2.2　Simulation model of the L-PBF printer

2.2.1　Problem description

Fig. 2-1 shows spatter particles during the L-PBF manufacturing process of a Ti-6Al-4V object. Laser power is 100 W，and laser scanning velocity is 0.5 m/s.After the laser is switched off，there are noticeable spatter particles around the part above the workbench and pollution on the powder bed. This indicates that the spattering of particles during the L-PBF process is a severe problem that needs to be addressed.

2.2.2　Geometric model of the L-PBF printer

Fig. 2-2 presents a geometric model of a self-developed L-PBF printer. The model is fabricated using two gas pipes，a main chamber and a workbench. The inlet pipe is comprised of a condenser and two intake ducts: Inlet1 and Inlet2. The size of the main chamber is 440×782×483 mm 3 ，with the planar size of the workbench being 255×255 mm 2 . The diameter of the inlet and outlet pipes is 72 mm while the height of Inlet1 and the outlet are 30 and 40 mm，respectively. The intake gas velocity of the L-PBF printer is chosen as 4 m/s.

Inert gases have a wide range of applications in industrial engineering because of their chemical inactivity. The laser in the L-PBF printer generates high-temperature areas during the printing process，which could cause metals to readily react with air and this adversely affects product quality. Therefore，the chamber should be filled with an inert gas such as argon. There are some assumptions for the model listed as follows:

（1） All spatter particles inject with the same initial velocity along each direction.

（2） The wall of the printer chamber is set as ideal trapping boundary conditions regardless of the particle velocity.

（3） The shielding gas in the chamber is incompressible.

（4） The effect of metal evaporation is not considered.

2.2.3　Numerical model of the L-PBF printer

It simulates the flow field motion and heat transfer inside the chamber during L-PBF process. The governing equations for the flow field in the printer are as follows:

Continuity equation

where ρ and are the density and velocity，respectively.

Momentum equation

where p is the pressure and μ is the dynamic viscosity.

Energy equation

where c p ， T f and k f are the specific heat，temperature，and thermal conductivity，respectively. It uses a Gaussian heat source model to simulate the laser ［14-15］ . The energy density of a Gaussian heat source can be calculated as follows ［16］

where P is the laser power， R is the diameter of the laser spot， r is the radial distance from a point on the powder bed to the center of the laser spot，and A is the energy absorption rate of the material，which is affected by the laser wavelength，surface conditions，and physical properties of the material. The laser parameters are presented in Table 2-1，while the physical properties of the materials used in the simulation are listed in Table 2-2. Environmental pressure is set as 1 atmospheric pressure （atm），and environmental temperature is 300 K. Radiation heat transfer and gravitational acceleration （ g ） are taken into consideration in this study. The g is assumed to be 9.81 m/s 2 along the Z -direction. The pressure outlet boundary condition is set at the end of the outlet pipe，and its gauge pressure is 1 atm.

The particle trajectory inside the printer is simulated using the Lagrangian tracking method in the discrete phase model （DPM） ［17］ ，which is commonly applied for the calculation of various transports and depositions in complex geometries. In the DPM，particle tracking in the flow field is stochastic，and particle-particle interactions are negligible，since the discrete phase （particles） is sufficiently dilute.

The first part of the right side of this equation represents drag force，which is a function of the relative velocity，the second part is gravity force，and the third part is additional forces，including Saffman lift and other （user defined）.

Considering the complex geometry of the L-PBF printer，both the structural mesh and the unstructured mesh can be used flexibly. Because of the cross-scale problem，the entire model cannot be separated simply by using a uniform grid. It is necessary to refine the grid at specific locations to ensure computational accuracy without excessively increasing the overall computational burden. This mainly increases the mesh density around the laser scanning path，inlets，the outlet，and pipeline walls. As shown in Fig. 2-3，the diameter of the laser spot is 200 μm. To accurately simulate the process of laser scanning，a cross-scale mesh from the micrometer-to-millimeter scale is generated in the middle of the workbench. The gas flow near the pipeline wall is more complex. Thus，the mesh density is increased at the pipeline boundary. A boundary layer with five layers and a 1.2-fold growth rate is used to increase the mesh density and thus ensure accurate calculations.

To verify grid independence，18 points （ A 1- C 3'） evenly distributed on the two planes above the workbench are selected，as shown in Fig. 2-4. The vertical （ Z direction） distance between the two planes is 230 mm，and the horizontal （ X and Y directions） distance between any two neighboring points on each plane is 110 mm.

Three cases with different node numbers are calculated，and the numbers in these cases （cases 1，2，and 3） are 4364587，6994691，and 7700697，respectively.The velocity of the gas flow in the middle of the chamber is apparently higher because the structure of the inlet is fan-shaped. The velocities of point A 1'- C 3' are lower because the gas flow from the inlets got dispersed and disorderly. The velocity deviations between case 2 and case 3 are less than 5%. Thus，it uses mesh generation of case 2 in this work.