Phase Change: Cooling and Solidification of Metal

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:

Model depicting 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:

Graph of the phase change function.

Phase transition function.

COMSOL software settings.

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:

Plot highlighting latent heat.
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.

Modeling with uniform mesh.

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

Results from applying an adapted mesh in the continuous casting process.

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

Further Reading

Comentários (13)

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Denis Maus
August 29, 2014


looks interesting. Can one use this technique to model a “liquid-to-gaseous” phase-change, like in the case of heating water over the saturated temperature?

Kind regards,

Tung Tran
March 27, 2016

That’s also my concern, but in 3D model.
Best regards,

Nancy Bannach
April 1, 2016

Dear Denis, dear Tung,

for modeling liquid to gaseous phase change I recommend to have a look at another model from our application library: Evaporative Cooling of Water (
Liquid water evaporates as water vapor into a gaseous phase, whereas the gaseous phase is usually a mixture of (dry) air and water vapor.

Best regards,

Anjana Thimmaiah
July 6, 2016

Dear Nancy,
Does the Apparent Heat Capacity Method have a shortcoming in modelling liquid-gas phase change? Why is it not recommended for liquid-gas phase change modelling?

I used the model to simulate evaporation of a liquid film of water and used level-set to track the interface of the film. The simulation results concurred with most correlations.

Kind regards,
Anjana Thimmaiah.

Nancy Bannach
July 13, 2016

Dear Anjana,
Basically, the Apparent Heat Capacity Method describes a sudden change in the material properties of one phase and accounts for the latent heat at this sudden change.
Usually, evaporation takes place between two materials (e.g. air and water) and not at a specific temperature, rather it depends on the saturation of the gas-phase (which depends on the temperature). If the gas phase is saturated with the liquid phase, no evaporation occurs.
To model this process (diffusion of the liquid phase into the gas phase, depending on the saturation) the apparent heat capacity method is not sufficient.

Of course, if the method works for you there is no need to change it. It may be suitable for some processes but in general, I would say, it is not.

Feel free to follow up on this discussion and/or contact us via

soy Noh
June 5, 2017

Hello, I have a question.
Can it be possible?
Gas jet includes solid particles, and those particles can be sublimated into gas phase (by contacting hot plate).

Best regards,
Noh soy.

soy Noh
June 5, 2017

Hello, I have a question.
Can it be possible?
Gas jet includes solid particles, and those particles can be sublimated into gas phase (by contacting hot plate)

Best regards,
Noh soy.

Nancy Bannach
June 22, 2017

Dear Noh soy,
to answer your question properly we need more detailed information about your simulation. You could add a heat source at the boundary (hot plate) that is proportional to the incoming particles, but much more effects can be involved that you want to consider and the apporach shown here may be not sufficient in your case.

I recommend to contact our support team for discussion on this topic:

Online Support Center:

Best regards,

Ying Zhao
October 5, 2018

Hi Nancy,

Thanks for publishing this blog, it is very useful to understand apparent heat capacity method. But I found in COMSOL heat transfer with phase change interface, the build-in equations are not the same as you listed in the blog. Cp=(theta*rho,phase1*Cp,phase1+(1-theta)*rho,phase2*Cp,phase2)/rho + L*d(alpham)/dT
The theta is the phase1 fraction which =1 in solid and =0 in liquid (I think it is 1-alpha as in your blog?) But the alpham in those equation is defined as “mass fraction”. I don’t quite understand where this term comes from, could you explain more based on the equations used in COMSOL phase change interface?


Nancy Bannach
October 8, 2018

Dear Ying,

Thank you for your feedback.
It is not sufficient to just use the phase indicator alpha, which is based solely on the temperature. For calculating the latent heat released during the phase transition, you also need to take into account the mass of each phase.
The “Theory for Heat Transfer with Phase Change” section in the Heat Transfer User’s Guide has outlined this very well. If you have further questions I recommend to contact our support team for a detailed discussion:

Best regards,

Anand Mohan
October 9, 2018

Hi Nancy

Is it possible to simulate welding operation say in Laser Welding where there are two change in phase that is solid to liquid(heating cycle) and liquid to solid (cooling cycle). Is it possible to simulate this with the same concept and using two heat transfer with phase change(each one for a phsae change ) or we have to do something else?

With kind regards,
Anand Mohan

Sunku Prasad J
Sunku Prasad J
December 27, 2019

Dear Nancy,
In the apparent heat capacity method, COMSOL is using the following equation for the density of the phase change material:
rho = theta*rho_phase1+(1-theta)*rho_phase2.
we know theta is a function of temperature.

If the density changes with the temperature, how the volume changes are accommodated?
How it is possible to change the density of the material without changing the volume (since the mass of the PCM is constant before and after melting)?

Kindly clarify.
Sunku prasad

Nancy Bannach
Nancy Bannach
January 9, 2020 COMSOL Employee

Dear Sunku Prasad,
you are right, the volume changes are assumed to be small enough so that their influence on the simulation result is negligible. In many cases this assumption provides sufficiently accurate results. If you want to take volume changes into account, a coupling with a so-called ALE interface is required. This allows you to calculate the deformation due to density changes by deforming the mesh.

A good starting point to learn more about this can be found here:

I hope this helps.

Best regards,