Population balance aggregation

Kumar, J., Peglow, M., Warnecke, G. Heinrich, S. 2008a The cell average technique for solving multi-dimensional aggregation population balance equations. Computers and Chemical Engineering 32,1810-1830. 

The general form of the population balance including aggregation and rupture terms was solved numerically to model the experimental particle size distributions. While excellent agreement was obtained using semi-empirical two-particle aggregation and disruption models (see Figure 6.15), PSD predictions of theoretical models based on laminar and turbulent flow considerations... 

The significance of this novel attempt lies in the inclusion of both the additional particle co-ordinate and in a mechanism of particle disruption by primary particle attrition in the population balance. This formulation permits prediction of secondary particle characteristics, e.g. specific surface area expressed as surface area per unit volume or mass of crystal solid (i.e. m /m or m /kg). It can also account for the formation of bimodal particle size distributions, as are observed in many precipitation processes, for which special forms of size-dependent aggregation kernels have been proposed previously. 

Hounslow, M.J., Ryall, R.L. and Marshall, V.R., 1988. A discretized population balance for nucleation, growth and aggregation. American Institution of Chemical Engineers Journal, 34, 1821-1832. 

Wright, H. and Ramkrishna, D., 1992. Solutions of inverse problems in population balance aggregation kinetics. Computers and Chemical Engineering, 16(2), 1019-1030. 

At the simplest level, the rate of flow-induced aggregation of compact spherical particles is described by Smoluchowski s theory [Eq. (32)]. Such expressions may then be incorporated into population balance equations to determine the evolution of the agglomerate size distribution with time. However with increase in agglomerate size, complex (fractal) structures may be generated that preclude analysis by simple methods as above. 

In this section, the population balance will be used to model batch and CSTR precipitators where aggregation is a competing growth mechanism. Figure 6.30 is an example of the aggregate microstructure in... 

Again the assumption on the aggregation rate are that the most frequent collisions are between the larger particle and the small particles. This partial differential equation can be approximated by an ordinary one by creating a new characteristic time variable, t (= t -RIG = R G], which is constant. With this variable change the population balance becomes... 

This chapter discusses four methods of gas phase ceramic powder synthesis by flames, fiunaces, lasers, and plasmas. In each case, the reaction thermodynamics and kinetics are similar, but the reactor design is different. To account for the particle size distribution produced in a gas phase synthesis reactor, the population balance must account for nudeation, atomistic growth (also called vapor condensation) and particle—particle segregation. These gas phase reactors are real life examples of idealized plug flow reactors that are modeled by the dispersion model for plve flow. To obtain narrow size distribution ceramic powders by gas phase synthesis, dispersion must be minimized because it leads to a broadening of the particle size distribution. Finally the gas must be quickly quenched or cooled to freeze the ceramic particles, which are often liquid at the reaction temperature, and thus prevent further aggregation. 

When the particles are formed at high temperatures the particles are often liquid droplets. These droplets stick together when they collide, altering the particle size distribution produced. Accounting for aggregation in the population balance in gas phase reactors is performed in the following way ... 

This approach to coagulation can also be used to construct population balances to describe the growth of aggregates containing k spheres and having a radius of 0. The resulting conservation equation [3, p. 276] is... 

Randolph [95] and Randolph and Larson [96], on the other hand, formulated a generic population balance model based on the generalized continuum mechanical framework. Their main concern was solid particle crystallization, nucleation, growth, agglomeration/aggregation and breakage. 

The fundamental derivation of the population balance equation is considered general and not limited to describe gas-liquid dispersions. However, to employ the general population balance framework to model other particulate systems like solid particles and droplets appropriate kernels are required for the particle growth, agglomeration/aggregation/coalescence and breakage processes. Many droplet and solid particle closures are presented elsewhere (e.g., [96, 122, 25, 117, 75, 76, 46]). 

Vanni M (2000) Approximate population balance eqnation for aggregation-brakage processes. Journal of Colloid and Interface Science 221(2) 143-160... 

Hounslow MJ, Ryall RL, Marshall VR (1988) A Discretized Population Balance for Nucleation, Growth, and Aggregation. AIChE J 34 1821- ... 

Alexopoulos, a. H. Kiparissides, C. 2007 Solution of the bivariate dynamic population balance equation in batch particulate systems combined aggregation and breakage. Chemical Engineering Science 62, 5048-5053. 

Chiney, a., Shivendra, S. C. Kumar, S. 2010 Pivofwise expanding solution of bivariate population balance equations for composition dependent aggregative mixing, in 4th International Conference on Population Balance Modelling, Berlin. 

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