Precipitators population balance

Using the SFM, the influence of micromixing and mesomixing on the precipitation process and properties of the precipitate can be investigated. Mass and population balances can be applied to the individual compartments and to the overall reactor accounting for different levels of supersaturation in different zones of the reactor. 

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. 

Schreiner etal. (2001) modelled the precipitation process of CaC03 in the SFTR via direct solution of the coupled mass and population balances and CFD in order to predict flow regimes, induction times and powder quality. The fluid dynamic conditions in the mixer-segmenter were predicted using CFX 4.3 (Flarwell, UK). 

Marchal, P., David, R., Klein, J.P. and Villermaux, I., 1988. Crystallization and Precipitation engineering - I An efficient method for solving population balance in crystallization with agglomeration. Chemical Engineering Science, 43(1), 59. 

In precipitation, particle formation is extremely fast due to high supersaturations which in turn lead to fast nucleation. At least in the beginning, size distributions are narrow with particle sizes around one 1 nm. Nanomilling in stirred media mills is characterized by relatively slow particle formation kinetics, particle sizes ranging from several microns down to 10 nm and high sohds volume concentrations of up to 40%. Large particles may scavenge the fine fractions. The evolution of the particle size distribution can be described for both cases by population balance equations (Eq. (7)),... 

Here we see an exponential size distribution is predicted by the population balance. (B(Lq) is also the birth term due to nucleation of particles of size Lq. It could also be used in the population balance substituting for B directly, but this approach requires a LaPlace transform solution, which also results in equation (3.14).) Many inorganic precipitations operate in this way with small supersaturation that is, nearly all the mass is precipitated in one pass through the precipitation. This equation for a well-mixed constant volume aystallizer will be discussed in further detail in Chapter 6. 

The material in this section draws heavily from an excellent book by Neilsen [2]. During precipitation new particles are bom into the size distribution by nudeation processes. The nudeation rate, which appears as a boundary condition at size L = L 0 in the population balance, generally has a dominating influence on the particle size distribution. Nudeation is also the least understood of the various rate processes in predpitation. There are three main categories of nudeation ... 

SIZE DISTRIBUTION EFFECTS—POPULATION BALANCE AND PRECIPITATOR DESIGN... 

Coupled with a mass balance, the population balance accounts for all of the particles of each size that are generated in a precipitator. The population balance was first formulated by Randolph [96] and Hulbert and Katz [97]. A general review is provided by Randolph and Larson [98]. The population balance, when performed on a macroscopic scale like that of the whole precipitator, is given by... 

The simplest continuous reactor to consider is that of a constantly stirred tank reactor (CSTR) or precipitator, also called a mixed suspension, mixed product removal crystallizer (MSMPR) [98], shown in Figure 6.23. This tsrpe of precipitator has a constant volume, V, with an input flow rate equal to its output flow rate, Q. The population iJofR) in the precipitator is that which leaves as product. In this case, the population balance is used at steady state (i.e., drjfjdt — 0) ... 

The width of the size distribution is often measured in terms of the coefficient of variation (c.v.) of the mass distribution. Randolph and Larson [98] have shown that the coefficient of variation d the mass distribution is constant at 50% for this type of precipitator. This coefficient of variation is usually too large for ceramic powders. Attempts to narrow the size distribution of particles generated in a CSTR can be made by classified product removal, as shown in Figure 6.24. The classification function, p(R), is similar to those discussed in Section 4.2 and can be easily added to the population balance as follows ... 

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... 

Particle growth in a batdi environment is more difficult to predict because the steady state assumption previously used for the CSTR case is no loiter applicable. For a batch precipitator, the simplified population balance becomes... 

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)). 

Marchisio, D. L. 2009 On the use of bi-variate population balance equations for modelling barium titanate nanoparticle precipitation. Chemical Engineering Science 64, 697-708. 

Schwarzer H, Schwertfirm F, Manhart M, Schmid H, Peukert W (2006) Predictive simulation of nanoparticle precipitation based on the population balance equation. Chem Eng Sci 61(1) 167-181... 

In many systems, particularly precipitating systems such as proteins, agglomeration is an important factor and cannot be overlooked. Unlike the simple case of crystal growth, it is necessary to write the population balance in volume coordinates (Hartel and Randolph 1986). 

Continuous MSMPR Precipitator. The population balance, which was put forward by Randolph and Larson (1962) and Hulbert and Katz (1964), provides the basis for modeling the crystal size distribution (CSD) in precipitation processes. For a continuous mixed-suspension, mixed-product-removal (CMSMPR) precipitator with no suspended solids in the feed streams, the population balance equation (PBE) can be written as (Randolph and Larson 1988)... 

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