3.2 Governing Equations of Coupled Model
67
ρC ρ
∂ T
∂t
+
− →
U · ∇
T
= ∇ · (k∇T )
(3.3)
where:
− →
U
Three-dimensional velocity vector;
μ l
Dynamic viscosity of fluid;
ρ
Density;
p
Pressure;
g
Three-dimensional gravity acceleration vector;
β
Coefficient of thermal expansion;
C p
Specific heat capacity;
k
Heat conductivity coefficient;
T
Temperature;
T re f Reference temperature;
K
Carman-Kozeny coefficient in mixed phase model, i.e., transmission coefficient, which is closely related to the liquid mass fraction f 1 of current grid
cell.
K can be determined by the following equation.
K =
f
2
l d
2
180(1 − f l )
2
(3.4)
where: d is closely related to the size of dendritic arms, which is usually a constant. For
laser welding, its size is above 10
–2 cm. In addition, C in the Eq. (3.2) is a nonreactivity
coefficient related to liquid mass fraction. It is calculated by the following equation.
C = 0.13 f
−3/2
l
(3.5)
In order to improve the value calculation speed, the liquid mass fraction may be
deemed to have a linear relation with the temperature, thus:
f l =
⎧
⎪ ⎨
⎪ ⎩
1
T > T l
T −T s
T l −T s
T l >= T >= T s
0
T < T s
(3.6)
where: T l and T s —Liquidus temperature and solidus temperature of welded alloy
material.