Heat Source Cs

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ABSTRACT. It is a key issue to establishan appropriate model of theheat sourceinthesimulation of keyhole plasmaarc weld-ing (PAW). It requires that the model ac-count for the keyhole effect and have the

characteristic of volumetric distributionalong the direction of the plate thickness.For available heat source models, neitherGaussian nor double ellipsoidal modes ofheat source is applicable to the keyholePAW process. Considering the force of thehigh-speed plasma jet and the associatedstrong momentum, a modified three-dimensional conical heat source model isproposed as the basis for the numericalanalysis of temperature fields in the key-hole PAW process. Further, a new heatsource model for quasi-steady state tem-perature field in keyhole PAW is devel-

oped to consider the bugle-like configu-ration of the keyhole and the decay of heatintensity distribution of the plasma arcalong the direction of theworkpiece thick-ness. Based on this heat source model, fi-nite-element analysis of the temperatureprofile in keyhole PAW was conductedand the weld geometry was determined.The results showed that the predicted lo-cation and locus of the melt-line in thePAW weld cross section are in good agree-ment with experimental measurements.

Introduction

Plasma arc welding (PAW) offers sig-nificant advantages over conventional gastungsten arc welding (GTAW) in terms ofpenetration depth, joint preparation, andthermal distortion (Refs. 1, 2). The arcused in PAW is constricted by a small noz-zle and has a much higher gas velocity(3002000 m/s) and heat input intensity

(1091010 w/m2) than that in GTAW (gasvelocity 80150 m/s, heat input intensity108109 w/m2) (Refs. 3, 4). As the plasmaarc impinges on the area where two work-pieces are to be joined, it can melt mater-

ial and create a molten liquid pool. Be-cause of its high velocity and theassociated momentum, the arc can pene-trate through the molten pool and form ahole in the weld pool, which is usually re-ferred to as a keyhole (Ref. 5). Moving the

welding torch and the associated keyholewill cause the flow of the molten metal sur-rounding the keyhole to the rear region

where it resolidifies to form a weld bead.Thekeyhole mode of welding is thepri-

mary attribute of high-power densitywelding processes (PAW, laser welding,and electron beam welding), which makes

them penetrate thicker pieces with a sin-gle pass. Compared to laser welding andelectron beam welding, keyhole PAW ismore cost effective and more tolerant of

joint preparation, though its energy is lessdense and its keyhole is wider (Ref. 4).

Thus, keyhole PAW has found applica-tions on the welding of many importantstructures (Refs. 614). Although keyholePAW has the potential to replace GTAWin many applications as a primary processforprecise joining (Ref.1), the stablestateof the keyhole is an important issue in ap-plying PAW (Ref. 15). In keyhole PAW, thequality of theweld depends on thekeyholestability, which itself depends on a largenumber of factors, especially the physicalcharacteristics of the material to be

welded and the welding process parame-ters to be used (Ref. 5). Thus, keyholePAW is susceptible to the variation of

welding process parameters, which makesit have a narrower range of applicable

process parameters for good weld qualityso that keyhole PAW is still limited in its

wide application in industry (Ref. 16). Thetemperature profile around the weld poolhas great influence on the formation andstability of keyhole. Through numericalsimulation of the temperature field andthe weld pool behavior in keyhole PAW,the process parameters can be optimizedfor obtaining high-quality of weld struc-ture. Therefore, it is of great significanceto model and simulate the temperaturedistribution andweld pool geometryin thekeyhole PAW process.

The key issue for numerical analysis oftemperature field in keyhole plasma arcwelding is how to develop a heat sourcemodel that reflects the thermo-physicalcharacteristics of the keyhole PAWprocess. Because of the complexity of thephenomena associated with the formationof a keyhole, only a limited number of the-oretical studies treating the PAW processhave been reported, each of varying de-grees of approximation, and each focusingon different aspects of the problem (Refs.11, 1721). General Gaussian or double-ellipsoidal heat source models usedwidelyin simulation of arc welding processes arenot suitable for keyhole PAW. As the firststep in a series of study, this paper focuseson developing a suitable heat sourcemodel for finite-element analysis of tem-perature profile in keyhole PAW.

Models of Heat Source inKeyhole PAW

As mentioned above, the key problemin FEM analysis of keyhole PAW is how tomodel the welding heat source. Most re-searchersemployeda Gaussiandistributionof heat flux (W/m2) deposited on the sur-

A New Heat Source Model forKeyhole Plasma Arc Welding in FEMAnalysis of the Temperature Profile

The model takes into consideration keyhole configurationand decay of heat distribution along the workpiece thickness

BY C. S. WU, H. G. WANG, AND Y. M. ZHANG

KEYWORDS

Finite Element AnalysisPlasma Arc WeldingKeyhole WeldingHeat TransferThermal Analysis

C. S. WU and H. G. WANG are with the Institutefor Materials Joining, Shandong University, Jinan,China. Y. M. ZHANG is with the Center for Man-ufacturing and Department of Electrical andComputer Engineering, University of Kentucky,

Lexington, Ky.


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face of the workpiece (Refs. 2224).Although such a surface mode of heatsource may be used for the shallow pene-tration arc welding processes like GTAW, itdoesnotreflecttheactionofarcpressureonthe weld pool surface so that it is not suit-able for modeling the welding processes

with deeper penetration like gas metal arcwelding(GMAW).Goldak proposeda dou-ble-ellipsoidalheat source model, which has

the capability of analyzing the thermal his-tory of deep penetration welds like GMAW(Ref. 25). However, the double-ellipsoidaldistribution of heat intensity (W/m3) is stillnot applicable to the high-density weldingprocesses with high ratio of the weld pene-tration to width, such as keyhole PAW (Ref.26). Keyhole PAW produces a weld withhigh ratio of depth to width. The cross sec-tion of the weld is of the bugle-like con-figuration. To consider the strong action ofthe plasma jet and the resulted bugle-like

weld configuration in keyhole PAW, newtypes of welding heat source models must

be proposed.

Three-Dimensional Conical Heat Source

Three-dimensional conical heat source(TDC) is a volumetric heat source thatconsiders the heat intensity distributionalong the workpiece thickness. As shownin Fig. 1, the heat intensity deposited re-gion is maximum at the top surface of

workpiece, and is minimum at the bottomsurface of workpiece. Along the thickness

of the workpiece, the diameter of the heatdensity distribution region is linearly de-creased. But theheat density at thecentralaxis (z-direction) is kept constant. At anyplane perpendicular to z-axis, the heat in-tensity is distributed in a Gaussian form.Thus, in fact, TDC is the repeated addi-tion of a series of Gaussian heat sources

with different distribution parameters andthe same central maximum values of heatdensity along the workpiece thickness. Inthis way, the heating action of the plasma

jet through the workpiece in keyhole PAWis considered.

At any plane perpendicular to the z-

axis, the heat intensity distribution may bewritten as

(1)where Q0 is the maximum value of heat in-tensity, r0 is the distribution parameter,and r is the radial coordinate. The keyproblem is how to determine the parame-ters Q0 and r0.

As shown in Fig. 1, the height of theconical heat source is H = ze zi, the z-coordinates of the top and bottom sur-faces areze andzi , respectively, and thedi-ameters at the top and bottom are re andri, respectively. The distribution parame-ter r0 is linearly decreased from the top tothe bottom surfaces of the conic region,and it can be expressed as

(2)

Through complete derivation (seeAppen-

dix), Equation 1 is of the following form

r z r r rz z

z ze e i

e

e i0 ( ) = ( )

Q r z Qr

rV , exp( ) =

0

2

02

3

Fig. 1 Schematic of TDC model. Fig. 2 Schematic of MTDC model.

Fig. 3 Schematic of keyhole and weld pool. Fig. 4 Schematic of QPAW model.


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(3)

where is theplasma arc power efficiency,U is the arc voltage, and I is the weldingcurrent.

Modified Three-DimensionalConical Heat Source

Although the three-dimensional coni-cal heat source takes consideration of theheat intensity distribution and decay alongthe workpiece thickness, its characteristicof linear decline is not appropriate.In key-hole PAW, a bugle-like weld configuration

was the result. To reflect this feature, amodified three-dimensional conical(MTDC) heat sourceis proposed,which is

shown in Fig. 2. For the MTDC heat

source, the distribution parameter r0 de-creases no longer as linearly as for TDC,but in a curvilinear way.

As shown in Fig. 2, the height of themodified three-dimensional conical(MTDC) heat source is H = ze z i, the z-coordinates of the top and bottom sur-faces areze andzi, respectively, and the di-ameters at the top and bottom are re andri, respectively. Let r0 represent the distri-bution parameter atz.r0 is decreased non-linearly and can be expressed as

(4)

After complete derivation and manipula-

tion (see Appendix), the heat intensity forMTDC can be written as

(5)where

(6a)

(6b)

(6c)

(6d)

Q r zUIe

e

z z r r r r

V

e i e e i i

,( ) =( )

( ) + +

9

1

1

3

3

2 2

(( )

exp

3 2

02

r

r

r zr r z

z z

r z

e i

e i

i e

0

1

1 1

1

( ) =( ) ( )

( ) ( )

+ ( )

n

n n

n rr z

z z

e i

e i

( )( ) ( )

1

1 1

n

n n

br z r z

z z

i e e i

e i

=

1 1

1 1

n n

n n

ar r

z z

e i

e i

=

1 1n n

Q r zUIe

e A

r

rV , exp( ) =

3

1

1 33

3

2

02

r z a z b0 1( ) = +n

A aH z H z

z z

a a b

i i

i i

=+( ) +( )

( )

22

2

1

1

2

n

n

HH z H z

z z Hb H

i i

i i

+( ) +( )

+

1

1

2n

n

Fig. 5 The grid system. Fig. 6 The thermal conductivity vs. temperature.

Table 1 The Parameters Used for Heat Source Modes

Heat Source Modes re (mm) ri (mm) ze (mm) zi (mm) Dh (mm)

TDC 7.0 2.6 9.5 2.2MTDC 7.2 1.05 10.7 1.2QPAW 7.2 1.05 10.7 1.2 6.8

Table 2 Keyhole PAW Welding Process Parameters

Case Welding Current Arc Voltage Welding Speed Plasma Gas Flow Shielding Gas Flow(A) (V) (mm/min) Rate (L/min) Rate (L/min)

1 250 31.7 120 4 102 240 31.2 120 4 10

Table 3 Comparison of the Predicted and Experimental Weld Width

Top Weld Width (mm) Bottom Weld Width (mm)Heat Source

Predicted Measured Predicted Measured

TDC 13.60 14.35 4.25 2.11MTDC 12.90 14.35 2.30 2.11QPAW 13.90 14.35 2.30 2.11


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Quasi-Steady State PAW Heat Source

As aforementioned, the arc used in thePAW process is constricted by a small noz-zle and has a much higher gas velocity andtemperature than that in GTAW. The highplasmagas velocity andthe associated mo-mentum force the plasma jet to penetratethe base metal, forming a keyhole in the

weld pool of complete penetration. Thus,the plasma jet emerges from the under-bead at the bottom of the workpiece Fig. 3. As the torch moves along the weld,this keyhole progressivelycutsthrough themetal with the molten metal flowing be-hind to form the weld bead. The size andshape of the keyhole produced by PAWdepend mainly on the pressure of the im-pinging gas. It was found that a typicalplasma arc keyhole in 6-mm-thick stain-less steel,3-mm nozzlebore diameter,and810 mm torch standoff is a conical hole

with a circumferential diameter of 5 mmat the top and about 1.52 mm at the bot-

tom end (Ref. 27). The forces that tend toform andmaintainthe keyhole include theplasma stream pressure, vapor pressure,and recoil pressure (Ref. 27). The exis-tence of keyhole, on one hand, makes theheat intensity from the plasma arc distrib-ute through the workpiece thickness, buton the other hand, causes some vaporiza-tion, which results in some heat losses.

Therefore, the net heat input is not de-posited on the workpiece totally. To con-sider this point, part of the heat density atthe upper section of the keyhole is ex-cluded. As shown in Fig. 4, the heat inten-sity is distributed within a domainbounded by the curves f0 (z) and f1 (z) atthe upper section of the keyhole, while atthe lower section of the keyhole, the heatintensity is distributed as a modified coni-cal heat source. This model of heat sourceis proposed for the quasi-steady statePAW process, so it is referred to as QPAWfor short. It can be expressed as

For the upper part, (ze Dh)z ze

(7)where

(8)

(9)

For the lower part, zi z (ze Dh), it isofthesameexpression as Equations 5 and6.

FEM Analysis

After establishing a suitable heat

f zr

Dz z De

he h1 ( ) = +

f zr r z

z z

r z

e i

e i

i e

0

1

1 1

1

( ) =( ) ( )

( ) ( )

+ ( )

n

n n

n rr z

z z

e i

e i

( )( ) ( )

1

1 1

n

n n

Q r zIUe

e A

r

f z

f

V , exp ,( ) =( )

( )

3

1

33

3

2

02

11 0z r f z( ) ( )

Fig. 7 The heat transfer coefficiency and specific heat vs. temperature. Fig. 8 Comparison of predicted PAW cross section with experimentalresults.

Fig. 9 The cross section of a plasma arc weld under welding condition Case 1. A The predicted result based on model QPAW; B macrograph.

AB


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source model for keyhole PAW, the tem-perature profile and weld pool geometryare able to be determined by the finite el-ement method (FEM). In a moving coor-

dinate system o-xyz in which the origin co-incided with the intersecting pointbetween the torch centerline and the bot-tom surface of workpiece, x-axis along the

welding direction and z-axis normal to theworkpiece surface, the quasi-steady statetemperature field during keyhole PAW isgoverned by the following equation

(10)

where is density, CP is specific heat, v0 iswelding speed, Tis temperature, k is ther-mal conductivity, and QV is volumetricheat source. The boundary conditions forEquation 10 are as follows:

On the workpiece surface,

(11)

where T is the ambient temperature, andT is the combined heat transfer coeffi-cient.

At the symmetric plane xoz,

(12)

So only half of the workpiece is taken asthe calculation domain. The workpiece di-mension is 200 mm in length, 80 mm in

width, and 9.5 mm in height.The half domain is divided into 8-node

hexahedrons. As aforementioned, part ofheat density at the upper section of keyholeis excluded because of evaporation loss andkeyhole effect. To reflect this characteristicand match the distribution mode of heat

source QPAW, some el-ements in the uppersection of the keyholeare treated as dead.

The discrete grids areshown in Fig. 5.

Results

The workpiece ma-terial is stainless steel304. Its thermal con-ductivity k, specificheat CP, and the com-bined heat transfer co-efficient T are showninFigs.6 and 7.Itsden-sity is = 7860 (kg

m3

), and ambient tem-perature is T = 300 K.The parameters used for describing heatsource modes are given in Table 1. Key-hole PAW experiments are conductedunder two welding conditions (Table 2).The arc power efficiency takes a value of0.66 based on the literature (Ref. 3). After

welding, a macrograph of the weld is madeto show its cross section.

Firstly, two kinds of volumetric heatsource models, i.e., three-dimensionalconical (TDC) and modified three-dimen-sional conical (MTDC), are used to pre-

dict the plasma arc weld geometry. Figure8 shows the comparison of the experimen-tal results with the predicted ones basedon different models. Table 3 gives the dataof weld width. From the point of view ofagreement, the calculation precision ofTDC is poorer. It can be seen that TDC isnot suitable for determining keyholeplasma arc weld dimensions. Though theprecision of MTDC is improved com-pared to TDC, especially theweld width atboth top and bottom surfaces, the calcula-tion precision for the location and locus ofthe melt-line in the weld cross section isstill lower.

Secondly, the developed heat sourcemodel for quasi-steady state keyhole PAW,i.e., QPAW, is employed to calculate the

weld geometry. Figures 912 show thecomparison of experimental results withthepredicted ones based on QPAW modelfor welding conditions Case 1 and Case 2,respectively. In Figs. 9A and 11A, the dot-ted lines are drawn along the isotherm ofmelting temperature. It can be seen thatthe predicted weld geometry agrees well

with experimental measurements. Sincethe model QPAW depicts the character ofthe keyhole PAW process through consid-ering the bugle-like configuration of thekeyhole and the decay of heat intensitydistribution of theplasma arcalong thedi-rection of the workpiece thickness, thecalculation precision of the weld geometryat the cross section is quite satisfactory.

Conclusion

Because of thekeyhole effect, a plasmaarc weld has a large penetration-to-widthratio. For numerical analysis of weldingtemperature profile in keyhole PAW, an

C vT

x xk

T

x

yk

T

y

P

=

+

0

+

+z

kT

zQV

= ( )k

T

zT TT

=

T

y0

Fig. 10 Comparison between the predicted and measured weldcross section for Case 1.

Fig. 11 The cross section of a plasma arc weld under welding conditionCase 2. A The predicted result based on model QPAW; B macro-

graph.

A

B


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appropriate heat source model must bedeveloped to consider the high-speedplasma jet and the associated strong mo-mentum acted to the weld pool and thedistribution of heat intensity along the di-rection of the workpiece thickness. Be-cause neither the double-ellipsoidal modeof the heat source nor the three-dimen-sional conical mode of the heat source canbe appropriate for the finite elementanalysis of the keyhole PAW process, amodified three-dimensional conical heatsource model (MTDC) is put forward toreflect the nonlinear decay of heat inten-sity distribution along the direction of the

workpiece thickness. Although MTDCcan be used to calculate the weld width onboth the top and the bottom surfaces ofthe workpiece, for the location and locus

of the melt-line in the weld cross section,its calculating precision is lower.

A new heat source model for the quasi-steady state keyhole PAW, i.e., QPAW, isproposed, which depicts the characteristicof the keyhole PAW process quite well, be-cause it considers both the bugle-likeconfiguration of the keyhole, and thedecay of heat intensity distribution of theplasma arc along the direction of the

workpiece thickness. The QPAW heatsource model is applied in finite elementanalysis of the temperature field. The re-sults show that the calculated weld geom-

etry at the cross section is in good agree-ment with the experimental one.

Acknowledgments

The authors wish to thank the financialsupport for this research from the Na-tional Natural Science Foundation ofChina (Grant No. 50540420570) and theKey Project of Chinese Ministry of Edu-cation (Grant No.104109).

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Appendix 1Derivation for Heat IntensityDistribution Equations

For a volumetric distribution of heatintensity, the following assumptions aremade:

1) The heat intensity deposited regionis maximum at the top surface of work-piece, and is minimum at the bottom sur-face of workpiece.

2) Along the thickness of the work-piece, the diameterof the heat density dis-tribution region is decreased in some way.

But the heat density at the central axis (z-direction) is kept constant.

At any plane perpendicular to z-axis,the heat intensity distribution may be writ-ten as

(A1)

where Q0 is the maximum value of heat in-tensity, r0 is the distribution parameter,and r is the radial coordinate. The key

Q r z Qr

rV , exp( ) =

0

2

02

3

Fig. 12 Comparison between the predicted and measured weld cross sec-tion for Case 2.


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problem is how to determine the parameters Q0 when the decayrule ofr0 is known.

Because of thermal energy conservation, we have

(A2)where is the plasma arc power efficiency, U is the arc voltage,and Iis the welding current.

Three-Dimensional ConicalHeat Source (TDC)

As shown in Fig. 1, r0 decreases linearly for TDC. The heightof the conical heat source is H = ze z i, the z-coordinates of thetop and bottom surfaces are ze and zi , respectively, and the diam-

eters at the top and bottom are re and ri, respectively. The distri-bution parameter r0 can be expressed as

(A3)or

(A4)Since

(A5)Substituting A5 into A2, then

(A6)

Finally, substituting A6 into A1, we get

(A7)

Modified Three-Dimensional ConicalHeat Source

As shown in Fig. 2, the height of the modified three-dimen-sional conical (MTDC) heat source is H = ze zi, the z-coordi-nates of thetop andbottom surfaces areze andzi, respectively, andthe diameters at the top and bottom are re and ri, respectively. Letr0 represents the distribution parameter at z. r0 can be expressedas

r0(z) = a 1n z+b (A8)

Because

ri = a 1n zi+b

re = a 1n ze+b

thus

(A9)

(A10)

Substituting A9 and A10 into A8,

(A11)Let h = z zi. Since

(A12)Based on the Mathematics Handbook,

(A13)

Substituting A13 into A2

(A14)Substitute A14 into A1, we obtain

(A15)

Appendix 2 Nomenclature

a variable, defined inEquation A9

A variable, defined in

Q r zUIe

A e

r

rV , exp( ) =

3

1

33

3

2

02

UIQ e

r dhQ e

A

QUI

H=

( )=

( )

=

0

3

02 0

3

0

0

1

3

1

3

3 ee

A e

3

31

1 1

1 1 2 1 1

n n

n n n n2 2 2

udu u u u C

udu u u udu u u

= +

= =

( ) +

= +( ) +

2 1

1 2 102 2

0

u u u C

r dh a h z dh abiH

n

n n2

hh z dh b H

a h z h z h z

i

HH

i i i

+( ) +

= +( ) +( ) +

002

21 2n

2 (( ) +( )+ +( )

+ +( ) +( )

1 2

2 1

0n

n

h z h z

ab h z h z

i i

i i +( )

+

= +( ) +( )

h z b H

a H z H z z z

iH

i i i

0

2

2 1 1n n2 2 ii

i i i ia a b H z H z z z H b

( ) +( ) +( ) +2 1 1n n22

02 2 1 1

H

A r dh a H z H z z zi i i i

Let

n n2 2= = +( ) +( ) 00

22 1 1

H

i i i ia a b H z H z z z H b H

( ) +( ) +( ) +n n

r dh a z b dh a h z bH

i

HH

02 2

0

2

00

1 1= +( ) = +( ) +

n n ddh

a h dh ab h z dh b H iHH

= +( ) + +( ) +2 2

001 2 1n z n2 i

r zr r z

z z

r z

e i

e i

i e

0

1

1 1

1

( ) = ( )

( ) ( )

+ ( )

n

n n

n rr z

z z

e i

e i

( )( ) ( )

1

1 1

n

n n

br z r z

z z

i e e i

e i

=

1 1

1 1

n n

n n

ar r

z z

e i

e i

=

1 1n n

Q r zUIe

e

z z r r r r

V

e i e e i i

,( ) =( )

( ) + +(

9

1

1

3

3

2 2

))

exp

3 2

02

r

r

UIQ H e

r r r r

QUIe

e

e e i i=

+ +( )

=

0

3

2 2

0

3

3

1

9

9

11

1

2 2

+ +H r r r re e i i

UI Q r z rdrd dh

Q

V

rH

rH

= ( )

=

00

2

0

000

2

0

0

0

,

eexp

exp

=

3

3

3

2

02

00

0

2

r

rrdrd dh

Qr

r

r

H

0020

2

02

03

0

0 3

1

3

=( )

rd

r

r

dh

Q er

22

0

Hdh

r r r rh

Hh z zi e i i0

22

= + ( )

= ,

r z r r rz z

z ze e i

e

e i0 ( ) = ( )

r dh r r rh

Hdh

Hr r ri e i

H

e e i02

2

0

2

3= + ( )

= + + rri

H 2

0( )


8/8

WELDING RESEARCH

-s291WELDING JOURNAL

Equation A13b variable, defined in

Equation A10CP specific heatDh variable, defined in

Fig. 4e base of natural

lagarithmf0 function, defined in

Fig. 4

f1 function, defined inFig. 4

h z ziH ze z ik thermal conductivityQ0 maximum value of

heat intensityQv heat intensity distribu

tion functionr x2 + y2

r0 radial coordinate inheat source mode

re variable, defined inFigs. 1, 2, 4

ri variable, defined inFigs. 1, 2, 4

T temperatureT ambient temperature

U arc voltagev0 welding speedx, y, z coordinateze variable, defined in

Figs. 1, 2, 4zi variable, defined in

Figs. 1, 2, 4T heat loss coefficient

arc power efficiency angular variable density

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