A theoretical study of molten pool behavior and humping formation in full penetration high-speed gas tungsten arc welding

International Journal of Heat and Mass Transfer 132 (2019) 143–153

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

A theoretical study of molten pool behavior and humping formation in full penetration high-speed gas tungsten arc welding Xiangmeng Meng, Guoliang Qin ⇑ Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials, Ministry of Education, Shandong University, 250061 Jinan, PR China

a r t i c l e

i n f o

Article history: Received 19 August 2018 Received in revised form 23 October 2018 Accepted 2 December 2018

Keywords: Molten pool behavior Full penetration humping Sensitivity analysis Dimensional analysis High-speed GTAW

a b s t r a c t A clear insight into the full penetration humping (FP-humping) formation in high-speed gas tungsten arc welding (GTAW) is difficult owning to the complex and multi-coupled transport phenomena in molten pool. In this paper, sensitivity analysis and dimensional analysis are combined to study the molten pool behavior and FP-humping formation. Firstly, sensitivity analysis and an experimentally-verified numerical model are used to clarify the effect of driving forces on characteristic molten pool behaviors and defect formation quantitatively. The results show that both arc pressure and arc shear stress have considerable effects on promoting defect formation, and their significances are in the same order. The surface tension shows predominant role to suppress defect formation. Subsequently, dimensional analysis based on Buckingham p-theorem is performed to derive some physically meaningful dimensionless variables. The dimensionless humping frequency is a linear function of a dimensionless group containing characteristic molten pool variables and material properties. The reasonability of dimensional analysis result is tested by additional numerical data, and there is a good agreement. This study clarifies the physical origin of FP-humping defect in GTAW, and may also provide some fundamental guidelines for its suppression. Ó 2018 Published by Elsevier Ltd.

1. Introduction Due to the advantages of high arc stability, less spatter, superior weld quality and easy automation, gas tungsten arc welding (GTAW) is a critical joining technique in wide industrial fields for various metallic materials. The rapid development of modern manufacturing industry gives great demand for higher productivity, which requires welding processes to achieve higher welding speed without scarifying the resultant weld properties. However, continuous and smooth weld bead may not be obtained at high-current and high-speed category. When the welding current and welding speed exceed a critical value, the weld bead becomes undulated with alternate humped part and depressed part (or necked part), namely humping defect [1,2]. Apparently, it severely deteriorates the continuity of the weld bead, limiting the further improvement of welding productivity. Since the first report of humping in high-speed gas metal arc welding (GMAW) by Bradstreet, this defect has been observed in almost all kinds of fusion welding processes for both ferrous alloys and nonferrous alloys [3–8]. Based on final weld bead profile, transient molten pool images or analytical derivation, several

⇑ Corresponding author. E-mail address: [email protected] (G. Qin). https://doi.org/10.1016/j.ijheatmasstransfer.2018.12.004 0017-9310/Ó 2018 Published by Elsevier Ltd.

explanatory theories have been proposed to interpret the humping formation mechanism or to predict humping formation in various welding processes, such as capillary instability model [3,8], thermo-capillary model [9], hydraulic jump model [10], compound vortex model [11], arc induced model [12] and curved wall jet model [4]. For the GTAW without filler material, the arc induced model suggested that the molten pool free surface was deeply depressed under strong arc forces to form a gouging region with extremely thin liquid metal layer, and then the humping defect was initiated by the premature solidification of thin liquid layer [12]. The further study of Meng et al. found the important influence of lateral channel at periphery of gouging region on humping formation [13]. Apart from experimental or analytical investigations, many numerical efforts have also been made to give integrated descriptions of heat and mass transfer during humping formation. Computational fluid dynamic (CFD) models for high-speed GMAW were developed by Chen et al. [14], Cho et al. [15], Wu et al. [16] and Xu et al. [17] to calculate the heat transfer and fluid flow in humping formation. The capillary instability and rapid solidification of necked channel were found to be responsible for humping occurrence in GMAW. The suppression measures such as laser + GMAW or twin wire GMAW were also studied numerically [16,18]. Simulation works about humping phenomena in laser beam

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welding (LBW) were performed by Zhou et al. [19]. Recently, a multi-physical model was developed by Otto et al. to investigate humping effect in high-speed laser micro welding with base metal of 100 lm thickness [20,21]. Kumar and DebRoy applied KelvinHelmholtz instability theory combined with a numerical model to predict GTAW humping [22]. Meng et al. proposed a 3D transient CFD model to calculate the temperature field, fluid velocity, free surface deformation and solidified weld bead profile in highspeed GTAW, in which the detailed morphologies of humping was simulated [23]. However, the existing studies about humping, especially for GTAW, mainly focus on partial penetration condition for simplification. The base metal thickness is usually far larger than the GTAW penetration capacity. In fact, another type of humping defect may occur in full penetration condition. The full penetration humping (FP-humping) is composed of periodically or aperiodically spaced hump part and perforation part, and has distinct morphology compared with typical humping defect in partial penetration and melting-through defect in full penetration. The majority of weld metal is accumulated at the top side of hump part rather than the back side, and the perforation part without any solidified weld metal is through the thickness of base metal. It implies that unique formation mechanism may be involved in FP-humping formation. Albright et al. ascribed the FP-humping in LBW to the capillary instability phenomenon. The molten pool with top and bottom free surfaces was analogous to a cylindrical fluid jet under full penetration condition. When its length exceeded a critical value, it may spontaneously break into individual droplet under the surface tension force even if there’s only a small disturbance [24]. Deutsch et al. suggested that the FP-humping was an indication of excessive heat input in LBW of Al alloy, and formed due to keyhole instability [25]. In the double-sided welding of plasma arc welding + GTAW, Kwon et al. considered that the FP-humping occurred when the surface tension was insufficient to balance with arc force and gravity [26]. Parametric experiment research and visual inspection were performed by Meng et al. to study the formation mechanism of FP-humping in GTAW [13]. Similar to the partial penetration humping (PP-humping), the molten pool surface was also significantly depressed to generate a gouged region with thin liquid layer (less than 200 lm), and the lateral channel became the main transfer channel for liquid metal flowing backward. When the gouging region depth approached the base metal thickness, the thin liquid layer was easily disrupted to initiate the FP-humping formation procedure. A one-dimensional scaling model of heat conduction was developed to calculate the temperature and solidification of the lateral channel. The premature solidification of lateral channel was responsible for the subsequent defect formation. Recently, a numerical model was established by Meng et al. to simulate the formation procedure of FP-humping in GTAW. The thermal behavior and fluid flow during FP-humping formation were described quantitatively for the first time [27]. Although some preliminary researches about FP-humping have been reported, the research works are still relatively scarce. A clear insight into the FP-humping formation in high-speed GTAW is difficult because some important questions still remain unanswered. Firstly, the molten pool is an extremely complicated and highly multi-coupled heat transfer and fluid flow system which is driven by several driving forces with distinct physical essences including arc pressure, arc shear stress, electromagnetic force (EMF), thermo-capillary stress, viscous force, etc. The influence of different driving forces on molten pool behavior, i.e. the origin of FP-humping, is still not clear. Secondly, comprehensive thermal and flowing factors are involved in the FP-humping formation. The quantitative relevance between molten pool behavior and defect formation has not been clarified yet, and thereby, the FP-humping cannot be predicted.

In the present study, the effect of driving forces on characteristic molten pool behaviors during FP-humping formation in highspeed GTAW is investigated using a well experimentally-verified CFD model and local sensitivity analysis (LSA). The dominant driving forces promoting and suppressing defect formation are identified. Subsequently, several dimensionless numbers with explicit physical implications are derived based on Buckingham ptheorem. The mathematical relationship between characteristic molten pool behaviors and FP-humping formation (humping frequency) is formulated. 2. CFD model The transient heat transfer, liquid metal flow, free surface deformation as well as material melting and solidification are calculated in the 3D CFD model coupled with volume-of-fluid (VOF) method. The arc pressure, arc shear stress, EMF, capillary pressure, thermos-capillary stress, buoyancy and metallostatic pressure are implemented in the model, as shown in Fig. 1. The mathematical formulations of numerical model will be described briefly in the following sections, in which salient features of welding model for FP-humping are emphasized. More detailed descriptions about simplifying assumptions, governing equations, boundary conditions, mesh geometry and algorithm used etc. can be found in the authors’ previous works [23,28–30]. 2.1. Governing equations Continuity equation

!

r V ¼0

ð1Þ

! where V is the vector of velocity. Momentum conservation equation

q

! ! ! ! @V ! þ V  r V ¼ rp þ lr2 V þ bqðT  T 0 Þg~ k  qg~ k @t ! ! Sm  lK V þ F e þ ~

ð2Þ

where t is time, p is pressure, q is density, l is dynamic viscosity, b is the expansion coefficient, g is the gravity acceleration, K is attenuation coefficient from Carman–Kozeny equation, T0 is ambient ! temperature, F e is EMF, ~ Sm is other additional momentum source, ~ k is unit vector in Z direction. The third, fourth and fifth terms of right side of Eq. (2) represent buoyancy, gravity and velocity attenuation in the mushy zone, respectively. The liquid fraction fL is assumed to show a linear relationship with temperature. Energy conservation equation



q

  @h ! þ V  r h ¼ r  ðkrT Þ þ Sq @t

Fig. 1. Schematic of driving forces in GTAW.

ð3Þ

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where T is temperature, k is thermal conductivity, h is enthalpy, Sq is the additional energy source term. VOF continuity equation

!  @/ þr V/ ¼0 @t

ð4Þ

where / is volume fraction. 2.2. Boundary conditions of free surface The arc heat flux, convection dissipation and radiation dissipation are implemented on the top free surface of molten pool, as given below:

  @T k ! ¼ qa  hc ðT  T 0 Þ  re T 4  T 40 @n

ð5Þ

where qa is arc heat flux, hc is coefficient of convective heat transfer n (80 w/m2K), r is Stefan-Boltzmann constant, e is emissivity (0.3), ~ is normal vector of free surface. For the momentum boundary conditions of top free surface, arc pressure and capillary pressure are considered in the normal direction, and arc shear stress and thermo-capillary stress are considered in the tangential direction.

@V n p þ 2l ! ¼ pa þ cj @n

ð6Þ

@V t @ c @T l ! ¼ sa þ @T @ ! @n s

ð7Þ

X

qa ðx; yÞdS  Q ¼ 0

ZZ Q ¼ gUI  n

X0

ð12Þ

qa ðx; yÞdxdy

ð13Þ

where Q is the total heat input, aqf, aqr and bq are semi-axes of double-ellipse (4.8 mm, 6.2 mm and 4.8 mm) which are determined based on the work of Tsai et al. [31], vw is welding speed, g is thermal efficiency (0.7), I is welding current, U is arc voltage (17.2 V), X (x,y,t) is the free surface geometry function. The introduction of global controlling parameter Kq aims at conserving total heat input, and meanwhile, simplifying the complicated influence of free surface profile on heat flux distribution. X’ is the projection of perforation within the arc radius as shown in Fig. 2, and the RR n X0 qa ðx; yÞdxdy term is the energy dissipation through the perforation. Similarly, the arc pressure is also modelled varying with free surface profile and perforation geometry transiently, as expressed below:

6F 3ðx  v w t Þ2 3y2  exp  pa ðx; yÞ ¼ K p   2 a2 p apf þ apr bp bp  apf x P v w t a¼ apr x < v w t

!

ð14Þ

ZZ

where pa is arc pressure, sa is arc shear stress, Vn and Vt are normal and tangential velocity, respectively, c is surface tension coefficient, j is curvature of free surface, ! s is tangential vector of free surface. No external heat or momentum from arc is implemented on the bottom free surface. Therefore, the thermal and momentum boundary conditions can be expressed as follows:

  @T k ! ¼ hc ðT  T 0 Þ  re T 4  T 40 @n

ð8Þ

@V n p þ 2l ! ¼ cj @n

ð9Þ

X

pa ðx; yÞdS  F ¼ 0

F ¼ F  n

ð10Þ

ð15Þ

ZZ X0

pa ðx; yÞdxdy

ð16Þ

where apf, apr and bp are semi-axes of double-ellipse (4.3 mm, 5.6 mm and 4.3 mm) which are determined based on the study of Lin et al. [32], Kp is the global control parameter for arc pressure, F is total arc force, F⁄ is the total arc force before perforation forming. The dependence of arc force on perforation geometry can be calculated from Eq. (16) [27]. A double-ellipse arc shear stress model is used in this paper because of the distorted arc shape in high-speed welding. The mathematical expressions are shown below:



@V t @ c @T l ! ¼ @T @ ! @n s

sa ðx; y; zÞ ¼ smax g a 4:05I0:267



  1 r þ 1 r  2 d d R

smax ¼ 0:069I1:5

2.3. Welding models

(

The molten pool free surface is deeply depressed in highcurrent GTAW, which bring complex impact on the distribution of arc heat flux. Moreover, compared with the PP-humping, the thin liquid layer in the gouging region is disrupted during FP-humping formation, and some efflux plasma forms at the back of base metal. Not only the distribution pattern, but also total heat input will be influenced. To make the problem tractable, only the influence of perforation geometry on total heat input is considered. It is assumed that part of the arc energy will be dissipated to decrease the total heat input. In this study, a self-adaptive double-ellipse heat source model is used, considering the influence of dynamic free surface profile and perforation geometry.

6Q 3ðx  v w t Þ2 3y2  exp  qa ðx; yÞ ¼ K q   2 a2 p aqf þ aqr bq bq  aqf x P v w t a¼ aqr x < v w t

ZZ

g a ðr Þ ¼

ð17Þ ð18Þ

0:363r 3 þ 0:034r 2 þ 1:322r  0:0128 ðr 6 1:53mmÞ  r  3:46 exp  1:02 þ 0:0345 ðr P 1:53mmÞ ð19Þ

!

ð11Þ

Fig. 2. Schematic of heat source after disruption of thin liquid layer.

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where r ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx  v w t Þ2 þ y2 . smax is the maximum value of arc shear

stress which is a function of welding current. The term ga in Eq. (17) is a normalized double-ellipse distribution function, in which the welding current term makes the distribution region enlarge with increment of welding current, and the term in square brackets is a stretching factor, transforming centrosymmetric distribution into double-ellipse distribution [23,28]. A double-elliptical EMF model is applied here for the highspeed GTAW based on the study of Cho et al. [33]. An effective radius re is defined as follows to modify the current density and induced electromagnetic field.

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 bp apf re ¼ ðx  v w t Þ2 þ y 2 a ¼ 2 a apr

Jz ¼

I 2p

Jr ¼

I 2p

Bh ¼

Z

1

0

Z

1

0

lm I 2p

Z

x P vwt x < v wt

ð20Þ

  sinh fk½z þ H  Xðx; y; tÞg dk kJ 0 ðkr e Þ exp k2 r2j =2 sinh ðkHÞ ð21Þ   cosh fk½z þ H  Xðx; y; tÞg kJ 1 ðkr e Þ exp k2 r2j =2 dk sinh ðkHÞ ð22Þ 1

0

  sinh fk½z þ H  Xðx; y; t Þg dk J 1 ðkr e Þ exp k2 r2j =2 sinh ðkHÞ ð23Þ

F ex ¼ 

bp x  vwt J  Bh r a z

F ey ¼ J z  Bh F ez ¼ J r  Bh

y r

 a¼

apf apr

x P v wt x < v wt

ð24Þ ð25Þ ð26Þ

where Jz and Jr are the axial and radial current density in the cylindrical coordinates, Bh is the angular component of the magnetic field, rj is the distribution parameter of current density

(1.88 mm), J0 and J1 are the zero order and one order Bessel function respectively, H is the base metal thickness, Fex, Fey and Fez is the EMF component in x, y, z direction, respectively. 2.4. Numerical consideration The dimension of simulation domain is 70  5  4.5 mm. Two phases, steel and argon gas, are included for the implementing of VOF method. A 1.5 mm base metal, a 2 mm top gas layer and a 1 mm bottom gas layer are initialized in the model, as shown in Fig. 3. Therefore, both the top free surface and melt out region at back side can be tracked. Hexahedral cells with uniform dimension of 0.2 mm are meshed at weld central zone (0  y < 3 mm), and relatively coarse cells expanding from 0.2 mm to 0.6 mm are used in the other region. The governing equations of continuity, momentum, energy and VOF are solved by the commercial software ANSYS Fluent. PISO algorithm is chosen to solve the pressure-velocity coupling. About twenty hours is needed to simulate more than 1.5 s real-time GTAW process with average time step of 3  105 s. The material type of base metal is AISI 409L ferritic stainless steel, and its thermophysical properties are taken from the references of [34,35]. A representative combination of welding parameters for FP-humping, welding current of 350 A and welding speed of 2 m/min, is used as an example for the following numerical analysis, LSA and dimensional analysis. 2.5. Typical molten pool behaviors during FP-humping formation Fig. 4 gives the longitudinal section of gouging region. The depth of gouging region increases with welding process proceeding. More and more liquid metal is displaced to the rear region of melt pool. When the base metal is fully penetrated, the thin liquid layer at bottom of gouging region which is marked by the red arrow in Fig. 4(c) becomes unstable and is easily disrupted. The back side of molten pool is still flat at the initial stage, but it will become convex under gravity when quasi-steady state is reached [27]. After the bottom liquid layer is disrupted, the lateral channel becomes the only transfer channel for liquid metal flowing back-

Fig. 3. Meshed geometry of simulation domain.

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The temperature gradient will decrease, which weakens the effect of thermo-capillary stress. LSA is employed in this study to overcome this problem. This methodology can provide quantitative evaluation of input (driving forces) significance on output (molten pool behavior) in a mathematical model (molten pool system) [40]. The mathematical model can be either regressive, analytical or numerical. The numerical model with complicated mathematical expression and high computational intensity, however, is not suitable for direct LSA. The relationship between characteristic molten pool behavior and driving forces can be simplified into a regressive model based on proper statistical design. Five driving forces are selected as independent variables including arc pressure (pa), arc shear stress (ss), EMF (fe), capillary pressure (pca) and viscous force (fm). The thermo-capillary stress, buoyance and gravity are excluded from the sensitivity analysis due to their minor effects on molten pool behavior in high-current and high-speed GTAW [12,29]. Uniform design which is a quasi Monte Carlo method is applied for statistical sampling. The basic principle of uniform design is to minimizes the discrepancy between the design points (empirical uniform distribution) and a theoretical uniform design [41]. Simple and precise estimate can be obtained for an unknown function. The uniformity of design matrix can be evaluated by the discrepancy (DP) calculated by the following equation.

Z DP ¼ Fig. 4. Longitudinal section of gouging region (I = 350 A,

vw = 2 m/min).

ward. As the arc heat source moves forward, the lateral channel as well as the disrupted region are elongated, as shown in Fig. 5(b). Subsequently, premature solidification occurs at the transition region between lateral channel and rear region to prevent liquid metal flowing backward, and two lateral accumulations are formed, as shown in Fig. 5(c). With more and more liquid metal flowing backward from the leading wall of molten pool, the volume of lateral accumulation increases gradually. Ultimately, the two accumulations merge with each other to form a new hump part, as shown in Fig. 5(d). It should be noted that although there are some similarities between FP-humping and PP-humping, apparent differences can also be recognized. The first and the most intuitive aspect is that these two types of defects actually occur at distinct process spaces (3 m/in vs. 2 m/min at 350 A for 1.5 mm stainless steel plate according to [13]). Besides, the FP-humping is initialized by the disruption of bottom liquid layer, while elongation of the gouging region is considered as the initial requirement for PP-humping [23]. Fig. 6 gives the calculated morphologies of FP-humping in highspeed GTAW and the comparison with experimental results. A good agreement can be observed. It indicates that the CFD model is well verified to provide accurate data of molten pool behavior for the following sensitivity analysis and dimensional analysis.

3. Sensitivity analysis The effect of individual driving force on molten pool behaviors was studied by applying them separately in the numerical model [36–39]. Some criteria such as maximum fluid velocity were selected to evaluate the relative significance. Semi-quantitative results can be obtained by this method, but the complicated interactions between driving forces, as described in Fig. 7, are ignored. For instance, if strong arc forces exist, the spatial temperature distribution will be homogenized by the enhanced fluid convection.

1=p p jF n ðxÞ  F ðxÞj dx

ð27Þ

C

where Fn(x) is the empirical distribution function and F(x) is uniform distribution function on sampling domain C. Smaller discrepancy means better uniformity. The driving forces are first normalized as dimensionless variables to eliminate the influence of their different units and orders of magnitudes on regression, and denominated as bj.

bi1 ¼

pia pa

i

bi2 ¼

i fl sia f pi ; bi3 ¼ e ; bi4 ¼ ca ; bi5 ¼    sa pca fe fl

ð28Þ

where the superscript i represents the ith level and superscript ⁄ represents the reference value which is the actual value at a given welding condition. A schematic explaining the normalized driving force is given in Fig. 8. For example, the b1 of 0.5 means that the peak value of arc pressure applied in the model is only 50% of the reference value, but they have the same distribution pattern. Generally, the level number should be more than 3 times of factor number, so each factor has 20 levels in the sampling domain, and ranges from 0 to 2 (±100%). Based on the previous experimental and numerical studies, the significant free surface deformation and vigorous backward flow of liquid metal are crucial for FP-humping formation in high-speed GTAW. Hence, the objective functions (characteristic molten pool variables) are chosen as maximum velocity of backward flow (vbmax), global deformation of free surface (x) and humping frequency (f). The global deformation of free surface can be calculated as follows:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 XX x¼ ½Xðx; yÞ  z0 2 N

ð29Þ

where N is the cell number on molten pool free surface, z0 is the position of base metal plane. The uniform design matrix and response values (numerically calculated variables) are listed in Table 1. An exponential form is chosen as the mathematical expression of regressive function.

Y ¼ a0

Y e aj bj j¼1

ð30Þ

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Fig. 5. Typical molten pool behaviors during FP-humping formation (I = 350 A,

v bmax ¼ 0:435e0:100b

Fig. 6. Comparison between simulated and experimental FP-humping defects in high-speed GTAW (I = 350 A, vw = 2 m/min).

where a0a5 are the regression coefficients and Y is the response value. Multiple nonlinear regression analysis is conducted to calculate the regression coefficients. The regression models of maximum backward velocity, free surface deformation, and humping frequency are given below.

1

vw = 2 m/min).

e0:443b2 e0:116b3 e0:0205b4 e0:147b5

ð31Þ

x ¼ 0:375e0:320b1 e0:238b2 e0:094b3 e0:353b4 e0:0648b5

ð32Þ

f ¼ 1:320e0:547b1 e0:481b2 e0:0414b3 e0:602b4 e0:185b5

ð33Þ

The comparison between actual value and predicted value is shown in Fig. 9. It can be seen that all the data points are near the diagonal line, which indicates that the regressions are adequate. Additionally, the adjusted R2 and root mean squared error (RMSE) also prove the adequacy of regression model. Therefore, the relationship between characteristic molten pool behavior and driving forces can be represented by Eqs. (31)–(33) although they are only phenomenological. The sensitivity equations are defined as partial differentiation with respect to independent variables. The dimensionless sensitivity equations normalized by reference values can be calculated as follows:

Sv j ¼

1

v b max

@ v bmax

@bj

b15 ¼1

ð34Þ

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Sxj

1 @ x

¼ 

x @bj

ð35Þ

b15 ¼1

1 @f

Sfj ¼ 

f @bj

ð36Þ

b15 ¼1

Fig. 7. Interactions between driving forces in GTAW.

3500 1

3000

=1.5

Arc pressure (N/m2)

2500 1

2000

=1.0

1500 1

1000

=0.5

500 0 -8

-6

-4

-2

0

2

4

6

8

Distance to arc center (mm) Fig. 8. Explanatory schematic for normalized driving force.

where x⁄, v b max and f⁄ are the reference value of free surface deformation, maximum velocity of backward flow and humping frequency, respectively, when b15 ¼ 1. x⁄ is 0.49 mm, v b max is 0.63 m/s and f⁄ is 2.97 Hz. Fig. 10 gives the sensitivity of driving forces on characteristic molten pool behaviors. Positive value means there is a positive correlation between independent variable and response, and conversely, negative value means negative correlation. Higher value of sensitivity equation indicates that the factor can show more significant influence on the system. Fig. 10(a) shows the sensitivity of driving forces on maximum backward velocity. The Sv value of arc shear stress is 0.413, which suggests that the arc shear stress is the dominant driving force to push liquid metal to the rear part of molten pool. The arc pressure shows a secondary positive effect, but its significance is only about one fourth of arc shear stress. The EMF and viscous force have nonnegligible influence to suppress liquid metal flowing backward considering their sensitivity equation values are 0.108 and 0.148, respectively. The effect of capillary pressure, however, can be ignored. The influence of driving forces on free surface deformation is shown in Fig. 10(b). The Sx values of arc pressure and arc shear stress are 0.292 and 0.216, respectively. It demonstrates that these two driving forces have considerable positive effects on free surface deformation, and their significances are in the same order. The arc pressure has been revealed to have only minor effect on free surface deformation in the formation of PP-humping in highspeed GTAW [12,29]. It implies that the driving forces may show different contributions on molten pool behavior under different penetration conditions. The capillary pressure is the predominant force to hinder free surface deformation (Sx = 0.322), and the EMF plays a minor negative impact (Sx = 0.0857). In addition, the free surface deformation is insensitive to the viscous force. Fig. 10(c) plots the sensitivity of driving forces on humping frequency. The arc pressure (Sf = 0.430) and arc shear stress (Sf = 0.378) are the main driving forces to promote FP-humping

Table 1 Design matrix and response values.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 DP = 0.045.

b1

b2

b3

b4

b5

vbmax (m/s)

x (mm)

f (Hz)

0.47 0.91 0.38 1.65 1.41 1.18 1.80 0.12 1.27 0.77 1.00 1.87 0.24 1.53 0.61 1.00 0.77 1.46 1.90 1.69

1.41 1.78 0.49 0.74 0.89 1.65 1.04 1.11 0.22 1.87 0.62 0.13 0.37 1.54 1.28 1.00 1.66 0.33 1.20 1.33

0.63 1.63 0.75 1.14 0.49 1.01 1.43 0.35 1.76 0.13 1.50 0.23 1.87 0.87 1.27 1.00 1.56 0.53 1.49 0.11

1.34 0.74 0.63 1.15 1.45 1.08 0.86 0.56 0.67 1.20 1.40 1.00 1.26 0.93 0.82 1.00 0.65 0.71 1.45 1.13

0.92 1.39 0.67 0.80 1.13 1.32 0.60 1.19 0.87 0.56 1.00 1.43 1.26 0.75 1.06 1.00 1.23 0.78 1.12 0.55

0.75 0.74 0.47 0.57 0.63 0.71 0.67 0.61 0.39 0.98 0.48 0.45 0.29 0.87 0.65 0.63 0.67 0.468 0.67 0.84

0.44 0.54 0.38 0.52 0.49 0.55 0.60 0.45 0.39 0.49 0.34 0.50 0.094 0.57 0.49 0.49 0.52 0.55 0.54 0.53

1.18 4.63 0 2.75 2.31 3.47 3.74 1.29 2.3 3.17 0 2.85 0 3.66 3.06 2.97 4.63 3.17 3.78 3.96

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0.3

Sensitivit on vbmax

Predicted vbmax (m/s)

0.413

(a)

1.0

0.8

0.6

Adjusted R2=0.96 RMSE=0.030

0.4

(a)

0.2

0.1 0.0932

0.0 Arc pressure Arc shear

EMF

0.0190

Viscous force

Surface tension

stress -0.108

-0.1 0.2 0.2

0.4

0.6

0.8

-0.148

1.0

0.3

Actual vbmax (m/s)

0.292

(b) 0.216

0.2 0.6

Sensitivity on ω

(b)

Predicted ω (mm)

0.5

0.4

0.1 0.0 -0.1

EMF

Surface tension

Arc pressure Arc shear stress

Viscous force -0.0592

-0.0857

-0.2

0.3

-0.3 -0.322

Adjusted R2=0.79 RMSE=0.051

0.2

-0.4 0.6

0.1 0.1

0.2

0.3

0.4

0.5

(c)

0.430

0.6

0.4

0.378

Sensitivity on f

Actual ω (mm) 5

(c) Predicted f (Hz)

4

0.2 0.0

0.145 EMF Arc pressure Arc shear stress

Surface tension Viscous force

-0.0325

-0.2 -0.4

3

-0.473

-0.6

Adjusted R2=0.78 RMSE=0.53

2

Fig. 10. Sensitivity of different driving forces on (a) maximum backward velocity; (b) free surface deformation; (c) humping frequency.

1 1

2

3

4

5

Actual f (Hz) Fig. 9. Comparison between actual value and predicted value: (a) maximum backward velocity; (b) free surface deformation; (c) humping frequency.

formation, and the defect is dominantly suppressed by capillary pressure (Sf = 0.473). The effect of EMF is quite negligible for FP-humping formation. It is very interesting that the viscous force is a non-ignorable facilitating factor for humping frequency

(Sf = 0.145). This is because that the viscous force can decelerate the backward liquid metal, and correspondingly, promote the lateral accumulations and their merging. Previous studies have suggested that the arc shear stress dominates the PP-humping formation, and arc pressure has relatively small influence in high-speed GTAW [12,29]. However, both arc pressure and arc shear stress are found to have remarkable influence on FP-humping in the present study. As shown in Fig. 5, the disruption of thin liquid layer is an important initiation factor for FP-humping formation. When the base metal is fully penetrated, the thin liquid layer without any solid metal support is more easily disrupted by normal arc pressure rather than tangential arc shear stress.

151

X. Meng, G. Qin / International Journal of Heat and Mass Transfer 132 (2019) 143–153

The sensitivity analysis also provides some fundamental guidelines for the suppression of FP-humping. Any measure which can reduce arc pressure and (or) arc shear will reduce the formation likehood of FP-humping, such as larger electrode tip angle, truncated electrode shape, hollow electrode and using helium as shielding gas. Fig. 11 shows the longitudinal molten pool profile when a larger electrode tip angle is used. Comparing with Fig. 4, the full penetration occurs at the middle part of molten pool. There is still some unmelted metal at the bottom of gouging region because of smaller arc pressure and arc shear stress. The onset of FP-humping can be prevented.

Fig. 11. Longitudinal section of molten pool with blunt electrode of 90 deg (I = 350 A, vw = 2 m/min).

5

(a)

Frequency f (Hz)

4

4. Dimensional analysis Both intuitive simulation results and theoretical sensitivity analysis show that complex molten pool behaviors are involved in the FP-humping. If FP-humping frequency is taken as ordinate axis, and maximum backward velocity or free surface deformation is taken as abscissa axes, as shown in Fig. 12, no clear relationship can be observed. Therefore, it may be difficult to describe FPhumping tendency or predict FP-humping features, e.g. frequency, by one specific molten pool variable. In this paper, dimensional analysis is employed to establish the quantitative correlation between molten pool behaviors and FPhumping frequency. According to Buckingham-p theorem, if a physical system can be defined by p variables which can be expressed by k fundamental units, the system can be described properly by pk dimensionless numbers [42,43]. On the basis of sensitivity analysis, the variables in dimensional system will not be chosen arbitrarily. Six variables are selected to define the molten pool system in high-speed GTAW, as given in Table 2. The q and vbmax represent inertia force originating from arc pressure and arc shear stress, l represents the viscous force, and c and x represent the capillary pressure from surface tension. The EMF is excluded from the dimensional system because of its minor effect. These six variables can be expressed by three fundamental units i.e. mass (kg), length (m) and time (s). Therefore, three proper dimensionless numbers are needed. The derivation of dimensionless number is usually tedious and cumbersome, and many trail-and-error efforts are needed until appropriate dimensionless numbers are obtained. A simple methodology from Arora et al. is used in this paper [44]. By choosing q, l and c as principal variables, the three dimensionless numbers can be obtained using the following equations:

3

½p1  ¼

2

½p2  ¼

1

½p3  ¼

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Maxi. backward velocity vbmax (m/s) 5



kg m3



kg m3



kg m3

a1 

b  c kg 1 kg  m2 1 hmi 0 ¼ kg m0 s0 ms s s2 c b  kg 2 kg  m2 2 0 ½m ¼ kg m0 s0 ms s2

ð37Þ

a2 

c   b  kg 3 kg  m2 3 1 0 ¼ kg m0 s0 2 ms s s

ð38Þ

a3 

ð39Þ

After calculating, the dimensionless numbers can be written as

p1 ¼

q2 cv bmax l3

ð40Þ

p2 ¼

l2 x qc

ð41Þ

p3 ¼

q3 c3 f l5

ð42Þ

(b)

Frequency f (Hz)

4 3

Obviously, p1, p2 and p3 are dimensionless maximum backward velocity, free surface deformation and humping frequency, respectively.

2 1

Table 2 Variables and their units used in dimensional analysis.

0 0.35

0.40

0.45

0.50

Free surface defromation

0.55

0.60

(mm)

Fig. 12. Relationship between FP-humping frequency and (a) maximum backward velocity of liquid metal, (b) free surface deformation.

Variables

Units

Density q Viscosity l Surface tension coefficient c Maxi. backward velocity vbmax Surface deformation x Frequency f

kg m3 kg m1 s1 kg m2 s2 m/s m s1

X. Meng, G. Qin / International Journal of Heat and Mass Transfer 132 (2019) 143–153

If these dimensionless numbers are reasonable, a clear correlation should be found between p1, p2 and p3, namely, a function of p3 = f (p1, p2). The most direct strategy is to take p1  p2 as abscissa and take p3 as ordinate. The numerical data in Table 1 is used to plot the dimensionless curve, as shown in Fig. 13. Compared with the scattered data points in Fig. 12, a strong positive correlation is between p4 and p3. It can be fitted as

p3 ¼ 6:20  1022 exp ðp4 =202:2Þ þ 1:74  1023

1 2 3

b1

b2

b3

b4

b5

vbmax (m/s)

x (mm)

f (Hz)

1.37 1.00 1.23

1.89 0.64 1.47

1.89 1.36 0.44

0.83 0.59 0.94

0.72 0.89 1.06

0.838 0.508 0.787

0.59 0.55 0.57

4.56 3.43 4.27

ð43Þ

ð44Þ

Fitting data Verification data Fitting line

12

f/µ5 (×1024)

q3 cv 2b max x l4

14

10

4

3 3

p4 is the Reynolds number which is a ratio of inertia force to viscous force, so the relative balance between inertia force and viscous force plays important role on FP-humping frequency. The capillary pressure which is also a crucial factor for FPhumping formation, however, is not included in p4. Therefore, the p4 should be modified by some other dimensionless variable containing surface tension term c. In this study, p1 is introduced to p4 in order to establish a more complete correlation between molten pool behavior and humping frequency. The modified p4 is denominated as p5.

p5 ¼ p1  p4 ¼

Table 3 Additional simulation cases for verification.

π3=

152

8 6

2

Fig. 14 is plotted as p5 versus p3, in which a quite good linear relation between p5 and p3 can be observed. It can be fitted as

0 0

14

5

10

15

20

25

30

35

2 π5=π1×π4= 3γ vbmax ω /µ4 (×1017)

12

π3=

3 3

f/µ5 (×1024)

Fig. 15. Verification of the dimensional analysis using additional simulation cases.

10 8

π 3 =6.20 × 10 22 exp (π 4 202.2 ) +1.74 ×10 23

Adjusted R2=0.94

p3 ¼ 4:04  106 p5 þ 6:16  1022

6 4 2 0 200

400

600

800

1000

1200

π4=π1×π2= vbmaxω/µ=Re Fig. 13. Relationship between dimensionless variables p4 and p3.

16

5. Conclusions A theoretical study about the molten pool behavior and humping formation in high-speed GTAW is conducted in this paper based on local sensitivity analysis and dimensional analysis. The main conclusions are drawn below:

Adjusted R2=0.97

10

3 3

8 6 4 2 0 -2

The left side of Eq. (45) is a dimensionless humping frequency and the right side is composed of all important molten pool variables and material properties. Now the quantitative relationship between molten pool behaviors and FP-humping frequency is formulated. To further test the reliability of dimensional analysis result, three additional simulation cases are conducted, as shown in Table 3. All the variables are chosen randomly within 0–2. The verification data, original fitting data and the fitting line are all plotted in Fig. 15. All the verification data are distributed near the linear fitting line. It proves the reasonability of the dimensional analysis result, and the humping frequency can be predicted by Eq. (45) if only some characteristic molten pool variables and material properties are known.

π 3 =4.04 × 106 π 5 +6.16 × 10 22

12

π3=

f/µ5 (×1024)

14

ð45Þ

0

5

10

15

π5=π1×π4= γ 3

20

v2bmaxω /µ4

25

30 17

(×10 )

Fig. 14. Relationship between dimensionless variables p5 and p3.

35

(1) The arc shear stress dominates the backward flow of liquid metal. Both arc shear stress and arc pressure show considerable promoting effect on free surface deformation in the same order, and the capillary pressure plays the main negative effect. (2) Both arc shear stress and arc pressure are important factors to facilitate GTAW FP-humping formation, while the arc pressure only have minor impact on humping formation in partial penetration condition. The capillary pressure shows predominant role to suppress defect formation.

X. Meng, G. Qin / International Journal of Heat and Mass Transfer 132 (2019) 143–153

(3) Two dimensionless numbers p3 and p5 which have clear physical meanings are developed using dimensional analysis. p3 is a dimensionless humping frequency, and p5 contains characteristic molten pool variables and important material properties. (4) p3 is a quite good linear function of p5, by which FPhumping frequency can be predicted if only some important molten pool variables are known. The results of dimensional analysis are verified by the additional simulation cases.

Acknowledgement This work is supported by the National Natural Science Foundation of China (Grant No. 51575317) and the Shandong Provincial Natural Science Foundation (Grant No. 2014ZRE27072). Conflict of interest statement We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work. There is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled ‘‘A theoretical study of molten pool behavior and humping formation in full penetration high-speed gas tungsten arc welding”. References [1] T.C. Nguyen, D.C. Weckman, D.A. Johnson, H.W. Kerr, High speed fusion weld bead defects, Sci. Technol. Weld. Join. 11 (6) (2006) 618–633. [2] E. Soderstrom, P. Mendez, Humping mechanisms present in high speed welding, Sci. Technol. Weld. Join. 11 (5) (2006) 572–579. [3] B. Bradstreet, Effect of surface tension and metal flow on weld bead formation, Weld. J. 47 (7) (1960) 314–322. [4] T. Nguyen, D. Weckman, D. Johnson, H. Kerr, The humping phenomenon during high speed gas metal arc welding, Sci. Technol. Weld. Join. 10 (4) (2005) 447– 459. [5] W.F. Savage, E.F. Nippes, K. Agusa, Effect of arc force on defect formation in GTA welding, Weld. J. 58 (7) (1979) 212–224. [6] S. Alfaro, J. Vargas, G. Carvalho, G. Souza, Characterization of ‘‘humping” in the GTA welding process using infrared images, J. Mater. Process. Technol. 223 (2015) 216–224. [7] S. Tsukamoto, H. Irie, M. Inagaki, Effect of focal position on humping bead formation in electron beam welding, Trans. Natl. Res. Inst. Metals 25 (2) (1983) 62–67. [8] U. Gratzke, P.D. Kapadia, J. Dowden, J. Kroos, G. Simon, Theoretical approach to the humping phenomenon in welding processes, J. Phys. D. Appl. Phys. 25 (11) (1992) 1640–1647. [9] K. Mills, B. Keene, Factors affecting variable weld penetration, Int. Mater. Rev. 35 (1) (1990) 185–216. [10] T. Yamamoto, W. Shimada, A study on bead formation in high speed TIG arc welding at low gas pressure, in: Y. Gakkai (Ed.), The Second International Symposium of the Japan Welding Society on Advanced Welding Technology; 1975 Aug 25–27, Japan Welding Society, Osaka, Japan, 1975, pp. 321–326. [11] M. Lin, T. Eagar, Influence of arc pressure on weld pool geometry, Weld. J. 64 (6) (1985) 163–169. [12] P.F. Mendez, T.W. Eaga, Penetration and defect formation in high-current arc welding, Weld. J. 82 (10) (2003) 296–306. [13] X. Meng, G. Qin, X. Bai, Z. Zou, Humping phenomena in high-speed GTAW of different weld penetrations, Weld. J. 95 (9) (2016) 331S–339S. [14] J. Chen, C.S. Wu, Numerical simulation of forming process of humping bead in high speed GMAW, Acta Metall. Sin. 45 (9) (2009) 1070–1076.

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