A Comparison of the Thermo-Fluid Properties of Ti-6Al-4V Melt Pools Formed by Laser and Electron-Beam Powder-Bed Fusion Processes

Powder-bed fusion (PBF) process is a relatively new additive manufacturing (AM) technology where the thermal energy of a computer-controlled heat source is used for selective melting and sintering of regions of a powder-bed [1]. Once a layer of an object is completed, the building platform is lowered, and more powder is spread over (usually, rolled on) the build area for a new scan. The process ends with a postprocessing step of removing all the unbound powder [2] from the fabricated object. The conventional manufacturing technologies (e.g., casting and forging) used for fabricating medical implants and components for automotive, aerospace, and space applications constrain the customization of complex geometries and consume a significant amount of material and time. The PBF process overcomes these limitations by providing the advantage of cost-effective customization with reduced assembly [2] and thus, becomes a superior AM technology in the present era. Two of the most common types of PBF processes are the laser PBF (L-PBF) and electron beam PBF (E-PBF) processes, which have brought about a revolution in the field of metal AM technology. The L-PBF process uses finely focused monochromatic coherent photons, i.e., laser, while the E-PBF process uses a beam of electrons as the heat source for melting the powder-bed. Consequently, when the laser or the electron beam scans the top surface, it melts a selective region of the powder-bed to form a liquid volume, known as the “melt pool,” which is rapidly cooled and solidified either in a vacuum (for E-PBF) or an inert gas (for L-PBF) environment [1,2]. A wide variety of materials, including the alloys of titanium, copper, nickel, aluminum, and chromium, are being used in PBF technology where the material properties significantly influence the process behavior and quality of the processed part. Titanium alloy, especially, Ti-6Al-4V, is one of the prime choices while selecting PBF raw material because of its unique properties and extensive usability in aerospace, automotive, microelectronics, and biomedical applications.

In recent years, significant growth of interest in AM technologies, especially, in the PBF processes has been marked by numerous studies on the L-PBF and E-PBF methods [3–9]. In most cases, the focus has been on the application-based experimental studies such as build-part microstructure, powder metallurgy, morphological features, and mechanical properties of the material [4–11]. Besides these experimental analyses, numerical modeling is also conducted by many to determine the thermo-physical properties and melt-pool dynamics [12,13] for investigating the process envelope, part quality, reliability, and energy efficiency. However, studies on the comparison between the two processes under equivalent process parameters are very limited [13]. Instead of studying the L-PBF and E-PBF methods separately, a comparative study of the process parameters, melt-pool geometry, and part microstructures offers more valuable information to characterize suitable applications of the processes [13]. Numerical modeling, especially, thermal modeling is, by far, one of the most convenient and cost-effective methods to conduct a robust comparative analysis between the L-PBF and E-PBF processes [13,14]. The comparison can provide an important guide to select the appropriate technology to be used in the AM industry.

Development of the thermal models for L-PBF and E-PBF requires the understanding of complex heat transport, material phase change, and intricate relations among the thermal, mechanical, and metallurgical phenomena [13,14] which make it extremely challenging to implement. The E-PBF process requires preheating of the material at a high vacuum that needs to be characterized accurately in the simulation. On the other hand, modeling of L-PBF requires the incorporation of convection with the inert gas environment at room temperature. Consequently, understanding the correlation between the process parameters and the process outcomes without costly experimentation requires comprehensive numerical modeling. While developing a robust thermal model, it is important to find a convenient numerical scheme that can accurately estimate the melt-pool geometry and determine the temperature distribution in the processed part by taking into consideration the heat source parameters and material properties. Many researchers have developed thermal models using finite difference (FD) and finite element (FE) methods at various length and time scales [15–28]. As a common practice, the heat source in a PBF process is modeled as a conical volumetric heat flux under the surface of the powder-bed due to the resultant keyhole formed by an incident laser or electron beam. Qi et al. [15] used a controlled-volume FD method to develop a self-consistent model for studying the heat transfer, phase change, and fluid flow within the melt pool in the laser-based PBF process. Moraitis and Labeas [16] developed a thermo-mechanical FE model to investigate the residual stresses and distortions for aluminum lap joints in the laser beam welding process. Wang et al. [17], Roberts et al. [18], Liu et al. [19], Yang et al. [20], Lacki and Adamus [21], Shen and Chou [22], Cheng et al. [23], Chen et al. [24], and Zäh and Lutzmann [25] developed FE models incorporating the Gaussian heat source, powder porosity, and thermal properties to simulate the transient heat transfer in the PBF processes. Recent progress on FE analyses to investigate the effect of process parameters in PBF can be attributed to the studies of Andreotta et al. [26], Sadowski et al. [27], and Ladani et al. [28]. All these studies included numerical and/or experimental analyses of either the L-PBF or E-PBF process, but the comparison between the two processes, which could facilitate the selection of the suitable one in the industry, was not outlined.

Thermo-fluid models based on the computational fluid dynamics (CFD) and finite volume (FV) method become more effective than the FE thermal models when fluid flow and heat convection in the melt pool are dominant factors in the process outcomes [29]. Studies show that thermo-fluid models using CFD can effectively provide quantitative information about the part geometry, thermal cycle, cooling rate, and solidification process with the same accuracy as the FE models [29,30]. Wang et al. [30] developed a three-dimensional (3D) volume-of-fluid method to measure the real-time melt-pool shape and obtained the distribution of temperature in laser keyhole welding. Cho et al. [31] studied the effect of fluid motion in the melt pool using thermo-fluid simulation. Rai et al. [32,33] and Li et al. [34] showed that fluid convection inside the melt pool resulted in an increase in heat transfer and gave a better correlation between numerical and experimental results of the melt-pool geometry. Chahine [35] studied the effects of the current and exposure time of an electron beam on the temperature distribution and fluid flow of a melt spot using the CFD technique. Yuan and Gu [36] used FV simulation and laser experiments to investigate the melt-pool evolution and thermal behavior of TiC/AlSi10Mg powder-bed in the L-PBF process. Rahman et al. [37–42] conducted CFD-based thermo-fluid modeling of the Ti-6Al-4V [37–41] and Cu-Cr-Zr [42] melt pools to study the thermal features and melt-pool dynamics in the PBF processes. Jamshidinia et al. [43,44] developed 3D thermal and fluid flow models of E-PBF, where the influence of fluid convection on the melt-pool geometry was investigated and the effects of changing process parameters were studied numerically and experimentally. However, a comparison between the L-PBF and E-PBF processes based on thermo-fluid modeling is yet to be studied rigorously.

The study aims to compare the thermo-fluid properties of Ti-6Al-4V alloy while undergoing the L-PBF and E-PBF processes by developing a 3D multiphysics CFD model for each method. The effects of the laser and electron beam parameters on the temperature distribution, melt-pool geometry, melting, and solidification criteria are studied and compared under similar heat source specifications. The laser and the electron beam are considered as the Gaussian heat sources to perform simulations for the L-PBF and E-PBF models separately. The CFD simulations are conducted in ansys fluent 18.2 by setting appropriate solver specifications and utilizing the user-defined functions (UDFs) for the heat source and the material properties of Ti-6Al-4V. Results obtained from the CFD simulations are validated by comparing with analytical and experimental results to corroborate the comparative study of the L-PBF and E-PBF processes.

Ti-6Al-4V provides a unique combination of physical and mechanical properties including lightweight, high strength-to-weight ratio, and excellent resistance to corrosion and fatigue. However, the thermo-physical properties of metallic powder materials are significantly different from those of the corresponding solid bulk material [45,46], especially, the thermal conductivity, melting point, specific heat capacity, and density. When the phase change occurs from solid to liquid and vice versa, properties of the liquid material also have significant effects on the process outcomes. Table 1 shows the temperature-dependent properties of Ti-6Al-4V in solid, powder, and liquid states [39,40,46] that are used in both the L-PBF and E-PBF models as the UDFs.

A finite volume based 3D model is developed in ansys fluent for the transient thermo-fluid simulation of the PBF process. The geometry of the 3D PBF model is shown in Fig. 1, where the physical domain consists of a solid rectangular block of Ti-6Al-4V, known as the “substrate” and a layer of Ti-6Al-4V powder on top of the substrate. The dimensions of the substrate and the powder layer are considered as 14 mm × 4 mm × 4 mm and 14 mm × 4 mm × 0.07 mm, respectively [13], which implies that the powder layer thickness is 0.07 mm. The laser, or the electron beam, scans the top surface of the powder-bed in the y-direction. A single-track scan is considered for comparing the L-PBF and E-PBF models. The PBF model is first used to simulate the L-PBF process by assigning the laser as the heat source and setting the boundary and initial conditions for the L-PBF process. The same PBF model is then converted to simulate the E-PBF process by assigning the electron beam as the heat source and setting the appropriate conditions for the E-PBF process. The spot size and scanning speed for both the electron beam and the laser are kept the same (0.4 mm and 330 mm/s, respectively). They scan the top surface of the domain starting from (0, 2 mm, 0) to the endpoint at (0, 12 mm, 0) as shown in Fig. 1.

The maximum heat of the laser or electron beam is located at the center of a target surface, and the intensity varies radially from the center of the heat source. The powder-bed medium is homogeneous and continuous. The heat source is applied at the top of the powder-bed in the form of heat flux density obeying the Gaussian distribution. The top surface of the melt pool is assumed to be flat and all the nodes remained in both vertical and horizontal positions. During the melting and solidification processes, the thermo-physical properties including density, specific heat, thermal conductivity, and viscosity are considered as temperature-dependent parameters. Heat transfer by convection on the top surface is neglected in E-PBF due to the vacuum environment. The thermal boundary conditions applied in the simulations of L-PBF and E-PBF are similar, but unlike E-PBF, L-PBF included convection heat transfer on the top surface. The convective heat transfer coefficient between the powder-bed and surrounding gas in L-PBF is taken as a constant. The top surface is exposed to radiation heat transfer with an ambient temperature of 298 K for both the L-PBF and E-PBF models. The sidewalls and bottom of the domain in E-PBF are adiabatic at a temperature of 1003 K. As L-PBF does not require preheating, the side walls and bottom of the domain are considered adiabatic at a temperature of 298 K.

The laser and electron beams are modeled as conical volumetric heat sources with the Gaussian distribution as shown in Fig. 2, where the maximum power intensity is at the center and the intensity decreases with the increase of the depth and width.

Since the E-PBF process includes preheating of the entire domain to a temperature of 1003 K, the initial conditions are at t = 0, u = v = w = 0, and T = T_preheat = 1003 K, in the entire domain. The top surface in the E-PBF model is exposed to radiation only at 298 K while the side walls and the bottom of the substrate are considered as adiabatic surfaces. For the L-PBF model, the initial temperature is 298 K and boundary conditions are same as the E-PBF model except for the top surface, which is exposed to radiation and convection at 298 K. The temperature coefficient of surface tension (Marangoni coefficient) is set as −2.6 × 10⁻⁴ N/m K at the top surface for both cases [40,44].

Transient thermo-fluid analyses of the 3D configuration are performed numerically using ansys fluent 18.2. The thermal properties and the specifications of the moving heat source are assigned as UDFs to simulate the transient melting and solidification for both the L-PBF and E-PBF models. The mass, momentum, and energy conservation equations are discretized and solved using the control volume method with appropriate boundary conditions. ansys Design Modeler is used to create the geometry, Mesh tool is used to generate the structured mesh, and the mathematical model is followed to define the boundary types of the 3D computational domain. Under the pressure-based solver, the “Coupled” algorithm was selected for solving the conservation equations. The Pressure Staggering Option (PRESTO) is selected for pressure discretization, while density, momentum, and energy are discretized with the second-order upwind method. The transient formulation is done by choosing the first-order upwind method. The Courant number is set equal to 1 and all the residual criteria are set to be 10⁻⁵. The time-step size is restricted to be 0.001 s or lower to satisfy the convergence criteria while the maximum iterations per time-step are taken as 50/s for running the calculation. After the solution, the postprocessing steps begin to observe and save the results.

The 3D computational domain considered for the analysis is discretized using a structured mesh with hexahedral cells as shown in Fig. 4. The structured mesh is formed by biasing the grid in the powder layer region and around the scanning path of the moving heat source to have a very fine mesh in the target zone.

A mesh convergence (i.e., the mesh sensitivity) study is conducted for the structured mesh of the 3D domain considering the variation of melt-pool temperature with the increase of the number of nodes. The temperature at location $(x,y,z)$ = (0, 5 mm, 0.03 mm) is monitored for several different mesh densities at 0.009 s when the beam spot size is 0.4 mm, scanning speed is 330 mm/s, and the effective power is 216 W. The value of temperature inside the melt pool converges to 2571 K with the increase of the number of nodes in the domain for the E-PBF model. Figure 5 depicts the results for the mesh convergence study where the temperature at the fixed point remains unchanged after 200,889 nodes corresponding to 190,040 hexahedral cells. Results for both the L-PBF and E-PBF models are obtained for the converged mesh.

The validation of the fluent result is conducted by predicting the motion of the liquid–solid interface during the melting of pure Ti. The parameters shown in Table 2 are used in the ansys fluent simulation.

The simulation results of the change in interface position with respect to time during the melting of pure Ti show a good agreement with the analytical results. The liquid fraction contours obtained from the simulation at t = 0.37 s and t = 1.8 s are shown in Fig. 7, where the melt front moves forward with the increase of time.

Figure 8 shows the comparison between the analytical and simulation results for the melt-front position with respect to time considering x = 1 mm.

Results for temperature distribution also show a good match between the analytical and simulation results. At x = 1 mm, the comparison between the analytical and simulation results for centerline temperature at three different times is shown in Table 3.

The numerical results for the melt-pool geometry are also validated by comparing with the experimental results. The experimental procedure for the E-PBF process, conducted by Jamshidinia et al. [44] with Ti-6Al-4V, is followed to validate the proposed multiphysics model. Jamshidinia et al. [44] compared the results for the variation of the average melt-pool width and depth with the change in scanning speed. Using a constant electron beam spot size of 0.4 mm, a beam current of 14 mA, and a voltage of 60 kV, they applied three levels of scanning speed, namely, 100 mm/s, 300 mm/s, and 500 mm/s to measure the average melt-pool width and depth. The differences between their modeling results and experimental results ranged from −3.5% to +3% for the melt-pool width, and from +2.1% to +3.5% for the melt-pool depth.

Following a similar approach, the simulated results for the melt-pool geometry from the proposed multiphysics CFD model are compared with the experimental results presented by Jamshidinia et al. [44] using the converged mesh of 190,040 hexahedral cells connected by 200,889 nodes. The comparison gave a good agreement between the simulation results and the experimental results of Jamshidinia et al. with a maximum deviation of 3.73%, which indicates a good accuracy of the proposed multiphysics CFD model. The comparison of results for the melt-pool width and depth is illustrated in Fig. 9.

Results for the thermo-fluid properties and melt-pool geometry are generated at similar irradiation conditions to make a valid comparison. The simulations for the L-PBF and E-PBF models are conducted using the UDFs and the parameters shown in Table 4.

The heat source scans the top surface in the y-direction in both the L-PBF and E-PBF models. The temperature contours at the top surface for L-PBF and E-PBF are shown in Fig. 10.

The images for the temperature contours are captured when both the laser and electron beams are at y = 8.0 mm (t = 0.018 s). The melt region is longer in the contour for E-PBF than that in L-PBF. The position y = 8.0 mm at the top surface is chosen to show the complete tailing effect during melting and consolidation in the two processes. A cross section is considered at y = 8.0 mm (t = 0.018 s) along the xz-plane to show the comparison of the results in the form of a two-dimensional representation as shown in Figs. 11 and 12. The temperature contours at the cross section for L-PBF and E-PBF are shown in Fig. 11.

As the temperature in the domain increases, density and viscosity decrease following the temperature-dependent equations given in Table 1. The results for density and viscosity are not represented here but they show similar patterns as presented by Rahman et al. in Refs. [39] and [42], respectively. The density of the powder-bed is less than that of the solid substrate due to its porosity. The values of other thermal properties including thermal conductivity, specific heat capacity, and enthalpy increase with the increase of temperature in the domain. Before scanning by the heat source, the thermal conductivity of the powder material is lower than that of the solid substrate as the porosity of the powder reduces the thermal conductivity. When the heat source is applied and the powder material is melted, the thermal conductivity of the liquid melt pool becomes very high due to the high temperature. Contour plots for thermal conductivity in L-PBF and E-PBF are shown in Fig. 12. All the results are obtained at an energy density of 26 J/mm³ and an interaction time of 1.212 ms for both the laser and electron beams. Any increase in energy density due to the increase of power causes a temperature rise in the domain, and eventually affects the temperature-dependent properties significantly.

The evolution of the melt pool depends on several factors including the material properties, processing parameters, energy absorption, and thermo-fluid interactions. Under the same energy density of 26 J/mm³, the results for the maximum length, width, and depth of penetration of the melt pool at y = 8.0 mm for L-PBF and E-PBF are shown in Table 5.

The melt-pool volume is larger in E-PBF than that in L-PBF as obtained from the simulation results. The high absorption rate of electron beam, preheating condition, and lack of heat dissipation due to convection result in the formation of a larger volume of the melt pool in E-PBF. Figure 13 depicts the comparison of the length and width of the melt pool when the laser and the electron beam are at y = 8.0 mm.

A parametric study on the effects of processing parameters on the evolution of the melt pool is also conducted with the numerical simulations of the L-PBF and E-PBF processes. First, the effect of increasing the power of the laser and electron beams on the depth of penetration of the melt pool is investigated while keeping the same spot size of 0.4 mm and scanning speed of 330 mm/s. As expected, the depth of the melt pool increases with the increase of beam power. The comparison of the melt-pool depth in the L-PBF and E-PBF processes are shown in Fig. 14.

In contrast to the beam power, the depth of the melt pool decreases as the scanning speed increases at a given power and a spot size of the laser or electron beam. The simulation results for melt-pool depth versus scanning speed for L-PBF and E-PBF at a power of 240 W and a spot size of 0.4 mm are shown in Fig. 15. Results show that the melt-pool depth is more sensitive to the change in beam power as compared with the change in scanning speed at a given spot size. For instance, due to an increase of 127.27% of the scanning speed from 330 mm/s to 750 mm/s in the L-PBF model, the percentage of decrease in the melt-pool depth is 55% (as shown in Fig. 15). However, a power increase of 87.5% from 240 W to 450 W in the L-PBF model results in a 433.33% increase in the melt-pool depth, which is calculated from the values shown in Fig. 14. Therefore, variation in the melt-pool depth due to the change in beam power is significantly higher than the variation caused by the change in scanning speed.

The simulation results for the velocity distribution inside the melt pool obtained from the L-PBF and E-PBF models are also compared under the same energy density of 26 J/mm³ and heat source interaction time of 1.212 ms. The velocity of liquid inside the melt pool in L-PBF is higher than that in E-PBF due to the greater convection in L-PBF. Along the yz-plane corresponding to the origin (i.e., the longitudinal section), the velocity contours inside the melt pool when the laser and electron beam are at y = 8.0 mm are shown in Fig. 16. The maximum velocity for the L-PBF model is found in the middle of the melt pool, whereas the maximum melt-pool velocity in the E-PBF model is detected toward the tail end from the center.

The temperature variation leads to a surface tension gradient, which causes the Marangoni flow from low surface tension area to high surface tension area of the melt pool as described by Yuan and Gu [36]. The cooler liquid near the edge of the melt pool having higher surface tension tends to pull the liquid away from the melt-pool center. However, the magnitudes of the maximum velocity in the melt pool for the L-PBF and the E-PBF models are about 18.6 mm/s and 15.4 mm/s, respectively, which confirm that the values of the Reynolds number Re(= ρu_iΦ/μ) are very low and the flow is laminar in both the L-PBF and E-PBF cases.

The cooling rates for the Ti-6Al-4V melt pool in L-PBF and E-PBF are found very fast due to the combined heat transfer. For the given specifications of the laser and electron beam, a point on the top surface at y = 5 mm along the scan is considered to observe the temperature variation with respect to time. Any temperature above the liquidus temperature (1938 K) indicates a complete liquid state of that point. If the temperature of that point is below the solidus temperature (1878 K), it specifies the point to be in the solid state [39,41]. The material is in the mushy zone when it remains between the liquidus and solidus temperatures. The heating and cooling of the point at $(x,y,z)$ = (0, 5 mm, 0) with respect to time for L-PBF and E-PBF are shown in Figs. 17(a) and 17(b), respectively. The variation of heating and cooling rates (in K/µs) with respect to time [22] for the L-PBF and E-PBF models are shown in Figs. 18(a) and 18(b), respectively, where the vertical axes values are multiplied by (−1). In both cases, the time count starts when the heat source strikes the point. The liquid melt pool takes 4.5 ms to cool down from the maximum temperature to the solidus temperature in L-PBF, yielding an average cooling rate of 1.5 K/µs. Under the same irradiation condition, the liquid melt pool takes about 8 ms for cooling in E-PBF, which gives an average cooling rate of 0.74 K/µs. These results are obtained for an energy density of 26 J/mm³ and the interaction time of 1.212 ms of the laser and electron beams.

Numerical modeling for the transient thermo-fluid properties of a Ti-6Al-4V powder-bed in the PBF process is conducted to determine the differences between the L-PBF and electron beam PBF (L-PBF) processes. The unique features of the two processes are characterized by implementing physical assumptions. A moving heat source with the Gaussian distribution is applied as a user-defined function and the thermo-fluid properties of the melt pool are obtained using the finite volume method. Modeling results for the temperature distributions, thermo-physical properties, melt-pool geometries, and solid–liquid–solid phase changes are compared at similar operating conditions. Based upon the comparison, the following conclusions can be drawn:

• A similar irradiance condition is considered for both the laser and the electron beam to make a valid comparison. The absorption efficiency in the Ti-6Al-4V powder-bed is determined by adjusting the powder porosity, layer thickness, optical absorbance, optical penetration depth, and efficiency of the heat source.

• For an irradiance level of 26 J/mm³ and interaction time of 1.212 ms, simulation results show that the liquid melt pool in E-PBF cools down in 8 ms whereas the melt pool in L-PBF cools down 4.5 ms. Thus, the cooling rate in L-PBF is faster than that in E-PBF. The lower volume of melt pool, absence of preheating, and convection heat transfer at the top surface make the cooling process faster in L-PBF.

• At the same scanning speed and beam spot size, the depth of the melt pool in E-PBF is higher than that in L-PBF. The lack of penetration of laser causes shallow melt-pool depth in L-PBF. Although the length and depth are significantly different, the width of the melt pool is quite similar in both L-PBF and E-PBF under similar operating conditions.

• The melt-pool volume is larger in E-PBF than that in L-PBF under the same energy density of 26 J/mm³ and interaction time of 1.212 ms. The melt-pool volume increases with the increase of beam power but decreases with the increase of scanning speed as evinced by the simulation results and experimental validation.

• The porosity of Ti-6Al-4V powder is taken as 50% for both L-PBF and E-PBF to keep the consistency of comparison. However, if the powder porosity is increased, the maximum temperature in the melt pool becomes higher due to the lower density of powder. This eventually makes the cooling rate slower than the current cases.

• Due to greater convection, the melt-pool velocity in the L-PBF process is higher than that in the E-PBF process. The fluid flow is laminar in both cases.

• A comparative study, differentiating the effects of the laser and electron beams under similar irradiation conditions, provides a thorough understanding of the physics involved in the two processes. The study facilitates the design for correct experiments prior to the actual production by giving room to optimize the process parameters and control the energy transfers in the L-PBF and E-PBF processes.

Numerical simulation of thermal behavior and the melt-pool dynamics as a result of the interaction between the moving heat source and powder zone for a single scan is the foundation for obtaining feedback of laser or electron beam processing parameters in the PBF process. The residual stress analysis of the processed part and shape optimization of the melt zone also depend on the thermal history and melt-pool evolution during the process. Therefore, the thermo-fluid models presented in this study are focused on characterizing the thermal history and melt-pool dynamics of the two significant PBF processes and offer a wide range of results with good accuracy. The model can incorporate various materials and operating conditions for further analyses of the material and process behavior in the powder-bed fusion additive manufacturing process.

• This research is funded by the National Science Foundation (award number OIA-1541079) and the Louisiana Board of Regents.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

a =

absorption coefficient of laser (1/μm)

h =

convective heat transfer coefficient (W/m² K)

k =

effective thermal conductivity (W/m K)

s =

melt-front position with respect to time

t =

time (s)

z =

distance in the direction of penetration (mm)

A =

optical absorbance

B =

a computational constant to avoid division by zero

C =

a constant for viscosity calculation

H =

total enthalpy (J/kg)

P =

pressure (Pa)

R =

molar gas constant (kcal/mol K)

S =

penetration depth (µm)

T =

temperature (K)

V =

acceleration voltage of the electron beam (kV)

c_p =

specific heat capacity (J/kg K)

c_p,l =

specific heat capacity of liquid material (J/kg K)

f_L =

liquid fraction

g_z =

gravitational acceleration (m/s²)

k_l =

thermal conductivity of liquid Ti (W/m K)

l_t =

powder layer thickness (mm)

t_i =

heat source interaction time (s) or (ms)

v_s =

beam scanning speed (mm/s)

x_i =

distance along the Cartesian coordinates (mm)

x_s =

instantaneous position of heat source in the x-direction (mm)

y_s =

instantaneous position of heat source in the y-direction (mm)

A_w =

atomic weight (g/mol)

C_M =

a constant regarding mushy zone morphology

E_D =

energy density (J/mm³)

I_b =

electron beam current (mA)

I_xy =

the Gaussian surface intensity profile

I_Z =

beam penetration function

L_f =

latent heat of fusion (kJ/K)

P_H =

power of the heat source (W)

$Q˙(x,y,z)$ =

absorbed heat flux (W/m²)

$Sgz$ =

source term due to gravity

T_L =

liquidus temperature (K)

T_m =

melting temperature (K)

T_preheat =

preheat temperature (K)

T_ref =

reference temperature (K)

T_S =

solidus temperature (K)

T_w =

wall temperature (K)

u, v, w =

velocity components in the x-, y-, and z-directions, respectively (m/s)

u_i, u_j =

velocity along the Cartesian coordinates (m/s)

Ste =

Stefan number

α_l =

thermal diffusivity of liquid (m²/s)

β =

coefficient of volume expansion

ɛ =

emissivity

η =

efficiency of laser or electron beam

λ =

parameter in interfacial melt-front equation

μ =

absolute viscosity (N·s/m²) or (Pa·s)

μ_m =

viscosity of liquid metal (N·s/m²) or (Pa·s)

ρ =

density of the material (kg/m³)

ρ_l =

density of liquid Ti (kg/m³)

σ =

Stefan–Boltzmann constant (W/m² K⁴)

Φ =

laser or electron beam spot size (mm)

Φ_V =

viscous dissipation term