### On the role of numerical diffusion on QMOM and DQMOM simulation of fluidized beds

**Mazzei, L; Marchisio, DL; Lettieri, P ***(UCL)*

Friday 09 January 2009, 09:40-10:05

Seminar Room 1, Newton Institute

#### Abstract

When the behaviour of fluidized polydisperse powders is simulated by using an Eulerian-Eulerian framework, the population balance must be solved along with other governing equations. Different formulations for the population balance model exist, and in this work only particle size dependencies will be considered. Under this hypothesis, particles with the same size are assumed to move with the same velocity, calculated with specific momentum balance equations, leaving to the population balance model the tast of tracking changes in particle size only. The population balance equation can be conveniently solved with computational fluid dynamics (CDF) codes by using quadrature methods, such as the quadrature method of moments (QMOM) and the direct quadrature method of moments (DQMOM). QMOM has been mainly used in the past assuming that all the particles move with the same velocity, resulting in strong limitations, for example, when it comes to the description of particle segregation by size. DQMOM has been presented as an evolution of QMOM, being able to track the differences in particle size as well as those in particle velocity (and therefore able to descibe segregation). In this work, QMOM and DQMOM are formulated in a new form in terms of a volume density function (VDF), rather than the original number desnity function (NDF). Both methods use a quadrature approximation of order N, coupled with a multifluid model including N dispered phases plus a continuous phase. For the first time in this work QMOM has been implemented in such a form in a commercial CDF code (ie Fluent), by means of user-defined functions and scalars for the solution of the 2N transport equation for the moments, which must be convected with the proper velocity. Also DQMOM has been implemented in the same commerical CFD code, resulting in the same formulation already reported by other authors. Although the two methods are theoretically equivalent (as it is very easy to show by simple manipulation of the governing equations) they lead to completely different results because of numerical diffusion. This is demonstrated on a very simple test case, a two-dimensional fluidised bed containing two polydispersed powders, characterised by a very wide particle size distribution ranging from 70 to 400 micrometers. The system is fluidised under very different operating conditions, resulting in segregated and completely mixed configurations and the performances of QMOM and DQMOM are compared. Eventually, simulations are also compared with experimental results. Results clearly show that, in this particular case where no specific particulate processes occur (no particle aggregation nor breakage), because of numerical diffusion DQMOM leads to the wrong solution, and that for these cases it could be more convenient to solve the problem with QMOM, directly tracking the moments.

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