## 3 Results
The initial melting and solidification are experienced when the unit length is deposited from the nozzle, and the second melting and solidification are experienced when another unit length is deposited on top of the previously deposited unit length (\(x\)). This is experienced repeatedly by subsequent layers until the intended build (in this paper, a cube) is completed. This process of re-melting is accounted for by the thermal conductivity of the intermetallic alloys, which allows for the conduction of adequate heat energy from the melt pool to be transferred to the previously deposited layer during deposition [PERSON], 2019; [PERSON] et al. (2016). The total thermal energy transferred from the laser to a single unit length of deposited length is displayed in Equation (1):  
\[\Delta H=2.304\times LEPUL, \tag{1}\]  
where \(LEPUL\) is the Laser Energy Per Unit Length ([PERSON], 2019). The \(LEPUL\) was calculated using Equation (2):  
\[Laser\ Energy\ Density\left(\frac{\mathrm{J}}{\mathrm{mm}^{2}}\right)\times Spotsize (\mathrm{mm})=LEPUL\left(\frac{\mathrm{J}}{\mathrm{mm}}\right). \tag{2}\]  
In addition, the total amount of heat energy absorbed by a single deposited unit length during LENS deposition is the product of the integral of the specific heat with respect to temperature speed and the moles of \(Ti\) and \(Nb\) in the resultant phases. The moles of \(Ti\) and \(Nb\) are used since \(Ti\) is a base metal with a high melting temperature, and \(Nb\) had the highest melting point temperature among all elements used in this study. Thus, the minimum energy required to facilitate the melting of deposited metal powders must be high enough to melt \(Ti\) and \(Nb\) powder ([PERSON], 2019; [PERSON] et al., 2016). Heat energy can be calculated as displayed in Equations (3) and (4):  
\[\Delta H=n_{Ti}\times C_{p}\Delta T, \tag{3}\]  
\[\Delta H=n_{TiNb}\times C_{p}\Delta T, \tag{4}\]  
Figure 6: Stacked graph showing the effects of niobium on aluminium solubility (**top**) and heat loss (**bottom**). Heat energy can also be calculated as displayed in Equations (5) and (6):  
\[\Delta H=n_{Ti}\times\int_{298}^{T_{M}}C_{p}dT, \tag{5}\]  
\[\Delta H=n_{TiNb}\times\int_{298}^{T_{M}}C_{p}dT, \tag{6}\]  
where \(n\) is the total amount of \(Ti\) and \(Nb\) moles [23] in the build, \(C_{p}\) is the specific heat (J/mol-K), and \(T\) is the temperature velocity (K/cm) since the deposition nozzle moves at a velocity [23; 25]. The temperature values of the specific heat \(C_{p}\) for various phases are displayed in Table 5.  
The Equations listed in Table 5 were substituted into Equations (7) and (8). This was performed to calculate the unknown solidification temperature velocities depicted as \(T_{S}\) (K/cm) [23; 29]. The values for \(T_{S}\) were obtained in Microsoft EXCEL using SOLVER. Below are the equations used to solve for \(T_{S}\):  
For various builds, \(T_{S}\) is obtained by solving Equations (7) and (8) for \(T_{S}\). \[\Delta H=n_{Ti}\bigg{[}\int_{T_{S}}^{T_{L}}\frac{C_{p(Ti,l)}}{M_{Ti}}dT+\int_{ 298}^{T_{S}}C_{p(Ti,AlNb,\gamma)}dT\bigg{]}, \tag{7}\]  
\[\Delta H=2.304\times LEPUL, \tag{1}\]0  
where \(M\) depicts molar mass, which allows for unit conversion during calculation. It was assumed that the minimum temperature required from the laser should be high enough to melt \(Ti\) and \(Nb\) powders; thus, \(T_{M}\) = 1943 K and 2742 K [30] were used. Values obtained for \(T_{S}\) allowed for the calculation of the cooling rates per unit length during deposition until solidification was complete. The thermal gradient of a singularly deposited unit length until the completion of solidification with respect to time is displayed in Equation (9) as the derivative of temperature with respect to time that is inversely proportional to cooling rates and solidification rates as displayed in Equation (10) [25].