(155a) Predictive simulation of nanoparticles-precipitation in a T-mixer by coupling direct numerical simulation with population balance equations

Continuous precipitation in a static T-(micro)-mixer is a promising way for the economic production of commercial amounts of nanoscaled particles. This method is fast, inexpensive and works in water as well as in organic solvents at a wide range of different process parameters.

The challenge is to tailor the product properties of the precipitated particles, i.e. to control the particle size distribution (PSD). Particle dynamics depend on competing kinetics of various interacting parallel and subsequent steps. These are macro-, meso- and micro-mixing, chemical reactions, nucleation, growth, aggregation and stabilization. The coupling of the fluid dynamics with the kinetics of the solid formation is the key for predicting the PSD.

The modeling of the polydisperse particulate process is based on the numerical solution of a one-dimensional population balance equation (PBE), see e.g. Schwarzer & Peukert, 2004. The supersaturation which is the thermodynamically driven force of the solid formation process is calculated considering the formation of aqueous barium sulphate complex, the incomplete dissociation of sulfuric acid and the activity coefficients. Under the investigated high supersaturations homogeneous nucleation described by the classical nucleation theory and transport-controlled growth can be considered as dominant mechanisms. The complex population balance equation is solved by the commercial software tool PARSIVAL from CiT.

Two different approaches which are described in detail by Schwarzer et al. 2005 are applied to consider the influence of the fluid dynamics. First a global approach is used, which assumes plug-flow through the mixer and that mixing is totally micromixing-controlled. The mixing model which is based on the Engulfment model of micromixing by Baldyga & Bourne (1999) describes the temporal evolution of a mixing zone in which solid formation takes place. The kinetic parameter in the model is the so-called Engulfment parameter which is calculated from the specific power input. It is assumed that 50 % of the power which is determined experimentally dissipates in the first 10 % of the mixer.

The mean particle sizes can be predicted successfully by this model but not the width of the PSD. The reason for that is the neglect of the temporal and spatial concentration fluctuations which influence the individual mixing-history of each volume element in the mixer. Therefore another approach was applied calculating full flow profile by direct numerical simulation (DNS). The code of the DNS uses a finite-volume discretization of the Navier-Stokes-Equation and a scalar transport equation for incompressible flow on a Cartesian grid with 5.5 x 10⁶ cells. Due to this grid the flow field can be resolved down to the smallest vortex length (Kolomogoroff length) at a Schmidt number of 1. The calculated concentration field of the mixer is visualized in fig. 1 for a Reynolds number of 500.

The DNS is validated by optical measurements. Particle-Image-Velocimetry (PIV) and Laser-Induced-Fluorescence (LIF) are used to investigate the flow and concentration field in the mixer at a high spatial and temporal resolution. A special feature of the LIF-method is that structures smaller than the Batchelor-length can be measured.

Lagrangian Particle Tracking couples the flow field simulation by DNS with the micromixing model and the kinetics of the solid formation. Thus the provided informations (local specific power input and local instant concentration) along the paths of finite volumes through the mixer enter into the micromixing model. At this approach the Engulfment model is modified so that besides the aspect of micromixing at length scales smaller than Kolmogoroff-length it also considers the influence of slow macromixing. Therefore the passive scalar is supplied to the micromixing model in such a way that it is ensured that micromixing is always slower than the resolved mixing process in the DNS simulation. Due to this approach one resulting PSD can be determined for every path by solving the PBE. By averaging a stochastically sufficient number of paths (700 independent paths) weighted by the fraction of the total mass flow through the corresponding inflow position, it is possible to predict quantitatively the measured PSDs for different process parameters.

Additionally to the resulting PSDs lots of different characteristic parameter about the ongoing mixing and solid formation processes can be determined by the Lagrangian Particle Tracking approach. Due to this method a 3D-volume field of the micromixing rate, of the supersaturation distribution as well as of the nucleation rate can be depicted by interpolating the calculated data. These simulations give a deep insight about the temporal evolution and local distribution of the running subprocesses. A special interest is the local area of supersaturation built-up and reduction and consequently the position of nucleation as well as the regions in the mixer where micromixing is ended. This information helps for a better understanding of the processes in the mixer and thus the operating parameters and the feed composition can be optimized to synthesize nano-particles with a narrow particle size distribution. Baldyga J. & Bourne J.R., (1999) Turbulent mixing and chemical reactions. John Wiley. Chichester. United Kingdom.

Schwarzer H.C. & Peukert W., (2004) Combined experimental / numerical study on the precipitation of nanoparticles. AIChE Journal, 50(2004).

Schwarzer H.C., Schwertfirm F., Manhart M., Schmid H.-J. & Peukert W., (2005) Predictive simulation of nanoparticle precipitation based on the population balance equation, Chem. Eng. Sci., (in press)