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Advanced Scaling Techniques for the Modeling of Materials Processing

Patricio F. MendezColorado School of Mines

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Goals

• For people less familiar with scaling– will show how scaling is especially helpful for materials

processes

• For people familiar with scaling– will show a new relationship that permits to automate part

of the scaling process

• The reasoning applies to almost all materials processes

• For clarity, I’ll use a particular welding problem as an example, but the approach is valid beyond welding

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Materials Processes are “Multiphysics” and Coupled

• Welding example: free surface depression of weld pool. Can induce defects and lower productivity

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Materials Processes are “Multiphysics” and Coupled

• Multiphysics in the weld pool (12)

weld pool

substrate

solidified metal

arc

electrode

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Materials Processes are “Multiphysics” and Coupled

• Multiphysics in the weld pool– Inertial forces

weld pool

substrate

solidified metal

arc

electrode

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Materials Processes are “Multiphysics” and Coupled

• Multiphysics in the weld pool– Inertial forces– Viscous forces

weld pool

substrate

solidified metal

arc

electrode

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Materials Processes are “Multiphysics” and Coupled

• Multiphysics in the weld pool– Inertial forces– Viscous forces– Hydrostatic

weld pool

substrate

solidified metal

arc

electrode

gh

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Materials Processes are “Multiphysics” and Coupled

• Multiphysics in the weld pool– Inertial forces– Viscous forces– Hydrostatic– Buoyancy

weld pool

substrate

solidified metal

arc

electrode

ghT

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Materials Processes are “Multiphysics” and Coupled

• Multiphysics in the weld pool– Inertial forces– Viscous forces– Hydrostatic– Buoyancy– Conduction

weld pool

substrate

solidified metal

arc

electrode

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Materials Processes are “Multiphysics” and Coupled

• Multiphysics in the weld pool– Inertial forces– Viscous forces– Hydrostatic– Buoyancy– Conduction– Convection

weld pool

substrate

solidified metal

arc

electrode

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Materials Processes are “Multiphysics” and Coupled

• Multiphysics in the weld pool– Inertial forces– Viscous forces– Hydrostatic– Buoyancy– Conduction– Convection– Electromagnetic

weld pool

substrate

solidified metal

arc

electrode

J

BB

J×B

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Materials Processes are “Multiphysics” and Coupled

• Multiphysics in the weld pool– Inertial forces– Viscous forces– Hydrostatic– Buoyancy– Conduction– Convection– Electromagnetic– Free surface

weld pool

substrate

solidified metal

arc

electrode

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Materials Processes are “Multiphysics” and Coupled

• Multiphysics in the weld pool– Inertial forces– Viscous forces– Hydrostatic– Buoyancy– Conduction– Convection– Electromagnetic– Free surface– Gas shear

weld pool

substrate

solidified metal

arc

electrode

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Materials Processes are “Multiphysics” and Coupled

• Multiphysics in the weld pool– Inertial forces– Viscous forces– Hydrostatic– Buoyancy– Conduction– Convection– Electromagnetic– Free surface– Gas shear– Arc pressure

weld pool

substrate

solidified metal

arc

electrode

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Materials Processes are “Multiphysics” and Coupled

• Multiphysics in the weld pool– Inertial forces– Viscous forces– Hydrostatic– Buoyancy– Conduction– Convection– Electromagnetic– Free surface– Gas shear– Arc pressure– Marangoni

weld pool

substrate

solidified metal

arc

electrode

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Materials Processes are “Multiphysics” and Coupled

• Multiphysics in the weld pool (12)– Inertial forces– Viscous forces– Hydrostatic– Buoyancy– Conduction– Convection– Electromagnetic– Free surface– Gas shear– Arc pressure– Marangoni– Capillary weld pool

substrate

solidified metal

arc

electrode

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Materials Processes are “Multiphysics” and Coupled

Hydrostatic

Buoyancy

Electromagnetic

Free surface

Capillary

Gas shear

Arc pressure

Marangoni

Inertial forcesViscous forces

ConductionConvection

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Materials Processes are “Multiphysics” and Coupled

Hydrostatic

Buoyancy

Electromagnetic

Free surface

Capillary

Gas shear

Arc pressure

Marangoni

Inertial forcesViscous forces

ConductionConvection

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Materials Processes are “Multiphysics” and Coupled

Hydrostatic

Buoyancy

Electromagnetic

Free surface

Capillary

Gas shear

Arc pressure

Marangoni

Inertial forcesViscous forces

ConductionConvection

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Materials Processes are “Multiphysics” and Coupled

Hydrostatic

Buoyancy

Electromagnetic

Free surface

Capillary

Gas shear

Arc pressure

Marangoni

Inertial forcesViscous forces

ConductionConvection

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Materials Processes are “Multiphysics” and Coupled

Hydrostatic

Buoyancy

Electromagnetic

Free surface

Capillary

Gas shear

Arc pressure

Marangoni

Inertial forcesViscous forces

ConductionConvection

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Materials Processes are “Multiphysics” and Coupled

Hydrostatic

Buoyancy

Electromagnetic

Free surface

Capillary

Gas shear

Arc pressure

Marangoni

Inertial forcesViscous forces

ConductionConvection

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Materials Processes are “Multiphysics” and Coupled

Hydrostatic

Buoyancy

Electromagnetic

Free surface

Capillary

Gas shear

Arc pressure

Marangoni

Inertial forcesViscous forces

ConductionConvection

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Disagreement about dominant mechanism

• Experiments cannot show under the surface• Numerical simulations have convergence

problems with a very deformed free surface

Proposed explanations for very deformed weld pool• Ishizaki (1980): gas shear, experimental• Oreper (1983): Marangoni, numerical• Lin (1985): vortex, analytical• Choo (1991): Arc pressure, gas shear, numerical• Rokhlin (1993): electromagnetic, hydrodynamic,

experimental• Weiss (1996): arc pressure, numerical

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Scaling of a high current weld pool• Goals:

– Identify dominant phenomena:• gas shear? Marangoni? electromagnetic? arc pressure?

– Relate results to process parameters• materials properties, welding velocity, weld current

– Estimate characteristic values:• velocity, thickness, temperature

thickness

velocity

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Scaling of a high current weld pool• Governing equations (9):

U

z’

xz

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Scaling of a high current weld pool• Boundary Conditions:

at free surface at solid-melt interface

far from weld

free surface

solid-melt interfacefar from weld

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Scaling of a high current weld pool• Variables and Parameters

– independent variables (2)

– dependent variables (9)

– parameters (18)

from other models, experiments

with so many parameters Dimensional Analysis is not effective

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Classical Scaling Approach

1. Scale variables and differential expressions

2. Assume a set of dominant driving forces

3. Normalize equations

4. Solve for the unknown terms

5. Verify self-consistency

6. If not self-consistent, return to 3.

Roughly, this is the approach suggested by Dantzig and Tucker, Bejan, Kline, Denn, Deen, Sides, Chen, Astarita, and more

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Classical Scaling Approach

unknown characteristic values (9):

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Classical Scaling Approachgoverning equation

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Classical Scaling Approachgoverning equation

scaled variables

OM(1)

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Classical Scaling Approachgoverning equation

scaled variables

OM(1)normalized equation

output inputinput

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Classical Scaling Approach

output inputinput

two possible balances

B1

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Classical Scaling Approach

output inputinput

two possible balances

B1 B2

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Classical Scaling Approach

output inputinput

two possible balances

B1 B2

balance B1 generates one algebraic equation:

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Classical Scaling Approach

output inputinput

two possible balances

B1 B2

balance B1 generates one algebraic equation:

balance B2 generates a different equation:

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Classical Scaling Approach

output inputinput

two possible balances

B1 B2

balance B1 generates one algebraic equation:

balance B2 generates a different equation:

self-consistency: choose the balance that makes the neglected term less than 1

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Classical Scaling Approachtwo possible balances

balance B1 generates one algebraic equation:

balance B2 generates a different equation:

self-consistency: choose the balance that makes the neglected term less than 1

TWO BIG PROBLEMS FOR MATERIALS PROCESSES!

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Classical Scaling Approachtwo possible balances

balance B1 generates one algebraic equation:

balance B2 generates a different equation:

self-consistency: choose the balance that makes the neglected term less than 1

TWO BIG PROBLEMS FOR MATERIALS PROCESSES!

?

??

?

?

1 equation2 unknowns

1 equation3 unknowns

1. Each balance equation involves more than one unknown

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Classical Scaling Approach

1. Each balance equation involves more than one unknown

2. A system of equations involves many thousands of possible balances

two possible balances

balance B1 generates one algebraic equation:

balance B2 generates a different equation:

self-consistency: choose the balance that makes the neglected term less than 1

TWO BIG PROBLEMS FOR MATERIALS PROCESSES!

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Scaled equations (9)

all coefficients are power lawsall terms in parenthesis expected to be OM(1)

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Scaled BCs (6)

Boundary conditions

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Iterative process

• Simple scaling approach involves 334098 possible combinations

• There are 116 self-consistent solutions– there is no unicity of solution– we cannot stop at first self-consistent solution– self-consistent solutions are grouped into 55

classes (1- 6 solutions per class)

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Automating iterative process

• Power-law coefficients can be transformed into linear expressions using logarithms

• Several power law equations can then be transformed into a linear system of equations

• Normalizing an equation consists of subtracting rows

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Matrix of Coefficients

9 equations

6 BCs

one row for each term of the equation

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9 equations

6 BCs

one row for each term of the equation

18 parameters 9 unknown charact. values

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Calculation of a Balance1. select 9 equations2. select dom. input

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Calculation of a Balance1. select 9 equations2. select dom. input3. select dom. output

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Calculation of a Balance1. select 9 equations2. select dom. input3. select dom. output4. build submatrix of

selected normalized outputs

18 parameters 9 unknown charact. values

[No]P’ [No]S 9x9

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Calculation of a Balance

18 parameters 9 unknown charact. values

[No]P’ [No]S 9x9

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Calculation of a Balance

18 parameters 9 unknown charact. values

[No]P’ [No]S 9x9

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Calculation of a Balance

18 parameters 9 unknown charact. values

[No]P’ [No]S 9x9

incompatible

power law estimation

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Calculation of a Balance

incompatible

power law estimation

9 unknowns 18 parametersMatrix [S]

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Calculation of a Balance

9 unknowns 18 parametersMatrix [S]

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Self consistency

• can be checked using matrix approach

• checking the 334098 combinations took 72 seconds using Matlab on a Pentium M 1.4 GHz

secondary terms submatrices of normalizedsecondary terms

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Scaling resultsμm50cK100cT

m/s 1cU

c=

36

m

Uc

c

c

2/12 cc UD

kqT ccc

cc UDU 2

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force drivinggroups essdimensionl provide termsSecondary

Scaling results

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.

plasma shear causes crater

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Summary

• Materials processes are “Multiphysics” and “Multicoupled”

• Scaling helps understand the dominant forces in materials processes

• Several thousand iterations are necessary for scaling

• The “Matrix of Coefficients” and associate matrix relationships help automate scaling

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Approaches to the high current weld pool problem

• Experimental

Ishizaki, 1962,1980. Hammer blow, water droplets. Savage 1978, blank shot

Shimada, 1982

Force Balance:

Lin and Eagar, 1985

Savage, 1979

Adonyi, 1992

… and many more

very depressed weld pool become a “film”

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Approaches to the high current weld pool problem

• Numerical

Kumar A, Zhang W, DebRoy T, JOURNAL OF PHYSICS D, 2005

Lee, Welding Journal, 1997

Chen, 1998

Kim, Welding Journal, 1992

Tsai, Int. J. Num. Meth. Fluids, 1989Zacharia, Welding Journal, 1988

Most numerical models based on recirculating flows

Wei and Giedt, Welding Journal 1985

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Approaches to the high current weld pool problem

• Scaling: Focused on recirculating flowsOreper & Szekely, J. Fluid Mech.

1984, TWR 1986

DebRoy & David

Rev. Modern Phys

1995

Rivas & Ostrach,

Int. J. Heat Mass Transfer

1992

Chakraborty & Dutta

STWJ, 1992

No velocity BL

No thermal BL

Velocity BL

No thermal BL

Velocity BL

Thermal BL

T, R, D come from numerical calculations, experiments

T comes from scaling, R, D from numerical calculations, experiments