Multidimensional population balances

PBMs developed thus far are one-dimensional in nature, that is, they can handle only one internal coordinate (e.g. aggregate volume). Besides volume, aggregate porosity and surface area also change during aggregation. To predict the evolution of all these properties, it is necessary to formulate multidimensional population balances. Moreover, current PBMs treat single component systems only. Colloidal suspensions encountered in natural systems and in industrial processes are usually multicomponent in nature and formulation of PBMs for such systems should be an important area of research. 

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Braumann, a., Kraft, M. Wagner, W. 2010 Numerical study of a stochastic particle algorithm solving a multidimensional population balance model for high shear granulation. Journal of Computational Physics 229, 7672-7691. 

Buffo, A., Marchisio, D. L. Vanni, M. 2012 Multidimensional population balance model for the simulation of turbulent gas-liquid systems in stirred tank reactors. Chemical Engineering Science 74, 31 14. 

Chakraborty, J. Kumar, S. 2007 A new framework for solution of multidimensional population balance equations. Chemical Engineering Science 62,4112 125. 

Gunawan, R., Fusman, 1. Braatz, R. 2004 High resolution algorithms for multidimensional population balance equations. AIChE Journal 50, 2738-2749. 

PuEL, F., Fevotte, G. Klein, J. P. 2003a Simulation and analysis of industrial crystallization processes through multidimensional population balance equations. Part 1 a resolution algorithm based on the method of classes. Chemical Engineering Science 58, 3715-3727. 

There is a growing interest in what is, somewhat misleadingly, called multidimensional population balance models. One example of a 2D PB model is the description of a granulation process where not only the particle size distribution with time, but also the fractional binder content is predicted by the model. The binder (liquid) content of the granules governs the agglomeration process. 

This differential equation is the fundamental population balance. This equation together with mass and energy balances for a system form a dynstmic multidimensional accounting of a process where there is a change in the particle size distribution. This equation is completely general and is used when the particles are distributed along both external and internal coordinate space. External coordinate space is simply the position x, y, and z in Cartesian coordinates. Internal coordinates Xj are, for example, the shape, chemical composition, and the size of the particles. More convenient and more restrictive forms of the population balance will be subsequently developed. 

The population balance approach is employed for the description of droplet dynamics in various flow fields. A significant advantage of the method is that a vehicle is provided to include the details of the breakage and coalescence processes in terms of the physical parameters and conditions of operation. A predictive multidimensional particle distribution theory is at hand which, in the case of well-defined droplet processes, can be employed for a priori prediction of the form and the magnitude of the particle size distribution. The physical parameters which affect the form... 

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