Equation 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 general population balance equation requires numerical methods for its solution and several have been proposed (e.g. Gelbard and Seinfeld, 1978 Hounslow, 1990a,b Hounslow etai, 1988, 1990), of which more later. Fortunately, however, some analytic solutions for simplified cases also exist. 

Nicmanis, N. and Hounslow, M.I., 1998. Finite-element methods for steady-state population balance equations. American Institution of Chemical Engineers Journal, 44(10), 2258-2272. 

Ca ], can be balanced using the general macroscopic population balance equation as follows... 

At the simplest level, the rate of flow-induced aggregation of compact spherical particles is described by Smoluchowski s theory [Eq. (32)]. Such expressions may then be incorporated into population balance equations to determine the evolution of the agglomerate size distribution with time. However with increase in agglomerate size, complex (fractal) structures may be generated that preclude analysis by simple methods as above. 

Hence, the population balance equation for a restricted-in-space agglomera tion process can be written as... 

Then, since in any system accumulation is the net result of both evolution and birth/death processes, and since any latex particle can be characterized by a set of physical quantities which will fully specify a given particle or class of particles, one can obtain the following population balance equation (33) ... 

Venneker et al. (2002) used as many as 20 bubble size classes in the bubble size range from 0.25 to some 20 mm. Just like GHOST , their in-house code named DA WN builds upon a liquid-only velocity field obtained with FLUENT, now with an anisotropic Reynolds Stress Model (RSM) for the turbulent momentum transport. To allow for the drastic increase in computational burden associated with using 20 population balance equations, the 3-D FLUENT flow field is averaged azimuthally into a 2-D flow field (Venneker, 1999, used a less elegant simplification )... 

The importance of chemical-reaction kinetics and the interaction of the latter with transport phenomena is the central theme of the contribution of Fox from Iowa State University. The chapter combines the clarity of a tutorial with the presentation of very recent results. Starting from simple chemistry and singlephase flow the reader is lead towards complex chemistry and two-phase flow. The issue of SGS modeling discussed already in Chapter 2 is now discussed with respect to the concentration fields. A detailed presentation of the joint Probability Density Function (PDF) method is given. The latter allows to account for the interaction between chemistry and physics. Results on impinging jet reactors are shown. When dealing with particulate systems a particle size distribution (PSD) and corresponding population balance equations are intro-... 

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 some areas, e.g. aerosol physics and crystallisation, population balance models are used in situations when a number balance equation is required as well as conventional mass and energy balances. Randolph and Larson review this theory as it applies specifically to particulate systems [15], whilst Froment and Bischoff [16] present population balance equations in the context of an extension of classical RTD theory. 

What is a birth-death (or population balance) equation ... 

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)),... 

The above simplified analysis was intended to provide a feel for the relative importance of the processes that govern carbon loading, and therefore carbon combustion efficiency. More complete treatments of AFBC s are available which consider the detailed population balance equations for the char particles coupled with an oxygen balance (41-50). These treatments have given results which parallel observations on operating AFBC s but... 

Let i /dadV be the number of particles in volume dV within the age interval a, a + da. The total number of particles in the reactor is N = ja v ij/dadV and obeys the well known population balance equation... 

Tavlarides presents a sophisticated model for representing coalescence and breakage of droplets in liquid-liquid dispersions. The model relies on the population balance equation and still requires the adjustment of 6 parameters. The solution of such equations is difficult and requires the use of Monte-Carlo methods... 

Use of Saturation. Because of the potential for simplification of the population balance equations, much recent work has concentrated on studying saturation phenomena. First proposed by Piepmeier (9), and elaborated on by Daily (10), saturation in atomic species can lead to complete elimination of the need to know any collisional rates, and in molecular species may provide substantial simplification of the balance equation analysis. 

N. M. Faqir, Numerical solution of the two dimensional population balance equation describing the hydrodynamics of interacting liquid-liquid dispersions. Chem. Eng. Sci., 2004, 59 (12), 2567-2592. 

If random scission occurs together with end-chain scission (but at a different rate) then defining x by dx/dt = k(i the population balance equations (PBEs) for these combined processes are... 

Madras, G. and McCoy, B. Numerical and similarity solutions for reversible population balance equations with size-dependent rates. Journal of Colloid and Interface Science 2002 246 356. 

The evolution of the size distribution of aerosols due to coagulation can be described by the population balance equation... 

In these population balance equations, the volume change term is zero and the internal velocity term is also zero and therefore are not included. When the population balance is performed on a number basis, it is necessary to have a conservation equation, which for grinding is usually the total particle volume. 

For a batch miU, the summation of the left-hand size of the above equation is zero. In this case, an analytical solution can be obtain to the population balance (equation 4.29) ... 

The solution to the population balance equation is shown in Figure... 

A number of numerical approaches have been applied to the solution of the general population balance equation ... 

Feltham s analysis [88], using the same population balance equation. 

Herbst et al. [International J. Mineral Processing, 22, 273-296 (1988)] describe the software modules in an optimum controller for a grinding circuit. The process model can be an empirical model as some authors have used. A phenomenological model can give more accurate predictions, and can be extrapolated, for example from pilot-to full-scale application, if scale-up rules are known. Normally the model is a variant of the population balance equations given in the previous section. 

---