Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach to Model Building and the Art of Approximation

Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach to Model Building and the Art of Approximation

Volumn , Issue , 2006, Pages 1-529

  Krantz, William B ^a,b,c  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^b UNIVERSITY OF CINCINNATI   (United States)

^c UNIVERSITY OF COLORADO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84890596938     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9780470121931     Document Type: Book

References (90)

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9. Jean Le Rond d'Alembert (1717-1783) studied experimentally the drag force on a sphere in a flowing fluid. He expected that the force would approach zero as the viscosity of the fluid approached zero. However, the drag force observed converged on a nonzero value as the viscosity became very small. The disappearance of the viscous drag force for very high Reynolds number flows is known as d'Alembert's paradox.
10. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd ed., Wiley, Hoboken, NJ, 2002, p. 137.
11. H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, 1960, pp. 116-124.
12. F. Janssen, J. Fluid Mech., 3, 329 (1958).
13. Note that we choose ts = 2π/ω rather than 1/ω since the oscillatory motion is characterized by the time it takes to complete one cycle.
14. The vorticity is defined to be the curl of the velocity field
15. u; if the curl of the equations of motion is taken, the vorticity appears as a diffused quantity, the transport coefficient being the kinematic viscosity; for this reason we refer to the diffused quantity in the equations of motion as the vorticity.
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17. Bird et al., Transport Phenomena, 2nd ed., Wiley, Hoboken, NJ, 2002, Problem 2D.2, pp. 73-74.
18. Examples of complex fluids include the flow of microemulsions, proteins, micellar solutions, and suspensions, as well as other nonhomogeneous liquids.
19. Bird et al., Transport Phenomena, 2nd ed. p. 186.
20. Section 3.3 for determining the conditions required for creeping flow.
21. The lubrication-flow approximation for this flow can also be justified using scaling analysis; this is considered in practice Problem 3.P.11.
22. The rotating disk electrode apparatus was first exhibited at the Brussels World's Fair in 1958. The application of scaling analysis for using the rotating disk to study mass transfer is considered in Chapter 5.
23. T. von Kármán, Z. Angew. Math. Mech., 1, 244-247 (1921).
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27. The ideal or inviscid flow equations correspond to an infinite Reynolds number, which implies no viscous effects whatsoever; in the case of hydrodynamic boundary-layer flows, the flow region outside the boundary layer is described by the ideal flow equations.
28. This simplified set of describing equations has been solved in Bird et al., Transport Phenomena, 2nd ed., pp. 93-95
29. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, Wiley, New York, 1960 pp. 126-130.
30. Note that scaling analysis could be used to determine when surface-tension and curvature effects can be neglected; the latter were considered in Section 3.7.
31. R. J. Ray, A Rayleigh free convection compliant ice front model for sorted patterned ground, M.S. thesis, University of Colorado, Boulder, Co, 1981.
32. This effect has been analyzed by D. B. Thiessen and W. B. Krantz, Rev. Sci. Instrum., 63(9), 4200-4204 (1992).
33. H. L. Langhaar, Trans. ASME, 64, A55 (1942).
34. Scaling was used to determine the criterion for making this assumption in Example Problem 3.E.1.
35. the example in Section 3.7 as a guide to scaling this problem.
36. Note that scaling analysis could be used to determine when surface-tension and curvature effects can be neglected; the latter were considered in Section 3.7.
37. L. R. Ingersoll, O. J. Zobel, and A. C. Ingersoll, Heat Conduction with Engineering and Geological Applications, McGraw-Hill, New York, 1948, p. 197.
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39. The infinite series in equation (4.2-29) converges rather slowly; hence, 50 terms in this series were retained in determining the dimensionless temperature predicted by this exact solution.
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41. R. B. Bird, W. E. Stewart, and E. L. Lightfoot, Transport Phenomena, 2nd ed., Wiley, Hoboken, NJ, 2002, pp. 703-708.
42. Bird et al., Transport Phenomena, 2nd ed., p. 439.
43. Bird et al., Transport Phenomena, 2nd ed., pp. 388-390.
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45. J. S. Turner, Buoyancy Effects in Fluids, Cambridge University Press, Cambridge, England, 1973.
46. R. B. Bird, W. E. Stewart, and E. L. Lightfoot, Transport Phenomena, Wiley, New York, 1960, pp. 349-350.
47. Bird et al., Transport Phenomena, 2nd ed., pp. 346-349
48. Bird et al., Transport Phenomena, 2nd ed., p. 440.
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52. The Fourier number in heat transfer was introduced in Section 4.3.
53. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd ed., Wiley, Hoboken, NJ, 2002, pp. 704-706.
54. Bird et al., Transport Phenomena, 2nd ed., pp. 706-708.
55. Bird et al., Transport Phenomena, 2nd ed., pp. 545-549.
56. Mass rather than molar concentrations are used for polymer systems, due to their very high molecular weight relative to the diffusing solute, which would result in extremely small mole fractions.
57. A free-convection model for the avoidance phenomenon is developed in J. J. Pellegrino, R. L. Sani, and R. I. Gamow, J. Theor. Bio., 105(1), 77-90 (1983).
58. R. R. Bilodeau, R. J. Elgas, W. B. Krantz, and M. E. Voorhees, U.S. patent 5,626,759, issued May 6, 1997.
59. Bird et al., Transport Phenomena, 2nd ed., pp. 672-675.
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61. The dimensionless group corresponding to the Sherwood number defined here is sometimes referred to as the Nusselt number for mass transfer.
62. Note that Example Problem 5.E.1 considers scaling to determine when the bulk-flow effect can be neglected when an equation of state for the mass density is known.
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72. Anonymous-attributed to a very practical engineer.
73. R. R. Bilodeau, R. J. Elgas, W. B. Krantz, and M. E. Voorhees, U.S. patent 5,626,759, issued May 6, 1997.
74. The criterion for ignoring this term is considered in Practice Problem 7.P.1.
75. The criterion for ignoring transient flow effects is considered in Practice Problem 7.P.2.
76. The region of influence associated with an oscillating boundary was discussed in Section 3.5.
77. An O(1) scaling analysis to obtain this result is considered in Practice Problem 7.P.4.
78. R. L. Jones and G. E. Keller, Sep. Process Technol., 2, 17 (1981).
79. E. M. Kopaygorodsky, V. V. Guliants, and W. B. Krantz, A.I.Ch.E. Jl., 50(5), 953 (2004).
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83. D. Li, A.R. Greenberg, W.B. Krantz, and R.L. Sani J. Membrane Sci., 279, 50 (2006).
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90. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd ed., Wiley, Hoboken, NJ, 2002, p. 526.