2.2.3　Numerical model of the L-PBF printer

We simulate the flow field motion and heat transfer inside the chamber during the 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 cp, Tfand kf are the specific heat, temperature, and thermal conductivity, respectively.We use 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 materials.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/s2along 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 mesh.It is necessary to refine the mesh 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, 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 ensures accurate calculations.

To verify mesh independence, 18 points(A1-C3') 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 A1'-C3' 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, we use mesh generation of case 2 in this work.