state and into a gaseous state. You have probably seen this transformation process at a rock concert, where blocks of solid carbon dioxide, or dry ice, create the “smokey” atmosphere up on stage. This ability for a solid to skip the liquid phase and pass directly into the gaseous phase can be explained by a phase diagram, where a substance’s state under different pressure and temperature conditions is described. As is shown in a phase diagram, at very low pressures and temperatures, a solid can pass directly into the gaseous stage without passing through the intermediary liquid phase.
Freeze-drying consists of two main steps. In the first step, a heat-sensitive substance such as food, blood plasma, or antibiotics is frozen in a vial or chamber. Next, the pressure is lowered in the chamber and enough heat is slowly added to allow the frozen substance, usually water, to sublime. A simulation of this process is shown below, where ice sublimes into vapor while in a vial.
Simulating Sublimation Using a Moving Mesh
When simulating the freeze-drying process, a moving mesh must be used in order to account for the two material phases and the change in shape that occurs as the ice sublimes. This can be modeled according to a heat and mass balance analysis defined at the solid-gas boundary using the Deformed Geometry interface. The moving mesh method allows for a very detailed description of the physics phenomena at the interface between the two phases. The geometry of the model is shown below:
Geometry of the container used in the freeze-drying process.
To make the simulation more interesting, a heater is placed under the vial so that it covers only half of the bottom of the container, producing uneven heating. In the beginning of the simulation (shown below), there is ice on the bottom of the container, while the remaining space in the container is composed of vapor that has been created by sublimation.
Initial state of the water in the vial.
As the ice continues to sublime, the temperature and flux distribution in the model can be measured. After a period of t = 1800 s, the melting ice has the proportions shown below:
Geometry of the melting front after 30 minutes.
You can see from this visualization that the melting front has a concave curved shape that is higher around the outside edges of the vial wall. Additionally, because the heating is uneven at the bottom of the vial, the interface level is slightly lower, near x > 0. As the ice becomes a vapor, the mass of the solid decreases and therefore the mesh of the solid must change as well. The image below shows the moving mesh at the initial time (left), and after 30 minutes (right). As you can see, the swept mesh has changed to follow the receding freezing front.
The moving mesh at initial time (left), and after 30 minutes has passed (right).
- Download the model: A Transient Analysis of Freeze-Drying
- Related paper from the COMSOL Conference 2011: Finite Element Modeling of Freezing of Coffee Solution
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