Evaluation of weighted residual methods for the solution of a population balance model describing bubbly flows: The least-squares, Galerkin, tau, and orthogonal collocation methods

Industrial and Engineering Chemistry Research

Volumn 52, Issue 45, 2013, Pages 15988-16013

  Solsvik, Jannike ^a   Jakobsen, Hugo A ^a  

^a NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

Author keywords

[No Author keywords available]

Indexed keywords

BUBBLE SIZE DISTRIBUTIONS; COMPUTATIONALLY EFFICIENT; LEAST-SQUARES TECHNIQUES; ORTHOGONAL COLLOCATION; ORTHOGONAL COLLOCATION METHODS; POPULATION BALANCE EQUATION; POPULATION BALANCE MODELING; WEIGHTED RESIDUAL METHOD;

BUBBLE COLUMNS; BUBBLES (IN FLUIDS); GALERKIN METHODS; INTEGRODIFFERENTIAL EQUATIONS; LATEXES;

LEAST SQUARES APPROXIMATIONS;

EID: 84887605292     PISSN: 08885885     EISSN: 15205045     Source Type: Journal    
DOI: 10.1021/ie402033b     Document Type: Article

References (139)

1. Deckwer, W. D. Bubble Column Reactors; John Wiley & Sons: New York, 1992.
2. Jakobsen, H. A. Chemical Reactor Modeling: Multiphase Reactive Flows; Springer: Berlin, 2008.
3. Jakobsen, H. A.; Sannæs, B. H.; Grevskott, S.; Svendsen, H. F. Modeling of vertical bubble-driven flows Ind. Eng. Chem. Res. 1997, 36, 4052-4074
4. Jakobsen, H. A.; Lindborg, H.; Dorao, C. A. Modeling of bubble column reactors: Progress and limitations Ind. Eng. Chem. Res. 2005, 44, 5107-5151
5. Dorao, C. A. High order methods for the solution of the population balance equation with applications to bubbly flows. Ph.D. Thesis, Norwegian University of Science and Technology (NTNU), 2006.
6. Ramkrishna, D. Population balance: Theory and Applications to Particulate Systems in Engineering; Academic Press: San Diego, 2000.
7. Ramkrishna, D.; Mahoney, A. W. Population balance modeling. Promise for the future Chem. Eng. Sci. 2002, 57, 595-606
8. Ramkrishna, D. The status of population balances Rev Chem Eng 1985, 3, 49-95
9. Randolph, A. D.; Larson, M. A. Theory of particulate processes. Analysis and techniques of continuous crystallization, 2 nd ed.; Academic Press: San Diego, 1988.
10. Sporleder, F.; Borka, Z.; Solsvik, J.; Jakobsen, H. A. On the population balance equation Rev. Chem. Eng. 2012, 28, 149-169
11. Rigopoulos, S. Population balance modelling of polydispersed particles in reactive flows Prog. Energy Combust. Sci. 2010, 36, 412-443
12. Ramkrishna, D. Analysis of population balance-IV: The precise connection between Monte Carlo simulation and population balances Chem. Eng. Sci. 1981, 36, 1203-1209
13. Mahoney, A. W.; Ramkrishna, D. Efficient solution of population balance equations with discontinuities by finite elements Chem. Eng. Sci. 2002, 57, 1107-1119
14. Dorao, C. A.; Jakobsen, H. A. The quadrature method of moments and its relationship with the method of weighted residuals Chem. Eng. Sci. 2006, 61, 7795-7804
15. Singh, P. N.; Ramkrishna, D. Solution of population balance equations by MWR Comput. Chem. Eng. 1977, 1, 23-31
16. Dorao, C. A.; Jakobsen, H. A. A least squares method for the solution of population balance problems Comput. Chem. Eng. 2006, 30, 535-547
17. 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
18. Marchisio, D. L.; Fox, R. O. Solution of population balance equations using the direct quadrature method of moments J. Aerosol Sci. 2005, 36, 43-73
19. Marchisio, D. L.; Pikturna, J. T.; Fox, R. O.; Vivil, R. D.; Barresi, A. A. Quadrature method of moments for population-balance equations AIChE 2003, 49, 1266-1276
20. Marchisio, D. L.; Vigil, R. D.; Fox, R. O. Quadrature method of moments for aggregation-breakage processes J. Colloid Interface Sci. 2003, 258, 322-334
21. Tambour, Y.; Seinfeld, J. H. Sectional representations for simulating aerosol dynamics J. Colloid Interface Sci. 1980, 76, 541-556
22. Yeoh, G. H.; Tu, J. Y. Two-fluid and population balance models for subcooled boiling flow Appl. Math. Model. 2006, 30, 1370-1391
23. Dorao, C. A.; Fernandino, M.; Patruno, L. E.; Dupuy, P. M.; Jakobsen, H. A.; Svendsen, H. F. Macroscopic description of droplet-film interaction for gas-liquid systems Appl. Math. Model. 2009, 33, 3309-3318
24. Vafa, E.; Shahrokhi, M.; Abedini, H. Solution of population balance equations in emulsion polymerization using method of moments Chem. Eng. Commun. 2013, 200, 20-49
25. Balakin, B. V.; Hoffman, A. C.; Kosinski, P. Population balance model for nucleation, growth, aggregation, and breakage of hydrate particles in turbulent flow AIChE J. 2010, 56, 2052-2062
26. Gerstlauer, A.; Mitrović, A.; Motz, S.; Gilles, E. D. A population balance model for crystallization process using two independent particle properties Chem. Eng. Sci. 2001, 56, 2553-2565
27. Poon, J. M. H.; Immanuel, C. D.; Doyle, F. J.; Lister, J. D. A three-dimensional population balance model of granulation with a mechanistic representation of the nucleation and aggregation phenomena Chem. Eng. Sci. 2008, 63, 1215-1329
28. Celnik, M.; Patterson, R.; Kraft, M.; Wagner, W. Coupling a stochastic soot population balance to gas-phase chemistry using operator splitting Combust. Flame 2007, 148, 158-176
29. Fredrickson, A. G.; Mantzaris, N. V. A new set of population balance equations for microbial and cell cultures Chem. Eng. Sci. 2002, 57, 2265-2278
30. Solsvik, J.; Tangen, S.; Jakobsen, H. A. On the constitutive equations for fluid particle breakage Rev. Chem. Eng. 2013, 29, 241-356
31. Silva, L. F. L. R.; Rodrigues, R. C.; Mitre, J. F.; Lage, P. L. C. Comparison of the accuracy and performance of quadrature-based methods for population balance problems with simultaneous breakage and aggregation Comput. Chem. Eng. 2010, 34, 286-297
32. JunWei, S.; ZhwoLin, G.; Yun, X. X. Advances in numerical methods for the solution of population balance equations for dispersed phase systems Sci. China Ser. B, Chem. 2009, 52, 1063-1079
33. Mantzaris, N. V.; Daoutidis, P.; Srienc, F. Numerical solution of multi-variable cell population balance models. II. Spectral methods Comput. Chem. Eng. 2001, 25, 1441-1462
34. Attarakih, M.; Al-Zyod, S.; Abu-Khader, M.; Bart, H. J. PPBLAB: A new multivariate population balance environment for particulate system modelling and simulation Proc. Eng. 2012, 42, 1445-1462
35. Braumann, A.; Man, P. L. W.; Kraft, M. Statistical approximation of the inverse problem in multivariate population balance modeling Ind. Eng. Chem. Res. 2010, 49, 428-438
36. Ramachandran, R.; Barton, P. I. Effective parameter estimation within a multi-dimensional population balance model framework Chem. Eng. Sci. 2010, 65, 4884-4893
37. Sajjadi, B.; Raman, A. A. A.; Shah, R. S. S. R. E.; Ibrahim, S. Review on applicable breakup/coalescence models in turbulent liquid-liquid flows Rev. Chem. Eng. 2013, 29, 131-158
38. Borka, Z.; Jakobsen, H. A. Evaluation of breakage and coalescence kernels for vertical bubbly flows using a combined multifluid-population balance model solved by least squares method Proc. Eng. 2012, 42, 623-633
39. Borka, Z.; Jakobsen, H. A. 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 Proc. Eng. 2012, 42, 1270-1281
40. Solsvik, J.; Borka, Z.; Becker, P. J.; Sheibat-Othman, N.; Jakobsen, H. A. Can. J. Chem. Eng., DOI: 10.1002/cjce.21876.
41. Nguyen, V. T.; Song, C. H.; Bae, B. U.; Euh, D. J. Modeling of bubble coalescence and break-up considering turbulent suppression phenomena in bubbly two-phase flow Int. J. Multiphase Flow 2013, 54, 21-42
42. Han, L.; Gong, S.; Li, Y.; Ai, Q.; Luo, H.; Liu, Z.; Liu, Y. A novel theoretical model of breakage rate and daughter size distribution for droplet in turbulent flows Chem. Eng. Sci. 2013, 102, 186-199
43. Raikar, N. B.; Bhatia, S. R.; Malone, M. F.; Henson, M. A. Experimental studies and population balance equation models for breakage prediction of emulsion drop size distributions Chem. Eng. Sci. 2009, 64, 2433-2447
44. Mallikarjunan, V.; Pushpavanam, S.; Immanuel, C. D. Parameter estimation strategies in batch emulsion polymerization Chem. Eng. Sci. 2010, 65, 4967-4982
45. Jildeh, H. B.; Attarakih, M.; Mickler, M.; Bart, H. J. Can. J. Chem. Eng., DOI: 10.1002/cjce.21892.
46. Singh, K. K.; Mahajani, S. M.; Shenoy, K. T.; Ghosh, S. K. Population balance modeling of liquid-liquid dispersions in homogeneous continuous-flow stirred tank Ind. Eng. Chem. Res. 2009, 48, 8121-8133
47. Laakkonen, M.; Alopaeus, V.; Aittamaa, J. Validation of bubble breakage, coalescence and mass transfer models for gas-liquid dispersion in agitated vessel Chem. Eng. Sci. 2006, 61, 218-228
48. Laakkonen, M.; Moilanen, P.; Alopaeus, V.; Aittamaa, J. Modelling local bubble size distributions in agitated vessels Chem. Eng. Sci. 2007, 62, 721-740
49. Becker, P. J.; 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
50. Becker, P. J.; Puel, F.; Chevalier, Y.; Sheibat-Othman, N. Can. J. Chem. Eng., DOI: 10.1002/cjce.21885.
51. Lucas, D.; Krepper, E.; Prasser, H. M. Development of co-current air-water flow in a vertical pipe Int. J. Multiphase Flow 2005, 31, 1304-1328
52. Prasser, H. M.; Beyer, M.; Carl, H.; Gregor, S.; Lucas, D.; Pietruske, H.; Schutz, P.; Weiss, F. P. Evolution of the structure of a gas-liquid two-phase flow in a large vertical pipe Nucl. Eng. Des. 2007, 237, 1848-1861
53. Lucas, D.; Beyer, M.; Kussin, J.; Schütz, P. Benchmark database on the evolution of two-phase flows in a vertical pipe. 2008; XCFD4NRS, Experiments and CFD code applications to nuclear reactor safety.
54. Duan, X. Y.; Cheung, S. C. P.; Yeoh, G. H.; Tu, J. Y.; Krepper, E.; Lucas, D. Gas-liquid flows in medium and large vertical pipes Chem. Eng. Sci. 2011, 66, 872-883
55. Krepper, E.; Lucas, D. Population balance model for the CFD simulation of adiabatic and diabatic two phase gas liquid flows. 2012. Unpublished work.
56. Dorao, C. A.; Lucas, D.; Jakobsen, H. A. Prediction of the evolution of the dispersed phase in bubbly flow problems Appl. Math. Model. 2008, 32, 1813-1833
57. Zhu, Z.; Dorao, C. A.; Lucas, D.; Jakobsen, H. A. On the coupled solution of a combined population balance model using the least-squares spectral element method Ind. Eng. Chem. Res. 2009, 48, 7994-8006
58. Maaß, S.; Gäbler, A.; Zaccone, A.; Paschedag, A. R.; Kraume, M. Experimental investigations and modelling of breakage phenomena in stirred liquid/liquid systems Chem. Eng. Res. Des. 2007, 85, 703-709
59. Zaccone, A.; Gäbler, A.; Maaß, S.; Marchisio, D.; Kraume, M. Drop breakage in liquid-liquid stirred dispersions: Modeling of single drop breakage Chem. Eng. Sci. 2007, 62, 6297-6307
60. Maaß, S.; Kraume, M. Determination of breakage rates using single drop experiments Chem. Eng. Sci. 2012, 70, 146-164
61. Buffo, A.; Vanni, M.; Marchisio, D. L.; Fox, R. O. Multivariate quadrature-based moments methods for turbulent polydisperse gas-liquid systems Int. J. Multiphase Flow 2013, 50, 41-57
62. Buffo, A.; Vanni, M.; Marchisio, D.; Fox, R. O. Comparison between different methods for turbulent gas-liquid systems by using multivariate population balances. 2012; 8th International Conference on CFD in Oil & Gas, Metallurgical and Process Industries, SINTEF/NTNU, Trondheim Norway.
63. Bhole, M. R.; Joshi, J. B.; Ramkrishna, D. CFD simulation of bubble columns incorporating population balance modeling Chem. Eng. Sci. 2008, 63, 2267-2282
64. Hagesaether, L.; Jakobsen, H. A.; Svendsen, H. F. Modeling of the dispersed-phase size distribution in bubble columns Ind. Eng. Chem. Res. 2002, 41, 2560-2570
65. Sha, Z.; Laari, A.; Turunen, I. Multi-phase-multi-size-group model for the inclusion of population balances into the CFD simulation of gas-liquid bubbly flows Chem. Eng. Technol. 2006, 29, 550-559
66. Chen, P.; Dudukovic̀, M. P.; Sanyal, J. Three-dimensional simulation of bubble column flows with bubble coalescence and breakup AIChE J. 2005, 51, 696-712
67. Chen, P.; Sanyal, J.; Dudukovic̀, M. P. CFD modeling of bubble columns flows: implementation of population balance Chem. Eng. Sci. 2004, 59, 5201-5207
68. Krepper, E.; Lucas, D.; Frank, T.; Prasser, H.-M.; Zwart, P. J. The inhomogeneous MUSIG model for the simulation of polydispersed flows Nucl. Eng. Des. 2008, 238, 1690-1702
69. Sanyal, J.; Marchisio, D. L.; Fox, R. O.; Dhanasekharan, K. On the comparison between population balance models for CFD simulations of bubble columns Ind. Eng. Chem. Res. 2005, 44, 5063-5072
70. Morud, J. Implementation of the sectional quadrature method of moments in Fluent. 2011.
71. Zhu, Z. The least-squares spectral element method solution of the gas-liquid multi-fluid model coupled with the population balance equation. Ph.D. Thesis, Norwegian University of Science and Technology (NTNU), 2009.
72. Patruno, L. E. Experimental and numerical investigations of liquid fragmentation and droplet generation for gas processing at high pressures. Ph.D. Thesis, Norwegian University of Science and Technology (NTNU), 2010.
73. Sporleder, F. Simulation of chemical reactors using the least-squares spectral element method. Ph.D. Thesis, Norwegian University of Science and Technology (NTNU), 2011.
74. Nayak, A. K.; Borka, Z.; Patruno, L. E.; Sporleder, F.; Dorao, C. A.; Jakobsen, H. A. 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
75. Dorao, C. A.; Jakobsen, H. A. Numerical calculation of the moments of the population balance equation J. Comput. Appl. Math. 2006, 196, 619-633
76. Lafi, A. Y.; Reyes, J. N. General particle transport equation. Final report 1994.
77. Lasheras, J. C.; Eastwood, C.; Martinez-Bazan, C.; Montanes, J. L. 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
78. Lathouwers, D.; Bellan, J. Yield optimization and scaling of fluidized beds for tar production from biomass Energy Fuels 2001, 15, 1247-1262
79. Dorao, C. A.; Jakobsen, H. A. Least-squares spectral method for solving advective population balance problems J. Comput. Appl. Math. 2007, 201, 247-257
80. Dorao, C. A.; Jakobsen, H. A. Time-property least-squares spectral method for population balance equations J. Math. Chem. 2009, 46, 770-780
81. Dorao, C. A.; Jakobsen, H. A. hp -adaptive least squares spectral element method for population balance equations Appl. Num. Math. 2008, 58, 563-576
82. Dorao, C. A.; Jakobsen, H. A. Time-space-property least squares spectral method for population balance problems Chem. Eng. Sci. 2007, 62, 1323-1333
83. Patruno, L. E.; Dorao, C. A.; F, S. H.; Jakobsen, H. A. Analysis of breakage kernels for population balance modelling Chem. Eng. Sci. 2009, 64, 501-508
84. Zhu, Z.; Dorao, C. A.; Jakobsen, H. A. Solution of bubble number density with breakage and coalescence in a bubble column by least-squares method Prog. Comput. Fluid Dyn. 2009, 9, 436-446
85. Zhu, Z.; Dorao, C. A.; Jakobsen, H. A. A least-squares method with direct minimization for the solution of the breakage-coalescence population balance equation Math. Comput. Simul. 2008, 79, 716-727
86. Zhu, Z.; Dorao, C. A.; Jakobsen, H. A. Mass conservative solution of the population balance equation using the least-squares spectral element method Ind. Eng. Chem. Res. 2010, 49, 6204-6214
87. Sporleder, F.; Dorao, C. A.; Jakobsen, H. A. 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
88. Borka, Z.; Jakobsen, H. A. Least squares higher order method for the solution of a combined multifluid-population balance model: Modeling and implementation issues Proc. Eng. 2012, 42, 1121-1132
89. Solsvik, J.; Jakobsen, H. A. A numerical study of a two property catalyst/sorbent pellet design for the sorption-enhanced steam-methane reforming process: Modeling complexity and parameter sensitivity study Chem. Eng. J. 2011, 178, 407-422
90. Solsvik, J.; Jakobsen, H. A. Modeling of multicomponent mass diffusion in porous spherical pellets: Application to steam methane reforming and methanol synthesis Chem. Eng. Sci. 2011, 66, 1986-2000
91. Solsvik, J.; Tangen, S.; Jakobsen, H. A. On the consistent modeling of porous catalyst pellets: Mass and molar formulations Ind. Eng. Chem. Res. 2012, 51, 8222-8236
92. Van den Bosch, B.; Padmanabhan, L. Use of orthogonal collocation methods for the modeling of catalyst particles-I. Analysis of the multiplicity of the solutions Chem. Eng. Sci. 1974, 29, 1217-1225
93. Lefévre, L.; Dochain, D.; Feyo de Azevedo, S.; Magnus, A. Optimal selection of orthogonal polynomials applied to the integration of chemical reactor equations by collocation methods Comput. Chem. Eng. 2000, 24, 2571-2588
94. Alhumaizi, K.; Henda, R.; Soliman, M. Numerical analysis of a reaction-diffusion-convection system Comput. Chem. Eng. 2003, 27, 579-594
95. Nigam, K. M.; Nigam, K. D. P. Application of Galerkin technique to diffusion problems in tubular reactor Chem. Eng. Sci. 1980, 35, 2358-2360
96. Roussos, A. I.; Alexopoulos, A. H.; Kiparissides, C. Part III: Dynamic evolution of the particle size distribution in batch and continuous particulate processes: A Galerkin on finite elements approach Chem. Eng. Sci. 2005, 60, 6998-7010
97. Shafi, M. A.; Joshi, K.; Flumerfelt, R. W. Bubble size distribution in freely expanded polymer foams Chem. Eng. Sci. 1997, 52, 635-644
98. Solsvik, J.; Jakobsen, H. A. On the solution of the population balance equation for bubbly flows using the high-order least-squares method: Implementation issues Rev Chem Eng 2013, 29, 63-98
99. Coulaloglou, C. A.; Tavlarides, L. L. Description of interaction processes in agitated liquid-liquid dispersions Chem. Eng. Sci. 1977, 32, 1289-1297
100. Prince, M. J.; Blanch, H. W. Bubble coalescence and break-up in air-sparged bubble columns AIChE J. 1990, 36, 1485-1499
101. Morel, C.; Ruyer, P.; N, S.; Lavièville, J. M. Comparison of several models for multi-size bubble flows on an adiabatic experiment Int J Multiphase Flow 2010, 36, 25-39
102. Proot, M. M. J. The least-squares spectral element method. Ph.D. Thesis, Delt University of Technology, The Netherlands, 2003.
103. Jiang, B.-N. The least-squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics; Springer: Berlin, 1998.
104. Bochev, P. B.; Gunzburger, M. D. Finite element methods of least-squares type SIAM Rev. 1998, 40, 789-837
105. Bochev, P. B.; Gunzburger, M. D. Least-squares finite-element methods for optimization and control problems for the Stokes equations Comput. Math. Appl. 2004, 48, 1035-1057
106. Bochev, P.; Gunzburger, M. D. Least-Squares Finite Element Methods; New York: Springer, 2009.
107. Bolton, P.; Thatcher, R. W. On mass conservation in least-squares methods J. Comput. Phys. 2005, 203, 287-304
108. Pontaza, J. P. Least-squares variational principles and finite element method: theory, formulation, and models for solid and fluid mechanics. Ph.D. Thesis, Texas A&M University, 2003.
109. Pontaza, J. P. A least-squares finite element formulation for unstady incompressible flows with improved velocity-pressure coupling J. Comput. Phys. 2006, 217, 563-588
110. Pontaza, J. P. A new consistent splitting scheme for incrompressible Navier-Stokes flows: A least-squares spectral element implementation J. Comput. Phys. 2006, 225, 1590-1602
111. Pontaza, J. P.; Reddy, J. N. Spectral/hp least-squares finite element formulation for the incompressible Navier-Stokes equation J. Comput. Phys. 2003, 190, 523-549
112. Pontaza, J. P.; Reddy, J. N. Space-time coupled spectral/hp least squares finite element formulation for the incompressible Navier-Stokes equation J. Comput. Phys. 2004, 190, 418-459
113. Pontaza, J. P.; Reddy, J. N. Least-squares finite element formulations for viscous incompressible and compressible fluid flows Appl. Mech. Eng. 2006, 195, 2454-2494
114. Proot, M. M. J.; Gerritsma, M. I. A least-squres spectral element formulation for Stokes problem J. Sci. Commut. 2002, 17, 285-296
115. Proot, M. M. J.; Gerritsma, M. I. Mass- and momentum conservation of the least-squares spectral element method for the Stokes problem J. Sci. Commut. 2006, 27, 389-401
116. Solsvik, J.; Jakobsen, H. A. 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. 2012, 39, 1-21
117. Ritz, W. Uber eine neue Methode zur Lsnung gewisses Variationsprobleme der mathematishen Physik J. Reine. Angew. Math. 1908, 135, 1-61
118. Rayleigh, J. W. In finding the correction for the open end of an organ-pipe Phil. Trans. 1870, 161, 77
119. Galerkin, B. G. Series solution of some problems in elastic equilibrium of rods and plates Vestn. Inzh. Tech. 1915, 19, 897-908
120. Villadsen, J. V.; Steward, W. E. Solution of boundary-value problems by orthogonal collocation Chem. Eng. Sci. 1967, 22, 1482-1501
121. Villadsen, J. Selected approximation methods for chemical engineering problems; København: Danmarks Tekniske Højskole, 1970.
122. Villadsen, J.; Michelsen, M. L. Solution of Differential Equation Models by Polynomial Approximation; Prentice-Hall: Englewood Cliffs, NJ, 1978.
123. Michelsen, M. L.; Villadsen, J. Polynomial solution of differential equations. 1981; In Foundations of Computer-Aided Chemical Process Design; Mah, R S H; Seider, W D, Eds.; Engineering Foundation: New York, 341-368.
124. Finlayson, B. A. The Method of Weighted Residuals and Variational Principles; Mathematics in Science and Engineering; New York: Academic Press, 1972; Vol. 87.
125. Rice, R. G.; Do, D. D. Applied Mathematics and Modeling for Chemical Engineers; John Wiley & Sons: New York, 1995.
126. Grienberger, J. Untersuchung und Modellierund von Blasensäulen. Ph.D. Thesis, Der Technishen Fakultät der Universität Erlangen-Nürnberg, 1992.
127. Kostoglou, M.; Karabelas, A. J. On sectional techniques for the solution of the breakage equation Comput. Chem. Eng. 2009, 33, 112-121
128. Liao, Y.; Lucas, D. A literature review of theoretical models for drop and bubble breakup in turbulent dispersions Chem. Eng. Sci. 2009, 64, 3389-3406
129. Liao, Y.; Lucas, D. A literature review on mechanisms and models for the coalescence process of fluid particles Chem. Eng. Sci. 2010, 65, 2851-2864
130. Lasheras, J. C.; Eastwood, C.; Martínez-Bazán, C.; Montañés, J. L. A review of statistical models for the break-up of an immiscible fluid immersed into fully developed turbulent flow Int J Multiphase Flow 2002, 28, 247-278
131. Ferziger, J. H.; Peric̀, M. Computational Methods for Fluid Dynamics, 3 rd ed.; Berlin: Springer, 2002.
132. Patruno, L. E.; Marchetti, J. M.; Jakobsen, H. A.; Svendsen, H. F. Modeling of droplet entrainment in annular two-fluid three-phase dispersed flow. The 13th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-13), N13P1056, 2009.
133. Solsvik, J.; Tangen, S.; Jakobsen, H. A. Evaluation of weighted residual methods for the solution of the pellet equations: The orthogonal collocation, Galerkin, tau and least-squares methods Comput. Chem. Eng. 2013, 58, 223-259
134. Rout, K. R. A study of the sorption enhanced steam methane reforming process. Ph.D. Thesis, Norwegian University of Science and Technology (NTNU), 2012.
135. Szegö, G. Orthogonal Polynomials; American Mathematical Society: New York:, 1939.
136. Golub, G. H.; Welsch, J. H. Calculation of Gauss quadrature rules Math. Comput. 1969, 23, 221-230
137. Golub, G. H. Some modified matrix eigenvalue problems SIAM Rev. 1973, 15, 318-334
138. Crandall, S. H. Engineering Analysis; McGraw-Hill; New York, 1956.
139. Vichnevetsky, R. Use of functional approximation methods in the computer solution of initial value partial differential equation problems IEEE Trans. Comput. C-18 1969, 18, 499-512