A Gaussian quadrature method for solving batch crystallization models

AIChE Journal

Volumn 57, Issue 1, 2011, Pages 149-159

  Qamar, Shamsul ^a,b   Mukhtar, Safyan ^b   Ali, Qasim ^b   Seidel Morgenstern, Andreas ^a  

^a MAX PLANCK INSTITUTE FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS   (Germany)

^b COMSATS Institute of Information Technology ^*  (Pakistan)

Author keywords

Crystal growth; Crystallization; Nucleation; Numerical solutions; Particulate flows

Indexed keywords

ALTERNATIVE APPROACH; ANALYTICAL SOLUTIONS; BATCH CRYSTALLIZATION; CRITICAL SIZE; FINITE VOLUME SCHEMES; GAUSSIAN QUADRATURES; LOWER ORDER MOMENTS; MOMENT CALCULATION; NUMERICAL RESULTS; NUMERICAL SOLUTION; ORTHOGONAL POLYNOMIAL; PARTICULATE FLOWS; POPULATION BALANCE MODELS; QUADRATURE METHOD OF MOMENTS; TEST CASE; TEST PROBLEM;

AGGLOMERATION; CRYSTAL GROWTH; DISSOLUTION; METHOD OF MOMENTS; NUCLEATION;

CRYSTALLIZATION;

EID: 78650069639     PISSN: 00011541     EISSN: 15475905     Source Type: Journal    
DOI: 10.1002/aic.12264     Document Type: Article

References (37)

1. Hulburt HM, Katz S. Some problems in particle technology. Chem Eng Sci. 1964; 19: 555-574.
2. Randolph A, Larson MA. Theory of Particulate Processes, 2nd ed. San Diego, CA, Academic Press, Inc., 1988.
3. Barrett JC, Jheeta JS. Improving the accuracy of the moments method for solving the aerosol general dynamic equation. J Aerosol Sci. 1996; 27: 1135-1142.
4. Madras G, McCoy BJ. Reversible crystal growth-dissolution and aggregation breakage: numerical and moment solutions for population balance equations. Powder Technol. 2004; 143-144: 297-307.
5. Marchisio DL, Vigil RD, Fox RO. Quadrature method of moments for aggregation-breakage processes. J Colloid Interface Sci. 2003; 258: 322-334.
6. McGraw R, Nemesur S, Schwartz SE. Description of aerosol dynamics by the quadrature method of moments. Aerosol Sci Tech. 1997; 27: 255-265.
7. Lim YI, Lann J-ML, Meyer LM, Joulia L, Lee G, Yoon ES. On the solution of population balance equation (PBE) with accurate front tracking method in practical crystallization processes. Chem Eng Sci. 2002; 57: 3715-3732.
8. Qamar S, Warnecke G, Elsner MP. On the solution of population balances for nucleation, growth, aggregation and breakage processes. Chem Eng Sci. 2009; 64: 2088-2095.
9. Rawlings JB, Witkowski WR, Eaton JW. Modelling and control of crystallizers. Powder Technol. 1992; 69: 3-9.
10. Smith M, Matsoukas T. Constant-number Monte Carlo simulation of population balances. Chem Eng Sci. 1998; 53: 1777-1786.
11. Tandon P, Rosner DE. Monte Carlo simulation of particle aggregation and simultaneous restructuring. J Coll Int Sci. 1999; 213: 273-286.
12. Kumar S, Ramkrishna D. On the solution of population balance equations by discretization-I. A fixed pivot technique. Chem Eng Sci. 1996; 51: 1311-1332.
13. Dorao CA, Jakobsen HA. Numerical calculation of the moments of the population balance equation. J Comput Appl Math. 2006; 196: 619-633.
14. Dorao CA, Jakobsen HA. A least squares method for the solution of population balance problems. Comput Chem Eng. 2006; 30: 535-547.
15. Dorao CA, Lucas D, Jakobsen HA. Prediction of the evolution of the dispersed phase in bubbly flow problems. Appl Math Model. 2008; 32: 1813-1833.
16. Gunawan R, Fusman I, Braatz RD. High resolution algorithms for multidimensional population balance equations. AIChE J. 2004; 50: 2738-2749.
17. Qamar S, Elsner MP, Angelov I, Warnecke G, Seidel-Morgenstern A. A comparative study of high resolution schemes for solving population balances in crystallization. Comp Chem Eng. 2006; 30: 1119-1131.
18. McGraw R. Properties and evolution of Aerosol with size distributions having identical moments. J Aerosol Sci. 1998; 29: 761-772.
19. Marchisio DL, Pikturna JT, Fox RO, Vigil RD, Barresi AA. Quadrature method of moments for population balance equations. AIChE J. 2003; 49: 1266-1276.
20. John V, Angelov I, Öncül AA, Thévenin D. Technique for the reconstruction of a distribution from a finite number of its moments. Chem Eng Sci. 2007; 62: 2890-2904.
21. Grosch R, Briesen H, Marquardt W, Wulkow M. Generalization and numerical investigation of QMOM. AIChE J. 2006; 53: 207-227.
22. Diemer RB, Olson JH. A moment methodology for coagulation and breakage problems: Part I-analytical solution of the steady-state population balance. Chem Eng Sci. 2002; 57: 2193-2209.
23. Gordon RG. Error bounds in equilibrium statistical mechanics. J Math Phys. 1968; 9: 655-663.
24. Rosner DE, Pyykonen JJ. Bivariate moment simulation of coagulating and sintering nanoparticles in flames. AIChE J. 2002; 48: 476-491.
25. Wright DL, McGraw R, Rosner DE. Bivariate extension of the quadrature method of moments for modeling simultaneous coagulation and sintering of particle populations. J Coll Int Sci. 2001; 236: 242-251.
26. Barrett JC, Webb NA. A comparison of some approximate methods for solving the aerosol general dynamic equation. J Aerosol Sci. 1998; 29: 31-39.
27. Fan R, Marchisio DL, Fox RO. Application of the direct quadrature method of moments to polydisperse gas-solid fluidized beds. Powder Technol. 2004; 139: 7-20.
28. Su J, Gu Z, Li Y, Feng S, Xu XY. An adaptive direct quadrature method of moment for population balance equations. AIChE. 2008; 54: 2872-2887.
29. Gimbun J, Nagy ZK, Rielly CD. Simultaneous quadrature method of moments for the solution of population balance equations, using a differential algebraic equation frameworks. Ind Eng Chem Res. 2009; 48: 7798-7812.
30. Diemer RB, Ehrman SH. Pipeline agglomerator design as a model test case. Powder Technol. 2005; 156: 129-145.
31. Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: the Art of Scientific Computing, 3rd ed. New York: Cambridge University Press, 2007.
32. Qamar S, Mukhtar S, Seidel-Morgenstern A, Elsner MP. An efficient numerical technique for solving one-dimensional batch crystallization models with size-dependent growth rates. Chem Eng Sci. 2009; 64: 3659-3667.
33. Miller SM, Rawlings JB. Model identification and quality control strategies for batch cooling crystallizers. AIChE J. 1994; 40: 1312-1327.
34. Su J, Gu Z, Li Y, Feng S, Xu XY. Solution of population balance equation using quadrature method of moments with an adjustable factor. Chem Eng Sci. 2007; 62: 5897-5911.
35. Scott WT. Analytic studies of cloud droplet coalescence I. J Atmos Sci. 1968; 25: 54-65.
36. Smit DJ, Hounslow MJ, Paterson WR. Aggregation and gelation. I. Analytical solutions for CST and batch operation. Chem Eng Sci. 1994; 49: 1025-1035.
37. Ziff RM, McGrady ED. The kinetics of cluster fragmentation and depolymerisation. J Phys A Math Gen. 1985; 18: 3027-3037.