Population balance formulation

In the work of Fleischer et al [23] and Hages ther et al [28] one dimensional population balance formulations and closure laws very similar to those of Luo and Svendsen [74] were employed. In the bubbly flow simulations by Hagesffither et al [29, 30] and Wang et al [119, 120] the macroscopic population... 

In this section the macroscopic population balance formulation of Prince and Blanch [92], Luo [73] and Luo and Svendsen [74] is outlined. In the work of Luo [73] no growth terms were considered, the balance equation thus contains a transient term, a convection term and four source terms due to binary bubble coalescence and breakage. 

The Microscopic Continuum Mechanical Population Balance Formulation... 

For flows where compressibility effects in a gas are important the use of the particle mass as internal coordinate may be advantageous because this quantity is conserved under pressure changes [11]. In this approach it is assumed that all the relevant internal variables can be derived from the particle mass, so the particle number distribution is described by the particle mass, position and time. Under these conditions, the dispersed phase flow fields are characterized by a single distribution function /(m, r,t) such that f m,r,t)drdm is the number of particles with mass between m and m+dm, at time t and within dr of position r. Notice that the use of particle diameter and particle mass as inner coordinates give rise to equivalent population balance formulations in the case of describing incompressible fluids. 

For bubbly flows most of the early papers either adopted a macroscopic population balance approach with an inherent discrete discretization scheme as described earlier, or rather semi-empirical transport equations for the contact area and/or the particle diameter. Actually, very few consistent source term closures exist for the microscopic population balance formulation. The existing models are usually solved using discrete semi-integral techniques, as will be outlined in the next sub-section. 

Particulate processes are characterized by properties like the paxticle shape, size, surface area, mass, and product purity. In crystallization the particle size and total number of crystals vary with time. Thus, determining particle size distribution (PSD) is important in crystallization. A population balance formulation describes the process of crystal size distribution with time most effectively. Thus, modeling of a batch crystallizer involves use of population balances to model the crystal size... 

The formulation of a population balance requires defining growth rate as the rate of change of the characteristic dimension... 

Mote quantitative relationships of the CSD obtained from batch operations can be developed through formulation of a population balance. Using a population density defined in terms of the total crystallizer volume rather than on a specific basis (n = nU), the general population balance given by equation 42 can be modified in recognition of there being no feed or product streams ... 

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. 

Note that the RANS formulation used in (B.44) and (B.45) can easily be extended to the LES, as outlined in Section 5.10. Moreover, by following the same steps as outlined above, DQMOM can be used with the joint velocity, composition PDF transport equation. Finally, the reader can observe that the same methodology is applicable to more general distribution functions than probability density functions. Indeed, DQMOM can be applied to general population balance equations such as those used to describe multi-phase flows. 

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. 

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... 

Therefore, the conservation equation plays an important role in the population balance by placing limits on the population. In general, these conservation equations are sometimes coupled into the definitions of the birth and death functions, B and D, in the population balance, thus assuring both the conservation of a property and the balance of the population simultaneously, without the necessity of two separate differential equations—population balance and property conservation—to be solved simultaneously. Such birth and death formulations will be discussed in Chapter 4 for grinding. 

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... 

Because of the large number of mechanistic processes opo-ative in emulsion polymerizations, complete theories for the PSD are necessarily complex. Nevertheless, they can be formulated by a population balance approxch. Much remains to he done, however, to clarify the basic colloid science that underpins the nucleation process in Interval I. The experimental challenge in evaluating the predictions of the theory for PSDs resides not only in the attainment of agreement with experiment but also in showing that such agreement is not merely fortuitous but arises from the correct mechanistic scheme. Considerable experimental work will be required to establish the validity of mechanistic assumptions for any particular monomer. 

Delichatsios and Probstein (D4-7) have analyzed the processes of drop breakup and coagulation/coalescence in isotropic turbulent dispersions. Models were developed for breakup and coalescence rates based on turbulence theory as discussed in Section III and were formulated in terms of Eq. (107). They applied these results in an attempt to show that the increase of drop sizes with holdup fraction in agitated dispersions cannot be attributed entirely to turbulence dampening caused by the dispersed phase. These conclusions are determined after an approximate analysis of the population balance equation, assuming the size distribution is approximately Gaussian. 

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... 

The work discussed in this section clearly delineates the role of droplet size distribution and coalescence and breakage phenomena in mass transfer with reaction. The population balance equations are shown to be applicable to these problems. However, as the models attempt to be more inclusive, meaningful solutions through these formulations become more elusive. For example, no work exists employing the population balance equations which accounts for the simultaneous affects of coalescence and breakage and size distribution on solute depletion in the dispersed phase when mass transfer accompanied by second-order reaction occurs in a continuous-flow vessel. Nevertheless, the population balance equation approach provides a rational framework to permit analysis of the importance of these individual phenomena. 

Besides, to understand the basic principles of kinetic theory, granular flows and population balances we need to widen our knowledge of classical mechanics. Newton s mathematical formulation of the laws of motion is perhaps the most intuitive point of view considering familiar quantities like mass, force, acceleration, velocity and positions and as such preferred by chemical engineers. However, this framework is inconvenient for mathematical generalizations as required describing the motion of large populations of particles for which it is necessary to take into account the constraints that limit the motion of the... 

The main challenge in formulating these equations is related to the definition of the collision operator. So far this approach has been restricted to the formulation of the population balance equation. That is, in most cases a general transport equation which is complemented with postulated source term formulations for the particle behavior is used. Randolph [80] and Randolph and Larson [81] used this approach deriving a microscopic population balance equation for the purpose of describing the behavior of particulate systems. Ramkrishna [79] provides further details on this approach considering also fluid particle systems. 

In a series of papers Lathouwers and Bellan [43, 44, 45, 46] presented a kinetic theory model for multicomponent reactive granular flows. The model considers polydisersed particle suspensions to take into account that the physical properties (e.g., diameter, density) and thermo-chemistry (reactive versus inert) of the particles may differ in their case. Separate transport equations are constructed for each of the particle types, based on similar principles as used formulating the population balance equations [61]. 

Randolph [95] and Randolph and Larson [96], on the other hand, formulated a generic population balance model based on the generalized continuum mechanical framework. Their main concern was solid particle crystallization, nucleation, growth, agglomeration/aggregation and breakage. 

Lee et al [66] and Prince and Blanch [92] adopted the basic ideas of Coulaloglou and Tavlarides [16] formulating the population balance source terms directly on the averaging scales performing analysis of bubble breakage and coalescence in turbulent gas-liquid dispersions. The source term closures were completely integrated parts of the discrete numerical scheme adopted. The number densities of the bubbles were thus defined as the number of bubbles per unit mixture volume and not as a probability density in accordance with the kinetic theory of gases. 

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