0　Introduction

Gas metal arc welding (GMAW) using the arc composed of an electron flow and electrically neutral arc plasma under the shielding gas to melt the moving wire and work-piece has lots of advantage like low cost, easy operation and good adaptability and so on. In GMAW, the tip of the welding wire melts to be droplet and falls into the weld pool when the sum of the resultant forces overcomes the resistance of surface tension. This whole process is called metal transfer which plays an important role in GMAW. Mckelliget and Szekely[1] investigated temperature distribution of electrode and weld pool with a two-dimensional theoretical model earlier but steady and without consideration of dynamics of droplet. This steady model also was used to study the properties of free-burning arc columns and the cathode by Lowke et al.[2]. Zhao and Chung[3] proposed a model with phase field model and numerical studied the transition of metal transfer from globular to spray mode in GMAW. They showed the higher precision of phase field model than the VOF method. Coupled the model proposed by Zhao and Chung[3] with energy equation, Jiang and Li[4] derived the continuous and discrete energy law of GMAW and showed the accuracy of numerical method and the energy law by comparing the numerical solution of horizontal and vertical diameters with the data of high-speed photography. In narrow gap welding, Xu et al.[5] proposed a three-dimensional model to investigate the sing arc and they used this model to calculate the temperature field and fluid flow in transient state of weld pool. Zhou et al.[6] used VOF to track free surface of molten pool and droplet and developed a three dimensional weak coupling modeling method to simulate the arc and molten pool. Zhao and Chung[7] develop an unified numerical model to clarify the metal transfer phenomena during a VP-GMAW process considering the interaction between the arc plasma and the moving droplet. A dynamic model for metal transfer considering electromagnetic force, surface tension and arc plasma with VOF was presented by Cheng, Wu and Lian[8]. Also, some numerical researches for fume formation and vapor in GMAW are carried on[9-11].

百米林带中乡土树种占22科27属29种,外来树种47科75属94种,外来树种的数量远高于原生的乡土树种[13],比例超过了3∶1 (94∶29),这与上海地区外来树种和乡土树种的比例基本一致(547∶174),反映了高度人工化和城市化是上海地区人工植被的普遍特征。

Actually, the previous work did not consider the influence of multiple parameters on the geometry of the droplet and free surface, for example, the influence of temperature. According to the research in multi-phase fluids, thermocapillary effect popular in Marangoni convection[12] would be an important factor when the surface tension becomes dominant[13-15]. Surface tension makes the droplet tend to be a ball and stops the droplet detaching from the molten wire playing an important role in metal transfer[16] which leads to that the theoretical model should consider the thermocapillary effect especially for GMAW having a temperature gradient with a large scale.

GMAW is a mixture system and its mixture energy has an effect on the phase transfer and interface change[17-19]. Different phases in phase field model are treated as one phase[20,21] and the physical property parameters like density, capacity and conductivity change continuously from one phase to the other phase. In fact, there is no thickness between different phases in the mixture or the thickness of the interface is nanometer scale. It is difficult to compute the result of phase field model when the thickness is nanometer scale, so the sharp interface limit[19] for moving contact line is needed to support the theory of phase field model.

In this paper, we present a new theoretical model for metal transfer in GMAW combined with energy equation. In section 1, we introduce the model, boundary conditions and the weak formulation. The fully discretized GMAW system with a continuous finite element method is presented in section 2. We compute numerical exam-ples with pulse and constant current and show the numerical result in section 3. section 4 is conclusion.

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1　New model and weak form

Fig.1 shows a general physical model of GMAW and the welding region is defined as Ω with a boundary which is called Γ.

Fig.1　The simple structure of GMAW system

The governing equations for GMAW is shown as follows:

▽·(ρv)=0

(1)

▽·(ρv⊗v)=-▽p+▽·(η(▽v+▽vT))-▽·(λf(▽f⊗▽f))+μ▽f-ρG+J×B+ρGβT(T-T0)

(2)

▽·(ρvf)=▽·(M▽μ)

(3)

(4)

▽·(ρcpvT)=▽·(k▽T+Mμ▽μ)▽T

(5)

▽·(σe▽Φ)=0,J=-σe▽Φ,ΔA=-μ0JB=▽×A

(6)

+η▽▽

▽·v=0

(7)

▽p+ηΔv-▽·(λf(▽f⊗▽f))+μ▽f-ρG+J×B+ρGβT(T-T0)

(8)

▽

(9)

▽T=▽

▽T

(10)

▽·(σe▽Φ)=0,J=-σe▽Φ,ΔA=-μ0J,B=▽×A

(11)

A positive constant c is introduced to rewrite Eq.(9) as[12, 18]

▽f=MΔ(ω+cf),ω+cf=μ

(12)

We compare the numerical result of droplet size with the data of high-speed photography (Fig.4) to identify the validity of the new model. We choose the horizontal and vertical diameter at the time just before the droplet touches the weld pool, respectively. The result in Table 2 shows that the result of new model agrees well with the data of high-speed photography. Also, this new model gives a higher precision in predicting the size of the detaching droplet than VOF.

Please see more details in references[13, 18] for the finite element space and then we have

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∀

(13)

+ρG·u-(J×B)·u-ρGβT(T-T0)·u-(ω+cf)▽

∀u∈W1,3

(14)

∀γ∈W1,3

(15)

-λfε▽f·▽χdX=0　∀χ∈W1,3

(16)

∀θ∈W1,3

(17)

∀α,φ,S,L∈W1,3

(18)

2　Finite element scheme and fully discretized GMAW system

A finite difference scheme in time and a conformal C0 finite element method in space is applied into Eq.(13)-(18). Δt>0denotes the time step size and is the approximation at time tn+1=(n+1)Δt. The discretized formulation reads:

(19)

where βT, M,ε, ρ, η, k, cp, kb, e, σe are the thermal expansion coefficient, phonological mobility coefficient, thickness of interface, density, viscosity, thermal conductivity coefficient, specific heat, Stefan-Boltzmann constant, electronic charge and electrical conductivity, respectively. μ is called chemical potential. v,J are the velocity and current density. f is the phase field parameter for defining different phases of the mixture (f=1: fluid and f=-1: metal) and T represents temperature. Φ,B are the electrical potential and self-induced electro-magnetic field. A,T0 are magnetic vector and initial temperature. λf is function of temperatureλf=ηεσ0-ηεσT(T-T0). The boundary conditions and computing domain in the previous work of Hu and Tsai[15] are used in this paper. The additional boundary conditions for f and μ are ∂nf=0, ∂nμ=0.

综上所述，苏轼与王安石之争，恰是王安石的“国本主义”与苏轼是一位“民本主义”的分歧所在。显然，这种思想认识上的分歧是无法调和的。而他在《上神宗书》中的冒死相争就成为一种必然。

随着地球的自转，星星也会慢慢滑过画面，“500法则”可以帮助我们计算出能够使用的最长焦距。用500除以镜头的等效焦距，因此对于24mm镜头，最长的曝光时间是20秒。对于更长时间的曝光，就需要借助星轨仪了，或者利用免费的Sequator堆栈软件将多张焦段曝光时间相同的照片合成在一起。

(20)

(21)

(22)

▽▽θ

(23)

(24)

(25)

(26)

(27)

3　Numerical examples and result

In this section, we use the discrete system Eq.(19)-Eq.(27) to compute numerical examples: pulse and constant current. We choose stainless steel with 1.2 mm diameter as the electrode and the shielding gas is pure argon. Table 1 gives the pulse current and Fig.2 simulates the wave of pulse current (this pulse current comes from the real welding machine of Beijing Time Technologies Co., Ltd.) and we choose four points (0 ms, 1.377 ms, 4.116 ms, 7.231 ms in one period) to show the numerical result. From Fig.3a to Fig.3b, the amount of molten metal in wire increases and the droplet grows up as the current in peak time increases quickly. The necking effect appears at the root of the attaching droplet according to the numerical result. Compared with Fig.3b and Fig.3c, with the contribution to attaching droplet in radial direction made by electro-magnetic force and the gravity in axial direction, the necking effect becomes more clearly. When the strain could not be held any more by the resistance like surface tension, in other words, the droplet would break up when the sum of gravity, electro-magnetic force and arc pressure overcome the resistance of surface tension (Fig.3d).

Table 1　The time and current in one period of pulse

We also apply the penalty formulation[18] to Eq.(7) with▽·v+δp=0 where δ is small penalty.

▽

Fig.2　The wave of pulse current (Large: One period; Small: Three periods)

Fig.3　Evolution of interface structure of droplet and wire at (a) t = 0, (b) t =1.377 ms, (c) t =4.116 ms, (d) t =7.231 ms

Fig.4　Metal transfer of high-speed photography

Table 2　Comparison among high-speed photography new model and VOF

We also compute the GMAW with 1.2 mm stain-less steel electrode with a constant current 140 A and present the interface structure in Fig.5. Compared with the result in Fig.3, we can see the fusion zone in globular transfer would be longer than spray transfer and the detaching droplet would be bigger than the droplet in the last numerical example. Still, we compare the numerical solution with the data of high-speed photography in Table 3. We can see that the new model has a higher precision in predicting the size of detaching droplet than VOF method.

Fig.5　The attaching and detaching process of metal transfer with constant current 140 A

Table 3　Comparison among high-speed photography, new model and VOF with constant current

4　Conclusion

(1) The metal transfer in GMAW with pulse and constant current are examined with this phase field model as the numerical result fits the theory of dynamics of droplet in metal transfer well.

(2) Compared with the data of high-speed photography, the validity of this new model has been validated due to the relative error of droplet diameter below 12%.

事故当时，系统最高电压等级为220kV，主系统仍然以110kV为主，系统与那曲地区电网间只有一条110kV 联络线即当那线（110kV当雄变—110kV那曲变），那曲地区电网主要由一座110kV那曲变、安多变、35kV查龙水电厂及部分35kV变形成，由于那曲地区电网仅有一座查龙水电厂（4台机组，4×2.7兆瓦），所以那曲地区电网供电主要依靠系统通过唯一一条 110kV当那线输送，尤其事故当时处冬季枯水期，因水库来水问题，查龙水电厂只能保证单台机运行（事故当时#3机组）运行，110kV那曲变那曲变两台主变（#1、#2）其中规定正常方式安排#1主变中性点接地运行。

(3) We also show that the new model can make a higher precision in predicting the droplet size than the previous work making the numerical simulation closer to the reality.

(4) This new model and method could provide guiding function in predicting the geometry of droplet during the metal transfer which plays an important role in control of GMAW.

References

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