# Analytical Prediction of Balling, Lack-of-Fusion and Keyholing Thresholds in Powder Bed Fusion
## 2 Analytical Modeling
Then, the temperature drop due to the thermal radiation and convection [5] can be expressed as  
\[\theta_{loss}(x,y,z)=\sum_{i=1}^{n}\frac{A_{i}}{2\pi kR_{i}(T_{m}-T_{0})} \left[h(T_{i}-T_{0})+\varepsilon\sigma\Big{(}T_{i}^{4}-T_{0}^{4}\Big{)}\right] \tag{4}\]  
In this equation, the \(i\) represents the index of heat sinks on the top surface of the part, and \(n\) is the total number of the sinks. The total temperature distribution in the part can be obtained by the superposition of the linear heat input solution and linear heat loss solution, which can be expressed as the following equation  
\[\theta(x,y,z)=\theta_{laser}-\theta_{loss} \tag{5}\]  
The molten pool dimensions were determined by comparing the temperature profile with the melting temperature of the material. The effect of powder bed porosity on the thermal properties of the powders was also considered in the prediction of temperature profile and molten pool geometries. An advancing front approach was employed to compute the powder bed porosity based on the information of powder size distribution and powder packing. Several assumptions are enforced in this approach. First, the shape of the powders is assumed to be a circle. Second, the powder bed is assumed to have the maximum packing density in the manufacturing process. Third, this approach calculates the packing porosity in a 2D domain. Figure 2 shows the strategy of the advancing front approach. Three circles are generated first as the initial front. The following circles are generated based on the initial front along the positive directions of the blue vectors, as shown in Figure 2. It should be noted that the positive direction of the vector is along the anti-clockwise direction. The generated circles show the densest packing pattern in this 2D domain [22]. The effect of powder bed porosity on the thermal conductivity of the powder material was developed in [23], which can be expressed as  
\[k_{p}=\frac{k_{s}(1-\varepsilon)}{1+\psi\frac{k_{s}}{k_{s}}} \tag{6}\]  
Figure 1: Mechanisms of heat transfer in LPBF. The red arrow represents the heat source, the blue arrows represent the heat loss through convection and radiation to the ambience. The purple arrow shows the scanning direction of the laser. The small squares are heat sinks on the top surface [5; 18]. In this equation, \(k_{p}\), \(k_{s}\) and \(k_{g}\) represent the thermal conductivity of powders, solid material and gas, respectively. For Argon, \(k_{g}=0.0176\) (W/mK). \(\varepsilon\) represents the void fraction of the powder bed, and \(\psi\) denotes a factor in the calculation process [24]. Lack-of-fusion defect is induced by incomplete melting of powders. The threshold for the lack-of-fusion defect was first developed by [PERSON] et al. in [6], with the assumption of dual half-ellipse molten pool shapes. This threshold can be expressed as  
Figure 1.0  
In this equation, \(H\) and \(L\) represent the hatching space and layer thickness, respectively. \(W\) and \(D\) represent the molten pool width and depth, respectively. When the equation is not satisfied, the lack-of-fusion defect will occur in the manufacturing process. The balling defect usually happens in the low energy density regime, and it is caused by the poor wettability of the molten pools to the solid substrate. The threshold for the occurrence of balling behavior was proposed in [25], which can be expressed as  
Figure 1.1  
where \(w\) and \(l\) represent the molten pool width and length, respectively. The balling defect will occur when the equation is not satisfied. The keyholing defect is related to the evaporation of materials during the manufacturing process. A criterion for the generation of the keyholing defect was developed in [26], which can be expressed as  
Figure 1.2  
Figure 1.3  
In this equation, \(\frac{\Delta H}{h_{s}}\) represents normalized enthalpy, with \(\Delta H\) as the specific enthalpy and \(h_{s}\) as enthalpy of melting. \(P\) and \(u\) are the process conditions, which are the laser power and scanning speed.