# Population balance modelling for poly-disperse two-phase flows

A large number of two-phase flows occurring in nature and technology are characterized by the fact that one phase, referred to as continuous, occupies a coherent region of space, while the other so-called *disperse* phase is present in the form of bubbles, drops or particles. Examples of such disperse two-phase flows are boiling of water in power plants, the combustion of fuel sprays or the synthesis of nanoparticles in flame reactors. The efficiency of the exchange of mass, energy and momentum depends critically on the particle size. It is frequently distributed, i.e. the particles phase features a size distribution which develops with the flow. Collisions between particles can lead to coalescence or agglomeration and shear forces may lead to breakup. As soon as it features a particle size distribution, the flow is referred to as *polydisperse*.

The typically very high number concentration makes it infeasible to consider individual particles in simulations of technical processes. Rather, an additional equation that describes the spatial and temporal evolution of the size distribution function is solved - the population balance equation. Since the associated solution variable is not a scalar or vector property, but a function, special solution approaches are required. One technique which is adopted and further developed by HZDR is the method of classes. Here the particle population is subdivided into a series of size classes, each representing a characteristic particle size. As a result, a set of coupled transport equations is solved for the respective particle number concentrations, which include source terms that describe coalescence and breakup. A key advantage of the class method is the direct availability of the particle size distribution. By comparison against experimentally determined size distributions, coalescence and breakup models can be validated and calibrated systematically.

The implementation created at HZDR and described by Lehnigk et al., AIChE J, 2021, Vol. 68, e17539 was contributed to the development line of the OpenFOAM Foundation software in 2017. Basic elements of the technique and its application are described in the Guide to CFD for Polydisperse Flows.

## Computation on Graphics Processing Units

A disadvantage of class methods is the high computational cost associated with the pairwise calculation of coalescence and breakup frequencies as well as the assembly of long source terms. Fortunately, the computation can be parallelized and lends itself to the use of graphics processing units (GPU). GPUs are characterized by a large number of cores and, correspondingly, a high throughput. Based on the CUDA programming framework of Nvidia, an implementation is being developed at HZDR which allows a part of the computational work to be outsourced to GPUs. The time for the calculation of coalescence and decay frequencies could be reduced by a factor greater then 10 in a demonstration case. The corresponding implementation is provided through the Multiphase Code Repository by HZDR.

## Applications

The publication of the source code already enabled a number of different groups outside of HZDR to carry out simulations of polydisperse two-phase flows. The following pictures give impressions from different applications.