(458c) Direct Quadrature Method of Moments for Turbulent Aggregation of Fine Particle Populations
AIChE Annual Meeting
2006 Annual Meeting
Particle Technology Forum
Population Balance Modeling for Particle Formation Processes II: Nucleation, Aggregation and Breakage Kernels
Wednesday, November 15, 2006 - 3:55pm to 4:15pm
The computational fluid dynamics (CFD) modeling of nanoparticle synthesis reactors operated in the turbulent regime requires explicit knowledge of the particle size distribution (PSD) and the interaction between turbulence and mixing. Quadrature method of moments (QMOM) tracks the evolution of selected moments in space and time, overcoming the difficulty of discretizing the population balance equation (PBE) governing the number density function (NDF). However, QMOM approaches suffer from inverting the weights and abscissas given moments when extended to handle a bi-variate PBE.
In this work, the direct quadrature method of moments (DQMOM) is applied firstly to a univaritate then a bivariate PBE in which the NDF depends on the volume and area of the particles. The interaction by exchange with the mean (IEM) model is used to close sub-grid mixing between two fluid environments, each of which has its own NDF, and thus its own set of NDF moments (weights and abscissas). Given the initial distribution and the boundary conditions, the weights and abscissas of each environment and thus the Reynolds-averaged mean are found directly from the resulting DQMOM-IEM-PBE rather than through the QMOM-IEM-PBE in which the moments of the PSD are modeled. Nevertheless, QMOM and DQMOM are essentially consistent since the former can be exactly recovered from the latter.
For univariate cases, we show that QMOM-IEM and DQMOM-IEM yield identical results (M weights, M volumes and 2M moments) for homogeneous and one-dimensional inhomogeneous turbulent flows with micromixing and aggregation. For example, we consider the application of a two-environment (N=2), four-node (M=4) DQMOM-IEM-PBE in a 1-D periodic domain. The model equations are solved by a time-splitting method. The zero-order NDF moments decrease gradually as a result of the decreasing number density due to aggregation. The Reynolds first-order NDF moment remains constant, indicating that the total volume of the particles does not change with mixing or aggregation. Due to micromixing, the NDF moments in the environments approach the Reynolds-averaged NDF moments at large times.
The implementation of the DQMOM-IEM-PBE can be extended to 3-D inhomogeneous reactive flow through user-defined functions hooked to a CFD code. The nanoparticle size distribution in a confined impinging-jet reactor is investigated and compared with the existing experimental results for model validation purposes. The DQMOM-IEM-PBE approach has the attractive property that it predicts the evolution of the weights and abscissas directly and takes micromixing into consideration at the same time at a relatively low computational cost.