Page 64 - MSAM-4-1

Materials Science in Additive Manufacturing In situ electromagnetic field manipulation during LMD

The high-energy beam formed by semiconductor lasers 2.3.4. Buoyancy equation for heat
is unevenly distributed, exhibiting a trend of high energy Thermal buoyancy is typically generated by density
density in the center that gradually decreases toward the variations caused by temperature changes in liquid metals.
periphery. The Gaussian distribution laser heat source The density gradient is related to the expansion of the
model is adopted and expressed as follows: 25 liquid metal, commonly represented by Boussinesq. It

is assumed that the density of the fluid is constant in
  ( 2 2  
−
P
ωη  ω 2   xv t) + y    the time derivative and convective terms, with only the
s
Q laser = 2 r π l exp  − r 2  (IX) buoyancy term affected by density fluctuations. The

l

 b  relevant equations are simplified, and the spatial variation
of density is confined to the buoyancy term. The equation
for buoyancy is as follows: 32
Where P represents the laser power, ω is the Gaussian
2
distribution coefficient of the laser, ɳ is the laser utilization F buoyancy = ρ gβ(T-T) (XIV)
l
l
l
efficiency, and r is the effective laser radius. Where ρ represents the density of the metal at the
b
l
Convective heat transfer exists in the four lateral surfaces liquidus temperature T, and β is the thermal expansion
l
of the substrate, while both convective and radiative heat coefficient of the metal.
transfer occurs on the substrate surface. Heat dissipation
through convection and radiation is represented by the 2.3.5. Electromagnetic force equation
following equation: 28 The movement of conductive particles in an electromagnetic
field produces current density during the scanning process
h TT− ) +⋅ (
4
4
T
Q losses = Q con + Q rad =⋅( 0 εσ b ⋅ T − ) (X) of the laser head. The electromagnetic force is generated
0
under the influence of an externally applied longitudinal
electromagnetic field, and the electromagnetic force in the
Where h represents the convective heat transfer
coefficient, T is the real-time wall surface temperature, T molten pool can be expressed by the following equation:
0
is the ambient temperature, ε is the surface emissivity of F = J×B (XV)
the material, and σ is the Stefan-Boltzmann constant. Where F represents the Lorentz force, J is the current
2.3.3. Surface force equation density vector, and B is the magnetic flux density vector.
The numerical model includes two surface tensions: (i) the 2.4. Material thermophysical parameters
surface tension f generated due to the curvature between The thermophysical parameters of the Ti-6Al-4V
S,n
argon gas and the liquid molten pool interface, which acts alloy used in the simulation are listed in Table 1. The
perpendicular to the surface, and (ii) the Marangoni shear parameters are determined by commercial material
stress f arises from the uneven thermal distribution on
S,t
the molten pool surface, resulting in larger temperature Table 1. Physical parameters used in the simulation
gradients tangent to the free surface. The specific equation
for surface forces is presented as follows: 29 Property Value Unit
F = f + f = σ.κ.n+∇ σ (XI) Density (ρ) 4440 kg∙m -3
t
S,t
S
S,n
-1
-3
Where σ represents surface tension, κ is surface Dynamic viscosity (μ ) 2.6×10 (1878 K) kg∙m ∙s -1
0
curvature, and n is a vector perpendicular to the surface. Solidus temperature (TS) 1878 K
The equations for surface tension and Marangoni shear Liquidus temperature (TL) 1928 K
stress are as follows: 30,31 Specific heat capacity (cp) 550 (297 K) J∙kg ∙K -1
-1
-1
Thermal conductivity (k) 5.74 (297 K) W∙m ∙K -1
∇γ
n = (XII) Latent heat of fusion (L) 2.92×10 5 J∙kg -1
∇γ Thermal expansion coefficient (β) 1.6×10 -4 K -1
Surface tension (σ) 1.65 (1928 K) N∙m -1
dσ dσ
T
∇ σ = dT ∇ T = dT  Tn ⋅∇ )   (XIII) Surface tension coefficient   dσ   −2.44×10 -4 N∙m ∙K -1
∇− ⋅(n
-1

t
t
dσ  dT 
Where represents the temperature coefficient of -8
dT Stefan-Boltzmann constant (σS) 5.67×10 K
surface tension. Radiation emissivity (ε) 0.8 -
Volume 4 Issue 1 (2025) 5 doi: 10.36922/msam.8332