Modeling emulsion structures with STAR-CD

Fabien Jousse, Foods Research Centre, Unilever R&D Vlaardingen, Netherlands

As the rheology of an emulsion depends on the droplet size and volume fraction of the dispersed phase, itself evolving with time under local flow defined by local rheology, the simulation needs to reproduce this closed-loop coupling. With this aim, we define population balance equations for droplet size distribution, taking into account transport, droplet break-up, and coalescence (see Fig. 2). The governing equations are expressed in terms of the distribution moments and, as the moments are discrete quantities, the treatment is simplified. In theory, the precision of the distribution depends on the number of moments used in the simulation. In practice, however, it is generally found that the droplet size distribution follows a log-normal behaviour; hence, only 2 independent moments are required to characterise the distribution. The rheology of the dependence on the volume fraction and droplet size can be described with suitable correlations. Alternatively, an Eulerian 2-phase approach can give the flow conditions directly for the two phases, without using correlations as an intermediate step.

Fig 4: Calculated droplet size evolution in a model pin-stirrer at high phase volume fraction (a=0.8). The inserts show the distribution of droplet size along the simulation.

Both approaches were applied to the study of liquid-liquid emulsion. For low phase volume fractions (a<0.1), the simulation results showed an excellent agreement with experiments. This is exemplified in Fig. 3, for the evolution of the droplet size in a model pin stirrer. At low phase volume fraction however, the emulsion rheology is only weakly dependent on the droplet size and therefore the typical flow pattern observed in the stirrer does not depend on droplet size distribution. The simulation is different at high phase volume fraction, where the volume fraction and droplet size change the emulsion rheology. Simulations performed in a model pin stirrer at phase volume fractions up to 80% showed the profound effect of the droplet size on the velocity field: initially homogeneous drop size quickly became inhomogeneous, depending on the location within the stirrer, to finally equilibrate to a homogeneous value throughout the mixer. The latter was anticipated because of the small density difference between the two fluids.

Understanding the generation and structure of emulsions continues to present exciting challenges. Luckily, modern CFD tools allow one to model the creation of liquid-liquid emulsions in a reliable way. This becomes especially useful for process optimisation and scale-up.

Fig 1: Micro-picture of mayonnaise, with lipids in green and proteins in yellow

Fig 2: Typical droplet size distribution observed in an emulsion. Two approximations are often made to represent the actual distribution: 1) representation of the population as bins of a given width; 2) decomposition as basic log-normal distribution. The S_ approach represents an alternative way to formulate population balance equations in term of the successive moments of the distribution

Fig 3: Average droplet size in a model pin stirrer at low volume fraction. Points are the experimental observations for two different rotation speed (N) and continuous phase viscosity (m), while the lines are the corresponding simulations with STAR-CD.