(517c) Direct Numerical Simulation of Liquid-Liquid Droplet Interactions

Due to its lower costs regarding money and time compared to performing experiments, computer simulations grow in importance when developing industrial separation processes. On plant scales, process simulators are widely applied and on particular scales, Computational Fluid Dynamics (CFD) calculations can support design. However, those CFD simulations often do not incorporate information about the thermodynamic equilibrium that constitutes the achievable efficiency of any separation process. Regarding extraction processes, it is usual to use experimentally adjusted droplet population balances [1] or phase field models based on non-thermodynamic expressions for the free energy (e.g. f = (x-1/2)²(x+1/2)², [2]) in multi-phase simulations. On the one hand, droplet population balance based models do not consider the microscopic interfacial behavior and on the other hand, phase-field models that rely on an arbitrary two-minimum potential will not achieve the thermodynamic equilibrium.

Regarding mixtures, applying a phase building component’s concentration as phase field [3] together with a so-called influence parameter coming from Density Gradient Theory (DGT) [4] leads to a full physical representation of the interfacial behavior. To adjust the influence parameter, the model requires only one experimental interfacial tension. The DGT accounts for the impact of concentration gradients on chemical potentials and thereby defines the value of chemical potentials in the non-stable phase diagram region. The non-steady DGT [5] together with the momentum balance equations form the system of Cahn-Hilliard/Navier-Stokes equations. However, the fact that CFD requires a closure in form of a relation between density and pressure still holds. When considering liquid-liquid extraction, it is convenient to assume a mixture of two viscous, incompressible liquids, where the density just depends on concentration but not on pressure. Hence, the component balance determines also the overall density. The task then is, to calculate the pressure field in a way that mass conservation holds. Such a mixture is usually referred to as “quasi-incompressible”.

In this work, we apply this model in two dimensions and additionally consider the energy balance. Here we account for the enthalpy of mixing that is an outcome of the thermodynamic model and is calculated from chemical potentials. This procedure results in the necessity to solve the energy balance in terms of specific energy, where the solution of a non-linear equation determines temperature from the specific energy. This equation must be solved at each grid node at each time step. As thermodynamic model for equilibrium, we use the Non-Random-Two-Liquid (NRTL) model. Due to the DGT, the interfacial tensions are inherent in this model.

Having this framework implemented, we are able to investigate influences of transport properties on decomposition behavior as well as influences of temperature and convection. The model allows calculating droplet formation, coalescence and even break-up in a thermodynamic consistent approach without any simplifying assumptions. In our model calculations, we consider phase separation of binary and ternary mixtures with various mixing gaps and we can induce temperature gradients or boundary heat flows (temperature quenching) or mechanical disturbances.

In our presentation, we will show a variety of dynamic simulations of binary and ternary mixtures that have a single mixing gap. Besides investigating the influence of temperature (and therefore the influence of tie line length) on coalescence we also show, how interfacial enrichment affects mass transport across the interface. With wall boundary conditions on the bottom and top of the simulation domain, we are also able to include gravitation and study its effects on e.g. falling droplets in combination with various density and viscosity ratios.

Those and further researches may help to build large-scale fluid dynamic models that incorporate more physics rather than empiric findings. For example, in our model it is possible to measure coalescence time parallel to interfacial tensions, which then can be utilized as parameters in population balance based models.

[1] V. Alopaeus, J. Koskinen, K.I. Keskinen, J. Majander, Chem. Eng. Sci., 57, 2002, 1815.

[2] D. Jacqmin, J. Comput. Phys., 155 (1999), 96.

[3] E. B. Nauman, D. Q. He, Chem. Eng. Sci., 56 (2001), 1999

[4] J. W. Cahn, J. W. J. Chem. Phys., 42, 1965, 93.

[5] K. Kruber, M. Krapoth, T. Zeiner, Fluid Phase Equilib., 440, 2017, 54.