Continuous population balance

Although evidence exists for both mechanisms of growth rate dispersion, separate mathematical models were developed for incorporating the two mechanisms into descriptions of crystal populations random growth rate fluctuations (36) and growth rate distributions (33,40). Both mechanisms can be included in a population balance to show the relative effects of the two mechanisms on crystal size distributions from batch and continuous crystallizers (41). 

Population balances and crystallization kinetics may be used to relate process variables to the crystal size distribution produced by the crystallizer. Such balances are coupled to the more familiar balances on mass and energy. It is assumed that the population distribution is a continuous function and that crystal size, surface area, and volume can be described by a characteristic dimension T. Area and volume shape factors are assumed to be constant, which is to say that the morphology of the crystal does not change with size. 

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. 

The CSD from the continuous MSMPR may thus be predicted by a combination of crystallization kinetics and crystallizer residence time (see Figure 3.5). This fact has been widely used in reverse as a means to determine crystallization kinetics - by analysis of the CSD from a well-mixed vessel of known mean residence time. Whether used for performance prediction or kinetics determination, these three quantities, (CSD, kinetics and residence time), are linked by the population balance. 

Given expressions for the crystallization kinetics and solubility of the system, the population balance (equation 2.4) can, in principle, be solved to predict the performance of both batch and of continuous crystallizers, at either steady- or unsteady-state... 

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. 

Hounslow, M.J., 1990a. A discretized population balance for continuous systems at steady state. American Institution of Chemical Engineers Journal, 36, 106-116. 

In conventional radical polymerization, the chain length distribution of propagating species is broad and new short chains are formed continually by initiation. As has been stated above, the population balance means that, termination, most frequently, involves the reaction of a shorter, more mobile, chain with a longer, less mobile, chain. In living radical polymerizations, the chain lengths of most propagating species are similar (i.e. i j) and increase with conversion. Ideally, in ATRP and NMP no new chains are fonned. In practice,... 

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 author s work in the area of CFD analysis of chemical reactors has been supported nearly continuously for the last 15 years by the U.S. National Science Foundation. The work on gas-solid multiphase flows and population balances was funded by the U.S. Department of Energy. The author would also like to acknowledge support from several companies, including Air Products and Chemicals, BASF, BASELL, BP Chemicals, Dow Chemical, DuPont Engineering, and Univation Technologies. Last, but not least, the author wishes to acknowledge his many collaborators over the years who are many in number to name them individually. 

As discussed in Chapter 15, the size distribution of particles in an agglomeration process is essentially determined by a population balance that depends on the kinetics of the various processes taking place simultaneously, some of which result in particle growth and some in particle degradation. In a batch process, an equilibrium condition will eventually be established with the net rates of formation and destruction of particles of each size reaching an equilibrium condition. In a continuous process, there is the additional complication that the residence time distribution of particles of each size has an important influence. 

Crystal nucleation and growth in a crystalliser cannot be considered in isolation because they interact with one another and with other system parameters in a complex manner. For a complete description of the crystal size distribution of the product in a continuously operated crystalliser, both the nucleation and the growth processes must be quantified, and the laws of conservation of mass, energy, and crystal population must be applied. The importance of population balance, in which all particles are accounted for, was first stressed in the pioneering work of Randolph and Larson1371. ... 

Growth and nucleation interact in a crystalliser in which both contribute to the final crystal size distribution (CSD) of the product. The importance of the population balance(37) is widely acknowledged. This is most easily appreciated by reference to the simple, idealised case of a mixed-suspension, mixed-product removal (MSMPR) crystalliser operated continuously in the steady state, where no crystals are present in the feed stream, all crystals are of the same shape, no crystals break down by attrition, and crystal growth rate is independent of crystal size. The crystal size distribution for steady state operation in terms of crystal size d and population density // (number of crystals per unit size per unit volume of the system), derived directly from the population balance over the system(37) is ... 

In formulating a population balance, crystals are assumed sufficiently numerous for the population distribution to be treated as a continuous function. One of the key assumptions in the development of a simple population balance is that all crystal properties, including mass (or volume), surface area, and so forth are defined in terms of a single crystal dimension referred to as the characteristic length. For example, Eq. (19) relates the surface area and volume of a single crystal to a characteristic length L. In the simple treatment provided here, shape factors are taken to be constants. These can be determined by simple measurements or estimated if the crystal shape is simple and known—for example, for a cube area = 6 and kY0 = 1. 

A population balance can be used to follow the development of a crystal size distribution in batch crystallizer, but both the mathematics and physical phenomena being modeled are more complex than for continuous systems at steady state. The balance often utilizes the population density defined in terms of the total crystallizer volume, rather than on a specific basis n = nVj. Accordingly, the general population balance given by Eq. (51) can be modified for a batch crystallizer to give ... 

The discussion presented here has focused on the principles associated with formulating a population balance and applying simplifying conditions associated with specific crystallizer configurations. The continuous and batch systems used as examples were idealized so that the principles... 

Simplify the macroscopic population balance to describe the particle size distribution in a continuous constant volume isothermal well-mixed crystallizer with mixed product removal operating at steady state. Assume the crystallizer feed streams are free of suspended particles, that the crystallizer operates with ne igible breakage, and that agglomeration and crystallization cause no change in the volume of the system. 

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 birth and death functions now have the same units as the population balance. To attempt a solution, an integral or continuous approach will be used in place of this discrete summation. This suggests that there is a continuous distribution of particle sizes (i.e., the sizes of interest for the population balance are much larger than that of singlets, doublets, etc.). Some key substitutions for this integration are necessary ... 

Gas phase ceramic synthesis is the subject of several review papers. The treatment here is analogous to that in Magan [1], Friedlander [2], and Pratsinis and Kodas [3] but instead of using the traditional aerosol nomenclature, this chapter uses the nomenclature developed in Chapter 3 on population balances for educational continuity. Each of the gas phase powder synthesis methods is summarized in Table 7.1. The maximum temperatures are also listed. The adiabatic flame temperature is the maximum possible temperature achieved in flame synthesis and will depend on the concentration of reactants in the feed. Powder synthesis in a furnace uses conduction, convection, and radiation, giving a maximum temperature of 2300 K. A plasma is an ionized gas. High velocity electrons remove other electrons from the neutral gas molecules present in the plasma, thereby producing ions and electrons that sustain the plasma. 

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