Particle size distribution population balance model

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... 

Leblanc and Fogler developed a population balance model for the dissolution of polydisperse solids that included both reaction controlled and diffusion-controlled dissolution. This model allows for the handling of continuous particle size distributions. The following population balance was used to develop this model. 

The importance of chemical-reaction kinetics and the interaction of the latter with transport phenomena is the central theme of the contribution of Fox from Iowa State University. The chapter combines the clarity of a tutorial with the presentation of very recent results. Starting from simple chemistry and singlephase flow the reader is lead towards complex chemistry and two-phase flow. The issue of SGS modeling discussed already in Chapter 2 is now discussed with respect to the concentration fields. A detailed presentation of the joint Probability Density Function (PDF) method is given. The latter allows to account for the interaction between chemistry and physics. Results on impinging jet reactors are shown. When dealing with particulate systems a particle size distribution (PSD) and corresponding population balance equations are intro-... 

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. 

For different values of n ing(i ) = i2", other kinetic expressions can be developed. Figure 8.10 [18] shows the type of powder produced on spray diydng a solution that consists of metal salts of barium and iron in the ratio 1 12 (i.e., barium ferrite). Here we see the remains of the spherical droplets with a surface that consists of the metal salt precipitates, which form a narrow size distribution of platelet crystals (see Figure 8.10(a) and (b)). This narrow crystal size distribution is predicted by the population balance model if nudeation takes place over a short period of time. When these particles are spray roasted (in a plasma gun), the particles are highly sintered into spherical particles (see Figure 8.10(c)). 

Vilchis et al. [81] presented a new idea to achieve better control of the particle size distribution by the synthesis in situ of a water-soluble copolymer of acrylic acid-styrene as suspension stabilizer without additional inorganic phosphate. Publications describe increasing the particle formation by using a physical (population balance, Maxwell fluid, power law viscosity, compartment mixing) modeling approach [22,60,98,105]. 

An example of the use of the population balance method to predict reaction in particulate systems is presented in the work of Min and Ray (M16, M17). The authors developed a computational algorithm for a batch emulsion polymerization reactor. The model combines general balances, individual particle balances, and particle size distribution balances. The individual particle balances were formulated using the population balance... 

These models require information about mean velocity and the turbulence field within the stirred vessels. Computational flow models can be developed to provide such fluid dynamic information required by the reactor models. Although in principle, it is possible to solve the population balance model equations within the CFM framework, a simplified compartment-mixing model may be adequate to simulate an industrial reactor. In this approach, a CFD model is developed to establish the relationship between reactor hardware and the resulting fluid dynamics. This information is used by a relatively simple, compartment-mixing model coupled with a population balance model (Vivaldo-Lima et al., 1998). The approach is shown schematically in Fig. 9.2. Detailed polymerization kinetics can be included. Vivaldo-Lima et a/. (1998) have successfully used such an approach to predict particle size distribution (PSD) of the product polymer. Their two-compartment model was able to capture the bi-modal behavior observed in the experimental PSD data. After adequate validation, such a computational model can be used to optimize reactor configuration and operation to enhance reactor performance. 

Crowley, T.J. Meadows, E.S. Kostoulas, E. Doyle, F.J. Control of particle size distribution described by a population balance model of semibatch emulsion polymerization. J. Process. Control 2000, 10 (5), 419-132. 

The main contribution from the work of Luo [95, 96] was a closure model for binary breakage of fluid particles in fully developed turbulence flows based on isotropic turbulence - and probability theories. The author(s) also claimed that this model contains no adjustable parameters, a better phrase may be no additional adjustable parameters as both the isotropic turbulence - and the probability theories involved contain adjustable parameters and distribution functions. Hagesaether et al [49, 50, 51, 52] continued the population balance model development of Luo within the framework of an idealized plug flow model, whereas Bertola et al [13] combined the extended population balance module with a 2D algebraic slip mixture model for the flow pattern. Bertola et al [13] studied the effect of the bubble size distribution on the flow fields in bubble columns. An extended k-e model was used describing turbulence of the mixture flow. Two sets of simulations were performed, i.e., both with and without the population balance involved. Four different superficial gas velocities, i.e., 2,4,6 and 8 (cm/s) were used, and the superficial liquid velocity was set to 1 (cm/s) in all the cases. The population balance contained six prescribed bubble classes with diameters set to = 0.0038 (m), d = 0.0048 (m), di = 0.0060 (m), di = 0.0076 (m), di = 0.0095 (m) and di = 0.0120 (m). 

Another objective in the study of the application of CFD in crystallization is to simulate the particle size distribution in crystallization. In order to solve this problem, the simulation should take into account the population balance. The internal coordinates of the population balance make it difficult to utilize it in the CFD environment. In addition, different-sized particles have different hydrodynamics, which causes further complications. Wei and Garside [42] used the assumption of MSMPR and the moments of population balance to avoid the above difficulties in the simulation of precipitation. In the CFX commercial application, the MUSIC model offers a method for solving the population balance equation in CFD and defines the flow velocity of different-sized particles... 

Despite the commercial importance of PVC particle morphology to its end-use applications, there has been little work done on the development of quantitative models relating the size evolution of primary particles in terms of process conditions. Kiparissides [57] developed a population balance model to describe the time evolution of the primary particle size distribution as a function of the process variables, such as temperature and ionic strength of the medium. However, for the solution of the population balance model, the coalescence rate constant between the primary particles needs to be known. This, in turn, requires the calculation of electrostatic and steric stabilization forces acting on these particles. 

Kiparissides et al. [65] developed a comprehensive mathematical model for the quantitative prediction of the evolution of primary particle-size distribution during the free-radical polymerization of VCM. The population balance equation, describing the evolution of the primary particles in bulk or suspension polymerization, has the following general form ... 

The term /3(m, v) represents the coalescence rate constant of two colloidal particles of volume u and v. Note that the initial particle growth occurs mainly by particle aggregation and, to a smaller extent, by polymerization of the adsorbed monomer in the polymer-rich phase [58]. Thus, knowledge of analytical expressions for the coalescence rate constant is of profound importance to the solution of the population balance model (Equation 4.46), describing the time evolution of the primary particle size distribution. Such expressions have been derived by Kiparissides et al. [57, 59]. 

Kotoulas, C., Kiparissides, C., 2006. A generalized population balance model for the prediction of particle size distribution in suspension polymerization reactors. Chem. Eng. Sci. 61, 332-346. 

There are examples in the literature of fitting parameters to single particle models in both aggregation and breakage processes until an experimentally measured equilibrium particle size distribution is closely matched by the solution to the population balance equation. The rationality of such a procedure is much in question, as it is clearly not sensitive to the time scales of breakage and aggregation. 

Mahoney, A. W. Investigation of Population Balance Models towards Control of Particle Size Distribution, PhD. Thesis, Purdue University, West Lafayette, In, 2000. 

One manner of using the presented results is to incorporate them in the traditional way of tackling fluid bed granulation theoretically, namely population balance modeling. This can be achieved by expanding the population balance to more internal coordinates than just particle size (see Volume 1 of this series. Chapter 6, Section 6.9.1). The additional property in the case of the present example would be wet a lomerate composition, defined either by the mass fraction of solids within one particle or the binder/solid ratio. The latter can be further spht up to account for the spatial distribution - and, thus, accessibUity - of the hquid binder and for the thickness of the binder layer on the outer surface of the agglomerate. Alternatively, discrete models of agglomeration (see Section 7.7) could be expanded to account for non-spherical primary particles. 

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