2.2 Basic Models of Quasi-Steady Laser Welding
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2.2.1 Governing Equations of the Flow and Heat Transfer
in Quasi-Steady Laser Welding
The governing equations of the main variables (velocity, temperature, etc.) that need
to be solved in the flow and heat transfer problems can be expressed in the following
general forms:
∂(ρϕ)
∂t
+ div
ρ
− →
U ϕ
= div
ϕ gradϕ
+ S ϕ
(2.1)
where: ϕ—General variable;
− →
U —Cartesian coordinate velocity component u in the x direction, v in the y direction,
and w in the z direction;
ϕ —General diffusivity;
S ϕ —General source term, including Darcy source term of the mushy zone of
momentum conservation equations, Boussinesq buoyancy source term, internal heat
source term and latent heat source term in the energy equation;
ρ—Density of material.
Suppose m l represents the mass component of chemical composition l. If we have
speed
U , the conservation equation of m l can be expressed as follows:
∂(ρm l )
∂t
+ div(ρm l
U ) = div(( l gradm l ) + R l
(2.2)
where: R l —Productivity of composition l per unit volume, (kg/s m
3 );
ϕ —Diffusion coefficient of composition l.
In Eq. (2.1), the expressions of Darcy source term and buoyancy source term are
as follows:
S ϕ = −
μ
K
− →
U + ρ 0 βg(T − T 0 )
(2.3)
where: K—Permeability, calculated by Karman-Kozeny equation;
ρ 0 —Density of material at reference temperature;
β—Thermal expansion coefficient of representative material;
g—Gravitational acceleration;
T 0 —Reference temperature.
Karman-Kozeny equation is expressed as follows:
K =
( f l )
3
D l (1 − f l )
2
(2.4)
where: f l —Liquid fraction;