![]() A time-varying temperature is applied at one side. We are interested in computing the temperature as a function of time and position through the thickness and can reduce this to a one-dimensional model to get started. ![]() One side wall is perfectly insulated and the wall on the other side is held at a known temperature that varies periodically over time. ![]() Here, we will look at a simple example model of a phase-change material within a thin-walled container. Implementing Thermal Hysteresis in COMSOL Multiphysics Let’s now look at how to implement this in COMSOL Multiphysics. The only additional requirement for modeling thermal hysteresis is to introduce a switch to determine which path to follow. In fact, the existing Application Gallery example, Cooling and Solidification of Metal, makes use of such a temperature-dependent specific heat, albeit without hysteresis. This temperature-dependent specific heat data can be put directly into the governing equation for heat transfer and, along with an appropriate set of boundary conditions, can be solved in COMSOL Multiphysics. The specific heat is the derivative of the enthalpy with respect to temperature and is different if the material is heated or cooled. The specific heat is constant except for a small region around the melting and freezing temperatures. The above plot will also give us the specific heat, which is the derivative of enthalpy with respect to temperature. Since the material is assumed to be incompressible, the enthalpy depends only on the temperature. The plot below shows a gradual smoothing, but this transitional zone can be made very narrow to better approximate the behavior if we really did have a perfectly pure material.Ī smoothed enthalpy curve is more amenable to numerical analysis. Note that we have centered the smoothing around the nominal melting and freezing temperatures, so the fully molten state is at a temperature slightly higher than the nominal melting temperature and the fully solid state is slightly below the freezing temperature. Only once the material is fully outside of the transition zone will it switch over to following the other curve. The physical interpretation of this is that the material changes phase over some finite temperature and in the intermediate range, the material is a mixture of both solid and liquid. However, if we introduce a small transition zone over which the enthalpy varies smoothly, then we have a model that is much more amenable to numerical analysis. It is also a bit impractical for computational modeling purposes since this immediate transition between states represents a discontinuity that is quite difficult to solve numerically. Now, the above curve represents a bit of an idealized case that would only occur in the real world if we had a perfectly pure material. The latent heat of melting and solidification is the jump in these curves.Įnthalpy versus temperature for an idealized incompressible material. In the completely molten or completely solid state, the two enthalpy curves overlap. If the material is then heated back up, it will follow the bottom path, and so on. Once the freezing temperature is reached, the material becomes completely solid. When this material is subsequently cooled in the liquid state, it will follow the upper path, thus the material remains liquid at temperatures below the melting temperature. ![]() As the material passes the melting temperature, it becomes completely liquid. When the material is in the solid state and is being heated, the enthalpy is given by the bottom curve or path. We will begin by considering a representative material with hysteresis that is incompressible and plot out the enthalpy of the material as a function of temperature, as shown below. We won’t concern ourselves here with the exact physical mechanisms by which thermal hysteresis happens, but rather focus on how to model it. Such materials have applications in heat sinks and thermal storage systems and are even used by living organisms, such as fish and insects living in cold climates. What Is Thermal Hysteresis and How Is It Modeled?Ī material with thermal hysteresis will exhibit a solidification temperature that is different from the melting temperature. We can implement this behavior in COMSOL Multiphysics via the Previous Solution operator and a little bit of equation-based modeling. Such behavior can be modeled by introducing a temperature-dependent specific heat function that is different if the material has been heated or cooled past a certain point. In today’s blog post, we will introduce a procedure for thermally modeling a material with hysteresis, which means that the melting temperature is different from the solidification temperature.
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