l y 7000 K f o r a 1000 K
anode.
The c o n v e c t i v e c o n t r i b u t i o n c o u l d b e
f o u n d b y a p p l y i n g a correlation f o r a
stagnation p o i n t f l o w . H e r e , a m o d i f i e d
version o f the a p p r o a c h taken b y M c K e l liget and Szekely (Ref. 13) is used. The
same c o n v e c t i o n correlation w a s taken
f r o m the literature (Refs. 30, 31).
Nuv
= 0.515
PePe
(12a)
MwPw
Ul
t/i
Nuw
z
\ (h e -
hv
Pr„
ui
S
a,
O
>
ui
as
r —• 0
and
as
, t h e n Equation 12a can be m a n i p u -
•»>
x
o
a:
<
— j — both
r —• oo, and assumes a reasonable f o r m
i n - b e t w e e n . Additionally, if the Reynolds
n u m b e r , R e w , is assumed t o b e g i v e n b y
F~
V
dure
(12b)
MwPw-^T
T h e radiative heat flux t o the a n o d e is
calculated using the f o l l o w i n g relationship—Fig. 5:
'<> = J 477?cos ^ d V i
(14)
V|
a
<r:
Ul!
z
ui
£
a
Fig. 6 — Temperature
profile for a 200-A
arc and 100-mm arc
length. Predictions
compared with
experimental results
of Pfender, et al.
(Ref. 12).
The a b o v e e q u a t i o n m a y b e w r i t t e n as
an elliptic integral in the 9 d i r e c t i o n , the
solution of w h i c h can b e f o u n d f r o m an
integral table (Ref. 32). It is t h e n integrated
numerically in the r a n d z directions, o v e r
the entire calculation d o m a i n . T h e ass u m p t i o n e m p l o y e d h e r e is that the v o l u m e o f the cells used in the calculation is
small e n o u g h t o a p p r o x i m a t e a differential
volume.
In this analysis, heat transfer d u e t o va-
Radial Position ( m m )
>
x
_l
ui
>
ui
0
-
a
o
ce
<
UJ
(/>
Experiment
Ul
ce
3 5 0 - s | SEPTEMBER
1990
200
260
280
300
1.0
1.2
1.8
4.0
19 X 37
19 X 38
19 X 41
19X43
Domain
Dimensions
(mm)
10.95
10.95
10.95
10.95
X
X
X
X
11.7
11.9
12.5
14.7
p o r i z a t i o n o f the a n o d e metal has b e e n
n e g l e c t e d . This assumption is realistic if
w a t e r - c o o l e d anodes are used. In actual
w e l d i n g c o n d i t i o n s , v a p o r i z a t i o n of t h e
a n o d e material c o u l d b e c o m e i m p o r t a n t
and m a y significantly change b o t h t h e
c o n d u c t i v i t y a n d the radiation losses o f
the plasma near the a n o d e . A n o d e v a p o ration m a y b e r e p r e s e n t e d a p p r o x i m a t e l y
using a mass-transfer analogy t o c o n v e c tive heat transfer. A n o d e v a p o r a t i o n is n o t
c o n s i d e r e d in this analysis.
The heat flux t o the a n o d e surface can
t h e n b e w r i t t e n as,
(15)
T h e a n o d e is assumed t o b e a p e r f e c t
c o n d u c t o r relative t o the plasma so that a
constant value of <fr is used f o r the b o u n d ary c o n d i t i o n at the a n o d e surface.
T h e g o v e r n i n g equations and b o u n d a r y
conditions w e r e solved using a finite-volu m e a p p r o a c h d e s c r i b e d b y Pun a n d
Spalding (Ref. 32) w h i c h w a s i m p l e m e n t e d
using a m o d i f i e d version o f t h e 2/E/FIX
c o d e . The difference equations w e r e
solved by iteration until t h e y w e r e satisfied w i t h i n 99.9%. Several variable meshes
w e r e used d e p e n d i n g o n the arc length
d e p t h of the a n o d e depression. These are
listed in Table 2. T h e 300-A case takes
a b o u t t w o hours CPU t i m e o n a VaxStation II.
Results and Discussions of the Welding Arc
Modeling
—
o
o
Grids
(radial x
axial)
Solution Technique
Ci
CJ
Surface
Depression
(mm)
Q a (r) = Q e (r) + Q c (r) + Q r (r)
lated i n t o the f o r m o f Equation 13.
Q
Arc
Current
(A)
Figures 6 a n d 7 s h o w a c o m p a r i s o n o f
the theoretical predictions w i t h e x p e r i mental measurements of Hsu, et al. (Ref.
12), a n d N e s t o r (Ref. 34), f o r the t e m p e r ature profiles, and t h e c u r r e n t density a n d
heat flux, respectively. T h e g o o d agreem e n t p r o v i d e s a calibration f o r t h e c o m putational a p p r o a c h .
Figures 8 - 1 1 s h o w the current density
flux, heat flux, t e m p e r a t u r e profile and
velocity p r o f i l e , respectively, f o r w e l d
pools w h i c h h a v e b e e n depressed by 1.0
m m (0.04 in.) (200 A), 1.2 m m (0.05 in.)
(260 A), 1.8 m m (0.07 in.) (280 A), a n d 4.0
m m (0.16 in.) (300 A). Their c o r r e s p o n d i n g
calculated arc parameters are s h o w n in
Table 3. W e n o t e that the w e l d p o o l d e -
80 I
_, , , , 1 ,
'—
CO
ta
a
2
—
:
200 AMP ABC
L = 6.3 MM
x — Calculation
Voltage - 16.4
Efficiency = 0.59
Experiment
(Nestor)
\
\
\
~
;
^
i i -i—i—i—i—r
—
200 AMP ARC
L = 8.3 MM
r - Total Flui
* - Qrad
+ - Qelectron
o — QconT
Experiment
(Nestor)
—
—
•
\\
i
^^^^
i i 1 i i i i rr^r^-rrH—i—*mt
2
i
4
6
RADIAL LOCATION (MM)
8
~
"
nir"
«*-*•
10
2
10
8
4
6
RADIAL LOCATION (MM)
Fig. 7— Current density and heat flux distribution at anode for a 200-A arc. Predictions compared with experimental results of Nestor (Ref. 34).
pression itself was not calculated from the
model, but reflects experimental measurements reported by Lin and Eagar (Fig.
2) (Ref. 5).
In Figs. 8 and 9, a bimodal distribution
(only one-half is seen because of symmetry) is observed for the heat and current
fluxes. This is in sharp contrast with Figs. 6
and 7 in which a Gaussian distribution is
observed for a flat anode. As the current
increases, the bimodal distribution becomes more pronounced due to the
deeper surface depression. The effect of
this surface depression is quite dominant.
For the 280-A case, we see that the peak
current density is four times that at the
center of the pool, while the 300-A case
is almost ten times that of the center. Since
about 70% of the thermal energy transferred to the anode is carried by the electrons, this change in the heat flux may
have a marked effect on both shape of
the melt-solid boundary and on convection within the weld pool.
flux, but the ratio of the maxima to the
minima is not as large because the deformed pool affects the convective contribution much less than it does the electronic contribution. As the current increases, the gas jet velocity also increases;
so that we see a rather flat heat flux profile in the middle region of the weld pool
(r < 1.5 mm) as convection continues to
add some heat to the center of the pool.
The radiative contribution to the total heat
flux is generally small ( ~ 10%). The important finding represented by these results is
The heat flux to the anode shows a bimodal distribution similar to the current
n
55
W
Q
2
4
6
RADIAL LOCATION (MM)
2
. " " I
4
6
RADIAL LOCATION (MM)
1
1
!
1
|
T
1
/-*
f
3 —
—
-
1
I
l
I"
1
T • "I" T
1
4 m m DEPRESSION
T
'
"-
—
Voltage = 18.4
\ Efficiency = 0.66
1
T
—
r - Calculation
\
1
D;
300 AMP ARC
\
/
T
Q
—
*T7.
2
4
6
RADIAL LOCATION (MM)
,1 ,
i
2
i
i
1
i
i
i
i
1
i
4
6
RADIAL LOCATION (MM)
i
i
i
1
X-r
10
Fig. 8-Anode current density for the four cases studied. A-200 A; B-260 A; C-280 A; D-300 A.
WELDING RESEARCH SUPPLEMENT | 351-s
Table 3—Calculated Arc Parameters for the
Four Cases Studied'3'
280
Arc current (A) 200
260
300
369
402
Max. velocity
298
421
(m/s)
21,800 23,100 23,600 23,:
Max.
temperature
K
Max. anode
934
1080
565
1140
pressure
(Pa)
Max. current
5.5
3.9
4.2
3.6
density
(A/mm 2 )
Max. heat tlux 35.0
48.3
39.9
30.5
(W/mm2)
Arc voltage
16.4
17.6
18.0
18.4
(V)
59
Arc efficiency
59
58
56
(%)
(a) The arc length is defined al 6.3 m m which is the distance
f r o m the tip of the electrode to the level surface of the w o r k piece.
that the actual heat flux to the anode may
markedly depend on the shape of the free
weld pool surface.
Although the total heat flux for the 280A case (Fig. 9C) appears larger than the
300-A case (Fig. 9D), this is due to the fact
that the surface area of the depressed
pool for the 300-A case (4.0 mm/0.16 in.
depression) is much larger than the 280-A
2
case (1.8 mm/0.07 in. depression). In fact,
the total heat flux received by the 300-A
case is about 10% larger than the 280-A
case.
The temperature contours shown in
Fig. 10 and the maximum temperatures
given in Table 3 illustrate the nature of the
temperature field in an arc adjacent to a
depressed pool. The isotherms are very
close near the cathode because the cathode must be maintained at 3000 K — near
the melting temperature of tungsten. Additionally, the isotherms tend to be parallel to the anode surface because it is
assumed to maintain a constant temperature of 1000 K. The other main limitation
of the current model is that vaporization
is neglected at the anode.
Figure 11 shows the corresponding velocity vectors and the nature of the pool
depression. As expected, the maximum
velocity increases as the arc current increases (Table 3). It is interesting to note
that the maximum velocity occurs near
the tip of the cathode rather than near the
anode surface. This is because the primary
driving force for the cathode jet is the
) X B term. Figure 12 shows a typical vector plot of the current density, in this
instance for the 200 A case. The strong
divergence of current can be seen near
the cathode tip which gives rise to the
cathode jet. The relatively high conduc-
4
6
RADIAL LOCATION (MM)
2
tivity of the anode adjacent to the plasma
causes the current density vectors to be
nearly perpendicular to the deformed
anode surface. Consistent with this observation, the anode surface is very nearly
a line of constant electric potential.
Figure 13 shows the surface shear
stresses generated by the plasma gas as it
passes radially outward over a flat anode
surface. The maximum shear stress shown
is for a 6.3-mm (0.25-in.) long 300-A arc
which is seen to be nearly 100 N/m 2 . This
shear stress is of the same order of magnitude as those caused by the surface
tension gradients and may be a significant
factor in determining the free surface
shape. Previous investigators (with the
exception of very recent work by Matsunawa, Ref. 35) have neglected the gas
shear stress and this may be the reason
previous models have been unable to
predict the surface depressions based on
arc pressure alone (Ref. 5).
Modeling of Weld Pool Circulation with a
Deformed Free Surface
Figure 14 shows the computational domain employed in simulating the weld
pool. The welding arc strikes the surface
of the workpiece, adding heat to it via the
radiative, convective and electronic contributions. The heat absorbed will raise the
4
6
RADIAL LOCATION (MM)
^^1
-i—i—i—i—I—i—i
\~
1—i—r -
D
60 —
300
4.0
x » + -
40 —
AMP ARC
m m DEPRESSION
Total Flux
CJrad
Qelectron
20
2
4
6
RADIAL LOCATION (MM)
2
Fig. 9 - Anode heat flux for the four cases studied. A - 200 A; B-260 A; C - 280 A; D- 300 A
352-s | SEPTEMBER 1990
4
6
RADIAL LOCATION (MM)
temperature of the surface, and a molten
pool develops. This pool will grow until
the heat input equals the heat loss by radiation, convection, conduction and vaporization. Flow in the pool is driven by a
combination of buoyancy, electromagnetic, surface tension and impinging
plasma arc forces. The transport processes of importance are listed below:
1) Current and heat flux distributions
due to the arc.
2) Interaction of the arc with surface
(surface shape).
The complete modeling effort requires
a dynamic coupling between the arc and
the weld pool. In this first attempt, the free
surface has been assigned on the basis of
the studies of Lin and Eagar (Ref. 5). The
primary focus of the present effort is to
investigate the type of flow field genera'ed by the bimodal heat source. Work is
currently in progress to represent the
two-way interaction between the weld
pool and the welding arc.
Basic assumptions for the model are:
1) Laminar flow is assumed since the
3) Convective heat transfer due to fluid
flow in the molten weld pool.
4) Thermal conduction into the solid
workpiece.
5) Convective and radiative heat losses
from surface.
6) Heat and mass losses due to vaporization.
In addition, t w o other processes that
are of concern include (1) transient solidification and melting of the pool and (2)
the free surface behavior as a result of
surface ripples or surface deformation.
2
UJ
s
a.
o—I
UJ
>
UJ
a
o
ce
<
UJ
to
Radial Position (mm)
Radial Position (mm)
10.
io.
2
3
4
5
6
7
8
-
2
3
4
5
6
7
8
9
12000 K
13000 K
14000 K
15000 K
17000 K
19000 K
21000 K
-
12000
13000
14000
15000
17000
19000
21000
23000
Q.
o
>
ui
1 - 11000 K
1 - 11000 K
2
Q
K
K
K
K
K
K
K
K
X
o
oc
<
UJ
co
UJ
cr
2
a
O
-i
UJ
>
Ul
a
i
oDC
<t
ui
V)
Ul
280 A
200 A
ce,
»•«
i-
?.
0.
10.
f
0.
4.
•
i
-
1 - 11000 K
>
X
TJ
O
• 6.
2
3
4
5
6
7
8
9
-
12000
13000
14000
15000
17000
19000
21000
23000
B
K
K
K
K
K
K
K
K
>
X
ES'
'
>-
4^
JWW\
/ | \\\
TJ
o
o
.3
4^
fi
./ttll\\\
l\Yv
'J
7
6.
•
i
1
2
3
4
-
a.
'
i
11000
12000
13000
14000
z
HT
•
K
K
K
K
1
UJ
D
15000KK
65 - 17000
"
7 -
19000 K
8 - 21000 K
9 - 23000 K
O
—i
UJ
>
UJ
a
—
o
<
u
V)
u
a
^
i-
Z
s
8.
TJ.
2
a.
T \ ^"^5"
>
,)
a
o
-i
UJ
>
u
a
oia
<
12.
260 A
14.^ f
300 A
12.
Fig. 10 - Temperature contours for the four cases studied. A - 200 A; B - 260 A; C - 280 A; D - 300 A.
WELDING RESEARCH SUPPLEMENT 1 353-s
ui
Ui
a
Radial Position (mm)
Radial Position (mm)
3.
>
X
>
x
T3
o
00
T3
o
o
00
0
o
3
O
1\5
CS
3
u
280 A
2
1 '
n ,
t
4
1
fi
1
'
1
a
i
'
p
1
D
B
>
>
~
o
2
4
o
00
«:<^ ' 4.-77, /
'
c
'
fi
s
-
•
1
j
j
|
•
> ( ' < • • •
t ,
•
<
V
•
Hi? I I1' 1' . <'
[ 1
5'
V
o
p
1
'/
GO
>—.
vim
< •
•
p.
Wi J
3
CO
?
^ r r : "'.'.'.
IP
Hi
3
12
f
00
300 A
14
260 A
3
- if > ^r
Fig. 11 - Velocity plots for the four cases studied. A - 2 0 0 A; B-260 A; C-280 A; D-300 A.
size of the pool is small.
2) The surface is assumed to be a gray
body.
3) All transport, physical and electrical
properties of the liquid and solid are
assumed constant, independent of temperature, with the exception of the surface tension, which is assumed to be a linear function of temperature. These assumptions may be readily relaxed without
undue computational difficulty.
4) Vaporization is simulated, as a first
approximation, by putting an upper
boundary on liquid temperature, which is
500 K below the boiling point (Refs. 1,36).
for the system depicted in Fig. 14 are:
conservation of mass
d(rur)
dr
J3u r
For incompressible laminar flow, the
relevant governing transport equations
354-s I SEPTEMBER 1990
dur
dP
•
-
5r '
dr
dUr
<WJr
~7- + r)Z
- K u
2
3uz[
1 du,
dP
<92uz I
(18)
- Tr) + JrB„ - Kuz
conservation of thermal energy
fr3T
(17)
2
u
Prgl3(l
3u r l
d-Ur
r
3uz
I <92uz
(16)
conservation of radial momentum
ar
Governing Transport Equations
duz
dz
f<5u2
a
\d2l
5T ,
1dT
+
d2T [
|5? rlF 5?J_
AH Mi
cn
+
3T]
(19)
at
r
conservation of axial momentum
The buoyancy forces are calculated via
the Boussinesq approximation while the
Lorentz forces are calculated in a manner
identical to that described for the arc, except that the electrical conductivity is assumed constant for the workpiece.
Note that in the weld pool modeling,
the origin is taken to be the bottom of the
workpiece, while in the welding arc modeling the origin is taken to be the electrode.
Radial Position (mm)
200 A
>
X
Special Source Terms and Boundary
Conditions
o
The enthalpy method is used to simulate the melting phenomena. Here, a drag
term is included in the momentum equation to inhibit flow when the temperature
falls below the liquidus temperature. A latent heat term is added to the energy
equation for phase change. In particular,
the method proposed by Hirt (Ref. 37) is
chosen for its simplicity.
The temperature-dependent drag term
is incorporated into the momentum equation via — Ku r and — Ku z where
-v
O
3
co
CO
+
o
3
3
*
#
T > TL; k = 0
Fig. 12 —Current
density vector plot
for the 200-A case
(1-mm depression).
Ts < T < TL;
k = k
(JL
- T)
TL-TS
(20)
T < T s ; k = co
which, by using the Darcy-type relationship, i.e., flow through porous media,
represents flow phenomena in the mushy
region. For convenience of comparison,
the K term is defined as a function of drag
via DRC
DRG =
1
1.0 + A t • K
than Fig. 15B.
The latent heat term is added to the
energy equation via (AHi/C p ) • (<3f|/3t)
where f|_, the fraction of liquid, has been
linearized for simplicity. It can be used to
simulate true volume fraction liquid if the
phase diagram of the alloy is known. Here,
it is given as:
= + ffX)
(23a)
df
dr
- • ( * ) - - ( » ) •
T > TL; fL = 1
(S)
Ts < T < TL;
(21)
Thus, DRG = 1.0 for pure liquid and 0.0
for solid state. The DRG term is also a
function of the time step; for At = 10 _3 s,
Kmax is chosen as 104. The effect of DRG
as a function of T for t w o different values
of Kmax is shown in Fig. 15. Figure 15A
represents drag in the mushy zone better
dur
(T - T5)
fL —
(22)
TL-TS
T < Ts; fL = 0
The other t w o driving forces, surface
tension and the plasma arc, are added as
boundary conditions. For surface tension:
(23b)
where dy/dl
has been defined as the
surface tension coefficient. The arc force
is the stress imposed upon the free surface
to deform it. It may be calculated from the
J X B term in the plasma jet or obtained
from experimental measurements (Ref.
38). In this study, the effect of arc force is
not included since w e have assumed a
Arc Heat Flux
100
Vaporization
V
Convection and Radiation
3
2
J3
2
4
8
RADIAL LOCATION (UM)
Fig. 13 - Shear stress distribution for a flat anode.
Convection and Radiation
Fig. 14 —Modes of heat transfer in the workpiece.
WELDING RESEARCH SUPPLEMENT I 355-s
1.0
•
0.9
0.8
-
Kmax = lo 3
0.7
0.6
o
13
CC
a
IT
Q
0.5
0.4
0.3
-
0.2
0.1
B
0.0
1523
1723
TT
1623
To
Temperature (K)
Fig. 15 —Drag as a function of temperature between
p r e d e f i n e d p o o l surface shape. W e h a v e
also assumed that the f r e e surface shear is
d u e only t o surface tension. T h e effect of
m o m e n t u m transfer of the gas jet m a y b e
i m p o r t a n t a n d an initial estimate has indic a t e d that d e p e n d i n g u p o n the arc curr e n t , the gas shear stress can b e in t h e
same o r d e r of m a g n i t u d e as the surface
tension shear stress. Thus, o n e may n o t
readily ignore the gas jet as has b e e n d o n e
in t h e past.
•
i
1523
Tc
the liquidus and solidus temperature.
This drag is used to model flow in the mushy zone.
It should b e r e m a r k e d (Ref. 39) that f o r
l o w current values ( < 2 0 0 A) the w e l d p o o l
depression calculated f r o m purely m o m e n t u m considerations agrees w e l l w i t h
measurements (Ref. 5). H o w e v e r , f o r
higher current values, b o t h magnetic pressure a n d the arc pressure c a n n o t account
f o r t h e experimentally o b s e r v e d d e e p
depressions. Since the arc pressure, surface shear stress, shear stress d u e t o surface tension ( w h i c h seeks t o resist the d e -
f o r m a t i o n ) , and the metallostatic head are
all of the same o r d e r of m a g n i t u d e , o n e
m a y speculate that a p r o p e r a c c o u n t i n g
f o r the effect o f gas-induced surface shear
may p r o v i d e the explanation for the d e e p
depressions o b s e r v e d at high current levels.
T h e input heat and current fluxes are
o b t a i n e d f r o m the calculations described
previously. T h e y serve as b o u n d a r y c o n ditions f o r t h e free surface. It should b e
stressed that b y this p r o c e d u r e w e h a v e
b e e n able t o p r o v i d e an i m p o r t a n t linkage
b e t w e e n t h e arc a n d the w e l d p o o l b e havior. This is t h e first t i m e that this has
b e e n d o n e explicitly. T h e c o n v e c t i v e heat
loss at surfaces o t h e r than the w e l d p o o l
is given by the standard e q u a t i o n
o,out
200 A
(a)
1723
1623
Temperature (K)
<b)
—
hc(TSLJrf
T a m b)
(24)
w h e r e h c is calculated f r o m an empirical
correlation f o r natural c o n v e c t i o n . This is
given as (Ref. 40):
vertical w a
hc=
1.42
hxrzis&y
(25a)
b o t t o m wall
hc=
(
1.59
Ta A M
Tsurf
(25b)
L
In a d d i t i o n , the usual b o u n d a r y c o n d i tions are z e r o velocities at all solid surfaces, a n d t e m p e r a t u r e a n d velocity gradients are z e r o at the axis of s y m m e t r y .
T h e physical p r o p e r t i e s of steel are listed
in Table 4.
Solution Technique
(c)
(d)
Fig. 16 — Temperature profile in weld pool of the solidus line (1523 K) as a function of time for the
four cases studied. A - 200 A;B-260A;C—280
A; D — 300 A. The axis of symmetry is on the righthand side with the origin at the bottom right-hand corner. The arc is at the top right and the nature
of the blockage cells is indicated. The steps match those given in Fig. 8. The dimensions of the plots
are given in Table 4. The numerals on the contour lines indicate the time in seconds.
3 5 6 - s I SEPTEMBER
1990
The g o v e r n i n g equations and b o u n d a r y
conditions w e r e solved b y the c o m m e r cial fluid a n d heat f l o w numerical p a c k a g e
called PHOENICS (Ref. 41). This uses the
f i n i t e - v o l u m e a p p r o a c h . Four variable
meshes are used t o simulate t h e f o u r arc
currents previously d e s c r i b e d . T h e y are
listed in Table 5.
\N -
? \YV\VCvJ
280 A
Fig. 17 — Velocity plots of the weld pool. The
coordinate configuration is similar to that described in Fig. 15. A-t = 3.0 s (200 A):
um3X=12
cm/s; B-t = 3.0 s (260 A):
Umax" 12 cm/s; C-t = 3.0 s (280 A):umax
= 11 cm/s; D-t = 7.0 s (300 A): umax = 11
cm/s; F — temperature plot for 300-A arc at
t = 3.0s <1> 1523K;<2> 1700K;<3> 1900K;
<4> 2 WOK. The dimensions for A, B and E are
given in Table 5. Figures C and D are partial
plots and their dimensions are 10.95 X 6.3 mm
and 10.95 X 6.9 mm, respectively.
3
The time step used is 10" s. PHOENICS
uses the fully implicit method and is thus
inherently stable. Although a limit of 30
sweeps was set for each step, the residual
has been defined at 10 - 8 , thus the number of sweeps needed to satisfy the residual may be less than 30. It has been noted
the maximum residuals when the 30th
sweep is reached is no larger than 10~3.
Results and Discussions of the Weld Pool
Modeling
The calculation starts by postulating a
given (deformed) solid shape; then the
computed heat and current flux given in
Figs. 8 and 9 are used to calculate the
temperature and the velocity field as
shown in Figs. 16 and 17. Here the solidus
temperature (1523 K) is plotted as a function of time. In all cases, dy/dl is assumed
to be + 10" 4 N/m-K. Because of the
somewhat arbitrarily imposed initial conditions, these computed results are meaningful only for larger times (i.e., t > 3 s);
indeed, only these results are given here.
From Fig. 16 it seems clear that the
hump is heated more quickly than the
center of the pool because of the higher
heat flux there. As a result, this region will
reach peak temperature (2500 K) first and
surface tension flow may be reduced.
Previous models (Refs. 2, 42) have indicated that the highest velocities are encountered near the edge of the pool. The
reduced flow can affect the geometry of
the weld bead. This may have important
Table 4—Physical Properties of the
Workpiece
implications depending on how long the
anode is heated for a spot weld or how
long a point is heated in a moving weld.
Figure 16 also indicates the surface profile
of the weld pool.
Figure 17 shows the velocity plots of all
four cases at t = 3.0 s except Fig. 17D
which is shown at t = 7.0 s so as to depict
the nature of the flow more clearly. A
similar result is obtained at t = 3.0 s for the
300-A case, but the flow field is not so
pronounced. Figure 17B shows a larger
molten pool than that shown in 17A since
a higher current is used, which will provide
more melting. However, in contrast to
17D, Fig. 17C shows a deeper pool penetration at the center even though the
Cp
g
hc
Kmax
k
TL
Ts
=
=
=
=
=
=
753 J/kg-K
9.81 m/s 2
80 W / m 2 - K
104 S- 1
15.48 W / m - K
1723 K
1523 K
/3
7
AH L
(
n
p
Tamb
=
=
=
=
=
=
=
10" 4 K" 1
1.2 N/m
247 k|/kg
0.4
0.006 kg/m-s
7200 k g / m 3
300K
surface is only depressed by 1.8 mm (0.07
in.). In addition, Fig. 17D has been heated
for 7.0 s and has a surface depression of
4.0 mm (0.16 in).
An important feature of these computed results is how the shape of the weld
bead geometry affects the flow field.
Since 5Y/<9T is positive, the flow is directed toward the direction of increasing
surface temperature (normally toward the
center of the pool). In Figs. 17A-C, the
usual inward flow is obtained. The bimo-
WELDING RESEARCH SUPPLEMENT | 357-s
dal heat flux at the free surface causes a
bimodal surface temperature distribution
which can be seen in Fig. 17E. This will
produce three loops; one directly below
the hump, a second near the center of the
pool, and a third in-between. This is seen
in Fig. 17D. Notice that in Fig. 9D the maximum is t w o times the minimum heat flux.
The consequence is the three velocity
loops formed due to the strong surface
tension-driven flows. Furthermore, in contrast to Figs. 17 A - C , here, the strong surface flows are able to bring the hotter fluid
from the outside inward and a single loop
is formed. It also implies that the hump is
not heated fast enough (the maximumminimum heat flux ratio < 1.6 for the
three cases) so to as to produce bimodal
surface temperature profile. Instead, a
unimodal surface temperature is produced
and thus the three isothermal loops are
not seen in Figs. 1 7 A - C
The results in Fig. 17 indicate the importance of surface tension-driven flow in
controlling pool circulation. Since these
flows are driven by the temperature gradient, the ability to effectively describe the
surface temperature is critical. In this
model, the peak surface temperature has
been taken empirically at 500 K below the
boiling point of the workpiece, which is
2500 K (Refs. 1, 36). This implies that there
are certain regions at the weld pool
surface where 6T/5r equals zero and thus
(mm)
16
r
14
12
10
fc:
Cb)
(d
(d)
te)
Marangoni-driven flow is absent. This is
one of the limitations of this model. Although this limitation can be overcome by
assuming simple vaporization kinetics such
as that due to Langmuir, this method tends
to overpredict the rate of heat loss by a
factor of ten. A more accurate vaporization model, which is currently under development, takes into account both Langmuir vaporization and mass transfer across
the anode boundary layer. It is important
to note that the latter model can only be
developed provided the arc characteristics (plasma velocity and temperature) are
known. This model will be incorporated in
subsequent studies.
While the specific nature of these computed results (i.e., the absolute values of
the melt velocity, the number of recirculating loops and the like) will, of course,
depend on the values assigned to the
properties of the system, some very important general conclusions may be drawn
from these computed results. These are
the following:
1) For high current levels, surface tension
and gas shear may play equally important
roles in affecting melt circulation.
2) In the case of significantly deformed
weld pools, quite complex circulation patterns may develop due to the nature of
the (bimodal) heat flux.
3) The final, solidified shape of the weld
bead may be also markedly affected by
these phenomena.
Modeling of Transient Weld Pool Collapse
One of the intriguing free surface problems in welding is the collapse of the weld
pool once the arc is extinguished. This is a
somewhat simpler situation than the establishment of the transient free surface
shape for continuous welding operations,
ih)
tg>
Ij)
Fig. 18 — Transient collapse of finger penetration 3 mm deep and 2 mm in diameter. The origin is
at the bottom-left-hand corner, and the computational domain is 6 mm radial (12 grids) by 76 mm
axial (32 grids). The liquid surface is at 10 mm. A—Oms, umax = 0 cm/s; B-5 ms, umax = 68 cm/s;
C— 10 ms, umax = 32 cm/s; D— 15 ms, umax = 48 cm/s; E—20 ms, umax = 125 cm/s; F— 145 ms,
Umax = 69 cm/s; C— 150 ms, um 56 cm/s; H— 160 ms, umax = 43 cm/s; I— 165 ms, umax = 35
cm/s; j - 180 ms, umax = 122 cm/s.
Table 5 --Variable Mesh for the Cases Tested for the Weld Pool Modeling
*
Arc Current
(A)
Surface
Depression
(mm)
Grids<a>
(radial X axial)
Domain
Dimensions
(mm)
Real time
simulated
(s)
CPU time
(h)
(MicroVax II)
200
260
280
300
1.0
1.2
1.8
4.0
17X22
17 X 22
17 X 35
17 X 30
10.95 X 6.2
10.95 X 6 . 2
10.95 X 12.3
10.95 X 12.5
3.0
3.0
3.0
7.0
42.6
44.4
52.9
150.0
(a) The number of grids is less than those given in Table 2 because PHOENICS automatically includes the nodes at the walls.
358-s | SEPTEMBER 1990
Fig. 19 — Schematic representation of liquid
metal filling in the crater of a finger penetration
weld after the arc is extinguished (Ref. 45).
Fig. 20-Surface depression at 300 A of a stationary arc weld on Type
304 stainless steel. A — Top surface during welding showing deep narrow
depression; B — cross-section of same weld. The porosity at the bottom
of the finger penetration indicates that the depression penetrated to the
very bottom of the weld (Ref. 5).
nonetheless, it is an important intermediate step in reaching this goal.
In order to illustrate the nature of this
problem, let us illustrate the transient collapse of a keyhole. The material considered was steel, under isothermal conditions. This may seem an absurd oversimplification, but was thought to be a
necessary first step in understanding the
physical phenomenon of collapse.
Governing Transport Equations and Solution
Technique
The governing transport equations include continuity (Equation 16) and momentum (Equations 17, 18). In addition,
the volume of fluid equation (Ref. 43) employed to handle free surface is given by
df
3t +
3F
Ur
5r +
SF
Uz
dz = °
ing assumptions made. The behavior
shown in Fig. 18 is remarkable because it
depicts the formation of a liquid droplet
and the transient establishment of an occluded gas space. This behavior mimics
very well the experimental observations
of Lin and Eagar (Refs. 5,45), shown in Figs.
19 and 20.
The purpose of modeling the collapse
of the weld pool is to determine whether
the pool surface will freeze before it flattens out. Such analysis may provide some
insight into ripple formation and the uneven surface morphology frequently
found in welding.
Work is currently proceeding to represent the combined effect of heat transfer
and weld pool collapse and such a sequence is shown in Fig. 21; these results
are preliminary at this stage, but are indicative of the types of novel modeling
efforts currently underway. Somewhat
similar work, albeit with much smaller deformation and 0.2 mm (0.008 in.) pool radius dimension, recently has been published by Paul and DebRoy (Ref. 18). The
way in which the deformation was handled was not discussed in detail.
IRAN'SIENT fOOL BEHAVIOR WITH BUOYANCY AND P O S I T I V E ^
(26)
where
F =
1
fluid cell
0.0 < F < 1.0 surface cell
0.0
void cell
(27)
In addition to the boundary conditions
given previously, there are additional
boundary conditions needed to handle
the free surface such as surface pressure
due to curvature and the effect of wall
adhesion. These are discussed in detail in
Nichols, et al. (Ref. 43), and will not be reiterated here.
t=0.0s ; u m a x = 5 8cm/s
t=0.02s ; Umax='»4 cm/s
t=0.08s; u m a x 5 ? c m / s
t=0.5s, un-,ax=2 8 cm/s
Fig. 21— Transient
collapse of deep
depression with
solidification. The
domain is 15 X 14
mm with a
maximum axial and
radial depression of
6.0 and 2.5 mm,
respectively.
dy/6T= 10~4
N/m-K.
Results and Discussions of Weld Pool
Collapse
These calculations used the SOLA-VOF
(Ref. 43) algorithm, and employing a coarse
12 X 32 grid ( 6 X 1 6 mm) required about
t w o hours on a CYBER 205 supercomputer. Figure 18 shows the computed results
indicating the very strong free surface disturbances and wave motion. Calculations
not actually depicted here showed that a
flat surface would be established after
about 0.5 s. W e note that an order of
magnitude analysis, using an expression
cited by Batchelor (Ref. 44) would give a
result within a factor of 3, which seems
quite reasonable, in view of the simplify-
WELDING RESEARCH SUPPLEMENT | 359-s
Conclusions
A n extensive set of calculations has
b e e n p r e s e n t e d t o describe the b e h a v i o r
o f the arc w e l d p o o l interface in arc
w e l d i n g o p e r a t i o n s . T h e ultimate o b j e c tive of this w o r k is t o p r o v i d e a full r e p r e sentation of the t w o - w a y interaction b e t w e e n the w e l d i n g arc a n d t h e w e l d p o o l .
T h e results r e p r e s e n t e d h e r e p r o v i d e a
partial a t t a i n m e n t of this goal.
Calculations are p r e s e n t e d t o describe
the b e h a v i o r of a w e l d i n g arc w h i c h is
m a d e t o i m p i n g e o n the t o p o f b o t h flat
a n d d e f o r m e d w e l d p o o l surfaces. T h e
c o m p u t e d results f o r a flat a n o d e agree
w e l l w i t h measurements r e p o r t e d in t h e
literature, b o t h regarding the heat flux a n d
t h e current distribution. (The arc pressure
is also w e l l represented).
Calculations p e r f o r m e d f o r d e f o r m e d
a n o d e surfaces h a v e s h o w n that the current and heat flux distributions m a y b e
v e r y m a r k e d l y a f f e c t e d b y the shape o f
t h e free surface of the w e l d p o o l . I n d e e d ,
it has b e e n s h o w n that t h e generally postulated Gaussian-type distribution may b e
t r a n s f o r m e d t o a b i m o d a l distribution
c u r v e . T h e v e r y i m p o r t a n t finding of this
w o r k is that, particularly f o r high-current
o p e r a t i o n s , t h e heat flux falling o n t h e
w e l d p o o l surface m a y n o t be specified
i n d e p e n d e n t l y o f the w e l d p o o l b e h a v ior — as has b e e n d o n e b y virtually all p r e vious investigators.
Calculations p e r f o r m e d t o describe t h e
shape a n d circulation of d e f o r m e d w e l d
pools have relied o n the previously c o m p u t e d heat flux a n d current distribution as
t h e b o u n d a r y c o n d i t i o n s ; thus p r o v i d i n g a
first-order c o u p l i n g b e t w e e n the arc a n d
w e l d p o o l calculations. H o w e v e r , the d e f o r m e d w e l d p o o l shape still had t o b e
d e d u c e d f r o m previously r e p o r t e d experimental data. These calculations h a v e
s h o w n that t h e melt circulation in these
systems m a y b e m a r k e d l y a f f e c t e d b y
b o t h the f r e e surface shape of the w e l d
p o o l and also b y t h e nature of the incident
heat flux. These findings suggest that t h e
f l o w patterns in w e l d pools m a y b e e v e n
m o r e c o m p l e x than previously t h o u g h t .
Finally, calculations h a v e b e e n p r e sented describing t h e transient collapse o f
a w e l d p o o l , o n c e the arc has b e e n extinguished. In these preliminary results, isot h e r m a l c o n d i t i o n s have b e e n p o s t u l a t e d
as a first, rather a p p r o x i m a t e representat i o n ; nonetheless, e v e n these preliminary
results w e r e able t o c a p t u r e s o m e essential features of experimentally o b s e r v e d
w e l d p o o l collapse, including t h e e n t r a p m e n t of gases a n d the expulsion of liquid
droplets.
It is t h o u g h t that t w o i m p o r t a n t issues
have b e e n raised b y the w o r k d e s c r i b e d
here. O n e , is that f o r t h e first t i m e it has
b e e n possible t o bring a b o u t a c o u p l i n g
b e t w e e n t h e b e h a v i o r o f the w e l d i n g arc
a n d the w e l d p o o l , such that t h e heat flux
360-S | SEPTEMBER 1990
falling o n t h e p o o l is directly calculated
f r o m t h e arc properties a n d that the arc
b e h a v i o r is a f f e c t e d b y t h e shape o f the
f r e e surface of the w e l d p o o l .
The s e c o n d , rather m o r e i m p o r t a n t
point, is that this interaction m a y i n v o l v e
quite substantial effects. M o r e specifically,
w h e n the w e l d p o o l is d e f o r m e d , t h e arc
characteristics m a y b e m a r k e d l y m o d i f i e d ;
i n d e e d , the previously e m p l o y e d p r o c e d u r e o f specifying t h e heat flux falling o n
the p o o l i n d e p e n d e n t l y o f t h e p o o l characteristics, is p r o v e n t o b e questionable
f o r m a n y situations. By the same t o k e n ,
f r e e surface d e f o r m a t i o n m a y m a r k e d l y
m o d i f y b o t h the shape of the melt-solid
interface a n d the circulation w i t h i n t h e
weld pool.
It should b e stressed that these c o m p l e x
p r o b l e m s are far f r o m being fully s o l v e d .
A great deal m o r e w o r k will b e r e q u i r e d
t o p r o v i d e a faithful representation o f the
d y n a m i c c o u p l i n g b e t w e e n the w e l d i n g
arc a n d the w e l d p o o l surface, particularly
t h e f r e e surface d e f o r m a t i o n and the f r e e
surface instabilities.
Acknowledgments
T h e a u t h o r s w i s h t o thank the U.S. D e p a r t m e n t o f Energy, Basic Energy Sciences
Division, f o r financial s u p p o r t of this investigation u n d e r Grant N o . DE-FG0287ER45289, a n d in part t o Florida State
University S u p e r c o m p u t e r C o m p u t a t i o n s
Research Institute, w h i c h is partially f u n d e d
by the U.S. D e p a r t m e n t of Energy t h r o u g h
C o n t r a c t N o . DE-FC-85ER25000.
References
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Appendix
B
e
F
g
fl
h
he
he
K
Kma:
k
kb
L
Nu w
P
2
Magnetic flux density (Wb/m )
Azimuthal magnetic field (Wb/m 2 )
Heat capacity ()/kg)
Electronic charge (C)
Void fraction (-)
Gravitational acceleration (m/s 2 )
Fraction of fluid (—)
Plasma enthalpy (J/kg)
Heat transfer coefficient (W/m 2 -K)
Plasma enthalpy at the edge of the
boundary layer (J/kg)
Plasma enthalpy at the anode temperature (J/kg)
Current (A)
Current density (A/m 2 )
Anode current density (A/m 2 )
Cathode current density (A/m 2 )
Radial current density (A/m 2 )
Axial current density (A/m 2 )
Drag index in source term (kg/m 3 s)
Maximum drag index (kg/m 3 s)
Thermal conductivity (W/m-K)
Boltzmann constant (J/K)
Characteristic length (m)
Nusselt number at the anode (—)
Pressure (Pa)
Prw
Qa
Qc
Qe
Qioniz
Qr
Q r ij
qout
Rc
Re w
r
t\ j
Sr
SR
T
Tamb
Te
Teiec
Tf
TL
Tr
Ts
Tsurf
Tw
t
ur
ure
U2
V arc
Vc
ve
Plasma Prandtl number at the anode temperature (—)
The anode heat flux (W/m 2 )
Convection contribution to the anode heat flux (W/m 2 )
Electron contribution of the anode
heat flux (W/m 2 )
Heat source from the cathode fall
to ionize the plasma (W/m 2 )
The radiative contribution to the
anode heat flux (W/m 2 )
Heat flux received at surface S, from
volume Vj located the distance r r j
away (W/m 2 )
heat loss from anode (W/m 2 )
Cathode spot radius (m)
Reynolds number at the anode (—)
Radial coordinate (m)
The direction vector from Sj to Vj
(m)
The differential surface in the radiation view factor relation (J/m2s)
Radiation source (W/m 3 )
Temperature (K)
Ambient temperature (K)
Plasma temperature at the edge of
the boundary layer (K)
Electron temperature (K)
Film temperature (K)
Liquidus temperature (K)
Reference temperature for Boussinesq approximation (1523 K)
Solidus temperature (K)
Surface temperature (K)
Wall temperature (K)
Time (s)
Radial component of velocity (m/s)
radial velocity at the edge of the
boundary layer (m/s)
Axial component of velocity (m/s)
Arc voltage calculated (includes approximate cathode fall voltage) (V)
Cathode fall voltage (V)
Plasma velocity parallel to the anode surface at the edge of the
boundary layer (m/s)
The differential volume in the radiation view factor relation (m3)
Work function of the anode material (V)
Axial coordinate (m)
Q.
o
>
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Q
X
Greek Symbols
a
fi
y
AH
At
A
fi
Mw
Me
Mo
P
Pe
Pr
Pw
a
Tzx
8
*c
4>
4>6
*
dy
Thermal diffusivity (m 2 /s)
Coefficient of thermal expansion
(K- 1 )
Surface Tension (N/m)
Latent heat of fusion (J/kg)
Time step (s)
Del operator
Viscosity (kg/m-s)
Viscosity at the anode temperature
(kg/m-s)
Viscosity at the edge of the boundary layer (kg/m-s)
Magnetic permeability of free space
(H/m)
Density (kg/m 3 )
Density at the edge of the boundary layer (kg/m 3 )
Reference density for Boussinesq
approximation (7200 kg/m 3 )
Density at the anode temperature
(kg/m 3 )
Electric conductivity (ohm-m) n
Shear stress (Pa)
Azimuthal direction (radian)
Work function of the cathode material (V)
Electric potential (V)
Thermal diffusion coefficient for
electrons (A/m-K)
The angle ry makes with the normal
to surface > (radian)
Surface tension coefficient (N/m-K)
5T
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WRC Bulletin 349
December 1989
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This bulletin contains two reports that evaluate the PWHT cracking susceptibility of several Cr-Mo steels
and several HSLA pressure vessel and structural steels.
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Postweld Heat Treatment Cracking in Chromium-Molybdenum Steels
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By C. D. Lundin, J. A. Henning, R. Menon and J. A. Todd
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Postweld Heat Treatment Cracking in High-Strength Low-Alloy Steels
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By R. Menon, C. D. Lundin and Z. Chen
Publication of this report was sponsored by the University Research C o m m i t t e e of the Welding Research
Council. The price of WRC Bulletin 349 is $35.00 per copy, plus $5.00 for U.S., or $10.00 for overseas,
postage and handling. Orders should be sent with payment to the Welding Research Council, 345 E. 4 7 t h
St., Room 1 3 0 1 , New York, NY 10017.
WELDING RESEARCH SUPPLEMENT 1361-s
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