# Powder Bed Selective Laser Processing of Alumina: Scanning Strategies Investigation
## 2 Methodology
### Numerical Procedures  
When the laser beam scans the powder with specific power and speed, a part of the laser beam power is back-scattered into the printer chamber, and the powder absorbs the remaining laser beam to form the melt pool. The powder's melting is considered a complex process, and it is a challenge to be mathematically modeled. Therefore, the following assumptions were considered: (1) the laser beam has a uniform heat distribution, (2) the molten pool has a flat top surface, (3) the powder bed has a homogeneous and continuous medium, (4) no heat loss due to evaporation. In PBSLP, there are three heat transfer mechanisms: thermal conduction, radiation, and convection. The developed thermal model considered all these mechanisms.  
Numerical Model Development  
When the laser beam scans the powder, heat is transferred to the powder, as described previously. The energy equation (Equation (1)) was used to model this heat transfer according to [34], where \(\rho\), \(C_{p}\), \(T\), k, and t are the density, specific heat, temperature, thermal conductivity, and time, respectively.  
\[\rho C_{p}\frac{\partial T}{\partial t}=\
abla.(\mathrm{k}\ \
abla\ \mathrm{T})+S_{h} \tag{1}\]  
The term \(S_{h}\) in Equation (1) refers to the laser heat source per unit volume (W/m\({}^{3}\)) and a User-Defined Function (UDF) was developed, compiled, and solved using ANSYS FLUENT 2020R1 to model the laser source as described by Equations (2) and (3) according to [34].  
\[S_{h}=\beta I_{0}\pi\ \mathrm{exp}(-\alpha z) \tag{2}\]  
Figure 3: Six scanning strategies investigated in this study.  
\[\left(\text{x}\,-\,\text{xs}\,-\,\text{v}\,\text{t}\right)^{2}+\left(\text{y}\,-\, \text{ys}\,-\,\text{v}\,\text{t}\right)^{2}\,\,<=R^{2} \tag{3}\]  
where \(\beta\), \(I_{o}\), \(\alpha\), \(v\), and \(R\) are the material absorptivity, laser intensity, absorption coefficient, scanning speed, and laser radius, respectively. The xs and ys parameters in Equation (3) are the laser spot initial location on the powder bed and were used to control the laser beam movement on the powder bed (scanning strategies). The absorption coefficient was estimated according to [35]. The laser intensity can be calculated according to [34] as follows:  
\[I_{o}=\frac{p}{\pi\,R^{2}} \tag{4}\]  
To model the nature of the laser modulated mode in the LB-PBF 125 printer, the developed UDF considered this mode where the scanning speed was calculated using the exposure time (exp_t) and the point distance (\(pd\)) according to Equation (5).  
\[v=\frac{pd}{\exp\_t} \tag{5}\]  
The initial and boundary conditions applied in this study are shown in Figure 4 and described by Equations (6) and (7):  
\[T(x,y,z)_{t=0}=T_{O} \tag{6}\]  
\[-\text{k}(\frac{\partial T}{\partial z})=\dot{S_{h}}-h_{cov}(T_{a}-T_{s})- \sigma\varepsilon\Big{(}T_{a}^{4}-T_{s}^{4}\Big{)} \tag{7}\]  
where \(T_{a}\) is the ambient temperature, \(T_{s}\) is the powder surface temperature, \(\sigma\) is the Stefan-Boltzmann constant, \(\varepsilon\) is emissivity, \(h_{cov}\) is the convection coefficient.  
Alumina material properties (the effect of colloidal graphite on the material properties was ignored since it is only 0.1 vol%) as a function of temperature (as described in Table 1) and other constants, according to [36], were considered in the developed numerical model and included in the developed UDF. Additionally, the UDF considered the phase transformation from solid to liquid and liquid to solid based on the latent heat of melting. User Defined Memory (UDM) available in ANSYS FLUENT was used to monitor the molten region during the movement of the laser.  
Figure 4: The boundary and initial conditions considered in the numerical modeling.