composition change of stainless steels during …...of stainless steel, 204 (204 ss) and 304 (304...

Introduction Laser welding is widely used in aero-space, automotive, and other manufac-turing industries owing to its deep pen-etration, excellent precision, high pro-ductivity, and amenability to automa-tion. Because of the high power densityused, the weld metal is rapidly heated tovery high temperatures and a keyholefilled with metal vapors is formed di-rectly under the laser beam. The forma-tion of the keyhole is accompanied bysignificant vaporization of variousvolatile alloying elements, not all ofwhich vaporize at the same rate. Apartfrom the keyhole, vaporization also oc-curs from the surface of the weld pool.The loss of alloying elements results inan undesirable change in the weld metalcomposition as well as degradation of

the mechanical and corrosion proper-ties of the weld. In the electronics in-dustry, components are often processedin clean rooms where emission of metalvapors is not acceptable. Selective vaporization of alloyingelements from the weld pool at rela-tively low power densities have beenstudied before because of its detri-mental effects. For example, the hard-ness of aluminum alloy weld metal wasfound to be lower than that of the basemetal due to vaporization of magne-sium and the reduction of solid solu-tion strengthening (Ref. 1). Most ofthe previous research on alloying ele-ment vaporization was focused on ei-ther isothermal systems or at relative-ly low laser power densities where thekeyhole did not form. For example,Anisimov and Rakhmatulina (Ref. 2)

and Knight (Refs. 3, 4) derived a set ofgas dynamic equations for the vaportemperature, density, velocity, and theextent of the condensation across theKnudsen layer by solving the equa-tions of conservation of mass, mo-mentum, and energy for the vaporiza-tion from isothermal surfaces. Mundra and DebRoy (Ref. 5) pro-

posed a model to calculate the vapor-ization rate and composition changeof stainless steel in conduction modelaser welding. Similarly, Zhao and DebRoy (Ref. 6) and He et al. (Ref. 7)computed vaporization rates and com-position changes in aluminum alloy andstainless steel during conduction modeNd:YAG laser welding, respectively. As aresult of the previous work (Refs. 1–13),the composition change and its detri-mental effects are now well recognizedfor conduction mode laser welding.However, very little work has been doneon the alloying element loss and compo-sition change due to selective vaporiza-tion of alloying elements during keyholemode laser welding. Keyhole mode laser welding ismuch more complex than conductionmode welding. Pronounced vaporiza-tion of alloying elements is a require-ment for the stability of the keyhole.The distribution of power density,laser beam radius and profile, weldingspeed, specimen thickness, and the na-ture of the alloy affect the geometry ofthe keyhole, its stability, weld pooltemperature distribution, and the re-circulating flow of liquid metal withinthe moving weld pool. The vaporiza-tion rates of alloying elements dependon the shape and size of the keyhole aswell as its wall temperature. They arealso affected by welding parameters

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WELDING JOURNAL / JULY 2017, VOL. 96258-s

Composition Change of Stainless Steels duringKeyhole Mode Laser Welding

The experimental and theoretical study of composition changed during keyhole mode welding of stainless steels

T. LIU, L. J. YANG, H. L. WEI, W. C. QIU, AND T. DEBROY

ABSTRACT Vaporization of alloying elements during laser beam welding adversely affects weldmetal composition and properties. Alloying element vaporization and weld metal composition change during partial and complete joint penetration keyhole mode Nd:YAG laserwelding of two stainless steels containing different concentrations of manganese wereexamined experimentally and theoretically. The keyhole and weld pool geometry and thetemperature field were computed from a welltested, threedimensional heat transferand fluid flow model, and the results were used in a model for the calculation of weldmetal composition change based on the principles of transport phenomena, kinetics, andthermodynamics. The results showed that vaporization from the keyhole was the mostpronounced source of alloying element loss for all welding conditions. The weld metalcomposition was spatially homogeneous, but the concentration of manganese decreasedwhile the concentration of iron slightly increased for all welding conditions. The changein the weld metal composition was much more pronounced for the higher manganesesteel. The change in the weld metal composition was much more sensitive to laser powerthan welding speed for both partial and complete joint penetration welds.

KEYWORDS • Composition Change • Vaporization • Keyhole • Laser Welding • Stainless Steel

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and alloy properties. Because of thesecomplexities, several simplifying as-sumptions have been made in the pastto understand the alloying element va-porization during keyhole mode weld-ing. For example, Dilthey et al. (Ref.14) developed a model to calculate thecomposition change in keyhole modeCO2 laser welding. The keyhole wasconsidered as a cylinder for simplicityaround which the liquid metal circulat-ed within the weld pool. Jandaghi etal. (Refs. 15, 16) studied the composi-tion change in 5754 aluminum alloyand 316 stainless steel during pulsedNd:YAG laser welding. However, thetemperature fields were obtained froma simplified heat conduction modelthat ignored convective heat transfer.It is well known that this practice re-sults in significant errors (Ref. 17). The keyhole shape and size, thegeometry of the weld pool, and thetemperature field in the weld pool af-fect the vaporization behavior. Vapor-ization occurs from both within thekeyhole and the top surface of theweld pool. Rigorous calculations ofboth are necessary to quantitativelyunderstand the alloying element lossduring keyhole mode laser welding. Adetailed experimental and theoreticalstudy of the vaporization behaviorduring keyhole mode laser welding hasnot been reported.

Here we presentan experimentaland theoretical in-vestigation of theloss of alloying ele-ments and weldmetal compositionchange during par-tial and completejoint penetrationkeyhole modeNd:YAG laser welding of two varieties ofstainless steel containing different con-centrations of manganese. A well-test-ed, comprehensive, keyhole mode heattransfer and fluid flow model was usedto understand the keyhole shape andsize as well as temperature field. Spatialvariations of the concentrations of iron,manganese, chromium, and nickel inthe weld metal were examined by elec-tron probe microanalysis (EPMA) afterwelding. The experimentally deter-mined weld pool geometry and thechange in the composition of the alloy-ing elements for various welding speedswere compared with the correspondingmodel predictions.

Experimental Procedure

A schematic diagram of the experi-mental setup is shown in Fig. 1. Bead-on-plate Nd:YAG laser welds on 3.0-

mm-thick stainless steel plates werestudied. The compositions of two typesof stainless steel, 204 (204 SS) and 304(304 SS), containing different concen-trations of manganese and other alloy-ing elements are presented in Table 1.The plate surface was polished withsand paper to remove the oxide skinand cleaned in ethyl alcohol beforewelding. Keyhole mode welding was per-formed using a continuous waveNd:YAG laser (GSI JK2003M) having awavelength of 1064 nm. The laser beamwas focused on the specimen surfaceusing a 300-mm focal length lens. Thefocal spot of the laser beam was 0.8 mmin diameter. The laser beam parameterswere from the manual of the equip-ment. The defocusing distance was 0mm for all the experiments. Laterallyinjected argon was supplied as theshielding gas through a 6-mm-diametergas nozzle at a flow rate of 20 L min–1 to

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A

BFig. 1 — Schematic representation of the experimental setup.

Fig. 2 — Schematic representation of the locations where thechemical composition was determined by EPMA: A — Completejoint penetration welds; B — partial joint penetration welds.

Table 1 — Compositions of the Stainless Steels wt%

Alloying Mn Cr Ni Si C P S FeElements

204 SS 10.5 14.0 1.3 0.45 ≤0.05 ≤0.04 ≤0.03 balance 304 SS 1.2 19.2 6.5 0.77 ≤0.05 ≤0.04 ≤0.03 balance

DebRoy Supplement-201708.qxp_Layout 1 6/8/17 4:27 PM 59

avoid oxidation. As shown in Fig. 1, theangle of the gas nozzle was 45 deg tothe horizontal plane, and the distancebetween the laser spot and the nozzlewas about 20 mm. The variables studiedwere laser power in the range of 1200 Wto 1350 W and welding speed between 6to 12 mm s-1. After welding, the specimens werecut to obtain the transverse cross sec-tions using wire electrical dischargemachining, and prepared using thestandard polishing technique. Theconcentrations of manganese, chromi-um, nickel, and iron in both the basemetal and the weld metal were deter-mined by EPMA. The EPMA was per-formed by using a CAMECA SXFivewith an accelerating voltage of 20 KeVand a beam current of 100 nA. The Klines of manganese, chromium, nickel,and iron were measured with the stan-dards of manganese metal, chromiummetal, nickel metal, and cast iron andNIST 363 steel, respectively. The concentrations of alloying ele-ments were determined before and af-ter the experiments in at least 11 loca-tions on the transverse cross sectionsas shown in Fig. 2. The average con-centrations of each alloying elementand their standard deviations areshown in Table 2.

Mathematical Modeling

Heat Transfer in the Weld Pool

The temperature and velocityfields, keyhole geometry, and fusionzone geometry were calculated from awell-tested, 3D, steady-heat transferand fluid flow model (Refs. 18–20).The data used for the calculations(Refs. 5, 7) are presented in Table 3.The assumptions, governing equa-

tions, boundary conditions, and theirsolution for the keyhole mode weldinghave been described in previous pa-pers (Refs. 18–22). The salient fea-tures of the model are provided in Ap-pendix A. The computed geometry ofthe keyhole and the weld pool and thetemperature field were used for thecalculation of the alloying element va-porization and composition change asdescribed below.

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Table 2 — EPMA Results of Elements Concentration in Various Conditions

Material Laser Power/W Welding Speed/mm s1 Average Weight Percent (deviation) Mn Ni Cr Fe

204 SS base metal 10.437 1.288 14.064 72.992 (0.261) (0.047) (0.059) (0.283) 1200 8 9.112 1.326 14.210 74.247 (0.217) (0.001) (0.036) (0.046) 1250 8 9.265 1.314 14.203 74.041 (0.059) (0.012) (0.045) (0.132) 1300 8 9.489 1.324 14.201 73.656 (0.078) (0.008) (0.021) (0.095) 1350 8 9.549 1.305 14.174 73.868 (0.166) (0.007) (0.041) (0.079) 1300 6 9.350 1.319 14.164 73.875 (0.079) (0.012) (0.005) (0.382) 1300 8 9.489 1.324 14.201 73.656 (0.078) (0.008) (0.021) (0.095) 1300 12 9.442 1.308 14.178 73.825 (0.102) (0.010) (0.001) (0.123)

304 SS base metal 1.205 6.526 19.225 72.525 (0.043) (0.300) (0.158) (0.254) 1250 8 1.080 6.569 19.046 72.793 (0.024) (0.004) (0.054) (0.055)

Table 3 — Data Used for Calculations

Property/Parameter 204 SS 304 SS

Density of liquid metal (kg m–3) 7.80 103 7.20 103

Absorption coefficient 0.22 0.22 Effective viscosity (kg m–1 s) 0.1 0.1 Solidus temperature (K) 1787 1697 Liquids temperature (K) 1811 1727 Enthalpy of solid at melting point (J kg–1) 1.2 106 1.2 106

Enthalpy of liquid at melting point (J kg–1) 1.26 106 1.26 106

Specific Heat of solid (Jkg–1 k) 710.6 711.8 Specific Heat of liquid (Jkg–1 k) 836.0 837.5 Thermal conductivity of solid (J m–1 s K) 19.26 19.26 Thermal conductivity of liquid (J m–1 s K) 20.93 20.93 Temperature coefficient of surface tension (N m–1 K) 4.3 10–4 4.3 10–4

Coefficient of thermal expansion (K–1) 1.96 10–5 1.96 10–5


Driving Forces for Vaporization

The metal vapors exit from the key-hole to keep it open during the weld-ing process. The pressure inside thekeyhole is slightly higher than the am-bient pressure. The overpressure pro-vides the main driving force for themetal vapor to escape from the key-hole. Furthermore, the concentrationsof the alloying elements in the vaporwithin the keyhole are considerablyhigher than those outside the keyhole.Therefore, the flow of metal vapor outof the keyhole is driven by the concen-tration gradients. In addition to thekeyhole, vaporization also occurs fromthe surface of the weld pool similar toconduction mode laser welding. If equilibrium is attained on thekeyhole surface, the keyhole wall tem-perature needs to slightly exceed theboiling point of the alloy for the equi-librium vapor pressure on the keyholewall to be greater than the ambientpressure. The excess pressure providesa driving force for the vapor to moveout from the keyhole wall. Close to thesurface of the keyhole wall, there is aspace of several mean-free-paths

length, known asthe Knudsen lay-er (Ref. 3). Theoretical calculations ofthe vaporization rates in the Knudsenlayer is based on treating this regionas a gasdynamic discontinuity. Anisi-mov and Rakhmatulina (Ref. 2) andKnight (Ref. 4) derived expressions forthe vapor temperature, density, veloci-ty, and the extent of the condensationacross the Knudsen layer by solvingthe equations of conservation of mass,momentum, and translational kineticenergy. A brief account of the calcula-tion of the vaporization flux, Jp, owingto the pressure gradient on the poolsurface is provided in Appendix B. Thevaporization rate can be calculatedfrom the following expression (Refs.5–10):

(1)

where v is the density of the vapor atthe edge of the Knudsen layer, and Mand S are the Much number and speedof sound in the vapor, respectively.The vaporization flux of an alloying el-

ement i, JP,i, from the pressure gradi-ent can be obtained from the followingrelation (Ref. 10):

(2)

where ai is the activity of the element iin the liquid metal, which is deter-mined by its concentration in the weldpool. In this calculation, the mole frac-tion of the element i was used as thevalue of ai because alloys behave ideal-ly close to their boiling points. Pi

0 isthe equilibrium vapor pressure of ele-ment i over pure liquid, Pl is the equi-librium vapor pressure over the alloy,Mi and Mv are the molecular weight ofelement i, and the average molecularweight of the vapor, respectively. The concentrations of metal vaporsat the keyhole wall and weld pool sur-face differ from those in the bulk gasphase. These differences provide driv-ing forces for diffusive flux of alloyingelements, Jc,i, which is given by the fol-

Jp,i = aiPi0

Pl

MiMv

Jp

Jp =�vMS

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Fig. 3 — A flow chart for the calculation of vaporization ratesand composition change during keyhole mode laser welding.

Fig. 4 — Computed temperature and velocity fields for different welding parameters: A — P = 1200 W, v = 8 mm s1; B — P = 1250 W, v = 8mm s1; C — P = 1300 W, v = 8 mm s1; D — P = 1350 W, v = 8 mm s1; E— P = 1300 W, v = 6 mm s1; and F — P = 1300 W, v = 12 mm s1.

BA

EF

DC


lowing expression (Ref. 10):

(3)

where Kg,i is the mass transfer coeffi-cient of element i, R is the gas con-stant, Tl is the temperature at the edgeof the Knudsen layer, Ci

b is the concen-tration of element i in the shieldinggas. The concentrations of alloying ele-ments in the shielding gas, Ci

b is negli-gible. The mass transfer coefficient,Kg,i, between the weld pool surface andthe shielding gas can be deduced fromthe graphical results of Schlunder andGnielinski (Ref. 23) for a jet impingingon a flat surface:

(4)

where d is the diameter of the nozzle, ris the radial distance on the pool sur-face, Di is the average diffusivity of theelement i in the shielding gas, Re is theReynolds number at the nozzle exit,and Sci is the average Schmidt numberof the element i and is defined as theratio of kinematic viscosity of theshielding gas and the average diffusivi-ty of element i in the shielding gas.

Composition Change in the Keyhole Mode Weld Pool

In this calculation, the temperatureof the keyhole wall is assumed to beconstant and slightly higher than theboiling point, which is iterated to ob-tain good agreement with the experi-mental data. For the conditions of allexperiments, a value of 10 K for 204stainless steel and a value of 9 K for304 stainless steel above the boilingpoints resulted in good agreement ofthe computed and the experimentallydetermined weld metal compositions.The vapor flux from the keyhole, Jk,i, isthe sum of the diffusion driven fluxJc,i, and the pressure driven flux, Jp,i,and is given by

(5)

However, the vapor flux from theweld pool surface, Js,i, is only driven bythe diffusion, which is given by

(6)

The geometry of the keyhole walland the surface of the weld pool can becomputed from the heat transfer andfluid flow model. Then, the vaporiza-tion rate of element i, Gi is obtained by

integrating the vapor flux from boththe keyhole wall and the weld pool sur-face. The total vaporization rate, G, isgiven by

(7)

where Sk and Ss denote the internalsurface area of the keyhole wall andthe area of the weld pool top surfaceoutside the keyhole, respectively. The final concentration of element i

Kg ,i =2Sci

0.42Re0.5Did

1+ Re0.55

200

�

��

��

0.5

0.483�0.108rd+7.71�10�3

rd

��

��

2�

��

�

��

Jk,i = Jp,i + Jc,i

Js,i = Jc,i

G = Gii=1

n

�i=1

n

� Jk,i dxdy+Sk��i=1

n

�

Ss�� Js,idxdy

Jc ,i = Kg ,i MiaiPi

0

RTl�Ci

b�

��

��

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Fig. 5 — Experimental and calculated weld pool transverse cross sections for variouswelding parameters: A — P = 1200 W, v = 8 mm s1; B — P = 1250 W, v = 8 mm s1; C — P= 1300 W, v = 8 mm s1; D — P = 1350 W, v = 8 mm s1; E — P = 1300 W, v = 6 mm s1; F— P = 1300 W, v = 12 mm s1.

BA

E F

DC


after welding, Ci, is given by:

(8)

where is the density of the liquidmetal, v is the welding speed, A is thetransverse cross section area of theweld, and Cb,i is the concentration ofelement i in the base metal. Then thecomposition changes are calculated by:

(9) The steps involved in the calcula-tion of the vaporization rate and com-position change during keyhole modelaser welding are presented as a flowchart in Fig. 3.

Results and Discussion

Temperature Field and WeldPool Geometry

Figure 4 shows the computed tem-perature and velocity fields at the topsurface and the symmetry plane of thewelds for various laser powers andwelding speeds. The boiling point con-tours mark the keyhole boundarywhereas the solidus isotherms indicatethe weld pool periphery. On the topsurface of the molten pool, the moltenmetal flows outward from the middleto the periphery driven by theMarangoni force, which results fromthe spatial variation of the surfacetension owing to the temperature gra-dient. The velocities decrease and theweld pool narrows below the top sur-

face. Due to the motion of the laserbeam, the temperature contours arecompressed in the front of the heatsource and elongated behind it. Note that complete joint penetrationwelds are achieved for the power andspeed indicated in Fig. 4B, C, D, and E,while partial joint penetration welds areobtained for the welding conditions inFig. 4A and F. For the complete jointpenetration welds, the weld pool ex-tends to the bottom surface, and a flowpattern of liquid metal similar to that atthe top surface is attained because ofthe Marangoni force. The weld pool iswider both on the top and bottom sur-faces and narrower in the middle. The maximum velocities of the liq-uid metal in the molten pool for vari-ous welding conditions shown in Fig. 4are roughly 150 mm s-1, and at thesevelocities convection is the dominantmechanism of heat transfer. The im-portance of heat transfer by convec-tion relative to heat transfer by con-duction can be determined by thePeclet number, Pe, which is given by

(10)

where is the characteristic velocity, is the density, Cp is the specific heat, L isthe characteristic length, and k is thethermal conductivity. Using = 0.15 ms-1, = 7800 Kg m-3, Cp = 836.0 J kg-1 K-1,L = 1.0 × 10-3 m, and k = 20.93 J m-1 s-1

K-1, Pe = 46.7. This value of Peclet num-ber indicates that convective heat trans-fer is the dominant mechanism of heattransfer within the weld pool. Figure 5 shows the experimental andcalculated weld pool transverse sectionsfor various welding parameters. A widefusion zone on the top surface is pro-duced due to the convective heat trans-fer within the molten pool. For the com-plete joint penetration welds, the fluidflow pattern on the bottom surface issimilar to that on the top surface, andthe widening of the fusion zone is alsoobserved. The computed weld poolgeometry is in fair agreement with thecorresponding experimental results forall the welding conditions.

Vaporization and CompositionChange

Figure 6A shows the computed va-

Ci =�vACb,i �Gi

�vA�G

Pe =μ�CpL

k

�Ci =Ci �Cb,i

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Fig. 6 — Equilibrium vapor pressures of the various alloying elements over the following: A — Respective pure liquids; B — liquid 204 stainless steel; C — liquid 304 stainlesssteel; D — total vapor pressures as a function of temperature.

B

C D

A


por pressures of various alloying ele-ments over their pure liquids at vari-ous temperatures. It can be seen thatthe equilibrium vapor pressure ofmanganese is higher than those ofchromium, iron, and nickel. Assumingthat the molten steels behave as idealsolutions, the equilibrium vapor pres-sures of alloying elements over the204 stainless steel liquid steel is ob-tained by multiplying the vapor pres-sures of Fig. 6A by their respectivemole fractions. These values areshown in Fig. 6B as a function of tem-perature. The equilibrium vapor pres-sure of iron is higher than that ofmanganese when the temperate ex-ceeds 3100 K because of its high molefraction of iron. The boiling points of the alloys aredetermined by computing the equilib-rium vapor pressures of the alloyingelements over the liquid alloys as afunction of temperature and deter-mining the temperature at which thesum of the equilibrium vapor pres-sures of all alloying elements add up to1 atmosphere. Figure 6C and D showthe equilibrium vapor pressures of thevarious alloying elements over 304stainless steel and the total equilibri-um vapor pressures over the two liquid

alloys 204 and 304 stainlesssteels, respectively. In calcu-lating the equilibrium vaporpressures over the alloys, itwas assumed that the metal-lic solutions behaved ideallyand their activities were equalto their respective mole frac-

tions. The boiling point of the two al-loys were found to be 2844 and 3051K for 204 and 304 stainless steels, re-spectively. Figure 7 shows the computed tem-perature field and vapor flux on thetop surface of the weld pool. As can beseen, the spatial distribution of vaporfluxes is similar to that of the temper-ature profiles with larger fluxes origi-nating from regions of higher temper-atures. The temperature inside thekeyhole is slightly higher than theboiling temperature of the alloy anddecreases sharply outside the keyholefrom 2844 to 2200 K in a small circu-lar region of approximately 0.3 mm di-ameter. Since the temperature outsidethe keyhole is lower than the boilingtemperature of the alloy, the vaporfluxes driven by the concentration gra-dient are much lower than the fluxesfrom inside the keyhole, which isshown in Fig. 7B and C. As anticipated,the vaporization is most pronouncedin the keyhole zone. The total vaporflux from the keyhole is larger thanthat from outside the keyhole becauseof the large internal surface area of thekeyhole and the pressure driven flux. Figure 8 shows the typical concen-tration profiles of manganese and

chromium along a horizontal monitor-ing line across the transverse sectionof the weld. The concentration ofmanganese decreases significantly inthe fusion zone compared with that inthe based metal. The loss of man-ganese concentration is consistentwith the high equilibrium vapor pres-sure shown in Fig. 6B. Apart from theusual scatter, the decrease in man-ganese concentration in the fusionzone does not show any trend of spa-tial variation, which indicates that theweld pool was well mixed because ofthe recirculating flow of the liquidmetal. An interesting feature of Fig. 8 isthat the concentration of chromium inweld metal is higher than that in thebase metal. Although chromium va-porizes during welding, more pro-nounced loss of mass of the other al-loying elements, particularly iron, re-sults in an increase of chromium con-centration. This behavior is consistentwith the mass balance expressed byEquation 8, which indicates that theconcentration of an element may in-crease because of the more pro-nounced loss of other alloying ele-ments. Similarly, a small increase inthe concentrations of nickel and ironcan be observed in Fig. 9. The calculated changes in composi-tion of the various alloying elementsagree well with the correspondingmeasured values. In the high-man-ganese stainless steel, the concentra-tion of manganese decreases by about1.3 wt-% for a laser power of 1250 W.

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Fig. 7 — Temperature field and the vapor fluxes onthe top surface of the weld pool for 204 stainlesssteel. P = 1250 W, v = 8 mm s1.

Fig. 8 — Concentration profiles of manganese and chromium across the fusion zone in the transverse section of 204 stainless steel weld. The solid line isthe computed result. P = 1250 W, v = 8 mm s1.

A

B

C


The concentrations of chromium andiron increase by about 0.2 and 1.2 wt-%, respectively. The change in the con-centration of nickel was insignificant,indicating that the loss of nickel wassimilar in proportion to the overallloss of all alloying elements by vapor-ization. Figure 9 shows the comparison ofthe alloying element loss in the twostainless steel welds with differentmanganese concentrations. As antici-pated, the decrease in the concentra-tion of manganese was much morepronounced in 204 stainless steel thanthat in 304 stainless steel because ofthe difference in the manganese con-centrations of the two steels. All alloy-ing elements are lost from both steelsduring welding. However, the extentof loss of an individual alloying ele-ment such as chromium relative to the

total loss of all alloying elements de-termine its concentration in the weldmetal. Although the change in the concen-tration of chromium was small in bothsteels, its concentration decreased in304 whereas the chromium concentra-tion was slightly higher than that inthe 204 steel base metal. The total rateof vaporization of all alloying elementswas much more pronounced in the204 stainless steel. Furthermore, thevaporization rate of chromium wasmore pronounced from the 304 stain-less steel because of its higher concen-tration in this steel. The higher vapor-ization rate of chromium comparedwith the total vaporization rate of allalloying elements resulted in a loss ofchromium concentration in 304 stain-less steel. In contrast, in 204 stainlesssteel, the vaporization rate of chromi-

um was lower than that from 304steel. In addition, the overall vaporiza-tion rate from 204 steel was higherthan that from 304. As a result, thechromium concentration showed a de-crease in the weld metal.

Influence of WeldingParameters

Figure 10A shows the influence oflaser power on the compositionchange of 204 stainless steel. The ex-perimental data show considerable de-crease of manganese concentration inall laser powers in the range of 1200 to1350 W. The change in concentrationof manganese is slightly higher at low-er laser power. It is instructive to re-view the causative factors for this be-havior based on Equation 8, whichshows that the composition change is

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Fig. 9 — Comparison of the calculated and the experimentally determined composition change at a laser power of 1250 W and weldingspeed of 8 mm/s for the following: A — 204 stainless steel ; B — 304 stainless steel.

Fig. 10 — Influence of laser power on the following: A — Composition change; B — vaporization rate and transverse cross section area ofthe fusion zone during laser welding of 204 stainless steel at a welding speed of 8 mm s1.

BA

BA


affected by the vaporization rate andthe melting rate from which the va-pors are lost. At a constant welding speed, themelting rate is the product of density, ,the welding speed, v, and the area of thetransverse cross section of the fusionzone, A. Both the vaporization rate andthe area of the fusion zone in the trans-verse section increases with the increasein laser power from 1200 to 1250 W.The mode of welding changes from par-tial to complete joint penetration andboth the keyhole and the weld pool be-come deeper and wider. Figure 10B shows that when thelaser power increases from 1250 to1350 W, the cross section area increas-es while the vaporization rate does notchange significantly due to the in-significant change of the keyhole size— Fig. 4. Therefore, in this powerrange, the composition change be-comes less pronounced as the laserpower increases as shown in Fig. 10A. As shown in Fig. 11A, the influenceof welding speed on the compositionchange of 204 stainless steel is negligi-ble. The variation of melting rate andthe vaporization rates with weldingspeed are plotted in Fig. 11B becausethey are important in determining thecomposition change as indicated byEquation 8. It can be observed that boththe vaporization and melting rates de-crease with the increase in weldingspeed. The vaporization rates are higherfor complete joint penetration welds atwelding speeds of 6 and 8 mm s-1 thanfor the partial joint penetration welds ata welding speed of 12 mm s-1. The key-

hole size is significantly larger in com-plete joint penetration welds as shownin Fig. 4. Furthermore, the higher weld-ing speed reduces the cross section areaof the fusion zone as shown in Fig. 5. Asa result, the composition change is notsignificantly affected by the weldingspeed.

Summary and Conclusions Loss of alloying elements because ofvaporization and the resulting composi-tion change of the weld metal duringkeyhole mode laser welding of two

stainless steels were examined experi-mentally and theoretically. Concentra-tion profiles of alloying elements in thewelds were determined for both partialand complete joint penetration welds. Awell-tested numerical model was usedto compute the keyhole and weld poolgeometry and the temperature fields forvarious laser powers and weldingspeeds. These simulations were com-bined with a model developed to calcu-late weld composition change based onthe principles of transport phenomena,kinetics, and thermodynamics. Thegood agreement of the composition

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Fig. 11 — Influence of welding speed on the following; A — Composition change; B — vaporization rate and melting rate during laserwelding of 204 stainless steel for a laser power of 1300 W.

Fig. 12 — Schematic diagram of the velocity distribution functions in the Knudsen layerand in adjacent regions. Tboiling is the boiling temperature of the alloy; Tl , l , and Pl , arethe temperature, density, and pressure of the vapor on the liquid surface, respectively; Tv ,v , and Pv ,are the temperature, density, and pressure of the vapor at the edge of theKnudsen layer, respectively, T2 , 2 , and P2 are the temperature, density, and pressure ofthe vapor behind the wavefront, respectively; and Tg , g , and Pg are the temperature,density, and pressure of the vapor in front of the wavefront, respectively. R is the gas constant, is the velocity component normal to the vaporizing surface, u is the mean velocityof the local gas, and is the proportionality coefficient.

A B


change of alloying elements betweenthe calculated result and the correspon-ding experimental data indicates thatthis model is capable of predicting com-position change in keyhole mode laserwelding for various conditions. Themain conclusions are as follows: 1. In all cases of partial and completejoint penetration keyhole mode linearwelding of two stainless steels, theshape, size and the internal surface areaof the keyhole, the surface area of themolten pool, and the temperature fieldaffected the vaporization rates of all al-loying elements. These variables wereaffected by the laser power, power den-sity, and the welding speed. The changein the composition of the weld metalwas affected by the relative rates of va-porization of various alloying elementsand the melting rates of the base metal.Because of the interdependence ofmany variables, the role of welding vari-ables on the final chemical compositionof the alloying elements could not bepredicted intuitively. However, the pro-posed computational model could cor-rectly predict the roles of laser powerand welding speed on the compositionchanges. 2. An increase in laser power keep-ing the welding speed constant result-ed in an expected larger weld pool andhigher rates of vaporization of alloyingelements. However, the faster vapor-ization did not result in more pro-nounced changes in composition sincethe vaporization of alloying elementsaffected the chemical composition of asignificantly larger volume of weldmetal at higher laser powers. Higherlaser power resulted in lower composi-tion change of alloying elements dueto the significant increase of the melt-ing rate compared with the modest in-crease of the vaporization rate. 3. Higher welding speed resulted inboth a lower vaporization rate and areduced melting rate for both partialand complete joint penetration welds.The composition change was not sen-sitive to the welding speed because ofthe compensating effects of these twoparameters. 4. Vaporization of iron, chromium,manganese, and nickel took place fromthe keyhole for partial and completejoint penetration welds for both 204and 304 stainless steels. Pronouncedvaporization of iron and chromium oc-curred for all laser powers and welding

speeds investigated for both steels.Rate of vaporization of manganesewas much higher for 204 stainlesssteel than for 304 steel. 5. The vaporization of alloying ele-ments from the keyhole was driven byboth concentration and pressure gra-dients while outside the keyhole therates of vaporization were influencedby the concentration gradients of met-al vapors. 6. As expected, pronounced vapor-ization of alloying elements occurredfor both partial and complete jointpenetration welds mainly from thekeyhole due to high local tempera-tures. A keyhole wall temperature ofabout 10 K higher than the boilingpoints of the alloys could correctly ex-plain the rates of vaporization of alloy-ing elements for both alloys studied inpartial and complete joint penetrationkeyhole welds. 7. The concentration of manganesein the weld metal was significantlylower than that in the base metal forthe welding of 204 stainless steelmainly because of its high concentra-tion of manganese. In contrast, for thewelding of 304 stainless steel, the de-crease in the concentration of chromi-um in the weld metal was greater thanthat of manganese. Because of thehigher boiling point and higher con-centration of chromium in 304 than204 stainless steel, the loss of chromi-um was more pronounced in 304stainless steel.

The authors would like to thank S.L. Gao and R. X. Yang for performingthe laser welding experiments, as wellas James S. Zuback for assisting withthe metallography. This research wassupported by the Natural ScienceFoundation of Tianjing (grant no.16JCZDJC38700).

1. Zhan, X., Chen, J., Liu, J., Wei, Y.,Zhou, J., and Meng, Y. 2016. Microstruc-ture and magnesium burning loss behaviorof AA6061 electron beam welding joints.Materials & Design 99: 449–458. 2. Anisimov, S. I., and Rakhmatulina, A.K. 1973. The dynamics of the expansion ofa vapor when evaporated into a vacuum.

Soviet Physics — JETP 37(3): 441–444. 3. Knight, C. J. 1982. Transient vapor-ization from a surface into vacuum. AIAAJournal 20(7): 950–954. 4. Knight, C. J. 1979. Theoretical mod-eling of rapid surface vaporization withback pressure. AIAA Journal 17(5):519–523. 5. Mundra, K., and DebRoy, T. 1993.Calculation of weld metal compositionchange in high-power conduction modecarbon-dioxide laser-welded stainless-steels. Metallurgical Transactions B 24(1):145–155. 6. Zhao, H., and DeBroy, T. 2001. Weldmetal composition change during conduc-tion mode laser welding of aluminum alloy5182. Metallurgical and Materials Transac-tions B 32(1): 163–172. 7. He, X., DebRoy, T., and Fuerschbach,P. W. 2003. Alloying element vaporizationduring laser spot welding of stainless steel.Journal of Physics D-Applied Physics 36(23):3079–3088. 8. He, X., DebRoy, T., and Fuerschbach,P. W. 2003. Probing temperature duringlaser spot welding from vapor compositionand modeling. Journal of Applied Physics94(10): 6949–6958. 9. Khan, P. A. A., and DebRoy, T. 1984.Alloying element vaporization and weldpool temperature during laser-welding ofAisi 202 stainless-steel. MetallurgicalTransactions B 15(4): 641–644. 10. Mundra, K., and DebRoy, T. 1993. To-ward understanding alloying element vapor-ization during baser beam welding of stain-less steel. Welding Journal 72(1): 1-s to 9-s. 11. Collur, M. M., and DebRoy, T. 1989.Emission-spectroscopy of plasma duringlaser-welding of Aisi-201 stainless-steel.Metallurgical Transactions B 20(2):277–286. 12. Collur, M. M., Paul, A., and DeRoy,T. 1987. Mechanism of alloying elementvaporization during laser-welding. Metal-lurgical Transactions B 18(4): 733–740. 13. Sahoo, P., Collur, M. M., and De-bRoy, T. 1988. Effects of oxygen and sulfuron alloying element vaporization rates dur-ing laser-welding. Metallurgical Transac-tions B 19(6): 967–972. 14. Dilthey, U., Goumeniouk, A.,Lopota, V., Turichin, G., and Valdaitseva, E.2001. Development of a theory for alloy-ing element losses during laser beam weld-ing. Journal of Physics D—Applied Physics34(1): 81–86. 15. Jandaghi, M., Parvin, P., Torkamany,M. J., and Sabbaghzadeh, J. 2008. Alloyingelement losses in pulsed Nd : YAG laser weld-ing of stainless steel 316. Journal of PhysicsD: Applied Physics 41(23): 235503. 16. Jandaghi, M., Parvin, P., Torkamany,M. J., and Sabbaghzadeh, J. 2009. Measure-ment of the composition change in Al5754alloy during long pulsed Nd : YAG laser weld-

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JULY 2017 / WELDING JOURNAL 267-s

Acknowledgments

References


ing based on LIBS. Journal of Physics D: Ap-plied Physics 42(20): 205301. 17. Bag, S., Trivedi, A., and De, A. 2008.Use of a multivariate optimization algo-rithm to develop a self-consistent numeri-cal heat transfer model for laser spot weld-ing. The International Journal of AdvancedManufacturing Technology 38(5-6):575–585. 18. Rai, R., Elmer, J. W., Palmer, T. A.,and DebRoy, T. 2007. Heat transfer andfluid flow during keyhole mode laser weld-ing of tantalum, Ti–6Al–4V, 304L stainlesssteel and vanadium. Journal of Physics D:Applied Physics 40(18): 5753–5766. 19. Rai, R., Kelly, S. M., Martukanitz, R.P., and DebRoy, T. 2008. A convective heat-transfer model for partial and full penetra-tion keyhole mode laser welding of a struc-tural steel. Metallurgical and MaterialsTransactions A 39(1): 98–112. 20. Rai, R., Roy, G. G., and DebRoy, T.2007. A computationally efficient model ofconvective heat transfer and solidificationcharacteristics during keyhole mode laserwelding. Journal of Applied Physics 101(5):054909. 21. Ribic, B., Rai, R., and DebRoy, T.2008. Numerical simulation of heat trans-fer and fluid flow in GTA/Laser hybridwelding. Science and Technology of Weldingand Joining 13(8): 683–693. 22. Ribic, B., Tsukamoto, S., Rai, R., andDebRoy, T. 2011. Role of surface-active ele-ments during keyhole-mode laser welding.Journal of Physics D: Applied Physics 44(48):485203. 23. Schlunder, E., and Gnielinski, V.1967. Heat and mass transfer between sur-faces and impinging jets. Chemie IngenieurTechnik 39: 578–584. 24. Zhao, H., and DebRoy, T. 2003.Macroporosity free aluminum alloy weld-ments through numerical simulation ofkeyhole mode laser welding. Journal of Ap-plied Physics 93(12): 10089–10096. 25. Rai, R., Burgardt, P., Milewski, J. O.,Lienert, T. J., and DebRoy, T. 2009. Heattransfer and fluid flow during electronbeam welding of 21Cr–6Ni–9Mn steel andTi–6Al–4V alloy. Journal of Physics D: Ap-plied Physics 42(2): 025503. 26. Honig, R. E., and Kramer, D. A.1970. Physicochemical Measurements in Met-al Research. 505, New York, NY, Inter-science Publishers. 27. Hultgren, R., Desai, P. D., Hawkins,D. T., Gleiser, M., Kelley, K. K., and Wag-man, D. D. 1973. Selected Values of the Ther-modynamic Properties of the Elements. Mate-rials Park, Ohio, ASM International. 28. Yaws, C. L. 1994. Handbook of VaporPressure. Houston, Tex., Gulf Publishing Co. 29. Alcock, C. B., Itkin, V. P., and Horri-gan, M. K. 1984. Vapor pressure equationsfor the metallic elements. Canadian Metal-lurgical Quarterly 23(3): 309–313.

Appendix A1. Keyhole Geometry

The keyhole geometry was calculat-ed from local heat balance on the key-hole wall where temperature is as-sumed to be the boiling point of the al-loy (Ref. 24). Since the orientation ofthe keyhole is almost vertical, the heattransfer occurred mainly along thehorizontal plane. The local keyholewall angle with the vertical direction is given by (Refs. 24, 25):

(A1)

where Ic is the heat flux conducted intothe keyhole wall, Ia is the locally ab-sorbed beam energy flux, and Iv is theevaporative heat flux on the keyholewall. The value of Ic is calculated froma two-dimensional temperature fieldin an infinite plate with the line heatsource (Ref. 24):

(A2)

where (r, ) indicates a location in theplate with the line source as the origin,T is the temperature, and is the ther-mal conductivity. The two-dimension-al temperature field, T(r, ), can be cal-culated conduction heat from the key-hole wall into the plate (Ref. 24):

(A3)where Ta is the ambient temperature,P’ is the power per unit depth, K0( ) isthe zero-order second kind modifiedBessel function, and = v/(2), wherev is the welding speed and is thethermal diffusivity. The locally absorbed beam energyflux, Ia, on the keyhole wall is obtainedby considering the Fresnel absorptionduring multiple reflections and theplasma absorption (Ref. 24):

(A4)where is the plasma attenuation co-efficient, L is the average path of thelaser beam in plasma before it reachesthe keyhole wall, is the absorptioncoefficient of the workpiece, ̅is the

average angle between the keyholewall and the initial incident beam axis,and I0 is the local beam intensity,which varies with depth from the sur-face and radial distance from the beamaxis, and given by (Ref. 24):

(A5)where Ip is the peak intensity at the fo-cal point, given by 2P/(r0

2), P is thelaser power, r0 is the beam radius atthe focal point, and rf is the local beamradius, which is calculated by (Ref. 24):

(A6)where z, and z0, are the depth and thebeam defocusing, respectively. l is thebeam focal length, and db is the beamdiameter on the laser focusing lens. The vaporization flux, Iv, on thekeyhole wall is obtained from Ref. 24:

(A7)where Ji is the vaporization flux of ele-ment i, Hi is the heat of vaporizationof element i, and n is the total numberof alloying elements in the alloy. Forsimplicity, the vaporization flux of ele-ment i was calculated by using theLangmuir equation (Refs. 12, 13)

(A8)

where Pi is the vapor pressure of ele-ment i over the alloy, Mi is the molecu-lar weight of element i, Tb indicates theboiling temperature of the alloy, and Ris the gas constant. The factor 7.5 ac-counts for lower vaporization rates at1 atmosphere pressure compared withthat in vacuum-based experiments. The keyhole profile was calculatedfor various horizontal planes from topto bottom. Initially, on the top surface,the local angles of the front and rearwalls of the keyhole were calculatedfrom Equation A1. The asymmetric key-hole boundary was determined by botha maximum x value and a minimum xvalue for each y value where x is thewelding direction and y is the width di-rection. The keyhole wall positions in

T r ,�( )=Ta +P'2��

K0 �r( )e��r cos�

Ia = e��L 1� 1��( )1+�/4�( )I0

I0 = Ipr0rf

�

��

��

2

exp � r2

rf2

�

��

�

��

rf = r0 1+ z+ z02r0l / db

��

��

2�

��

�

��

1 2

Ic r ,�( )= �� T r ,�( )�r

tan �( )= IcIa � Iv

Iv = Ji�� ii=1

n

�

Ji =Pi7.5

Mi2�RTb

WELDING RESEARCH



the plane right below the top surfacecould be calculated from the current po-sitions and the local angles. Similarly,the keyhole positions in various planesdownward were determined. Using thecalculated position of front and rearwalls of the keyhole, the strength of theline source P’ at any given depth werecalculated. Using the calculated strengthand the location of the line source, thetemperature field in the x-y plane ateach depth were determined. Subse-quently the temperature field of the en-tire calculation domain was computedconsidering the energy transferred fromthe keyhole wall. The temperature andvelocity fields in the vapor within thekeyhole were not calculated. Heat trans-fer and fluid flow in the base plate out-side the keyhole was computed as fol-lows.

2. Heat Transfer and Fluid Flow in theWeld Pool

The temperature and velocity fieldsin the liquid region and the tempera-ture field in the solid region were cal-culated by solving the equations ofconservation of mass, momentum,and energy in three dimensions. It isassumed that the molten metal is in-compressible, laminar, and Newtonianfluid. The momentum conservationequation of the fluid flow in the weldpool is given as (Ref. 19):

(A9)where is the density, t is the time, xi

is the distance along the i th (i = 1, 2,and 3) orthogonal direction, uj is thevelocity component along the j direc-tion, is the effective viscosity, and Sj

is the source term for the j th momen-tum, equation due to frictional dissi-pation in the mushy zone, buoyancysource, and the relative motion be-tween the laser source and the workpiece. The pressure field can be obtainedby solving the momentum equation inconjunction with the following conti-nuity equation (Ref. 19):

(A10)

The thermal energy transportationin the weld workpiece can be ex-

pressed by the following modified en-ergy equation (Ref. 19):

(A11)

where k is the thermal conductivity,and Sh is the source term due to the la-tent heat content. The calculationprocess of the source term Sj and Sh iswell documented in the literature(Refs. 18–20), and not discussed here.

3. Boundary Conditions

The calculation is based on a 3DCartesian coordinate system. Since theweld is symmetrical about the center-line, only half of the workpiece is con-sidered. Further discussion of theboundary conditions are representedas follows: A. Top surface — On the top sur-face, the weld pool except the keyholeregion is assumed to be flat. The veloc-ity components along the x and y di-rections, u and v, are determined fromthe Marangoni effect. However, thevelocity components along the z direc-tion, w, is 0 due to the negligible out-ward flow at the top surface. There-fore, the velocity boundary conditioncan be obtained as the following equations:

(A12)

where fL is the liquid fraction, andd/dT is the temperature coefficient ofsurface tension. The heat flux at the top surface canbe derived from the heat input, whichfollows a Gaussian heat distribution,the heat loss by radiation, and the heatloss by convection. The expression isgiven as follows:given as:

(A13)where rb is the beam radius, f is thepower distribution factor, Q is the to-tal laser power, is the absorptivity, is the Stefan–Boltzmann constant, hc

is the heat-transfer coefficient, and Ta

is the ambient temperature. B. Symmetric plane — The velocitycomponents and heat flux along the ydirection across the symmetric surfaceare zero. As a result, the boundaryconditions can be defined as:

(A14) C. Keyhole surface — In this model,mass flux due to convection at thekeyhole surface is neglected. As a re-sult, the velocity components perpen-dicular to the keyhole surface are as-signed 0. Since the temperature of thekeyhole wall is treated as boiling pointof the alloy, h is equal to the sensibleheat of the boiling temperature:

(A15)

D. Bottom surface — For partial jointpenetration welds, the velocities arezero at the bottom surface, and the con-vective heat transfer is considered asthe boundary condition. For completejoint penetration welds, the velocityboundary conditions are similar withthat at the top surface under theMarangoni effect, and are given as thefollowing:

(A16) E. Solid surfaces — The velocities

μ �u�z

= ƒLd�dT

�T�x

μ �v�z

= ƒLd�dT

�T�y

w=0

�

�

��

�

��

k��z top=

ƒQ��rb

2 exp �ƒ x2 + y2( )

rb2

�

��

�

��

� T4 �Ta4( )�hc T �Ta( )

��uj�t

+�� uiu j( )�xi

= ��xi

μ�uj�xi

��

��+Sj

� �ui( )�xi

=0

�u�y

=0

v =0�w�y

=0

�h�y

=0

�

�

��

�

��

h= hboil

��h�t

+�� uih( )�xi

= ��xi

kCp

�h�xi

�

��

�

� +Sh

μ �u�z

= fLd�dT

�T�x

μ �v�z

= fLd�dT

�T�y

w=0

�

�

��

�

��

WELDING RESEARCH



are set to be 0, and the temperaturesare set at ambient temperature at allsolid surfaces far away from the heatsource.

Appendix BVapor Flux Due to Pressure Gradient

For welding at atmospheric pres-sure, when the local temperature onthe liquid surface exceeds the boilingpoint of the alloy, the local equilibriumpressure exceeds the ambient pressureand vaporization occurs due to pres-sure gradient. Knight (Ref. 4) deriveda set of equations that describe the va-porization rate under these condi-tions. The derivation of these equa-tions is not repeated here. Instead, thesalient features and the final expres-sions are indicated. The vapor molecules that escapefrom the liquid surface, their only pathis away from the liquid. As a result, thevelocity distribution function is half-Maxwellian as shown schematically inFig. 12. In the Knudsen layer, which is a thinlayer close to the interface having athickness of a few mean-free paths, thevelocity distribution can vary from to . Many of the molecules that va-porize from the surface, recondenseback on the surface and the net vapor-ization rate is the difference betweenthe two rates. Based on these concepts,the temperature Tv, the density v, thepressure Pv, and the mean velocity u ofthe vapor at the edge of the Knudsenlayer can be related to the temperatureTl, density l, and pressure Pl of the va-por on the liquid surface. The derivedjump conditions across the Knudsenlayer are given (Refs. 2, 4):

(B1)

(B2)

(B3)

where the dimensionless velocity, m, isgiven by m = u/√2Rv Tv, Rv = R/Mv, R isthe gas constant, Mv is the average mo-lecular weight of the vapor, v is the ra-tio of specific heat of the vapor, whichis treated as a monatomic gas (v=5/3),erfc is the complimentary error func-tion, erfc(m) = (2/)me–v2dv, and isthe condensation factor. The equilibri-um vapor pressure Pl at the interface isobtained from the equilibrium vaporpressure-temperature relationship ofthe various alloying elements:

(B4)

where Pi is the equilibrium vapor pres-sure of element i over the alloy, ai is theactivity of the element i in the liquidmetal, which is determined by its con-centration in the weld pool. In this cal-culation, the mole fraction of the ele-ment i was used as the value of ai be-cause alloys behave ideally close to theirboiling points. Pi

0 is the equilibrium va-por pressure of element i over pure liq-uid, and n is the number of alloying ele-ments. The equations used to calculatePi

0 are presented in Table 4.

The average molecular weight ofthe vapor, Mv, is given by:

(B5)

where Mi is the molecular weight of el-ement i. The relationship of the pressure atthe edge of the Knudsen layer and theambient conditions is obtained by ap-plying the Rankine-Hugoniot relation:

(B6)where Pg and P2 are the pressures infront of and behind the wavefront, re-spectively, P2 = Pv, g is the ratio of thespecific heats for the shielding gas,=vRvTv/gRgTg, and M is the Machnumber, which is related to dimen-sionless velocity, m, by the equation:

(B7)

The Mach number M and the densi-ty v obtained by solving equationsB1–7, can be used to calculate the va-porization flux. The unknown vari-able, u, can be obtained by solvingthese equations. Jp, owing to the pres-sure gradient on the keyhole wall sur-face as follows:

(B8)where S is the speed of sound in thevapor at temperature Tv.

Pl = Pii=1

n

� = aiPi0

i=1

n

�

Mv = Mii

n

� aiPi0

Pl

PlPg

P2Pl

=1+ � gM�

� g +14

M�+ 1+� g +14

M��

��

2

�

��

�

��

m=M� v2

�= 2m2 +1( )�m � TlTv

�

��

�

e

m2 �l�v

TlTv

�v�l

= TlTv

m2 + 12

��

�� e

m2erfc m( ) m

��

��

+ 12TlTv

1 mem2erfc m( )�

� ��

�v�l

=1+� � v �1

� v +1m2

��

�

2

� � � v �1� v +1

m2

�

��

�

�

��

2

Jp =�vMS

WELDING RESEARCH


Table 4 — Equilibrium Vapor Pressures of Various Elements in Atmospheres as a Function of Temperature in K (Refs. 26–29)

Mn log P0 = 5.58 10–4T1.503 10–4/T + 16.615 Ni log P0 = 6.66620765/T Cr log P0 = 13.505 103/T+33.65 logT9.29 10–3T+ 8.381 10–7 T291.083 Fe log P0 = 11.55491.9538 104/T0.62549logT2.7182 10–9T +1.9086x10–13T22.881

T. LIU, L. J. YANG, and W. C. QIU are with the School of Materials Science and Engineering, Tianjin University, Tianjin, China. T. LIU, H. L. WEI andT. DEBROY ([email protected]) are with the Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pa.


composition change of stainless steels during …...of stainless steel, 204 (204 ss) and 304 (304...

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