Mathematical Modelling of GTAW and GMAW

Mathematical Modelling of Gas Tungsten Arc Welding

(GTAW) and Gas Metal Arc Welding (GMAW) Processes

A thesis submitted in conformity with the requirernents for the degree of Doctor of Philosophy

Department of Metallurgy and Materials Science University of Toronto

@ Copyright by Massoud Goodarzi 1997

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Abstract ii

Mathematical h,fodelling of Gas Tungsten Arc Welding (GTAW) and Cas Metal Arc Welding (GMAW) Processes

Doctor of Philosophy, 1997

Massoud Goodarzi

Depanment of Metallwgy and Materids Science University of Toronto

GTA W Process

Models for the GTAW arc and the weld pool were developed. In the arc

model, the shape of the electrode tip was addressed and the effect of the elrcnode tip

angle in the range of 10 to 150 degrees on the arc properties. especially on the anode

current density, the heat tlux and the gas shear stress over the weld pool, was studied.

It was found that by increasing the rlecuodr tip ansla. the anode spot at the weld

pool surface tended to be more iocalized. This led to h i ~ h e r rnavimum heat tlux and

anode current density. On the other hand, the gas shear stress increased with decreas-

ing electrode tip angle. The variation in the anode current density, the heat tlux and

the gas shear stress with the elecuode tip angle has signifrcant effect on the weld pool

properties. By developing a model for the weld pool the variation in the weld pool

due to variation in the electrode tip angle was studied.

In the weld pool model, by considering the four tlow driving forces into the

weld pool. it was round bat, in determining the flow pattern into the weid pool and

hence i ts shape, the gas shear stress and surface tension arc dominant forces.

GMAW Process

A model for GMAW arc is developed. ln this model a cathode spot detïned

Abst ract i ii

such chat the electrons with a ccnstant current density emit ti-om the cathode surface.

It was found that the radius of the cathode spot is a tùnction of applied current. arc

length. eleçuode diameter and shielding p. This mode1 can successtùlly predict the

arc properties. such as temperature. gas velocity and gas pressure at the surface of the

calhode in both argon and helium for 2-10 mm arc of applied currents from 100 to

250 A.

Thesis Supervisors:

Professor James M. Toguri and Dr. Tuck Chow (Roland) Choo

Acknowledgements iv

ACKNOWLEDGEMENTS

I would likc to offer rny sincere thanks to Prof. James M. Toguri for provid-

ing me with support h m the tint sreps. and Dr. Roland Choo for his guidance during

the course of rny study.

M m y thanks go to al1 members of the Chernical Metallurgy Rcsearch group

for their Mendship during rny staying in Toronto. Special thanks to Dr. Tomio Ta-

kasu for his elaborating comments on my thesis.

I am sxuemely grateful to my wife. ÿnd my children for rheir patient during

years far from their heloved people and land. Without efforts of my wife to keep cve-

rything organized. doing this study was hardly possible.

1 am always thankfui to my parents. for their moral support since day one.

Withciut them, I would not be here today.

Finülly I would likc to mknowledge the Ministry of Culture and Higher Edu-

cation of Islamic Rcpublic of Iran for providing me the opportunity to continue m y

study abroud.

Table of contents v

ACKNOWLEDGEMENTS

LIST OF TABLES

LIST OF FIGURES

TABLE OF CONTENTS

CHAPTER ONE: TNTRoaucnorq 1.1 GAS TLWGSTEN ARC WELDING (GTAW) 1.2 GAS METAL AKC WELDING (GMAW) 1.3 THE ELECTRIC ARC

1.3.1 The Cathode Region 1.3.3 The Arc Column 1.3.3 The h o d c Region

1.4 DEFINITION OF THE PROBLEM AND OBJECTIVES OF THIS STUDY

1 - 4 1 Detinition of the Problem 1 A.2 Ohjectivcs of the Present Investigation

1.5 LITERATIJRE REVIEW OF MAITEMAp[ICAL MODELLING OF THE ARC'

REFERENCES

3.1 WTROD~~CTION 2.2 MODEL DEVELOPMENT 2.3 GOVERNING EQUATIONS

2.4 BOUNDARY CONDITTQNS AND

2.4.1 Boundary Conditions 2.4.3 Numerical Method

2.5 R E S U L U AND DISCUSSION

2.5. i Arc Properties

W E R I C A L METHOD

2 - 5 1 Anode cu ren t Density 2.5.3 Anode Heat Flux 2.5.4 G u Shear Stress on the Anode Surface

2.6 SUMMARY REFERENES

xix

Table of contents vi

CHAPTER THREE: ~ C T OF THE C-E Tll?

m 70

3.2 hd0DEL DESCRIPTION 70 3.3 RESULTS A N D DISCUSSION 74

3.3.1 Arc Prop~rties 76 3.3.2 Anode Cument Density and Heat Flux 86 3.3.3 Shear Stress 94

1 . 1 INTROD~JLTION 1.2 WELU POOL MODEL

4.7.1 Goveming Equations 102 4.2.2 iMclting and Solidification Modelling 105 4.2.3 B o u n d q Conditions 106 4 . Matcrial Properties for the Workpiecc 108 4.3.5 Numerical Method 1 O8

4.3 RES!LTS AND DISCUSSION 109 4.3.1 Electromagnetic Farce 4.32 Buciyancy 4.3.3 Gas Shear Stress 4.3.4 Surfacc Tension 4.3.5 Combination of Driving Forces 1.3.6 Corn parison with Experimcntal Data

5.3 GOVERNING EQIJATIONS AND BOWDARY CONDITIONS 134 5.4 RESULTS AND DISCUSSION 138

5.1.1 Cathode Spot Radius 5.4.2 Mode1 Veritkation 5.4.3 Arc Properties 5.1.4 Effkct of the Shielding Gris 5.4.5 Effect of the Electrode Tip Shape

Table of contents vii

6.1 CONCLL~S~ON 6.1.1 GTAW Process 6.1.2 GMAW Process

6.2 SL~C~C~ESTIONS FOR FUTURE STUDIE!

6.3.1 GTAW Process 6 - 2 2 GMAW Process

APPENDIX r I 186

List of tables viii

LET OF TABLES

CHAPTER TWO

Table 2- L : Boundary Conditions.

Table 2-2: The Corresponding Quantities for the Diffcrent ConservaUon

Equa tions.

Table 2-3: Arc Parameters for Different Arc Lengths and Applied Currents.

A Cornparison with a Sarnple of Other Numerical Results. (T,,.

(K): u,,, (rn.s*% A@. (VI; 4, (Pa)).

Table 2-1: Maximum Anode Current Density and Maximum Heat Flux for

Different Currents and Arc Lengths. t J,. ,,, (~.rnrn- ') ; q,. ,,. I: ~ . r n r n - ' ) ) .

Table 2-5: Ctmtribution of Different Heat Transfer Mechanisms to the An-

ode Heat Flux. (q,. %: q,. %: q, %).

Table 3- 1:

Table 3-2:

Table 3-3:

Table 3-4:

Boundary Conditions.

Arc Paramr ters for Differen t Arc Lengths. Applied Cunents md

Elecuode Tip Angles. (Tm,. (K): u,, ( m s - l ) ; A@. (V): AP,,

(Pa)).

Maximum Anode Current Density. Maximum Heat Flux and

Maximum Temperature of the Gas on the Anode Sudace for Dis-

lerent Elecuode Tip Angles, Arc Lengths and Applied Currents.

(.Jas (~-rnm- ' ) ; qa, mm, (W-mm-'): Ta. mm. (KI 1.

Total Heat Transferred to the Anode and Contributions of' the

Three Mechanisms for Different Situations. (Qa. a. Q,. Q,, (W)).

CHAPTER FOUR

Table 4- 1 : Boundary Conditions.

Table 4-2: Data Used for the Surface Tension Calculation [ IO].

List of tables ix

Table ;!-3: The Corresponding Qumtities for the Different Conservation

Equa tions.

Table 5- 1:

Table 5-2:

Table 5-3:

Table 5-3:

Table 5-5:

Table 5-6:

Boundary Conditions.

Maximum Temperature. Maximum Velocity and Eleçtric Poten-

tial Difference far a 250 A Arc o f Different Cathode Spot Radius.

The Corresponding Cathode Spot Radius with the Minimum

Electric Potential Difference for Different Cases.

Arc Properties for Different Arc kngths and Applied Currents.

(Tm,, (KI: u,, (ms-'1: A@. (V; R,, (mm)).

The Ratio of Droplrt Contribution to Radiation luid Convection

C(mtributian.

Arc Proprnies for Different Arc Len~ths and Applied Currenw in

Heliurn. (Tm=. (K); u,,, (m.al?: Rc. (mm)).

List of figures x

LIST OF FIGURES

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 1.5

Figure 1.6

Figure 1.7

Figure 1 .X

Schematic diagram of the GTAW process.

Schematic diagram of the GMAW process.

Variation in volume and transfer rare of drops with welciing cur

rent (steel electrode). [41

Voltqe distribution dong an arc shown schematicaily.

Temperature for thermionic emission at various çurrent density

lcvels [7]. hdividual points show boiling point o f pure elements

versus work function.

Mrasured and calculated isotherms of a tiee-burning argon arc. (1

= 200 A. L, = 10.0 mm. P = 1 atm). [491

Speçies temperature distribution in the anode boundary layer. (Jc

= 9% 1 Oh ~ . r n - ~ . P = I O 0 kPa. Gas = Arson). [ 101

Calculated cathode surface temperatures for a 200 A arc in argon

at 1 atm compared with expenmental measurements of Haidar

and Farmer [34. [62]

C H A ~ R TWO

Figure 2.1 Schematic diagram of the welding ürc (GTAW).

Figure 2.7 Caiculation domain for GTAW (scheinatic).

Figure 2.3 Contïguration of radiation view factors.

Figure 2.4 Isothens of 10.0 mm arcs in cornparison with the experimentd

results of Hsu et. al. [Ml. For cdculated results, the cathode cur-

rent drnsity is 6 . 5 ~ 1 0 ~ ~.rn-'. a: 1 = 100 A; b: 1 = 200 A: c: 1 =

300 A.

Figure 2.5 Typical velocity profile in GTAW arcs of two different Iengths.

Ic = 6 . 5 ~ 1 0 ~ ~ . r n - " 1 = 220 A. a: L, = 2.0 mm; b: L, = 10.0

mm.

List of figures xi

Figure 2.6

Figure 2.7

Figure 2.8

Figure 3.9

Figure 2-10

Figure 2.1 1

Fisure 2.12

Figure 2.13

Fisure 2.14

Figure 3.15

Figure 3-16

Figure 2.17

Figure 2.18

Distribution of axial velocity dong the ais of symrneüy for LOO

A arc of 10.0 mm length.

Variation of axial velocity at the cenue line of the arc for differ-

rnt arc kngths. Numbers are lrngth of arcs in mm. I = 300 A.

Typical pressure distribution in a GTAW arc. I = 2013 A. L,, =

10.0 mm. Jc = 6 . 5 ~ 107 ~ . m - ' .

Variation of the gas pressure at the surface of the anode with the

arc Irngth and applied current. Numbers are appiied current in A.

Variation of rlccuomagnetic force at the surface of the anode

with arc lensth. Numbers are the arc lengths in mm.

Variation of arc voltqe wih applied cumnt and arc kngth in

çomparism with experimental and se lected cdculated results.

Anode current density for d i f f e ~ n t applied currents compared

with experimental [24], and theoretical [ l I l results. a: I = 100 A:

h: 1 = 200 A: c: I = 300 A. L, = 6.3 mm. Jc = 6.5~10' ~ . m - ' .

Anode current density for arcs with different lengths in cornpari-

son with experimental [NI results. Numbers are arcs lengths in

mm. 1 = 200 A.

Variation of maximum anode çurrent density with an: length in

cornparison with the rxperimrnral [25] values. 1 = IO0 A.

Anode hzar tlux for different applied current in çomparison with

the rxprrimental [24] and theoretical [ 1 1 ] results. a: 1 = 100 A: b:

1 = 100 A: c: 1 = 300 A. L, = 6.3 mm.

Anode hrat flux for different applied currents in cornparison with

oxprrimental [27] values. a: I = 50 A; b: 1 = 100 A; c: I = 150 A.

La, = 5.0 mm.

Anode hrat flux for arcs with difkrent kngths in cornparison

with experimental [XI results. Numbers are an: longhs in mm.

= 200 A.

Variation of maximum anode heat flux with arc length and ap-

plied current in cornparison with the experimental [25, 261 values.

List of figures xii

Figure 2.19

Figure 2.20

Figure 2.21

Figure 2.22

Figure 2.13

Figure 7.24

Figure 2.25

Contribution of the three rnechanisms in the anode heat tlux in

corn parison with other theoretical estimation [ 1 1 1. a: I = 100 A:

h: 1 = 200 A: c: I = 300 A. L, = 6.3 mm.

Variation of temperature at the vicinity of the anode surface with

applied current and radial distance.

Variation of Electrical conductivity of Argon with temperature

[91-

Variation of three mechanisms of heat trmster contribution into

the anode heat flux with the arc length. I = 200 A. a: Electron: b:

Convection; c: Radiation.

Variation of radiai velocity with the radial dismce and the arc

lèngth. I = 200 A-

Variation of shear stress with radial distance and applied current

in cornparison wirh othrr theoretical estimation [ I l ] . La, = 6.3

mm, a: C = 100 A; b: I = 200 A: c: I = 300 N400 A.

Variation of die shear stress with radial distance and arc length in

cornparison with other theoretical estimation [ L 11. I = 200 A. a:

L,, = ?.O m d 3 . 2 mm: b: L, = 6.3 mm: c: La, = 10.0 md17 .7

mm.

CHAPTER THREE

Figure 3.1 Effeçt of rlecuode tip _oeometry on the weld pool shape and size

111. Fisure 3.2 Clllçulation domain for tapered rlecuode GTAW (schematic).

Figure 3.3 Efftxt o f the electrodr tip ansle on the maximum temperature and

iis position. Jc = 108 A .rn-?.

F i 3.4 Vilnation in the area of the cathode surface covered by plasma as

a function of the cathode tip angle 131. 1 = 200 A. L, = 5.0 mm.

Figure 3.5 Variation of the cathode surface area with the elrctrode tip angle.

Figure 3.6 Variation of plasma temperature at 1.5 mm from the tip of the

tungsten elecuode with the electrodt: tip angle and applied cur-

List of figures xiii

Figure 3.7

Figure 3.8

Figure 3.9

Figure 3.1 O

Figure 3.1 1

Figure 3.12

Figure 3.1 3

Figure 3.14

Figure 3.15

Figure 3.1 6

Figure 3.17

rent. Numbers are applied current in A.

Isotherrns of 10.0 mm arcs in cornpaison with the experirnental

results of Hsu et. ai. [ 1 11. a: 1 = LOO A; b: 1 = 200A.

Radial distribution of temperature at 1.5 mm from the cathode tip

in cornparison with experimental data [31. 1 = 200 A. L, = 5.0

mm. a: a = 20.71 deg (Exp.: a = 18 deg); b: a = 60 deg.

Variation of the axial distribution of the axial velocity with elec-

trode tip angle for 200 A and 5.0 mm arcs. Numbers are rlec-

trode tip angles in deg.

Variation of the radial distribution of the axial velocity with elec-

uode tip angle. 1 = 200 A. Numbrrs are the electrode tip angle in

deg .

Variation of the radial distribution of pressure at the anode sur-

face with electrodr tip ansle. 1 = 100 A. Numbers are rlectrode

tip angle in deg.

Disüibution of pressure contours for three electrode tip angles. I

= 200 A. L, = 5.0 mm. a: a = Y. 18 deg; b: a = 37.33 deg; c : a

= 60.00 des.

Variation of axiai distribution of pressure with zlectrode tip an-

sir. Numbers are electrode tip angle in deg.

Variation of arc voltage with electrode tip angles and applied cur-

rent. Numbers are applied cument in A.

Variation in the arc volüige with arc length and applied çurrent in

cornparison with experimentd data [ l ] . Numbers are arc length in

mm.

Distribution of rlectnc potential in the arc for different elecuode

tip angles. The isotherm of 10000 K shows the domain that the

results are diable. I = 200 A. L, = 5.0 mm. a: a = 9.18 deg; b:

a = 60.00 deg, c: a = 100.06 deg.

The variation of the voltage drop into the electrode with the elec-

trode tip angle. Numbers arc applied cunent in A.

List of figures xiv -

Figure 3.18 Variation of a: maximum anode current density, and b: maximum

cas temperature on the anode surface. with the ebctrode tip angle C

and applied current for a 5.0 mm arc. Numbers are the applied

current in A.

Figure 3.19 Variation of maximum anode current density with electrode tip

an# and arc length. 1 = 200 A. Numbers are an: length in mm.

Figure 3.20 Variation of the maximum anode heat tlux with rlectrode tip an-

ale for arcs of different applied currents. Numbers are arc iength - in mm. a: I = 100 A; b: 1 = 200 A; c: i = 250 A.

Figure 3.21 Trmsferred heat to the anode versus the rlrctrode tip angle for

200 A arcs with different arc lengths. a: L,, = 2.0 mm: b: La, =

5.0 mm; c: L, = 10.0 mm.

Figure 3.22 Effecr of the rlectrode tip an_olr on the radial distribution of the

hèat tlux to the anode. 1 = 200 A. L, = 5.0 mm. a: Total heat

t'lux; b: Electron contribution; c: Convection contribution; d: Ra-

diation contribution.

Figure 3.23 Variation of radiai velocity of the gas with the rlectrode tip an-

de. a: Effeçt of arc length. I = 200 A. Numbers are arc length in L

mm. b: Effect of applied current Luc = 5.0 mm. Numbers are ap-

plieci current in A.

Figure 3.24 Effect of the rlrcirode tip ansle on the distribution of shear stress

in arcs with dii'frrent lengths. I = 200 A. a: L, = 2.0 mm: b: L,

= 5.0 mm; c: L, = 10.0 mm.

CHAPTER FOUR

Figure 4.1 Schcmatic representation of the GTAW arc in the weld pool. The

various physical phciioffiena occurring in the workpiece (the nght

side) and the calcuiation domain (the left side) are indicated. The

orisin of the calculation domain is located at point A.

Figure 4.2 Variation of apparent viscosity with solid volume fraction (used

in this study).

List of figures xv

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 1.7

Figure 4.8

Figurc 4.9

Figure 4.1 O

Fisurc 4.1 1

Fisure 1.12

Figure 4.1 3

Figure 4.14

Figure 4.1 5

Radial distribution of the heat flux for 9.18 and 60.0 degree elec-

trode tip angles. 1 = 200 A. Lare = 2.0 mm.

Radial distribution of the current density for 9.18 and 60.0 degee

clectrode tip angles. 1 = 100 A. L, = 2.0 mm.

Radial distribution of the shear stress for 9.18 and 60.0 degree

clecuode tip angles. 1 = 200 A. L, = 2.0 mm.

Current density distribution into the workpiece for a: 9.18 and b:

60.0 degree elecuode tip angle. 1 = 200 A. L, = 2.0 mm.

Electromagnetic force into the weld pool for a: 9.18 md b: 60.0

degree c1ectrode tip angle. 1 = 200 A. L, = 2.0 mm.

Velocity pattem into the weld pool and liquidus and solidus lines

for a: 9.18 and b: 60.0 degree electrode tip angle due to electro-

magnetic force. 1 = 200 A. L, = 2.0 mm.

Buoyancy dnving force into the weld pool for a: 9.18 and h: 60.0

degrce electrode tip angle. 1 = 200 A. L, = 2.0 mm.

Vzlticity pattern into the weld pool and liquidus and solidus lines

for a: Y. 18 and b: 60.0 degree electrode rip angle due to buoyancy

driving force. 1 = 200 A. L, = 2.0 mm.

Vclocity pattem into the weld pool and liquidus and solidus lines

for a: 9.18 and b: 60.0 degree electrode tip angle due to gas shear

stress. I = 200 A. L,, = 2.0 mm.

Variation of maximum temperature in the weid pool for differen~

driving forcc as a Iùnction of the clecuode tip angle. I = 200 A.

L,, = 2.0 mm.

Variation of surface tension gradient as a function of temperature

for Fe-0.022 wr8 S. Required information was derived h m Sa-

hoo et. al. [ 101.

Velocity pattern into the weld pool and Liquidus and solidus lines

for a: 9.18 and b: 60.0 degree electrode tip mgle due to surface

tension. I = 200 A. L, = 2.0 mm.

Radial distribution of the Marangoni shear and surface tempera-

List of figures mi

ture for stainless steel AIS1 304 with the electrode tip angle as

the parameter. 119

Figun 4.16 Puddle shape in GTAW of stainless steel AlSI 304 with 2.0 and

5.0 mm arcs. The electrode tip angle changes from Y. 18 to 13 1.4 1

degrees. 1 20

Figure 4.17 Total surfacc stress (rS1 + rgaS ) and temperature at the weld pool

surface for stainkss steel AIS 304 after GTAW with a 2.0 mm

mes . arc. The electrode tip angle is 9.18, 60.00 and 13 1.11 de, L 23

Figure 4-18 Flow pattern into the weld pool for stainless steel AIS? 304 in the

case o f different e!ectrode tip angles and arc lengths. 133

Figurc 4.19 Variation in the weld pool width as a function of the elecrrode tip

angle. Cornparison with experimental results [20]. 135

Figure 4.20 Variation in the weld pool depth as a tùnction of the electrode tip

angle. Comparison with experimental results [?O]. 125

Figure 4.21 Variation in the depthlwidth ratio as a function of the zlectrode

tip angle. Cornparison with experimental results [2 11. 127

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figurc 5.5

Figure 5.6

Figure 5.7

Schema~ic of the welding arc (GMAW). 133

Calculrition damain for GMAW (schematic). 135

Variation in the maximum tempenturc with the cathode spot ra-

dius t'cir 750 A Ar ünd He arcs as a function of arc Iength. 141

Variation in the maximum velocity with the cathode spot radius

for 250 A Ar and He arcs as a function of arc Iength. 142

Variation in the elecüic potential differcnce with the cathode spot

radius for 250 A Ar and He arcs as a hnction of arc length. 143

The radial distribution of temperature at different distances from

the cathode plate in cornparison with the experimentril data [ l 11. 1

= 150 A. Distance from the cathode plate a: 7.5 mm; b: 5.0 mm;

c: 3.5 mm. 145

The radial distribution of temperature at different distances from

List of figures xvii

Figure 5.8

Figure 5.9

Figure 5.1 0

Figure 5.1 1

Figure 5.12

Figure 5.13

Figure 5.14

Figure 5. f 5

Figure 5.16

Figure 5.17

Figure 5.18

Figure 5.19

Figure 5.20

the cathode plate in comparison with the experirnental data [ l I l . I

= 250 A. Distance tram the cathode plate a: 7.5 mm; 6: 5.0 mm:

c: 2.5 mm.

Variation in Cas velocity with appiied current tor 10.0 mm arcs in

comparison with experimental results [Il] .

Variation in the total current at the tip of the electrode as a fùnc-

tion o f applied current for different electrode diameters. Numbers

are the electrode diarneter in mm.

Isotherms of 10.0 mm arcs with different applied currents. a: I =

150 A; b: 1 = 250 A; c: I = 350 A.

Variation in the arc voltage with applied current and arc lengrh.

Numbers are arc length in mm.

Variation of eleciric field with a: applied current of I 0 . 0 mm

arcs; b: arc length of 250 A arcs.

Variation in the thermal power transferred to the cathode due t«

a: radiation and b: convection with the applied cumnt as a func-

tion of arc length.

Variation in the transîèrred heat to the anode due to elecuons

condensation with the applied current.

Variation in the percentagc of current which reach the tip of the

rlecuode with the a: applied current; and b: electrode diametcr.

Effect of the applied current on the decîromagnetic force around

the slectrode tip. a: I = 150 A; b: 1 = 250 A: c: I = 350 A.

Schematic of elecîrons path when there is side current into the

consumable eiectrode.

Electrical conductivity of Ar and He as funclions of temperature.

151 Isotherrns of 10.0 mm helium arcs with different currents. a: I =

150 A; b: 1 = 250 A; c: I = 350 A.

Velocity profiles of the helium arcs of Figure 5.19. a: I = 150 A:

b: I = 250 A; c: 1 = 350 A.

List of figures xviii

Figure 5.2 1

Figure 5.23

Figure 5-23

Figure 5.21

Figure 5.25

Figure 5.26

Figure 5.27

Figure 5-28

Variation in the cumnt into the elecuodt: with the applied current

and distance fiom the electrode tip. a: [ = 150 A: b: 1 = 250 A; c:

I = 350 A. 16 1

Variation in the heat transferred to the workpiece or the electrode

in unit time with the applied current in helium and argon. a: Elec-

won heat transfer to the ekcuode; b: Convective heat transfer to

the workpiece; c: Radiâtive heat transfer to the workpiece. 163

Effeçt of the shielding gas on the geomrtry of the weld pool is

shown schematically. [23] 164

Variation in the eleciron contribution to the total heat tramferreri

to the weld pool in He and Ar as a function of applied current. 164

Schematic shape of the electrode tip in two different metal trans-

fer modes. a: GIobular; b: Spray. L 65

Isotherms ut' 10.0 mm. 200 A arcs for two rlectrode tip angles. a:

a = 10 deg; b: cc = 180 &_o. 167

Efkct of the rlecuode tip angles on the velocity pattern for 10.0

mm. 200 A arcs. a: a = 10 deg; b: cc = 180 deg. 168

Effect of the rlectrode tip mgle on the distribution of the pressure

in 10.0 mm. 200 A arcs. a: a = 10 deg; b: a = 180 deg. Numbrrs

are pressure difference in P a

--

A~

A* b

B o

1

C~

E

E Z

G~

Hf

1

I . J

Ij, arc

J

J a

I arc

Jc

'max

J Z

K

K scg

Nomenclature xix

Currcnt cross section near cathode [m-'1

Temperature coefficient of surface tension for iron [4.3x10-" N.(m.K)-'1

Azimutai component of magnetic field [Tl

Empirical constant appearing in E equation [ 1.441

Empirical constant appearing in e equation [1.92]

Specific heat at constant pressure [J.(kg.K)-II

Empiriçal constant appearîng in F~ equation [0.091

Eiecuic field vector [V.m-'1

Axial elcctric tïeld [ ~ . m - ' 1

Generation o f turbulent energy [kg.( m.s3)- '1

Latent heat of fusion of the metai [ J . k g l ]

Applicd current [A]

Total current in die rlecuode at slab j [A]

Total current in the arc at slab j [AI

Currcnt density vector [ ~ . m - ' ]

Anode surrent density [A.m-']

Input current density to the weld pool from the arc [~ . rn- ' ]

Cathode current density [~.rn-'1

Maximum çurrent density at the centre [ine of the arc [ ~ . m - ' ]

Radial current density [A.rn"l

Axial current density [ ~ . m - ' ]

Turbulent kine tic energy [m2.s2]

Equilibrium constant for segregation [- 1

Nomenclature xx

Arc length [ml

Pressure [Pa]

Maximum pressure at the centre Iine of the arc [Pa]

Turbulent Prandti number [- j

Prandtl number at Tw [-]

Anode total heat [Wl

Anode heat due to convection and conduction [W]

Anode heat due to electrons w] Volumetric heat source from the cathode fail to ionize the plasma [Lm-)]

Anode heat due to radiation w] Güs constant [8.3 144 mole.^ )-' ]

Cathode spot radius [ml

Elèctrode radius [m 1

Calcula tion domain radius [m 1

Cathode surface area [m21

The anode surface element. [ d l

Radiation e n e q y source of V. [n J

Radiation loss V . [J.s ' .m-'l J

Source rate of [ -1

Temperature [KI

Eleçiron temperature at the rdge of the plasma [KI

Elcctron temperature at O. 1 mm from the anode surface [KI

Gas temperature at inllow boundary [Kj

Liquidus temperature of AIS1 304 stainless steel [1733 KI

Mclting point of iron in y equation [MO9 KI

Melting point of the rlectrode material [KI

Nomenclature xxi -

T m. wp

T r

Ts

T water

v

vc

'j

"s b

C

*e

t2

l

O c'

h

hr

h W

k

1

k,

keff

q a

'la,,

9,

Melting point of the workpiece materiai [KI

Reference temperature [ 1523 K]

Solidus temperature of AISI 304 stainless steel [ 1523 KI

Temperature of wall [KI

Temperature of coohng water [KI

Arc voltage [VI

Cathode faIl voltage [VI

The plasma volume element [m3]

Activity of sulfur [-wt% S]

Experimental constant in equation ( 1-8)

Specd of light in vacuum [299,792.458 rn .s1]

Elcctrodt: diameter [mj

Elemrnülry charge [ 1.602~ 10-19 CI

Liquid fraction [-j

Standard acceleration of gravity l9.80665 m.s-?]

Enthalpy [J .kgl ]

Enthalpy of the gas at Te [J .kg l ]

Enthalpy of the gas at Tw kg-'1

Thermal conductivity [W.(m. K)-l]

Entropy Factor in y equation [0.00318 (wt%)-'1

Boltzmann constant [1.38x J . K 1 ]

Eflcctive thermal conduc tivity [W.(rn. K)"]

Turbulent component of thermal conduc tivity [W.(m. K)" j

Total anode heat flux [ ~ . m - ' ]

Hcat flux to the weld pool frorn the arc [~.rn- ' ]

Anode heat flux due to convection and conduction [ ~ . r n - ' l

Nomenclature xxii

9, Anode hrat tlux due to electrons [W-rn-']

q r Anode heat flux due to radiation [ ~ . r n - ' !

q, i Radiative heat tlux receive by Si y.m-']

r Radial coordinate, Radial distance [rn]

r. 1. j

Distance between Si and V . [ml I

Radius of Vj [ml

s Radius of Si [ml

rt Radius of current conducting zone [ml

u Axial velocity [ms-'1

v Velocity vector [rn-se']

v Radiai velocity [m.$ ' z Axial coordinate. Axial distance [ml

Greek syrnbols

r S

Surlace excess at saturation in y rquation [1.3 x 10-' rno1e.m-'1

Diftùsion coeftïcient o f @ [-]

A H Enthalpy of segregation in y equation [-LX8 x 105 .i.rnolr-'l

General çonserved property [- 1

The ansle that ri, makes with the nomai to the S i [rad]

Elcçuodr tip angle [deg]

Volume coefficient of thermal expansion [K.']

Surface tension [Nm-' 1

Surface tension of irori üt Tm in y rquation [1.943 ~ . r n - ' l

Dissipation rate of turbulence energy [rn2.s-'1

Elcctric potential [VI

Nomenclature xxiii

Work function of the anode matenal [VI

Work function of the cathode materid [v Themal diffusion coefficient of the ekctron [rn2.s-' j

Newtonian viscosity of tluid [kg.(rn.s)-' 1

Viscosity of the gas at Te [kg.(m-SI-' 1

Effective viscosity of tluid Ikg.(rn.s)-'

Turbulent componrnt of viscosity [kg.(rn.s)-l]

Viscosity of the gas at Tw F,o.<m.s)-'1

Permeability of free space [Jirx lW7 NA-?]

Dummy function [ml

Ruid density [kg.m-3]

Gas density at Te [ko_.mb31

Gas density at Tw [kg.m-31

Electrical conductivity [Sm-' 1

Elccuode electrical conductivity at slab j [S.m-']

Empirical constant appearing in K equation [ l .O]

Empirical constant appraring in E rquation [1.3 1

Shear stress [~ .m- ' ]

Gas shrar stress [Nrn-']

Shear stress due to surface tension [ ~ . r n - ? ]

O con- Today fusion welding is the most important technique used in weldin,

struction [ i 1. Fusion welding is definrd as a joining process in which the coalescence

of metais is accomplished by fusion. This process requires a heat source o f sufficirnt

intensity to maintain a molten liquid metal pool. Based on the heat source, fusion

welding c m be divided into three different categories; sas welding. arc welding and

hi$-eneqy bearn welding [2]. Most cornmon welding processes in the arc weldin?

çategory an be Iisted as hilows [ I I :

0 Shielded Metal Arc Welding (SMAW):

8 Gas Tungsten Arc Wrlding (GTAW):

8 G u Metal Arc Welding (GMAW):

Subrnrged Arc Welding (SAW):

Electroslag Welding (ESW).

There is a gentml trend to increast: producûvity and safety in the welding industries-

To realize this objective, there has bren a significant growth in automated welding

systilms. In this regard. more attention has been paid to GTAW and GMAW because

of their continuous nature and their relative merits [3]. As a result of these importcrnt

trends, GTAW and GMAW will be considered in this suidy. A brief introduction of

each process is now provided.

Cha~ter 1 Introduction 2

1.1 GAs TUNGSTEN ARC WELDWG (GTAW)

GTAW is a welding process that uses an arc between a tungsten (nonconsum-

able) electrode and the weld pool. A shielding gas, which is usually an inert gas such

as argon or helium, is also used to protect the elecvode and weld metal from atmos-

pheric contaminations. A sketch of this process is shown in Figure 1.1.

GTAW c m operate in either AC or DC mode. The direct current configura-

tion. in which the workpiece is connected to the positive terminal and the tungsten

electrode is connecred to the nesative terminal, is the most cornmon mode and is usu-

ûlly referred to as Direct Current Electrode Negative or DCEN. The decuons are

rmitted from the tungsten elrcirode and are accelerated to very high speeds while uav-

elling through the arc. Such high-speed electrons hit the workpiece and exert a strong

heatinz rt'fect on it [1]. If the tungsten rlectrode is connected to the positive terminal.

which is callcd Direct Current Electrode Positive or DCEP. then clcctrons will acceler-

ate to it. Thzreiore. thicker elecuodes must be u x d to prevent melting. The weld

pool depth in DCEP is also less than the weld pool depth in DCEN [ 2 ] . Because of

the cathodic çleaning action of the arc. this modc of operation is important for weld-

Shielding gas inlet -7 7

Contact tube

Figure 1.1 Schrmatic diagram of the GTAW process.

Chapter 1 Introduction 3

ing aluminium and magnesium. To increase the capability and to remove some disad-

vantages of this process, some other operating modes, such as square wave AC, pulsed

DC, and high frequency pulsed welding are also recommended [3,4].

1-2 GAS METAL ARC WELDING (GMAW)

GMAW is a welding process that uses an arc between a consurnable electrode

and weld pool. The process is used with shielding gas, again to protect the electrode

and weld metal apinst contamination. Argon, heliurn and a mixture of argon with ox-

ygen or carbon dioxide are among the most common gases used for shielding the

process. In this process the consurnable electrode acts as the filler metal as well.

therefore the deposition rate is high. A sketch of this process is shown in Figure 1.2.

As in GTAW, different operating modes have been used for GMAW. Because

of the difficulty in obtaining a smooth metal uansfer. DCEN and AC current are sel-

dom used [ 2 ] . To improve positional control and aiso tu obtain a spray-type rnetal

transkr with minimum spatter, the pulsed DC mode can be applied [3].

Electrode wire feeder

Shidding gas inlet

Welding torch - Power supply

Contact tube i -- a

a

~rkpiece

I 1 I

Figure 1.2 Schematic diagram of îhe GMAW process.


There are two basic modes of meml transkr h m the consumable electrode to

the weld pool; free-flight mode and dip or short circuiting transfer mode. In the free-

tlight transfer, an arc is rnaintained between the elecrrode and the workpiece and the

metal is transferred ricross the arc in the form of droplets. The most cornmon free-

flight modes are globular transfer and spray transfer. If the electrodr is fed toward the

workpiece at a speed which exceeds the rate at which the arc alone c m melt the wire.

it will eventually bridge the arc gap and dip into the pool.

The type of metal transfer is denrmined by a number of factors, the most im-

portant are the following [A]:

Magnitude and type of welding current:

Electrode diameter:

Electrode composition:

EIectrorie extension;

Shielding gas.

With a DCEP. globular uansfer takes place when the current is relatively low

regardless of the type of shielding gas. However. with carbon dioxide and helium

shiolding sas, this type o f transfer takes place at dl usablr welding currents. Globular

transkr is çharacteriz.ed by a drop diameter rhat is greatcr than the electrode diameter.

With argon-rich shielding gas, it is possible to produce a very stable. spatter-

free axial spray transfer mode. This requires the use of direct current and a positive

elecuode, and a cunent Ievel above a critical vaiue called the transition current. As

shown in Figure 1.3. below this current, metal transfer occurs in the globular mode at

a ratc o f a few drops per second. Above the transition current, the transfer occurs in

the form o f very smdl drops that are formed and detached at a ntc of hundrcds per

second.


C - al T:a nsi tion a

current 1 9

n I I

- 5

I

! I . O - I

O 1 O0 200 300 400 JO0 600

Current, A

F i ~ u r c 1.3 Variation in volume and uansfer rate of drops with wetding cvrrent (steel electrode). [JI

1.3 THE ELECTRIC ARC

In hoth GTAW and GMAW processes. an arc acts as the source of energy,

aïid the quality of weldment to a large extent depenus on the heat and electric çurrent

from the arc. It is therefore worthwhile to invesrigate the arc in morc detail.

An arc cm be detined as a discharge of cleçuicity between two clecmdes.

anode and cathode. in a gascous phase. in which (a) the voltage drop at the cathode is

o f the order OC the excitation potential of the electrode vapour (that is about 1 0 V)

and (b) the çurrent density can have any value above a minimum of about 106 ~ . m - '

[5] . Some invcstigators, for exampie Guile [71, put 1 A üs a lower limit for the çur-

rent in the arc, however it would be morc appropriate to determine the lower limit

hased on currcnt density.

It is found that the distribution of electric field dong an arc is not uniform

(Figure 1.4). By considering the voltage distribution. an arc can be divided into three

differcnt pans; the cathode resion, the arc çolurnn and the anode region which shall


' Anode fall voltage

1

Cotumn vottagc

Cathode fall voltage

Arc length

Anode surface

Cathode surface

Figure 1.4 Voltage distribution alone rin arc shown schematically.

now be described.

1.3.1 The Cathode Region

This region is the most critical part of an arc. When a cathode is heated to a

sufficiently hi$ temperature. elecuons are emitted with a current density Jc given by

the Richardson-Dushman equation 151

where A is a constant of about 6x105 A. (~ .K) - ' for most metds md h = eoc / k,

$= is the work function of the cathode material and is d e h e d as the energy that

must be supplied to an eleclron over and above its energy at absolute zero to make it

possible for the electron to overcome the surface forces [6]. The ternpenture of the

cathode surface is the main variable in the current density equation and it must be

high enough to reach currcnt densitics which arc rncountered in arcs. In this case the

emission of elecuons is thermionic and only materials with high boiling point (4000 K

Chapter t Introduction 7

or higher) can withstand these temperatures (Figure 1.5). In thermionic emission. the

current density is in the range 106 to 108 ~ . r n - ' and the cathode spot nomally occu-

pies a fixed position [7]. The boiling point of mosr metiils is less than 1000 K md in

these metals the emission of elecuons is non-thermionic. In GMAW in which the

workpiece is the cathode. the electron emission is non-thermionic. with current densi-

ties of 1 0 l 0 ~ . m - ' or more [7]. Aiso. the cathode spot is not stationary. Usually the

- - Ci i antalum

Zirconium O

t Uranium Carbon / Thonum + +

t + Tantalum dnnn carbide

Y Elernents for which therrnionic type emission has flot been observed: O

Elernents for whrch information 1s lacking: $. -

1000 l . l l l J t l l l l l l r l ! l p l

2 3 4 5 Thermionic work function. V

Figure 1.5 Temperature for thmnionic emission at various current density levels [71. Individual points show boiling point of pure elements versus work bc t ion .


mobility of Ihe cathode spot is not desirable in welding. If il occurs on the elecuode.

the arc wanders up and down the electrode and may damase the surrounding parts. If

it occlrrs on the workpiece, the arc heat may be too widely spread. It has been shown

in the case of aluminium that the presencc of a relatively thick oxide layer tends to

restrict rnovemcnt of the cathode. A similar èffect is used in the GMAW of steel.

Normiilly the gas metal arc process is operated with the elecuode positive and if the

shielding gas is pure argon, the arc wanders on the workpiece surface excessively [8].

In fact. it has been obsenred that arcs c m become unsrable and even extinguish for

lack of sufficient oxide or other inhornogenities and impunties. Probabiy the destnic-

tion of the oxide Iriyers at an emitting site by ion bombardment causes the emission

rfficirncy of that site to faIl to the point where other sites take over its share of the

ç u m n t [7 ] . Thttreforc an oxidizing gas c m help the arc to be stable. and it is corn-

mm to add such an oxidizing gris. typically 2% O, - o r 20% CO,. - to the shielding

gas (argon). in the GMAW proçess. (81.

1.3.2 The Arc Column

The main body of the arc is the arc column and i t c m be defined as that part

of the arc which is located between the cathode and the anode Mls [7] . The arc col-

umn is a piasma and consists of neuval particles. such as atoms and molecules (both

in the excited and non-excited states) and charged particles. such as rlectrons and ions.

At aunospheric and higher pressures, the arc column is elecuically neutral. Thus, the

numbers oc positive and negative alectricd charge carriers are alrnost the sarne.

For many years, it has been considered that the arc column is in local thermo-


dynaniic equilibnum ( L E ) . This requires that collision processes (not radiative proc-

esses) p v e m transitions and reactions in the plasma and that there will be a

microreversibility among the collision processes [9 ] . This definition is applicable for

optically thin and atmospherïc plasmas. One of the consequences of the LTE state is

that dmost compkte energy exchange behveen the different gas particles takes place.

The important finding is that the temperatures of the electrons, Te, and heavy parti-

cles (atoms and ions). Tu, are the same. In the case of the low pressure arcs. the C

mean k e path of elecuons between two consecutive collisions is quite large. During

this time interval they cm gain additional rnergy from the electric field. and so the

temperatures of electrons and heavy particles may not be equal [71.

Howcver. it was shown rhat the temperatures of electrons and heavy particles

in the atmosphcric arcs. very close to the electrodes [ 101 and al the fnngcs [ 1 11, arc

different. Furthemore. Fariner and Haddad [121 have shown by measuring tempera-

turcs of the outer boundary of a tice buming arc (stabilized by natunl convection) us-

ing thc Rayleigh scattering technique, that the iirc at its outer part is not optically thin.

From al1 of thcse investigations. it can be conçluded bat only the çore of the arc col-

umn c m be çonsidered as a thermal plasma under LTE condition.

1.3.3 The Anode Region

The third part is the anode region which receives electrons tiom the arc. Al-

though the influence of the anode on the arc is less than the cathode [7], in welding

in particular and in plasma processing in general. the anode in most cases is the objec-

tive of plasma heating. In this sense, the cnergy transport from the arc to the anode is

qui@ important,

Therc arc two anode modes found in arcs. These two modes arc the cathode

jet dominated [13] or the normal anode mode [7] and the anode jet dorninated [13] or

Chapter 1 Introduction 1 O

the anode spot mode [7]. In the anode spot mode, the anode is tixed to a specitic

point on the workpiece m d tnversing the eiectrode causes the anode to jump from

point to point. Constriction of the current path on the anode surface induces an anode

jet. This is the situation that can be observed in relatively long arcs with low currents.

By increasing the current. the anode spot tends to disappear. In this situation there is

no distinguishable anode spot and the anode is in its normal mode. The normal or

cathode jet dominated mode is usually observed in welding and in this case the arc

column is bell-shaped. stable and symmeuicaI [71. Even in GMAW. which normally

operatcs with a positive electrode and where the anode spot forms at the tip of elec-

trode, the electrode is in the normal mode as it is continuously being consumed.

The objective of most plasma processine operation is to describe the heat flux

to the anode. Although there an: three mechanisms (if heat transfer from the arc to

thc anode, radiation. convection and vlectron drift and condensation, not al1 of them

arc neccssuily present. In DCEN which is the most common operating mode for

GTAW. the anode k a t input is due to the condensation of clectrons plus the energy

cained in püssing through the anode drop zone. and heat convection and radiation b

liom the arc [7 . 141. In GMAW in which the DCEP is more common. the anode heat

input froni the arc does not include the convective part [71. Heat transfer from the arc

to the anode for thc case [if GTAW and GMAW. will be discussed in detail in the

fr~llowing chaptcrs.

1.4 DEFINITION OF THE PROBLEM AND OBJECTIVES OF THIS STUDY

1.4.1 Definition of the Problem

Welding of metal pieces is a major part of many manufactunng processes and

thc integrity and soundness of the final product depend on the strcngth of the weld-

ment. The quality of welds, on the othrr hand, is determined by metallurgical changes


in the weldment. including fusion zone and the heat affected zone and geometry of the

weld pool. I t has been shown that the puddle geometry. the temperature gradients, the

local cooling rates and the solidification structure can be significantly influenced by

heat and fluid tlow into the weld pool [15]. The geometry of the weld pool. in

GTAW specifically. can change with variation in the shape of the electrode tip [41. In

addition, there are some other weld characteristics such as weld penetration, undercut-

ting. surface smoothness, segregation pattern and gas porosity. which are likely to oc-

cur due to changes in flow pattem in the weld pool [16]. In the case of arc welding

processes. the heat and fluid flows in the weld pool are drtermined by elecuomagnet-

ic, buoyancy and surface tension forces 1171, and the impinging force of the arc plas-

ma [18]. Gas shear stress also can be a parameter affecting the flow pattern into the

weld pool, specially for high current arcs (191.

The mathematical modelling of weld pool developed very rapidly during the

1980's [20. 2). The need for industry to achieve higher quaiity and more consistent

wclds, on the one hand, and availability of computing facilities and software packages

and computational tluid mechanics. on the other hand. were two reasons for this tre-

mendous growth [2 11.

The description of the input energy and the electric current sources are basic

for every numcrical mode1 to sirnulate the weld, and the resultant output is affected by

these two sources. There are many numerical models for tluid and heat flows of the

weld pool in GTAW [17, 18, 20, 22-27] and in GMAW [28-3 13. In the majonty of

the GTAW models, heat tlux and electric current tiom the arc are defined to have

Gaussian distribution at the top of the weld pool. Recently there have been some ef-

forts to combine arc and weld pool models together [32, 331. Although it was indicat-

ed that the elecuode tip shape can affect the weld pool geomeuy [4] and even the

properties of the arc [341, none of the models consider the effect of the tungsten elec-


uode tip shape. Because of the continuous metal transkr tiom the consumable elec-

uode to the weld pool. the situation in GMAW is more complicated. For this process,

the# is no cornbined model available. In the only model available for the arc [35],

there is no effort made to evaluate the heat tlux to the weld pool.

1.4.2 Objectives of the Present Investigation

The focus of this study is on the mathematical modelling of the arc in both

GTAW and GMAW processes. Mathematical modelling h a been chosen because it

çan provide a powerful tool to study differcnt parameters and their effects on the arc

behaviour. Moreover in many cases, determining the properties of the arc experimen-

tdly is difficult if not impossible due to the very small dimensions and very high p-

dients of temperature and electnc potential in the arc column.

For the GTAW process. the focus of the investigation will be directed to the

rffects of the tungsten elecuode tip angle on the heat tlux to the workpiece and mode

current density. Tu obtain these objectives. the following steps will be undertaken:

Devcloping a mathematical mode1 for GTAW with tlat elecuode:

Calculating thc heat flux to the workpiece by taking the heat mnsfer rnechanisrns

from the arc to the workpiecc into account;

Cornpuin_« the çalculated results with the available experimcntal data:

Expanding tiîc mode1 to cover variation in the electrode tip angle;

Comparing the calculatcd results for angular rlec~ode model with available

rxpenmental data;

Cumbining the arc model with a simple model for weld pool to show the effects of

the electrode tip angle on the weld pool shape, and compiuing the resuits with the

available experimental results;

Providing sensitivity mdysis on the critical process paramelers.


For the GMAW, the tocus of investigation will be on the efft-cts of variation

in the çonsumable electrode tip shape from tlat to upered (conical). on the arc charac-

teristics. In this stage, the contributions of the arc and the droplet in the heat tlux to

the workpiece will be estirnated. The following steps will be pursued:

Developing a mathematical model for GMAW with tlat electrode:

Snidying the effects of the electrode material and also the shielding gas on arc

pro perties ;

Comparing the results with available experimental and other theoretical results:

Estirniiting the contribution of the arc and the droplets into the heat flux t the

workpiece:

Snidying the variation of the elrctrode tip shape on the arc properties.

The thesis consists of two sections. In section one. the GTAW process will be

considered. This section consists of three chapters: 2. 3. and 1. Chapter 2 will explain

the mathematical model formulation for the arc in GTAW with tlat electrode. The re-

sults {if calculation and cornparison with some othrr numerical and available experi-

mental values will be prexnted. Chapter 3 will hcus on the mathematical model

improvements in order to mat itngular electrodes. The effect of the eleçtrode tip ansLe

on the arc properties will be considered in this çhapter. In chapter -1. a simple model

for the weld pool will be presented. Information from the arc modelling will be used

in the weld pool model to show die effect of the electrode tip angle on the weld pool

shape.

Section two (chapter 3, will focus on the GMAW process. ln this chapter,

the mathematical model for the arc in GMAW with tlat electrode will be developed.

Effects of the elcctrode material and the shielding gris along with welding parameters

such as applied current and arc length (distance between the cathode and the anode)

and die effect of the variation of the electrode tip shape from tlat to tapered will be


studied in this chapter.

Finally, in chapter 6 conclusions and suggestions for future studies will be

presen ted.

1.5 LITERATURE REVTEW OF MATHEMATICAL MODELLING OF THE ARC

The objective of the present investigation is to evaluate the heat tlux to the

workpiecr in GTAW and GMAW throuzh mathematical modelling of the arc in these

processes. Therefore. it will be worthwhile to review the modelling of the arc for

welding purposes. In this review both analytical and numerical rnodels will be cov-

ered.

The main part of an arc is its colurnn which consists of almost neutral cornbi-

nation of at l e s t three différent particles (elecuons. positive ions and neutral atoms).

At atmospheric pressure. such a combination is collision dominated and can be created

as a continuum fluid [36], so that the conservation rquations c m bt: written for such

cui arc. However. plasma is an electricity conductor and interaction of electric cunent

with its magnetic field rnust be considered as a mornenturn source. Moreover, like

every othrr conductor. the passage of elrctricity through the plasma raises the temper-

ature due to the Joule's effect. The amount of this rnergy is proportional ro the

square of çurrent densities and is high enough to kerp the ionized -as stable.

As repartrd by Pfender [5] and Boulos [37], die modelling of electric arc be-

gan at 1935 with Elenbaas [381 and Heller [391. Ln their models. they sohed the mer-

gy conservation equation by considering Joule heating as a source of energy and

conduction as the main mechanisrn for heat uanskr. For m axisymmetric arc column,

their models c m be fomulated as foilows:

and EZ can be obtained throu, oh Ohm's law


where r. shows the outer radius of the arc. In their rnodels, they fomulated the

physical phenomena that take place in an arc in a very simple form.

In 1955. Maecker [40] considered the electromagnetic force and pressure Sra-

dient and derived a model for the arc.

mdyticaily. he obtained an approximate

r 2

L

In his model, Maecker assumed that the

By solving the flow and continuity equations

value for the maximum pressure as

( 1-4)

current density over each cross-section is con-

stant. Later Schoeck [41] moditïed Maecker's model by considering a parabolic cur-

rent density distribution in the t o m

where r, is the outer radius of the current conductin_o zone. Baxd on this he obtained an-

other expression for the maximum pressure. as follows:

- 5 1' - -- ' m a ~ ( s ) C ' A ~

Maecker [40] and Shoeck [-Il] did not consider the variation of gas properties

dur to temperature. but by introducing the main dnving forces for the plasma into the

arc column, they provided a basis for further investigation related to the arc.

As reported by Boulos [37] in 1967, Watson and Pegot [42] for the t-mt time

combined tlow and energy equallons in a two-dimensional rnodel and through solving

these equations numerically obtained accurate results for tlow and temperature fields.

Their model considered only the column of the arc. The main assumptions that they

us& c m be summarized as follows:

axisymmetric. steady state larninar tlow;


local thermodynamic equilibrium ( L n ) ;

neg lisible thermal diffusion. gravi ty and viscous dissipation e ffec ts:

optically thin plasma;

8 therrnodynamic and transport properties are only temperature dependent.

In 1979 Glickstein [43] and Chang et al. [a] presented two different models

for the arc in GTAW. Glickstein analytically solved the one-dimensionai conservation

equations for mass, momentum and energy. To evaiuate the electric fields, he sirnply

measured the arc voltage and by knowing the length of the arc cdculated the rlectric

field through the following equation:

Glickstein showed that any addition of minute amount of low ionization potentiai e k -

ment to the welding gas can cause significant changes in the configuration of the weld

head. Also by taking into account the heat loss due to radiation into die energy equa-

tion. hr showed that above 10000 K this term is important and can not be neglected.

Chang et ul. [U] presented a numerical solution for the two-dimensional rnass

and momentum conservation rquations, by assuming an isothermal plasma. They

quantitied the variation of the plasma velocity due to welding çurrent and arc length.

and sugested that this variation has a profound iniluence on the shape of the weld

pool.

At the sarne time, Lowke [45] presented his one dimensional mode1 arc. By

separating arcs into two cakgories of low current and high current. hr concluded that

in low current u c s the properties of the arc are conuolled by natural convection (con-

vection due only to gravity), whereas for high current arcs they are conrrolled by the

self-magnetic M d . He ais0 found that the propenies of the plasma at high current

arcs are dependent to some extent on the cathode current density. In his study he as-


sumed a cathode current density of 108 ~.m- ' . He aiso suggested that the cathode cur-

rent density is not a function of the applied cumnt for currents between 1 kA and 20

kA.

Ramakrishnan and Nuon [46], based on an integral formulation [471, present-

ed their one-dimensional mode1 for 100-500 A arcs and concluded that the overall arc

properties are insensitive &O the cathode current density.

By adding vixous body forces to the momentum transport equation and as-

suming a Gaussian form for the radial variation of the axid velocity. Allum [J81 pre-

dicted the gas tlow in GTAW. He solved the conservarion rquations for mass and

momentum, hy assuming a Gaussian distribution h r the axial current density and by

neglecting the variation o f the plasma properties with temperature. He found that the

huoyançy forces are dominant in the outrr part of the arc and increased in imp«nance

with pressure.

In 1983. Hsu rr al. [49] presented a cornplete model for a freee-burning high-

inti'nsity arc. In this model a comptete set o f two-dimensional conservation equation

of m a s . momentum and rnerpy dong with continuity of elrctric potential was solved

numrrically. The most crucial boundary condition was î'ound to be the cathode current

density. They dehned current dcnsity as a hnction of radius as follows:

J = J exp ( -br) z m a x

in which b is a constant and Jmax is obtained from

whrre r, is the radius of the hottest pan. of die cathode. For the anode boundary.

thry used the experirnentally measured values of temperanire at 0.5 mm frorn the an-

ode. They found that the flow field boundq conditions were not so critical. A sam-

ple of the predicted temperature field which is compared with experimental values is


shown in Figure 1.6 [49].

Later Hsu and Pfender [50] presented a two-temperature mode1 for the arc.

They solved two energy conservation rquations for electrons and heavy particles sepa-

rately. As a result of this study. it was found that diere was a large difference be-

tween temperatures of electrons and heavy particles at the kinges of the arc and very

close to the electrodes. This difference specificdly is shown by Dinulescu and Pfender

[ I O ] ior the anode boundary (Figure 1.7).

Kovitya and Lowke [SI] developed a two dimensional mode1 for free buming

arcs. In the mode1 they considered only the arc column. At the cathode they applied a

constant çurrent drnsity and at die anode considered the temperature to be LOO0 K. By

Figure 1.6 Measured and cdculüted isolhcrms of a free- burning argon m. (1 = 200 A, L,, = 10.0 mm, P = 1 aun).[491


-

T, - electron temperature

T, - heavy partrcle temperature

I 1 I 1 I 1 1 I ! I 1

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 8.8

Distance from anode, 10.' m

Figure 1.7 Species temperature distribution in the anode b o u n d q layer. t Jc = 9x 106 Am-'. P = 100 kPa, Gas = Argon). [ 101

comparing with the experimental values of Haddad and Farmer [52] , they found the

hest value for current density al the cathode to he lu8 A.m-2. In this study they ana-

lysed the cffects of differcnt ierms in the momentum and cnergy equations in detail.

One ycar later. in 1986, Kovitya with Cram [53) presented another mode1

which was based on the previous study 1511 with considering some modifications Sor

both Lhe cathode and the anode boundaries. They detined the cumnt density at the

cathode as a function of radius by,

where J,,, is the maximum current density and is considered to be 10%.m-'. For

the anode they detined the anode spot radius based on experimental rneasurements and

considered the temperature inside the anode spot as 10000 K and outside as 1000 K.

In both of his models, Kovitya calculated the argon plasma propenies.

The tïrst attempt to estimate the heat flux from the arc to the weld pool


rnathematicaily was carried out by McKelliget and Szekely [lit]. By calculating the

rnergy vansfer through convection. radiation and electrons tlow. they predicted the an-

ode heat flux. By using Mckelliget and Szekely's approach. Westhoff [54 presented a

more complete mode1 for GTAW arc. He showed chat the deformation of the anode

surface has significant effect on the current path in the arc. even for relatively small

deformations of about 1 millirnetre.

Although in GTAW a sharpened tungsten rod acts as the cathode. in most of

the mentioned models the cathode was considered to be a tlat tip rod. Tsai and Kou

[55] simulated a real GTAW torch to compare sharpened and tlat tip electrode and to

study the effect of the nozzle wall on the arc properties. They found that the conical

surface of the electrode tip facilitated die motion of =as through the creation of an

rlecuomagnetic force parallel to the conical surface. They also showed that heat

rransfer and tluid tlow in the arc plasma are quiet sensitive to the distribution of the

cathode çurrent density dong the elecuode tip. They found that the presence of the

shidding sas nozzle. under welding conditions, does nui have any significant eiièct on

the ve loci ty and tem perature fie Ids.

One of the problerns in modelling the arc is the cathode and the anode re-

cions. Many investigators attempted to develop mathematicai models to simulate these C

parts of the arc [lO. 55-581.

In 1990. Delalonûre and Simonin [601 presented the fvst mode1 of arcs which

includcd the electrode region. To avoid pre-dedning the cathode density at the cath-

ode boundary. they solved a ~eneral set of thennodynamic and electrornagnetic equa-

tions for the non-equilibrium boundary layer of the cathode. This procedure provided

the required boundary condition for the calculation of the plasma propenies in the arc

cdumn region.

Jog et (11. [6 1 1. in 199 1. considered a set of continuum conservation equations

Chapter 1 introduction 2 1

for the çharged particle densities and temperatures and Poisson's equation for the elec-

tric k l d . to drvelop a model for the welding arc. In addition. by utilizing the arc

properties they calculated the heat flux to the weld pool. In this model they consid-

ered Iow current arcs, hence they did not solve the flow equations.

Zhu et ul. [62] combined both arc and cathode by using a one dimensional

model for the rkcuode layer [59]. They divided the calculation domain into three

parts. electrode (cathode). cathode sheath and arc column. By defining two intemal

boundaries between these parts. they solved the relevant conservation equations for the

rlrcuode and the arc. n ie results of this model are in fairly good agreement with

some available experimentd data. For example, a cornparison of the calculated cath-

ode surfiicc temperature with Haidar and Farmer expenmrntal values [34] is shown in

Figure 1.8. Later Lowke et ul. [63] simplified the model by putting boundary condi-

tions for elrctric potentiai into the rod electrode. In their model, they solved the m a s

cmtinuity and consemation equations for momentum and cnergy for the =as only.

Thcy neglectcd the cathode sheath between the electrode and the arc. Although in this

model thcre is no need to pre-define the cathode çurrent density. the rcsults are de-

pendent to some extent on the cathode surfxe temperature.

Lowkct r t ul. [64 and Zhu et (11. [331 have extended their models LO include

arc column and bixh elrcirodes. In these models they divided the whole calculation

domain into iïve parts. including arc çolumn. cathode. anode and two electrode

shraths. For the internai boundaries they adjusted the temperature at each iteration

such that the energy conservation equation was satisfied. Lowke et <il. [64] focused

prirnarily on the cathode surface temperature, while Zhu et al. [331 concenuated their

attention on the k a t flux to the anode and the anode surface temperature.

Most recentiy Kaddani e t al. [65] developed a three-dimensional mode1 for

unsteady arcs. in this model. they considered only the arc colurnn. and found that the


Axial distance from the tip. m m

Figure k.8 Calculaied cathode surface temperatures for a 200 A arc in argon at 1 atm compared with experimental measurements of Haidar and Farrner [34]. [621

cathode current density played a very important role in stabilizing the arc.

The only mode1 for GMAW arc is presented by Jonsson et al. 1661. They

considered the workpiece as the cathode and the consumable clectrode as the anode,

and solved the relevant conservation equations and physical laws for a GMAW proc-

ess for aluminium which is shielded by argon gas.

From the above it can be concluded that while our knowledge of the mathe-

matical modelling of the welding arc in GTAW have improved considerably over the

last years there are still important areas that need much work to provide helpful infor-

mation to the welding society. These c m be identified as follows:


Evaluating the heat flux and the mode current density to the workpiece:

Studying the effect of elecirode geomeûy on the ÿrc properties and heat flux:

Modelling of arcs bumed into the mixed pases;

Deviation from LTE and optically thin behaviour.

Modelling of the arc in the GMAW process has only started recently and it

will be sometime before the appearance of a well developed model. Some of impor-

tant areas to study Tor GMAW can be listed as follows:

Non-thermionic cathode nature and behaviour in difirent grises;

El'lect of composi~ion of gas on the arc properties;

Consumable-arc interaction;

Drople t-arc interaction.

It is ohviaus that some of the topics in hoth GTAW and GMAW require

years to study. In the current study. the inputs Iiom the arc to the workpiece wi11 be

invcstigatcd.


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Chapter 1 introduction 28

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C hapter 1 Introduction 30

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1993, vol. 74. pp. 5997-6006.

2- 1 INTRODUCTION

In this çhapter a mathematical mode1 for the welding arc in the GTAW proc-

ess will be developed by solving the conservation equations and Maxwell's equations

sirnultaneously. To check the validity of the model, the obtained results will be com-

pared with available experimental data, Effects of the arc variables, i.e. arc length and

applied current on the properties of the arc will be investigated. Finally. the heat flux

from the arc to the workpiece will be evduated, by estimating the contributions of dif-

ferent heat trlinst't'r mechanisms.

2.2 MODEL DETELOPMENT

Figure 2.1 shows a schrmauc diagram of a DCEN gas tungsten welding arc.

A nonconsurnable tungsten electrode serves as the cathode with the workpiecr being

the anode. By applying an electnç potential drop between the electrodes and by ioniz-

ing the g u between them. for example by a pilot arc, an arc can be establishrd. Ther-

mdly emitted electrons from the cathode strike the neutral atoms in the arc gap and

ionizr hem and finally condense on the suriace of die anode. This combination of

free electrons, ions and neuual atoms is called a plasma. The plasma is a very good

conductor of elecuicity and so electrons can readily pass through the gap between the

elecuodes. The passage of the dectrons between die cathode and the mode produces

a magmtic field. In the arc region of variable cross section, the interaction of the cur-

Chapter 2 GTAW mode1 32

Figure 2.1 Schernatic diagram of the welding arc (GTAW.

rent with its self-induced magnetic field lrads to the phenornena of inducrd plasma

jets [ i l . - - - -

The j!kSage or eTeCtronsns tf%ou$ï die- ptasma puduces k a t ttircrugh 3mle

hearing. Since the cumnt density in the arc column is very high (106-10' Am-' [2]),

the amount of this hrat generated is sufticiently high to h e p die ionized gas srahle.

ThrmalIy emitted electrons uavel across the arc and eventually hit the work-

piece (anode), giving up rnergy which is proportional to the work function of the an-

ode material and the kinetic energy of the electrons. Other mechanisms taking part in

the heat tlux to the anode are convection and radiation from the arc [3I.

Although a plasma consists of at least three different panicles, ekctrons, ions

and neutral atoms, it c m be treated as a continuum tluid since a high intensity. high

pressure arc is collision dominated [4. This indicates that the mean free paths for

Chapter 2 GTAW rnodel 33

particle collisions for al1 species an: much smaller than the characteristic length xaie

of the macroscopic change. In this case. the equations which describe the global con-

servation of mas. momentum and energy for nonconducting tluids can be written for

the plasma.

Tagether with this fact. the following assumptions which are çornmonly am-

ployed in arc modelling are considered:

in two-dimensional cylindricai coordinates;

steady state sn ~ ! a t the It is assurned that the arc is in

parameters with time is eliminated;

It is assumed that the arc is in

It is assumed that the arc is axially symmetric so that the equations can be written

variation of different

local thermodynarnic cquilibrium (LTE). which

means that the transitions and reactions in the plasma are collision dorninated.

Therelim the temperatures of electrons and heavy particles arc almost equal. It is

shown that [5] this assumption is valid for most of the arc excepr in regions very

close to the cathode and the anode surtaces and in the fringes.

Ir is assumed thüt the plasma is optically thin so that the ndiated energy h m the

arc will not be absorbed by the plasma. Although Cram rr cri. [61 and F m e r and

Haddad [7] have shown that this assumption is not accurate at the outer part of the

arc, the corc of the arc c m still be considered as an optically thin plasma.

Laminar tlow is assumed. This assumption is justified by McKelliget and Szekely

[81 on the b a i s of larninar-turbulent transition for a tiee jet.

It is ÿssumed that the electrode is a tungsten rod with a tlat surface. This

üssumption simplifies the geometry. Later the effect of the electrode tip angle will

be investigated.


2.3 GOVERNINC EQUATIONS

Using the above assumptions. the conservation equations c m be written as

follows:

The iast terms in the momentum quaiions are due to interaction «f the current and its

self-induced magnetic field (Lorentz force). In welding. the clectromagnetic compo-

nent of the Lorentz force is usually much greater than die elecirosiatic çomponent [3],

thus the latter is neglected in the momentum equations.

The source tcrms in the energy equation an: Joule heating radiation loss and heat


trcuisfer due to elecvon drift.

Continuity of current çan be written in terms of electric potential as:

Since the arc is axi-symmetnc. the azimutha1 component of the magnetic field will be

die only component that acts on the plasma through the Lorentz force. The azimurhal

cornponent of the magnetic field c m be calculated by following relation from Am-

pere's Law:

To calculate the çurrent density. Ohm's law çan be used. The generaiized

Ohm's law may be simplifieci as

Physiçal properties. narnely density, viscosity. thermal conductivity. heat ça-

pacity and elecuical conductivity are uerited as functions of temperature. Ai1 these

properties are taken from the iabulated data of Boulos et (il. [91. Radiation l o s data

are &ken hom Evans and Tankin [ 1 O].

2.4 BOUNDARY CONDITIONS AND NUMERICAL METHOD

2.4.1 Boundary Conditions

The calculation domain for al1 variables is shown in Figure 3.2. A non-uni-

form grid point system is employed with finer gnd sizes near the cathode region. The

distance between the electrodes varies from 2.0 to 12.7 mm. The inflow boundary at

the top of the domain is taken at 3.1 mm above the elecuode face (cathode surface).


Figure 2.2 Calculation domriin for GTAW (schernatic).

The boundary at the side is 15 mm away from the mis of symmrtry. The corre-

sponding boundary conditions are given in Table 2-1.

The specification of the boundary condition for velocity is not as criticai. At

the solid boundaries, the face and side of the clectrode, and the surface of the work-

piece, the velocities are zero. At the mis of symmetry. the radial velocity will br zero

and there is no tlux for axial velocity. The radial velocity is zero at the top inîlow

and the axial velocity equals a constant value of 2.0 m . s l and proportional to the

shielding _pas inflow. The side boundary is located far enough so that there is no axial

and radial momentum fluxes at this boundüry. At the top and side boundÿnes. pres-

sure is fixed to a constant value to esse the convergence of the solution.

Chapter 2 GTAW model 37

Table 2-1: Boundary Conditions.

At these two houndaries pressure is tixrd to ri constant value.

Currents. axial and radial, at ail boundiines except at the cathode md the an-

ode are zero. Thus. the rlectric potential tlux at thosc boundaries is zero. For the en-

thalpy cquation. it is assumed that the temperature of the gas entering the calcularicin

domain through the top ruid side boundaries rquals a constant value. For the outîlow

part of the side boundary, the energy tlux is considered to be zero. At the axis of

symmetry, the flux of both die electric potential and the enthdpy rqual zero. Since at

the cathode and at the anode, LTE condition does not exist, and t'urther these two re-

gions üct as inflow and outflow boundaries for the electric potential. a special ueat-

ment will be required for both the cathode and the anode boundary rezions.

The Cathode Region

The cathode region is a very criticai boundary in solving transport equations


of the arc. In fact, very Little is known about the transition from the relatively cold

surface (electrode) to hot plasma column over c very small distance. Usually, an elec-

tncal sheath is considered between the arc column and the electrode. Strong electric

fields accelerate the electrons across this sheath towards the arc column. It is in this

layer that ionization of gas atoms occurs causing a substantiai potential drop.

It was shown by Hsu and Pfender [SI, that in the transition layer the plasma

is not in the LTE condition. While the temperature of the heavy particles decreases

rapidly to the cathode surface temperature as the electrode is approached. the electron

temperature remains rnuch higher, almost equal to the plasma temperature.

By assuming that the electrons are in a free fa11 across the cathode sheath.

McKelliget and Szekely [8] presented a mode1 to evaluate the cathode fa11 voltage,

where Te is the temperature of electrons in the vicinity of the cathode spot. By this

they suggested that the kinetic energy of the electrons at the interface of the cathode

shcath and the arc column is obtained by passing of the electrons across a cathode faIl

voltage, V C . B a x d on this voltage. a heat source for the arc column cm be consid-

ered as Iollows:

J v Qioniz = 1 C ( c (2-9)

There is another condition for the electrode surface. It is assurned that the electrode

surface is at a constant temperature close to the melting point of the electrode materi-

al.

For the ekctric potentiai, the boundary condition is approximated by 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;

McKelliget and Szekely [8] found that a single value of the cathode current density

gave good results, compared with experimental rneasurernents. for different values of - applied current and arc Iength. The value for the cathode current density, and there-

tore Cor the cathode spot radius was obtained from the Westhoffs study [ I l l .

The Anode Region

Since the electrical conductivity of the metal is much hisher than the plasma,

it çan be assumed tha~ dong the anode surface the electric potential is constant.

For the enthalpy boundriry condition, a heat flux from the arc to the anode

(workpiece) is considered. Conventionally the heat tlux to the anode c m be expressrd

by [31

where qc = the local heat iluxes by conduction and convection, qr = the radiation

heai uansfer and qc = the energy ~ansferred to the anode by the rlectrons. q, con-

sisu of three parts. The Tist part is the kinetic energy of electrons which is gained by

the rlectrons on passing across the arc çolumn. The second is the energy sained by

the electrons on passing across the electric field of the anode sheath. It is shown the-

oretically [12.13] and expenmentally [14] that the anode Cal1 voltage is negative and

does not contribute to the heat transkr to the anode. The l u t terrn is the condensation

heat of the elecuons which is proportional to the work function of the anode material.

In the electron contribution to the anode heat flux. the energy transfcrred because of

the axial gradients of the elecuon temperatures and density [12] must be considered.


Thus the elecuon contribution can be expressed as follows:

In this equacion qd is the thermal diffusion coefficient of the rlectrons. By consider-

in= an argon plasma with electron cmperatures of about 10000 K. the value of the

term in the parenthesis is 3.203. Therefore equation (2-17) simplifies to

It is assumed rhat the tempenture of the electrons rntering the anode shrath

equals the film temperature. Te = ( Te, O.l + T 1 / 2.0. Here Te, o. is the temper-

ature of the rlectrons a[ the edge of the arc which is at 0.1 rnillimetre trom the anode

surface, and Tw is the tempenture of the waii. assumed to be 1000 K.

ïhe first term in the right hand side of eq. (2-1 1). q , , consists of two parts:

convective and conductive. The convective contribution depends upon the arc current

and the separation of the electrodes. In the welding process short and high intensity

arc buming in the argon is encounered and the cathode jet dominates over the whole

Içngth of the arc such that the conductive part in cornparison with the convrçtive part

can be neglectcd.

To evaluate the convective heat transfer. McKrUiget and Szekely [81, and

Westhoff [ l l ] approach was usrd. From the literature [151 a correlation for the im-

pinging flow of argon was taken and modified [ I l ] as follows:

To calculate the convective part, as in the electron term, the temperature of the wail

was considered to be 1000 K.

The final contribution to the anode heat flux is radiation tiom the plasma.

Chapter 2 GTAW model 4 1

This part cm be calculated using the foUowing relationship (Figure 7.3) [8. L LI.

In the cylindricai coordinate system. eq. (2 - 15) c m be rewritten as

JjJ 'R". j (dz) 3 / 2 , , e 4~ ( A - B C O S ~ )

and ÿikinp advantage of radiai symmetry

7 2 2 where A = r; + rs + z and B = 2rrrs, and rr and rS are the radius of the anode

surface and plasma volume elrments, respectively. The above integral is isvailable in

tables [Ml. and cm be caiculated numerically over the whole of the plasma for rach

mode surface ekmènt.

In this model. sinçe a water coolrd copper plate is çonsidered as the anode.

the heat loss due to vaporization of the anode materiai is neglected. However ir

should be mentioned that in the real welding process. the heat loss due to vaporization

Figure 2.3 Contiguration of radiation

1 view factors.


of the metal from the weld pool results in a snonger cooling rffect on the arc and

therefore must be taken into account.

Finally. the total heat tlux to the anode can br evaluated by combining equa-

tions (2-13), (2-14) and (2-17). The heat loss due to radiation and the heat transfer

due to temperature gradients have already been considered in the energy equation. As

a result. only the convective heat transfer to the sudace and the work function terms

in the electron heat uansfer rquation are considered as the anode boundary conditions

for enthalpy.

2.4.2 Numerical Method

To solve the above equations. the PHOEMCS code was used. This code, de-

vrloped by Concentration. Heat and Momontum Ltd. (CHAM). provides solutions to

the discretized version of sets of differentid rquations having the general t o m [ 171:

or in steady state

The symbol stands for any conserved proprrty. such as enthalpy. momentum and mas .

The tïrst term on the left hmd side is the convective trrrn and the second term is the con-

ductive or diffusive term. The terni on the right hand side shows the source rate of 4). By

cornparing the general form. rq. (2- 19) with the conservation equations, eqs. (2- 1)-(2-4). the

associated quantities c m bt: exlracted. These quantities are listed in Table 2-2. Also, the

continuity of electric potenual c m be solved in this way and it can be treated as a conduction

problern without the convection term.


Table 2-2: The Corresponding Quantities for the Different Conservation Equations.

1 Equation

Axial momznturn 1 Radial mornen tum

2.5 RESULTS AND DISCESSION

This model has been simulated with a r p n as the shielding gas with two dif-

ferent current densitirs over the cathode spot radius for different dectrode seplirations

and applied currents. Arnong the arc properties, heat flux IO the workpiece and the an-

ode current dcnsity are rnostiy concemed. beçause these two determine the fluid and

hrat tlow into the weld pool. To verify the vdidity of the model. other arc propenies.

such as temperature profile, velocities. and electric potentiai must be exarnined. To do

this, the results of the caiculations are cornpared with available experimental and other

calçulated data.

2.5.1 Arc Properties

In Table (2-3). four major properties of the arc, i.r. maximum temperature.

maximum axial velocity, eleciric potential difference, and anode pressure difference. in

different cases are summuized. The results reported by Wrsthoff [L l ] arc also given.

From Table 2-3, the maximum temperature of the arc is slightly ditTerent frorn Wesr-

C hapter 2 GTAW model 44

Table 2-3: Arc Parameters for Different Arc Lengths and Applied Currents. A Cornparison with a Sample of Other Numerical Results. ( Tm,, , t Kk u,,, . (rns''): A$ . (V); AP, . (Pa)).

Applied CurGnt ( A )

:, c: Westhoff s results [ 1 I ]


h o r s results [ 151. This difference in al1 cases is less than 5%.

In Figure 2.4. the calculated isothems at 100, 200 and 300 ampere for arcs of

10 mm length are compared with the experimental results of Hsu et. al. [18]. In this

Figure the cathode current density for caiculated results is 6.5x10-' ~ .m- ' . These

graphs show that the agreement between the calculated and the experimental values,

specifically for the lower applied currents and except for the region very close to the

elecuodes, is very good, especiaily when the experimental enors are reported to be

110- 15% [ 191.

The velocity profiles for the 200 A arc at a length of 2.0 and 10.0 millimetres

are shown in Figure 2.5. In Figure 2.6. the calculated values of the velocities for both

cathode current densities for a 100 A arc of 10 rnillimetre length are compared with

the experimental values of Allum [20] and Seqer and Tiller [21]. As this graph

shows. the agreement between the numerical results of this work with the lower cath-

ode current density and Lhe experimental values of Seeger and Tiller [21] is reasonably

oood, rspecially for the maximum velocity and its position. The theoretical values of b

Allum [21] and Kovitya and Lowke [22] are also shown. The calculated values for

the lower cathode current density are in very good agreement with Westhoffs results

[ 1 1 ] (Table 2-3). This Table clearly indicates that arc lengths longer than 6.3 millime-

tre have no effect on the maximum velocity of the plasma. This is also shown in Fig-

ure 2.7 for 300 A arcs of different lengths with a cathode current density of 6.5~10'

~.rn ' ' .

As expected, the maximum pressures are located at the front of the tungsren

electrode and the weld pool (Figure 2.8). Compared to Westhoff s results [ I l ] , the an-

ode pressure differences in al1 cases are higher, which is the result of the finer g i d

size that is used in this study. By considering the variation of the pressure of the an-

ode with increasing arc length and applied current, it is observed that for the 200 and


1 11000K + 1 11000K + 2 12000 K -0- 2 12000 K -e 3 73000 K t 3 13000 K + 4 14000 K 0 4 14000 K * 5 15000 K + 5 15000 K + 6 17000K * 6 17000 K *

7 19000 K -F 8 20500 K 4 2 7 000 K

Radial distance. mm Radial distance, mm

Figure 2.4 Isolherms of 10.0 mm arcs in cornparison with the ex-

ptximcntal resu1t.s of Hsu et

al. [18]. For caiculated results, the cathode current density is 6.5~10' ~ . r n - ~ a: 1 = 100 A; b: 1 = 200 A; c: I = 300 A.

Cal. 1 ?1000K + 2 i2000 K -0- 3 13000 K + 4 14000K +3-

5 15000 K + 6 17000 K * 7 19000 K + 8 21000K + 9 22000 K +(23000

Radial distance. mm


4.0 2 .O O.

Radiai distance. mm

Figure 2.5 T-vpical velocity profile in GTAW arcs of two differ-

ent lenpths. lc = 6.5~10'

Am-'. 1 = 200 A. a: L, = 2.0 mm, b: L,, = 10.0

mm. 6.0 4.0 2.0 0.0

Radiai distance. mm

300 A arcs. the pressure reaches a maximum and then decreases (Figure 2.9). For the

200 A arc, this maximum occurs at about 3.2 mm but for the 300 A arc this is at

about 1.0 mm. As illustrated in Figure 2.10 this behaviour is due io an incrcase in the

rlectromagnetic force at the anode as a rcsult of a decreax in the separation of the

electrodes.

In Figure 2.1 1 the voltage of the arcs at differcnt applied currenls and arc

lengths is compared with Westhoff s results [ I l ] and the expenmental values [23].

The absolute values of the caiculated voltage of this work are much closer to the ex-

perimental results [231. However. the siope of the variation of arc voltage with ap-


- - - - Seeger b Tilier (Exp. 1979) 12 1 1 8 Allum (Exp.. 1981 1201

- - - - - -'Kovin/a b Lowke (Cal.. 1985) 1221

Axial distance. mm

Figure 2.6 Distribution of axial velocity dong the axis of symmeuy for 100 A arc of 10.0 mm Iength.

plied current for both cathode current densities is less than the cxperimental dam. On

the 0 t h hand. the values of the arc voltage are very close to the cxpenmental values

when difirent cathode current densities arc considerd for differcnc applied currents.

For the 2.0 mm arc the k s t agreement for the IO0 A arc is obtained at a çurrent den-

O .O 2 .O 4.0 6.0 8.0 10.0 12.0

Axial distance. mm

Figure 2.7 Variation of axial velocity at the centre line of the arc for differcnt arc lenglhs. Nurnkrs are length of arcs in mm. 1 = 300 A.


Figure 2.8 T-vpical pressure distri bution in a GTAW arc. 1 = 200 A. L,, = 10.0 mm.

10 Pa 100 Pa 400 Pa 600 Pa 750 Pa

7.5 6.0 4.5 3.0 1.5 0.0 Radial dtsrance, mm

sity of 6 . 5 ~ 10' ~.m-'. while for the 300 A arc. the best agreement occurs at a çurrent

density of 8.5~10' ~ . r n - ? . This is undersiandahle since the cathode çurrent density is a

iùnction of cathode surface temperature (Equarion (1- 1 )).

2.5.2 Anode Current Densi ty

The maximum anode current densities for diffcrcnt arc Iengths and currents

Arc iength. mm

Figure 2.9 Variation of the $as pressure at the surface of the anode with the arc length and applied current. Numbers are applied current in A.


Radial distance. mrn

8.0 6.0 4.0 2.0 0.0 6.3 ' I 1 1 1 l l

i

. . . . - - - - - - - - - . - - v 1

Figure 2.10 Variation of efeciro- 2.0 l

I I 1 I 1 magneric force at the

l

surface of the mode with chaneing the ruc Iength. Numbers are the atc lcngths in mm. - 2 . 0 ~ 1 O5 N

are summruized in Table 2-3. The catculated mode current densities from this study

are also cornparcd wirh the experimental data of Nestor [141, Tsai and Eagar [25] and

Lu and Kou [26] and Lhe calculated results of Westhoff [ I l ] .

---.- Exp. Welding handbook, 2.0 mm 1231 ---Exp. Welding handbook. 4.0 mm (231 +Calculated, 3.2 mm (Westhoff 1989) [ l 11 *This study. 2.0 mm. Jc = 6 .5~10 ' ~ . r n . ~ +This study. 4.0 mm. Jc = 6 . 5 ~ 1 ~ ' A.m +This study, 2.0 mm. Jc = 8.5~10' A-rn ' +This study. 4.0 mm, Jc = 8.5~10' ~ . r n - ~

9.0 1 I i 1 1 I 1

100 150 200 250 300 Applied current, A

Figure 2.11 Vÿnation of arc voltqe wilh applied current md arc length in cornparison with experimental and selectcd cdcuIated results.

Chapter 2 GTAW mode1 5 1

Table 2-4: Maximum Anode Current Density and Maximum Heat Flux for Different Cur- rems and Arc Lengths. ( J , ,,,. (~ .mm-') ; q, ,,, . (w.rnrm2)).

Applied current, (A)

ùi Figure 2.12. a cornparison of the anode current density distribution for a

6.3 mm arc ar h r e e different applied çurrents almg with the experimenüil data of

Nestor [21j is shown. The results of Nestor [24] are for a 1% thoriated tungsten elec-

trode having a pointed tip of 30" as the cathode and a water cookd copper plate as


Radial distance. mm Radial distance. mm Radial distance. m m

Figure 2.12 Anode current density for different applied çurrents compared with experirnental [24], and theoreticai [ I l ] results. a: 1 = LOO A; b: 1 = 200 A; c: I = 300 A. L, = 6.3 mm and Jc = 6 . 5 ~ 10' ~ . r n - ~ .

the anode. For 100 A and 300 A arcs the calçulated anode current drnsity is higher

than Nestor's data but for the 200 A case, it is aimost the samè. As shown, the ex-

penmental ma,ximum anode current densities of the anode for both the 200 A and the

300 A are alrnost rqud. Nestor [2J] related this to the lowr axial rlectrical field

smngth of the arc with higher applird current. On the other hmd. in the calculation it

is found that the electrical potential difference betwern the last node at the centre and

the anode surtàce are alrnost the same and the difference between the maximum anode

current densitics is related to the differcnce in temperatures and therefore the eleçuicai

conduçtivity of the gas in the vicinity of the anode surface. For the 200 A anrs with

different lengths. the anode current densities are compared with the experimental val-

ues of Nestor [24] in Figure 2.13, and shows that these two senes of data are in fauly

sood agreement with each other.

For the 100 A arc with different lengths, the maximum anode current densi-


Solid lines: This study

Dashed lines: Nestor. 1962 1241

Radial distance, mm

Figure 2.13 Anode current density for arcs with different lengths in comparison with experimental [% 1 results. Num- bers are arcs Iengrhs in millimetre. i = 200 A.

ues are higher than the experîmental values of Tsai and Eagar [25] (Figure 2.14). Tsai

and Eagar usrd Nestor's method to meuure the current density and by defining a dis-

uibution panmeter. expressed their results in a Gaussian form as follows:

Solia lins: This study Dashed I ine Tsai b Eagar. 1985 1251

I 1 I 1 I t

2.0 4.0 6.0 8.0 10.0 12.0 Arc length. mm

Figure 2.14 Variation of maximum anode current density with arc length in comparison with the experimental [25] values. 1 = 100 A.


where o, is thc distribution parameter and is a function of the applied current, arc

length. cathode geometry and gas composition. Lu and Kou's results [26] for the 100

A and 2.7 mm arc length are the only experimental data which are higher than the

calculated values. They have shown that the maximum current density is higher than

the one based on a Gaussian approximation.

In Figure 2.12, Westholfs results [ I l ] are also shown. It is clear that the two

series of calculated results arc different The reason is the stronger cooling effect of

the anode which was considered for the anode boundary condition in the present

study. In this study, in addition to convective heat transfer, the effect of the anode

work function was also considered.

2.5.3 Anode Heat Flux

The heat tlux lrom the arc to the workpiece have been measured experimen-

tally by Nestor [24], Tsai and Eagar [25], Lu and Kou [26] and Schoeck [27]. In Fig-

ure 2.15 the calculated results of this study are compared with Nestor's experirnental

results [24]. For the case of the 200 A. Westhoffs calculated heu ilux [ l l j , fit the

experimental data better than the present study. However the agreement between the

theoretical values of this study and the experimental values in the range of 100-300 A

is reasonable. Figure 2.16 shows a cornparison of the theoretical heat tlux with Sch-

oeck's values [27] for the 50 A. LOO A and 150 A arcs of 5 mm length. Except for

the 50 A arc, the calculated maximum heat flux for the two other arcs is very close to

the experimental values. but the variation of the heat flux with radial distance in both

cases is different, while the agreement between heat flux distribution of this study and

Ncstor's data (Figure 2.15) is much better. Both Nestor 1241 and Schoeck [27] used a


-Th~s study

Exp. Nestor. I ----- 1962 t241 Cal. Westhoff. l989[f I l

Radial distance. mm Radial distance, mm Radial distance, mm

Figure 2.15 Anode heat flux for different applied current in cornparison with experimental [24] ruid theorerical [ I l ] results. a: 1 = 100 A; b: 1 = 200 A; c: I = 300 A. Lm = 6.3 mm.

split plate as the anode but for the cathode Nestor used a 1 9 thoriated tungsten rod

with 3.175 mm (118 in.) diameter and 30 degrees tip angle. whilr Schoeck used a pure

tungsten rod of 6.35 mm (1/4 in.) with a 45 degrees tip aqle . The tungsten rod di-

ameter used in Nestor's experiments is almost equal to the diameter of the elecuodr

which is considered for the calculation. Anodier important parameter is the electrode

tip angle which will be studied in the next chapter.

Figure 2.17 shows the effect of the arc length on the heat flux to the anode

as obtained in the prexnt study and is compared fo the experimental values of Nestor

[24]. By decreîsing the an: length, the maximum heat flux at the centre of the anode


60 I I - This study

Radial distance. mm Radial distance. mm Radial distance. mm

Figure 2.16 Anode heat flux for different applied currents in cornparison with experimencal [27] values. a: 1 = 50 A; b: 1 = 100 A; c: 1 = 150 A. L, = 5.0 mm.

increases and is more localized. The agreement between the calculated results for arc

lensth 3.2 to 12.7 mm and at 200 A with the experimental values of Nestor is very

good.

In Figure 2.18 the maximum heat Cluxes for the arcs of different lengths with

arec different applied currents are compared with Tsai and Eagar [25] and Lu and

Kou [26] experimental values. The calculated values, are in good agreement with Lu

and Kou's values, except for one case, but compared to Tsai and Eagar's data the cal-

culated values are much higher.

To calculate the heat flux the three mechanisms, electron condensation, radia-

tion and convection, are considered. A summary of the contribution of these mecha-

nisms for different cases are given in Table 2-5. The contribution of the electron heat

transkr varies tiom 63.5% to 92% and is the dominant mechanism of heat transfer.

The electron contribution increases with decreasing arc length and applied current.

Both convective and radiative heat transfer show a slightly increase with the applied

current and a significant increase with arc length.


00 1 I l i

Solid iines: This study

Dashed lines: Nestor. 1962 124) -

-

-

-

0.0 2.0 4.0 6.0 8.0 1 C.0 Radial distance. mm

Figure 2.17 Anode heat tlux for arcs with different Iengths in cornparison with experimentai 1241 results. Numbers are arc lengths in mm. I = 200 A.

The only available estimation of the contribution of these three mechanisrns is

Wrsthoffs data [ L 11. In Figure 2.19. the results of this study for 6.3 mm arcs at dif-

lerent applied çurrents are compared with Westhoffs results. There is no difference in

the radiation contribution. however. the convcctive heat tlux of this study in d l cases

îs lrss than the others. This is due to the mentioned svonger cooling effeçt of the an-

ode and hencr lower gas temperature. that is a lower enthaipy at the vicinity of the

mode (rq. (2-14)). The rleçtron component increases with lower current and on in-

creasing the current its value decreases to Lrss than Wrsthoff's estimation. The srnail-

er v d u e for the electron contribution for the 300 A arc results because of the lower

rlectrical conductivity and lower anode current density (Figure 2 .12~) . In the case of

the 100 A arc, because of the smaller Joule effect. the temperature is lower. In the vi-

cinity of the mode the temperature is about 2000 K less than the 300 A case (Figure


--- 1 .Lu 6 Kou. 60 A. d,=?.d mm 1261 1

"\ Otu & Kou. 100 A. d,=2.4 mm

$Lt: 6 Kou. 100 A. de= 3.2 mm

OTsai & Eagar. 100 A {251

QTsai & Eagar. 190 A (251

QTsai b Eagar. 280 A 1251 Solid lines: T h ~ s study

O 1 1 1 I 1 1 ! I 0 2 4 6 8 10 12 14

Arc length, mm

Figure 2.18 VIuirttion of maximum anode heat ilux with arc length and applied current in cornparison with the experimen- ta1 [25, 261 values.

1.30). On the othrr hand. at temperatures less than 8000 K. the çonductivity of the ar-

oon plasma is very low (Figure 2.71) and this causes locdizing of the elecuon cur- .=

renrs in the centre part of the anode so that al1 current m u t p u s through a srnall area.

This leads to a signitiçant increase in the çurrent density (Figure 2.1%) and hence an

increÿx in the elrctron contribution (Figure 7.19a).

The variation in the contributions of the thrce mechanisrns of heat transfer at

the anode with arc lcngth are shown in Figure 2.22. This graph shows that the elec-

uon contribution is dominant. especially tor short ucs. At the shortest arc length. the

ekctron part (Figurc 2.22a) is reduced to a small area with the highrst maximum heat

flux. By incrcasing the length of the arc, the surface area of the rlectiron passage in-

creases and the maximum heat tlux decreases.

For the convective part (Figure 2.22b). the maximum value increases initially


Table 2-5 Contribution of Different Heat Transfcr Mechanisms to the Anode Heat FIux.

Applied current, (A)


Figure 2.19 Contribution of the threc mechanisms in the anode heat flux in cornparison with other theoretical estimation [ 1 1 1. a: 1 = 1IX) A; b: 1 = 200 A; c: 1 = 300 A. L,, = 6.3 mm.

i Solid iines: This study Dashed lines: Westhoff. 1989 11 11

I I - - - - -. ) Elecrron heat flux

,,,,,,,, )Convection heat flux 1 I )Radiation heat flux

Radial distance. mm

Radial distance, mm Radial distance, mm

Chapter 2 GTAW model 6 1

Radial distance, mm

Figure 2.20 Variation of temperature rit the vicinity of the anode surface with spplicd current and radial distance.

with increasing arc length due to the increase in the maximum radial velocity near the

anode (Fisure 2.33). Then, by decreasing the maximum radial velocity. the çonvective

part decreases.

In the case of the radiarive hrat ü-ansfer. increasing the arc length increases

Temperature.

Figure 2.21 Variation of Electrical çonductivity of &son with tcrnperature [9 1.


Radial distance. mm

Sadial distance. mm

Radial distance. mm

Figure 3.22 Variation of three rnechanisms of heat uansfer contribution into the anode heat tlux wiih the arc length. I = 200 A. a: Elecuon; b: Convection; c: Ra- diation.

Chapter 2 GTAW modei 63

Radiat distance. mm

Figurc 2.23 Variation of radiai velocity with the radial distance and tfie arc length. 1 = 200 -4.

the amount of heat (Table 2-5). In Figure 2 . 2 2 ~ i t is shown that the arc lengh c m

aiso change the distribution of radiative heat transkr to the workpiece. such that by

increasing the arc length the transferrcd heat tends to be distributed on the surface of

the workpicce more unilormly.

2-54 Gas Shear Stress on the Anode Surface

The rolc of the shear stress on the surface of the anode due to llow of the

plasma is the other panmeter which is imp«nant in ihe ettèct of arc on the weld pool

behaviour. To evaluate this parameter, WesthoKs [ I l ] approach was used. The shear

stress acting on the surfacc c m be caiculiitcd through the following aquation:

Figurc 2.24 shows the effect of the applied current on shear stress for a 6.3

mm arc length togerher with Wcsthoffs results [ I I ] . On increasing the applied cur-

rent, the differcnce between the two calculated results increascs. This probably results


Solid lines: This srudy Dashed line: Westhoff. 1989 (1 11

Radial distance. mm

Radial distance, mm

I L

7 - 4 6 Radial distance. mm

Figure 2.24 Variation of shear suess with radial distance ruid applied current in comp-xison with other theoretical estimation [ I I I . L, = 6.3 mm. ri: 1 = LOO A; h: i = 200 A; c: 1 = 300 AAOO A.

[rom the çooler gas ncar the anode. which is obtainrd through lhis study, and its ef-

fect on the viscosity of the gas. This cffect is also observed in the variation of the

shear suess distribution of the 200 A arc (Figure 2.25).

2.6 SUMMARY

in this chapter a mathematicai mode1 Cor the GTAW an: was developed. To

simplify the model. a cylindrical tungsten rod with a tlat tip was considered as the

cathode elecuodr. Rasults for the applied current range Born 100 A to 300 A and the

arc length range from 2.0 mm to 12.7 mm are prexnted. Cornparison with available

expenmentai results showed that the calculated temperature field was completely relia-

Chapter 2 GTAW mode1 6 5

I Solid Iines: This study Dashed Iine: Westhoff. 1989 11 1 1 1

Sadra! distance. mm

2 4 6 8 Radtal dtstance. mm

Radial distance. mm

Figure 2-25 Variation of the shear stress with radiai distance and arc length in cornparison with other theoretical estimation [ I l ] . 1 = 200 A. a: Lx, = 2.0 mm13.2 mm; b: Lu, = 6.3 mm; ç: L,, = 10.0 mm/12.7 mm.

ble. On the other hand. the information for the arc voltage indicated that a constant

cathode current density at different üpplied currents was not accurate and the cathode

current density mgsi increüse by increasing the applied current.

The anode currcnt density was calculaicd and it was iound that by decreasing


the arc length. the anode spot area decreases so that the anode current density increas-

es signitïcantly. Also. it wlis hund that the disuibution of the anode current density

changed with the applied current or in other words with gas temperature at the vicinity

of the anode surface. At lower currents or cooler gas. current density is more local-

ized at the centre part of the anode. By increasing the current and thereforc the tem-

penture of the gas near the anode. a wider m a is hot enough to let clectrons pass

through (Figures 7.20 and 2.31) and then the cumnt density at the anode spreads over

a wider area.

Calculatiun o f the anode hcat transfer showed that the dominant mechanism in

heat transfer to the workpiece is the electron contribution. The share of the electron

heat transfcr chmges from 63% for very long arc and high current to more than 92%

for short arc and low current, The share of the convection heat transfer increascs h m

-48 io -15% with applied currcnt and arc length. Also, the radiation contribution in-

creases h m -3.54 to - 2 2 1 with arc length and applied current. The rate of increase

with arc length for radiation is higher than that for convection such that in long arcs

the second important mechanism in heat transfer is radiation.

The sas shear stress applicd on the surface of the anode was also çaicuiated

for diffcrent currcnts and arc lengths and it was found that the increase in shear stress

with currcnt is signitlcant.

Although the rcsults of this modei lire in very good a p e m e n t with the exper-

imental data. this rnodel c m not predict the arc properties in the case of an electrode

wilh an angular tip. In the next chapter. the shape of the elecuode tip will be consid-

ered to obtain a mon: complete modei for GTAW arc and then the information ob-

tained from the arc modef will be used to develop the weld pool model.


REFERENCES

1- E. Pfender. "Electric Arcs and Arc Gas Heaters". in Gaseorts Electrunicx vol. 1,

edited by M. N. Hirsh and H. J. Oskam, Academic Press Inc.. 1978, pp. 291-398.

2- 1. F . Lancaster. The Physics of WUrL.ing. 2nd edn. Pergamon Press. 1986.

3- E. Pfender, "Energy Transport in Thermal Plasmas". Pure & Appl. Chem.. 1980.

vol. 52. pp. 1773-1800.

4- M. Mitçhener and C. H. Kruger Jr.. PurriLIIly ionized guses, John wiley & Sons.

Inc., 1992.

5- K. C. Hsu and E. Pfender. 'Twa-Temperature Modeiling of the Free-Burning

High-Intensity Arc". J. Appl. Phy~., 1983, vol. 54. pp. 4359 4366.

6- L. E. Cram. L. Poladian and G. Roumeliotis. "Depiinure fiom Equilibrium in a

Free-Buming Argon Arc". J. Phys. D: Appi. Ph'. 1988. vol. 2 1. pp 418-475.

7- A. J. D. F m e r and G. N. Haddad. "Rayleigh Scattenng iMeasuremcnts in a Free-

Burning Argon Arc", J. Ph- D: Appl. Ph-. 1988. vol. 2 1, pp. 42643 1.

8- J. McKelliget and J. Szekely, "Heat Transfer and Ruid Flow in the Welding Arc".

Merulfurgicul Trunsucriuns A. July 1986, vol. 17A. pp. 1 139- L 138.

9- M. 1. Boulos. P. Fauchais and E. Pkndcr. The~nia i plu.ss,nu. firndurnrntri1.s und up-

plicarions. vol. 1. Plenum Press. 1994.

10- D. L. Evans and R. S . Tankin. Measurement of Emission and Absorption of Radi-

ation by an Arpn Plasmi'. Phyics of Fluids. 1967, vol. 10, pp. 1 137-1 144.

I l - R. C. Westhoff, S.M Thesis, Dept. of Materials Sçiencc and Engineering. The

Massachusetts Institute of Technology, 1989.

12- H. A. Dinulrscu and E. Ptènder. "Analysis of the Anode Boundilry Layer of High

Intensity Arc". J. Appl. Phys.. 1980. vol. 5 1. pp. 3 149-3 157.

13- P. Zhu, J. J. Lowke. R. Morrow and J. Haidar, "Prediction of Anode Temperature

of free Buming Arcs", J. Phvs. D: Appl. Ph-.. 1995, vol. 28. pp. 1369- 1376.


14- N. A. Sanders and E. F%&der, "Measurement of Anode Falls and Anode Heat

Transfer in A~nosphenc Pressure Hi$ Intensity Arcs*', I. Agpf. P-. 1984. vol.

55. pp. 7 14-722.

15- W. M. Rosenhow and J. P. Hartnett, Handbook of Heat Transfer, Mc Graw-Hill,

1973. p. 8- 126.

16- 1. S. Gradshteyn and 1. M. Ryzhik, Table of Integrais. Series and Prodncts. 4th

edn, Academic Press. 1965.

17- CHAM, TR100- A Guide to the PHOENICS Input Languqe

18- K. C. Hsu, K. Etemadi and E. Pfender. "Study of Fee-Burning High Intensity Ar-

gon Arc'', J. Appf. Ph-., 1983, vol. 51. pp. 1293-1301

19- J. F. Couden and P. Fauchais, "The Influence of the Arc Fluctuations on the Tem-

penture Measurements in D.C. Plasma Jets", in n i e r m l Plasma Application in

Materials und Metallcrrgical Processing. rdited by N. El-Kaddah, TMS, 1992, pp.

75-83.

20- C. J. Allum. "Gas Flow in the Column of TIG Welding Arc". J. P h s . D: Appl.

Phu.s.. 1981. vol. 11, pp. 1041-1059.

21- C. J. Allum. "Power Dissipation in the Column of TIG Weldin~ Arc". J. Phus. D:

Appl. Ph?..s.. IY83. vol. 16. pp. 1149-2165.

22- P. Kovitya and J. J. Lowke, 'Two-Dimensional Analysis of Free-Buming Arcs in

Argon", J. Ph-. D: Appl. Ph-., 1985. vol. 18, pp. 53-70.

23- Wrlcling hundbook. 8rh rdn, vol. 2, American Welding Society, 199 1.

24- 0. H. Nestor, "Heat Intensity and Current Density Distributions at the Anode of

High Current. Inert Gas Arcs", J. Appl. Phys.. 1962. vol. 33, pp. 1638- 1648.

25- N. S . Tsai and T. W. Eapr, "Distribution of the Heat and Current Fluxes in Gas

Tungsten Arc". Metallurgical Transactions B. Dec. 1985. vol. 16B, pp. 841-846.

26- M. I. Lu and S . Kou, "Power and Current Distributions in Gas Tungsten Arc",


Welcing J.. 1988. vol. 67. pp. 2%-34.

27- P. A. Schoeck. "An Investigation of the Anode Energy Baimce of High [ntensity

Arcs in Argon*', in .+fodem Drvrloprnrnrr in H a t hn?~+Jer, cdited by W. Ibele.

Academic Press. 1963. pp. 353-4400.

C H A P T E R =7

3

3.1 INTRODUCTION

In chapter 7, a mathematical mode1 for GTAW was developed assuming a tlat

tip tungsten electrodr. In production different electrode tip configurations are used to

improve welding performance [ I l . The geometry of the electrode tip can significmtly

affect the shape and s i x of the weld pool [ 1.21 ris shown in Figure 3.1. Haidar and

F m e r [3] found diat the arc temperature varies significantly for different thoriated

tungsten cathodes with different tip mgles.

Tsai and Kou [4] studied the effect of the electrode shape on the arc proper-

ties. n i r y found that a çonical rlcctrode. when compared to a tlat tip çlecuode.

caused a highrr axial velocity ar the centre line of the arc and hence a signitkant out-

tlow near the anode suriace. and a higher pressure at the surface of the anode. They

dso showed that the configuration of the electrode çhanged the electron path around

the cathode. In some recent coupled cathode-arc and cathode-arc-anode models [S-81,

the cathode faces were not t lat But in al1 these models, a single configuration for the

cathode was considrred. Moreovcr. there is no information available regardins the ef-

fect of the electrode tip angle on the anode heat tlux and the mode current density.

These are very important in detemining the shape and properties of the weldment.

3.2 MODEL DESCRIPTION

The governing equations, including the conservation and Maxwell's equations

Chapter 3 GTAW, Effect of the cathode tip angle 7 1

Figure 3.1 Effect of electrode tip geometry on the weld pool shape and size [ I 1.

and the assumptions made have been presented in chapter 7. The source terms for the

conservation equations have ais0 been explained in detail in the previous chapter. The

main differences in the model for angular electrodes are the calculation domain and

the boundary conditions.

Figure 3.2 shows the calculation domain. h the modified model, the ningsten

electrode is added to the calculation domain for the electric potentiaL The other pa-

rameters are the same as those used previously. Another difference is the boundary

condition for the elecuic potential. Instead of assigning a constant current density to

the cathode surface, the cathode surface area is defmed as a function of the elecuode

Chapter 3 GTAW, Effect of the cathode tip angle 72

C

lnflow

# Workpiece (Anode). 15.0 mm

Figure 3.2 Cdculation domain for tapered electrade GTAW (schematic).

tip angle. The boundary conditions for this mode1 are summarized in Table 3-1.

The reason for both modif'ications was the result of the early calculations with

a constant curent density, or constant surface area. for the cathode. for different elec-

uode tip angles. By decreasing the angle of the electrode tip. the maximum tempera-

ture of the plasma tended to occur outside of the axis of symmetry (Figure 3.3), while

the available experimentril results of the plasma temperatures [3, 9- 151 show that the

maximum temperature always occur at the centre line of the arc, at least for a wide

range of the elecvode tip angles. Therefore the inflow of the electrons with uniform

disuibution placed somewhere into the electrode before the conical part (for example.

AB surface), and the cunent density distribution on the assigned cathode surface was

C hapter 3 GTAVJ, Effect of the cathode tip angle 73

Table 3- 1: Boundary Conditions.

At these two boundaries pressure is tixed to a constant value.

v

u

h

I

adjusteci so that the total entering current became equal to thc applied c u m n t An at-

trmpt was made to sirnulate the real case in which the current density is not a con-

stant value over rhe cathode spot surface (Richardson-Dashman equation. eq. ( 1- 1 )).

Haidar and Farmer [3] merisurcd the èlectrode surface area covered by the

plasma for a range o f the elcctrode tip angles. n ~ r variation of Lhis arca with the

rlectrode tip angles based on their measurement is shown in Figure 3.1. From this in-

formation. rhe cathode surface area (the rlectron rmission surface). for angles less than

60 degees is constant and above 60 degrees. up to 180 degrees, it increases lineariy

with the elrçtrodt: tip angle. The cathode surface area for the 60 degree angle and the

dope of the line were determined by compÿnng the calculakd arc temperatures and

the experirnenüil data of Haidÿr and F m e r ar 1.5 mm lrom the tip of the elecuode

[3]. Variation of the cathode surface area with the elecuode tip angle is shown in

Figure 3.5.

In these cdculations, the cathode rod was assumed to be pure tungsten and

the properties of the tungsten was obtained from Touloukian [Ki].

L

FG

O

O

T=3()00 K

Qionir=(JCI ' C

S , = f ( a )

BC1

0

u = Const.

T=10 K

= 0 z

EF I

O

au - & = O

t)h =

as a ; = O

AB

1 J z = 7

2=Rei,c

CD1

, = O av

- & = O au

htlow: T=lOOO Outtlow:

ah z=O ab

= O

GH

O

0

T=3000 K

J, = O J, = 0

DE

O

O

Eq. (2-L4)+

Jaba

0-CO"".

m

HB

O

O

T=3000 K

Jr = O

Chapter 3 GTAW. Effect of the cathode tip angle 74

Tip angle = 180 deg Tip angle= 106.3 deg

Radial distance, mm

Tip angle= 71.5 deg

Radial distance. mm

Tip angle= 33.2 deg

Radial distance. mm

I 1 1.5 1 .O 0.5 C

Radial distance. mm

Figure 3.3 Effcct of the eiccuode tip angle on the maximum temperature and

position. Ic = 10%.rn-~.

3.3 RESULTS AND DlSCUSSION

The GTAW mode1 for pure argon was simulated ror a range of elecuode tip

angles. Thnie ansle of the electrodr tip was varied tiom about 9 to 150 degrees. Also,

calculations were performed for arc lengths of 2.0, 5.0 and 10.0 mm and for applied


50 75 1 O0 Cathode tip angle. deg

Figurc 3.4 Variation in the area of the cathode surface covered by plasma as a tùnction o f the cathode tip angle [3]. 1 = 200 A. L,, = 5.0 mm.

I 1 1 I I I I

- -

- -

- -

- -

I I 1 l 1 I 1 O 2 O 40 60 80 1 O0 120 140 160

Eiectrode tip angle. deg

Figurc 3.5 Variation of the cathode surface m a with the electrode tip angle.


currents of LOO, 200 and 250 A. The et'fects of the anode heat tlux, anode current

density, shear stress and arc pressure on the weld pool due to plasma tlow are of pri-

mary interests. It is important to verit'y the results of the proposed model developed

from this study with published experirnental results. For this purpose the effect of the

electrode tip angles on the arc properties. especially anode heat tlux. anode current

density, arc pressure and plasma llow shear stress will be prcscnted.


Four major properties of the arc, i.e. maximum tempenture, maximum axial

velocity. electric poiential difference and anode pressure difference. for different cases

are given in Table 3-2. Figure 3.6 shows the variation of temperature at 1.5 mm be-

low the cathode tip with the electrode tip angle and applied çurrent. In this graph the

experirnental result of Hÿidar and F m e r 131 For the 200 A arc is also shown. Up to

120 degrees. the difference between the theoretical and the cxperimental vdues is less

than 5% and for the 150 degrees. the lürgest difference is about 12%. Thus it can be

concluded that for angles less than 120 degrces. the devcloped model for the cathode

surface area for the 200 A arc is vcry good. In Figure 3.7 the temperature profiles o f

the LOO A and 200 A arcs of 10 mm length and 60 d e p x rlcctrode tip angle are

cornpared with Hsu et a[. rcsults [ 1 LI. Although the tip anglc in Hsu cr ul. [ l 1 ] zxper-

iments is not specified, thc agreement between the theoretical and the experimental

data for the 200 A arc is excellent even at die cathode boundary. But in the case of

the 100 A the maximum temperature is about 1000 K under zstimated. Even in this

case, the agreerncnt in the outer part of the arc between theoretical and experimental

values is reuonable.

The radial distributions of temperature at 1.5 mm [rom the tip of the rlecvode

for two different cathode tip angles are compared with Haidar and Farmer experimen-

Chaater 3 GTAW, Effect of the cathode tip angle 77

Table 3-2: Arc Parameters for Different Arc Lengths, Applied Currents and Electrode Tip Angles

4pplied Current

(A)

Electrode Tip Angle (deg)

Chapter 3 GTAW, Efiect of the cathode tip angle 78

Solid lines: This stud ' I

Dashed line: ~ a i d a r l ~ a r r n e r , 1994 131 -

d - - - - _ -

-- - - - - - - 9 -

- - * -

Cathode tip angle. deg

Figure 3.6 Variation of plasma temperature at 1.5 mm from the tip of the tungsten electrode with the electrocle tip angle and applied current. Numbers are applied current in A.

tal results [31 in Figure 3.8. It can be observed that the agreement between the theo-

retical and the experimental values down to 10000 K is very good. However below

10000 K the theoretical values decreasr s h q l y to very low temperanires. This behav-

iour is probahly due to the absorption of energy. emitted from the central pan of the

arc, by the outer piut of the arc. in chapter 2 it w u pointed out that the opticdly thin

assumption t'or the arc is correct only for the central part of the arc. Therefore it cm

he concluded that the theoretical temperatures below 10000 K are not reliable.

Table 3-2 indicates that the maximum velocity occurred at about 37 degree tip

angle for difkrent âpplied currents. This is shown in Figure 3.9 for the 200 A and 5

mm arc for the axial distribution of axial velocity for different electrode tip angles. At

high (150 degrees) and low (9.18 degrers) tip angles. the maximum velocity is found

at longer distances from the elrctrode tip. The maximum velocity is closest to the

electrode tip at 60 degree angle.

The radial distributions of the axial velocity of the gas at 2.35 mm from the

rlectrode tip are shown in Figure 3.10 for different electrode tip angles. The peak ra-


Cal. Exp. 1 11000K * 2 12000 K -o 3 13000 K 4

4 14000K * 5 75000 K 4

17000 K O

Radial distance, mm Radial distance. mm

Figure 3.7 Isotherrns of 10.0 mm arcs in cornparison with the experimentrtl results of Hsu t?r d. [ 1 1 1. a: I = 100 A; b: 1 = 200 A.

dia1 velocity increases as the tip angle decreued from 150 to 37 degees. Bclow 37

degrees, the peak radiai velocity decreases with deçreasing tip angle and the radial dis-

tribution tends to become flatter.

Table 3-2 shows that the pressure at the mode increases with the applied cur-

rent. For the 100 A arc the pressure at the anode decreases with increasing arc length.

but for the 200 A and the 250 A arcs, the anode pressure rernains alrnost constant

when the arc length is increased from 2 to 5 mm. This is due to the significant in-

crease in the elecirornagnetic force at the anode surface when the arc length is de-

creased. The variation of the radial distribution of the pressure at Ihe anode with the

electrode tip angle is shown in Figure 3. L 1. By decreasing the cathode tip angle. the


- Solid h e : This study Dashed Itne: Haidar 6 Farmer 1994 131

w

Solid line: Thrs study Dashed line: Haidar Er Farmer 1 994 (31

5000 1 I 1 I 1 1 0.0 1 O 2.0 3.0 4.0 5.0

Radial distance from the arc axis. mm Radial distance from the arc axis. mm

Figure 3.8 Radial distribution of temperature lit 1.5 mm tiom the cathode tip is com- p m d with the experimental clritri [3 1. 1 = 200 A. L, = 5.0 mm. a: a = 20.74 dcg (Exp.: a = 18 deg); b: a = 60 deg.

anode pressure increaxs to a maximum at an angle of about 20 degrces. At tip angles

srnaller than 20 degrees, the pressure profile tends to be a little flatter. Increasing the

pressure at the anode cm change the contiguration of the weld pool surface. Westhoff

[17] has shown that cven a srnall depression of about L mm at the surface of the weld

Axial distance from the electrode tio, mm

Figure 3.9 Variation of the axial distribution of the axial velocity with electrode tip angle for 200 A and 5.0 mm arcs. Numbers are dcctrode tip angles in deg.


Radial distance from the arc axis. mm

Figure 3.10 Variauon of the radial distribution of the axial velocity wiîh clectrode tip angle. 1 = 200 A. Numbers are the electrode tip angle in deg.

pool c m change the electric current path significantly.

Typical pressure distribution in the arc for three different zlecuode angles is

shown in Figure 3.12. For low an$rs, the maximum cathode pressure is on the side of

the electrode cone. In tact. for very low angles thrre is a negative pressure at the tip

Figure 3.1 1 Variation of the radiai distribution of pressure at the mode surfsace with electrodt: tip angle. 1 = 200 A. Numbers arc electrode tip angle in deg.


Figure 3-12 Distribution of pressure contours for three electrode tip angles. 1 = 2 0 A; L, =5.0 mm. a: a = 9.18 deg; b: a = 37.33 deg; c: a = 60.00 deg.

Radial distance. mm

7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 Radial distance, mm

7.6 6.0 5.0 4.0 3.0 2.0 1.0 0.0 Radial distance. mm

of the electrode. The axial variation of pressure at the cenue line of the arc for differ-

ent angles is illustnted in Figure 3.13. %y increasing the cathode tip angle. a maxi-

mum in pressure occurs at the cenue which moves toward the rlectrode tip. Since the

flow is parallel to the surface of the electrode. changing the flow direction to a direc-

tion dong the axis of symmetry produces the maximum pressure, and by increasing

the angle, which conesponds to increasing the amount of reflection. the maximum

pressure increases (Figure 3.13).


Distance from the electrode rip. mm

Figure 2-13 Variation of axid distribution of pressure with elecuode tip angle. Numbers are elecuode tip angle in deg.

The variation of the arc voitage with the rlectrodr tip angle and the applied

current is shown in Figure 3.14. There are two panmeters that affect the elrctric po-

tential difference between the two electrodes, the mean diameter of the arc, or the

conductor media beween the electrodes, and the electrical conductivity of the cas

I I I I 1 I 1 0.0 30.0 60.0 90.0 120.0 150.0

Electrode tip angle, deg

Figure 3.14 Variation of arc voltage with electrode tip angles and applied current. Numbers are applied current in A.

Chapter 3 GTAW, Effed of the cathode tip angle 84

which is function of temperature. It will be shown later in this chapter that by in-

creasing the ekcuode tip angle. the arc shrinks. In other words. its mean diamrter de-

creases. Therefore. if no change in the conductivity of the gas occurs any increase in

the electrode tip angle increases the arc voltage. Since the conductivity of the gas is a

function of the temperature and the latter one itself is a function of the electrode tip

angle, then it is possible for the rlectric potential to decrease or increase. Figure 3.14

shows that up to 60 degrees. the arc voltage increÿses with the tip angle for 5 mm

arcs. Above 60 degrees. it depends on the arc temperature and hence the slope of the

variation of the elecuical conductivity of the plasma (Figure 2.2 1 ) and the arc voltage

increases or demeases.

The arc voltage for different applied currents and arc lengths for an elec~ode

with a 60 degree tip angle is compared with the experirnental values [ I l in Figure

3.15. This graph indicates that the calculated values for the 200 A arc are in good

azreernent with the rxperimentai data. The slope of the variation in both 2 mm and 5

- - -

1 - 1

Soitd lines: This study

Dashed Iines: Exoerimental data Il l I

8.0 1 I I I 1 1 50 100 1 50 200 250 300

Applied current. A

Figure 3.15 Variation in ihe arc voltage with arc length and applird cumnt in cornpiin- son with the rxperimentd data [ I I . Numbea are lrngth of arc in mm.


mm arcs is higher than the experimental values. By considering this and also the tem-

penture results (Figure 3.7a). it can be concluded that the cathode surface area chang-

es with the applicd current. The distribution of the electric potential for three

electrode tip angles is illusülited in Figure 3.16. The rlectrode tip angles cm also

change the voltage difference in die electrode significandy. The effect for different

currents is depicted in Figure 3.17. The effect of the angle on the voltage drop 1s due

to the variation in the cross seciional area of the elecirode.

Figure 3.16 Distribution of electric potential in the arc for different electrode tip angles. The isothem line of 10000 K shows the domriin that the results are reliable. 1 = 200 A, L, = 5.0 mm. a: a =9.18 deg; b: a = 60.00 deg; c: a = 100.06 deg.

Radial distance. mm Isoelectric potentials. 1 V intervals. Electric potential at the anode 1s zero

Radiai distance. mm Radial distance, mm



Figure 3.17 The variation of the voltage drop into the electrode with the eiectrode tip angle. Numbers are applied current in A.

3.3.2 Anode Current Density and Heat Flux

The maximum current density, the maximum heat flux and the maximum tem-

perature on the anode surface for different electrode tip angles, arc lengths and applied

currents are summarized in Table 3-3. In most cases the anode current density in-

creases with the electrode tip angle. It can be concluded that by increasing this angle.

the electrons path toward the anode become more concentrated at the centre part and

leads to a higher current density for larger electrode tip angles. The exceptions are for

the 5 and 10 mm arcs with I00 A current. In these cases the anode current density

reaches a maximum at 60 degrees and then decreases. Although the electric potential

gradient still increases with the electrode tip anple. the plasma temperature at the an-

ode surface is very low. therefore the electrical conductivity of the gas is exuemeiy

small, in the order of hundreds ~ . m - ' . Typical variation of the anode current density

with the electrode tip angle and applied current for the 5 mm arc is illustrated in Fig-

ure 3.18. The variation of the plasma temperature at the surface of the anode is also

shown. Temperature of the gas at the anode surface for the 100 A arc is much lower

and is below 9000 K in all cases. Typical variation of the anode current density with


Tale 3-3: Maximum ode Current Density, Maximum Heat Fiux and Maximum Temperature of the Gas on the Anode Surface for Different Elecuode Tip Angles, Arc Lengths and Applied Cur-

Applied Arc Current Length -l

(A) (mm) 9.18 r

6 . c W


the ekctrode tip angle and the arc length for the 200 A arcs is shown in Figure 3.19.

The effect of the electrode tip angle is more significant for short arcs.

The effect of the electrode tip angle. arc lrngth and applied current on the

maximum hcat tlux to the anode is shown in Figure 3.20. Aimost in al1 cases, the

maximum hrat tlux to the anode occurred between 30 and 60 degrees. The same ba-


l l I I l

30 60 90 120 150 Electrode tip angle. deg

Figure 3-18 Vaiation of a: maximum anode currcnt density, and b: maximum _sas temperature on thc anode surface. with the clectrode tip angle and applicd çurrcnt for the 5.0 mm WC. Numhers are the üpplied curent in A.

u

haviour was observed in the total heat tnnsferred to the anode. Variation of the total

heat and also the coniributions of the three different mechmisms with the eiectrode tip

angle and an: length is shown in Figure 3.21. It is obvious that the electron heat flux

is the major mechanism of heat uansfer io the anode. The absolute value of heat due

to electrons remains almost constant with change in the angle ot' the electrode tip. but

i 5000

*- L 13000

3 - C a r 1000

E c.

r/>

G 9000

its contribution to the total heat decreases with increasing arc length.

1 I I i I I I

- - 2 5 0 - - - - 200.-- 1 - -

- - - -

- -

- 7000 6

- 1 0 0 I I I I I

30 60 90 120 150 Elecrrode tip angle. deg


Figure 3.19 Variation of maximum anode current density with electrode tip angle and arc lengrh. 1 = 200 A. Numbers are arc length in mm.

20 P.

E E

>r % <n C

8 I O - C a'5: F L 3

5 - al -0 O C Q:

O

Convective heat transfer is the most sensitive mechanism to the electrode tip

angle. The variation in convection causes the variation in ~ h e total heat. The radiation

and electron heat transkr mechanisrns do not show significant change with the elec-

trode tip angle. On the other hand, the variation in the radiation heat transfer with arc

length is the most signifiant. For long arcs, the contribution of radiation is more than

convection (Figure 3.21). For very long arcs, radiation heat uansfer is dominant.

The total heat and the contributions of ail three mechanisms are given in Ta-

ble 3-4. This Table shows that the variation of the total heat and its components with

the electrode tip angle in the range of 100-250 A is similar but at different levels.

The effect of the electrode tip angle on the distribution of the total heat flux

and the three heat transfer mechanisms for the 200 A and 5 mm arc is shown in Fig-

ure 3.22. The most significant change is related to convection. By decreasing the an-

*le from 150 degrees to about 38 degrees (which is not shown in this graph), the t

maximum heat tlux due to convection increases about four h e s . The heat flux due to

convection spreads over a very small area at the central part of the anode, while the

I I 1 I I

2.0 y -

: 5.0 ,

10.0

I I I I I O 30 60 90 1 20 150

Electrode tip angle. deg


Efectrode tip angle, deg

Electrode trp angle, deg

Electrode tip angfe, deg

Figure 3.20 Variation of the maximum anode heat flux with elrcvodr tip mgle and arc Icngth for arcs of different applicd currents. Numbers arr: arc length in mm. a: 1 = 100 A; b: 1 = 200 A; c: 1 = 250 A.


Totai heat Eiectron contribution Convection contribution Radiation contributron

Electrode trp angle. deg m


I I j I 1

1 l 1 ! l

30 60 90 120 1 50 Electrode trp angle. deg

Figure 3.21 Trmsferred heat to ihe anode versus the electrode tip angle for 200 A arcs with differeni arc lengths. a: L,, = 2.0 mm; b: L m = 5.0 mm; c: L,, = 10.0 mm.

C hapter 3 GTAW, Effecî of the cathode tip angle 92

Table 3-4: Total Hear Transferred to the Anode and Contributions of the Three Mechanisms for Different Situations. (Q,, Q,, Q,, Q, (W)).



* - - - . - - . - - . 150.00 degrees

60.00 degrees - - - - - - - - - 20.74 deg rees

9.18 degrees

Radial distance. mm Radial distance, mm

Radial distance, mm Radial distance, mm

F i p r 3.22 Effect of the rlectrode tip mgle on îhr radial distribution of the heat flux to the mode. 1 = 200 A. L, = 5.0 mm. a: Total hent flux; b: Elecuon contribution; c: Convection contribution; d: Ra- diation contri bution.

Chapter 3 GTAW, Eff ect of the cathode tip angle 94

two other mechanisms are spread over a laser area.

3.3.3 Shear Stress

The shear stress applied to the surface of the anode is proportional to the ra-

diai velocity of the gas. The effect of the electrode tip angle. together with the ap-

plied current and the arc length. on the maximum radial velocity at the anode surface

is illustrated in Figure 3.23. The effect of the tip angle is the most significant for the

shortest arc. Figure 3.23a shows that for the sharp electrode, the maximum radial ve-

locity for the 2 mm arc is the highest but by increasing the tip angle it decreases to

about the maximum radial velocity for 10 mm arc. Figure 3.23b shows that for the

100 A arc. the radial velocity of the gas is very srnall. Thus, for t h e x arcs the shear

stress c m not be important.

The effects of the electrode tip angle and arc length on the distribution of the

shear stress at the anode surface of the 200 A arc are shown in Figure 3.24. As illus-

trated, a sharp eiectrode can increase the shear suess by 2.5 to more than 4 times. For

the 2 mm arc, althou~h the maximum radial velocity for the 28.36 degrees electrode

tip is the highest (Figure 3.23), the shear suess in the case of 9.18 degrees is higher

than that of the 28.36 desrees. From Figure 3.18b, it is known that the gas tempera-

ture for a 5 mm arc changes with the electrode tip angle. For the 2 mm arc. this vari-

ation is more significant. This observation explains the ditference in the shear stress

and the radial velocity distributions on the surface of the anode.

3.4 CONCLUSION

Althaugh the assumption of a constant current density. which is a function of

the applied cumnt, over the cathode spot gave good results for the tlat tip electrode.

this was not the case Cor the angular tip electrode. Instead, it was necessary to use a

variable cathode surface area. This resulted in a very good agreement of the calculat-



Figure 3.23 Variation of radial velocity of the gas with the electrode tip angle. a: Effect of arc length. 1 = 200 A. Numbers rire arc length in mm. b: Effect of applied current. L,, = 5.0 mm. Numbers are applied current in A.

200 - 2 SI50 - O O - 2 5 100 7 3 L

5 E .x 5 0 -

2 O

ed results with the experimental data, particularly for the 200 A arcs, for a wide range

of the electrode tip angles from 10 to about 150 degrees. This study indicated that the

calhode surface m a increased with the applied current.

Increasing the tip angle led to shrinkage in the arc diameter. Therefore the an-

ode current density and the heat flux due to the electrons increased with ihe tip angle.

! 1 I T 1

- -25

- r200 I 1 O0 1 1 I 1

O 30 60 90 1 20 150 Electrode tip angle, deg

Chapter 3 GTAW, Eifect of the cathode tip angle 96

- - - - - - - - - '1 50.00 degrees

60.00 degrees ----------- 28.36 degrees

------ 9.1 8 degrees

Radial distance. mm

Radial distance. mm

Radial distance. mm

Figure 3.24 Effect of the alectrode tip mglr on the distribution of shear stress in arcs with different lengrhs. I = 200 A. a: L, = 2.0 mm: b: L, = 5.0 mm; c: L, = 10.0 mm.


On the other hand. decreasing the electrode tip angle increased the sas velocity and

therefore the convective contribution to the anode heat tlux. The highest heat tlux oc-

curred for electrode tip angles in the range of 30 to 60 degrees.

Decreasing the electrode tip angle h m 150 to 10 degees caused the pressure

on the surface of the anode to increase by 50 to 180% dependinp on the applied cur-

orees. rent. The maximum pressure at the anode surface occurred at angles of 20-30 de,

The shear stress increased significantly with decrease in the elecuodr tip angle. In-

crease in both the pressure and the shear stress with the applied current was very con-

siderable. Thus, the surface of the weld pool tends CO be more unstable at hizh

currents dut: to the high pressure and the high shear stress. This suggested that for

high currents. it is better to use electrode with wider tip angles to obtain a smoother

wdding surface.


REFERENCES

1- Welding handbook. 8th rdn, vol. 2. Amencan Welding Society, 199 1.

2- S. Kou. Wrlding Mrtallurgy. John Wiley & Sons. 1987.

3- J. Haidar and A. J. D. Famer. "Large Effect of Cathode Shape on Plasma Tem-

penture in High-Current Free-Burning Arcs". J. Phys. D: Appl. P h . . 1994, vol.

27, pp. 555-560.

3- M. C. Tsai and S. Kou, "Heat Transfer and Fluid Flow in Welding Arcs Produced

by Sharprned and Hat Electrades", Int. J. Heur and Muss Transfer, 1990. vol. 33.

pp. 2089-2098.

5- C. Delalondre and 0. Simonin. "Modelling of High Intensity Arcs including a

Non-Equilibrium Description of the Cathode Sheath". Colïoq~ie Dr Ph?~siqur. 15

1990. Colloque CS. Supplement Au n 18. Tome 51. pp. 199-206.

6- P. Zhu. J. J. Lowke and R. Morrow. "A Unified Theory of Free-Buming Arcs.

Cathode Sheaths and Cathodes". J. Phys. D: Appl. Phys.. 1992. vol. 25. pp. 122 1 -

1230.

7- J. J. Lowke, P. Zhu, R. Morrow, J. Haidar, A. J. D. Farnier, and G. N. Haddad,

"Onset of Cathode and Anode Melting by Electnc Arcs". in Int. Svmp. on Pfusmcl

Chrm.. Loughborough, UK, Aug 1993.

8- P. Zhu, J. J. Lowke, R. Morrow and J. Haidar. "Prcdiction of Anode Temperature

of Free B urning Arcs". J. P h y . D: Appl. Phys., 1995, vol. 28, pp. 136% 1376.

9- H. N. Olxn, "Measurement of Argon Transition Probabilities Using the Thermal

Arc Plasma as a Radiation Source", J. Quant. S p e c ~ o x . Rudiat. fiansfer, 1963.

vol. 3, pp. 59-76

LO- L. Bober and R. S. Tankin. "Emission and Absorption Meiisurernents on a Suong-

ly Self-Absorbed Argon Atom Line". J. Quant. Specrrosc. Radiut. Trunsfer. 1969,

vol. 9, pp. 855-879.


11- K. C. Hsu. K. Etrmadi and E. Pfendrr. "Study of Free-Buming High-Intensity Ar-

@on Arc". 1. Appl. Phys.. 1983, vol. 54. pp. 1293- 130 1. s

12- G. N. Haddüd and A. I. D. F m e r . "Temperature Deteminations in a Frer-Burn-

ing Arc: 1. Experimental Techniques and Rrsults in Argon". J. Phys. D: Appl.

Phys., 1984. vol. 17. pp. 1189-1196.

13- A. J. D. F m e r and G. N. Haddad, "Rayleigh Scattering Measuremcno: in a Free-

Buming Argon Arc". J. Phys D: Appl. Phys., 1988. vol. 21. pp. 426-43 1 .

14- P. Vervisch, B. Chrron and I. F. Lhuissier, "Spectroscopie Analysis of a TIG Arc

Plasma", J. P h ~ s . D: Appl. P h y ~ . , 1990, vol. 23. pp. 1058-1063.

15- A. B. Murphy, "Laser-Scattering Temperature Measurements of a Frec-Buming

Arc in Nitrogen". J. Ph- Dr Appl. Phys.. 1994. vol. 27. pp. 1 - W - 1498.

16- Y. S. Touloukian. 7% r rniophxsicul Propri-tics of High Temperutrire Solid Mute ri-

d s , VOL 1: E1tirn~~nr.s. h1cMillm Co. 1967.

17- R. C . Wcisthoff, S.M nes i s . Dept. of Materials Science and Engineering. The

Massachusetts institute of Technology, 1989.

3.1 INTRODUCTION

In the two previous chapters. a model for the GTAW arc was developed and

die rffect of the electrode tip angle was investigated. The main objective of these cd-

culations was to provide the necessary information for a joint mode1 for WC and weld

pool. To check the accuracy of the obtained results, they were compared with availa-

ble experirnentd data but unfortunately the experimental data for the variation of the

anode current dcnsity and the hrat flux with the rlectrode tip angle were nor available

in the literature. In this chapter, the obtained data from the arc calculations are used

to develop a rather simple weld pool model. Using this model. the effect of different

welding parameters on the shape and size of the weld pool are investigated. For the

first tirne. the effwt of the elecuode tip angles on the weld pool proprrties will be

considered.

4.2 WELD POOL MODEL

Therc are severai models available for the weld pool and a detailed discussion

about different aspects of the weld pool rnodels c m be found in the litenture [ M l .

The GTAW process is shown schematically in Figure 4.1. By applying an

electric potential difference berween the cathode (the tungsten electrode) and the anode

(the workpiece). an arc c m be rsiablished between the two poles. This arc is the en-

ergy source in this process. The amount of heat that is absorbed by the workpiece is

Chapter 4 WeM Pool Model 101

O Electrode cip angle (a)

Vaporizarion. radiative and convective tosses

Rad~ative and convective losses

Figure 4.1 Schematic representation of the GTAW arc in the weId pool. The various physicai phenornena occurring in the workpiece (the right side) md the calçulation domain (the left side) are indiçat- ed. The origin of the cdculation domain is locrited rit point A.

high to melt it and develop a molten metal pool. This pool will grow until the heat

gained by the workpiece equals the heat loss by conduction. convection, radiation and

vaporization. Flows in the pool are driven by a combination of forces mentioned in

section 1.41 and are shown schrmatically in Fisure 1.1.

The main objective of this chapter is to providr a tool to determine the validi-

ty of the arc mode1 for the angular tip electrodes. The rnodels for the weld pool and

Chapter 4 Weld Pool Model 1 02

the arc c m be used to study the effect of the electrode tip angles and arc lengths on

the shape of the weld pool. These results will be considered in evaluating the validity

of the arc mode1 for the angular tip elrctrodes.

The mathematical model presrnted in this chapter describes the tluid tlow and

thermal rnergy transport for two dimensional tlow. lt is shown that the assumption of

lamina tlow is not very accunte [6], hence the model is based on the K-E turbulent

model. The caiculated data for the 200 A arc with lengths of 2.0 and 5.0 mm and dif-

ferent rlecuode tip angles have been used in this model. h these calculations it was

assumed that the free surface of the weld pool is ilat.

The driving forces that are considered in this model are as follow:

The elrctromagnetic force arising from the interaction of the diversent current md

its own rnagnetic tïeld:

The buoyancy force due to variation of density:

0 The surface tension gradient due to temperature gradient (Maranponi rffect );

0 The shear stress on the surface of the weld pool by the tas dow.

This model also includes the Joule heating into the workpirce and the mzlt-

ing/sdiditïcation phenornenon into the weld pool. Althoush some heat will be lost

from the free surface of the workpiece due to radiation. convection and vaponzation.

for simplicity these mechanisms are not considered in the present model. It has been

reponed that the vaporization in GTAW of stainless steel. for example. does not play

a critical rote in determinin: the temperature of the weld pool surface [ I l . Also. vis-

cous dissipation in the weld pool is assumrd negligible.

4.2.1 Coverning Equations

The relevant transport equations based on the mentioned assumptions are List-

ed below. Ic must be noticed that d l functions of flow. and accordingly functions of

Chapter 4 Weid Pool Model 1 03

temperature are time avenged variables.

Thc sourcc rems in Equation 14-21 arc due to electrnmagnetic hrces and to

Boussinesq's approximation tif the thermal buoyancy force. In Equation (4-3) the

sourcc term is only duc to clecmrnagnetic forces.

The source terms in Equation (4-4) are the Joule heating and the heat loss

due to mcliing of the metal.


To calculate the elecuomagnetic tïeld and çurrent densities, auxiliary equa-

tions based on Ampere's Law and Ohm's Law are needed (Equations (2-6) and (3-7)).

The fraction of liquid in Equation (4-4) is defined as:

T - T

C, = 1 for T 2 TI

For the turbulent model the standard two-equation K - E rnodel [7] is em-

ployed:

The effective viscosity. p, and the thermal conductivi ty, kef f , are calculat-

cd as the sum of their molecuIar and turbulent components:

Chapter 4 Weid Pool Model 1 05

keff

- h -

while

The turbulent viscosity in the K - E model is given by: 3

the turbulent thermal conductivity is derived Srom the turbulent Pnndtl number:

C ~ p , Pr, = - = 0.9 (4- 13)

The recomrnended valucs for the !ive empirical constants are:

C, = 0.09. C, = 1.U. C, = 1.92. bK = 1.0. oE = 1.3.

4.2.2 Melting and Solidification Modelling

[o the cnergy equation there an: thrce terms that detcrmine the amount or heat

lost due to mrlting. These terms are functions of the latent heat and variation of the

liquid fraction with timr and position. and are derived from the expression for the en-

thaipy as follows:

liquid fraction, TI, is defined by Equation (4-6).

In the momentum equations (Equations (4-2) iind (4-3)) there is no cxtra tcrm

for tlow in the mushy zone. It is shown that the apparent viscosity of dispersed sus-

pensions of uniform sphencal particles is a funciion o f the volume fraction of solid

particles [8]. Data for variation of the viscosity of iron with the solid content is not

available, thus a variation of the viscosity with the solid fraction similar to the v u a -

tion in the Sn-15% Pb system [9] was assumed to simulate the flow into the mushy

region. The variation of viscosity with the solid fraction. used in the present calcula-

tions, is shown in Figure 4.2.


O 10 2C) 30 40 50 S o l d volume fraction.%

Figure 4.2 Variation of apparent viscosity with solid volume fraction (used in this study).

4.2.3 Boundary Conditions

The calculation domain is shown in Figure 4.1. A non-uniform grid point

system is cmployed with tïner grid sizes near the weld pool region. The workpiece is

considered to bti a 4 0 ~ 4 0 ~ 1 2 . 5 mm watcr cooled stainless steel AISI 304 block and

the origin of the calculation is located a[ the centre of the hloçk. The corrcsponding

boundary conditions are summarized in Table 1- I .

The momrntum boundary conditions are given in the second and third rows

in Table 4-1. At the solid boundaries (BC. CD, DE, and EF), the velocities are zero.

At the axis of symmetry, FA. the radial velocity and the axial momentum flux arc zc-

ro. At the free surface, AB. there are two sources of radial momentum. The tïrst is

the shrÿr stress which is being üpplied by the flowing gas at the surface of the weld

pool, and rhe second one is the Marangoni elfect which is due to the variation of the

surface tension with temperriture and is described by:


Table 4- 1: Boundacy Conditions.

The relation between the surface tension of a binary solution with temperature and

composition is given by [IO] as follows:

The energy boundary conditions arc given in ihe fourth row o l Table 1-1. At

CD md DE boundaries. which are the watcr cooled wdls. the temperature cquals the

temperature of the water. By taking advantage of the axial symrnetry. it is çonsidered

thal the radial energy tlux is zero at the AE boundary. Finally, a heat flux calculateci

h m the arc modcl is considered at the t'rce surface. AC.

The electric potentiril boundaiy conditions are given in the tifth row of Table

4- 1. The bottom boundary is electrically isolated and hence the surrent density in this

boundÿry is zero. For the side boundary, since the electricd conductivity of the metal

is very high and this boundüry is lar enough from the weld pool surface, it is assumed

that the currcnt density at this surface is constant and independent of the distance from

the surface. For the î'ree surface there is a çurrent density that is dso obtained from


the arc model.

4.2.4 Material Properties for the Work piece

Calculations have bern performed for an AIS1 type 304 stainless steel. Ther-

modynamic and transport properiies of this alloy considered to be constant and are de-

t-ived tiorn Choo [ l ]. The surface tension is calculated based o n Equations (4- 1 Fia and

b), assuming that surface tension of steel is the same as pure iron and only changed

with temperature and the sulfur content. The terms in rhese equations and their values

are Iisted in Table 4-3 and are taken trom Sahoo et. al. [101.

3.2.5 Numerical blethod

To solvi: the conservation equations. the PHOENICS code was used. The gen-

Table 4-2: Data Used for the Surface Tension Cdculütion [ 101.

Value

1.943 ~ m - '

4.3 x I O ~ N ( m - K) -' 1809 K

8.3144 I ( m o l e . K) "

1.3 x 10-~rnole - me*

Equation (4-1 6 b )

- wt% S

0 3 1 8 t wi% )-'

5 -1.88 x 10 J - mole-'

Nomenclature

Surface tension of iron rit Tm

Tcmperature coettficiznt of surface tension for iron

Mclting point of iron

G-as constant

Surfacc excess at saturation

Equilibrium constant for segregation

Activity of sulfur

Entropy tactor

Enthalpy of segrcgation

Sym bol

'f m

*S

Tm

R

r~

%g-

" s

1

AH:

Chapter 4 Weld Pool Mode1 1 09

Table 4-3: The Corresponding Quantities for the Different Conservation Equations.

I A x i d momcntum

Radial mornçncurn

Energy

p- - -

Turbulent kinetic cnerg y

rra1 î k n of the rquations which c m be solved by this code was presented in çhapter

2 (Eqs. 3- 18 and 2- 19). The source ternis for each variable are given in Table 1-3.

4.3 RESULTS AND DISCUSSION

In this section the calçulated results of this study will be analysed by compar-

ing them with the relevant available exptximrnral information. The presrnted calcuiat-

ed results are for 2 seconds welding. As mentioned earlier, the main objective of this

chapter is the coupling of the weld pool rnodei with the arc model. Along with this

coupling, there is an opportunity to discuss the validity of the int'orrnation from the

arc model for the tapered electrodes. Therefore. in the calculation for the stainless

steel. as a workpiece material, previously predicted information from the 200 A arcs

of 2.0 and 5.0 mm arc lengths and different electrode tip angles have been uscd. The

electrode tip mgle varies from 9.18 to 13 1.41 degrees. The information that are used

Chapter 4 Weld Pool Mode1 110

include heat t'lux, current density and gas shear stress. In Figures 4.3-4.5 samples of

these data for 2.0 mm arcs and two different electrode tip angles of 9.18 and 60.0 de-

grees, are presnted.

There are different driving forces acting in the weld pool. nius, it is appro-

pnate to consider the effect of the electrode tip angles on these forces separately prior

to the combination of these forces and the final shape of the weld pool. Finally, the

predicted results wiil be compared with the experimental inloimation.

4.3.1 Electromagnetic Force

The electromagnetic force is directly related to the cumnt density to the weld

pool. In the previous chapter it was concluded that by decreasing the electrode tip an-

ale, the effective anode surface area on which the electrons from the arc condense e

tends to spread over the workpiece surface. This led to a lower maximum current

density for the sharper electrode tip. This effect is also shown in Figure 4.4. By con-

sidering the effect of the electrode tip angle on the hcat flux and current density distri-

bution over thc weld pool surface. i t can be predicted that the electromagnetic force

for the wider

piece and the

electrode Up is stronger. The current density distribution into the work-

tilcctromagnetic force into the weld pool for the electrode tip angles of

Radial distance, m m

Figure 4.3 Radial disiribution of the heat flux for 9.18 and 60.0 degree electrode tip angles. 1 = 200 A. L,, = 2.0 mm.

Chapter 4 Weld Pool Model Ill

Figure 4.4 Radiai distribution of the currenr density for 9.18 and 60.0 degret! electtde tip mgles. 1 = 200 A. L, = 2.0 mm.

9.18 and 60.0 degrers are shown in Figures 1.6 and 1.7. respectively. The tlow pat-

tern into the weld pool and h e liquidus and solidus lines for the two mrntioned rlec-

trode tip angles are shown in Figure 4.8. The suonger electromagnetic force produces

higher veloçity for the metal into the weld pool. and since the flow is toward the çen-

tre of the weld pool surface. the higher velocity causes deeprr penetration of the weld-

ment. It must be mentioned that the higher heat flux and curent density in the case of

Radial distance. mm

Figure 4.5 Radial distribution o f Lhe sheu suess for Y. 18 and 60.0 degree electrode tip angles. 1 = 200 A. L, = 2.0 mm.


c 4 . 0 5.0 4.0 3.0 2.0 1 .O 0.0

Radial distance, mm

Max. current density: 8.23 ~ . r n r n ' ~

4.0 5.0 4.0 3.0 2.0 1 .O 0.0

Radial distance, mm

Max. current denslty: 14.4 ~ . m r n - ~

Figure 4.6 Current dcnsity distribution into the workpiece for a: 9.18 and b: 60.0 de- eree electrode tip angle. 1 = 200 A. L, = 2.0 mm. +

the eltxuode tip angle of 60.0 dezgrees also act to increase the depih of prneîration.

4.3.2 Buoyancy

Variation of temperature o f the molten metai causes variation in die density of

the metal which produces tlow into the weld prml due to buoyancy. The present mod-

Max. €MF. 4.38E - 4 ~ . r n - ~

5.eE -4 bl.m3 -

Radial distance, mm

Max. EMF: 9.59E -4 ~ . r n ~ -

..--.-....--.-. I I I 1 I l 1

5.0 4.0 3.0 2.0 Radial distance. mm

Figure 4.7 Elecuomagnetic force into Ihe weld pool for a: Y. 18 and b: 60.0 degee elecuode tip angle. 1 = 200 A. L,, = 2.0 mm.

Chapter 4 Weld Pool Mode1 11 3

Max. velocity . 75.8 mms- ' Max. velocity: 135 mm .s- 1

1 00 m rn .s-' -

Radial distance. mm Radial distance. mm

Figure 3.13 Velocity pat .tem into the weIJ pool and Iiquidus and solidus Iine: s for a: 9.18 and b: 60.0 degree electrode tip angle due to electrornrignetic force. 1 = 200 A. L,, = 2.0 mm.

el takes rhis efkct into account by use of the Boussinesq's approximation. The rffect

of the electrode tip angle on the driving forci: due to buoyancy is shown in Fisure

4.9. For the 200 A arc of 2.0 mm length, the buoyancy driving force is one order of

magnitude less than the elecuomagnetic force, and is acting against it. The velocity

profile and liquidus and solidus Iines for the electrode tip angles of Y. 18 and 60.0 de-

grees are shown in Figure 4.10. It is obvious bat buoyancy has a minor contribution

in convection into the weld pool and the variation in the size of the weld pool is

mostly due to the variation of the heat flux with the ekctrode tip angle (Figure 4.3).

4.3.3 Gas Shear Stress

I t was found that the electrode tip angle has a significant effect on the gas


Max. BDF: 1.76E -4 ~ . m '

2.OE - 4 ~ . r n - ~

0.0

E 1.0 E

6 .................... U ................ C

2.0 ; . - -0 - a

3.0 2

4 .O 5.0 4.0 3.0 2.0 1.0 0.0

Radial distance. mm

Max. BDF: 2.44E - 4 ~.m'

( 1 1 l i l i i i i 1 4 . 0 5.0 4.0 3.0 2.0 1 .O 0.0

Radial distance, mm

Figure 4.9 Buoyancy driving force into the weld pool for a: 9.18 and b: 60.0 degree electrode tip angle. I = 200 A. Lu, = 2.0 mm.

shear stress on the top of the weld pool (Figure 4.5). This stress acts from the centre

of the weld pool surface towards the edge of the weld pool. and its magnitude increas-

es with decreasing angle of the electrode tip. This was shown in the previous chapter

Max. velocity: 18.7 rnms" Max. velociry: 19.5 .nm.sd'

Radial distance. mm

2 solid & Liquid 3 Liqu~d -

Radial distance. mm

Figure 1.10 Velocity pattern into the wrld pool and liquidus and solidus linrs for a: 9.18 and b: 60.0 degree electrode tip angle due to buoyancy drivuig force. 1 = 200 A. L, = 2.0 mm.

C hapter 4 Weld Pool Mode1 115

(Figure 3.24). This action tends to increase the width of the weld pool. On the other

hand, a higher heat flux in Lhe case of an elecîrode tip of 60.0 degrees promotes a

larger weld pool both in depth and in width. The overâll cffect of these two parame-

ters determines the size of the weld pool when only the shear stress is examined. The

velocity profile and the liquidus and solidus Lines for electmde tip angles of 9.18 and

60.0 degrees are shown in Figure 4.1 1. The velocity of the molten metal in cornpari-

son with the case of the alecuomagnetic force and the buoyancy drivine force is larger

by one and two orders of magnitude. respectively. The maximum temperature in the

weld pool decreases significantly by increasing the velocity. By applying individual

driving forces separately, the maximum temperature at the centre of the weld pool sur-

face for different rlectrode tip angles between 9.18 and 13 1.41 degrees were calculat-

ed and are shown in Figure 4.12. In the case of the shear stress, the maximum

temperature for al1 tip angles is between 2000 to 2300 K less chan the temperatures

calcuiated for the case of rlectromagnetiç force and tor the case of buoyancy this dif-

ference is even more. Accordingly. i t can be predicted that for the 200 A arcs. the

shrar stress plays a significant role in detemining the tlow pattern into the weld pool

Max. velociry: 337 mm.s7'

500.0 mrn.s-' - Max. veloctty: 389 mm.s- 1

5.0 4.0 3.0 2.0 1.0 0.0 Radial distance. mm Radial distance, mm

Solid Solid & Liquid Liquid

Figure 4. I l Velwity pattern into the weld pool and liquidus and solidus lines for a: 9.18 and b: 60.0 degree eiectrode tip mgie due to gas sherrr stress. 1 = 200 A. L,, = 2.0 mm.


+ Electromagnetic force -(3- Surface tension -#- S hear stress

I ' O ~ O ~ 20 40 60 80 1 0 0 120 140

l I I 1 1 l

Electrode tip angle. d q

Figure 4.12 V~Yiation of maximum temperature in the weld poal for different dnving force as a function of the elecirode tip an- de. 1 = 200 A. L, = 2.0 mm. -

and the shape and size of the weld pool. especially for short arcs and for the elecuode

tip angle smaller than 60 degrees.

4.3.3 Surface Tension

Surface tension and its variation with temperature and composition have a sig-

nificant elfect on the determination of the tlow pattern into the weld pool [11-171. For

iron. surface tcnsion decreaxs with increasing temperature. Most commercial krrous

alloys contain small amounts of surface active elements. such as oxygen and sulfur.

For these cases. the dope of the variation of surface tension with rempenture a n be

positive [IO]. Sllhoo et. al. [ L O I demonstrated that for dilute binary metai-surface ac-

tive solute systcms, the surface tension is a function of both temperature and composi-

tion (eq. 4- 16).

By using the rquation proposrd by Sahoo et. ul. [10], the effect of surface

tension on the tlow pattern in the weld pool and the shüpe and size of the weld pool

for diffcrent électrode tip angles was studied. Variation of the surface tension coeffi-

cient ( ay / aT) as a function of temperature for Fe-0.022 wt% S is shown in Figure

4.13. Figure 4.12 shows that the surface tension lowers the maximum tempenture at


Temperature, K

Figure 4.13 Variation of surface tension gradient as a tùnction of temperature for FeQ.022 wt% S. Required information was derived from Sahoo et. fll. [ IO].

the surface of the weld pool substantially compared with the electromagnetic and

buoyancy forces. The tlow patterns for 9.18 and 60.0 degrer rlecuode tip ansles is

shown in Figures 4.13. In this case the shapc: of the weld pool is comparable to the

case of the elecuomagnetic force (Figure 4.8). The pool velocity is at least one order

of magnitude higher and accordingly the maximum cmperature at the weld pool sur-

face is much lower. The Mwmgoni shrar along the surface of the pool for this case

is shown in Figure 1.15a and b. As the temperature gradient in this situation is nega-

tive. the sign of the surfice tension cortticient determines the direction of the resultant

Marangoni shear. For the sharp elrcirode. the maximum temperature at the surface of

the weld pool is kss than about 2250 K (Figure 4.15a). thus ay/ aT is positive (Fig-

ure 4.13) and the Marangoni force is always negative. However. for the electrode

with wider tip angles (60 degrces), the temperature is higher dian 2250 K at the centre

of the weld pool and the Marangoni force is positive. At a distance away from the

centre of the wrld pool, the temperature is less than 2350 K and similar to the s h a -


Max. velocity: 510 rnrn.5-'

Radial distance. mm

Max. velocrty: 41 5 mms-'

5.0 4.0 3.0 2.0 1 .O 0.0 Radial distance. mm

Figure 4.14 Velocity pattern into the weld pool and liquidus and soiidus lines for O: Y. 18 and b: 60.0 degree electrode tip angle due to surface tension. I = 200 A. L,, = 2.0 mm.

electrode case, the Marangoni forcc is positive. This contiguration of the Manngoni

force may produce çountercurrent tlows at the surtacc of the weld pool for wide elec-

mde tip angles.

4.3-5 Combination of Driving Forces

in 4.3.1-4.3.4. the individual role of the divins forces (electromagnetic force,

buoyancy, gas shear stress and surface tension) and their variations with the electrode

tip angle in determining the weld pool shape were discussed. In this section. the mm-

bination of these forces for the stainless steel will be considered. Figure 4.16 shows

the puddle shape for al1 considered cases. The variation of the puddle shape for the

2.0 mm arc is shown in the tlrst and third rows. By increasing the elecuode tip angle.

the depth of the weld pool increases [rom 1.1 mm for the 9.18 degrees to 3.55 mm

Chapter 4 Wekl Pool Model 11 9

a: 9.18 deg 400 I 1 I I T 2800

Li

Radial disrance, mm

Radial disrance. mm

b: 60.00 deg

Figure 4.15 Radiai distribution of the Mxangoni shex and surface temperature for stainless steel AIS1 304 with the elecuode tip mgle as the parameter.

400

200

O

-200

for the 131.4 degrces. On the ather hÿnd. the width of the weld pool decreases with

the elcçtrode tip anglc from 8.8 mm for the 9.18 dcgrces to 6.3 mm for the 13 L.41

I I 1 I T 2WO

-

6, O - - 1600 3 L

3 cn

degrees. This trend in the variation of depth and width of the weld pool can d s o be

d006 I

5 4 3 2 1 O

observed for the 5.0 mm arc (the second and forth rows). The surface tension obvi-

ously has an important role in deepening the weld pool, especially in welding with

electrodes with wide tip angles. This effect is more signitïcant in shorter arcs. By

considering the variation in the total suess (tst + on the surface of the weld e

pool. the observed behilviour cm be bettcr explained. The total stress on the weld

pool surface due to the gas shear stress and the surface tension stress for a 2.0 mm

Chapter 4 Weld Pool Model 120


arc for different tip angles (9.18, 60.00 and 131.41 degrees) is shown in Figure 1.17.

In this Figure, the radial distribution of tempenture at the surface of the weld pool is

also shown.

Since ay /aT beiow about 2250 K is positive (Figure 1.13), the surface ten-

sion stress (sa, ) is ne~ative close to the edge of the weld pool. Since the gas shcar

stress away from Ihc centre part of the weld pool surface is very small (Figure 4.3,

the total stress also will be negative and pushes the liquid metal towards the centre of

the weld pool. For wider tip angles the gas shear stress (which is always positive) is

much less than thar for the sharp electrode. Due to the surface tension, the negative

stress expands across a wider distance from the weld pool edge md its magnitude is

larger. çompcired with the positive stress due to the gas shear stress. The resultant

tlow pattern duc to the combination of the dnving hrces into the weld pool for the

stainless steel is shown in Figun: 1.18 for the three mcntioned electrode tip angles.

In çomparison with the gas shear suess ÿnd surface tension, the electromag-

netiç force and buoyancy for a 200 A an: play a minor role in the determination of

the wefd pool shape. As shown in Figures 4.17 and 4.18, for very sharp electrodes.

the cffrct of the gas shear stress is dominant. For elecuodes with wide tip angles. the

interaction of thc gas shear stress and surface tension and especially the direction OC

the surface tension determine the tlow pattem into the weld pool and thercfore its

shape and s ix . Finally, in al1 cases the heat tlux and the current density was çorre-

sponded to the relevant elecirode tip angles. Therefore the heat tlux and the current

density are the main 1'actors which determine the size of the weld pool and the tlow

dnving forces are important in the determination of the tlow pattem into the weld

pool and thercfore its shape.


Radial distance. mm b: 60.1 8 deg

400 I I I I I 2800

c: 131.41 dea Radial distance. mm

Radial distance, mm

Figure 4.17 Total surface stress (T,, + igas ) and temperature at the weld pool surface for stainless steel AIS1 304 after GTAW with a 2.0 mm arc. The electrode tip angle is 9.1 8, 60.00 and 13 1.4 1 degrees.


9.18 degree

60.00 degree

131.41 deçree

2.0 mm arc

Max. velacity, 379 m m s - '

1

l l 1 I 2 .O 5.0 4.0 3 0 2.0 1 .O 0.0

Radial disrance. mm

Max. velocity, 365 mms-'

Max. velocity. 302 mm.s- 1 n n

1114.0 5.0 4.0 3.0 2.0 1 .O 0.6

Radial disrance. mm

500 mm .s-' 5.0 mm arc

Max. velocity. 362 mm.s- 1

m 2.9.0 4.0 3.0 2.0 1 .O 0.0

Radial distance. mm

Max. velocity, 362 mm-s- 1

3%!0 4 0 3:o 210 1iO i . 0 Radial distance. mm

Max. velocity. 320 mm-s' 1

4.0 1 1 I I 1 5.0 4.0 3.0 2.0 1 .O 0.0

Radral distance. mm

Figure 4.18 Flow pattern into the weld pool for stainkss steel AIS1 304 and different electrode tip mgles and arc lengths.

4.3.6 Cornparison with Experirnental Data

[n çhapter 3 the computed arc properties were compared with the correspond-

ing experimental data. For the heat flux and current density to the weld pool. there


are very limited data available (1 8.191 which cuver the electrodt: tip angles more than

50 degrees. For gas shear stress thrre are no experimenial data available in the Litera-

ture. The lack of experimental data shows that rncasurin_o different arc parameters is a

very difficult task. if not impossible. This is one of the reasons that justify the modrl-

ling efforts in the arc welding processes, including GTAW. In spite of this problem,

the GTAW model with tapered electrodes must be veriîied. One way to check the va-

lidity of the arc modrl data is by applying rhem into a weld pool model. and by corn-

paring the caiculated results with the weld pool related experirnrntal dam. On the

other hand, in addition to the arc variables, the workpiece material iüelf has signifi-

cant rffect on the weld pool properties.

The rffect of the electrode tip angle on the weld pool was studied by Savage

et. al. [20] for a plain carbon steel with 0.16 wt& C and 0.022 w t l S. and Key [Xi

for an MSI type 304 stainless steel. ln the c a x of Savage e t uL 1.201, rhr arc length

was 1 .Y mm and the rlectrode tip angle changed from 30 to 120 degrees. The ekc-

uode was a 2 4 thoriated tungsten rod of 2.38 mm diameter. They found that increas-

ing the ekctrode tip angle from 30 to 120 degrees. decreased the weId pool width by

about 50% and increasrd the weld pool depth by about -15%. Key [ X I u x d an arc

length of L.0 mm. 150 A arc on the workpiece plate with an elrcuode tip angle of 15

to 180 degrees and uuncation diameters of 0.125 and 0.5 mm. There are differencrs

between the expenrnental conditions and the assumptions u x d in the calculation,

which are explained in section 4.2, nevertheless if is possible to qualitatively compare

the calculated with the experimrntal results. Certainly there is f ' e r work necessw

to have a quantiiatively comparable sets of experirnental and theoretical results.

The effect of the electrode tip angle on the weld pool width and depdi, re-

speçtively. are shown in Figures 4.19 and 4.20. The calculated resula of this snidy

me et. for stainless steel AISI 304 are compared with the experimenütl results of Sava,


+ AlSI 304, 200 A 6 2.0 mm arc Cal. + AlSI 304, 200 A b 5.0 mm arc Cal. * Steel.190A61.27mmarctExo.1 ' 4 011

Figure 4.19 Variation in the weld pool width as a function of the efecuode tip mgle. Cornparison with experimental results [ZO].

ai. [20]. In the experimental resulu for the electrode tip angles in the range of 30-120

degrees, the slope of the variation of the weld pool width with the electrode tip angle

is almost constant and çquals to about -1130 mm.degree" for the 1.27 mm and 190 A

arc. In the caiculatcd results of the 2.0 mm arc the dope of the variation of the weld

+ AlSI 304, 200 A b 2.0 mm arc (Cal. + AISI 304. 200 A b 5.0 m m arc (Cal. # Steei, 190 A & 1.27 mm arc (Exp. [


Figure 4.20 Variation in the weld pool deplh as a function of the electrode tip angle. Comparison with experimental results [20].


pool width with the electrode tip angle is 4/49 mm-degree-'. For the 5.0 mm arc, af-

ter a positive slope for very sharp electrodes. the weld pool width changes with the

elecuode tip angle with a negative slope of 4/67 mm.depree'l. These data show that

by increasing the arc length, the effect of the electrode tip angle on the weld pool

width becomes less important. The depth of the weld pool in comparison with the ex-

perimental data of Savage et. al. [20] is shown in Figure 4.20. For 2.0 mm arc there

is an increase in the weld pool depth with the electrode tip angle while for 5.0 mm

arc there is only a small increase for the electrode tip angles wider than 70 de, m e s .

The experimental results of Savage et. al. [20] which are for 1.27 mm arc show deep-

er weld pool especially for sharp electrodes. Considering the calculated and experi-

mental data suggests that the variation in the depth of the weld pool is mostly due to

variation in the configuration of the arc and especially distribution of the heat flux and

the current density over the top of the weld pool.

In comparison with Key's results [21] (Figure 3.1). the trend in the variation

of the puddle shape (Figure 4.16) is in relatively good agreement, especially for die

eIectrode with a truncation dimeter of 0.125 mm. The variation of depthlwidth of the

experirnental results [2 1 ] and the calculated values is compared in Figure 4.2 1. From

this figure, it appears that the slope of the variation of the depth/width ratio with the

electrode tip angle decreases with arc length increase. This is mostly due to variation

of the weld pool depth. For tip angles wider than 90 degrees, the slope of cdculated

results is positive, while for the experimental results is negative. To understand the

reason for this inconsistency more experimentd work is necessary.

4.4 CONCLUSION

In this chapter, a simple mode1 for the weld pool for the GTAW process is

developed. To handle the solidification phenornenon. the mode1 is based on solving


-+ AIS1 304. 200 A b 2.0 mm arc. Catculated . +- AlSi 300. 200 A 6 5.0 mm arc. kalcularedl., - AiSI 304 150 A 6 1 .O mm arc. 0.125 mm dia. truncation (Ex . 21 1). - - - AISI 304: ,?XI A b 1 .O mm arc. 0.5 mm dia truncation. (&Q. b111.

0.00 1 I I 1 I 1 I O 30 60 40 1 2C 1 50 180

Electrode trp angle. deg

Fisure 4.21 Variation in the depwwidth ratio as a function of the elecuocie tip angle. Cornparison with experimental resuIts [2I].

the temperature rquations. To simulate the flow in the mushy zone, the variation in

the visçosity of the overÿll tluid containing a solid fraction has been used. The re-

quired information to simulate the tlow into the weld pool is derived iiom the arc

model drveloped in chapter 3. Thrrefore the results of the weld pool model can also

br uxd to venfy the results of the arc mode1 for angular tip elecuodcs.

To rvaluate the relative importance of the driving forces into the weld pool.

includhg buoyançy. çlectrornagnetic force. surface tension and gas shear stress. each

parameter applied in the weld pool mode1 separately. Also. the variation in these fort-

es with the rlrcuode tip angle and their effects on the weld pool shape has bren ex-

arnined. It is found that for a 200 A arc. the buoyancy and slectromagnetic forces do

not play a major role in deteminhg the How pattern into the weld pool. compared

with the gas shear stress and surface tension. The flow pattem into the weld pool, on


the other hand. is determined by the relative magnitude of the gas shear stress and the

surface tension, and also sign of the surface tension. The gas shear stress is an arc

property and changes with the arc parameters. Variation in the gas shear stress with

arc parameters is discussed in chapter 3, section 3.3.3. On the other hand. the surface

tension, is a function of temperature of the weld pool surface and weld pool composi-

tion,

Comparing the results of this study with the experimentai results of Savage et.

al. PO] and Key [21] has shown that the arc and weld pool models together c m give

a realistic picture of the weld pool s h a p and size for a range of rlectrode tip angles

from 10 to 90 degrees. For the rlectrode tip angles more than 90 degrees. although

the predicted puddle shape is realistic but there is inconsistency in variation of depth/

width ratio between the calculated results and experirnental data.


REFERENCES

I - R. T. C. Choo. S c D 7hesi.s. Dept of Materials Science and Engineering. The Mas-


2- C . R. Heiple and .J. R. Roper, 'The Geometry of Gas Tungsten Arc, Gas Metal

Arc and Submerged Arc Weld Beads", in Welding: ??zeop und Practice. edited by

D. L. Olson. R. Dixon and A. L. Liby, Elsevier Science Publishee B. V.. 1990.

pp. 1-34.

3- T. Zacharia and S. A. David. "Heat and Fluid Flow in Welding" in Muthcmnrical

Modelling of Weil Phrnomena. edited by H. Cejak and K. E. Easterling, The In-

stitute of Materials. 1993. pp. 3-23.

4- T. De broy . "Weld Pool Surface Phenornena-A Perspective" in Mufhrman'cul Mod-

rlling of WeW Phenomrna. edited by H. Cejak and K. E. Easterling. The Institute

of Materials. 1993. pp. 24-38.

5- K. Mundra. T. Debroy. T. Zacharia and S. A. David, "Role of Thermophysical

Properties in Wrld Pool Modelling". Wdding J.. 1992. vol. 7 1. pp. 3 13s-320s.

6- R. T. C . Choo and J. Szekely. 'The Possible Role of Turbulence in GTA Weld

Pool Behaviour". Wrfding J.. 1994. vol. 73. pp. 2 5 3 1s.

7- B. E. Launder and D. B . Spalding, Murhemu~cul Mode1 of Turbulencr. Academic

Press. 1972.

8- D. G. Thomas. 'Transport Characteristics of Suspension: VIII. A Note on the Vis-

çosity of Newtonian Suspensions of Unifonn Sphencal Particles. J. of Culloid Sci-

ence. 1965. vol. 20. pp. 267-277.

9- D. B. Spencer. R. Mehrabian and M. C . Flemings, "Rheological Behaviour of Sn-

15% Pb in the Crystallization Range". Mrtullurgicul Transactions. 1972. vol. 3.

pp. 1925- 1932.

10- P. Sahoo, T. Debroy and M. J. McNallan. "Surface Tension of Binary Metal-Sur-

Chapter 4 Weid Pool Model 130

face Active Solute Systems Under Conditions Relevant to Welding Metailurgy"

Metulhrgicul Tran.wctions B. 1988. vol. 19B. pp. -GE!-@ 1.

11- C. R. Heiple and J. R. Roper, "Mechanism for Minor Element Effect on GTA Fu-

sion Zone Geomeuy. Welding J.. 1982, vol. 6 1. pp. 97s- 102s.

12- C. R. Heiple, I. R. Roper, T. Stmger and R. J. Aden. ''Surface Active Element

Effects on the Shape of GTA. Laser and Electron Beam Welds", Welding J.. 1983,

VOL 62, pp. 73-77s.

13- W. H. Giedt. X. C . Wci and S. R. Wei. "Effect of Surfacc Convection on Station-

ary GTA Weld Zone Ternperatures. Wrlding J.. 1984, vol. 63. pp. 376s-383s.

14- P. Burgardt and C. R. Heiple, "Interaction Between Impurities and Welding Varia-

bles in Determinin- GTA Weld Shape". Wrlding /.. 1986, vol. 65, pp. 150~45%.

15- R. E. Sundell. H. D. Solornon and S. M. Correa, "Minor Element Effects on Gas

Tungsten Arc (GTA) Weld Penetration- Weld Pool Physics", in Advances in Wrlrl-

irrg Science und Techntdogy TWR '86. edited by S. A. David. ASM. 1986. pp. 53-

57.

16- K. Ishizaki. "Dynamic Surface Tension and Surface Enthalpy Theory on Hear

Transfcr and Pcneuation in Arc Welding, IN Doc. No. 2 12- 736-89. July 1989.

17- T. Zacharia, S. A. David, J. M. Vitek and T. Debroy, "Weld Pool Developrnent

During GTA and Lascr Welding oT Type 304 Stainless Steel, Part iI: Experimental

Correlation". Wrlding J.. 1989, vol. 68, pp. 5 10s-5 1%.

L8- N. S. Tsai and T. W. Eagar. "Distribution of rtie Heat and Current Fluxes in Gas

Tungsten Arc". Metallrrrgical Transucrions B. 1985. vol. 16B, pp. 841-846.

19- M. J. Lu and S. Kou, "Power and Current Distributions in Gas Tungsten Arc".

Wrll i i~~g J.. 1988, vol. 67. pp. 29s-34s.

20- W. F. Savaye, S. S. Stmnck and Y. Ishikawa, 'The Effect of Electrode Geometry

in Gas Tungsten-Arc Welding". Welding J.. 1965. vol. 4. pp. 489~496s.

Chapter 4 Weld Pool Modsl 131

21- 5. F. Key. "AnodeKathode Geometry and Shielding Gas Interrelationships in

GTAW". Wrlding J.. 1980. vol. 59. pp. 364~470s.

5.1 INTRODUC~ION

Unlike the GTAW process, mathematical modeUing of arc in GMAW process

has only recently being developed [ i l . A non-thermionic cathode. a randomly rnoving

cathode spot on the surface of the workpiece. variation of the electrode tip shape dur-

ing the welding process, presence of metal droplets in CLight between anode and cath-

ode and gas mixture are some aspects of the GMAW process which complicate the

developrnent of a model. In this chapter, a mathematicai model for GMAW will be

presented. As indicated in section L.4.2, the main objective of modeiiing the arc in

the GMAW process is to rstimate the heat flux frorn the arc to the workpiece. This

needs a better understanding of the GMAW process. To obtain this objective, several

steps are necessary. In the present chapter a model for GMAW with a Bat tip elec-

trode is developed. By comparing the obtained results with available experirnental da-

a, the validity of the model is discussed. The chapter continues with snidying the

effects of the applied current, arc length, elecuode diameter and shielding gas on the

arc properties, and then an estimation of the contributions of metal droplets and arc

into the heat flux are given. Finally, by considering an angular tip elecuode, estima-

tion for the effect of the electrode tip shape on the arc properties is presented.

5.2 PHYSICAL DESCRIPTION

A schematic sketch of a DCEP gas metal arc welding is iliustrated in Figure

Chapter 5 GMAW m d e l 1 33

5.1. Tke DCEP configuration is preferred in the GMAW process because of the melt-

ing of the continuousIy fed elecuode. For this reason the workpiece is the cathode.

EIectrons, either emitted from the cathode (workpiece) or produced in the arc, con-

dense on the anode (electrode). The heat transferred by the elecuons dong with the

radiation heat and more importantly the Joule heating into the electrode raise the tem-

perature of the electrode to the meltïng point The heat content of the Calhg droplets

will be transferred to the weld pool. Other heat sources for the workpiece wiLl be ra-

diation and convection. In addition to these mechanisms, positive ions that neutralize

at the surface of the cathode transfer some amount of rnergy to the cathode (work-

piece) [2]. This is quite important in GMAW because of the non-thermionic nature of

the cathode [3]. The proportion of the ion current in the total current is not clearly

known and determining the theoretical heating rate due to ions needs much further in-

vestigation [4]. Therefore, in this study the heat due to ion bombardment is not taken

into account.

I

O ;

Molten droplet

l 1

Figure 5.1 Schematic of the welding arc (GMAW).

I

1 '

I I

1 I

I

Consumable electrade {Anode)

Chapter 5 GMAW madel 1 34

To establish a model for the arc in the GMAW the mode1 which was devei-

oped for the GTAW (chapter 2). must be modified. The assumptions made in chapter

2 remain applicable. Additional assumptions are introduced to sirnplify the model.

These assumptions are as follows:

Siqce the droplet spends only a very short time in flight it is assumed that there is

no droplet between the consumable electrode and the workpiece;

The short life of the droplet in the very high temperature zone combined with the

high velocity of the plasma about the axis of symrnetry lead to the assumption that

there is no metal vapour in the space between the electrodes. A limited amount of

metal vapour enters the plasma but it c m not diffuse radiaily into the outer parts.

Theretore a smdl amount of metal vapour is present near the axis of syrnmeuy,

very close to the droplets, but the effect of this on the whole arc is insignificant.

5.3 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS

In the GMAW process, there are elecirons emitted from the cathode which

paçs through the gap between the workpiece and the wire electrode to condense on the

anode (wire electrode). The interaction of the magnetic field produced by this cumnt

with the current itsell: produces an electromagnetic force which is the main driving

force for the plasma. This phenornenon is very similar to ihe case in the GTAW

process. Although in the GMAW, the electron current direction is opposite to that of

the GTAW, the change in the sign of the magnetic field results in the electromagnetic

force to act in the same direction as the case of the GTAW. The thermal effect of the

electric current which originates from the Joule heating. keeps the temperature high

enough for maintaining a stable plasma. The equations that explain the physical phe-

nomenon occurring in the arc are the sarne as those presented in chapter 2 section 2.2.

Also the same numericd method was used to solve the equations.

Chapter 5 GMAW mode1 135

The caiculation domain for dl variables is shown in Figure 5.2. A non-uni-

form grid point system is employed with finer grid sizes near the consumable elec-

trode. The distance between the electrodes is varied from 2.0 to 10.0 mm. The inflow

boundary at the top of the domain is taken at 20.0 mm above the etectrode face (an-

ode surface). The boundary at the side is 20.0 mm away from die axis of symmetry.

The corresponding bounday conditions are given in Table 5- 1.

The specification for the velocity boundary condition is not very critical. At

the solid surfaces (face and side of the electrode and surface of the workpiece) the ve-

lociùes are zero. At the mis of symmetry. the radial velocity will be zero and there is

no tlux for the axial velocity. Again the radial velocity is zero at the top inflow and

the axial velocity is constant at 2.0 r n d , which is typical of the shielding gas intlow

velocity. The side boundary is located tàr enough so that there is no axial momentum

Inflow -

Outflow +

C

t

'1 Z

___,

I Workpiece (Cathode), 20.0 mm

Figure 5.2 Calculation domain for GMAW (schematic).

C hapter 5 GMAW mode1 136

Table 5-1: Boundary Conditions.

At these boundaries, pressure is fixed to a constant value.

and mass thxes.

BC1

O

U = Const.

T=500

as , = O

For the enthalpy equation, it is assumed that the remperatures at the top in-

flow and the side boundary equal constant values of 500 K and 1000 K. respectively.

At the face of the electrode. the temperature of the eiectrode is assumed to be equal to

the melting point of the deetrode material. At the side, HB. it is considered that the

temperature decreases from the rnelting point of die metai at the face of the electrode,

point H, to the temperature of the gas at the inflow boundary. 500 K, at point B line-

arly. Therefore the temperature at the anode will be:

T = T m, elec ter z = 'riec

= T i n f ~ o w for z = O

T = A z + B fo r 0 < 2 c Zeleç

At the surface of the workpiece (the cathode), a cathode spot radius is defuied so that

the temperature within the cathode spot equals the rnelting point of the workpiece and

beyond that equals a constant value which is less than the metal melùng point. Thus

CD1

aprv .T = O

au a ; = o

Inflow: T = l W K Outnow:

a h a ; = O

a6 --&=O

DE

O

O

T=5ûûK

' = = O

EF

O

O

T = Tm.,,

I J c = - ,

KR;

FG

O

z = o au

ah - & = O

ab , = CI

GH

O

O

T = Tm el,,

Eq. (5-3

HB

O

O

Eq. (5-1)

Eq. (5-5)

Chapter 5 GMAW madel 137

the temperature at the cathode is:

At the axis of symmeuy. there will be no rnthalpy gradient

Currents. axial and radial. at ail boundaries, except at the cathode and the an-

ode, are zero. For the cathode. although there is no fixed cathode spot at the surface

of the workpiece and the arc root continuously wanders from one place to another. it

is considered that there is a constant current density within a predetlned area and zero

current density beyond that area. Therefore the current density at the surface of the

workpiece c m be detintid as hllows:

Ic = O for

For the anode, die elecuic potentiai at the AB boundary is presumed to be

zero and since the lttngth of the electrode is much longer than its diameter. it is con-

sidered that the elecvic potential into the elecuode is only a function of distance from

the top surface. To evaluate the rlectric potentiai at different points into the electrode.

Ohm's law has been used. The rquation for this purpose c m be written as foilows:

where 1. and o. are local values of current and electrical conductivity of the elec- J J

trode material resprctively. Ij in different points of the electrode c m be calculated as

Rdomain

t j = 1 - C I j, arc (5-5)

'elcc


Similar to enthalpy and axial velocity, there is no electric potential gradient at the axis

of symmetry.

The physical properties of the gases. namely density, viscosity, thermal con-

ductivity, heat capacity and rlectrical conductivity are treated as functions of ternpera-

ture. All rhese properties are taken from the abuiated data of Boulos et al. [5] . The

physical properties of helium for temperatures more than 24000 K are taken from Lick

and Emmons [6]. Radiation loss data for the q o n is taken from Evans and Tankm

171. Since there is no data available for radiation loss for helium. the data for argon

has been used for helium as well. The electrical conductivity of the electrode metal is

also ueated as a functian of temperature and the requîred data extracted h m Toulou-

han [SI-

5.4 ~ S U L T S AND DISCUSSION

in this section the results of calculations for different cases are presented. The

heat flux to the weld pool and the estimation of the contribution of different mecha-

nisms in heat transfer are oC most concern. Additionally the main properties of the arc

including the temperature distribution and velocity pattern of the plasma and rhe rlec-

uic potential difference are considered. The computed results are çompared with rx-

pcrimental data if they are available.

The çalçulations have been performed for both pure argon and pure helium

using currents nnging from 150 to 350 A. The electrode material has been chosen to

be aluminium with an elecuode diameter from 0.8 to 1.6 mm, and the elecuode sepa-

ration from 2.0 to 10.0 mm. These are the main process variables. Having knowledge

about the effects of these variables on the arc properties will help to gain a better un-

derstanding of the arc behaviour.

Chapter 5 GMAW mode1 t 39

5.4.1 Cathode Spot Radius

One of the problems in modelling the GMAW arc is the non-thermionic na-

ture of the cathode. The current density in a non-themionic cathode cm be as high

as 10" ~.rn- ' [9]. On the other hand. the cathode root splits into a number of sepa-

nte emitting areas [91, which in GMAW is clairned to be localized to the edge of the

rnolten part of the workpiece [IO]. To overcome this problem, a mean value is con-

sidered for the current density over a specified area as the cathode spot. Therefore. i t

is necessvy to examine the sensitivity of the cornputed results to the cathode current

density or in other words to the cathode spot radius.

In Table 5-2 results of the calculations for 250 A argon and helium arcs are

summarized. The variation of the maximum temperature and the maximum velocity of

the plasma and the elecvic potential difference for given arc lengths and electrode di-

ameters with cathode spot radius nnging from less than L.0 mm to more than 7.0 mm

are given. The electric potential difference can show the power consurnption of the

welding process. The maximum temperature and the maximum velocity of the plasma

give some information about the shape of the arc and the role of the anode and the

cathode mean current densities in rhe overail tlow of the plasma into the arc column.

The variations of the mentioned parameters with the cathode spot radius for 250 A ar-

oon and hrlium arcs For different arc lengths are also shown in Figures 5.3-5.5. ïhese Ci

Figures and Table 5-2 for both He and Ar plasmas show that the cathode spot radius

for different electrode diameters for relatively long arcs does not have any effect on

the maximum temperature and velocity of the gas. In very short arcs, both maximum

temperature and velocity of the gas change with the cathode spot radius. The reason

for the variations in maximum temperature and velocity in short arcs is the effect of

the cathode spot radius on the configuration of the arc close to the anode. Therefore

the cathode spot radius is insignificant as long as the maximum temperature and the


Table 5-2: Maximum Temperature, Maximum VeIocity and Electric Pokntial Difference for a 250 A Arc of Different Cathode Spot Radius.

I Cathode Spot Radius, mm; Max. Temperature, K; M a x Velocity, m d ; Electrk Potential Difierence, V

3.209 3.900 4.294 4.806 5.374 5.982 - 21080 21080 21080 21080 21090 21090 - - 396.6 396.8 397.0 397.0 3972 397.3 - 13.86 13.63 13.56 13.57 13.66 13.83 -

Chapter 5 GMAW model 141

Cathode spot radius. mm

m

Cathode spot radius, mm

Figure 5.3 Variation in the maximum temperature with the cathode spot radius for 250 A Ar and He arcs as a function of arc length.

maximum velocity of the plasma are concerned.

On the other hand, the electric potential difference is a function of the cath-

ode spot radius for ail 250 A arcs. There is dways a minimum in the elecuic poten-

tial difference-cathode spot radius curves. For the other two applied currents. 150 and

350 A. the sarne trend in the variation of the electric potential difference with cathode

spot radius has been observed. Therefore. the present calculations reveal that there is

an optimum cathode spot radius which is at least a function of the applied currenr. the

arc length and the shielding gas. The minimum electric potential and its comspond-

ing cathode spot radius for different cases are listed in Table 5-3. Accordingly it is

reasonable to consider that the cathode spot radius is related to the minimum electric

potential difference, which corresponds to the lowest energy consumption by the sys-

iem, as the optimum value. Thus for aii cases the cathode spot radius i s the value thar



Cathode spot radius, mm

Figure 5.4 Variation in the maximum velocity with ttie cathode spot radius for 250 A Ar and He arcs as a hction of arc length.

400

200

O00

800

400 200

5.4-2 Mode1 Verification

nie reliability of the predictions must be demonstrated before any discussion

and conclusions based on the results of the present calculation. The best way to deter-

mine the validity of the mathematical simulations is to compare the predictions with

the correspondhg experimental data. However, one reason for performing mathemati-

cal modelling, specifically in the present case. is to circurnvent the difficulties associ-

ated with the experirnental study of the GMAW process. Therefore obtaining accurate

experirnental data is not a simple task.

I 1 i I

- - O - - * - O

- - - -

600:/-- 1

0.0 1 O 2.0 3.0 4.0 5.0

Chapter 5 GMAW mode1 1 43


I I I I 1

1 I 1 I

1 .O 2.0 3. O 4.0 5.0 Cathode spot radius. mm

Figure 5.5 Variation in the electric potential ciifference with the cathode spot radius for 250 A Ar and He arcs üs a t'unction of arc length.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The latest experimentd measurement of temperature in a GMAW arc is re-

ported by Smârs and Acinger [ 1 11. In their study they measured the radial distribution

of the temperature at 2.5, 5.0 and 7.5 mm frorn the cathode plate in a GMAW arc

protected by argon as the shielding gas. Both electrodes were aluminium and the con-

sumable electrode diameter was 1.2 mm. In Figures 5.6 and 5.7 the radial distribution

of the temperature at different distances from the cathode plate, obiained from the cal-

culation for 150 A and 250 A arcs is compared with the experimental results [L 11. It

can be seen that the agreement between the two sets of results for 150 A arc is better

than for the 250 A arc. The largest difference in al1 cases is related to the cenual part


Table 5-3: The Corresponding Cathode Spot Radius wiih the Minimum Elecuic Potential Differe- nce for Different Cases.

Applied Elect rode Arc Minimum Electric Cathode Current Diameter Length Potential Difference Spot Radius

(A) (mm) (mm) (v) (mm)

0.8 10.0 1 1.88 3.664

of the arc, and this difference is greater for higher applied currents. The most proba-

ble reason for this is the existence of a local higher current density at the surface of

the droplets due to the large difference in electrical conductivity of metal and gas mix-

ture around the droplets. Also at temperature less than 10000 K, the expenmental and


Dashed Ilne: Experimental \

\

I 1 I I \

1 .O 2 .O 3.0 4.0 5.0 Radial distance, mm

20000 l i I I 1

hd 15000 - P

- 3 u lu g i o w ~ o - E aJ 4

'" 5000 -Solid line: Calculated G \

Dashed Iine: Experimental \ \

O I I 1 l '. I

0.0 1 .O 2.0 3.0 4.0 5 .O 6.0 Radial distance. m m

20VOO l i 1 I I 1

hd m

L' -

3 L

<O g 10000 - E a, - . Y \

5000 -Solid line: Calculated \ \

- u

Dashed Iine: Experimental \ \

O 0 0 I 1 I l I I '-

1 .O 2 .O 3 .O 4.0 5 .O 6.0 7.0 Radiai distance. mm

Figure 5.6 The radial distribution of temperature at different distances from the cathode plate in cornparison with the experimental data [ I I ] . 1 = 150 A. Dis- tance from the cathode plate a: 7.5 mm; b: 5.0 mm; c: 2.5 mm.

theoretical predictions differ significantly. The difference at this region can be due to

absorption of the radiated heat from the arc core. the heat released becaux of recom-

bination of ions and electrons and deviation from L E condition at the outer part of


-Solid line: Calculated \ \

\

Dashed line: Experimental \

I I 1 I \

\ ,

1 1 .O 2.0 3.0 4.0 5.0 Radial distance. mm

20000 È - , _ I I I 1 T - - - -- Y

oi 4 - -

3 Y

2-9- -9 - - _ % ioooo - z 0 w

m 5000 -Solid line: Calculated t I

- a \

Dashed line: Experimental s

1 \

O * I I I 1 '

0.0 1 .O 2.0 3.0 4.0 5.0 6.0 Radial distance, mm

Figure 5.7 The radiai distribution of temperature at different distances from the cathode plate in cornparison with the experimental data [ I l ] . 1 = 250 A. Dis- tance from ihe cathode plate a: 7.5 mm; b: 5.0 mm: c: 2.5 mm.

20000

LL 15000

2- - d -. m g 1 O000

Y - g 5000 u

the arc. Thus. some of the previous assumptions need to be revised.

The caiculated maximum Cas velocity at the centre üne of the arc is corn-

I I 1 1 I I I

- - - - - - _ _ -- - - El

- - - 9 - - - - - - -

9 - - - -_ - - - - - - - _ _ _ - _ _ _ _ - - -_ - \ \

d - \ \

s - -Soiid line: Calculated

8 \ \

Dashed Iine: Experimental \

l I I 1 \ \ L

O 0:o I I

1 .O 2 .O 3 .O 4.0 5.0 6.0 7 .O 8.0 Radial distance. mm


pared with the experimental results of Sm5s and Acinger [11] in Figure 5.8. The ex-

perimental values were estimated based on the drag force of the plasma sueam and by

knowing the acceleration and size of the droplets. The reported experirnental values

are in fact the mean values and in spite of their lower values compared with the

present theoretical predictions. which are the maximum velocities. it can be concluded

that the theoretical values are not far from the real gas velocities.


Three major properties of the argon arc. i.e. maximum temperature, maximum

velocity and electric potential difference, for different arc lengths and different currents

are summarized in Table 5-4. The corresponding caihode spot radius is also @en. It

c m bi: seen that by decreasing the elecirode diameter. the maximum an: temperature

and maximum gas velocity increase, which indicate that the anode current density has

a critical role in determining the arc properties. The anode current density depends on

the t o d current at the tip of the electrode. The variation of total current at the tip of

the electrode with the electrode diameter and applied current is illustrated in Figure

5.9. The importance of the total current at the tip of the electrode in determining the

Applied current. A

500

400 7

E 300

r O O - $ 200 m Q u

100

0

Figure 5.8 Variation in gas velocity with applied current for 10.0 mm arcs in cornparison with experimenlal results [ 1 11.

I I I I I I Solid line: Calculated

- Single dots: Experimental -

- -

- d

- d

O O

2 I I I I I I I

O 1 00 200 300 400


Table 5-4: Arc Properties for Different Arc Lcngths and Applied Currents. (T ,,,. (Kk u ,,,. (m-s"); A@.(V); Rc. mm)

Applied Current (A) 150 1 250 1 350

metd transfer mode wiU be discussed later.

Typical isotherms for 10 mm arcs of different currents are shown in Figure

5.10. In al1 cases the bell-shaped characteristic of the arc c m be seen. Unlike the

GTAW arcs the maximum tempenture occurs at the front of the anode. In cornpari-

son with the other calculated results [ l ] these temperatures are lower. The reason for

the difference in the two calculated rcsuits is not known. but by considering the exper-

imental [12,13] and theoretical [12,l4- 161 datri for the GTAW arc, the results of this

study are more reasonable.

Variation of the arc voltage with the arc length and current is shown in Fig-


Applied current. A

Figure 5.9 Variation in the total current at the tip of the electrode as a function of applied m e n t for different elecuode diameters. Numbers are the elecuode diameter in mm.

ure 5.11. It c m be seen that the arc voltage increases with both an: length and applied

current. The values presented in Figure 5.11 are the arc voltage plus the cathode fall

which are calculated from Equation (2-8). For 10 mm arcs of different currents the

cathode fall changes from 1.58 V for 150 A arc to 2.84 V for 350 A m. and for 250

A arcs its value changes h m 2.92 V for 2 mm arc to 2.52 V for 10 mm arc. By

considering the cathode Ml it c m be seen that the electric field increases with the

current and decreases with the arc length (Figure 5.12). On the other hand, it must be

mentioned that the cathode fail, in the case of the non-thermionic emission is found to

be between 10-20 V [9.17] which is much higher than the estimated values.

To extend the arc model to a joint model for the whole process including the

arc and the weld pool, the heat flux io the workpiece must be calculated. Several

mechanisms are involved in heat transfer from the arc to the workpiece. It is well un-

derstood that the ion impinging on the surface of the cathode at current densities as

high as log ~ . r n - ' can transfer energy in the order of 105 M W . ~ - ' to the cathode [4]-

Cha~ter 5 GMAW mode1 1 50

El El 1 9000 K

Interval 2000 K Intemal 2000 K

C

U] - 6.0 ü

10.0 8.0 6.0 4.0 2.0 0.0 10.0 8.0 6.0 4.0 2.0 0.0 Fladial distance, m m Radial distance, mm

El 7 900 K lnterval 2000 K

Figure 5.10 Isotherrns of 10.0 mm ruçs with different applied currents. a: 1 = 150 A; b: 10.0 8.0 6.0 4.0 2.0 0.0 1 = 250 A; c: i = 350 A. Radial distance, mm

In GMAW of aluminium, it is found that 12% of the total efficiency is due to the

cathode heating [18]. Essers and Walter [19] concluded that 65-754 of total heat in-

put to the workpiece is due to the arc, which includes radiation, convection and ion

cathode heating. The remainder is due to the molten droplet from the wire. These re-


Fisure 5.1 1 Variation in the arc voltage with appIied current and arc Iength. Numbers are arc Iength in m.

16

14

> 12 6 0 (O = IO O > 2 8 a 6

4

sults are in good agreement with the results of Lu and Kou [18]. Watkins e t ai. [20]

measured the heat trmsferred to the workpiece in machine welding of steel and report-

ed that the total heat input cfficiency was 85%; 40% due to droplets and 458 due to

arc.

- I I I I I

- -

- - - -

6 - -

- 2 - I I I I I

In the present study by usiq the same approach as the GTAW (chapter 2 ) . a

1 00 1 50 200 250 300 350 400 Applied current. A

Figure5.12 Variation of electric field with a: applied current of 10.0 mm arcs; b: arc length of 250 A arcs.

Applied current, A

Arc length. mm

Chapter 5 GMAW madel 1 52

fist order estimation of the heat llux from the arc to the workpiece and the wire elec-

trode was performed to determine the relative importance of the different heat transfer

mechanisms. The estimation made in this study includes convection and radiation to

the workpiece and electron heating uansferred to the anode. It is found that the heat

transferred to the workpiece due to radiation increases with both the arc length and the

current. Also, the contribution of convection increases with the current, while the vari-

ation in the arnount of heat due to convection with the arc length is not significant.

The variation in the heat transferred to the workpiece due to radiation and convection

with the current and the arc length is shown in Figure 5.13.

The heat transferred to the anode does not depend on the arc length but in-

creases with the applied current. Figure 5.14 illusuates the variation in the heat to the

Applied current. A

- - - - - -

1 O0 200 250 300 350 400 Applied current, A

Figure 5.13 Variation in the thermal power transferred to the cathode due to a: radiation and b: convection with the applied current as a function of arc length.


600 1 I I I l I 1 1 O0 1 50 200 250 300 350 400

Applied current. A

Figure 5.14 Variation in the transferred heat CO the anode due to elecuons condensation with the applied current.

anode from the arc per unit time as a function of the applied current. By considering

the total absorbed heat by the anode (wire electrode), dong with the transfer of drop-

lets to the weld pool. the ratio of droplet contribution to radiation and convection con-

tribution for different cases is cdculated and presented in Tabb 5-5. By comparing

these data with expenmenüil data [18-201, it cm be concluded that only part of the

absorbed heat by the anode wire transfers to the weld pool. However. the contribution

of the droplet to the total heat input is more than what is found expenmrntally. This

shows that the cathode heating due to the bombardmrnt of ions rnust have a signifi-

cant contribution to the heat transferred to the workpiece.

Table 5-5: The Ratio of Droplet Contribution to Radiation and Convection Contribution.

de (mm)

1.2

1.2

0.8

12

1.6

AppIied Current (A)

350

4.99

2.90

2.06

2.29

2 .52

150

10.64

5 .O6

3.4 1

3.73

4.08

250

6.5 1

3.69

2.53

2.82

3 .O2


The dectrons that mach the electrode (anode) can condense on both the side

and the tip of the electrode. It is found that in the globular mode of metal transfer,

the majority of the electrons condense on the tip of the electrode (anode). In other

words, the tapering of the electrode starts when the electrons condense on the side of

the electrode [21]. Based on the results of this study, the percentage of the applied

current that reaches the tip of the electrode decreûses with increasing applied current

(Figure 5.15). This leads to a higher arnount of heat entering the side of the electrode.

On the other hand, it is known that there are several forces involved in metal transfer

to the weld pool including gravitational force, aerodynamic drag force, electromagnetic

force, surface tension and vapour jet force [22]. Among these forces, the aerodynamic

drag force and the electromagnetic force usually act to detach the droplet and the sur-

Applied current. A

Eiectrode diameter. mm

Figure 5.15 Variation in the percentage of current which reach the tip of the electrode with the a: applied current; and b: electrode diameter.

C hapter 5 GMAW model 155

face tension and the vapour jet forces oppose the droplet detachment Depending on

the welding position, the gravitational force can have both effects. An electromagnetic

force parallel to the electrode (Figure 5.16) which can accelerate gas tlow in this re-

@on can be created by having a side entering current Hence by increasing the per-

centage of elecuon entry at the side of the electrode, there will be an increase in both

drag force due to hizher velocity, and electrornagnetic force due &O variation in the

electrons path (Figure 5.17). Therefore the higher side heating and the elecuomagne tic

force paralle1 to the electrode (anode) can be two major reasons for changing the

mode of metal trmsfer from globular to spray. A decrease in die diameter can also

increase the percentage of side heating of the electrode (Figure 5.15).

5.4.4 Effect of the Shielding Cas

The calcuiation has been performed for both argon and helium as the shield-

ing gas. Helium, becaux of its higher ionization potential than argon. produces some

1 .s 1 .O 0.5 0.0 Radial distance, mm

El T i - 1.5

1.5 1.0 0.5 0.0 Radial distance. mm

Figure 5.16 Effect of the applied current on the electromagnetic force around the elecaode tip. a: 1 = 150 A; b: 1 = 250 A; c: I = 350 A.

1.0

E 0.5 6

U c

- . . . . . .

. . - . . . - - * 0 , # l l

r u Y

a 0.0 g - s

- - \\\- 0.5 a r r . . -------<

l i 1 .O 1.5 1 .O 0.5 0.0

Radial distance. m m

-

-


Figure 5.17 Schematic of elecuons path when there is side curent into the consumable electrode.

problems in an: stability and arc initiation [23.24]. On the other hand. GMAW with

helium shielding gas produces a deep, broad and parabolic weld bead [23] and produc-

es additionai heat which makes it suitable for metals with higher thermal conductivi-

ties such as aluminium and copper (22.24.251. In this section. the effect of shielding

oas on the cathode spot radius, temperature into the arc, gas velocity, electrons path C

around the anode, and the heat uansfer to the weld pool will be studied.

It is obxrved that the cathode spot radius changes with the arc eiecrric poten-

tial using He shielding gas just as with Ar (Figure 5.5). In helium, due to its lower

electrical conductivity (Figure 5-18), the cathode spot radius corresponds to the mini-

mum elecuic potential and is smaller than in argon. Similar to the analysis performed

for Ar gas in section 5.4.1. the cathode spot radius which produces the minimum arc

voltage drop is adopted as the reference cathode spot radius on the basis of minimum

energy considerations. In Table 5-6 the computed arc properties for 2 and 10 mm arcs

at different currents for heliurn are presented. In c o m p ~ s o n with argon (Table 5-31,

the maximum temperature in helium is higher, due to the higher ionization potentid of

C hapter 5 GMAW mode1 1 57

4000 8060 12000 16000 20000 24000 Temperature, K

Figure 5.18 Elecuical conductivity of Ar and He as functions of temperature. [ 5 ]

helium. The velocity of gas in helium in most cases is two to three times higher than

that in argon due to the lower density of helium. Although the velocity in He is very

high, it is still well below the sound velocity. To calculate the sound velocity the fol-

towing equation c m be used [26],

Table 5-6: Arc Properties for Different Arc Lengths and Applied Currents in Helium. (Tm,,, tKk u ,,,, (rn.s''j: Rc.(mm))

L- (mm)

1

2.0

10.0

Arc Properties

T m umax

Rc

Tmp,

Um,

R c Tmax

" m a

Rc

Tma*

Umax

Rc

de (mm)

1.2

0.8

1.2

1.6

. Applied Current (A) 350

1

25120

1490

1 -400

35830

5602

2.57 1

25 120

2344

2.625 24230

1402

2.67 1

150

21 520

58.99

1 .O83

22550

823.4

1.316

2 1430

382.1

1.235

20080

238.0

1.273

250

236 i O

579.7

1.400

30630

3520

2.370

23910

1152

2.176

220 1 O

697.2

- 2.139 -


In this equation a is sound velocity, k is Cp CV . R is he universai Cas constant. M

is the molecular mass and T is temperature. For temperatures between 10000 to

35000 KT the sound velocity changes from 5000 to 10000 m d , which is at least two

Urnes higher than the gas velocities in the corresponding temperatures.

ln Figure 5.19 isotherms of the helium arcs of 10 mm with three different

currents are shown. The bel1 shape characteristic of the an: cm not be seen in the he-

lium arc, especially in the low current arc. This is due to the cathode jet in the heli-

um arc (Figure 5.20). In cornparison with argon plasma the conductivity of helium at

temperatures less than 20000 K is much lower (Figure 5.18). which leads to a srnaller

cathode spot radius and thus a higher mean current density. By increasing the current,

the anode jet becomes stronger and can overcome the cathode jet This trend is ex-

plicitiy shown in Figure 5.20. In this figure the velocity profiles for the helium arcs

of Figure 5.19 are shown.

With pure argon as the shielding gas. there is a transition current below

which the metal uansfer mode is globular while above it, a very stable, spatter-free

axial spray transkr mode can be obtained (Figure 1.3). However, arcs shielded only

by helium do not exhibit tme axial spray transfer at any current leveis [23]. It was

mentioned zarlier (section 5.4.4), that the elec~ode side current in the argon shielded

arcs c m play an important role in the transition of the metal transfer mode. In pure

helium, on the other hand, di the electrons condense at the tip of the electrode (an-

ode), in other words there is no current at the side of the electrode regardless of the

applied currents. The variation in the total current into the electrode with distance

from the tip of the elecuode for both argon and helium with 10 mm arc gap of differ-

ent currents is shown in Figure 5.21. The main reason for this is that the temperature


Radial distance, mm

Figure 5.19 Isotherms of 10.0 mm helium arcs with different currents. a: 1 = 150 A; b: 1 = 250 A; c: 1 = 350 A.

Radial distance, mm

Radial distance. mm

at the side of the elecûode (anode) never reaches the ionization temperature for He

which is more thm 8000 K (Figure 5.18). cornpared with the argon ionization temper-

d 2 ?3 P O

P g O

Axial distance. mm

a P P O O 8 O

Axial d~stance, mm


Distance from the electrode tip. mm

Distance from the electrode tip, mm

Distance from the electrode tip. mm

Figure 5.2 1 Variation in the current into the elecuode with ihe applied current and distance from the eiectrode tip. a: 1 = 150 A; b: I = 250 A; c: 1 = 350 A.

a t m of about 4500 K. regardles of the arc current Due to its high ionization poten-

tial. He gas acts as an insulator.

In cornparison with argon. in the case of helium the higher temperature of the


arc results in a higher thermal content being transferred to die workpiece (rhe cathode)

and electrode (the anode), dirough radiation. convection and electron condensation. As

an example, the heat transferred to the cathode and to the anode for 10 mm arcs of

different currents for both helium and argon is plotted in Figure 5.22. On the other

hand. it is found that helium cm significantly improve the shape of the weld pool

[23]. The shape of the weld pools for helium and argon shielded arc are shown in

Figure 5.23 schematicÿlly. This figure shows the bead shape in the case of argon as a

"fin@' type penetration whereas in the case of helium there is a deep, broad and par-

abolic weld bead. This difference is atuibuted to the higher thermal conductivity of

helium [23]. It has to be mentioned that the "finger" type penetration is the character-

istic of argon shielded welds. rspecially when the metd uansfer is in the spray mode.

In this case the kinetic enrrgy of droplets plays a very important role in determinhg

the shape of the wetd pool. Moreover, the elecfron contribution to the heat transfer to

the weld pool. which is associated with droplets. in argon shielded weld is higher than

the electron contribution in the helium shieided weld. The variation of the electron

contribution for the two shielding gases as a function of applied current is shown in

Figure 5.24. This graph shows that radiation and convection. which are more spread

over the weld pool surface play more imporîant roles in the heliurn shielded welds

than the argon shielded welds.

One of the problerns in using helium as the shielding gas is the large arnount

of spatter produced. The mechanism of formation of spatter is not clear. Eickhoff and

Eagar analysed the spatter formation mechanism in the GMAW of Ti in pure argon

with low currents [27]. They concluded that localhg the cathode spot at the top of

the droplet as it approaches the surface of the weld pool and before incorporating into

die weld pool is responsible for spat te~g in the case of Ti welding in argon. In this

case, since the electron emission mechanism is themionic. the cathode spot is small


oai 'E 2.z

Applied current, A

Applird current. A

1 I J I I 1 1 O0 1 50 200 250 300 350 400

Applied current. A

Figure 5.22 Variation in the heat wmsfcrred to the workpiece or the efectrode in unit Ume with the applied current in helium and argon. a: Elecuon heat transfer to the elecuode; b: Convective heat transfer to the workpiece; c: Radiative heat tmnsfer to the workpiece.

enough to be able to locate at the top of the droplet Then the Lorentz-type forces ex-

pel portion of the metal droplet from tbe weld pool. Because of its low melting point


Argon Helium

Fi,we 5.23 Effect of the shieiding gas on the geomeuy of the weld pool is shown schematically. [23]

aluminium does not show a thermionic cathode spot However, during welding under

the protection of helium. a mean cathode spot radius of 1.235 mm for 150 A and 10

mm arc has been observed. It should be kept in mind that in non-thermionic emission

there are severai very srnall ernitting sites which move on the surface of the cathode

and most probably into the mean cathode spot radius continuously. For a very smdi

cathode spot radius, it is quite possible for the cathode spot sites to be located at the

top of the unincorporated weld droplet as it touches down on to the weld pool. There-

fore the same mechanism of spattering as in the case of the welding of Ti in pure ar-

con is quite possible. By increasing the current, the cathode spot radius increases and C

Figure 5.24 Variation in the electroa contribution to the total heat transferred to the weld pool in He and Ar as a fùnction of appIied m e n t .

80

75

70

65

60

I I I - --- - - - *

- - - - - - - -* - - - -_ - - - - - - - - ---- Ar -

- - He

1 1 I 150 250 350

Applied current. A


the possibility of locating the emitting sites on the droplet decreases. However, the

velocity of the impinging droplets on the surface of the weld pool increases simultane-

ously. Both a decrease in the possibility of locating the emitting sites on the top of

the droplet and or increase in the velocity of the droplet impinging on the weld pool

surface resuIt in a decrease in the spatterïng.

5.4.5 Effect of the Electroàe Tip Shape

In previous sections. a model for GMAW process was developed and the ef-

fect of the arc variables on the arc propenies was discussed. In the model, the elec-

trode tip was considered to be flat It has also been shown that the electrode tip angle

c m change propeities of the arc in the GTAW process and even in its weld pool sig-

nificantly (chapters 3 and 4). n i e shape of the electrode c m not stay tlat d u ~ g the

welding process. Thus. it is worthwhile to snidy the effects of the variation in the

electrode tip shape on the arc propertïes.

The metal uansfer mode c m change from almosr semi-sphencal to conical de-

pending on the ÿn: current Figure 5.25 shows the shape of the electrode tip for two

metal iransfer modes. globular and spray. To simulate the arc in the case of the spray

Figure 5.25 Schematic shripe of the elecirode tip in two different metal transfer modes. a: Globular; b: Spray.


uansfer mode, a conical electrode tip shape with an angle of 10 degrees is considered.

Assuming a quasi-steady state, all the equations explained in section 5.3 with the s m e

boundary conditions as for the tlat tip electrode and including some modification at

the anode boundary conditions are used to handle the new configuration. To observe

the effect of the electrode tapenng more clearly. caiculations have been performed for

a flat tip electrode too. In both cases a 10 mm and 200 A arc was considered. The

electrode diameter is 1.2 mm and the cathode spot radius was considered to be 4.806

mm for both cases.

Figure 5.26 illusuates the effect of the electrode tip shape on the temperature

distribution into the arc. The clifference in the maximum tempenture for these two

cases is about 2% and this shows that the anode current density entering the electrode

does not change from one case to the other one. Therefore the anode surface is al-

most constant and in the case of the conicai electrode, the arc must move up to cover

enouph surface area of the electrode. On the other hand. the variation in the current

flow around the electrode tip. due to its shape, changes the electromagnetic force con-

tiguration which leads to a higher velocity for arcs with the conical electrode tip.

Figure 5.27 shows the velocity profile for the two cases. The maximum ve-

locity in the case of the conical electrode is more than three times higher than that in

the other case. This causes a higher heat convection. especially around the axis of

syrnmeuy, shown by the lengthened tempenture contours. Also. Figure 5.26 shows

that the temperature near the weld pool in the conical electrode case is higher, due to

more efficient convection. High velocity also produces higher pressures at the surface

of the weld pool. The gas pressure at the surface of the weid pool can increase by

more dian three times (Figure 5.28). This is important from the point of view of the

shape of weid pool surface.

Chapter 5 GMAW model 1 67

Radial distance. mm

Radiaf Distance, mm

Figure 5.26 lsotherms of 10.0 mm. 200 A arcs for two electrode tip angles. a: a = 10 deg; b: a = 180 deg.


Radial distance. mm Radial distance. mm

Figure 5.28 Effect of the electrode tip angle on the distribution of the pressure in 10.0 mm, 200 A arcs. a: a = 10 deg; b: a = 180 deg. Nurnbers are pressure di fference in P a

corne the problem of the non-thermionic nature of the cathode, a cathode spot is de-

fined such that inside the cathode spot the current density is a constant value and

outside that the current density is zero. Although the plasma temperature and velocity

are almost constant over a wide range of the cathode spot radius, especially for longer

arcs. it is observed that the elecvic potential between the electrodes highly depends on

the size of the cathode spot. The optimum value for the cathode spot radius is ob-

tained by considering the minimum energy consumption by the arc. This optimum

value is a function of applied cunent, arc gap, electrode diameter and the shielding

gas.

By cornparing the results of this study with the available experimental data, it

is found that the mode1 for temperatures higher than 10000 K is reiiable, although

there are sorne modifications that must be considered for future studies, such as the

interaction between plasma and metai droplet, continuous variation in the shape of the

electrode tip and presence of metal vapour in the shielding gas.


Evaluation of the heat transferred to the workpiece and the electrode shows

that there must be some other mechanisms, in addition to convection, radiation and

electron condensation, for the heat transfer to the workpiece. It is likely that one of

the rnost important mechanisms will be the impingement of the positive ions on to the

cathode surface. Amongst the three mechanisms, the electronic energy goes to the

electrode and along with the metal droplets will be transferred to the workpiece, and

the two others, convection and radiation, go directiy to the workpiece. In both aqon

and helium arcs, the electron condensation on the wirc electrode (the anode) is the

major contributor in heat tnnsfer to the weld pool. This portion of hear flux is associ-

ated with droplets falling into the weld pool. The contributions of radiation and con-

vection depend on the applied current and arc length. In pure argon, radiation

increases with both arc length and applied current. but convection only increases with

the current and remains alrnost constant with increasing arc length.

It is found that the anode spot in pure argon c m extend to the side of the

electrode. This leads to a side flow of electrons which c m have a signifiant role in

drtermining the metal transfer mode. In pure helium, in which there is no m e transi-

tion of the globulÿr mode of metal transfèr to the spray mode, even at cwents as high

as 350 A the side current density cm not be observed.

Due to a higher ionization potential and lower electrical conductivity. the

cathode spot radius is much smaiier in pure helium than in argon. This leads to a

higher cathode current density and consequently a cathode jet which injects the plasma

toward the anode. This produces an upward flow toward the anode in low current

arcs. By increasing the applied curent, since the anode jet is dominant at higher cur-

rents, the upward flow disappears.

By adopting a quasi-steady state, the effect of the electrode tip shape on the

arc properties in the GMAW process is addressed for the t-mt time. This preliminary


study shows that variation in the arc properties, especiaiIy dose to the elecuodes, due

to variation in the shape of the electrode tip is signitïcant. Therefore variation in the

shape of the electrode tip can not be neglected in future studies on th& subject.


REFERENCES

1- P. G. Ionsson, ScD niesis, Dept of Materials Science and Engineering, The Mas-

sachusetts Institute of Tec hnology, 1995.

2- J. D. Cobine, Gaseous Conductors, Dover Publication Inc., 1958.

3- A. Lesnewich, Tontrol of Melting Rate and Metal Transfer in Gas-Shielded Metal

Arc Welding, Part 1: Control of Electrode Melting Rate", Welding J., 1958, vol.

37, pp. 343s-353s.

4- W. L. Morgan, L. C. Pitchford and S. Boisseau, "The Physics of Ion Impact Cath-

ode Heating", J. Appl. Phys., 1993, vol. 74, pp. 6534-6537.

5- M. 1. Boulos, P. Fauchais and E. Pfender, T h e m l plasma, fundamentals and ap-

plications, vol. 1, Plenum Press, 1994.

6- W. J. Lick and H. W. Emmons, niermodynomic Properties of Helium, Harvard

University Press, 1962.

7- D. L. Evans and R. S. Tankin, "Measurernent of Emission and Absorption of Ra-

diation by an Argon Plasma*', Physics of Fluids. 1967, vol. 10, pp. 1 137- 1 144.

8- Y. S. Touloukian, Thermophysical Properties of High Temperature Solid Materi-

ais, vol. 1: Elernents. McMi1la.n Co. 1967.

9- J. F. Lancaster, The Phpics of Welding, 2nd edn, Pergamon Press, 1986.

10- P. Boughton and M. Amin Mian, "Aspects of Arc Root Behaviour in Welding".

2nd International Conference on Gus Discharges, 1972, 1 1 - 15 Sep., pp. 130- 13 1.

11- E. A. Smhs and K. Acinger, "Material Transport and Temperature Distribution in

Arc Between Melting Aluminium Electrodes", IZW Document No. 212-162-68,

1967.

12- K. C. Hsu, K. Etemadi and E. Pfender, "Study of Free-Buming High-Intensity Ar-

gon Arc", J. Appl. Phys.. 1983, vol. 54, pp. 1293-1301.

13- G. N. Haddad and A. J. D. F m e r , "Temperature Deteninations in a Free-Bum-

Chapter 5 GMAW rnodel 173

ing Arc: 1. Experimental Techniques and Results in Argon", J. P@s. D: Appl.

P@, 1984, vol. 17 pp. 1189- 1196.

14- P. Kovitya and J. I. Lowke. 'Two-Dimensional Analysis of Free-Burning Arcs in

Argon", J. Phvs. D: Appl. Phys., 1985. vol. 18, pp. 53-70.

15- C. Delalondrc and 0. Simonin. "Modelling of High Intensity Arcs hcluding a

Non-Equilibrium Description of the Cathode SheathT', Coiloque De Physique. 15

Sep. 1990, Colloque CS, Supplement Au n 18. Tome 5 1. pp. 199-206.

16- J. J. Lowke, P. Kovitya and H. P. Schmidt, "Theory of Free-Burning Arc Columns

Including the Influence of the Cathode", I. P h y . D: Appl. Phys.. 1992, vol. 25,

pp. 1600- 1606.

17- R. Hajossy and 1. Morva, "Cathode and Anode Fails of Arcs with Fusible Elec-

trodes", J. Phys. D: Appl. Ph-, 1994, vol. 27. pp. 2095-2101.

18- M. I. Lu and S. Kou, "Power Inputs in Gas Metal Arc Welding of Aiuminum-

Part II", Wei@ J., 1989. vol. 68, pp. 452s-456s.

19- W. G. Essers and R. Walter. "Heat Transkr and Peneuation Mechanisms with

GMA and Plasma-GMA Welding", Wefding J.. 198 1, vol. 60, pp. 37~42s.

20- A. D. Watkins, H. B. Smartt and C. J. Einerson, "Heat Transfer in Gas Metal Arc

Welding", in Recrnr Trends in Wefding Science and Technology TWR'89. edited

by S. A. David and I. M. Vitek. American Society for Metals, 1990, pp. 19-23.

21- Y. S. Kim and T. W. Eagar, "Analysis of Metal Transfer in Gas Metd Arc Weld-

ing", Welcling J., 1993, vol. 72, pp. 269s-278s.

22- J. Norrish, Advanced welding processes. institute of Physics Publishing, 1992.

23- Welding handbook. 8th edn. vol. 2. American Welding Society, 1991.

24- K. A. Lyttle and W. F. G. Stapon, "Selected the Best Shielding Gas Blend for the

Application", Welding J.. 1990, vol. 69, pp. 21-27.

25- W. Lucas, "Choosing a Shielding Gas- Part 2", Welding und Meral Fabrication,


1992, vol. 60, pp. 269-276.

26- M. H. Aksel and 0. C . Eralp, Gus clynamics, Prentice Hall. 1994.

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GMAW of Titanium Alloy Plate", Wrlding J.. 1990, vol. 69. pp. 382s-388s.

C H A P T E R 6

ON AND FUTURE STIJDM

6.1.1 GTAW Process

1 For the GTAW with a tlat tip electrode. a constant current density over the

cathode spot c m give reliable resulb for temperature, velocity and even heat

tlux and anode current density. This cathode current density increases with the

applied current. However for GTAW with angular tip electrodes. a constant

current density over the cathode spot gives a maximum temperature that is not at

the axis of symmetry for sharp electrodes.

2 A dominant rnechanism for the heat transfer to the workpiece is the electron

contribution. For arcs wiih a flat tip electrode, the electron contribution changes

from 63% for a 12.7 mm arc and 300 A current to more than 92% for a 2.0 mm

arc and 100 A current. The convection component increases from -48 to - 15%

with the applied current from 100 to 300 A and arc length Krom 2.0 mm to 12.7

mm. The increase in the radiation component with the current and the arc length

in the mentioned raqes is from -3.56 to -22%.

3 A model is proposed for the variation in the cathode spot surface area on the

tungsten e:ectrode with the electrode tip angle, based on experimental t-uidings of

other investigators. According to this model, the cathode spot surface area is

constant for electrode tip angles l e s than 60 degrees and then increases with the

Chapter 6 Conclusion and future studies 1 76

electrode tip angle linearly. The arc mode1 developed is based on the new

approach for the cathode spot surface area. It can give reliabie results for arc

voltages, gas velocities, temperature profiles (even very close to the cathode),

anode current densities, heat tluxes to the workpiece and pressures on the

surface of the weld pool, for 100-250 A arcs of 2.0 to 10.0 mm Iength.

4 It is found that the electrode tip angle has signif-ïcant effect on the arc shape.

The wider electrode tip angle gives more constrict arcs, such that the anode

surface area on the workpiece surface is srnail. By decreasing the electrode tip

angle (sharper electrode), the arc was expanded and accordingly the anode

surface area was increased. This increase in the surface area with decreasing

electrode tip angles leads to higher current densities for arcs with wider electrode

tip angles. Since the main contributor to the heat tlux to the workpiece is the

electron contribution, this variation in the angle of the electrode f ip can affect

the distribution of the heat flux significantly. It is observed that an increase in

the heat t7ux due to an increase in the electrode tip angle leads to a larger weld

pool.

5 It is also found that a decreâse in the electrode tip angle cm increase the axial

gas velocity two to three times, deprnding on the arc length. and the radial gas

velocity at the surface of the weld pool up to four times. hcrease in the radial

gas velocity c m be mslated into an increase in the gas shear stress applied to

the surface of the weld pool. The gas shear stress. among the tlow dnvuig

forces into the weld pool, can have significmt effect on the weld pool free

surface and accordingly on the flow patterns in the weld pool.

6 For the first time, the effect of the electrode tip angle on the heat flux from the

arc to the weld pool and its contributors, including electron contribution,

convection and radiation. is evaluated. It is found that, for example for a 5.0

Chapter 6 Conclusion and future studies 177

mm. 200 A arc. increasing the electrode tip angle from 10 to 60 degrees

increases the maximum heat transferred to the anode due to electron contribution

from -50 to more than 70 Wmm-'. Further increase to 150 degrees decreases it

by about 30%. The maximum heat transferred to the mode by convection

decreased more than three times when the electrode tip angle was increased from

10 to 150 degrees. The variation in the maximum heat due to radiation is about

20%. In spite of the significant variation in the maximum heat due to different

mechanisrns, the total heat transferred to the anode by a convection mechanism

changes only slightly with the elecuode tip angle.

7 A simple model is developed for the weld pool based on the variation of

viscosity of a mixture of varying fraction of solid and liquid. In this rnodel.

instead of using a pre-detked anode current density and heat flux, the data from

the arc model are used. Therefore, for the tkst time it was possible to study the

effects of the electrode tip angle on the weld pool shape quantitatively. The

estimations show that the arc and the weld pool models together c m give

reasonable results for tlow pattem in the weld pool, although there is some

inconsistency between calculated and experimental data for the elecuode tip

angle wider than Y0 degrees. Also, some modification regarding the free suface

of the weld pool is nrcessary, especially for very sharp electrodes.

8 The main tlow dnving forces into the weld pool in welding with a 200 A arc of

2 and 5 mm lengths are the gas shear stress and the surface tension. Direction

of the surface tension which is a function of the temperature and the

composition of the workpiece material. is very important in determining the flow

pattern in the weld pool and therefore its shape and size.


6.1.2 GMAW Process

1 By defining a cathode spot, a niodel for the arc in the GMAW process is

developed. For the t - i t tirne, it is found that there is an optimum radius for the

cathode s p o ~ for which the energy consumption of the arc is a minimum. This

optimum radius is a function of arc length, applied cunent, wire electrode

diameter and shielding $as. The model c m give reasonable information for

temperature distribution and gas velocity for arcs shielded by argon for

temperatures higher than 10000 K. For the cold parts of the arc at the edge of

the stable plasma. because of recombination of the çharged particles (the ions

and the rlecuons) and absorption of the energy emitted from the centre of the

arc column, the assumptions of the existence of the LTE condition and an

optically thin for the plasma are not valid.

2 It is found that the anode spot surface area for arcs shielded by argon increases

with the current, thus some fraction of the electrons enter the anode (wire

elecuode) through its side. For arcs shielded with helium, however, electrons

aiways enter the mode through its tip. This different behaviour for the gases is

due to their difference in ionization temperature and therefore in elecuical

conductivity. The transition in the metal trruisfer mode with increasing in the

current obxrved in GMAW under argon protection. can be atîributed to the

presence of side currents.

3 Evaluation of the heat transfer from the arc to the workpiece for the GMAW

process reveals that in addition to convection, radiation, and electron (which is

transferred to the weld pool through metal droplets) contributions, there must be

another mechanism(s) for heat transfer in this process. The positive ion

bombardment of the cathode (weld pool) is the most probable mechanism.

4 It is shown by simulating the arc in the GMAW process that the electrode tip


shape has a significant effect on die arc properties. For example, for a 10 mm,

200 A arc, the maximum axial velocity c m be increased by more than three

times. This shows that, although the model for GMAW with a flat tip elecuode

cm give very valuable information about the GMAW arc. the variation in the

shape of the electrode tip must be considered.

6.2 SUGGESTIONS FOR FüTURE STUDIES

6.2.1 GTAW Process

1 Although the angle of the electrode tip has been successfully taken into account

in the present model, diere are some possibilities to make the model more appli-

cable. For exmple. using a body-fitted coordinate to rernove the difficulties and

the uncertainties related to the stepwise electrode tip. By using a body-fitted

coordinate. it will be possible to extend the mode1 for angular etectrodes up to

the flat tip rlectrode. In the present model, the results related to angles wider

than 130 to 150 degrees is less reliable and for more thm 150 degrees there are

no results available, that are based on the mode1 for angular electrodes. Also, it

shiiuld be possible to consider the uuncation ar the tip of the electrode easily

and accurately.

2 The availability of experimental data for checking the validity of the results

obtained from the arc model is very poor, especiaily for different electrode tip

shapes. From the welding point of view. the important experimental data are

those which are uansferred to the workpiece. including anode current density.

heat flux and gas shear stress.

3 Variation in the shape of the weld pool surface in modelling both the arc and the

weld pool must be considered in more detail. The best way to handle this varia-

tion is by combining the arc and weld pool models. The interface of the arc and


the weld pool c m probably be determined by balancing the applied forces on the

top of weld pool surface and the variation in the metal volume due to the melt-

ing and to the increase in temperature. The balancing of the applied forces on a

liquid surface in not an easy task and mus& be developed from very simple

cases.

4 Apparent viscosity of mixtures of liquid and solid is a function of the relative

amounts of solid and solid panicle shapes. Therefore in order to use this prop-

erty in the modelling of a melting or solidifying body of metal which is under

the influence of liquid motion, it is necessary to obtain a more realistic expres-

sion for the viscosity variation by variables of the system.

6.2.2 GMAW Process

1 Variation in the shape of the electrode tip is continuously occumng dunng the

welding in the GMAW process. Thus, in order to obtain more reliable resu1t.s

from the theoretical calculations, development of a coupled model of the con-

sumable electrode and the arc is necessary. In this rnodel. Joule heating into the

wire. melting of the metal and fonnation of the metal droplet, detachment of the

metal droplet, heat transfer to the consumable electrode due to the elecaon bom-

bardment and the wire feed speed should be considered. In a more advanced

model, the effect of current on dl of these, and especially on the metal transfer

mode must be addressed. Regarding the arc, the presence of metal droplets in

flight between the electrodes must be considered. Heat transferred to the droplet

t'rom the arc, vaporization of metai from the droplet surface, distribution of metal

vapour into the plasma, and current density at the surface of the droplet should

be addressed.

2 One of the problems in modeIlhg the arc in the GMAW process is the


availability of the physical and transport properties of the shielding gas. Usuaily

a mixture of two or three gases is used to protect the welding parts. The

information about the properties of the gas mixtures and their variation with

temperature and gas composition (due to presence of metai vapour) is rarely

available in the published iiterature. Therefore to extend the applicability of such

models for more practical applications. there is a need for these information.

3 Evaluating the optimum cathode spot radius requires a very long time for

calculation and this limits the applicability of the model. [n order to extend the

model applications, a joint model for the arc and the weld pool and ultirnately

for the consumabk electrode. arc and weld pool is required. Regarding the wetd

pool part. there are severd things that must be investigated. Heat transfer frorn

the arc to the weld pool is one of the mosr important issues that needs to be

solved. For the Ume being, the information about heat transferred to the weld

pool (cathode) due to the positive ion bombardment is very limited. Extending

the reliablr and applicable information regarding the role of ions in heat trmsfer

requires knowledge of the contribution of the positive ions to the curent density

in non-thermionic cathodes. Also. it is very i m p o m t to know the magnitude of

the momentum that c m be transferred to the weld pool due to the impingement

of metal droplet. To have a realistic evaluation for this, it is necessary to snidy

the interaction of die metai droplets and the arc, and the momentum that can be

transferred to the droplets due to the electric tield in the plasma, the cwen t at

the surface of the metal droplets in flight and sven from the elecuomagnetic

forces at the moment of detachment.

Appendix I Solution Technique and lmpiementation 182

!%LUTION CH NIQUE AND LMPLEMENTATION

The PHOENICS code was employed for both arc md weld pool models. This

is a commercial control volume code based on SIMPLE algorithm [l], and has been

used to study plasma related phenornena and solidification problems [2-51. The de-

tailed of description of the PHOENICS c m be found in CHAM documents f6-91. In

this appendix some of the numerical settings imposed for different models presenred in

this thesis are explained. All computations were performed on a SUN SPARC work-

station.

1 Grid Configuration

In al1 cases of arc modelling, a non-uniform tïxed grid configuration was

used. The number of cells in the radial direction for GTAW tlat tip electrode, GTAW

tapered rlectrode and GMAW models were 50. 62 and 52 respectively. In the axial

direction the number of cclls is function of the arc length and the electrode tip angle

(in the case of GTAW with a tapered electrode). For the tlat tip electrode (GTAW)

the number of cells in the axial direction varied from 17 for a 2.0 mm arc to 52 for a

12.7 mm arc. For a GTAW tapered electrode, this number depended on the arc length

and the electrode tip angle and m g e d from 50 to 87. The number of cells in the axi-

al direction in the arc region is only a function of the arc length. For the GMAW

process. the number of cells chanpd from 25 to 48 with the arc length. In al1 cases

the finest grid was immediately below the electrode tip face.

For weld pool modelling, a single gnd configuration, 58x35, was used for all

cases. The finest cells were put in the liquid region of the calculation domain. This

was done by estimating the limits of the weld pool using very coarse grid size. For

weld pool also a uniform time grid was used with time intervals of 0.01 S.

Appendix I Solution Technique and lmplementation 1 83

II Initiai Conditions

For the weld pool case. the calculation staned from room temperature and ze-

ro velocities and zero elecuic potential into the workpiece.

In the case of arc modelling, since the conductivity of gases, argon and heli-

um. at room temperature is aimost zero, obtaining convergence of the results by start-

ing from room temperature, especially for the electric potential is very difficult.

Therefore to overcome this problem a predefined temperature field between the elec-

uodes was used which allowed the results for the elecuic potential field to converge

much faster. It must be emphasized that as long as there is a path way for electrons

between the electrodes, the condition of the predefined temperature field does not have

any effect on the results converging. Also, it is worthwhile to mention that the results

do not depend on the initial conditions.

III Convergence Criterion

To insure convergence, the residual values of al1 the variables were moni-

tored, and the sweep numbers were considered to be high enough such that the residu-

al values of al1 variables. with respect to the f is t sweep, becarne at least two orders

of magnitude smaller. Also. in al1 models, in some selected cases the spot values of

the variables were monitored to make sure that these values, by continuing the calcu-

Mons, remain constant.

IV Under Relaxation Factors

In arc modelling convergence of the results was obtained in dmost al1 cases

by using under relaxation technique. This was due to the very sharp temperature gra-

dient and also to significant changes in the gas properties, especially density. Momen-

tum and pressure fields were most sensitive to these variations. Therefore in order to

obtain convergence in the results, it was necessary to appiy variation of the different

Appendix I Solution Technique and Implementation 184

variables from one node to the another one only slightly. This was possible by in-

creasing the ce11 numbers and by using under relaxation, or by a combination of these

procedures. Usually, a combination of these techniques was necessary and the re-

quired combination was obtained by trial and error. As examples. in the appendices

[II and IV the q l and the gr0und.f files, respectively, for the argon shielded 100 A arc

of 2.0 mm length in the GTAW process with the tiat tip electrode are presented.

For the weld pool mode1 it was also necessary to use under relaxation. In

this case one of the main factors which led to difficulties in convergence was the par-

tial melting of the workpieçe followed by the initiation of flow. Another factor was

the variation of the surface tension coefficient with temperature. Remedies that cm be

applied are as follows;

8 increase the number of cells;

decrease the tirne intemals;

utiiize under relaxation technique.

Finding the best combination of these remedies to obtain convergence is possible by

trial and error.

Appendix l Solution Technique and lmplementation 185

References :

1- S. V. Patankar, Numerical Heat Trunsfer and Fluid Flow. Hemisphere Publishing

Corporation, 1980.

2- P. C. Huang, J. Heberlein and E. Pfender. "A Two-Fluid Mode1 of Turbulence for

a Thermal Plasma Jet", Plasma Chernistry and Plasma Processing, 1995, vol. 15,

pp. 25-46.

3- W. P. Hu and J. D. Lavers, "Coupled Electro-Thermal-Flow Mode1 for Very Long

Electric Arc", Accepted for publication in IEEE Transaction of Magne tics.

4- V. R. Voller and C. Prakash, "A Fixed Gnd Numerical Modellinp Methodology

for Convection-Diffusion Mushy Region Phase-Change Problems", ?nt. J. Heat

Mass Transfer, 1987, vol. 30, pp. 1709-17 19.

5- C. Prakash, M. Samonds and A. K. Singhal, "A Fixed Grid Numerical Methodolo-

gy for Phase Change Problems Involving a Moving Heat Source". Int. J. Heur

Mass Trunsfer. 1987, vol. 30. pp. 2690-2694.

6- CHAM, TR100- A Guide to the PHOENICS Input Language.

7- CHAM, PHOENICS Instruction Course, Unit 1, Introductory Course, Module 1- 1,

Introduction to PHOENICS.

8- CHAM, PHOENICS Instruction Course, Unit 1, Introductory Course, Module 1-2,

The Main Ingredients in a PHOEMCS Simulation.

9- CHAM, PHOEMCS Instruction Course, Unit 2, Advanced Course, Module A-2,

The Use of Subroutine GROUND.

Appendix Il Arc Modelling Assurnptbns 186

ARC MODELLING ASSUMPTIONS

1 Axial Symmetry

It is assumed that the arc is axially symmetric, so that the equations can be

written in two-dimensional cylindrical coordinates. The fust attempt to expand the arc

model to a three dimensional case was made by Kaddani et. al. [ l J. In their model

which is restricted to the arc column, they showed that as long as the arc is stable, re-

sults of the two dimensional and the three dimensional calculations are in good agree-

ment with each other. In fact the main advantage of the 3-D model is in studying the

conditions of having a stable arc and also studying the effects of the external factors

such as extemai magnetic field on the arc properties.

II Steady State

It is assumed that the arc is in steady state so that the variation of different

parameters with time is eliminated. To justify this assumption. it must be shown that

the residence tirne of the working gas in the calculation domain is much less than the

period of any fluctuation imposed upon the system due to the external variables. Tak-

ing a typical velocity of 100 m . d and an arc length of 10 mm, the residence time for

the gas is about 0.1 ms which is much shoner than any extemally imposed fluctua-

tions.

111 Local Thermodynamic Equilibrium and Optically Thin

It is assumed that the arc is in local thermodynamic equilibrium (LTE) and

the plasma is optically thin. This requires that the collision processes (and not radia-

tive processes) govem transitions and reactions in the plasma and that there will be a

microreversibility among the collision processes [2]. This definition is applicable for

Appendix Il Arc Modelling Assumptions 187

optically thin (which means chat the plasma does not absorb its own emitted energy)

and atmospheric plasmas. One of the consequences of the LTE state is that almost

complete energy exchange between the different gas particles iake place. The impor-

tant finding is that the temperatures of the elecuons. Te, and heavy particles (atoms

and ions). Tg, are the sarne. Aithough LTE conditions do not exist very close to the

electrodes [3] and ar the cold part of the arc [4-71, and therefore the plasma is not op-

tically thin, LTE occurs at the core of the arc and arc is optically thin at very high

tempentures. The results of this study also show the validity of this assumption for

tempentures higher than 10000 K.

IV Laminar Flow

I t is assumed that the tlow is Iarninar. McKeiIiget and Szekely [8] justified

this assumption based on the laminar-turbulent transition for a free jet. Taking the

typical velocity of gas as 200 m.s-', the characteristic length as 0.01 m and the kine-

matic viscosity of argon at 20000 K as 0.0045 m2.s-' lead to a Reynolds nurnber of

about 450. This value is much kss than a Reynolds number of about 100000 where

the rnnsition to turbulent tlow for a free jet occurs. On the other hand the agreement

of the calculated results of this study with experimental data presented in the thesis.

especially for the heat flux. is a suong proof to show that this assumption for the case

of the welding arc is valid.

Appendk Il Arc Modelling Assumptions 1 88

References :

1 - A. Kaddani, S. Zahrai, C. Delalondre and 0. Simonin, 'Three-Dimensional Model-

iing of Unsready High-Pressure Arcs in Argon", J. Phys. Dr Appl. Phys.. 1995.

vol. 28, pp. 2294-2305.

2- M. 1. Boulos, P. Fauchais and E. Pfender, 7?zermol plasma, fùndamenruls and ap-

plications, vol. 1, Plenum Press, 1994.

3- H. A. Dinulescu and E. Pfender, "Analysis of the Anode Boundary Layer of High

Intensity Arc", J. Appl. Phvs., 1980. vol. 5 1, pp. 3 149-3 157.

4- K. C. Hsu, PhD Thesis. Dept of Mechanical Engineering, University of Minneso-

ta, 1982.

5- A. J. D. Farmer and G. N. Haddad, "Rayleigh Scattering Measurements in a Free-

Burning Argon Arc", J. Ph- D: Appl. Phys., 1988. vol. 2 1, pp. 426-43 1.

6- L. E. Crarn, L. Poladian and G. Roumeliotis, "Departure from Equilibrium in a

Free-Buming Argon Arc", J. Phys. D: Appl. Phys. 1988, vol. 21. pp 418425.

7- A. J. D. F m e r and G. N. Haddad, "Rayleigh Scatterin; Measurements in a Free-

Buming &son Arc", J. Phys. D: Appi. Phys. 1988, vol. 2 1. pp. 426-43 1.

8- 1. McKeliiget and J. Szekely, "Heat Transfer and Huid Flow in the Weldinz Arc",

Metalliirgicul Transactions A. Juiy 1986. vol. 17A. pp. 1 139- 1 148.

Appendix I I I q l file, example 189

In this appendix, as an example, the q l fiie for the argon shieided 100 A arc

of 2.0 mm length in the GTAW process with the flat tip electrode, is presented.

*** QI File for GTAW arc with f l a t electrode TALK=F; RUN ( I,1) ; VDU=VGFMOUSE

* C o n s t a n t values R E A L ( PI) ; P I = 3 . 1 4 1 5 9 2 6 5 4 R E A L ( T O l , T 0 6 , T 0 3 ) + T01=1000;T02=3000;T03=10900 R E A C ( H O I , H O 2 , HO31 + H 0 1 = 3 . 6 5 2 4 E + 0 5 ; H 0 2 = 1 . - 1 0 5 9 E + 0 6 ; H O 3 = 5 . 9 6 6 3 E + 9 6 REAL (HMIN, HMF.X) + HMIN=HOl;HMAX=8.Z946E+07

* Applied c u r r e n t , A REAL (ACUR) ;ACUR=100. REAL (CCRD) ; CCRD=6.5 E + 0 7

** D i m e n s i o n of calculation domain, m + Radial direction

REAL ( PAHNA , ELSHOA , CFSHOA + PAHNA=l.SE-2;ELSHOA=l15E-3 + CASHOA=A.CUR/ ( PItCCRD 1 ; CASHOA=SQRT (CASHOAI

* Axial direction REAL (GHAD , ELGHF.D, GHOS 1 + ELGHAD=3.1E-3;GHOS=220E-3;GHAD=ELGffAD+GHOS REAL ( CATLAY, CATFAL, ANOLAY + CATLAY=l.E-4;CATFOL=f.E-3;.4NOLAAi=I.E-4

+* Grid n u n b e r s Radial direct ion

INTEGER ( M Y , MYE, MYC1 + Mtf=50;MYE=15;MYC=9

* Axial direction TNTEGER (MZ, MZE) + M2=17;M?E=7

GROUP 1. R u n title a n d other prelirninaries TEXT(GTAW, Ar, F . E . l . S i 4 , 2 . 0 , 1 0 0 . ( X )

GROUP S. Transience; time-step specification STEADY =T

GROUP 3 . X-direction grid specification CARTES = F 14X= 1

GROUP 4 . Y ( Radial 1 -direction grid speci f ication

Appendk III q l file, example 190

NY =MY SUBGRD(Y, 1, W C , CASHOA, 1.) SUBGRD(Y, MYC+l, MYE, ELSHOA-CASHOA, 1.) SUBGRD(Y, MYE+1, MY, PAHIJA-ELSHOA, 1.2)

GROUP 5 . ?(Axial)-direction g r i d specification NZ=MZ SUBGRD(Z, 1, MZE, ELGHAD, - 2 . ) SUBGRD(Z, MZE+l, MZE+3, S.*CATLAY, 1.1 SUBGRD(Z, MZE+4, MZE+7, CATFAL-S.*CATLAY, 1-15] SUBGRD(Z, MZE+8, MZ-1, GHOS-(CATFAL+2.*ANOLAY),l.) SUBGRD ( 2 , MZ, MZ, 2 . *P.EIOLAY, 1.1

GROUP 7. Variables stored, solved t named SOLUTN(Hl,Y,Y,Y,N,El,N) ;NAME(Hl)=EElTA " use harmonic averaging for POTA

SOLUTN(Cl,Y,Y,Y,N,M,Y) ;NAME(Cl) =POTA SOLUTN(PI,Y,Y,Y,N,N,N) SOLUTN(Vl,Y,Y,N,N,N,N) ;NAME(Vl) =RVEL SOLUTN(Wl,Y,Y,N,N,N,N) ;NAME('EJI)=AVEL

STORE(RHOl,TMPl,ENUL) STORE(CSfC3,C4) + NAME (CS) =RCR;NAME (C3) =RCZ;NAME ( C d ) =BTA STORE(CS8C6,C7,C8) + NAME(C5) =COEiQ;NFME(C61 =ELEQ;NAME(C7) =RADQ;EIFME(c8) =TOTQ STORE ( C g 1 + EIAME(C9) =NOOR

STORE(CIO,C11,ClS,C13,C14,C1S) + NAME ( C I O ) =DVIS;NAME (Cl1 1 =CP;NAME(C12) =EC + NPME(c13) =THDF;N.~ME(C~~) =THCO;NAME(C~~) =ELDF STORE(C25,C30) + NAME(c~~)=SHER;NFJIE(C~O)=ACD STORE(C21,C22) + N A M E ( C ~ I ) = E M F Y ; N M E ( C ~ ~ ) = E ~ ~ F Z

GROUP 8.Terms (in differential equations) & devices DENPCO=T DIFCUT=O.O TERMS(Pl,Y,Y,Y,N,Y,Y) TERMS(RVEL,Y,Y,Y,N,Y,Y) ;TERMSIAVEL,Y,Y,Y,N,YIY) TERMÇ(ENTA,N,Y,Y,N,Y,Y);TERMS(POTA,N,N,Y,N,N,N)

GROUP 9. Properties of the medium TMPlzGRND; RHOl=GRND; ENULzGRM PRNDTL ( ENTA) =-GRND; PRNDTL ( POTAI =-GRND

GROUP 11.Initialization of variable or porosity fields * * Blocking the electrode domain

CONPOR(ELECT,ZERO,CELL, l,NX,l,MYE, 1,MSE) * * Initial values

Appendix III q l file, example 191

IEIIADD=F FI INIT ( Pl ) =ZERO; FI INIT (RVEL) =ZERO; FI IMIT (AVEL) =ZERO FIINIT (TMP1) =TO3; F I IMIT (ENTA) =HO3 ; FI INIT ( POTA) = Z E R O

GROUP 13. Boundary conditions and special sources * * * AB Boundary - Electrode current input

PATCH(CLWl,LXALL,1,NX,l,MYC,MZE+1,MZE+l,l,I) COVAL (CLW1, ENTA, l.OE+O3, HO21 COVAL (CLW1, RVEL, l.OE+O3, ZERO)

PATCHtCATSA,LOW,l,NXI~8MYC,MZE+1,MZE+l,l,l~ COVAL I CATSA , POTA, FI XFLU , GRND 1 C0VP.L (CATSA, ENTA, FIXFLU, G R N D )

* * * BC Boundary - Electrode surface P~.TCH(CLW2,LWALL,1,~IX,M'~C+l,MYElM~E+l,MZE+l,~,~) COVAL(CLW2, EMTA, 1 .OEiO3 ,HO21 COVAL (CLWS , RVEL, 1.OE+O3, ZERO)

* * * CD Boundary - Electrode side wall PATCH(CSW,SWALL,I,NX,MYE+l,FlYE+l,l,MZE,l,I) COVAL(CSW, ENTA, 1 . O E + O 3 , HO21 COVAL (CSK, A V E L , l.OE+r)3, ZERO)

* * * DE Boundary - Top inflow PATCH(TOP,LOW, 1,NX,M'fE+18b!Y, 1,1,1,1) COVAL (TOP, Pl, 1. E+O3* FIXP, ZERO) COVAL (TOP, EMTA, ONLYMS, HO 1 1 COVAL(TOP,AVEL, 1.0,2-0)

*** EF Boundary - Side inflow/outflow PATCH(SIDE,NORTH, 1,[JX,Mff,MY8 1,MZ, 1,I) COVAL(SIDE,Pl,l.E+03*FIXP,7ERO) COVAL(SLDE,ENTA,ONLYMS,HOl)

* * * FG Boundary - Xeld pool (anode) PATCH ( APIBG, HWALL, 1, NX, 1, MY, MZ MZ, 1,l) COVAL(ANBG, POTA, 1.OE+03, ZERO) COVAL (ANBG, RVEL, 1 . O E + O 3 , ZERO)

PATCH(ANH,HICH,1,EIX,l,tIY,bI3,ML,l,l) COVAL (A.I\IH, ENTA, FIXFLU, GRbID)

* * * HA Boundary - Symrnetry axis phoenics default condition is symmetry conditions

* * SOURCE TERPliS * * RVEL and AVEL sources due to Lorentz Force

PATCH(STERM,VOLWE,l,NX,1,MY,l,MZ,I,l) COVAL(STERM,RVEL,FIXFLU,GRND);COVAL(STERM,AVEL,FIXFLU,GRND)

** RVEL viscous source term = -2.*emu*v/r"2 PATCH (SVlVIS, VOLUME, 1, EIX, 1 ,MY-1,1, MZ8 1,l) COVAL(SVIVIS,RVEL,CRNDISERO)

* * RVEL viscous stress terms PATCH(WISTRS,VOLUME8 1, NX, 1,MY-1, l,M7,1,1) COVAL ( W I S T R 2 , RVEL, FTIXFLU,GRND)

Appendix III ql file, example 192

** AVEL viscous stress terms PATCH(r~ISTRS,VOLIIME,l,EIX11,W~,11M7-L,l,I) COVAL ('ilrNISTRL, AVEL, FI:CFLU, GREID)

** ENTA. radiation sink PATCH(HRADIAT,VOLWE,l,FIX, L,M'i,l,MZ,l,l) COVAL ( HRADIAT , ENTA., GP.ND , ZERO 1

* * ENTA joule-heating source PATCH(HJOULE,'/OLUME, l,NX, l,MY,I,MZ,l, 1) COVAL (HJOULE, ENTA, FIXFLU, GRPJD)

** ENTA source due to electron drift PATCH ( ELECOENS , SOUTH, 1, EIX, 2, MY, 1, MZ ,1, 1) COVAL ( ELECOEIV S , ENTA, GRND , GRND) PATCH(ELECONVN,NORTH, l,NX, LIMY-1,1,M2,T, 1 ) COVAL ( ELECOEMV, EEITA, GRND, GRND) PATCH (ELECOETVL, LOV, 1, NX, 1, MY, 2 , M7,1,1) COVAL ( ELECOENL, ENTA, SREID, GRND P*TCH(ELECOEF/H,HIGH,l,EIX,1,MY,Lt~3-1,1,1) COVAL ( ELECOWH, EbITA, GRND, GREID 1

GROUP 15. Termination of sweeps LSXEEP=2500; LITER( POTA) =2S; LITER (ENT-5 RESREF(PI)=l.E-8;RESREF(RVEL)=1.€-8;RESREF(AVEL)=l.E-Q EESREF(ENTA)=l.E-2;REÇREF(POTA)=l.E-5 OVRRLX=1.7

GROUP 17. Under-relaxation devices PATCH1RELXT, PHASEM, l,l~iXI IIMYI I,MZ, Ill) CO71P.L (RELXT, RVEL, 2. E + 0 3 , SAME) COVAL (RELXT,Ai/EL, 2 . E+03 ,SAME) COVAL ( R E L X T , POTA, 2. E t O L , SAME)

RELAX(TMP1, LINRLX, 0.01) RELAX ( R H O I , LINkLX, O. 3 )

GEOUP 18. Limits on variables or increments to them VARM 1 N ( ENTA ) = KbI 1 El ; ' JAM4X ( EEITA 1 = HKaX

GROUP 19. Data communicated by satellite to GROUND RG ( 1 ) =PI ; RG ( 2 ) =ACUR; F.C ( 3 ) =APIOLAY RG ( 4 1 =PAHNA; RG ( 5 } =ELSHOF.; RG ( 6 1 =CASHOA RG(7)=GHAD;RG(8)=ELGHAD;RG(9)=GHOS RG(1O) =TOl;RG(11)=TOî;RG(12)=T03 RG(l3)=HOl;RG(14)=H02;RG(1s)=H03 RG(l6) =CfMIN;RG(17) =HMAX;RG(18) =CCRD

GROUP 20. Prel irninary print-out ECHO=F

GROUP 22. Spot value print-out

Appendix III ql file, example 193

GROUP 2 4 . Durnps f o r restarts STOP

Appendix IV ground-f file, example 194

G R O ~ D . F FILE

In this appendix, as an example, the ground-f file for the argon shielded 100

A arc of 2.0 mm length in the GTAW process with the flat ûp electrode, is presented.

c FILE NAME GROUND.FTN-------------------------------- 011093

SUBROUTINE GROUND INCLUDE 'lpS/d-includ/satear' INCLUDE 'lp2/d-includ/grdloc' INCLUDE 'lpS/d-includ/grdearl INCLUDE ' Lp2/d_includ/grdbfc '

C EQUIVALENCE ( 17, IZSTEP) CXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX USER SECTION STARTS: C

PARAMETER (NLG=20, NIG=20, NRG=100, NCG=10) PARAMETER(MZ=17,MY=5OII0l=1,J1=5OIJ2=28)

C DATA BK,ECH/1.38E-23,1.602E-I9/ COMMON/LGRND/LG(NLG)/IGRND/IG(NIG)/RGRND/RG(NRG)/CGRI4D/CG(NCG)

TXL=IABS ( I X L ) IF(IGR.YQ.1) GO TO 1 IF(IGR.EQ.9) GO TO 9 IF(IGR.EQ.13) GO TO 1 3 IF(IGR.EQ.19) GO TO 19 RETURN

C************t****t*********t******t******************************

c--- GROUP 1. Run title and other preliminaries C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

1 GO TO (1001,1002),ISC 1001 CONTINUE

C CALL MAKE ( DYG2 Dl CALL MAKE (DYV2 D 1 CALL MAKE ( R G S D ) CALL MAKE(RV2D)

C

Appendix IV gr0und.f file, example 195

IF (IGR. EQ. 1 .AND. ISC. EQ. 1 .AND. ,MOT. NULLPR THEN CALL h'RYT4O('GROUND file is GROUND-FTN of: 111093 ' 1

END1 F C

RETWRli 1002 CONTINUE

RETURN C********************************1t******t***********t************ c- - - GROUP 9. Properties of the medium (or media) C********+**************t**************************************t**

9 CONTINUE

C For TMP1 .LE.GE?ND--------- phase-1 temperature Index TEMPl IF(ISC.EQ. 101 THEN

LOHl =LOF (Hl 1 LOTMP=LOF (TEMP1) DO 60000 IY=l,MY

ENT=F (LOHl+IY 1 CALL DAMA(ENT,TEM, 101, JI) F(LOTMP+IY) =TEM

60000 CONTINUE RETURN

C * ------------------- SECTION 1 ---------------- C For RHO1.LE.GRND--- density for phase 1 Index DEN1

ELSE IF(ISC.EQ.1) THEN LOTMP=LOF (TEMPI LODEN=LOF(DENl) DO 70000 IY=l,MY

TEM=F(LOTMP+IY) CALL RHOAR(TEM,DENS,IOI,fl)


F ( LODEN+ IY) = D E W 30000 CONTINUE

RETURN C * ------------------- SECTION 6 --------------------------- C F o r EIJUL.LE.GRI.ID--- reference L a r n i n a r kinematic v i s c o s i t y C Index ' J ISL

ELSE IF(ISC.EQ.6) THEN LOTMP=LOF ( T E M P 1 ) LODEN=LOF ( DENl ) L O V I S L = L O F ( ' J I S L ) LODVIS=LOF ( C I O ) DO 90000 I Y = l , M Y

TEM=F(LOTMP+IY) CALL VISAR(TEM,VISVlIOl,J1) F ( L O D V I S + I Y ) = V I S V F(LOVISL+TY)=VISV/F(LODEN+IY)

80000 CONTINUE RETURI4

C * ------------------- S E C T I O N 7 --------------------------- C F o r PRNDTL ( 1 . L E . GRND--- l a r n i n a r PRANDTL nos., o r diffusivity C Index LAMPR

ELSE f F ( I S C . E G . 7 ) THEN LOTMP=LOF ( T E M P 1 ) LODEN=LOF ( DEI41 1 LODIF=LOF (LAMPR) IF(INDVAR.EQ.Hl1 THEN

LOC14=LOF ( C l 4 LOCP =LOF ( C l 1 ) LOC13=LOF (Cl3 1

CO 9 5 0 0 0 I ' f = I , MY TEM=F f LOTMP+IY) CALL TCAR (TEt4, TC'J, 101, JI CALL CPAR (TEM, CP'/, 1 0 1 , JI i F(LOCP+X'O =CPV F (LOC14+IY) =TCV F (LODI F+ IY) =TCV/ ( C PVtF ( LODEN+ IY 1 F(LOC13+IYl =F ( L O D I F + I Y )

85000 CONTINUE RETURN

END 1 F IF(INDVAR.EQ.Cl) THEN

LOEC=LOF ( C l Z ) LODPOT=LOF(C25) DO 88000 1 Y = l , M Y

TEM=F ( LOTMPt [Y) CALL ECAR(TEM, ECV, 101, JI1 F ( L O E C + I Y ) =ECV F(LODPOT+IY) =F(LOEC+IY) /F(LODEN+IY) F ( LODIF+ IY 1 = F ( L O D P O T + I Y

8 8 0 0 0 CONTIIJUE R ETUR1.I

Appendk IV gr0und.f file, example 1 97

ENDI F RETURN

ENDIF RETURN

C**************tt*t*****t**tt*t******ft***t******************t****

c--- GRGUP 13 . Boundary conditions and special sources

C i n d e x for Coefficient - CO

C Index f o r V a l u e - :/AL 13 COI-ITIIJUE

I F ( I S C . E Q . 1 ) THEN c------------------- SECTION 1 ------------- coefficient = G R N D

I F ( I N D V A R . E Q . V l 1 THEN IF(NPATCH.EQ. 'SVIVIS') THEN

LORG=LOF ( RG2D) LOCO=LOF ( C O )

LOVTSC=LOF ( C I O [jr, 13011 I T i = I Y F , I Y L '!IS=F ! LO1l ISC+ IY 1 F(LOCO+T' f ) = 2 . *'!IS, ( F ( L d R G + I ' i ) ' * Z + l . E-:O1

1 3 0 1 1 CONTI!>IUE END I F

ELSE I F ( 1 N D V A R . E Q . H l ) THEIi IF ( NPhTCH . EQ . ' HRADIAT' 1 THEN

LOTMP=LOF ( T E l 4 P l ) L O H l = L O F ( H l 1 LOCO=LOF ( C O )

LORr?D=tOF ( C 9 1

DO 1 3 0 1 2 I Y = I P F , IYL TEM=F (LùTMP+IY 1 CALL RADAh(TEM,AFUD, I O 1 , J Z ) F (LOC(>+ I ' f ) f F ( L O H l +Il i +1. E - 2 0 F I LÙRAD+ 1';) =ARAD

13012 CONTTIJUE ELSE 1 F (NPATCH. EQ. ' E L E C O W ' 1 TWEN

COEFH=2.StBKIECH LORCR=LOF ( C 2

LOCP=LOF ( C l 1 1 LOCO=LOF ( C O ) DO 13013 I Y = I Y F , IYL

F (LOCO+ 1'1') =COEFHfAMAXl I - F (LORCR+ I'i 1 , O . O 1 / 1 ( F ( L O C P + I Y ) + 1 . E - 2 0 )

1 3 0 1 3 COtU' I IJUE ELSE I F ( IJPATCH . EQ . ' ELECONVIJ ' 1 THEN

COEFH=S.5*BK/ECH LORCR=LOF ( C S LOCO=LOF ( C O )

LOCP=LOF ( C l 1 1 DO 1 3 0 1 4 I Y = I Y F , I Y L

F(LOCO+IY)=COEFH*AMAXl(FILORCR+TY+I),~I / ( F I L O C P + I Y ) + 1 . E - 2 0 )

CONTINUE


ELSE IF (IJPATCH . EQ. ' ELECONVL ' 1 THEN COEFH=2.SfBKiECH LORCZ=LOF(C3) LOCO=LOF (CO) LOCP=LOF (Cl 1 ) DO 13015 IY=IYF, IYL

F (LOCû+ IY) =COEFHtAMAXl ( -F (LORCZ+ I'i) , O . O 1 ;' (F(L0CPtIY) t1.E-20)

1 3 0 2 5 CONTI LIU E ELSE 1 F (NPATCI-1. EQ . ' ELECONVH ' THE14

COEFH=î.5'BK/ECH LORCZH=LOF ( M I G H ( C 3 1 1 LOCO=LOF (CO)

LOCP=LOF(Cll) DO 13010 IY=IYF,IYL

F ( LOCOc IY) =COEFH*AMAXI ( F (LORCZH+ IY 1 , 13 - 0 1 /'

L (F(LOCP+IY) + l . E - 2 0 )

IF ( IIeIDVAR. EQ . H l THEN 1F!NPATCH.EQ.'HJOULE8) THEN IF(ISWEEP.EQ.1) THEN

CALL FIJI ( ' JAL , O . O 1 ELSE

LORcR=LOF ( C 2 1 L O R C I = L O F ( C 3 LODVY=LUF(DYVSD) IF(I7.EQ.MI) THEM

LORCZH=LORCZ ELSE

LORCZH=LOF (HIGH (C3 1 END1 F LOEC=LOF(CZz) LOVAL=LOF (VAL DO 13121 IY=I1fF,MY


fF(IY.EQ.MYI THEN GRCRP=F (LORCR+IY)

ELSE GRCRP=(F(LORCR+IYI *FARQl(IY) +F (LORCh+I ' f+ l l "FARQL (Iï) 1

/F(LODVY+IY) END 1 F GRC7P=(F(LORC7tIY)*FAZQl(IZ)+F(LORCZH+I'~)*FAZQZ(IZ) 1

/ DZ IF(F(LOEC+IY).LT.l.E-10) THEN F ( LOVAL+ IY I =O. O

E L S E F ( LOVAL+IY) = (GRCRP*GRCRP+GRCZP*GRC3F) ,IF ( LOEC+ I ' i 1

END 1 F 1 3 1 2 1 COEIT 1 EJU E

ENDIF E L S E 1 F (EIPATCH. E Q . ' ELECOENS ' 1 THEEI

CALL FM0 (VF.L, SOUTH ( H l 1 ELSE I F (NPATCH. EQ. ' ELECONVEI ' t THEEI CALL FNO ( VAL, NORTH ( H l 1

ELSE 1 F (NP.A.TCH. EQ. ' E L E C O W L ' 1 THEN CALL FNO ( :;AL, LOX ( H 1 J

ELSE 1 F (NPATCH. EQ. ' ELECONVH' I THEN CALL FNO ( '/>.LI HIGH ( H I 1 1

E L S E IF (PIFATCH. EQ. 'ANHt 1 THEN LOVAL=LOF (VAL) LOCONQ=LOF ( C S 1 LOhCZ=LOF ( C 3 1 DO I 1 r = I Y F , I Y L

CLJEDEN=-F ( L : J R C Z c I Y 1 ELE=CURDEN*Z. E-O6*.I. 3 F ( LO1;AL+ I'i 1 =- ( ( F I LOCONQ+ IY) +ELE I ) ' 1 . E+O 6

ENDGO E t J D I F

ENDIF

Appendk IV gr0und.f file, example 200

TIEL=(F(LOV1+IY)+FiLOVI+IY-1))/2. E N D 1 F A3 =PMAXl ( V E L , O . O 1 A4=(5.7633E-5*0.48672*A3/F(LORC+IY) l t t ( 1 - 5 F ( L O C O N Q + I Y ) =(AlfASfA4* ( F ( L O H l + I Y ) -HO11 ) "1 . E - O 6

1 3 1 2 2 CONT T NUE END 1 F

C

I F (IJPATCH. EQ. ' S T E R M ' THE14 IF(ISWEEP.LE. L O O ) THEN

CALL FtJl ( ' J A L , O. O 1 E L S E

I F ( I I J D V A R . E Q . V l ) THEN LORCZ=LOF ( C 3 1 LOBTA=LOF ( C d )

L 0 E M F q i = L O F ( C 2 1 )

LOVAL=LOF ( ' /AL) 1FIII.EQ.MZ) THEN

LORCZH=LORCZ ELSE

L O R C X = L O F ( H I G H ( C ~ ) 1

ENDIF ENDLF

E N D I F


T E R M I = ( PAIN-BALA) / D Z IF(IY.EQ.1) THEN

SHEEB ( I'i) = F ( LOVliIY) / F ( LODVY+IY)

END 1 F IF(IY.EQ.MY) THEN

' J IÇCH=F ( LODVIS+IY) SHIBCH=SHEEB ( IY

E L S E VISCH=(F(LODVIS+IY)~FARQ~(IY+~)+F(LODVIS+IY+~~*

FARQl(1Y) /F(LODGY+IY)


S H I B C H = ( S H E E B ( I'{+l) *FP.RQl (IY) + S H E E B ( I ' I ) ' F .4hQ2 ( I Y + l ) 1 / F ( L O D G Y + I Y )

EldG I F I F ( f ' f . E Q . 1 ) T H E N

' I I S P A = F I L O D V I S + f Y SHIBRA=O . O

ELSE T~ISRA=(F(LODVIS+IY)*FARQ1 ( I Y - I ) + F ( L O D ~ f I S + ( I Y - 1 ) *

FARQ2 ( IY) 1 / F ( L O D G Y + I T f - 1 ) SHIMRA=(SHEEB(I'~)+FARQ~(IY-~)+SHEEB(IY-I)*€P.RQ~ ( I Y ) 1

/ F ( L O D G Y + I Y - 1 1 END I F CHAP=VISCH*ÇHIBCH*F(LORV+IY) I F ( I Y . E Q . 1 ) T H E N W.ST=O. O

E L S E F-A.ST=VISRA*SHIBRA*F (LORTf+I ' f-I 1

EPJD 1 F TEFMS=(CHAP-RAST) / ( F ( L O R G + I Y ) + F ( L O D ~ / ' f + I Y ) 1 F ( LOVAL+IY =TERMI+TERM~

CONTIbIUE EEID I F

E N D I F

I F (NPATCH. E Q . ' h V I S T R 2 ' ) THEN L O D T f I S = L O F ( C I O 1 L O D V I S L = L O F (LOW ( C l 0 1 L O D V I S H = L O F ( H I G H ( C I O ) 1 LOI11 =LOF i '11 LOV1L=LOF(LOW(VZ) L d W 1 = L O F ( h l 1 LO'rilL=LOF(LO'N'(:N'1) LOR1/=Li3F ( R 7 / S G ) L O R G = L O F i RGSD) LODVY=Li lF ( DYT/ZD1 LODGY=LOF ( G Y G S D ) LOVAL=LijF ( ' J A L ) DO 1 3 1 2 7 I q f = 1 Y F , M Y

I F ( I Z . E Q . 1 ) T H E N SHEE", (IY, 17) =0.O

E L S E Ç H E E z ( I Y , TZ) = ( F ( L O W l + I Y ) - F ( L O W 1 L + ( I 7 f ) ) !DZ

EEID 1 F I F ( I 3 . E Q . l ) THEN

V I S B = F ( L O D V I S + I Y ) ELSE

V I S B = I F ( L O D V I S L + I Y ) F A b Q 2 ( I Z ) + F ( L O D V I S + ( I Y 1 1 * F A Z Q l ( 12-1 ) 1 / D Z G L

E N D I F I F ( I Z . E Q . M 7 ) THEN

V I S P = F ( L O D V I S c I Y 1


Appertdix IV ground-f file, example 204

E N D I F CHAP=V1SCH+SHIRCH*F(LORV+I7f) IF(IY.EQ.1) T H E N R4ST=O. O ELSE RAST=VISRA*SHIRRA*F(LORV+IY-1)

ENDIF TERMî=(CHAP-RAST) / (F(LODVY+IY) *F(LORG+I1f 1 1

F(LOVAL+IY)=TERMl+TERK 13127 COI,JT 1 EIU E

ENDI F ENDIF

C RETURN

C*****~***t******"******t*t"**"****ttt************t*************

c--- GROUP 19. Special c a l l s to GROUND €rom EARTH 19 GO TO (191,19S,193,194,195,196,297,198,199,1910),1SC

1 9 1 CONTINUE c * ------------------- SECTION 1 ---- Sta r t of tirne s tep .

CALL GETZ ( DIGN7, DG7 , MZ CALL GETZ i DZ:2;147, DWZ , MZ 1 CALL GETZ ( 3 G W , ZGZ, MZ CALL GETZ ( 2hT17, ZWZ , MZ

C LORV =LOF ( R1f2D1 LORG =LOF (RG2D) DO 19011 IY=l,MY

FARQ1 (IY) =F(LORV+IY) -F(LORG+IY) IF(IY.EQ.1) THEN

FARQ2(IY)=F(LORG+IY) ELSE

F A R Q ~ ( I Y ) = F ( L O R G + I Y ) - F ( L O R V + M ) END 1 F

19011 CONT I NUE C

DO 19012 IZ=l,MZ FAZQl(I7) =ZWZ(IZ) -ZGZ(IZ) IF(IZ.EQ.1) THEN

FASQZ ( 1 1 ) =ZGZ(I?) E L S E

FAZQî(1I) =ZGZ(IZ) -ZWZ(Iz-l) ENDIF

19012 CONTINUE C

RETURN 192 CONTINUE

C " ------------------- SECTION 2 ---- Start of s w e e p .

RETURN 193 CONTINUE

c c ------------------- SECTION 3 ---- Star t of iz slab. RETURN


1 9 4 CONTINUE c " ------------------- S E C T I O N 4 ---- S t a r t of iterations over slab.

RETURM 199 CONTINUE

c e ------------------- S E C T I O N 9 ---- S t a r t o f solution sequence for

C a variable RETURN

1 9 1 0 CONTINUE c ------------------- S E C T I O N IO---- Finish of solution sequence for

C a variable RETURN

1 9 5 CONTINUE c * ------------------- SECTION 5 ---- Finish of itsrations over slab.

RETURN 196 CONTINUE

c * -------a----------- SECTION 6 ---- Finish of iz slab.

IF(INDVAR.EQ.9) THEN LORC"LOF(C3 1 L O R C Z H = L O F ( H I G H ( C 3 ) 1 LORCR=LOF ( C S 1 LOBTA=LOF i C 4 1 LO D W = LO F ( DYVS D 1 LODGY=LOF ( D Y C Z D ) LORG=LOF (RCZD) LOPOT = L O F ( C l I L O P O T L = L O F ( L O W ( C 1 ) LDEC =LOF ( C l 2 1 LOECL =LOF (LOW(C12) 1 LOTMP=LOF TLMPl

c calculate axial current density & self-induced magnetic field BT=iI. i> DO 1 9 0 6 1 I ï = I Y F , M Y

I F ( I Z . E Q . 1 ) THEN GFCFJl=F ( LOEC+ IY 1 SHI P=O . ù

ELSE 1FrIZ.EQ.MZ) THEN GEGAA=F ( LOEC+ IY 1 S H I P = - F ( L O P O T + I Y ) /ANOLAY

ELSE IF(IZ.NE.1.AND.IZ.NE.MI) THEN GRCAM= ( F ( L O E C L + I Y 1 * FA7Q2 ( 17.1 +

F ( LOECtIY) *FAZQl( I Z - 1 ) 1 /D7GL SHIP=(F(LOPOT+IY)-F(LOPOTL+lY))/D%GL

ENDIF GRCIN=-CRGAM*SHI P IF(F(LOTMPcIY).LE.4000.) GRCZN=O.O IF( IZ. L T . MIE+^) . A N D . I Y . L T . (MYE+I) 1 GRCZI\I=O .O I F ( 1 2 . E Q . M Z E + l .AND. I Y . LE-MYC) GRCZI?=-ACUR/ ( PI*CASHOA**S)

F (LORCC+ I Y ) =GRCZN I F ( 1 2 . E Q . M Z ) THEN

ZCUR=GRCZI.I ELSE

Appendk IV gr0und.f file, example 206

ZCUR= (GRCZN*FAZQl (IZ) + F (LORCZH+IY 1 *FAZQZ (IZ) 1 / G Z END I F BT=BT+ZCUR*F (LORG+IY) * F ( LODW+IY) F(LCBTA+Iqf =PI*4 .E-7*BT/F(LORG+IY)

19061 COEJT 1 NUE DO 19Q62 IY=l,MY

RADIO ( IY) = F (LORG+IY) FASY (IY) =F (LODVY+IY)

19062 CONTIbIüE IF(I7.EQ.MZ) THEN IF(ISXEEP.GE. 1 .OR. IS'VVEEP. EQ. LSKEEP) THEM

CURSUM=O.O DO 19903 IY=l,MY

AB=;. 'PI*F r LORCZ+IY) *FADIO( I'i *F?.Sqf ( 1.1') CURSülI=CURSüM+AB

19063 COEIT I IW E ENDIF

ENDI F c calculate radial current density c zero current on a x i s

F(LORCR+Z) =O. O DO 19054 IY=IYF+l, MY

GRCFJi= (F (LOEC+IY-1) *FARQ2 (IY) +

1 FILOEC+IY)*FARQl(IY-1) 1 ,'F(LC)DG'i+I'i-'1) GRCR4=-GRG;?M* (F(LOPOT+IY)-F(LOPOT+IY-1) 1 ;

1 F (LODGY+IY-1 i IF ( F (LOTMP+ IY) . LE. 4 0 0 GRCRM=O. O

IF(I~LT.(MZE+l).AND.IY.LT.(MYE+îI) GRCRl*I=F(LORCR+IY-1) F (LORCR+ ZY =GRCRM

19064 CONTIIWE ENDIF IF(ISWEEP.EQ.LSWEEP.PFbIDDI2.EQ.MZ) THEN

LOTMP=LOF (TEMP1) L O V 1 =LOF ( ' d l

L O D V I S = L O F ( C l 0 1 LORCZ=LOF ( C 3

LORG=LOF ( RG2D) LODVY=LOF ( D Y V 2 D ) LOSH=LOF ! C 2 S 1 LûACD=LOF ( C 3 O ) LOCONQ=LOF ( C 5 ) LOELEQ=LOF (Co 1 LORADQ=LOF (C7 ) LOTOTQ=LOF ( C 8 F A S S H A D - 3 G Z (M3 1 DO 19065 IY=I,MY TF(IY.EQ.1) THEN VELO=F(LOVl+IY) 1 '2 . ELSE V E L O = ( F ( L O V ~ + I Y ) + F ( L O V ~ + I Y - 1 1 ) / 2 -

END 1 F


V I S O = F ( LODVZS+I1i) TTVAL=F ( LOTMP+IqY) T F I L M = ~ T T V A L + l O O O . / 2 CUF.DEFI=-F ( LORCZ+IrY 1 F ( L O R A D Q + I Y ) = T A B E S H ( IY1 *1 . € - O 6 F(LOELEQ+IY)=CURDENt1.E-6t(443+34~03tBK

1 *TFILM/ ECH) F ( L G T O T Q + I ' ~ ) = F ( L O E L E Q + I Y ) + F ( L O C O N Q + I Y ) +F(LOP.ADQ+I ' l ) F(LOSH+IY)=VISO+VELO/FAS F ( L O A C D + I Y ) = - F ( L O R C ~ + T Y ) * ~ . E - O ~

19065 CONT 1 NUE ENDIF

C L O M D = L O F ( C 9 1 DO 19066 I ' f=L,MY AFTAB(TY, 1x1 =F(LOhAD+TY)

19066 CONTINUE c

ERTEFA ( 13 1 =GHAD-ZGZ IZ 1 FASZ ( 1 2 ) =D7

C RETURkI

1 9 7 CONT 1 NUE c * ------------------- SECTION 7 ---- Finish cf sweep.

DO 1 9 0 7 1 J Z Z = l , M S L O C l = L O F (ANv{= ( C l , J73)) LOTMP=LOF t MII'fZ ( TEPIIP1 , J 3 Z 1 1 DO 1 9 0 7 1 J Y 7 f = 1 , MY

I F ( F ( L O T M P + J 1 i Y ) .LE. . IOOO. 1 F ( L O C ~ + J Y Y ) = O . ( ) 1 9 0 7 1 CONTINUE

DAGH=O . O DO 19072 J32=1,MZ

LOTMP=LOF (AFIYI (TEMPI , J Z Z ) G T C L = F (LOTMP+ 1 ) DAGH=FJiL?..XI (GTCL DAGH

1 9 0 7 2 CONTINUE L O T M ~ L O F (ANYZ t T E M P L , ~3Ec2 GARM=F ( L O T M P c l )

C FAST=O . O LOV=LOF (ANYI ( V I , M7) 1 DO J Y Y = l , M Y

VEL=ABS ( F ( L O V + J Y Y 1 1 FAST=AEL4Xl( FAST, V E L 1

ENDDO

I F ( ISKEEP. EQ. L S X E E P ) THEbJ GWlMAX=O. 0 DO 1 9 0 7 3 J Z Z = L r M 7 LOWI=LOF(ANYZ(XI, J Z 7 )


GWlMF.X=AMAXI (GWCL, GWlt4AjO 19073 C O N T I W E C

LO P l = L o F (AIJYZ ( Pl, M X 1 C P A N O D = F ( L O P I + l ) L O P f = L O F ( A N Y 3 ( P l , M Z E + I ) GPCATH=F ( L O P 2 t l )

C LOPOT=L(lF (ANYZ (Cl ,MLE+11 1 L O E C = L O F ( A I ~ J Y I ( C l S , M L E + L ) DELPHI=-CATCD*DGZ (MZE+l) iF(LOEC+l) G V A R C = F ( L O P O T + l ) + D E L P H I CALL GETSOR('CATSA',HI,GHSORJ CALL GETSCR('CATSA',CI,CPSOR)

LOT=LiJF (.ANY? ( C a , M Y 1 LOR=LOF ( A I W Z ( C 7 , M 2 ) LOE=LOF (.AFJ'fT ( C 6 , MC 1 LOC=LOF ( AldYZ i C S , M 2 i 1

TOTHEAT=O . O

RADHEA r=O . O

ELEHEAC=O . O CONHEAT=O. O

DO 1 9 0 7 4 I Y Y = 1 , M Y TOTQT=F ( LOT+ IYY) *1. E6 hADQT=F ( L O R + I Y ' t ' 2 . E6 ELEQT=F ( L O E + I ' t Y ) *1. E 6 CONQT=F ( LOCc I Y Y 1 '1 . E 6 DSATH=S. *PI'RADIO ( IYY 1 *FF.SY (IYY) TOTHEAT=T(1'rHEAT+TciTQrï' DSATH RADHE3T=RADHEAT+RADQTfDSATH ELEHEF.T=ELEHEhT+ELEQT*DSATH COt\lHEAT=COt~IHEAT+CONQTf DSATH

29074 CONTINUE TOTPOWER=ACURfGVOLT E F F I C A B S (TOTHEAT*lOO/'TOTPOWER) RADPERC=RADHEATf IO0 /TOTHEAT ELEPERC=ELEWEAT*lOO/TOTHEAT COWPERC=CONHEAT'100 /TOTHEAT

CALL XRITSR ( ' TMAX ' , DAGH, ' TANOD ' , XANTM)


1 9 0 7 7 CONTINUE 1 9 0 7 6 CONTINUE

TABESH ( IYI i =T?.BID l 'AB1 D=O . O

1 9 9 7 5 CONTIIJUE c

RETURN 1 9 8 CONTI PJUE

C * ------------------- SECTION 8 ---- F i n i s h of time step.

SUBROUT II!IE ARGON DIMENSION RHOTPB ( 50 , VISTA0 ( 5 0 1 , TCTAB ( 5 0 DfMENSION ECT.A.BtSO) ,CPTAB(SO)

C L" - - - - - - - DENS I TY

DATA RHOTAB 1 j 9 . 7 3 5 3 E - 0 1 , 9 .7353E-(31 , 4 .8672E-01 , 3 . 2 4 5 0 E - 0 1 , 2.4338E-01,

2 1 . 9 4 7 1 E - 0 1 , 1 . 6 2 2 6 E - 0 1 , 1 -3908E-01 , 1 . 2 1 7 0 E - 0 1 , 1 . 0818E-01 ,

3 9 . 7 3 6 l E - 0 2 , 8 . 8 5 1 0 E - 0 2 , 8 .1133E-02 , 7 . 4 8 8 8 E - 0 2 , 6 . 9527E-02 ,

4 6 . 4865E-02 , 6 . 0 7 5 7 E - 0 2 , 5 .7083E-02 , 5 . 3 7 4 0 E - 0 2 , 5 .0639E-OS,

5 4 . 7 6 9 6 e - 0 2 , 4 . 4 8 3 8 E - 0 2 , 4 .S003E-02 , 3 . 9 1 4 5 E - 0 2 , 3 . 6243E-02 ,

6 3 . 3 3 1 0 E - 0 2 , 3 . 0 3 8 7 E - 0 2 , S .7556E-02 , 2 . 4 9 0 6 E - 0 2 , 2 . 2539E-02 ,

7 2 . 0510E-02 , 1.8133SE-OS, 1 .7477E-Of , 1 . 6 3 9 f E - 0 2 , 1 .5516E-OS,

8 1 . 4 7 9 5 E - 0 2 , 1 . 4 1 8 7 E - 0 2 , 1 . 3660E-02 , 1 . 3 2 0 i E - 0 2 , 1 . 2775E-02 ,

9 2 . 2 3 6 9 E - 0 2 , 1 . 1 9 7 7 E - 0 2 , 1 .1590E-02 , 1.12!11E-02, '1.0802E-OS.

+ 1 .039SE-02 , 9 . 9 6 9 3 E - 0 3 , 9 .5460E-03 , 9 . 1 1 0 3 E - 0 3 , 9 . 1103E-03 /

Appendix IV gr0und.f file, example 21 0

c - - - - - - - V I S C O S TTY DATA V I S T A B I / 3 - 4 S t J E - 0 5 , 3 .4224E-OS, 5 . 7 6 3 3 E - 0 5 , 7 - 3 9 6 0 E - 0 5 , 8 . 9 7 1 5 E - 0 5 , 2 1 .O5LlE-O4, 1 . 1 9 8 7 E - 0 4 , L . 3 3 8 9 E - 0 4 , 1 - 4 7 1 S E - 0 4 , 1 . 5 9 7 2 E - 0 4 , 3 1 . 7 1 6 9 E - 0 4 , 1 . 8 3 1 4 E - 0 4 , 1 . 9 4 1 5 E - 0 4 , 2 . 0 4 d l E - 0 4 , 2 . 1 5 2 7 E - O J , 4 2 . 2 5 S 9 e - 0 4 , 2 . 3 5 1 8 E - 0 4 , 2 . 4 4 8 0 E - 0 4 , S .540LE-04 , 2 . 6 2 5 2 E - 0 4 , 5 2 - 6 9 6 5 E - 0 4 , ï . 7 4 3 1 E - 0 4 , f . 7 4 8 9 E - 0 4 , 2 . 6 9 4 7 E - 0 4 , 2 . 5 0 4 3 E - 0 4 , 6 2 . 3 5 5 2 E - 0 4 , S . 0 8 1 1 E - 0 4 , L . 7 7 4 8 e - 0 4 , 1 . 4 7 1 5 E - 0 4 , 1 . S 0 3 8 E - 0 4 , 7 9 . 8 6 5 6 E - 0 5 , 8 . 2 2 5 8 E - 0 5 , 7 . 0 6 2 5 E - 0 5 , 6 . S d 3 8 E - 0 5 , 5 . 7 9 5 7 E - 0 5 , 3 5 . 5 1 7 7 E - 0 5 , 5 . 3 8 6 5 E - 0 5 , 5 . 3 5 3 7 E - 0 5 , 5 . 4 2 3 8 E - 0 5 , 5 . 4 7 7 9 E - 0 5 , 9 5 . 5 3 0 4 E - 0 5 , 5 . 5 5 3 7 E - 0 5 , 5 . S219E-05 , 5 . 4 1 4 0 E - 6 5 , 5 .2186E-OS, + 4.938LE-05, 4 . 5 8 9 6 E - 0 5 , 4.2296E-05, 3.825OE-95, 3.825OE-OS/

c - - - - - - - THERM3.L CONDUCTIVITY DATA TCTAB

1 /2 .6712E-OS, 2 .6712E-OS, 4 .4982E-OS, 5 . 7 Ï L 5 E - 0 2 , 7.00SSE-02, 2 8 .;O38E-OîI 9 . 3 5 6 1 E - 0 2 , 1 . 0 4 5 8 E - 0 1 , 1. Z682E-OLt 1 . 4 6 3 9 E - 0 1 , 3 2 . 8 1 0 5 E - 0 1 , 1 . 4 3 7 R E - 0 1 , 1 - 5 J 9 4 E - 0 1 , 1 .7 i l12E-01 , 1 . 3 1 9 8 E - 0 1 , 4 2 . 2 2 7 8 E - 0 1 , î . 6 5 5 5 E - 0 1 , 3 - 2 4 8 0 E - 0 1 , 4 .OSlSE-01 , 5 . 1 0 5 Z E - 0 1 , 5 6 . - l- l28E-OlI 8 . 0 7 5 9 E - 0 1 , 1 . 0 0 2 3 E + 0 0 , l . L ; l ~ E + O ~ l , L . 4 6 5 0 E + 0 0 , O l . 7239E+OO, 1 . 97S2E+00r 2 . 1 9 7 0 E + 0 0 , 2 . 3 6 3 3 E + 0 0 , S . 4 6 Z S E + 0 0 , 7 2 . 4 8 9 9 E + 0 0 , 2 . 4 6 7 1 E t d 0 , 2 . 4 2 6 2 E + 0 0 r 2 . 3 9 5 8 E + 0 0 , 2 . 3 9 2 0 E + 0 U t 3 2 . 4 1 9 7 E + 0 0 , 2 . 4 7 6 7 E + 0 0 , 2 .5582E+0O1 2 . 6 5 d 4 E + 0 0 , 2 . 7 7 4 0 E + 0 0 1 9 2 . 9 0 0 3 E + 0 0 , 3 . 0 3 5 0 E t 0 0 , 3 . 1 7 4 7 E + 0 0 , 3 .3L73E+0OI 3 . 4 6 1 0 E + 0 O I + 3 . 6 0 4 4 E + 0 0 , 3.74613E+d0, 3 .8895E100, 4 .0294E+O(j, -1.OS94E+OO/

c - - - - - - - SPECIFlC HE>-T

DATA clPT.4B 1 !5. 2 0 3 3 E ~ O 2 , 5 . 2 0 3 3 E + 0 2 , S . 2 0 4 1 E + 0 2 , 5 . 2 0 3 6 E + 0 2 , 5 . S 0 3 4 E + 0 2 , 2 5 .;033E+OL, 5 . 2 O 3 3 E + 0 î r 5 .2033E+OS, 5 . 2 0 3 3 E + 9 2 , 5 . S 0 3 4 E + 0 2 , 3 5 .2 (345E+02 , 5 . 2 0 9 5 E + O S I 5 . L S 7 6 E t 0 2 , 5 . 2 8 0 1 E + 0 2 , 5 . .CO9lE+OZ, 4 5.6859EtO2, 6. C133E+02, 7 . t 5 1 S E + O S I 8 .&'dSE+OS, 1 . 1 0 2 7 E + 0 3 , 5 1 . 4 5 7 7 E c 0 3 , 1. 3-169Ec03, 2 . 6 0 H 7 E + 0 3 , 3 . 3 8 5 4 E t 0 3 , 4 .3?4SE+U3, 6 5 . 7 4 3 0 E 4 0 3 , : . 0 6 2 5 E + 0 3 , Y - 3 3 3 6 E c 0 3 , 9.CRQHE+03, 9 . 7 1 1 8 E + O 3 , 7 9 . 4 4 4 7 E + 0 3 , 3 . 5 0 0 5 E + 0 3 , 7 . 3 1 1 9 E + 0 3 , 5 . 9 9 3 3 E t 0 3 , 4 . d 1 7 8 E + 0 3 , i3 3 . 8 8 4 6 E c 0 3 , 3 . 2 1 1 1 E + 0 3 , 2 . 7 7 7 7 E + 0 3 , .Z .3449E+03, 2 . S358E+O3, 9 2 . 7 1 1 6 E + 0 3 , 3 . i3901E+03, 3 . 6 8 3 7 E + O 3 , 4 .4994E+(33 , 5 . 5 2 7 5 E + ( 1 3 , + 6 . 7 3 0 9 € + 0 3 , ;3 .0393E+03 , 9 . 3 4 9 5 E c O 3 , 1 .0553E+0-1, 1 .0553E+C)4/

c - - - - - - - ELECTRliAL COI4DUCTf I T Y DATA ECT.3-B

1 / 3 . 0 7 8 4 E - 2 3 , 3 . 0 7 8 4 E - 2 3 , 3 . 9 2 1 6 E - 2 3 , 1. Ï575E-LB, 1 . 6 5 9 2 E - 1 1 , 2 2 . 3 9 5 6 E - 0 7 , 5 . 4 5 7 7 E - 0 5 , 2 . 9 0 7 3 E - 0 3 , 0 .2R85E-OS , 7 . 1 8 1 6 E - 0 1 , 3 5 . L 6 1 S E + 0 0 , 2 . 3 6 4 6 E + 0 l I 7 . 6 8 2 1 E + 0 1 , L ,8290E+O2, 3 . 5 8 0 8 E + 0 2 , 4 6 . 2 4 4 0 E + 0 S 1 9 . 9 0 3 4 E c 0 2 , 1 . 4 3 7 8 E i - 0 3 , 1 . 9 3 5 8 E + 0 3 , 2 . 4 5 8 4 E + 0 3 , 5 2 . 9 9 0 0 E + 0 3 , 3 . 5 S 3 1 E + 0 3 , 4 . 0 5 4 4 E + 0 3 , 4 . 5 8 2 S E + 0 3 , 5 . 1 0 4 9 E + 0 3 , 6 5 . 6 1 9 9 E + 0 3 , 6 . 1 2 4 3 E + 0 3 , 6 . 6 1 3 4 E + 0 3 , 7 . 0 8 4 4 E + 0 3 , 7 . 5 3 2 2 E + 0 3 , 7 7 . 9 5 6 1 E + 0 3 , 8 . 3 5 6 5 E + 0 3 , 8 . 7 3 6 1 E + 0 3 , 9 . 0 9 8 5 € + 0 3 , 9 . 4 4 7 0 E + 0 3 , 8 9 . 7 8 4 3 E + 0 3 , I . O l l l E + 0 4 , 1 . 0 4 2 8 E + 0 4 , 1 . 0 7 2 8 E + 0 4 , 1 . 1 0 1 4 E + 0 4 , 9 1 . 1 2 7 5 E + O 4 , 1 .15OSE+04, 1 . 1 6 8 7 E + 0 4 , 1 . 1 8 1 9 E + 0 4 , 1 . 1 8 9 3 E + 0 4 , + L . l 9 1 S E + 0 4 , 1 . 1 8 8 2 E + 0 4 , 1 . 1 8 1 3 E + 0 4 , 1 . 1 7 3 5 E + 0 4 , 1 . 1 7 3 5 E + 0 4 /

C

ENTRY RHOAP (TEM , DESV , 1 I JJ) CALL JOST(TEM,DESV,RHOTAB, 1, JI

Appendk IV gr0und.f file, example 21 1

RETURN C

ENTRY VTSAF(TEM,'JISV, II,JJ) CALL JOST(TEM,';IS'J,'IISTA.B, 1, JI RETrJP.N

C

ENTRY TCAR(TEM,TC'/, I I , JJ; CALL JOST ( TEM, TCV, TCTAB, T,J 1 RETURN

C

ENTRY CPAR (TEM, CPV, I I , 33) CALL JOST ( TEM, C PV , C PTAB, 1, J RETURN

C

EEITRY ECAh ( TEM, EC'J , 1 1, JJ CP-LL 30ST ( TEM, EC'I , ECTAB, 1, J RETUREI END

C * ~ * * + * + + * * t t + * + r * * * * * * t f ~ t t * * * * * * * * * X * t * X * * * * t * + * * * * * * * * * * * * * * *

SUBROUT IME JOST ( TEM, VALUE, YTAB? JBOT , JTOP DIMENSION YTAB(5O) ,TTAB(501

TEMPEFS-TURE DATA TTAB

1 / l .e-20, 5 0 0 , 1 0 0 0 . , 1 5 0 0 . , 2 0 0 0 . , 2 2 5 0 0 . , 3 ( 3 0 0 . , 3 5 0 0 . , -1000. . 4 5 0 0 . , 3 5 O O O . , 5 . O . , 6 5 0 0 . , ~ 0 0 0 . , 4 7 5 0 0 . , 0 0 , S S O f l . , 9 0 0 0 . , 9 5 0 0 . , 5 O , L O O . , Lli300. , 1 1 5 0 0 . , L Z O O O . , 6 1 2 5 0 0 . , 13000., 3 S O O 1 4 0 0 0 . , 1 4 5 0 O . , 7 1 5 0 0 0 . , 1 5 5 0 0 . , 16OùO., 1 6 5 0 0 . , 1 7 0 0 0 . , 8 1 7 5 0 0 . , L O O . 1 8 5 0 0 . , 1 9 0 0 0 . , 1 9 5 O O . , 9 ïC30(>0., 2 0 5 ~ > 0 . , l O O . , 2 1 5 0 0 . , 2 2 0 0 0 . , + 225OO. , O , 235OO. , î4OOO. , 1. e;O/

JBOT= 1 JTOP=LSO

1 0 J = ( J T O P + J B O T ) ) 2 IF(TEM.LT.TT>-B(J) 1 THEN

JTOP=J E L S E

JBOT=J END1 F fF(JT0P-JBOT.EiE. 1) GO TO 1 0 VALUE=YTAB(JBOT)+(YTAB(JTOP) - Y T A B ( J B O T ) ) *(TEM-T'TAB(JBOT1 1 /

L ( T T A B ( J T O P ) -TTAB ( JBOT) 1 RETURN END

C*****t**+*********t*r*********************~***************+****

SUBROUT I N E DARAJEH DIMENSION TThB(Sd1


DATA TTRB L / ÇOO., S U O . , lOOO., 1 5 0 0 . , 2 2 5 0 0 . , 3O('iO. , 3 S O O . , 4000.. 3 5 0 , 5 0 0 0 0 0 0 . ~ 0 5 0 0 . ~ 4 7 5 0 0 . , 8 0 0 0 . , i3500 . , 9 0 0 0 . , 5 1 0 0 0 0 . , 1 0 5 0 0 . , LlOOO., L 1 5 0 0 . , 6 1 2 5 0 0 . , 1 3 0 0 0 . , 1 3 5 0 0 . , 1 4 0 0 0 . , 7 1 5 0 0 0 . , 1 5 5 0 0 . , 1 6 0 0 0 . , 1 6 5 0 0 . , 3 1 7 5 0 0 . , 1 8 0 0 0 . , 1 8 5 0 0 , , 1 9 0 0 0 . , 9 2 0 0 0 0 . , Z 0 5 0 0 . , 2 1 0 0 0 . , 2 1 5 0 0 . , + 2 2 5 0 0 . , S 3 i i O O . , ; 3 5 0 0 . , 2 4 0 0 0 . ,

C

ENTRY DAMA (EEIT, TEM, I I , JJ) CALL GARD ( E N T , TEM, TTAB, 1 , JI RETUREJ END

SUBROUT 1 PIE EEITIlAL P'iI DIMENS TOP1 ENT.3.B ( 5 0

c-- - - - - - EFITHALPY

DATA EPiTAE3 L /1 .O499E+OS, 1 . 0 4 9 9 E + 0 5 , 3 . 0 5 2 4 E + 0 5 , 6 . î S 4 2 E + 0 5 , 8 .8559E+0Çr 2 1 . 1 4 5 d E c Ù 8 , 1 . 4 0 5 9 E + 0 6 , 1 . 6 6 6 1 E t 0 6 , 1 . 9 2 b 3 E + O 6 , î . 1 8 6 4 E + 0 6 , 3 ; . 4 4 6 a E t O S , 2 . 7 0 7 0 E + 0 6 , 2 . 4 6 7 9 E + 0 6 , 3 . 2 3 0 7 E + 0 6 , 3 . 4 9 8 1 E + 0 6 , 4 3 . 7 7 6 1 E c O 6 , 4 . 0 7 5 0 E + 0 6 , 4 . 4 1 î O E + 0 6 , 4 . 8 1 2 7 E + 0 6 , 5 . 3 1 4 1 E + O 6 , 5 5 - 9 6 0 3 E + i l b , o . d 3 2 3 E + 0 6 , 7 . 9 9 3 7 E + 0 6 , 9 . 5 3 Z 6 E + 0 6 , 1 . 1 5 5 8 E + 0 7 , 6 1 - 4 1 7 6 E + O 7 , L . 7 - 4 S l E t 0 7 , S . 1 3 4 1 E + 0 7 , 2 . 5 7 7 7 E t t I 7 , 3 . 0 5 7 5 E + 0 7 , 7 3 . 5 3 7 9 E + 0 7 , 3 . 9 8 5 6 E + 0 7 , 4 . 3 7 7 0 E + 0 7 , 4 . ; l ISSE+07 , 4 . 9 6 6 4 E + 0 7 , 8 5 - 17YLE+07, 5 . 3 5 2 3 E + 0 Ï 1 5 . 4 9 7 9 E + 0 7 , 5 . 6 2 7 3 E t 0 7 , 5 . 7 5 3 7 E + 0 7 , 9 5 . 8 8 4 9 E + 0 7 , 8 . 0 3 1 0 E + 0 7 , 6 . 2 0 2 5 E + 0 7 , O . J l O Z E t O 7 , 8 . 6 0 5 3 E + 0 7 , + 6 . 9 7 7 1 E + 0 7 , 7 . 3 5 2 7 E + 0 7 , 7 . 7 9 0 3 E t 0 7 , 1 . S 9 4 8 E + 0 i I â . 2 9 4 6 E + 0 7 /

(3

EE1TP.Y GARMP. ( TEM, ENT, 1 1, JJ 1 CALL SOAL ( TEM, ENT , Ei.JTA.8, 1 , 3 1 RETURN END

C*********tt****t*tt******t***f******************************t**

SUBROUT INE SOAL ( TEM , 'JALUE , ENTA6 , JBOT, JTOP 1

DATA TTAB 1 / 5 0 0 . , 5 0 0 . , 1 0 0 0 . , 1 5 0 0 . , 2 0 0 ù . , - L 2 5 0 0 . , 3 0 0 0 . , 3SOO., 4 0 0 0 . , 4 5 0 0 . , 3 5 0 0 0 . , 5 5 0 0 . , 6 0 0 0 . , 6 5 0 0 . , 7 0 0 0 . . 4 7 5 0 0 . , 8 0 0 0 . , 8 5 0 0 . , 9 0 0 0 . , 9 5 0 0 . , 5 1 0 0 0 0 . , 1 0 5 0 0 . , 1 1 0 0 0 . , 1 1 5 0 0 . , 1 2 0 0 0 . , 6 1 2 5 0 0 . , 1 3 0 0 0 . , 1 3 5 0 0 . , 1 4 0 0 0 . , 1 4 5 0 O . , 7 1 5 0 0 0 . , 1 5 5 0 0 . , 1 6 0 0 0 . , 1 6 5 0 0 . , 17000 . , 8 1 7 5 0 0 . , l8OOO., 1 8 5 0 0 . , 1 9 0 0 0 . , 1 9 5 0 0 . , 9 2 0 0 0 0 . . 2 0 5 0 0 . , î l O O O . , LlSOO., 2 2 0 0 0 . ,

+ 2 2 5 0 0 . , 2 3 0 ( ! 0 . , S 3 5 0 0 . , 2 4 0 0 0 . , 2 4 0 0 0 . / C


JTOP=50 JBOT= 1

1 0 0 J = ( J T O P + J B O T ) i S I F ( T E M . L T . T T . W I J ) ) THEN J T O P = J ELSE JBOT=J ENDI F IF(JTOP-JBOT.NE.1) GO TO 100 '/ALUE=ENT.A.B ( J B O T ) + ( EI.IT.4B ( J T O P ) -ENTAI3 ( JBOT i " ( TEM-TTAB ( J B O T )

1 I/(TTAB(JTOP)-TTXB(2BOT)i RETURN END

Ctt***t***+*t********t*Ct*t**t***tt*****************************

SUBROUTII4E GAiID(EIJT, ' J .ALUE,TT.~.B~ JBOT, J T O P ) D I M E N S I O N TTAB ( 50 1 , ENTA-6 ( 5 0 DF-TA EîJTAB

1 / 1 . 0 0 0 0 E - ~ ~ , 1 . 0 4 9 9 E + 0 S 1 3 . 6 5 2 4 € + 0 5 , 6 .2542E+\2SI 8 . 8 5 5 9 E + 0 5 , 2 l . l 4 S 8 E + O 6 , 1. J 0 5 9 E + 0 6 , 1 . 6 6 6 1 E c 0 6 , 1 . 3 2 6 3 E + n 6 , 2.2864E+06, 3 î . . 1 4 6 0 E + 0 6 , 2 . ÏO70E+O6, 2 .9679E+06, 3 . 2 3 0 Ï E t 0 6 , 3 . .1982E+06, 4 3 .7761E+0G1 - l .O750E+06, 4 .41SOE+06, 4 . 8 1 2 Ï E c 0 6 , S . 3 2 4 1 E + 0 6 , S 5 . 9 6 6 3 E + 0 6 , 0 . 3 3 2 3 E + 0 6 , ' 7 . 9 9 3 7 € + 0 6 , 9 . 5 3 2 6 E c 0 6 , 1 .1558E+07 , 6 I . 4 E o E + O 7 , L.74SlE+O7, 2 . 1 3 4 2 E + 0 7 , 2 . 5 7 7 7 E c 0 7 , 3 . 0 5 7 5 E + 0 7 , 7 3 . 5379E+Oi , 3 .9858EcO7 , 4 . 3 7 7 0 E + 0 7 , 4 . 7 0 S t 3 E t 0 7 , 4.9664E+r)7 , 8 5 . L ï 8 2 E + O I , 5 . 3 5 1 3 E + 0 7 , 5 . 4 9 7 9 E + r ) 7 , S . 6 2 7 3 E t 0 7 , 5 .7537E+(37, 9 S . YH4?E+07, 6 . 0 3 1 0 E + 0 7 , 6 .SOSSE+07, 6 . -4102Et07 , a . 6 6 5 3 E + 0 7 ,

+ 6 . 9 7 7 1 E t 0 7 , 7 . 3 5 2 7 E t 0 7 , 7 . 79(13E+07, 8.2946Et07, l.Oi)l)OE+20/ C

J T O P = 5 0 JBOT= 1

100 J= (JTOPtJBOT) , S

KF(EI\IT.Lr.ElJT.r?B(J) 1 THEIJ JTOP=J ELSE J B O T = J END 1 F IF(JT0P-JBOT.[JE. 1) GOTG L O O VALUE=TTAB ( JBOT 1 + ( T T A B ( J T O P -TTAB ( J B O T ' ( €147'- ENTAI3 ( JBOT 1

1 / ( E I \ I T A B ( J T O P ) -EI \ ITA.B(JBOT) )

RETURN END

IV

SUBROUTINE RADP-RGOIJ DIMENSION RADTAB ( S 8 )

C c - - - - - - - RADLbTIOI\I

DATA RADTAB 1 / O . O O e c ) O , O . OOeOO, 0 .(3Oe(30, 0 . 0 0 e 0 0 , 0 . 0 0 e 0 0 , O . OOe00, 0 . 0 0 e 0 0 , 2 0 . 0 0 e 0 0 , 8 . 0 0 e 0 6 , 1 . 8 0 e 0 7 , 4.50e07, 2 . S8e08, 3 . 7 0 e 0 8 , 8 . 8 0 e 0 8 , 3 2.26e09, 3 . 3 7 e 0 9 , 4 . 6 5 e 0 9 , 5.40e09, 5 . 5 3 e 0 9 , 5 . 9 1 e 0 9 , 6 . 5 4 ~ 2 0 9 ,

Appendix IV ground-f file, example 21 4

4 8.17e09, 1.01e10, 1.19e10, 1.26e10, 1.32e10, 1.35e10, l.E20/ C

ENTRY RADAR (TEM, RADV, I I , JJ) CALL JOSTRAD ( TEM , RADV , RADTAB , i , J 1 RETURN

C

END P

SUBROUTINE JOSTRAD(TEM,VALUE,YTAB,JBOT,JTOP) DIMENSION YTAB(28),TTAB(28)

c------ TEMPERATURE

DATA TTAB 1 /l.e-20, SOO., 1000., 2000., 3000., 4000., SOOO., 2 6000., 7000., 8000., 9000., lOOOO., 11000., 12000., 3 13000., 14000. , 15000., 16000-, 17000. , 18000., 19000,, 4 20000., 21000., 22000., 23000., 2 4 0 0 0 . , 25000 . , l.eZO/

JBOT=l JTOP=28

10 J= (JTOP+JBOT} / S IF(TEM.LT.TTAB(J)) THEM

J T O P = J E L S E

JBOT=J ENDIF IF(JT0P-JBOT.NE.l) GO TO 10 VALUE=YTAB ( JBOT) + (YTAB (JTOP) -YTAB ( JBOT) ) **(TEM-TTAB (JBOT) } /

1 (TTAB(JTOP1-TTAB(JB0T)) RETURN END

IMAGE NALUATION TEST TARGET (QA-3)

APPLIED - IMAGE. lnc = 1653 East Main Street --- Rochester. NY 14609 USA -- ,-= Phone: 71 6/482-0300 --= , - Fax: i l 6/288-5989

O 1993. Appiwd Image. Inc.. NI Righis Resewed

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Mathematical Modelling of GTAW and GMAW