The three-dimensional heat transfer mathematical model of numerical simulation of solidification process of sand casting is unsteady heat conduction partial differential equation:
Where:
T = (x, y, Z, t) – temperature, which is a function of space and time
ρ — Density
C – specific heat
λ x, λ y, λ Z – thermal conductivity in X, y and Z directions
QL – latent heat of crystallization
Here is only isotropy, i.e（ λ x= λ y= λ z= λ) The above formula is described in the case of. The formula can be written as:
According to different problems, the formula can give the following boundary conditions:
1) The first type of boundary condition: specify the temperature value on the boundary. The simplest case of this kind of boundary condition is that the given boundary temperature value remains unchanged, that is, TW is a constant. For the unsteady heat conduction problem, the temperature boundary condition can also be specified as a function of space and time on the boundary, that is:
2) The second kind of boundary condition: the heat flux value on the given boundary. The simplest form of this kind of boundary condition is that the heat flux on a given boundary does not change, that is, QW = constant. For a given boundary condition, i.e. the density of heat flow in space and time:
3) The third type of boundary condition: the surface heat transfer coefficient between the object on the specified boundary and the surrounding fluid, and the temperature boundary of the surrounding fluid. Such boundary conditions can be expressed as:
In the above three kinds of boundary conditions The first kind of boundary conditions are mandatory boundary conditions, and the second and third kinds of boundary conditions are natural boundary conditions. In the process of sand casting, the heat transfer process between casting and sand mold, between sand casting and cold iron, and between mold and air are the third kind of boundary conditions.
In sand casting, the initial temperature of sand casting, sand mold and chill is constant, the initial temperature of sand casting is pouring temperature, and the initial temperature of sand mold and chill is preheating temperature.
The above heat conduction partial differential equation, the initial conditions and boundary conditions for solving the heat conduction partial differential equation. Together, they constitute a three-dimensional heat transfer mathematical model of sand casting solidification process.