Transcript Modeling of Welding Processes through Order of Magnitude Scaling

Modeling of Welding Processes
through
Order of Magnitude Scaling
Patricio Mendez, Tom Eagar
Welding and Joining Group
Massachusetts Institute of Technology
MMT-2000, Ariel, Israel, November 13-15, 2000
What is Order of Magnitude Scaling?
• OMS is a method useful for analyzing
systems with many driving forces
What is Order of Magnitude Scaling?
• OMS is a method useful for analyzing
systems with many driving forces
Weld pool
What is Order of Magnitude Scaling?
• OMS is a method useful for analyzing
systems with many driving forces
cathode region
A B
R

boundary layer
E F
ZS
cathode
Re
RC
H G
h
arc
Z
D
C
anode
Ra
Weld pool
Arc
What is Order of Magnitude Scaling?
• OMS is a method useful for analyzing
systems with many driving forces
cathode
Re
RC
Dw

E F
wire
w
cathode region
R
ZS
boundary layer
A B
H G
convection
through core
h
a
arc
anode
La
arc
Da
Dd
Z
D
C
thermal
boundary
layers
Regime I
anode
Ra
Weld pool
Arc
Electrode tip
Outline
• Context of the problem
• Simple example of OMS
• Applications to Welding
• Discussion
Context of the Problem
Context of the Problem
Engineering
Science
Arts
Philosophy
Context of the Problem
Engineering
Science
~1700
Engineering
Science
Arts
Philosophy
Arts
Philosophy
Context of the Problem
Engineering
Engineering
Science
~1700
Engineering
Science
Arts
Philosophy
Arts
Philosophy
~1900
Science
Applications
Fundamentals
Context of the Problem
Applications
Engineering
Engineering
Science
~1700
Engineering
Science
~1900
Science
~1980
Gap is
getting
too large!
Arts
Philosophy
Arts
Philosophy
Fundamentals
Example: Modeling of an Electric Arc
• Very complex process:
– Fluid flow (Navier-Stokes)
– Heat transfer
– Electromagnetism (Maxwell)
6
4
2
0
1945
cathode region
R

A B
E F
ZS
boundary layer
Lee 1996
Kim 1997
Choo, 1990
McKelliget 1986
Hsu 1983
cathode
Re
RC
H G
arc
h
8
Shercliff 1969 (analytical)
12
Maecker 1955 (approximate)
14
Ramakrishnan 1978
Glickstein 1979
16
Squire 1951(analytical)
number of dimensionless groups (m )
18
Yas'ko 1969 (dimensional analysis)
Lowke 1997
20
10
It is very difficult to
obtain general conclusions
with too many parameters
Z
D
C
anode
Ra
1950
1955
1960
1965
1970
1975
year of publication
1980
1985
1990
1995
2000
number of dimensionless groups
associated with geometry ( mg)
4
3
2
1
5
1
2
3
4
5
6
Choo 1990
Lee 1996
Kim 1997
McKelliget 1986
Hsu 1983
Yas'ko 1969
(dimensional analysis)
Squire 1951 (analytical)
Maecker 1955 (approximate)
Squire 1951 (analytical)
Shercliff 1969 (analytical)
Example: Modeling of an Electric Arc
Lowke 1997
Ramakrishnan 1978
Glickstein 1979
availability of
digital computers
7
number of dimensionless groups associated with the physics ( mp)
Complexity of the physics increased substantially
Generalization of problems with OMS
Fundamentals
Generalization of problems with OMS
Differential equations
Fundamentals
Generalization of problems with OMS
Asymptotic analysis
(dominant balance)
Differential equations
Fundamentals
Generalization of problems with OMS
Engineering
Asymptotic analysis
(dominant balance)
Differential equations
Fundamentals
Generalization of problems with OMS
Engineering
Dimensional analysis
Asymptotic analysis
(dominant balance)
Differential equations
Fundamentals
Generalization of problems with OMS
Engineering
Dimensional analysis
Matrix algebra
Asymptotic analysis
(dominant balance)
Differential equations
Fundamentals
Generalization of problems with OMS
Engineering
Artificial Intelligence
Dimensional analysis
Matrix algebra
Asymptotic analysis
(dominant balance)
Differential equations
Fundamentals
Generalization of problems with OMS
Engineering
Artificial Intelligence
Dimensional analysis
Matrix algebra
Asymptotic analysis
(dominant balance)
Differential equations
Fundamentals
Order of Magnitude Reasoning
Generalization of problems with OMS
Engineering
Artificial Intelligence
Dimensional analysis
Order of Magnitude Reasoning
Matrix algebra
Order of
Magnitude
Scaling
Asymptotic analysis
(dominant balance)
Differential equations
Fundamentals
OMS: a simple example
X  P1  P2  0
• X = unknown
• P1, P2 = parameters (positive and constant)
Dimensional Analysis in OMS
X  P1  P2  0
• There are two parameters: P1 and P2:
– n=2
Dimensional Analysis in OMS
X  P1  P2  0
• There are two parameters: P1 and P2:
– n=2
• Units of X, P1, and P2 are the same:
– k=1 (only one independent unit in the problem)
Dimensional Analysis in OMS
X  P1  P2  0
• There are two parameters: P1 and P2:
– n=2
• Units of X, P1, and P2 are the same:
– k=1 (only one independent unit in the problem)
• Number of dimensionless groups:
– m=n-k
– m=1 (only one dimensionless group)
– P=P2/P1 (arbitrary dimensionless group)
Asymptotic regimes in OMS
X  P1  P2  0
• There are two asymptotic regimes:
– Regime I: P2/P1 0
– Regime II: P2/P1 
Dominant balance in OMS
X  P1  P2  0
• There are 6 possible balances
 3
– Combinations of 3 terms taken 2 at a time:    6
 2
Dominant balance in OMS
X  P1  P2  0
• There are 6 possible balances
 3
– Combinations of 3 terms taken 2 at a time:    6
 2
• One possible balance:
X  P1  P2  0
balancing dominant secondary
Dominant balance in OMS
X  P1  P2  0
• There are 6 possible balances
 3
– Combinations of 3 terms taken 2 at a time:    6
 2
• One possible balance:
X  P1  P2  0
balancing dominant secondary
X
P2
1  0
P1
P1
Dominant balance in OMS
X  P1  P2  0
• There are 6 possible balances
 3
– Combinations of 3 terms taken 2 at a time:    6
 2
• One possible balance:
X  P1  P2  0
balancing dominant secondary
X
P2
1  0
P1
P1
P2/P1 0 in regime I
Dominant balance in OMS
X  P1  P2  0
• There are 6 possible balances
 3
– Combinations of 3 terms taken 2 at a time:    6
 2
• One possible balance:
X  P1  P2  0
balancing dominant secondary
X  P1 in regime I
X
P2
1  0
P1
P1
P2/P1 0 in regime I
Dominant balance in OMS
X  P1  P2  0
• There are 6 possible balances
 3
– Combinations of 3 terms taken 2 at a time:    6
 2
• One possible balance:
X  P1  P2  0
balancing dominant secondary
X  P1 in regime I
X
P2
1  0
P1
P1
“natural”
dimensionless group
P2/P1 0 in regime I
Properties of the natural dimensionless
groups (NDG)
• Each regime has a different set of NDG
• For each regime there are m NDG
• All NDG are less than 1 in their regime
• The edge of the regimes can be defined by
NDG=1
• The magnitude of the NDG is a measure of
their importance
Estimations in OMS
• For the balance of the example:
X
P2
1  0
P1
P1
• In regime I:
estimation
Xˆ
1  0
P1
Xˆ  P1
Corrections in OMS
Corrections
• Dimensional analysis states:
X 1 Xˆ 1  f ( P2 P1 )
correction
function
Corrections in OMS
Corrections
• Dimensional analysis states:
X 1 Xˆ 1  f ( P2 P1 )
correction
function
• Dominant balance states:
X 1 Xˆ 1  1
when P2/P10
Corrections in OMS
Corrections
• Dimensional analysis states:
X 1 Xˆ 1  f ( P2 P1 )
correction
function
• Dominant balance states:
X 1 Xˆ 1  1
when P2/P10
f ( P2 P1 )  1
when P2/P10
• Therefore:
Properties of the correction functions
Properties of the correction functions
– The correction function is  1 near the asymptotic
case
– The correction function depends on the NDG
– The less important NDG can be discarded with
little loss of accuracy
– The correction function can be estimated
empirically by comparison with calculations or
experiments
Generalization of OMS
• The concepts above can be applied when:
– The system has many equations
– The terms have the form of a product of powers
– The terms are functions instead of constants
• In this case the functions need to be normalized
xi 
X i  Ai
Bi  Ai
f j (x)  
F j ( X )  F j (A)
F j ( B)  F j ( A )
Application of OMS to the Weld Pool at
High Current
• Driving forces:
–
–
–
–
–
–
–
Gas shear
Arc Pressure
Electromagnetic forces
Hydrostatic pressure
Capillary forces
Marangoni forces
Buoyancy forces
electrode
rim
go
ug
in
g
re
tr a
gi
i
g
lin
s
on
re
gi
id
ol
on
ifi
ed
m
et
al
base metal
molten metal
channel
• Balancing forces
(b)
in s
go
ug
g
gi
c
re
ar
in
t
en
ic i
u ff h e a t
– Inertial
– Viscous
(a)
on
frozen gouging
region
tr a
il i
ng
re
so
gi
l id
on
ifi
ed
m
et
base metal
al
Application of OMS to the Weld Pool at
High Current
• Governing equations, 2-D model (9) :
–
–
–
–
–
–
–
conservation of mass
Navier-Stokes(2)
conservation of energy
Marangoni
Ohm (2)
Ampere (2)
conservation of charge
Application of OMS to the Weld Pool at
High Current
• Governing equations, 2-D model (9) :
–
–
–
–
–
–
–
conservation of mass
Navier-Stokes(2)
conservation of energy
Marangoni
Ohm (2)
Ampere (2)
conservation of charge
• Unknowns (9):
–
–
–
–
–
–
–
Thickness of weld pool
Flow velocities (2)
Pressure
Temperature
Electric potential
Current density (2)
Magnetic induction
Application of OMS to the Weld Pool at
High Current
• Parameters (17):
– L, r, a, k, Qmax, Jmax, se, g, n, sT, s, Pmax, tmax, U, m0, b, ws
• Reference Units (7):
– m, kg, s, K, A, J, V
• Dimensionless Groups (10)
– Reynolds, Stokes, Elsasser, Grashoff, Peclet, Marangoni, Capillary,
Poiseuille, geometric, ratio of diffusivity
Application of OMS to the Weld Pool at
High Current
• Estimations (8):
– Thickness of weld pool
– Flow velocities (2)
ˆ
Uˆ , Wˆ
– Pressure
P̂
– Temperature
Tˆ
– Electric potential
̂
– Current density in X
Ĵ X
– Magnetic induction
B̂
Application of OMS to the Weld Pool at
High Current
1/ 2
ˆ


 C  2 mU  D t max
TˆC  Qmax ˆC k
electrode
rim
T*
Uˆ C  2 U  D ˆC
tr a
*
gouging
region
ˆC  50mm
TˆC  100K
Uˆ C  1 m/s
rim
U*
n
ili
g
s
re
gi
id
ol
on
i fi
ed
m
et
al
base metal
rim
0.9
0.8
0.7
0.6
0.5
0.2
0.1
diff.=/diff.^
Marangoni / gas shear
0.03
7.E-05
capillary / viscous
0.03
buoyancy / viscous
hydrostatic / viscous
electromagnetic / viscous
0.06
0.03
arc pressure / viscous
0.07
3.E-04
0
0.08 convection / conduction
0.3
0.34 inertial / viscous
0.4
gas shear / viscous
1
1.00
Application of OMS to the Weld Pool at
High Current
Relevance of NDG (Natural Dimensionless Groups)
Application of OMS to the Arc
• Driving forces:
– Inertial
– Viscous
cathode region
A B
R
E F
H G
arc
h
• Balancing forces
boundary layer
RC

• Radial
• Axial
cathode
ZS
– Electromagnetic forces
Re
Z
D
C
anode
Ra
Application of OMS to the Arc
• Isothermal, axisymmetric model
• Governing equations (6):
–
–
–
–
conservation of mass
Navier-Stokes(2)
Ampere (2)
conservation of magnetic field
• Unknowns (6)
–
–
–
–
Flow velocities (2)
Pressure
Current density (2)
Magnetic induction
Application of OMS to the Arc
• Parameters (7):
– r, m, m0 , RC , JC , h, Ra
• Reference Units (4):
– m, kg, s, A
• Dimensionless Groups (3)
– Reynolds
– dimensionless arc length
– dimensionless anode radius
Application of OMS to the Arc
• Estimations (5):
– Length of cathode region
– Flow velocities (2)
Ẑ S
VˆRS ,VˆRZ
– Pressure
P̂
– Radial current density
Ĵ R
Application of OMS to the Arc
 m IJ
VˆZS   0 C
 4r



Zˆ S  15 mm
PˆS  1000 Pa
VˆZS  350 m/s
2

1
1
2
P
ZS
 I 
ˆ

Z S  
 4J C 
m IJ
PˆS  0 C
2
Z
VZ
Application of OMS to the Arc
• Comparison with numerical simulations:
Application of OMS to the Arc
• Correction functions
f Z  0.88 Re 0.058 h RC 
0.34
fVR  0.22 Re 0.026 h RC 
0.086
fVZ  0.55 Re 0.073h RC 
0.0068
f P  0.13 Re 0.17 h RC 
0.057
Application of OMS to the Arc
-0.1
VR(R,Z)/VRS
1
200 A
10 mm
-0.1
-0.2
1
2160 A
70 mm
-0.3
Z RC 2 
0.88 Re.058 h RC 0.34
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
difference in
cathode boundary
layer thickness
z
0
-0.1
0
0
-0.9
-0.7 -0.8
-0.6
-0.5
-0.4
-0.2 -0.3
r  R RC
1
Conclusion
• OMS is useful for:
– Problems with simple geometries and many
driving forces
– The estimation of unknown characteristic values
– The ranking of importance of different driving
forces
– The determination of asymptotic regimes
– The scaling of experimental or numerical data