Population balances

The mathematical description of a crystal size distribution and of its change in space and time makes use of the conservative character of the number of particles in space and state (i.e., particle size L). In the respective number balance, the particle size distribution is represented by the number density, see Hulburt and Katz (1964) and Randolph and Larson (1988), 

the quantity N is the amount of crystals per unit volume of crystal suspension and L is the crystal size. The number density n L) represents the number of crystals in a size interval AZ per volume of suspension  

the solution fed into a continuously operated crystallizer is free from crystals and only a single volume flow V is drained continuously which is containing a particle size distribution representing the distribution in the crystallizer. In this case, the population balance can be further simplified to 

Since the ratio V/ V of the volume flow V and the volume V is equal to the inverse of the average residence time t of the suspension the following expression is obtained for a cooling crystallizer  

This extremely simplified relation for the number density balance is only valid for the so-called MSMPR (mixed suspension mixed product removal) crystallizers. Integration with the integration constant as the nnmber density at grain size Z = 0 leads to 

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Correlations of nucleation rates with crystallizer variables have been developed for a variety of systems. Although the correlations are empirical, a mechanistic hypothesis regarding nucleation can be helpful in selecting operating variables for inclusion in the model. Two examples are (/) the effect of slurry circulation rate on nucleation has been used to develop a correlation for nucleation rate based on the tip speed of the impeller (16) and (2) the scaleup of nucleation kinetics for sodium chloride crystalliza tion provided an analysis of the role of mixing and mixer characteristics in contact nucleation (17). Pubhshed kinetic correlations have been reviewed through about 1979 (18). In a later section on population balances, simple power-law expressions are used to correlate nucleation rate data and describe the effect of nucleation on crystal size distribution. 

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

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. 

Consider the crystallizer shown in Figure 11. If it is assumed that the crystallizer is well mixed with a constant slurry volume FTp then, as shown (7), the following partial differential population balance can be derived ... 

If the crystallizer is now assumed to operate with a cleat feed (n = 0), at steady state (dn jdt = 0), and if the crystal growth rate G is invariant and a mean residence time T is defined as then the population balance can be written as... 

If the magma volume Gj, is allowed to vary in the system on which equation 42 is based, the population balance becomes... 

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 ... 

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... 

TABLE 18-5 Common Equations for Population-Balance Calculations... 

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-... 

Extent of Noninertial Growth Growth by coalescence in granulation processes may be modeled by the population balance. (See the... 

Coalescence Coalescence is the most difficult mechanism to model. It is easiest to write the population balance (Eq. 20-71) in terms of number distribution by volume n v) because granule volume is conserved in a coalescence event. The key parameter is the coalescence kernel or rate constant P(ti,i ). The kernel dictates the overall rate of coalescence, as well as the effect of granule size on coalescence... 

Analytical Solutions Solution of the population balance is not trivial. Analytical solutions are available for only a limited number of special cases, of which some of examples of practical importance are summarized in Table 20-59. For other analytical solutions, see general references on population balances given above. 

TABLE 20-59 Some Analytical Solutions to the Population Balance ... 

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). 

The term macromixing refers to the overall mixing performance in a reactor. It is usually described by the residence time distribution (RTD). Originally introduced by Danckwerts (1958), this concept is based on a macroscopic lumped population balance. A fluid element is followed from the time at which it enters the reactor (Lagrangian viewpoint - observer moves with the fluid). The probability that the fluid element will leave the reactor after a residence time t is expressed as the RTD function. This function characterises the scale of mixedness in a reactor. 

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 micromoment population balance is derived from the general population balanee (Lagrangian framework) and ean be written as... 

The macromoment population balance ean be obtained from the maerodis-tributed population balanee (2.94). Making the transformation... 

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 population balance is a partial integro-differential equation that is normally solved by numerical methods, except for special simplified cases. Numerical solution of the population balance for the general case is not, therefore, entirely straightforward. Ramkrishna (1985) provides a comprehensive review. 

Hounslow etal. (1988), Hounslow (1990a), Hostomsky and Jones (1991), Lister etal. (1995), Hill and Ng (1995) and Kumar and Ramkrishna (1996a,b) present numerical discretization schemes for solution of the population balance and compute correction factors in order to preserve total mass and number whilst Wojcik and Jones (1998a) evaluated various methods. 

This relationship, of course, only gives the total mass of solids formed. To reveal how that solid matter is distributed across a crystal population, the other conservation equation considered in Chapter 2 viz. the population balance must be invoked. Firstly, however, the crystal yield is considered a little further. 

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... 

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