# Analytical Prediction of Balling, Lack-of-Fusion and Keyholing Thresholds in Powder Bed Fusion
## 2 Analytical Modeling  
This study proposed analytical modeling methods for the prediction of defects generation in laser powder bed fusion additive manufacturing. The occurrence of lack-of-fusion, balling and keyholing defects were predicted under various process conditions and validated by experimental investigation of Ti6 Al4V, Inconel 718 and 316L stainless steel. The molten pool dimensions were first predicted by a temperature prediction model with closed-form solutions. The heat input from power source, boundary heat loss, and the effects of powder bed porosity on the thermal properties were considered in the calculation process. The powder bed porosity was calculated by an advancing front approach based on the information of powder size distribution and powder packing pattern. The predicted molten pool dimensions were then employed into the calculation of thresholds for the lack-of-fusion, balling and keyholing defects. The analytical thermal model employed in this study consists of a linear heat input solution and a linear heat loss solution. A point moving heat source model is employed as the heat input solution, which was developed by [PERSON] and [PERSON] based on the assumption of semi-infinite medium [21]. The heat input solution was used to calculate the temperature increase due to the heat input from power source; it can be expressed as  
\[\theta_{laser}(x,y,z)=\frac{P\eta}{2\pi kR(T_{m}-T_{0})}exp\left(\frac{-V(R+x )}{2\kappa}\right) \tag{1}\]  
where \(T_{m}\) denotes the melting point, \(T_{0}\) represents the room (initial) temperature, and \(\theta_{laser}=\frac{T-T_{0}}{T_{m}-T_{0}}\) represents the dimensionless temperature. \(P\) and \(V\) represent the process conditions, which are laser power and scanning speed. \(\eta\) is the absorptivity of the laser power. \(\kappa=\frac{k}{\rho c}\) is the thermal diffusivity of material, which can be calculated with the information of density \(\rho\), specific heat \(c\) and thermal conductivity \(k\). \(R=\sqrt{x^{2}+y^{2}+z^{2}}\) represents the distance from the calculated location to the laser source. The heat loss solution was used to consider the temperature drop due to the boundary heat transfer, which was derived from the point moving heat source model with equivalent power loss and zero moving velocity. The mechanisms of boundary heat loss in the process of laser powder bed fusion additive manufacturing are shown in Figure 1. The blue arrows above the part represent the thermal radiation and convection from the part to the ambience, which has obvious influence on the temperature profile in the part. The red arrow denotes the laser power source. The purple arrow on the part represents the scanning direction of the heat source. To calculate the heat loss at boundaries, the boundary surfaces are discretized into plenty of small rectangular sections, as shown in Figure 1. Each section can be considered as a small heat sink with equivalent heat loss as a stationary point heat source at its center. Thus, the equivalent heat loss from thermal radiation and convection [5] can be expressed as  
\[Q_{conv}=Ah(T-T_{0}) \tag{2}\]  
\[Q_{rad}=A\varepsilon\sigma\left(T^{4}-T_{0}^{4}\right) \tag{3}\]where \(Q_{conv}\) is the boundary heat loss through thermal convection, and \(Q_{rad}\) is the boundary heat loss through thermal radiation. \(h\) is the heat transfer coefficient, and \(\varepsilon\) is the emissivity. \(\sigma\) represents the Stefan-Boltzmann constant, of which the value is \(5.67\times 10^{-8}\) W/(m\({}^{2}\cdot\)K\({}^{4}\)). \(T_{0}\) represents the initial temperature, \(T\) is the temperature of each heat sink center, which can be obtained through the heat input solution. \(A\) is the surface area of each heat sink.