Population balance techniques

The past 30 years have seen great advances in our understanding of the fundamentals of crystallization and has resulted in improved crystallizer design and operation. A dominant theme during this period was the analysis and prediction of crystal size distributions in realistic industrial crystallizers. This led to the development and refinement of the population balance technique which has become a routine tool of the crystallization community. This area is best described in the book of Randolph and Larson (1) which has been an indispensable reference and guide through two editions. 

The development and refinement of population balance techniques for the description of the behavior of laboratory and industrial crystallizers led to the belief that with accurate values for the crystal growth and nucleation kinetics, a simple MSMPR type crystallizer could be accurately modelled in terms of its CSD. Unfortunately, accurate measurement of the CSD with laser light scattering particle size analyzers (especially of the small particles) has revealed that this is not true. In mar cases the CSD data obtained from steady state operation of a MSMPR crystallizer is not a straight line as expected but curves upward (1. 32. 33V This indicates more small particles than predicted... 

D. Type 1 Interaction Models Population Balance Techniques. 238... 

The actual value of mean activity can be calculated usmg population balance technique, (see Joshi P.A., PhD thesis, Dept Chemical Hn neering, HT, Bombay, 19SS)... 

In this section we have described two methods to determine the kinetics governing the nucleation process. The first method, which utilizes the width of the metastable zone, is easy to use and gives an apparent order of nucleation. The second method uses the induction time to predict the mechanism and order of nucleation processes. A third method, which employs population balance techniques and an MSMPR crystallizer, will be described in Chapter 4 of this volume. 

The population balance technique was also successfully applied to the run-seed, triple-jet precipitation of silver halide crystals (Wey 1990). In addition to silver nitrate and halide salt solutions, a third solution containing stable silver halide seed crystals was simultaneously introduced into the precipitation vessel. For a seed-crystal solution that contains monodisperse seed... 

A final topic concerns two phase liquid-liquid reactors, where droplet breaking and coalescence is of great importance. This complicated area cannot be covered here for a recent useful reference, see Coulaloglou and Tavlarides [86]. The methods are based on the use of population balance techniques, to be discussed in Sec. 12.6. 

Dispersed phase interaction models (1.Population balance techniques,2.Monte Carlo simulation models and 3.Models using macromixing and micromixing concepts)... 

Kumar, S. and Ramkrishna, D., 1996a. On the solution of population balance by discretization I. A fixed pivot technique. Chemical Engineering Science, 51, 1311-1332. 

Larson, M.A. and Garside, J., 1973. Crystallizer design techniques using the population balance. Chemical Engineer, London, June, p. 318. 

When solid particles are subject to noncatalytic reactions, the effects of the reaction on individual particles are derived and then the results are averaged to determine overall properties. The general techniques for this averaging are called population balance methods. They are important in mass transfer operations such as crystallization, drop coagulation, and drop breakup. Chapter 15 uses these methods to analyze the distribution of residence times in flow systems. The following example shows how the methods can be applied to a collection of solid particles undergoing a consumptive surface reaction. 

Analytical solutions of the self-preserving distribution do exist for some coalescence kernels, and such behavior is sometimes seen in practice (see Fig. 40). For most practical applications, numerical solutions to the population balance are necessary. Several numerical solution techniques have been proposed. It is usual to break the size range into discrete intervals and then solve the series of ordinary differential equations that result. A geometric discretization reduces the number of size intervals (and equations) that are required. Litster, Smit and Hounslow (1995) give a general discretized population balance for nucleation, growth and coalescence. Figure 41 illustrates the evolution of the size distribution for coalescence alone, based on the kernel of Ennis Adetayo (1994). 

Tavare and Garside ( ) developed a method to employ the time evolution of the CSD in a seeded isothermal batch crystallizer to estimate both growth and nucleation kinetics. In this method, a distinction is made between the seed (S) crystals and those which have nucleated (N crystals). The moment transformation of the population balance model is used to represent the N crystals. A supersaturation balance is written in terms of both the N and S crystals. Experimental size distribution data is used along with a parameter estimation technique to obtain the kinetic constants. The parameter estimation involves a Laplace transform of the experimentally determined size distribution data followed a linear least square analysis. Depending on the form of the nucleation equation employed four, six or eight parameters will be estimated. A nonlinear method of parameter estimation employing desupersaturation curve data has been developed by Witkowki et al (S5). 

The process inputs are defined as the heat input, the product flow rate and the fines flow rate. The steady state operating point is Pj =120 kW, Q =.215 1/s and Q =.8 1/s. The process outputs are defined as the thlrd moment m (t), the (mass based) mean crystal size L Q(tK relative volume of crystals vr (t) in the size range (r.-lO m. In determining the responses of the nonlinear model the method of lines is chosen to transform the partial differential equation in a set of (nonlinear) ordinary differential equations. The time responses are then obtained by using a standard numerical integration technique for sets of coupled ordinary differential equations. It was found that discretization of the population balance with 1001 grid points in the size range 0. to 5 10 m results in very accurate solutions of the crystallizer model. 

Attarakih M, Bart HJ, Faqir NM. An approximate optimal moving-grid technique for the solution of discretized population balances in batch systems. Proceedings of ESCAPE 12 European Symposium on Computer-Aided Process Engineering, The Hague, 2002. 

The SAXS/TGA approach has been demonstrated to be a useful technique for time-resolution of porosity development in carbons during activation processes. Qualitative interpretation of the data obtained thus far suggests that a population balance approach focusing on the rates of production and consumption of pores as a function of size may be a fruitful approach to the development of quantitative models of activation proces.ses. These then could become useful tools for the optimization of pore size distributions for particular applications by providing descriptions and predictions of how various activating agents and time-temperature histories affect resultant pore size distributions. 

Coalescence and redispersion models applied to these reaction systems include population balance equations, Monte Carlo simulation techniques, and a combination of macromixing and micromixing concepts with Monte Carlo simulations. Most of the last two types of models were developed to... 

Numerical Solutions For many practical applications, numerical solutions to the population balance are necessary. Several numerical solution techniques have been proposed. It is usual to break the... 

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. 

---