## 1 Introduction
Researchers' experience and assumptions on the shape, size and flux of the two models play a critical role to accurately display the heat and force effect in the welding process. A unified model of the plasma arc and weld pool is required to avoid these different subjective assumptions of heat source and  
Figure 1: A keyhole-mode plasma arc welding. arc pressure. A lot of unified models of gas tungsten arc welding (GTAW) [20; 21; 22] have been developed, but there is a problem with these models in that they are not able to calculate the keyhole-mode welding phenomenon as they assumed the weld pool surface is flat, which is quite different to the keyhole phenomenon. Just in recent years, several researchers began to construct unified models of PAW to analyze the keyhole effect. According to experimental observations, [PERSON] et al. [23] pre-set a constant keyhole and established a unified keyhole-mode PAW model to display heat transfer from the keyhole to the weld pool. Their model shows there are two opposite circular flows in the weld pool and about 10% of plasma arc outflows from the bottom keyhole. [PERSON] et al. [24] also constructed a 2D unified model with a pre-set keyhole to study physical characteristics of the plasma arc after full penetration. They demonstrate the plasma arc will be compressed within the keyhole since the keyhole radius gradually decreases. [PERSON] et al. [25] displayed heat transfer and fluid flow in a unified GTAW model, in which a constant keyhole was pre-set based on experimental observation. [PERSON] et al. [26] developed a 3D PAW model to analyze the energy propagation mechanism in the keyhole-mode welding process when the weld pool was at a quasi-steady state. Meanwhile, a pre-constructed keyhole is also required in their model. The above models managed to show some physical mechanisms in the keyhole-mode welding process, but they can only show the phenomena after keyhole formation and with a stable keyhole. [PERSON] et al. [27; 28] established a unified model with VOF to display the whole welding process from arc ignition to open keyhole formation in PAW. They demonstrated that the arc shape, temperature profile and current density distribution all vary dynamically as the keyhole increases. However, the predicted open keyhole time is shorter than the experiment. [PERSON] et al. [29] proposed a model including tungsten, plasma and an anode to analyze the heat transfer and fluid flow in a keyhole PAW process. They also used the VOF to display the dynamic keyhole process and calculated energy input, transformation, transfer and dissipation in the process. However, their model uses a simplified Ampere's circumference law to calculate the magnetic field, which is only suitable in a 2D structure. What is more, since the VOF method is used in the above models to track the keyhole surface, it costs too much computational time, and the predicted weld pool geometry does not coincide enough with the experiment now. In view of the convenience for industrial application as well as good accuracy, [PERSON] et al. [30] proposed a new unified model to describe the keyhole-mode heat transfer characteristics and predict weld pool geometry more conveniently. The model directly calculates heat transfer from the plasma arc to the workpiece instead of depending on assumed heat source models, meanwhile it predicts more accurate arc pressure distribution rather than empirical arc pressure models. The heat and force effect calculated from the arc region is transported to the workpiece region, and the unified model simulates a heat transfer phenomenon in each individual region. In this way, it simplifies the original two-phase problem into a single-phase problem in two different regions. So, the model takes quite less computational time than the VOF-unified models, meanwhile it is theoretically more accurate than the assumed heat source models and arc pressure models. As the previous model is just suitable to a 2D spot-welding process, we extended it into a 3D mobile model in the paper.