On the solution of the population balance equation for bubbly flows using the high-order least squares method: Implementation issues

Reviews in Chemical Engineering

Volumn 29, Issue 2, 2013, Pages 63-98

  Solsvik, Jannike ^a   Jakobsen, Hugo A ^a  

^a NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

Author keywords

Bubbly flow; Gas liquid dispersion; Implementation issues; Least squares method; Population balance equation; Weighted residual method

Indexed keywords

BUBBLY FLOW; GAS-LIQUID DISPERSION; IMPLEMENTATION ISSUES; LEAST SQUARES METHODS; POPULATION BALANCE EQUATION; WEIGHTED RESIDUAL METHOD;

CARRIER CONCENTRATION; LINEAR ALGEBRA; METHOD OF MOMENTS; NUMERICAL METHODS;

LEAST SQUARES APPROXIMATIONS;

EID: 84881511617     PISSN: 01678299     EISSN: None     Source Type: Journal    
DOI: 10.1515/revce-2012-0018     Document Type: Article

References (83)

1. Attarakih MM, Bart H-J. Solution of the population balance equation using the sectional quadrature method of moments (SQMOM). Chem Eng Sci 2009; 65: 742 - 752.
2. Becker PJ, Puel F, Henry R, Sheibat-Othman N. Investigation of discrete population balance models and breakage kernels for dilute emulsification systems. Ind Eng Chem Res 2011; 50: 11358 - 11374.
3. Bernardi C, Maday Y. Spectral methods. In: Ciarlet PG, Lions JL, editors. Handbook of numerical analysis. Vol. V. Techniques of scientific computing (part 2). Elsevier, Amsterdam, 1997: 209 - 256.
4. Bochev PB, Gunzburger MD. Finite element methods of leastsquares type. SIAM Rev 1998; 40: 789 - 837.
5. Borka Z, Jakobsen HA. Evaluation of breakage and coalescence kernels for vertical bubbly flows using a combined multifluidpopulation balance model solved by least squares method. Procedia Eng 2012a; 42: 623 - 633.
6. Borka Z, Jakobsen HA. Least squares higher order method for the solution of a combined multifluid-population balance model: modeling and implementation issues. Procedia Eng 2012b; 42: 1121 - 1132.
7. Borka Z, Jakobsen HA. On the modeling and simulation of higher order breakage for vertical bubbly flows using the least squares method: Application for bubble column and pipe flows. Procedia Eng 2012c; 42: 1270 - 1281.
8. Canuto C, Hussaini MY, Quarteroni A, Zang TA. Spectral methods in fluid dynamics. Berlin: Springer, 1988.
9. Chang CL, Nelson JJ. Least-squares finite element method for the Stokes problem with zero residual of mass conservation. SIAM J Numer Anal 1997; 34: 480 - 489.
10. Chen M-Q, Hwang C, Shih Y-P. A wavelet-Galerkin method for solving population balance equations. Comput Chem Eng 1995; 20: 131 - 145.
11. Coulaloglou CA, Tavlarides LL. Description of interaction processes in agitated liquid-liquid dispersions. Chem Eng Sci 1977; 32: 1289 - 1297.
12. Crandall SH. Engineering analysis. New York: McGraw-Hill, 1956.
13. Davis PJ, Rabinowitz P. Methods of numerical integration, 2nd ed. Orlando, FL: Academic Press, 1984.
14. Deang JM, Gunzburger MD. Issues related to least-squares finite element methods for the Stokes equations. SIAM J Sci Comput 1998; 20: 878 - 906.
15. Dorao CA. High order methods for the solution of the population balance equation with applications to bubbly flows. PhD thesis, Norwegian University of Science and Technology (NTNU), 2006.
16. Dorao CA, Jakobsen HA. Application of the least-squares method for solving population balance problems in ?1. Rd Chem Eng Sci 2006a; 61: 5070 - 5081.
17. Dorao CA, Jakobsen HA. A least squares method for the solution of population balance problems. Comput Chem Eng 2006b; 30: 535 - 547.
18. Dorao CA, Jakobsen HA. Numerical calculation of the moments of the population balance equation. J Comput Appl Math 2006c; 196: 619 - 633.
19. Dorao CA, Jakobsen HA. The quadrature method of moments and its relationship with the method of weighted residuals. Chem Eng Sci 2006d; 61: 7795 - 7804.
20. Dorao CA, Jakobsen HA. Least-squares spectral method for solving advective population balance problems. J Comput Appl Math 2007a; 201: 247 - 257.
21. Dorao CA, Jakobsen HA. Time-space-property least squares spectral method for population balance problems. Chem Eng Sci 2007b; 62: 1323 - 1333.
22. Dorao CA, Jakobsen HA. hp -Adaptive least squares spectral element method for population balance equations. Appl Numer Math 2008; 58: 563 - 576.
23. 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.
24. Dorao CA, Lucas D, Jakobsen HA. Time-property least-squares spectral method for population balance equations. J Math Chem 2009; 46: 770 - 780.
25. Dunn EL, Stein FM. Families of Strum-Liouville systems. SIAM Rev 1961; 3: 54 - 65.
26. Edwards CH, Penny DE. Calculus. New Jersey: Prentice-Hall, Inc., 2002.
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. Ferziger JH, Peric M. Computational methods for fluid dynamics, 3rd ed. Berlin: Springer, 2002.
29. Fibert F, Laurencot P. Numerical simulation of the Smoluchowski coagulation equation. SIAM J Sci Comput 2004; 25: 2004 - 2028.
30. Gelbard F, Seinfeld JH. Numerical solution of the dynamic equation for particulate systems. J Comput Phys 1978; 28: 357 - 375.
31. Golub GH. Some modified matrix eigenvalue problems. SIAM Rev 1973; 15: 318 - 334.
32. Golub GH, Welsch JH. Calculation of gauss quadrature rules. Math Comput 1969; 23: 221 - 230.
33. Heys JJ, Lee E, Manteuel TA, McCormick SF. An alternative least-squares formulation of the Navier-Stokes equations with improved mass conservation. J Comput Phys 2007; 226: 994 - 1006.
34. Hildebrand FB. Introduction to numerical analysis. New York: McGraw-Hill, 1956.
35. Hulbert HM, Katz S. Some problems in particle technology. Chem Eng Sci 1964; 19: 555 - 574.
36. Jakobsen HA. Chemical reactor modeling: multiphase reactive flows. Berlin: Springer, 2008.
37. Jiang B-N. The least-squares finite element method: theory and applications in computational fluid dynamics and electromagnetics. Berlin: Springer, 1998.
38. Karniadakis GE, Sherwin S. Spectral/hp element methods for computational fluid dynamics. Oxford, UK: Oxford University Press, 2005.
39. Kumar S, Ramkrishna D. On the solution of population balance equations by discretization - I. A fixed pivot technique. A fixed pivot technique Chem Eng Sci 1996; 51: 13311.
40. Kumar J, Peglow M, Warnecke G, Heinrich S, Morl L. Improved accuracy and convergence of discretized population balance for aggregation: the cell average technique. Chem Eng Sci 2006; 61: 3327 - 3334.
41. Kumar J, Warnecke G, Peglow M, Heinrich S. Comparison of numerical methods for solving population balance equations incorporating aggregation and breakage. Powder Technol 2009; 189: 218 - 229.
42. Lafi AY, Reyes JN. General particle transport equation. Final report. Tech. Rep. OSU-NE9409 Oregon: Department of Nuclear Engineering, Oregon State University, 1994.
43. Lasheras JC, Eastwood C, Martinez-Bazan C, Montanes JL. A review of statistical models for the break-up of an immisible fluid immersed into a fully developed turbulent flow. Int J Multiphase Flow 2002; 28: 247 - 278.
44. Lathouwers D, Bellan J. Yield optimization and scaling of fluidized beds for tar production from biomass. Energy Fuels 2001; 15: 1247 - 1262.
45. Mantzaris NV, Daoutidis P, Srienc F. Numerical solution of multivariable cell population balance models. II. Spectral methods. II Spectral methods Comput Chem Eng 2001; 25: 1441 - 1462.
46. Marchisio DL, Fox RO. Solution of population balance equations using the direct quadrature method of moments. J Aerosol Sci 2005; 36: 43 - 73.
47. Marchisio DL, Pikturna JT, Fox RO, Vivil RD, Barresi AA. Quadrature method of moments for population-balance equations. AIChE J 2003a; 49: 1266 - 1276.
48. Marchisio DL, Vigil RD, Fox RO. Quadrature method of moments for aggregation-breakage processes. J Colloid Interface Sci 2003b; 258: 322 - 334.
49. Mazzei L, Marchisio DL, Lettieri P. New quadrature-based moment method for the mixing of inert polydisperse fluidized powders on commercial CFD codes. AIChE J 2012; 58: 3054 - 3069.
50. McGraw R. Description of aerosol dynamic by the quadrature method of moments. Aerosol Sci Technol 1997; 27: 255 - 265.
51. Morel C, Ruyer P N S, Lavieville JM. Comparison of several models for multi-size bubble flows on an adiabatic experiment. Int J Multiphase Flow 2010; 36: 25 - 39.
52. Nayak AK, Borka Z, Patruno LE, Sporleder F, Dorao CA, Jakobsen HA. A combined multifluid-population balance model for vertical gas - liquid bubble-driven flows considering bubble column operating conditions. Ind Eng Chem Res 2011; 50: 1786 - 1798.
53. Nicmanis M, Hounslow MJ. Finite-element methods for steady-state population balance equations. AIChE J 1998; 44: 2258 - 2272.
54. Patruno LE, Dorao CA, Svendsen HF, Jakobsen HA. Analysis of breakage kernels for population balance modelling. Chem Eng Sci 2009a; 64: 501 - 508.
55. Patruno LE, Dorao CA, Svendsen HF, Jakobsen HA. On the modelling of droplet-film interaction considering entrainment, deposition and breakage processes. Chem Eng Sci 2009b; 64: 1362 - 1371.
56. Pontaza JP. Least-squares variation principles and finite element methods: theory, formulations, and models for solid and fluid mechanics. PhD thesis, Texas A&M University, 2003.
57. Pontaza JP, Reddy JN. Spectral/hp least-squares finite element formulation for the incompressible Navier-Stokes equation. J Comput Phys 2003; 190: 523 - 549.
58. Pontaza JP, Reddy JN. Space-time coupled spectral/hp least-squares finite element formulation for the incompressible Navier- Stokes equation. J Comput Phys 2004; 190: 418 - 459.
59. Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical recipes in FORTRAN 77: the art of scientific computing, 2nd ed. Vol. 1. Cambridge, UK: Cambridge University Press, 1992.
60. Prince MJ, Blanch HW. Bubble coalescence and break-up in air-sparged bubble columns. AIChE J 1990; 36: 1485 - 1499.
61. Proot MMJ, Gerritsma MI. Mass- And momentum conservation of the least-squares spectral element method for the Stokes problem. J Sci Comput 2006; 27: 389 - 401.
62. Ramkrishna D. Population balance: theory and applications to particulate systems in engineering. San Diego, CA: Academic Press, 2000.
63. Randolph AD, Larson MA. Theory of particulate processes. Analysis and techniques of continuous crystallization, 2nd ed. San Diego: Academic Press, 1988.
64. Rout KR, Jakobsen HA. Reactor performance optimization by the use of a novel combined pellet reflecting both catalyst and adsorbent properties. Fuel Process Technol 2012; 99: 13 - 34.
65. Rout KR, Solsvik J, Nayak AK, Jakobsen HA. A numerical study of multicomponent mass diffusion and convection in porous pellets for the sorption-enhanced steam methane reforming and desorption processes. Chem Eng Sci 2011; 66: 4111 - 4126.
66. Smoluchowski MV. Drei vortr age uber diffusion, brownsche bewegung und koagulation von kolloidteilchen. Physik Z 1916; 17: 557 - 585.
67. Solsvik J, Jakobsen HA. Effect of Jacobi polynomials on the numerical solution of the pellet equation using the orthogonal collocation, Galerkin, tau and least squares methods. Comput Chem Eng 2012a; 39: 1 - 21.
68. Solsvik J, Jakobsen HA. Solution of the pellet equation by use of the orthogonal collocation and least squares methods: effects of different orthogonal Jacobi polynomials. Int J Chem Reactor Eng 2012b; 10: A40.
69. Solsvik J, Tangen S, Jakobsen HA. Evaluation of weighted residual methods for the solution of the chemical reactor pellet model equations. In: Topping BHV, editor. Proceedings of the Eight International Conference on Engineering Computational Technology. Paper 128. Stirlingshire, Scotland: Civil-Comp Press, 2012. doi:10.4203/ccp.100.
70. Sporleder F. Simulation of chemical reactors using the least-squares spectral element method. PhD thesis, Norwegian University of Science and Technology (NTNU). Trondheim, Norway: Tapir Uttrykk, 2011.
71. Sporleder F, Dorao CA, Jakobsen HA. Simulation of chemical reactors using the least-squares spectral element method. Chem Eng Sci 2010; 65: 5146 - 5159.
72. Sporleder F, Dorao CA, Jakobsen HA. Model based on population balance for the simulation of bubble columns using methods of the least-squares type. Chem Eng Sci 2011; 66: 3133 - 3144.
73. Sporleder F, Borka Z, Solsvik J, Jakobsen HA. On the population balance equation. Rev Chem Eng 2012; 28: 149 - 169.
74. Subramanian G, Ramkrishna D. On the solution of statistical models of cell populations. Math Biosci 1971; 10: 1 - 23.
75. Szego G. Orthogonal polynomials. New York: American Mathematical Society, 1939.
76. Tanaka H, Inaba S, Nakaza K. Steady-state size distribution for the self-similar collision cascade. ICARUS 1996; 123: 450.
77. Vichnevetsky R. Use of functional approximation methods in the computer solution of initial value partial differential equation problems. IEEE Trans Comput C 1969; 18: 499 - 512.
78. Wulkow M, Gerstlauer A, Nieken U. Modeling and simulation of crystallization processes using parsival. Chem Eng Sci 2001; 56: 2575 - 2588.
79. Yuan C, Fox RO. Conditional quadrature method of moments for kinetic equations. J Comput Phys 2011; 230: 8216 - 8246.
80. Zhu Z. The least-squares spectral element method solution of the gas-liquid multi-fluid model coupled with the population balance equation. PhD thesis, Norwegian University of Science and Technology (NTNU). Trondheim, Norway: Tapir Uttrykk, 2009.
81. Zhu Z, Dorao CA, Jakobsen HA. On the coupled solution of a combined population balance model using the least-squares spectral element method. Ind Eng Chem Res 2009a; 48: 7994 - 8006.
82. Zhu Z, Dorao CA, Jakobsen HA. Solution of bubble number density with breakage and coalescence in a bubble column by leastsquares method. Prog Comput Fluid Dyn 2009b; 9: 436 - 446.
83. Zucca A, Marchisio DL, Vanni M, Barresi AA. Validation of bivariate DQMOM for nanoparticle processes simulation. AIChE J 2007; 53: 918 - 931.