Particle population balance

Models for emulsion polymerization reactors vary greatly in their complexity. The level of sophistication needed depends upon the intended use of the model. One could distinguish between two levels of complexity. The first type of model simply involves reactor material and energy balances, and is used to predict the temperature, pressure and monomer concentrations in the reactor. Second level models cannot only predict the above quantities but also polymer properties such as particle size, molecular weight distribution (MWD) and branching frequency. In latex reactor systems, the level one balances are strongly coupled with the particle population balances, thereby making approximate level one models of limited value (1). 

The particle population balance can be expressed as follows (a zero-one condition for the radical distribution, at most one growing radical in a particle) ... 

Chen, Z., Pauer, W., Moritz, H.-U., Priiss, J., Wamecke, H. J., Modeling of the suspension polymerization process using a particle population balance, Chem. Eng. Technol. 22 (1999) 699. 

The early kinetic model by Smith and Ewart was based on Harkin s mechanistic understanding of the batch process. The particle population balances were written for a stationary state assuming that the rate of formation of particles with n radicals equals the rate of their disappearance (see equation at the bottom of this page). Where / , is the rate of radical entry into a particle (m /sec) is the rate constant for radical exit (m/sec) S is the particle surface area (m ) ktp is the rate constant for bimolecular termination in the particles (m /sec) and o is the particle volume. According to Smith and Ewart three limiting cases can be identified ... 

By selecting process parameters, such as nature of the precursor, temperature, time, reactant state, and/or reactor geometry, product quality can be influenced. However, measurements in gas-phase reactors are a problem as times are extremely short, temperatures very high, and atmospheres often aggressive. Therefore, numerous models for process simulation, all based on particle population balances [B.91,11.1], have been developed as useful tools to better understand particle formation and support product and process optimization. 

We will see that the techniques that have been developed to handle these questions utilize notions of distributions of properties of the reacting fluids in the vessel, in the sense of probability theory. These properties can be the residence times of elements of flowing fluid(s), catalyst activity of particles, crystal size in a crystallizer, and others. The first type is usually termed a residence time distribution (RTD) and the others are handled by particle population balances that are more general and can be used for properties other than just residence time. The more widely developed and utilized RTD methods will first be discussed, followed by the general population balances. 

Table V. Particle Population Balance ANOVA Summary...

We will see later that techniques developed to address these issues using concepts of fluid properties distribution in systems based on probabiUty theory. We will also see the concepts of RTD for the reactions and fluid flow, and other properties, such as distribution of solid particles treated by particle population balance. 

The assessment on the basis of ehanges in the particle size distribution considers the left-hand side of Eq. (5), i.e., the absolute change in mass Am,- is measured. But the individual contributions that led to these changes, i.e., the terms on the right-hand side of Eq. (5), are not considered by this type of assessment. It may be possible to take the contributions of feed and system loss by means of measurements into account, but it is not possible at all to unravel the internal transfer masses rrif j and between the size intervals just by observing the changes in the particle size distribution. For this reason, it is not possible to extrapolate the description to a different duration of the attrition process or even to transfer it to other initial particle size distributions. For this purpose a description via particle population balances is needed. 

Figure 27 The particle population balance model of a fluidized bed system. (Reppenhagen and Werther, 2001.)...

As there were no experimental data available to evaluate the entire above-derived particle population balance, a plausibility check with a fictitious industrial scale system has been made instead. The chosen data of the system are given in the list here ... 

For this system, the particle population balance is solved in discretized time steps At. At the beginning of each time step, except the very first one, the fresh material and the material from the return line are fed and mixed with the bed material. It follows the calculation of jet-induced and bubble-induced attrition. Afterwards the particular material mass fractions are determined that are entrained from the bed in the course of this time step. These material fractions are then subjected to the combination of attrition and gas-solid separation inside the cyclone section. Finally, the catch of the cyclone is added to the return line, which in the next time step feeds its excess mass to the bed. At the starting time t = 0, the particle size distributions in both the bed and the return line are identical to that of... 

Polymer Particle Population Balance (Particle Size Distribution)... 

We will now derive an equation of change for the particle number density function n(rp) by developing a particle population balance in a small control volume of dimensions Ax, Ay and Az (Figure 6.2.1) ... 

This macroscopic particle population balance equation is general and used most often. But the form in which the change with respect to the particle growth coordinate directions is utilized is simpler than V - (u p n rpj. Usually,... 

For a steady-state ciystallizer receiving sohds-free feed and containing a well-mixed suspension of ciystals experiencing neghgible breakage, a material-balance statement degenerates to a particle balance (the Randolph-Larson general-population balance) in turn, it simplifies to... 

The energy laws of Bond, Kick, and Rittinger relate to grinding from some average feed size to some product size but do not take into account the behavior of different sizes of particles in the mill. Computer simulation, based on population-balance models [Bass, Z. Angew. Math. Phys., 5(4), 283 (1954)], traces the breakage of each size of particle as a function of grinding time. Furthermore, the simu-... 

In this ehapter, the transport proeesses relating to partiele eonservation and flow are eonsidered. It starts with a brief introduetion to fluid-particle hydrodynamics that deseribes the motion of erystals suspended in liquors (Chapter 3) and also enables solid-liquid separation equipment to be sized (Chapter 4). This is followed by the momentum and population balances respeetively, whieh deseribe the eomplex flows and mixing within erystallizers and, together with partieulate erystal formation proeesses (Chapters 5 and 6), enable partiele size distributions from erystallizers to be analysed and predieted (Chapters 7 and 8). 

Particle conservation in a vessel is governed by the particle-number continuity equation, essentially a population balance to identify particle numbers in each and every size range and account for any changes due to particle formation, growth and destruction, termed particle birth and death processes reflecting formation and loss of particulate entities, respectively. 

The population balance accounts for the number of particles at each size in a continuous distribution and may be thought of as an extension of the more familiar overall mass balance to that of accounting for individual particles. 

General solution of the population balance is complex and normally requires numerical methods. Using the moment transformation of the population balance, however, it is possible to reduce the dimensionality of the population balance to that of the transport equations. It should also be noted, however, that although the mathematical effort to solve the population balance may therefore decrease considerably by use of a moment transformation, it always leads to a loss of information about the distribution of the variables with the particle size or any other internal co-ordinate. Full crystal size distribution (CSD) information can be recovered by numerical inversion of the leading moments (Pope, 1979 Randolph and Larson, 1988), but often just mean values suffice. 

The population balance in equation 2.86 employs the local instantaneous values of the velocity and concentration. In turbulent flow, there are fluctuations of the particle velocity as well as fluctuations of species and concentrations (Pope, 1979, 1985, 2000). Baldyga and Orciuch (1997, 2001) provide the appropriate generalization of the moment transformation equation 2.93 for the case of homogeneous and non-homogeneous turbulent particle flow by Reynolds averaging... 

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 model is able to predict the influence of mixing on particle properties and kinetic rates on different scales for a continuously operated reactor and a semibatch reactor with different types of impellers and under a wide range of operational conditions. From laboratory-scale experiments, the precipitation kinetics for nucleation, growth, agglomeration and disruption have to be determined (Zauner and Jones, 2000a). The fluid dynamic parameters, i.e. the local specific energy dissipation around the feed point, can be obtained either from CFD or from FDA measurements. In the compartmental SFM, the population balance is solved and the particle properties of the final product are predicted. As the model contains only physical and no phenomenological parameters, it can be used for scale-up. 

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. 

Employing two co-ordinates of overall particle size, L, and degree of agglomeration, S (which is, of course, proportional to the mean primary particle size) to define the population density, n S, L, t), the population balance during precipitation with agglomeration is described as ... 

The reaction engineering model links the penetration theory to a population balance that includes particle formation and growth with the aim of predicting the average particle size. The model was then applied to the precipitation of CaC03 via CO2 absorption into Ca(OH)2aq in a draft tube bubble column and draws insight into the phenomena underlying the crystal size evolution. 

At the crystallization stage, the rates of generation and growth of particles together with their residence times are all important for the formal accounting of particle numbers in each size range. Use of the mass and population balances facilitates calculation of the particle size distribution and its statistics i.e. mean particle size, etc. 

Gertlauer, A., Mitrovic, A., Motz, S. and Gilles, E.-D., 2001. A population balance model for crystallization processes using two independent particles properties. Chemical Engineering Science, 56(7), 2553-2565. 

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