2.1 Numerical model

SLM process is complicated, involving heat transfer, evaporation, melting and solidification, re-melting and re-solidification, shrinkage and other thermophysical behaviors. In the numerical calculation model, in order to simplify the complicated physical process, the following assumptions need to be made: (1) The molten pool liquid is assumed as laminar and incompressible Newtonian fluid. (2) Mushy zone is treated as an isotropic permeability of porous medium in solid-liquid phase change. (3) Powder size is Gaussian distribution with sphere shape.

### 2.1.1 Establishment of randomly packed powder bed

A DEM-based randomly packed powder model was established by commercial platform PFC [13, 14]. In DEM, the contact between powder particles is regarded as a linear model while the mini deformations are allowable between the particles [15]. The linear force induced by the mini deformation could be regarded as the force exerted by a linear spring. The spring has a constant normal and shear stiffness, Kn and Ks [16]. In Figure 1, Fi, the contact force vector, is decomposed into two subvectors as Fn and Fs in the normal and shear direction, respectively. The stacking of the powder particles is accomplished by setting gravitational force until the powder particles reaches equilibrium. The powder bed porosity is set as 0.45, which is close to 0.5 which is the theoretical packing density. More details about the establishment of powder bed can be found in Refs. [13, 14].

As shown in Figure 2, two dense packing of Gaussian-sized spherical particles with dimensions of 500 ˜ 200 ˜ 35 and 600 ˜ 300 ˜ 35 μm were obtained by PFC with the parameters: the layer thickness of particles of 35 μm, the height of substrate of 30 μm. Then the powder bed model was converted to STL format for subsequent numerical simulation.

Heat and Mass Transfer of Additive Manufacturing Processes for Metals DOI: http://dx.doi.org/10.5772/intechopen.84889

#### Figure 1.

Contact forces generated by interaction between two spherical particles x1/x<sup>2</sup> and r1/r<sup>2</sup> are the center and radius of particle 1 and particle 2, respectively.

Figure 2. Model of randomly packed AlSi10Mg powder bed. (a) Single track simulation, (b) multi-tracks simulation.

#### 2.1.2 Conservation equations

The melt fluid flow is governed by N-S equations. The fluid free surface is captured employing the volume of fluid (VOF) method. The melt flow could be solved by the conservations of mass, momentum and energy, given by Eqs. (1)–(3), respectively.

Mass

$$\nabla \cdot \vec{\nabla} = \mathbf{0} \tag{1}$$

Momentum

$$\frac{\partial \vec{V}}{\partial t} + \left(\vec{V} \cdot \nabla\right) \vec{V} = -\frac{1}{\rho} \nabla P + \mu \nabla^2 \vec{V} + \vec{\mathcal{g}} \left[1 - \beta (T - T\_m)\right] \tag{2}$$

Energy

$$\frac{\partial H}{\partial t} + \left(\overrightarrow{V} \cdot \nabla\right) H = \frac{1}{\rho} (\nabla \cdot k \nabla T) + \mathbb{S}\_U \tag{3}$$

! where V is velocity of the melt, ρ is the liquid metal density, P represents ⇀ hydrodynamic pressure, μ is liquid viscosity, g is the gravitational acceleration, β is volumetric thermal expansion coefficient of the material,T represents the fluid temperature,Tm represents the melting temperature of AlSi10Mg.

The VOF method as shown in Figure 3 was employed to track the free surface of the particles model obtained by DEM as they are melted, and it defined a function of the fraction of fluid by the following equation [17].

$$\frac{\partial F}{\partial t} + \nabla \cdot \left(\overrightarrow{V}F\right) = 0\tag{4}$$

where F is the volume fraction of the liquid in a cell. When the cell is filled with liquid, F = 1; when the cell is void, F = 0. The value of F is between 0 and 1 when both the void and liquid are in the cell.

In this work, the laser energy of the AlSi10Mg powder bed was defined as 0.18 [18].

#### 2.1.3 Boundary conditions

The heat-flux boundary condition at fluid free surface was given by [19].

$$
\kappa \frac{\partial T}{\partial \mathbf{z}} = q(r) - h\_\varepsilon (T - T\_0) - \varepsilon\_r \sigma\_\varepsilon \left(T^4 - T\_0^4\right) - q\_{ev} \tag{5}
$$

where hc is the heat transfer coefficient, ε<sup>r</sup> is the emissivity, σ<sup>s</sup> is Stefan-Boltzmann constant, T0 is ambient temperature, and q is the heat loss by melt ev evaporation.

In SLM process, the liquid metal evaporation is given by the equation [20]

$$q\_{ev} = 0.82 \frac{\Delta H^\*}{\sqrt{2\pi MRT}} P\_0 \exp\left(\Delta H^\* \cdot \frac{T - T\_{lv}}{RTT\_{lv}}\right) \tag{6}$$

where M represents the molar mass, R is the ideal gas constant, P0 represents the ambient pressure,Tlv is the boiling point of the metal melt, and ΔH\* is the effective enthalpy of loss metal vapor.

Figure 3. Schematic diagram of VOF.

Heat and Mass Transfer of Additive Manufacturing Processes for Metals DOI: http://dx.doi.org/10.5772/intechopen.84889

In order to simulate the Marangoni effect induced by the temperature gradient of the molten pool fluid, the shear stress should be balanced with boundary condition at fluid free surface, as given by [21]:

$$\begin{aligned} \star \mu \frac{\partial u}{\partial z} &= \frac{\partial \gamma}{\partial T} \frac{\partial T}{\partial x} \\ \star \mu \frac{\partial v}{\partial z} &= \frac{\partial \gamma}{\partial T} \frac{\partial T}{\partial y} \end{aligned} \tag{7}$$

where ∂γ=∂<sup>T</sup> is the surface tension gradient. The surface pressure boundary condition including the surface normal force was given by [22]:

$$-P + 2\mu \frac{\partial \overrightarrow{v\_n}}{\partial n} = -P\_r + \sigma \left(\frac{1}{R\_\mathbf{x}} + \frac{1}{R\_\mathbf{y}}\right) \tag{8}$$

! where vn represents the normal velocity vector, Pr is the recoil pressure, σ means the surface tension, Rx and Ry represent the principal radius of surface curvature.

$$P\_r = 0.54 P\_0 \exp\left(L\_{lv} \cdot \frac{T - T\_{lv}}{RT T\_{lv}}\right) \tag{9}$$

#### 2.2 Material physical properties and numerical simulation

The AlSi10Mg powder (Felcon, China) used in SLM was produced by gas atomization. The chemical composition of the AlSi10Mg alloy is shown in Table 1. Drying the powder before laser melting by the drying ovens at temperature of 373 K can help reduce the humidity and the oxygen content within the powder. The scanning electron microscope (SEM) morphology of the AlSi10Mg powder is shown in Figure 4a, showing the morphology of powder particles is almost spherical. The powder particle size distribution was obtained by laser particle size analyzer (Sympatec, HELOS, Germany). In Figure 4b, the powder particle size is from 0 to 45 μm with the average size of 26.53 μm.

The thermophysical properties of AlSi10Mg and laser processing parameters are shown in Figure 5 and Table 2. The temperature-dependent surface tension can be expressed as [23].

$$
\sigma = 1000.726 - 0.152T \quad \text{When } T > T\_l. \tag{10}
$$

In this work, the final meshing of the model ensured the convergence of the simulation with the cell size of 2 μm. The minimum time step was defined as e �15 second while the maximum time step was defined as e �8 s. Implicit method was selected for the solvers of heat transfer, viscosity and surface tension. Explicit method was selected for free surface pressure solver. Numerical simulations were carried out on the commercial CFD platform Flow3D [13, 14].


Table 1. Composition of AlSi10Mg (wt.%).

Figure 4. (a) Microstructure of the AlSi10Mg powder, (b) the particles size distribution.

Figure 5. Thermal material properties of AlSi10Mg: (a) thermal conductivity and specific heat, (b) density and viscosity.

Heat and Mass Transfer of Additive Manufacturing Processes for Metals DOI: http://dx.doi.org/10.5772/intechopen.84889


#### Table 2.

SLM-processing conditions and material parameters used in this work.

#### 2.3 Results and discussion

#### 2.3.1 Model verification

Single melting track can used to validate the correction of the model and available of software. At experiment, the laser power and powder layer thickness were fixed at 180 W and 35 μm, and the laser scanning speed changed from 600 to 1600 mm. The experimental measurement criteria of the melt depth and melt width are as shown in Figure 6a. The melt depth is taken vertically from the free surface of the molten pool to the maximum depth of the melt boundary while the melt width is taken horizontally between the edges of the melt boundary. Figure 6a shows the micrograph of pool on cross section caused by Gaussian laser irradiation in the simulation and experiment for P = 180 W and v = 1000 mm/s. It is obviously that the calculated morphology of pool agree well with the experiment one. Meanwhile, Figure 6b shows the range and averaged experimental melt depth and width results at different laser scanning speed v with a fixed laser power P (P = 180 W,

#### Figure 6.

(a) Morphology of molten pool at P = 180 W and v = 1000 mm/s, (b) depth and width of molten pool at different scanning speed with a fixed P = 180 W.

v = 600, 1000, 1600 mm/s) with their corresponding simulated values. Compared the numerical depths and widths of pool with experimental results, the simulated melt depths and widths all fall within their corresponding experimental range. Due to the existence of the inherent experimental error and numerical error, the errors between experiment and simulation are inevitable. So experimental results and numerical results demonstrate the proposed numerical model can provide good predictions on shape of molten pool and, can be provided to predict the effect of layer thickness and hatching spacing on the morphology of scan track.
