Packed-Bed Membrane Reactors

Modeling of Process Intensification

Volumn , Issue , 2007, Pages 99-148

  Tóta, Ákos ^a   Hlushkou, Dzmitry ^a   Tsotsas, Evangelos ^a   Seidel Morgenstern, Andreas ^a  

^a OTTO VON GUERICKE UNIVERSITY   (Germany)

Author keywords

Distributed dosing of reactants (PBMR); Modeling; One dimensional modeling; Packed bed membrane reactors; Process intensification; Three dimensional modeling; Two dimensional modeling

Indexed keywords

EID: 47949127242     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9783527610600.ch5     Document Type: Chapter

References (111)

1. Abe, T., 1997. Derivation of the Lattice Boltzmann Method by Means of the Discrete Ordinate Method for the Boltzmann Equation. Journal of Computational Physics, 131(1): 241-246.
2. Al-Juaied, M.A., Lafarga, D. and Varma, A., 2001. Ethylene Epoxidation in a Catalytic Packed-Bed Membrane Reactor: Experiments and Model. Chemical Engineering Science, 56(2): 395-402.
3. Alfonso, M.J., Julbe, A., Farrusseng, D., Menendez, M. and Santamaria, J., 1999. Oxidative Dehydrogenation of Propane on V/Al2o3 Catalytic Membranes. Effect of the Type of Membrane and Reactant Feed Configuration. Chemical Engineering Science, 54(9): 1265-1272.
4. Alfonso, M.J., Menendez, M. and Santamaria, J., 2001. Coupling of Consecutive Reactions in a Two-Layer, Flow-through Catalytic Membrane. Industrial & Engineering Chemistry Research, 40(4): 1058-1064.
5. Aris, R. and Amundson, N.R., 1957. Some Remarks on Longitudinal Mixing or Diffusion in Fixed Beds. AIChE Journal, 3(2): 280-282.
6. Assabumrungrat, S., Jhoraleecharnchai, W., Praserthdam, P. and Goto, S., 2001. Application of Palladium Membrane Reactor for Dehydrogenation of Propane. Journal of the Chinese Institute of Chemical Engineers, 32(4): 295-301.
7. Bauer, R. and Schlünder, E.U., 1978. Effective Radial Thermal-Conductivity of Packings in Gas-Flow.2. Thermal-Conductivity of Packing Fraction without Gas-Flow. International Chemical Engineering, 18(2): 189-204.
8. Bear, J., 1972. Dynamics of Fluids in Porous Media. Environmental Science Series. American Elsevier Publishing Company, New York, 748pp.
9. Beavers, G.S. and Joseph, D.D., 1967. Boundary Conditions at a Naturally Permeable Wall. Journal of Fluid Mechanics, 30: 197.
10. Benenati, R.F. and Brosilow, C.B., 1962. Void Fraction Distribution in Beds of Spheres. AIChE Journal, 8(3): 359-361.
11. Bey, O. and Eigenberger, G., 1996. Determination of Flow Distribution and Heat Transfer Parameters in Packed Pipes. Chemie Ingenieur Technik, 68(10): 1294-1299.
12. Bey, O. and Eigenberger, G., 1997. Fluid Flow through Catalyst Filled Tubes. Chemical Engineering Science, 52(8): 1365-1376.
13. Bey, O. and Eigenberger, G., 2001. Gas Flow and Heat Transfer through Catalyst Filled Tubes. International Journal of Thermal Sciences, 40(2): 152-164.
14. Bird, R.B., Stewart, W.E. and Lightfoot, E.N., 2002. Transport Phenomena, Wiley, New York.
15. Bouwmeester, H.J.M., 2003. Dense Ceramic Membranes for Methane Conversion. Catalysis Today, 82(1-4): 141-150.
16. Brinkman, H.C., 1947. A Calculation of the Viscous Force Exerted by a Flowing Fluid on a Dense Swarm of Particles. Applied Scientific Research Section a-Mechanics Heat Chemical Engineering Mathematical Methods, 1(1): 27-34.
17. Capannelli, G., Carosini, E., Cavani, F., Monticelli, O. and Trifiro, F., 1996. Comparison of the Catalytic Performance of V2O5/Gamma-Al2O3 in the Oxidehydrogenation of Propane to Propylene in Different Reactor Configurations: 1) Packed-Bed Reactor, 2) Monolith-Like Reactor and, 3) Catalytic Membrane Reactor. Chemical Engineering Science, 51(10): 1817-1826.
18. Cheng, P. and Vortmeyer, D., 1988. Transverse Thermal Dispersion and Wall Channeling in a Packed-Bed with Forced Convective Flow. Chemical Engineering Science, 43(9): 2523-2532.
19. Coronas, J., Menendez, M. and Santamaria, J., 1995. Use of a Ceramic Membrane Reactor for the Oxidative Dehydrogenation of Ethane to Ethylene and Higher Hydrocarbons. Industrial & Engineering Chemistry Research, 34(12): 4229-4234.
20. D'Orazio, A., Corcione, M. and Celata, G.P., 2004. Application to Natural Convection Enclosed Flows of a Lattice Boltzmann Bgk Model Coupled with a General Purpose Thermal Boundary Condition. International Journal of Thermal Sciences, 43(6): 575-586.
21. Dalmon, J.A., 1997. Catalysis in Membrane Reactors. In: G. Ertl (Editor), Handbook of Heterogeneous Catalysis Wiley-VCH, Weinheim.
22. Delmas, H. and Froment, G.F., 1988. A Simulation-Model Accounting for Structural Radial Nonuniformities in Fixed-Bed Reactors. Chemical Engineering Science, 43(8): 2281-2287.
23. Diakov, V., Lafarga, D. and Varma, A., 2001. Methanol Oxidative Dehydrogenation in a Catalytic Packed-Bed Membrane Reactor. Catalysis Today, 67(1-3): 159-167.
24. Diakov, V. and Varma, A., 2004. Optimal Feed Distribution in a Packed-Bed Membrane Reactor: The Case of Methanol Oxidative Dehydrogenation. Industrial & Engineering Chemistry Research, 43(2): 309-314.
25. Dittmeyer, R., (editor) 2005. Catalysis in Membrane Reactors. Catalysis Today, 104(2-4): 101-371.
26. Dittmeyer, R., Svajda, K. and Reif, M., 2004. A Review of Catalytic Membrane Layers for Gas/Liquid Reactions. Topics in Catalysis, 29(1-2): 3-27.
27. Dixon, A.G., 2001. Analysis of Intermediate Product Yield in Distributed-Feed Nonisothermal Tubular Membrane Reactors. Catalysis Today, 67(1-3): 189-203.
28. Dixon, A.G., 2003. Recent Research in Catalytic Inorganic Membrane Reactors. International Journal of Chemical Reactor Engineering, Vol. 1.(R6).
29. Ergun, S., 1952. Fluid Flow through Packed Columns. Chemical Engineering Progress, 48(2): 89-94.
30. Ferziger, J.M. and Peric, M., 2001. Computational Methods for Fluid Dynamics. Springer, Berlin, 423 pp.
31. Freund, H., Bauer, J., Zeiser, T. and Emig, G., 2005. Detailed Simulation of Transport Processes in Fixed-Beds. Industrial & Engineering Chemistry Research, 44(16): 6423-6434.
32. Fuller, E.N., Schettle. Pd and Giddings, J.C., 1966. A New Method for Prediction of Binary Gas-Phase Diffusion Coefficients. Industrial and Engineering Chemistry, 58(5): 19.
33. Giese, M., Rottschafer, K. and Vortmeyer, D., 1998. Measured and Modeled Superficial Flow Profiles in Packed Beds with Liquid Flow. AIChE Journal, 44(2): 484-490.
34. Goodling, J.S., Vachon, R.I., Stelpflug, W.S., Ying, S.J. and Khader, M.S., 1983. Radial Porosity Distribution in Cylindrical Beds Packed with Spheres. Powder Technology, 35(1): 23-29.
35. Goyeau, B., Lhuillier, D., Gobin, D. and Velarde, M.G., 2003. Momentum Transport at a Fluid-Porous Interface. International Journal of Heat and Mass Transfer, 46(21): 4071-4081.
36. Greiner, L. et al., 2003. Membrane Aerated Hydrogenation: Enzymatic and Chemical Homogeneous Catalysis. Advanced Synthesis & Catalysis, 345(6-7): 679-683.
37. Haidegger, E., Vortmeyer, D. and Wagner, P., 1989. Simultaneous Solution of Energy, Mass, and Momentum Equations for Wall-Cooled Fixed-Bed Chemical Reactors. Chemie Ingenieur Technik, 61(8): 647-650.
38. Hamel, C., Thomas, S., Schädlich, K. and Seidel-Morgenstern, A., 2003. Theoretical Analysis of Reactant Dosing Concepts to Perform Parallel-Series Reactions. Chemical Engineering Science, 58(19): 4483-4492.
39. Harold, M.P. and Lee, C., 1997. Intermediate Product Yield Enhancement with a Catalytic Inorganic Membrane.2. Nonisothermal and Integral Operation in a Backmixed Reactor. Chemical Engineering Science, 52(12): 1923-1939.
40. He, X.Y. and Li, N., 2000. Lattice Boltzmann Simulation of Electrochemical Systems. Computer Physics Communications, 129(1-3): 158-166.
41. He, X.Y. and Luo, L.S., 1997. Lattice Boltzmann Model for the Incompressible Navier-Stokes Equation. Journal of Statistical Physics, 88(3-4): 927-944.
42. Himmelblau, D.M., 1974. Basic Principles and Calculations in Chemical Engineering. Prentice Hall, New Jersey, 542 pp.
43. Hunt, M.L. and Tien, C.L., 1990. NonDarcian Flow, Heat and Mass-Transfer in Catalytic Packed-Bed Reactors. Chemical Engineering Science, 45(1): 55-63.
44. Hussain, A., Seidel-Morgenstern, A. and Tsotsas, E., 2006. Heat and Mass Transfer in Tubular Ceramic Membranes for Membrane Reactors. International Journal of Heat and Mass Transfer, 49(13-14): 2239-2253.
45. Hwang, S.W. and Smith, R., 2004. Heterogeneous Catalytic Reactor Design with Optimum Temperature Profile I: Application of Catalyst Dilution and Side-Stream Distribution. Chemical Engineering Science, 59(20): 4229-4243.
46. Itoh, N., Tamura, E., Hara, S., Takahashi, T., Shono, A., Satoh, K. and Namba, T., 2003. Hydrogen Recovery from Cyclohexane as a Chemical Hydrogen Carrier Using a Palladium Membrane Reactor. Catalysis Today, 82(1-4): 119-125.
47. James, D.F. and Davis, A.M.J., 2001. Flow at the Interface of a Model Fibrous Porous Medium. Journal of Fluid Mechanics, 426: 47-72.
48. Julbe, A., Farrusseng, D. and Guizard, C., 2001. Porous Ceramic Membranes for Catalytic Reactors-Overview and New Ideas. Journal of Membrane Science, 181(1): 3-20.
49. Kee, R.J., Coltrin, M.E. and Glarborg, P., 2003. Chemically Reacting Flow. John Wiley & Sons, New Jersey, 928 pp.
50. Klose, F., Joshi, M., Hamel, C. and Seidel-Morgenstern, A., 2004a. Selective Oxidation of Ethane over a VOx/Gamma-Al2O3 Catalyst-Investigation of the Reaction Network. Applied Catalysis a-General, 260(1): 101-110.
51. Klose, F., Wolff, T., Thomas, S. and Seidel-Morgenstern, A., 2003. Concentration and Residence Time Effects in Packed Bed Membrane Reactors. Catalysis Today, 82(1-4): 25-40.
52. Klose, F., Wolff, T., Thomas, S. and Seidel-Morgenstern, A., 2004b. Operation Modes of Packed-Bed Membrane Reactors in the Catalytic Oxidation of Hydrocarbons. Applied Catalysis a-General, 257(2): 193-199.
53. Krishna, R., 2002. Reactive Separations: More Ways to Skin a Cat. Chemical Engineering Science, 57(9): 1491-1504.
54. Krishna, R. and Wesselingh, J.A., 1997. Review Article Number 50-the Maxwell-Stefan Approach to Mass Transfer. Chemical Engineering Science, 52(6): 861-911.
55. Krishna, R. and Wesselingh, J.A., 2000. Mass Transfer in Multicomponent Mixtures Delft Univ. Press, Delft.
56. Küfner, R. and Hofmann, H., 1990. Implementation of Radial Porosity and Velocity Distribution in a Reactor Model for Heterogeneous Catalytic Gas-Phase Reactions (Torus-Model). Chemical Engineering Science, 45(8): 2141-2146.
57. Kürten, U., 2003. Modeling of Packed Bed Membrane Reactors: Impact of Oxygen Distribution on Conversion and Selectivity in Partial Oxidation Systems. Thesis (Ph. D.), Universiteit Twente (The Netherlands), Enschede.
58. Kürten, U., van Sint Annaland, M. and Kuipers, J.A.M., 2004. Oxygen Distribution in Packed-Bed Membrane Reactors for Partial Oxidations: Effect of the Radial Porosity Profiles on the Product Selectivity. Industrial & Engineering Chemistry Research, 43(16): 4753-4760.
59. Kwong, S.S. and Smith, J.M., 1957. Radial Heat Transfer in Packed Beds. Industrial and Engineering Chemistry, 49(5): 894-903.
60. Lafarga, D. and Varma, A., 2000. Ethylene Epoxidation in a Catalytic Packed-Bed Membrane Reactor: Effects of Reactor Configuration and 1, 2-Dichloroethane Addition. Chemical Engineering Science, 55(4): 749-758.
61. Lallemand, P. and Luo, L.S., 2003. Theory of the Lattice Boltzmann Method: Acoustic and Thermal Properties in Two and Three Dimensions. Physical Review E, 68(3).
62. Levenspiel, O., 1999. Chemical Reaction Engineering John Wiley & Sons., New Jersey, 668 pp.
63. Lowe, C.P. and Frenkel, D., 1995. The Super Long-Time Decay of Velocity Fluctuations in a 2-Dimensional Fluid. Physica a-Statistical Mechanics and Its Applications, 220(3-4): 251-260.
64. Lu, Y.P., Dixon, A.G., Moser, W.R. and Ma, Y.H., 1997a. Analysis and Optimization of Cross-Flow Reactors for Oxidative Coupling of Methane. Industrial & Engineering Chemistry Research, 36(3): 559-567.
65. Lu, Y.P., Dixon, A.G., Moser, W.R. and Ma, Y.H., 1997b. Analysis and Optimization of Cross-Flow Reactors with Distributed Reactant Feed and Product Removal. Catalysis Today, 35(4): 443-450.
66. Lu, Y.P., Dixon, A.G., Moser, W.R. and Ma, Y.H., 1997c. Analysis and Optimization of Cross-Flow Reactors with Staged Feed Policies-Isothermal Operation with Parallel Series, Irreversible Reaction Systems. Chemical Engineering Science, 52(8): 1349-1363.
67. Mallada, R., Pedernera, M., Menendez, N. and Santamaria, J., 2000. Synthesis of Maleic Anhydride in an Inert Membrane Reactor. Effect of Reactor Configuration. Industrial & Engineering Chemistry Research, 39(3): 620-625.
68. Martin, H. and Nilles, M., 1993. Heat-Transport to the Wall of Packed Tubes. Chemie Ingenieur Technik, 65(12): 1468-1477.
69. Mason, E.A., Malinaus. Ap and Evans, R.B., 1967. Flow and Diffusion of Gases in Porous Media. Journal of Chemical Physics, 46(8): 3199.
70. Merks, R.M.H., Hoekstra, A.G. and Sloot, P.M.A., 2002. The Moment Propagation Method for Advection-Diffusion in the Lattice Boltzmann Method: Validation and Peclet Number Limits. Journal of Computational Physics, 183(2): 563-576.
71. Molerus, O., 1982. Feststoff-Strömungen: Strömungsverhalten Feststoffbeladener Fluide und kohäsiver Schüttgüter Springer, Berlin, 262 pp.
72. Ozdemir, S.S., Buonomenna, M.G. and Drioli, E., 2006. Catalytic Polymeric Membranes: Preparation and Application. Applied Catalysis a-General, 307(2): 167-183.
73. Papavassiliou, V., Lee, C., Nestlerode, J. and Harold, M.P., 1997. Pneumatically Controlled Transport and Reaction in Inorganic Membranes. Industrial & Engineering Chemistry Research, 36(11): 4954-4964.
74. Patil, C.S., Annaland, M.V. and Kuipers, J. A.M., 2005. Design of a Novel Autothermal Membrane-Assisted Fluidized-Bed Reactor for the Production of Ultrapure Hydrogen from Methane. Industrial & Engineering Chemistry Research, 44(25): 9502-9512.
75. Pena, M.A., Carr, D.M., Yeung, K.L. and Varma, A., 1998. Ethylene Epoxidation in a Catalytic Packed-Bed Membrane Reactor. Chemical Engineering Science, 53(22): 3821-3834.
76. Press, W.H., Teukolsky, S.A., Vetterling, W. T. and Flannery, B.P., 2002. Numerical Recipes in C++; the Art of Scientific Computing. Cambridge University Press, Cambridge.
77. Ramos, R., Menendez, M. and Santamaria, J., 2000. Oxidative Dehydrogenation of Propane in an Inert Membrane Reactor. Catalysis Today, 56(1-3): 239-245.
78. Reid, R.C., Prausnitz, J.M. and Poling, B.E., 1987. The Properties of Gases and Liquids McGraw-Hill Books in Chemical Engineering. McGraw-Hill, New York 741 pp.
79. Reyes, S.C., Iglesia, E. and Kelkar, C.P., 1993. Kinetic-Transport Models of Bimodal Reaction Sequences.1. Homogeneous and Heterogeneous Pathways in Oxidative Coupling of Methane. Chemical Engineering Science, 48(14): 2643-2661.
80. Rohde, M.P., Unruh, D. and Schaub, G., 2005. Membrane Application in Fischer-Tropsch Synthesis Reactors-Overview of Concepts. Catalysis Today, 106(1-4): 143-148.
81. Salomon, M.A., Coronas, J., Menendez, M. and Santamaria, J., 2000. Synthesis of Mtbe in Zeolite Membrane Reactors. Applied Catalysis a-General, 200(1-2): 201-210.
82. Sanchez Marcano, J.G. and Tsotsis, T.T., 2002. Catalytic Membranes and Membrane Reactors. Wiley-VCH, Weinheim, 251 pp.
83. Saracco, G., Neomagus, H.W.J.P., Versteeg, G.F. and van Swaaij, W.P.M., 1999. High-Temperature Membrane Reactors: Potential and Problems. Chemical Engineering Science, 54(13-14): 1997-2017.
84. Schäfer, R., Noack, M., Kölsch, P., Stohr, M. and Caro, J., 2003. Comparison of Different Catalysts in the Membrane-Supported Dehydrogenation of Propane. Catalysis Today, 82(1-4): 15-23.
85. Schlünder, E.U., 1966. Wärme-und Stoffübertragung in mit Schüttungen gefüllten Rohren. Chemie Ingenieur Technik, 38(11): 1161.
86. Seidel-Morgenstern, A., 2005. Analysis and Experimental Investigation of Catalytic Membrane Reactors. In: K. Sundmacher, A. Kienle and A. Seidel-Morgenstern (Editors), Integrated Chemical Processes. Wiley-VCH, Weinheim.
87. Sirkar, K.K., Shanbhag, P.V. and Kovvali, A. S., 1999. Membrane in a Reactor: A Functional Perspective. Industrial & Engineering Chemistry Research, 38(10): 3715-3737.
88. Stanek, V. and Szekely, J., 1974. 3Dimensional Flow of Fluids through Nonuniform Packed-Beds. AIChE Journal, 20(5): 974-980.
89. Struis, R.P.W.J. and Stucki, S., 2001. Verification of the Membrane Reactor Concept for the Methanol Synthesis. Applied Catalysis a-General, 216(1-2): 117-129.
90. Sullivan, S.P., Sani, F.M., Johns, M.L. and Gladden, L.F., 2005. Simulation of Packedbed reactors Using Lattice Boltzmann Methods. Chemical Engineering Science, 60(12): 3405-3418.
91. Tellez, C., Menendez, M. and Santamaria, J., 1997. Oxidative Dehydrogenation of Butane Using Membrane Reactors. AIChE Journal, 43(3): 777-784.
92. Tellez, C., Menendez, M. and Santamaria, J., 1999. Simulation of an Inert Membrane Reactor for the Oxidative Dehydrogenation of Butane. Chemical Engineering Science, 54(13-14): 2917-2925.
93. Thomas, S., Pushpavanam, S. and Seidel-Morgenstern, A., 2004. Performance Improvements of Parallel-Series Reactions in Tubular Reactors Using Reactant Dosing Concepts. Industrial & Engineering Chemistry Research, 43(4): 969-979.
94. Thomas, S., Schafer, R., Caro, J. and Seidel-Morgenstern, A., 2001. Investigation of Mass Transfer through Inorganic Membranes with Several Layers. Catalysis Today, 67(1-3): 205-216.
95. Tonkovich, A.L.Y., Jimenez, D.M., Zilka, J.L. and Roberts, G.L., 1996a. Inorganic Membrane Reactors for the Oxidative Coupling of Methane. Chemical Engineering Science, 51(11): 3051-3056.
96. Tonkovich, A.L.Y., Zilka, J.L., Jimenez, D. M., Roberts, G.L. and Cox, J.L., 1996b. Experimental Investigations of Inorganic Membrane Reactors: A Distributed Feed Approach for Partial Oxidation Reactions. Chemical Engineering Science, 51(5): 789-806.
97. Tota, A. et al., 2004. Theoretical and Experimental Investigation of Concentration and Contact Time Effects in Membrane Reactors. Chemical Engineering Research & Design, 82(A2): 236-244.
98. Tsotsas, E., 2002. Wärmeleitung und Dispersion in durchströmten Schüttungen, VDI-Wärmeatlas. Springer, Berlin, pp. 1366.
99. Tsotsas, E. and Schlünder, E.U., 1988a. On Axial-Dispersion in Packed-Beds with Fluid-Flow. Chemical Engineering and Processing, 24(1): 15-31.
100. Tsotsas, E. and Schlünder, E.U., 1988b. Wärmeübertragung in Festbetten, durchmischten Schüttgütern und Wirbelschichten Thieme Verlag, Stuttgart, 259 pp.
101. van de Graaf, J.M., Zwiep, M., Kapteijn, F. and Moulijn, J.A., 1999. Application of a Zeolite Membrane Reactor in the Metathesis of Propene. Chemical Engineering Science, 54(10): 1441-1445.
102. Veldsink, J.W., Vandamme, R.M.J., Versteeg, G.F. and Vanswaaij, W.P.M., 1995. The Use of the Dusty-Gas Model for the Description of Mass-Transport with Chemical-Reaction in Porous-Media. Chemical Engineering Journal and the Biochemical Engineering Journal, 57(2): 115-125.
103. Vortmeyer, D. and Schuster, J., 1983. Evaluation of Steady Flow Profiles in Rectangular and Circular Packed-Beds by a Variational Method. Chemical Engineering Science, 38(10): 1691-1699.
104. Wang, L.S., Murata, K. and Inaba, M., 2003. Production of Pure Hydrogen and More Valuable Hydrocarbons from Ethane on a Novel Highly Active Catalyst System with a Pd-Based Membrane Reactor. Catalysis Today, 82(1-4): 99-104.
105. Wieland, S., Melin, T. and Lamm, A., 2002. Membrane Reactors for Hydrogen Production. Chemical Engineering Science, 57(9): 1571-1576.
106. Winterberg, M. and Tsotsas, E., 2000a. Correlations for Effective Heat Transport Coefficients in Beds Packed with Cylindrical Particles. Chemical Engineering Science, 55(23): 5937-5943.
107. Winterberg, M. and Tsotsas, E., 2000b. Impact of Tube-to-Particle-Diameter Ratio on Pressure Drop in Packed Beds. AIChE Journal, 46(5): 1084-1088.
108. Winterberg, M., Tsotsas, E., Krischke, A. and Vortmeyer, D., 2000. A Simple and Coherent Set of Coefficients for Modeling of Heat and Mass Transport with and without Chemical Reaction in Tubes Filled with Spheres. Chemical Engineering Science, 55(5): 967-979.
109. Yoshino, M. and Inamura, T., 2003. Lattice Boltzmann Simulations for Flow and Heat/Mass Transfer Problems in a Three-Dimensional Porous Structure. International Journal for Numerical Methods in Fluids, 43(2): 183-198.
110. Ziaka, Z.D., Minet, R.G. and Tsotsis, T.T., 1993. Propane Dehydrogenation in a Packed-Bed Membrane Reactor. AIChE Journal, 39(3): 526-529.
111. Zou, R.P. and Yu, A.B., 1995. The Packing of Spheres in a Cylindrical Container-the Thickness Effect. Chemical Engineering Science, 50(9): 1504-1507.