Estimation of the Characteristic Properties of the Weld Pool during
High Productivity Arc Welding
Dr. Patricio Mendez
Prof. Thomas W. EagarMassachusetts Institute of Technology
October 4th, 1999
2
High Productivity Welding
• high current• high speed
Good welds
Bad welds
increase
d producti
vity
Good Weld
Bad Weld (humping)
Top view Cross sectionbeginning end
Top view
Cross sections
beginning end
(undercutting)
Sav
age
et a
l., 1
979
3
Outline
• Description of the problem and current understanding
• Methodology• Formulation of the problem• Results and discussion
4
Geometry of the Problem
• The productivity limiting defects are associated to a very depressed weld pool. (Bradstreet, 1968; Yamamoto, 1975; Shimada, 1982; Savage, 1979)
gouging region
trailing regiontrailing region
gouging region
rim
5
Possible Causes for Depression
• Marangoni Forces: they are dominant at lower currents (Heiple and Roper, 1982; Oreper and Szekely, 1984; etc.).
• Electromagnetic Forces: they increase with current.
• Arc Pressure: exerts a direct action on the free surface (Weiss et al., 1996; Lin and Eagar, 1983; Rokhlin and Guu, 1993).
• Gas Shear on the Surface: increases with current (Ishizaki, 1962; Choo and Szekely, 1991).
How to determine the dominant force in such a complicated geometry?
6
Methodology: Order of Magnitude Scaling
• Features:– Acts as a bridge between dimensional analysis
and asymptotic considerations.– Includes all of the desired driving forces.– Uses the governing differential equations.– Previous insight into the problem is especially
relevant.– Output:
• set of order of magnitude scaling factors for the solution of the problem
• determination of the relative importance of different driving forces and effects
• generalization of results from calculations or experiments
7
Elements of Order of Magnitude Scaling
• Normalization• Functional requirements (Domain partition)• Asymptotic considerations
This is applied to the governing equations
8
• This normalization generates dimensionless functions of the order of magnitude of one.
Normalization
0 10
1A B
F(A)
F(B)
X
x
F(X)
f(x)
ii
ii
i AB
AXx
)()(
)()()(
AB
AXx
FF
FFf
9
• The second derivatives must be of the order of one
• This condition assures that the first derivatives are also of the order of one.
0
1
0 1x
f(x)
0
1
0 1x
f(x) 5.2f
20f
OK not OK
This is a new conditionnot mentioned in other references
)1(O2
jixx
F
Limitation: equations ofsecond order or less
Functional Requirements
10
Functional Requirements: Domain Partition
• Choosing the appropriate domain the second derivatives are of the order of one.
• The size of the partition might be initially unknown.
domain of problem
subdomain for scaling
Limitation: many subdivisions may makethe process more difficult instead of simpler
11
• The unknown functions can be replaced by functions of the order of one with unknown scaling factors:
x
u
X
U
X
U
C
C
unknownfunction
unknownscalingfactor
dimensionlessfunction 1
Asymptotic Considerations: Extraction of Algebraic Equations from Differential Ones
12
• Dominant balance is used for the normalization of the differential equations .
• The normalized differential equations are transformed into algebraic equations:
0
Y
V
X
U
0
y
v
YU
XV
x
u
CC
CC
1 differential equation
1 algebraic equation: 1CC
CC
YU
XV
1 1
Asymptotic Considerations: Extraction of Algebraic Equations from Differential Ones
13
MatrixAlgebra
DimensionalAnalysis
DifferentialEquations
AsymptoticConsiderations
inspectional analysis
similarity
Szirtes 1998,Chen 1971,Barr 1987
Be
nd
er
an
d O
rsza
g,
19
78
dom
inan
t bal
ance
char
acte
ristic
val
ues
Denn,
198
0
inte
rmed
iate
asy
mpt
otics
Chen,
199
0; B
aren
blat
t, 19
96
Related Techniques
14
Formulation of the Problem
• 2-dimensional formulation, quasi-stationary traveling weld.
• Focus on depressed part of weld pool.• Driving forces included:
– Gas shear on the free surface– Arc pressure– Hydrostatic pressure– Capillary forces– Marangoni forces– Buoyancy forces
15
Formulation of the Problem
• 9 Unknowns:– (X), U(X,Z), W(X,Z), P(X,Z), T(X,Z) (X,Z), JX(X,Z), JZ(X,Z), B(X,Z)
• 8 Estimations– *, U*, W*, P*, T*, *, J*, B*
• 9 Equations:– mass conservation, Navier-Stokes(2), energy
conservation, Marangoni.– Ohm (2), Ampere (2), charge conservation.
16
Formulation of the Problem
• 17 Parameters:– L, , , k, Qmax, Jmax, e, g, , T, , Pmax, max, U, 0, ,
s
• 7 Reference Units:– m, kg, s, K, A, J, V
• 10 Dimensionless Groups– Reynolds, Stokes, Elsasser, Grashoff, Peclet,
Marangoni, Capillary, Poiseuille, geometric, ratio of diffusivity
17
Results: estimations
2/1
max
* 2 DU
kQT *
max
*
** 2 DUU
μm50*
K100* T
m/s 1* U
U*
*
*
18
Results: gas shear is the dominant driving force
1.00
0.34
0.08
0.07
0.06
0.03
0.03
0.03
7.E
-05
3.E
-04
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
arc
pres
sure
/ vi
scou
s
elec
trom
agne
tic
/ vis
cous
hydr
osta
tic
/ vis
cous
capi
llar
y / v
isco
us
Mar
ango
ni /
gas
shea
r
buoy
ancy
/ vi
scou
s
gas
shea
r / v
isco
us
conv
ecti
on /
cond
ucti
on
iner
tial
/ vi
scou
s
diff
.=/d
iff.
N2 N5N8N24N26N6N7N27N151
19
Discussion
• The driving forces previously suggested as possible causes for the big depression include:– Electromagnetic forces: they are not dominant,
because they tend to raise the surface instead of creating a depression (Tsai and Kou, 1989)
– Arc pressure: is not dominant, because it is too small to create the observed depression (Lin and Eagar, 1985; Rokhlin and Guu 1993)
– Marangoni: experiments were conducted on 304 stainless steel with high (230 ppm) and low (6 ppm) sulfur content.
20
Marangoni effect is of little importance
21
Conclusions
• In the high productivity regime:– Arc shear on the free surface is the dominant
driving force in the weld pool.– The weld pool degenerates into a thin liquid film.– The observed depression is approximately equal
to the weld penetration.– The order of magnitude of the dimensionless
groups obtained suggests that some terms in the governing equations could be simplified in more detailed calculations.
