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Phase Change: Cooling and Solidification of Metal

Nancy Bannach | August 12, 2014

Modeling phase change is important for many thermal processes, ranging from the food industry to the metal processing industry. The Heat Transfer Module offers a dedicated interface for modeling the characteristics of phase change. It uses the apparent heat capacity method, which we introduce here.

Example: Continuous Casting

Phase change is a transformation of material from one state of matter to another due to a change in temperature. Phase change leads to a sudden variation in the material properties and involves the release or absorption of latent heat. We can use the Heat Transfer Module to model this type of phase change. Let’s start with an example.

In the continuous casting process, liquid metal is poured into a cooled mold and starts to solidify. As the metal leaves the mold, the outside is solidified completely, while the inside is still liquid. To further cool down the metal, spray cooling is used. When the metal is completely solidified, it can be cut into billets. This is a stationary, time-invariant, process. The rate at which the metal enters and leaves the modeling domain does not vary with time, and neither does the location of the solidification front.

Here is an illustration of the continuous casting process:

Sketch of a continuous casting process.

In order to optimize and improve this process, we can turn to simulation. With COMSOL software, we can predict the exact location of the phase interface.

Modeling Phase Change with the Apparent Heat Capacity Method

COMSOL Multiphysics and the Heat Transfer Module together offer a tailored interface for modeling phase change with the Apparent Heat Capacity method. The method gets its name from the fact that the latent heat is included as an additional term in the heat capacity. This method is the most suitable for phase transitions from solid to solid, liquid to solid, or solid to liquid. Up to five transitions in phase per material are supported.

When implementing a phase transition function, \alpha(T), a smooth transition between phases takes place, within an interval of \Delta T_{1\rightarrow2} around the phase change temperature, T_{pc, 1\rightarrow 2}. Within this interval, there is a “mushy zone” with mixed material properties. The smaller the interval, the sharper the transition.

The below figure shows the phase change function for the continuous casting model:

Phase transition function.

COMSOL Multiphysics settings for phase change. Keep in mind that phase 1 is below T_{pc, 1\rightarrow 2} and phase 2 is above.

The material properties for the solid and liquid phase are specified separately. These values are combined with the phase transition function so that there is a smooth transition from solid to liquid. The heat capacity of the material is expressed as C_p=C_{p,solid}\cdot(1-\alpha(T))+C_{p,liquid}\cdot\alpha(T), and similarly for the thermal conductivity and density. For a pure solid, \alpha(T)=0, and for a pure liquid, \alpha(T)=1. Within the transition interval, the material properties vary continuously.

The latent heat is included by an additional term in the heat capacity. Let us take a look at the derivative of the phase transition function:

Derivative of the phase transition function.

Integrating this function over \Delta T_{1\rightarrow2} gives 1 and multiplying by the latent heat L_{1\rightarrow 2} gives the amount of latent heat that is released over \Delta T_{1\rightarrow2}.

Consider the stationary heat transfer equation with a convective term, of the form:

\rho C_p\cdot \nabla T=\nabla\cdot\left(k\nabla T\right)

The Apparent Heat Capacity method uses the following expression for the heat capacity:

C_p=C_{p,solid}\cdot(1-\alpha(T))+C_{p,liquid}\cdot\alpha(T)+L_{1\rightarrow 2}\frac{d\alpha}{dT}

The advantage of this method is that the location of the phase interface does not need to be known ahead of time.

Application to the Continuous Casting Process

With the help of the Heat Transfer with Phase Change interface, the implementation is straightforward. Axial symmetry is assumed, and the model is reduced to a 2D domain. The casting velocity is constant and uniform over the modeling domain.

To get a sharp transition and thereby the exact location between the solid and liquid phase, we need a small transition interval, \Delta T_{1\rightarrow2}. Resolving such a small interval properly requires a fine mesh. However, we do not know the location of the solidification front in advance, so we first solve the model with a gradual transition interval, and then use adaptive mesh refinement to get better resolution of the solidification interface. The transition interval can then be made even smaller.

The results are compared below for two different transition intervals. As the transition interface is made smaller, the model better resolves the transition between liquid and solid. This information can be used to improve the continuous casting process, and this same approach can be used for similar applications involving phase change.

Phase, temperature, and latent heat for \Delta T_{1\rightarrow2}=300K on the uniform mesh.

Phase, temperature, and latent heat for \Delta T_{1\rightarrow2}=25K on the adapted mesh.

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