Formulation of Population Balance Models

Either of equations (2.10.6) and (2.10.7) must be considered simultaneously with the continuous phase equation (2.9.1). 

12 The notation of the double dot inner product used in Eq. (2.10.6) is consistent with that generally used in transport phenomena. For example, 

Consider a well-stirred vessel initially containing a given mass (MJ of a solid present as a population of polydispersed particles in a liquid in which it is soluble. Assume that mass transfer controls the dissolution of each particle and that the heat of dissolution is negligible. The particles may all be assumed to be spherical and distributed according to their mass x. 

The rate of change of mass of a particle of mass x by dissolution, X(x, Y can be described by 

The population balance equation in the number density function / (x, t) is given by (2.7.6) with the right-hand side set equal to zero. Thus, 

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Finally, Chapter 7 also presents some formulations of population balance models applicable to biological systems in which correlated or anticorrelated behavior between siblings and between parent and offspring can be accommodated. Examples of applications pervade throughout the different chapters in the book introduced primarily as an aid to understanding the different aspects of population balance modeling. 

Chapter 7 is concerned with the statistical foundation of population balance models. The chapter deals with master density formulations leading to mean field equations for the average behavior of the system and fluctuations about average behavior. This represents the subject of stochastic population dynamics applicable to small systems the relevance of which to engineering is discussed. Departures of the mean field equations from population balance equations are demonstrated. The mean field equations so obtained suffer from lack of closure. Closure approximations are presented suggesting more complex mean field equations than population balance along with applications. 

The attribute of the foregoing model is its remarkable simplicity and ability to assess the effect of drop mixing on conversion. Of course a drop population can have a broad spectrum of sizes. There have been attempts to improve this feature by incorporating a size distribution as measured experimentally and to view the coalescence of a pair of unequally sized droplets to result in the same pair of droplets except for the mixing of their contents It cannot be said that this viewpoint is an improvement, for the assumption of such memory in redispersion is less realistic than that of uniform size. However, it is of interest to see whether a uniformly distributed redispersion event can predict a size distribution that is anything like what is observed. We discuss this as another example in the formulation of population balances. 

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]. 

Luo [73] and Luo and Svendsen [74] extended the work of Coulaloglou and Tavlarides [16], Lee et al [66] and Prince and Blanch [92] formulating the population balance directly on the macroscopic scales where the closure laws for the source terms were integrated parts of the discrete numerical scheme used solving the model equations. 

The multi-fluid model framework is required to simulate chemical processes containing dispersed phases of multiple sizes. Two different designs of the multi-fluid model have emerged over the years representing very different levels of complexity. For dilute flows the dispersed phases are assumed not to interact, so no population balance model is needed. For denser flows a population balance equation is included to describe the effects of the dispersed phases interaction processes. Further details on the multi-fluid model formulations are given in chap 8 and chap 9. 

If a flow system is visualized as consisting of a large number of fluid elements that collide, coalesce, and then reform into two new elements a population balance model similar to that for immiscible droplet interaction can be formulated. The latter was devised by Curl [118] also see Rietema [119] for a review of this complex area. 

The formation of gas bubbles bypasses reactant away from contact with the catalyst phase, although this effect is alleviated to some extent by exchange of reactant via mass transfer as well as bulk flow across the bubble surface. The bubbles coalesce to large sizes in their ascent through the bed. Our interest is in calculating the conversion in the reactor, which is assumed to occur under isothermal conditions. The following considerations are extremely important in the formulation of the population balance model. 

In the above equation, the reaction term features the bed height at minimum fluidization, which gives the volume of the dense phase per unit cross-section. The mutually coupled Eqs. (3.3.25) and (3.3.26) represent the mathematical formulation of the population balance model. Since the reaction rate is linear it is possible in this case to solve explicitly for q to obtain... 

Tab. 9.5. Full 3D set of population balance equations for radical polymerization of vinyl acetate in living and dead chain concentration variables Rn,i,k and P j k- (for indices, see Table 9.4) summation over the branching index k, yielding a 2D formulation of exactly the same form, provides the basis for the TDB classes model.

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