Cooling rate control

Unfortunately, ANSYS 11.0 does not have a calculation feature for the rate of a parameter.

Therefore, the specification of the cooling rate demands special formulation within the simulation. One way to control cooling rate is by imposing boundary conditions in the small time steps and determining the cooling rate of each grain in a post processing calculation. Then the decision for the next step is a function of the temperature and cooling rate computed for the grain. Figure 20 illustrates this method in detail.

Figure 20 Illustration of the flow chart of the program for the cooling control method defined in a multi time steps.

Preprocessing contains a data base for material properties, a processor to define geometry of the microstructure and predefined process for the boundary conditions. Solver module solves the problem for time step i. Post-processing is a decision box which determines the temperature of the grain and the cooling rate. If the cooling rate does not satisfy the simulation requirements, the solution data should be manipulated to fulfill the cooling rate requirements. If the temperature is at the solid-solid phase transformation range, the feedback process changes the

material properties to the corresponding austenite phase transformation products based on the method illustrated in Figure 20. If the temperature is below or above the eutectoid temperature range the program solves the next time step with the material properties defined for the temperature in subcritical, supercritical or intercritical temperature range. The final results are saved in output files for future analysis.

The multi time step process is very cumbersome because of the difficulties to program the feedback bridge and defining suitable solution parameters. It also requires a large amount of the memory of the computer which may cause the program to crash for big size problems. A more efficient solution of controlling the cooling rate is to introduce the cooling rate via the behavior of the thermal properties of the material. A material control rate of heat flux is proposed in the algorithm based on the conductivity equation. The energy generating term in Equation 7 can be ignored in the cooling slab process since no heat is generated because of chemical reactions or internal friction raised by external forces. Therefore, the heat conductivity governing equation is simplified to ‎[226]; obtain an exact solution for the heat distribution within the body for heat flux by conduction and convection by deploying a non-dimensionalizing integration method. The variation of density is small because atoms in the solid solution have short range diffusion. Because of the poor heat conductivity of steel, the temperature changes make little deviation in the value of the heat conductivity coefficient. Therefore, density and heat conductivity are considered at two extreme points of the temperature range of the phase appearance during the cooling process. The specific

heat capacity, C, at constant pressure, in J/(mole.K) for molar mass is defined by the next temperature dependent. This can be concluded from the next empirical expression for specific heat capacity ‎[313];

) 2

(T abT cT^

C_P (Eq-43)

The coefficients a, b and c are material dependent constants measured experimentally at constant pressure condition. Table 4 gives the measured values for a, b and c at one atmosphere pressure and above room temperature for some components in the steel microstructure ‎[313].

Table 4 The coefficient of empirical expression of the specific heat capacity defined for substances in the steel microstructure.

Equations 42 and 43 can be measured for the specific heat capacity at constant volume as well but the value of the specific heat would be less than the one calculated by Equation 42 at constant pressure. ANSYS evaluates with the same method described for the thermal expansion

coefficient for the whole range of the temperature in the simulation. If the divergence of the gradient of temperature (Laplacian of T) between adjacent points is small enough in which in the computation can be assumed to be constant, then the value of the specific heat capacity has a direct relationship to the rate of the changes in the temperature or, in other words, the cooling rate. This is the case where the thermal conductivity of material is high enough to establish such a low thermal gradient within the material. Figure 21 shows the relationship between the thermal expansion coefficient and the cooling rate.

Figure 21 The specific heat capacity of the steel can determine the value of the cooling rate through a direct relationship in the numerical calculation.

Although another table can be managed for each of material properties as well, in many cases constant value can be considered for the density and the thermal conductivity coefficient of the steel except where increased accuracy of the computation is required. The general case is to consider the convection heat coefficient of the film of the cooling agent at the vicinity of the surfaces where the heat flux leaves the bulk material. This case is more realistic since the cooling rate in the laboratory experiments and industrial applications is controlled by the material properties of the cooling agent. The trigger of heat flux is the temperature difference between the cooling agent and the hot steel. Although thermal conduction processes such as formation of gas

and vacuum at the surface of solid with fluid may affect the transient condition of the heat transfer ‎[294]. The governing equation for general heat transfer can be derived by plugging Equations 1, 2, 3 and 7 into the equation of balance of volumetric energy. This gives the following nonlinear, first order, non-homogenous, ordinary differential equation ‎[294];

T

Parameters A, V are the surface and volume of the body. The exact solution for the above equation does not exist unless some simplifications are applied on the equation. But the FEM solution can approximate the solution with acceptable accuracy. Note that in the as-cast slab cooling the term heat generating is zero. The cooling rate is defined in the equation by defining parameters h, , , and c. To increase the accuracy of the FEM approximation, the volume and surface of the as-cast steel are calculated as a function of temperature using Equations 5 and 6, respectively. Equation 43 gives the temperature dependent value of heat capacity. Hence the effect of cooling agent on cooling rate is controlled by the film convection coefficient, h, whose value can be constant or a function of non-steady process (e.g. bubble formation for different temperature gradient at convection surface).