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Scaling weld or melt pool shape induced by thermocapillary convection P.S. Wei ⇑, H.J. Liu, C.L. Lin Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 7 October 2011 Received in revised form 12 January 2012 Accepted 18 January 2012 Available online 20 February 2012 Keywords: Laser welding Electron beam welding Thermocapillary convection Moving boundary Scale analysis

a b s t r a c t The molten pool shape and transport variables induced by thermocapillary force during welding (or melting) of workpieces, such as pure iron, titanium, high speed steel and stainless steel alloys, can be self-consistently predicted from scale analysis. Determination of the molten pool shape and transport variables is crucial due to their close relationship with the microstructures, strength and properties of the fusion zone. In this study, the pool excludes a strongly wavy bottom and the surface velocity proﬁle has two peaks valid for Prandtl numbers lying between 0.3 and unity. The surface tension coefﬁcient is negative and suitable for all pure liquid metals and alloys containing minor surface active solutes, giving rise to an outward surface ﬂow. In view of high Marangoni number, the domain of scaling is divided into the hot and cold corner regions, boundary layers on the solid–liquid interface and ahead of the melting front. The results ﬁnd that the width and depth of the pool, peak and secondary peak surface velocities, and maximum temperatures in the hot and cold corner regions can be explicitly and separately determined as functions of working variables, or Marangoni, Prandtl, Peclet, Stefan, and beam power numbers and solid-to-liquid thermal conductivity ratio. The scaled results agree with numerical data and available experimental data. This work has academic and practical importance. Successful scaling not only reveals physical mechanisms, but also provides quantitative predictions of the fusion zone shapes and transport variables prior to melting or welding. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Microstructure and properties of workpieces after welding, melting, crystal growth, etc. are strongly determined by convection induced by thermocapillary force [1–6]. Thermocapillary ﬂow is the ﬂow driven by a temperature-dependent surface tension gradient along the free surface. Provided that surface temperature decreases in outward directions, energy transport induced by thermocapillary surface ﬂow with a negative surface tension coefﬁcient (dc/dT < 0) directs ﬂow from the center to the edge of the pool, leading to a shallow and wide molten pool. On the other hand, the pool becomes narrow and deep for a positive surface tension coefﬁcient. Even though the pool shape and transport variables induced by thermocapillary force have been extensively studied in recent decades [7–12], physical mechanisms and formation of the fusion boundary are still not well understood. Limmaneevichitr and Kou [13] observed the effects of Prandtl and Peclet numbers on the molten pool shape and Marangoni convection in stationary laser welding of gallium and sodium nitrate. The Peclet number, deﬁned by the product of the pool surface width and maximum surface velocity divided by thermal diffusivity, was actually the Marangoni number. During welding of gal⇑ Corresponding author. Tel.: +886 7 5254050; fax: +886 7 5254214. E-mail addresses: [email protected] (P.S. Wei), [email protected] (H.J. Liu), [email protected] (C.L. Lin). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2012.01.034

lium, low Peclet numbers promoted conduction down into the pool, and resulted in a concave bottom. For welding of sodium nitrate, however, high Peclet numbers promoted outward convection, and the pool bottom was shallow and ﬂat. In the case of a small beam radius the fast outward surface ﬂow turned and penetrated downward at the pool edge, resulting in a convex pool bottom. The molten pool shapes with small Prandtl numbers can be roughly identiﬁed by several regions [14]: (i) the molten pool has a hemispherical shape for Maf < 100, (ii) the bottom of the pool is convex near the centerline for 0.1 < Pr < 1 and 100 < Maf < 105, (iii) the bottom is slightly convex near the centerline of the shallow pool for 0.3 < Pr < 1 and Maf > 105, and (iv) the bottom exhibits a strong concave shape with concavity depth as high as one-half of the pool width for Pr < 0.1 and Maf > 100 and a concave shape for 0.1 < Pr < 0.3 and Maf > 105. Except for very small Prandtl numbers (for example, Pr = 0.06), leading to high pool depth, Wei et al. [15] showed that as Marangoni number increases in a range less than 103 the decrease in the pool depth is insensitive to the variation of Prandtl number. A further increase in Marangoni number results in the pool depth to continuously decrease and then increase. The minimum pool depth occurs at Pr = 1. The corresponding pool width, however, increases and then decreases as Marangoni number increases. Therefore, the width-to-depth ratio of the pool exhibits a rapid increase and then decrease by increasing Marangoni number. The width-to-depth ratio also decreases as Prandtl number decreases. The surface

P.S. Wei et al. / International Journal of Heat and Mass Transfer 55 (2012) 2328–2337

2329

Nomenclature b1, b2 h k L Ma, Maf Pew, Per q Q, Q⁄ Ste U u, v w x, y, z

empirical constants in Eq. (5) molten pool depth thermal conductivity latent heat for melting Marangoni numbers, deﬁned in Eqs. (32) and (13) Peclet numbers, deﬁned in Eqs. (6) and (32) incident ﬂux dimensional and dimensionless beam power, deﬁned in Eq. (8) Stefan number, deﬁned in Eq. (6) scanning speed horizontal and vertical velocity component molten pool width coordinate, as illustrated in Fig. 1

Greek letters c, dc/dT surface tension and surface tension coefﬁcient thicknesses of free surface layer, momentum and therd, D mal boundary layers, as illustrated in Fig. 2

temperature reveals the hot, intermediate and cold corner regions due to irradiation in the hot region and cooling near the edge of the pool. Surface temperature thus exhibits signiﬁcant drops, leading to peak surface velocities near the edge of the hot region and in the cold corner region, respectively. However, only one peak surface velocity occurs in the cold corner region for Prandtl number less than 0.3 [14]. Surface temperature in the hot region decreases with increasing Prandtl and Marangoni numbers. Irrespective of Prandtl number, the peak surface temperature increases with decreasing Peclet number or welding speed, and increasing beam power. Physical mechanisms of the pool shape and transport variables induced by thermocapillary force can be quantitatively and systematically revealed from scale analysis often used in heat transfer and ﬂuid mechanics ﬁelds [16]. Ostrach [17] was one of the earliest researchers to study thermocapillary ﬂow from scale analysis. By considering tangential shear stress to be the same order of magnitude as thermocapillary and inertial forces in the shear layer beneath the free surface, the maximum surface speed was found to be proportional to the surface tension coefﬁcient to the 2/3 power or Ma2/3. Chen [18] reviewed the scaling of thermocapillary convection in material processing for different Prandtl numbers. Scale analysis of thermocapillary convection was conducted in distinct regions in rectangular cavities subject to a centrally applied distributed heat ﬂux and differential wall temperatures, respectively. In order to satisfy momentum, energy and mass balances, the former included a hot corner, a free surface layer, a cold corner, a side wall boundary layer and core regions. It was proposed that scaling of the hot corner and free surface layer can be from Ostrach [17], the cold corner region from Zebib et al. [19], the core region from Cowley and Davis [20], and the boundary layer on the wall from conventional boundary layer theory, respectively. Comparisons of the scaled variables with numerical predictions and experimental data, however, were not presented. Kamotani et al. [21] also scaled the effects of constant ﬂux heating modes near the axisymmetric axis on thermocapillary ﬂows in a cylindrical container. For small Prandtl number and high Marangoni number the peak surface velocity and the beam power divided by temperature difference are proportional to the surface tension coefﬁcient to the 2/3 power and the 1/3 power, respectively. Kamotani and Ostrach [22] reconsidered the model from Cowley and Davis [20] by further accounting for a cold wall and

r g

energy distribution parameter melting efﬁciency

Superscript ⁄ dimensionless quantity 0 secondary 00 along solid–liquid interface Subscripts c, C cold corner f free surface ‘ liquid m melting s solid T thermal v viscous 1 surroundings r energy distribution parameter

thermocapillary ﬂow driven by temperature drop near the hot corner region for high Prandtl numbers. The thermocapillary pool shape induced by incident ﬂux, however, has not been presented in the literature. In this study, unknown fusion zone shapes coupled with thermocapillary convection for low Prandtl numbers during lowpower-density-beam welding or melting are self-consistently scaled. Different from previous studies, this work deals with a free boundary problem coupling with distinct boundary layers in different regions. This work has academic and practical importance. Successful scaling not only reveals physical mechanisms, but also provides quantitative predictions of the fusion zone shapes and transport variables prior to melting or welding.

2. System model Welding is conducted with a distributed low power-density laser or electron beam moving at a constant speed U along the joint line, as illustrated in Fig. 1(a) [23]. A typical ﬂow ﬁeld and fusion zone shape on a transverse x–y cross-section is sketched in Fig. 1(b). For a negative surface tension coefﬁcient thermocapillary surface ﬂow accompanying energy transfer is from the center to the edge of the pool, as shown in Fig. 1(c). High temperature gradient thus occurs near the pool edge. The major assumptions are described in a previous work [23]. That is, (1) The use of low power-density beam implies the absence of the keyhole in the molten pool. In the case of high-powerdensity beam welding, a keyhole is produced due to strong surface ﬂow induced by thermocapillary force and recoil pressure directs in outward and upward directions [24–26]. (2) The surface of the pool is ﬂat. This can be conﬁrmed by a small capillary number (Ca (dc/dT)DT/cm) [27]. Typical metals surface tension coefﬁcient dc/dT 104 N/m–K, temperature difference across the free surface DT 1000 K, and surface tension at the melting point cm 1 N/m. The capillary number is 0.1. More relevant estimation for deformation of the free surface can refer to the scale law provided by Wei [6]. It indicated that surface deformation can be neglected if surface velocity, density and molten pool width reduced and surface tension increased.

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Fig. 1. Schematic sketch for (a) three-dimensional welding process, (b) its equivalence to transient two-dimensional model, and (c) calculated ﬂuid ﬂow and temperature ﬁelds.

(3) The bottom of the pool is ﬂat or not deformed too much. This is applicable for Prandtl numbers lying between unity and 0.3 valid for pure iron, titanium, high speed steels, carbon and stainless steels, etc. The proﬁle of surface velocity also exhibits two peaks [14]. (4) The temperature-dependent surface tension can be expressed as c = cm + (dc/dT)(T Tm).The negative surface tension coefﬁcient, which is valid for all pure liquid metals and alloys containing minor surface active solutes, indicates that the surface tension increases with decreasing temperature. (5) Incident ﬂux absorbed is of a time-dependent Gaussian distribution [23], which is a function of beam power, energy distribution parameter, and scanning speed in the direction perpendicular to the transverse x–y cross-section (see Fig. 1(a)). (6) The transient two-dimensional model can be used to simulate the steady-state three-dimensional melting process [14]. This is because the melting rate in the depth direction is identical to that for melting the incoming solid along the scanning direction in a time scale of r/U [28]. That is,

rw

dh h rw ¼ whU dt r=U

ð1Þ

where the ﬁrst term represents volumetric melting rate with velocity dh/dt through the cross-sectional area rw in the depth direction, whereas the last term stands for volumetric melting rate with scanning speed through the cross-sectional area wh in the scanning direction. Transport variables to be scaled thus refer to those at the moment when the area of the pool is maximal.

Thermocapillary force responsible for the ﬂuid ﬂow is balanced by viscous stress along the free surface

l

@u dc @T ¼ @y dT @x

ð2Þ

The incident ﬂux is balanced by conduction into liquid at the free surface

q ¼ k‘

@T @y

ð3Þ

Heat conduction is removed by convection on the outer surfaces. 3. Scale analysis The domains for scale analysis in the molten pool with a high Marangoni number can be divided into hot and cold corner regions, boundary layers along the solid–liquid interface and ahead of the melting front, as illustrated in Fig. 2. In view of irradiation by the incident ﬂux and cooling from the cold solid, the peak surface temperature Tf occurs at the center of the pool, whereas a maximum temperature Tc takes place at the edge of the cold corner region. The signiﬁcant drops of surface temperature thus result in the peak surface velocity uf to occur near the edge of the incident ﬂux, and a secondary peak velocity u0f in the cold corner region. Since Prandtl number is less than unity, each region also contains a thicker thermal boundary layer than the momentum boundary layer. Incident energy is transferred via the surface shear layer from the hot, intermediate and corner regions, through the thermal boundary layer along the solid–liquid interface, and dissipated into the solid. Scale analysis of transport processes among the regions is described as follows:

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2331

In Eq. (5) empirical constants b1 = 1.4 and b2 = 0.3. The two terms in the parentheses in the denominator of Eq. (5) represent transverse heat conduction weighting between stationary and moving heat sources. The square root term refers to the side length of the fusion boundary. Melting efﬁciency therefore decreases with increasing the ratio between the area of the fusion boundary and transverse cross-sectional area of the fusion zone. It means that an increase in the side area of the fusion boundary enhances conduction in transverse directions and reduces energy for heating and melting the incoming solid. In view of energy transfer in directions other than the scanning direction, melting efﬁciency is always less than unity. Equation (4) in a dimensionless form becomes

gQ 1 þ

1 Pew Ste

ð7Þ

where the dimensionless beam power per unit depth is deﬁned as

Q

Q ks hðT m T 1 Þ

ð8Þ

Eq. (7), however, contains two unknowns, the width and depth of the molten pool. The other equation is thus needed and can be found from the scaling of ﬂuid ﬂow and heat transfer in the molten pool. 3.2. Hot region This region is irradiated by the incident ﬂux. Scaling of Eq. (2) for the balance between tangential and thermocapillary stresses in the shear layer beneath the free surface yields

l

dc T f T m uf dv H dT r

ð9Þ

where the shear layer thickness in the hot region dvH is determined by considering viscous stress as the same order of magnitude of inertia force. That is,

q Fig. 2. Schematic sketch of surface velocity and temperature, and distinct regions for analysis.

u2f

r

l

sﬃﬃﬃﬃﬃﬃ

Conservation of energy between incident energy and the energy used to heat and melt the incoming solid in the scanning direction is given by Wei et al. [29]

gQ q½cps ðT m T 1 Þ þ LUwh

ð4Þ

where the melting efﬁciency g [30] represents the fraction of beam power used to heat and melt the incoming solid. The introduction of melting efﬁciency in this work is important for unifying different expressions for predicting fusion zone shape suffered by experimental scattering in the literature [31–33]. It can be evaluated from total energy balance in the triangular slab model proposed by Hashimoto and Matsuda [31]. They accounted for energy required for heating, melting and evaporating the incoming solid and heat conduction transverse to the welding direction by using the simpliﬁed point source model. Since the heat conduction in the scanning direction was accounted twice, we propose a modiﬁed melting efﬁciency

1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ g¼ w 2 1 Ste Þ ðPew þ b2 Þ Steþ1 1 þ b1 1 þ ð2h

as

; Ste

cps ðT m T 1 Þ L

dv H

mr

ð11Þ

uf

Eq. (9) by substituting Eq. (11) leads to

uf w

m

2=3 Ma2=3 f Pr

w1=3

r

ð12Þ

where Marangoni number is deﬁned as

dc ðT f T m Þw Maf dT la‘

ð13Þ

The scaling of Eq. (3) for energy balance between incident energy and conduction to the liquid at the free surface is given by

Tf Tm gQ k‘ r2 h or in a dimensionless form

gQ r rk‘ ðT f T m Þ h

ð14Þ

ð5Þ 3.3. Cold corner region

where Peclet and Stefan numbers in Eq. (5) are deﬁned as

Uw

ð10Þ

d2v H

which gives

3.1. Solid region

Pew

uf

ð6Þ

The width of the cold corner region is identical to the thickness of the thermal boundary layer, which contains the viscous boundary layer on the wall (see Fig. 2). In view of the action of thermocapillary

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force, the surface velocity increases to a peak value at the edge of the viscous boundary layer and drops to zero at the wall. Therefore, the maximum temperature and secondary peak surface velocity occur at the edges of thermal and viscous boundary layers, respectively. 3.3.1. Thermal boundary layer The secondary peak of surface velocity at the edge of the viscous boundary layer can be scaled by considering the balance between shear stress and thermocapillary force

l

dc T f T m dv C dT w u0f

ð15Þ

Similar to Eq. (11), the momentum boundary layer thickness is given by

dv C

sﬃﬃﬃﬃﬃﬃﬃ mw u0f

ð16Þ

Substituting Eq. (16) into (15) leads to

u0f w

m

2=3 Ma2=3 f Pr

ð17Þ

Comparing Eq. (12) with Eq. (17) indicates the peak surface velocity in the hot region is always greater than that in the cold corner region. 3.3.2. Viscous layer Surface velocity driven by thermocapillary force is retarded by viscous stress due to rapid decrease to zero at the solid wall. Replacing ou/oy by ov/ox in the shear stress equation, Eq. (2) can be scaled as

l

v 0f Dv C

dc T c T m dT D

Fig. 3. Scale analysis based on energy balance in the solid surrounding molten pool.

where the terms on the left and right-hand sides, respectively, represent the vertical outﬂow and horizontal inﬂow through the cold corner region. The width of the cold corner region can be determined by momentum balance between inertia force and viscous stress in the horizontal direction

Dv C ð18Þ

vC

m

ð20Þ

u0f

Combining Eqs. (18)–(20) leads to

The conservation of mass is also required

v 0f Dv C ¼ u0f dv C

ð19Þ

u0f w

m

2=3 Ma2=3 f Pr

Tc Tm Tf Tm

2=3 ð21Þ

Table 1 Computed data used for scaling. dc/dT(N/m-K)

w(m) 103

h(m) 104

uf(m/s)

u0f (m/s)

Tf(K)

Tc(K)

g

l(kg/m-s) 102

0.0001 0.0003 0.0005 0.001 0.0001 0.0003 0.0005 0.001 0.0001 0.0003 0.0005 0.001 0.0001 0.0003 0.0005 0.001 0.0001 0.0003 0.0005 0.001 0.0001 0.0003 0.0005 0.001 0.0001 0.0003 0.0005 0.001

5.81 6.25 6.40 6.59 5.64 6.12 6.38 6.51 5.50 6.06 6.28 6.46 5.41 5.98 6.20 6.38 5.34 5.88 6.10 6.32 5.26 5.83 6.03 6.28 5.18 5.76 5.94 6.23

6.79 5.73 5.29 4.75 6.91 5.82 5.38 4.83 7.00 5.90 5.45 4.89 7.07 5.96 5.50 4.94 7.13 6.01 5.55 4.98 7.19 6.06 5.59 5.02 7.23 6.09 5.63 5.05

0.43 0.73 0.94 1.31 0.38 0.68 0.85 1.20 0.34 0.62 0.80 1.12 0.33 0.57 0.76 1.07 0.31 0.56 0.71 1.01 0.29 0.52 0.69 0.96 0.26 0.50 0.63 0.90

0.28 0.46 0.57 0.77 0.27 0.42 0.52 0.72 0.26 0.40 0.50 0.68 0.23 0.38 0.49 0.65 0.23 0.37 0.47 0.62 0.21 0.35 0.44 0.61 0.21 0.34 0.41 0.55

4010 3370 3110 2850 4100 3390 3170 2910 4180 3480 3230 2900 4250 3570 3280 2920 4330 3590 3340 2990 4380 3640 3390 3020 4410 3710 3420 3030

2190 2060 2010 1930 2210 2070 2010 1950 2230 2080 2020 1960 2250 2090 2030 1970 2260 2100 2050 1980 2270 2110 2050 1980 2280 2120 2060 1990

0.20 0.17 0.16 0.14 0.20 0.17 0.16 0.15 0.20 0.17 0.16 0.15 0.21 0.18 0.16 0.15 0.21 0.18 0.17 0.15 0.21 0.18 0.17 0.15 0.21 0.18 0.17 0.15

1.43 1.43 1.43 1.43 1.91 1.91 1.91 1.91 2.39 2.39 2.39 2.39 2.87 2.87 2.87 2.87 3.34 3.34 3.34 3.34 3.82 3.82 3.82 3.82 4.30 4.30 4.30 4.30

Q = 800 W, U = 6 103 m/s, r = 103 m, q = 5710 kg/m3, Tm = 1546 K, T1 = 300 K, ks = 97 W/m-K, k‘ = 43 W/m-K, as = 1.9 105 m2/s, a‘ = 8.4 106 m2/s, cps = cp‘ = 890 J/ kg-K, L = 3.1 105 J/kg.

P.S. Wei et al. / International Journal of Heat and Mass Transfer 55 (2012) 2328–2337

3.4. Boundary layer along the solid–liquid interface The total incident energy is dissipated through the thermal boundary layer along the entire solid–liquid interface into the solid. This gives

T T gQ k‘ c 0 m hr DT

ð22Þ

2333

where the thermal boundary layer thickness is obtained by considering transverse or horizontal conduction as the same magnitude of stream-wise or downward convection. This gives

D0T

sﬃﬃﬃﬃﬃﬃﬃﬃ a‘ h

v 00f

ð23Þ

Fig. 4. Scale analysis based on (a) momentum balance in the hot region, (b) energy balance in the hot region, (c) momentum balance near the cold corner region, (d) momentum balance in the cold corner region, and (e) total energy balance from incident energy to energy dissipated into solid.

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where v 00f is the stream-wise velocity in the thermal boundary layer. In the thermal boundary layer, velocity can be considered as the secondary peak velocity

v 00f u0f

ð24Þ

Eq. (22) by substituting Eqs. (23) and (24) leads to

gQ

rk‘ ðT c T m Þ

!1=2 u0f h

m

Pr1=2

ð25Þ

which conserves the total energy from the beam to solid. 4. Transport variables explicitly in terms of working variables Eqs. (7), (12), (14), (17), (21), and (25) can be simultaneously solved to give explicit expressions for the molten pool shape and transport variables as functions of working variables. They are, respectively,

1=12 1 k Ma1=6 Pr 1=12 ðgQ r Þ1=12 Per 1 þ Ste s r

ð26Þ

11=12 1 k Ma1=6 Pr1=12 ðgQ r Þ13=12 Per 1 þ Ste s

ð27Þ

h

w

r

1=18 1 k Ma5=9 Pr13=18 ðgQ r Þ11=18 Per 1 þ Ste s

ð28Þ

1=12 Tf Tm 1 k Ma1=6 Pr1=12 ðgQ r Þ11=12 Per 1 þ Ste s Tm T1

ð29Þ

uf r

m

u0f r

m

1=4 1 k Ma1=2 Pr3=4 ðgQ r Þ1=4 Per 1 þ Ste s

1=12 Tc Tm 1 k Ma1=6 Pr1=12 ðgQ r Þ11=12 Per 1 þ Ste s Tm T1

increase in magnitude of the surface tension coefﬁcient increases surface peak velocities and pool width, and decreases surface peak temperatures, pool depth and melting efﬁciency. On the other hand, an increase in dynamic viscosity increases pool depth, surface peak temperatures, and decreases pool width and surface peak velocities. Digital data and inter-substitutions among the scaled equations, and available experimental data are also used for comparing the scale analysis results. The dimensionless beam power per unit depth as a function of a dimensionless parameter involving Peclet and Stefan numbers obtained from numerical computations [23,34], experimental data [31,35], and scale analysis from Eq. (7) is shown in Fig. 3. Provided that the scale analysis is relevant, the relationship between abscissa and ordinate, respectively, chosen as the terms on two sides of Eq. (7) should be linear. Substituting numerical and experimental data of the fusion zone depth and width in the literature and from this work for different Prandtl numbers, the dimensionless beam

ð30Þ

ð31Þ

where Marangoni and Peclet numbers are, respectively, deﬁned as

dc ðT m T 1 Þr Ma ; la‘ dT Q r

Q ; k‘ rðT m T 1 Þ

Per

ks

ks k‘

Ur

as

;

Fig. 5. Relationship between the surface peak temperature and temperature at the edge of the cold corner region.

ð32Þ

The depth and width of the fusion zone shape, peak surface velocity and temperature, and maximum velocity and temperature near the edge of the cold corner region are, respectively, determined by Eqs. (26)–(31), which are functions of speciﬁed working variables on the right-hand side. 5. Results and discussion In this work, the molten pool shape and transport variables affected by thermocapillary force in melting and welding of pure iron, titanium, carbon steel, and stainless steel alloys are parametrically and systematically scaled. This is a complicated two-dimensional solid–liquid phase change problem involving recirculation ﬂow driven by temperature-dependent surface tension. In order to verify scale analysis, numerical computation of an unsteady two-dimensional model is used to simulate the steady-state three-dimensional problem, as discussed previously. The numerical method was a control-volume formulation, fully implicit, staggered ﬁnite difference method [23]. With appropriate working variables beam power Q = 800 W, U = 6 103 m/s, r = 103 m, the computed data are shown in Table 1. It can be seen that an

Fig. 6. Relationship between the peak and secondary peak surface velocities.

P.S. Wei et al. / International Journal of Heat and Mass Transfer 55 (2012) 2328–2337

power per unit depth versus Peclet number lies on a straight line, conﬁrming relevancy of the scale analysis. Eq. (7), unfortunately, contains unknown width and depth of the fusion zone. In order to separately determine the width or depth of the molten pool, a further study of thermocapillary ﬂow in the pool is required. The dimensionless peak surface velocity as a function of a parameter involving Marangoni and Prandtl numbers is shown in Fig. 4(a). Introducing the computed peak surface velocity, temperatures and molten pool width, the dimensionless peak surface velocity versus Marangoni number divided by Prandtl numbers, as derived from

2335

Eq. (12), is found to be linear for different high Prandtl numbers. The peak surface velocity thus is conﬁrmed to be driven by thermocapillary force from the rapid drop of surface temperature at the edge of the beam radius. The peak surface temperature predicted from Eq. (14) in the hot region is shown in Fig. 4(b). The relevancy is conﬁrmed by linear relationship between the dimensionless beam power and the energy distribution parameter-to-molten pool depth ratio. Eqs. (17), (21), and (25) derived for the scaling of the secondary peak surface velocity and maximum temperature at the edge of the cold corner region are shown in Fig. 4(c)–(e), respec-

Fig. 7. Scaled transport variables for (a) molten pool depth, (b) molten pool width, (c) peak surface velocity, (d) peak surface temperature, (e) secondary peak surface velocity, (f) temperature near the edge of the cold corner region, which are uncoupled and functions of speciﬁed working variables.

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tively. It can be seen that the results obtained from scale analysis also agree quite well with numerical computations. The secondary peak surface velocity can be predicted by either Eqs. (17) or (21), which is, respectively, related to the peak temperature in the hot region, or the maximum temperature in the cold corner region. Combining these equations leads to a linear relationship between the differences in the peak and melting temperatures and temperature at the cold corner edge and melting temperature, as can be seen in Fig. 5. Furthermore, dividing Eq. (12) by Eq. (17), the peak-to-secondary peak surface velocity ratio is found to be linearly proportional to the 1/3 power of the pool width-to-energy distribution parameter ratio, as shown in Fig. 6. Interestingly, this physical mechanism can be interpreted from conservation of momentum in the shear layer within the intermediate region between the hot and cold corner regions. Since surface temperature is comparatively uniform in the absence of the incident ﬂux in the intermediate region, thermocapillary force is small. Inertial force therefore maintains a relative constant value across the shear layer. That is,

sﬃﬃﬃﬃﬃﬃ

qu2f dv H qu02f dv C or qu2f

mr uf

qu02 f

sﬃﬃﬃﬃﬃﬃﬃ mw u0f

ð33Þ 6. Conclusion

which indicates that the peak-to-secondary peak surface velocity ratio is linearly proportional to the 1/3 power of the pool widthto-energy distribution parameter ratio. Eq. (33) fails if Prandtl number is very small. In view of high thermal diffusivity, surface temperature monotonically decreases from the hot to cold corner regions [28]. Thermocapillary force therefore cannot be neglected in the intermediate region, leading to only one peak of surface velocity near the edge of the pool. Solving Eqs. (7), (12), (14), (17), (21) and (25) simultaneously to ﬁnd the pool depth and width and transport variables in terms of speciﬁed working variables is of practical and academic importance. The proposed scale analysis can also be checked through manipulation of inter-substitutions among these equations. Fig. 7(a) shows that the algebraic expression of Eq. (26) for the molten pool depth agrees quite well with numerical data. The quantitative relationship between the pool depth and working parameters can be found by ﬁtting the data with a straight line Y = KX + C. This gives the equation

1=12 1 k ¼ 2Ma1=6 Pr 1=12 ðgQ r Þ1=12 Per 1 þ Ste s r h

ð34Þ

It is reasonably found that the pool depth decreases with increasing Marangoni and Peclet numbers, solid-to-liquid thermal conductivity ratio, and decreasing Stefan number. The increase in Peclet number represents an increase in welding speed, indicating a decrease in incident energy per unit length along the joint line. Since heat conduction from the liquid to the melting front is balanced by absorption of latent heat and dissipation of heat conduction to the solid, the pool depth is decreased by reducing Stefan number and increasing solid-to-liquid thermal conductivity ratio. It is interestingly found that the pool depth slightly decreases with increasing the beam power to a power of 1/12. Interpretation for this is that more energy is used to melt and expand the pool width so that the energy used to penetrate the pool depth is reduced, as can be seen later. The pool width as a function of different working parameters governed by Eq. (27) is shown in Fig. 7(b). Their linear relationship obtained by ﬁtting computed data is found to be

w

r

13=12

¼ 3Ma1=6 Pr1=12 ðgQ r Þ

11=12 1 k Per 1 þ 24:5 Ste s

zone is proportional to the beam power (see Eq. (4)), the signiﬁcantly increased fusion zone width reduces the fusion zone depth as the beam power increases. Fig. 7(c)–(f) show the peak surface velocity and temperature, maximum surface velocity and temperature in the cold corner region can be successfully scaled and represented as functions of working parameters, respectively. All these transport variables increase with increasing beam power and decreasing Prandtl number. An increase in Marangoni number reduces the maximum surface temperatures in the hot and cold corner regions, and increases the peak and secondary peak surface velocities. An increase in the product of Peclet number, the parameter 1 + 1/Ste with solid-to-liquid thermal conductivity ratio reduces the peak surface velocity and maximum temperatures in the hot and cold corner regions. This is attributed to reduced incident energy per unit length along the joint line used to heat, melt and raise temperature of the incoming solid with high latent heat. On the other hand, the increased secondary peak surface velocity with welding speed results from a decreased pool width, which increases temperature gradient and thermocapillary force.

ð35Þ

It can be seen that the pool width increases signiﬁcantly with the beam power. Since the transverse cross-sectional area of the fusion

In this study, thermocapillary convection and the shape of the molten pool irradiated by an incident ﬂux during welding or melting have been scaled. The Prandtl numbers lie between 0.3 and 1, indicating that the peak and secondary surface velocities occur in a pool with a slightly wavy bottom. The conclusions drawn are the following: 1. Correlation or approximate equations appearing in the literature cannot be used to separately predict the width or depth of the fusion zone. The other equation accounting for conservation of mass, momentum and energy in the molten pool is thus required and found by scaling of transport processes in the pool. 2. In view of high Marangoni number, the scale analysis is conducted by dividing the pool into distinct regions: the hot and cold corner regions, and boundary layers along the solid–liquid interface and ahead of the melting front. Each region also accounts for the thicker thermal boundary layer than the momentum boundary layer for low Prandtl number. There have six unknown variables: the pool depth and width, the peak velocity and temperature, and maximum temperature and velocity in the cold corner region on the free surface. After simultaneously solving six equations obtained from scale analysis, explicit expressions of the pool shapes and transport variables in terms of speciﬁed working variables are found. 3. The scaling results compare well with numerical data, different inter-substitutions among the scaled equations, as well as available experimental data. 4. An increase in beam power can reduce the depth of the pool. In view of enhanced thermocapillary convection due to increased temperature gradient, energy used to melt the pool in the horizontal direction is greater than that in the depth direction. 5. Physical mechanisms of transport variables and pool shape induced by thermocapillary force during melting, welding and various heat treatment processes are successfully and quantitatively revealed.

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