Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-18T14:56:16.352Z Has data issue: false hasContentIssue false

7 - The population balance equation

Published online by Cambridge University Press:  05 July 2015

Alison Lewis ,
Marcelo Seckler ,
Herman Kramer  and
Gerda van Rosmalen
Alison Lewis
Affiliation:
University of Cape Town
Marcelo Seckler
Affiliation:
Universidade de São Paulo
Herman Kramer
Affiliation:
Technische Universiteit Delft, The Netherlands
Gerda van Rosmalen
Affiliation:
Technische Universiteit Delft, The Netherlands
Get access

Summary

Why this chapter is important

The prediction of the product quality in industrial crystallization processes is not simple and requires detailed knowledge of the crystal size distribution (CSD). To obtain such knowledge, the population balance equation (PBE) has to be solved, describing the evolution of the CSD in the crystallizer.

In this chapter PBE-based models of the crystallization process will be discussed in detail, describing the formulation of the PBE based on either the length or the volume of the crystals, explaining the different terms of this important equation and their relation to the different kinetic processes in the crystallizer. Growth- and nucleationdominated systems and processes that are predominantly agglomeration and breakage will be discussed separately, as they require a different formulation. The coupling of the population to the enthalpy and material balances, to obtain a realistic description of the process is discussed, as well as solution methods for a PBE-based crystallizer model. Multi-compartment models are also introduced that can be used to describe the effects of profiles in the process conditions on the performance of an industrial crystallizer, due to the large volume or complex geometry. A few examples of such multi-compartment models are given.

Having a crystallization model is extremely valuable for the development, optimization and design of a process that meets specific product requirements. To be more precise, having a PBE-based crystallization model will allow the user

  1. • to predict the resulting particle size distribution even at a different scale of operation, on the basis of the known kinetic models;

  2. • to estimate the parameters of a kinetic model based on an experimentally determined particle size distribution knowing the process conditions under which it was obtained;

  3. • to optimize the process conditions to obtain a desired particle size distribution given a validated kinetic model.

Evolution of the crystal size distribution in a crystallizer

A crystalline product consists of a distribution of crystals (or of agglomerates of crystals that are often referred to as particles) of different sizes.

Type
Chapter
Information
Industrial Crystallization
Fundamentals and Applications
 , pp. 151 - 177
Publisher: Cambridge University Press
Print publication year: 2015

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bennett, R. C. 1993. Crystallizer selection and design. In Handbook of Industrial Crystallization, Myerson, A. S. (ed.), Butterworth–Heinemann.Google Scholar
Bermingham, S. K. 2003. A Design Procedure and Predictive Models for Solution Crystallization Processes, PhD thesis, Delft University of Technology.Google Scholar
Bermingham, S. K., Neumann, A. M., Muusze, J. P., Kramer, H. J. M. and Verheijen, P. J. T. 1998. Modelling the settling, dissolution and non–uniform nucleation kinetics in a 150–litre forced circulation crystallizer.Particle and Particle Systems Characterization, 15, 56–61.3.0.CO;2-U>CrossRefGoogle Scholar
Bermingham, S. K., Neumann, A. M., Kramer, H. J. M. et al. 2000. A design procedure and predictive models for solution crystallisation processes. AIChESymposium Series 323, 96, 250.Google Scholar
Bermingham, S. K., Verheijen, P. J. T. and Kramer, H. J. M. 2003. Optimal design of solution crystallization processes with rigorous models.Chemical Engineering Research and Design, 81, 893–903.CrossRefGoogle Scholar
Bird, R. B., Steward, W. E. and Lightfoot, E. N. 1960. Transport Phenomena, John Wiley & Son.
Daudey, P. J., van Rosmalen, G. M. and de Jong, E. J. 1990. Secondary nucleation kinetics of ammonium–sulfate in a CMSMPR crystallizer. Journal of Crystal Growth, 99, 1076–1081.CrossRefGoogle Scholar
Fakatselis, T. E. 2002. Residence time optimization in continuous crystallizers. Crystal Growth and Design, 2, 375–379.CrossRefGoogle Scholar
Kramer, H. J. M., Bermingham, S. K. and Van Rosmalen, G. M. 1999. Design of industrial crystallizers for a required product quality. Journal of Crystal Growth, 198/199, 729.CrossRefGoogle Scholar
Kumar, S. and Ramkrishna, D. 1996. On the solution of population balance equations by dis–cretization. 1. A fixed pivot technique. Chemical Engineering Science, 51, 1311–1332.Google Scholar
Kumar, S. and Ramkrishna, D. 1997. On the solution of population balance equations by discretization. 3. Nucleation, growth and agglomeration of particles. Chemical Engineering Science, 52(24), 4659–4679.CrossRefGoogle Scholar
Mesbah, A., Kramer, H. J. M., Huesman, A. E. M. and Van den Hof, P. M. J. 2009. A control oriented study on the numerical solution of the population balance equation for crystallization processes. Chemical Engineering Science, 64, 4262–4277.CrossRefGoogle Scholar
Mullin, J. W. 1993. Crystallization, Butterworth–Heinemann.Google Scholar
Mersmann, A. 1995. Crystallization Technology Handbook, Marcel Dekker, Inc.Google Scholar
Nývlt, J. 1992. Design of Crystallizers, CRC Press.Google Scholar
Kumar, S. and Ramkrishna, D. 1996. On the solution of population balance equations by dis–cretization. 1. A fixed pivot technique. Chemical Engineering Science, 51, 1311–1332.Google Scholar
O'Meadhra, R., Kramer, H. J. M. and Van Rosmalen, G. M. 1996. Model for secondary nucleation in a suspension crystallizer. AICHE Journal, 42, 973–982.CrossRefGoogle Scholar
Palosaari, S., Sha, Z. L. and Toyokura, K. 1996. Crystallization design chart and computer algo–rithm. Acta Polytechnica Scandinavica, Civil Engineering and Building Construction Series, 234, 1–49.Google Scholar
Qamar, S., Elsner, M. P., Angelov, I. A., Warnecke, G. and Seidel–Morgenstern, A. 2006. A comparative study of high resolution schemes for solving population balances in crystallization. Computers and Chemical Engineering, 30, 1119–1131.CrossRefGoogle Scholar
Qamar, S. and Warnecke, G. 2007. Numerical solution of population balance equations for nucleation, growth and aggregation processes. Computers and Chemical Engineering, 31, 1576–1589.CrossRefGoogle Scholar
Ramkrishna, D. 2000. Population Balances: Theory and Applications to Particulate Systems in Engineering, Academic Press.Google Scholar
Randolph, A. D. and Larson, M. A. 1988. Theory of Particulate Processes, 2nd edn., Academic Press.Google Scholar
Randolph, A. D. 1962. Size Distribution Dynamics in a Mixed Crystal Suspension, PhD Thesis, Iowa State University of Science and Technology.Google Scholar
Wulkow, M., Gerstlauer, A. and Nieken, U. 2001. Modeling and simulation of crystallization processes using parsival.Chemical Engineering Science, 56, 2575–2588.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×