Analytical treatment of fragmentation‐diffusion population balance

Kostoglou, Margaritis; Karabelas, A.J.

doi:10.1002/aic.10286

Cited by 7 publications

(9 citation statements)

References 32 publications

(34 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The case of constant velocity (u(r) ) u) with the diffusivity dependent only on particle size (D(x,r) ) D(x)) and spaceindependent breakage functions (b(x,r) ) b(x), p(x,y;r) ) p(x,y) has been examined in detail for the cylindrical geometry (among other geometries), using the separation of variables technique. 7 These authors have argued that the aforementioned form of the problem parameters (velocity, diffusivity, breakage rate, and kernel) is the most general one that allows application of either the separation of variables technique, or eigenfunction expansion, or finite integral transform, or linear operator expansion; in fact, all these procedures should lead to the same result. The final result of that work 7 is a series solution for f(x,r,z), where the r and z dependencies are expressed in terms of standard functions but the x dependence is given in terms of functions that must be constructed through recursive integral relations.…”

Section: On the Analytical Solution Of The Breakage Problem In Turbul...mentioning

confidence: 99%

On the Breakage of Liquid−Liquid Dispersions in Turbulent Pipe Flow: Spatial Patterns of Breakage Intensity

Kostoglou

Karabelas

2007

Ind. Eng. Chem. Res.

Self Cite

View full text Add to dashboard Cite

The breakage of droplets in turbulent pipe flow is a subject of great technological interest. The purpose of this work is to take advantage of state-of the-art theoretical developments and obtain quantitative predictions enhancing our physical understanding of this problem. From this perspective, the subject of the “exact” solutions to an appropriate mathematical model, being recently an issue in the literature, is clarified. Furthermore, the simplest possible mathematical model of the breakage process is derived that retains the complete process parameter dependencies. To implement the model, an appropriate algebraic approach is proposed for the prediction of the radial profiles of key turbulent flow field parameters. This mathematical model adequately relates the breakage pattern in the pipe with the operating process parameters. The results suggest that droplet breakage occurs almost exclusively in an annular region at the periphery of the pipe. Droplets larger than a certain critical size initially have a tendency to break quite rapidly in that region, whereas turbulent diffusion seems to be comparatively slow and cannot spread the fragments into the bulk. This interplay of breakage and diffusion mechanisms seems to lead to initially nonuniform lateral droplet number concentration profiles with rather strong axial (or equivalent time) dependence, over distances of practical significance for process plants. This work clearly suggests that the often-made assumption of uniform conditions in the pipe cross section, regarding size distribution of the droplets that are undergoing breakage, should be reconsidered.

show abstract

Section: On the Analytical Solution Of The Breakage Problem In Turbul...mentioning

confidence: 99%

On the Breakage of Liquid−Liquid Dispersions in Turbulent Pipe Flow: Spatial Patterns of Breakage Intensity

Kostoglou

Karabelas

2007

Ind. Eng. Chem. Res.

Self Cite

View full text Add to dashboard Cite

show abstract

“…A combination of the Lax‐Wendroff and Crank‐Nicholson methods was proposed by Bennett and Rohani22 having observed that these methods by themselves have not proved satisfactory. Kostoglou and Karabelas23, 24 carried out comparative studies of low‐order methods in computing growth and breakage processes, and presented an analytical treatment of breakage‐diffusion population balance 25. Efficient improvements of discretization methods were achieved by Kumar and Ramkrishna26, 27 developing the fixed and moving pivot techniques.…”

Section: Introductionmentioning

confidence: 99%

Dynamic simulation of crystallization processes: Adaptive finite element collocation method

Ulbert¹,

Lakatos²

2007

AIChE Journal

View full text Add to dashboard Cite

in Wiley InterScience (www.interscience.wiley.com).An adaptive orthogonal collocation on finite elements method with adaptively varied upper bound of the relevant size interval is developed for numerical solution of population balance equation of crystallizers. Nucleation producing monosized and heterosized nuclei, size-dependent crystal growth, seeding, classified product removal and fines removal with dissolution are included into the model. Adaptation of the number and length, as well as the distribution of finite elements over the variable length computational interval is carried out forming a number of adaptation rules, based on ordering the finite elements of size coordinate according to the maxima of first derivatives of the population density function. The approximation is obtained using the Lagrange interpolation polynomials. The method is used for solving the mixed set of nonlinear ordinary and partial differential equations, forming a detailed dynamical model of continuous crystallizers with product classification and/or fines removal. The program can be used efficiently for simulation of stationary and dynamic processes of crystallization systems, computing either transients or long-time oscillating steady states generated by different nonlinear phenomena, and internal and external feedbacks. AIChE J, 53: 3089-3107, 2007 Keywords: crystallization, population balance model, numerical solution, orthogonal collocation on finite elements, dynamic simulation American Institute of Chemical Engineers IntroductionThe population balance model is an adequate mathematical description of crystallization processes. This model consists of a mixed set of ordinary and partial integrodifferential equations even in the simplest case of MSMPR (mixed suspension, mixed product removal) crystallizers, and the crucial point of using this modeling approach is the numerical solution of the population balance equation.A number of numerical methods have been proposed for solving this equation, focusing on different modeling aspects of disperse systems, among which the weighted residual methods have received considerable attention. Subramanian and Ramkrishna, 1 considering microbial populations, were the first to use the method of weighted residuals for population balance equation, showing that the Laguerre functions were suitable trail functions on semi-infinite interval [0, 1). Singh and Ramkrishna 2 proposed problem-specific polynomials to obtain solutions to dynamic population balance problems. Lakatos et al. 3 applied the generalized Laguerre functions for simulation of batch crystallization processes, solving the population balance equation by means of orthogonal collocation, and making possible of modifying the distribution of collocation points along the size coordinate by varying the scale factor. Witkowski and Rawlings, 4 investigating identification and control issues of solution crystallizers, applied the Laguerre polynomials for computing the crystal-size distribution.Correspondence concerning this article shou...

show abstract

“…The case of constant velocity ( u ( r ) = u ) with the diffusivity being dependent only on particle size ( D ( x , r ) = D ( x )) and space-independent breakage functions ( b ( y ; r ) = b ( y ) and p ( x , y ; r ) = p ( x , y ) has been examined in detail for the cylindrical geometry (among other geometries), using the separation of variables technique, by Kostoglou and Karabelas . These authors argued that the aforementioned form of the problem parameters (velocity, diffusivity, breakage rate, and kernel) is the most general form that allows application of the separation of variables technique, or eigenfunction expansion, or finite integral transform, or linear operator expansion; all these procedures should lead to the same result.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Nere and Ramkrishna 4 considered a case more general than that of Kostoglou and Karabelas, concerning droplet breakage and diffusion in a turbulent pipe flow. The radial fluid velocity profile and the radial distribution of energy dissipation rate ε are obtained from a turbulent k −ε model.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, the breakage rate is r -dependent through its dependency on ε. In addition, the breakage functions considered are based on a particular physical system that has been studied previously. , The generalization, with respect to the work of Kostoglou and Karabelas, is that the velocity is r -dependent, the diffusivity is r -dependent (although the size dependency has been omitted), and the breakage rate is r -dependent (having a product-type dependence on its parameters, i.e., b ( x ; r ) = b 1 ( r ) b 2 ( x )). The resulting equation takes the form with a boundary condition of zero flux at the wall (no deposition).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the Evolution of Particle Size Distribution in Pipe Flow of Dispersions Undergoing Breakage

Kostoglou¹

2006

Ind. Eng. Chem. Res.

Self Cite

View full text Add to dashboard Cite

The problem of a flowing dispersion in a pipe with particles undergoing breakage and radial diffusion is considered in this note. This problem has been addressed in an earlier work [Kostoglou and Karabelas, AIChE J., 2004, 50, 1746] where it is solved analytically for the case of radially uniform parameters, while the case of turbulent flow (with radially dependent parameters) was treated by other researchers [Nere and Ramkrishna, Ind. Eng. Chem. Res. 2005, 44, 1187]. Here, simplified versions of the extended mathematical problem in the limits of small and large diffusion influence are presented. The conditions under which self-similar particle size distributions can be developed are determined. Finally, a closed-form solution for the total particle number concentration for linear breakage rate with radial dependence and radial uniform velocity and diffusivity is derived.

show abstract

Analytical treatment of fragmentation‐diffusion population balance

Abstract: The quest for advanced models to realistically simulate various processes involving populations of "particles" (aerosols, dispersions of all kinds, minerals, cultures of microorganisms)

Cited by 7 publications

References 32 publications

On the Breakage of Liquid−Liquid Dispersions in Turbulent Pipe Flow: Spatial Patterns of Breakage Intensity

On the Breakage of Liquid−Liquid Dispersions in Turbulent Pipe Flow: Spatial Patterns of Breakage Intensity

Dynamic simulation of crystallization processes: Adaptive finite element collocation method

On the Evolution of Particle Size Distribution in Pipe Flow of Dispersions Undergoing Breakage

Contact Info

Product

Resources

About