Table of Contents
ISRN Chemical Engineering
Volume 2013, Article ID 539209, 8 pages
http://dx.doi.org/10.1155/2013/539209
Research Article

Derivation of a Multiparameter Gamma Model for Analyzing the Residence-Time Distribution Function for Nonideal Flow Systems as an Alternative to the Advection-Dispersion Equation

1Department of Civil & Environmental Engineering, Tennessee State University (TSU), Nashville, TN 37209, USA
2United States Geological Survey (USGS), Nashville, TN 37211, USA
3Massie Chair of Excellence, Tennessee State University (TSU), Nashville, TN 37209, USA

Received 27 September 2012; Accepted 19 October 2012

Academic Editors: D. A. Drew and J. A. A. González

Copyright © 2013 Irucka Embry et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. B. MacMullin and M. Weber, “The theory of short-circuiting in continuous-flow mixing vessels in series and kinetics of chemical reactions in such systems,” Transactions of American Institute of Chemical Engineers, vol. 31, no. 2, pp. 409–458, 1935. View at Google Scholar
  2. H. S. Fogler, Elements of Chemical Reaction Engineering, Prentice Hall PRT, Upper Saddle River, NJ, USA, 3rd edition, 1999.
  3. O. Levenspiel, Chemical Reaction Engineering, John Wiley & Sons, New York, NY, USA, 3rd edition, 1999.
  4. P. V. Danckwerts, “Continuous flow systems. Distribution of residence times,” Chemical Engineering Science, vol. 2, no. 1, pp. 1–13, 1953. View at Publisher · View at Google Scholar
  5. R. Shinnar and P. Naor, “Residence time distributions in systems with internal reflux,” Chemical Engineering Science, vol. 22, no. 10, pp. 1369–1381, 1967. View at Publisher · View at Google Scholar · View at Scopus
  6. C. G. Hill Jr., An Introduction to Chemical Engineering Kinetics & Reactor Design, John Wiley & Sons, New York, NY, USA, 1977.
  7. G. Tchobanoglous, F. L. Burton, H. D. Stensel, and Metcalf & Eddy, Inc., Wastewater Engineering: Treatment and Reuse, McGraw-Hill, New York, NY, USA, 4th edition, 2003.
  8. L. D. Schmidt, The Engineering of Chemical Reactions, Oxford University Press, New York, NY, USA, 1998.
  9. V. K. Pareek, R. Sharma, C. Cooper, and A. Adesina, “Solids residence time distribution in a three-phase bubble column reactor: an artificial neural network analysis,” The Open Chemical Engineering Journal, vol. 2, no. 1, pp. 73–78, 2008. View at Publisher · View at Google Scholar
  10. J. Čermáková, F. Scargiali, N. Siyakatshana, V. Kudrna, A. Brucato, and V. MacHoň, “Axial dispersion model for solid flow in liquid suspension in system of two mixers in total recycle,” Chemical Engineering Journal, vol. 117, no. 2, pp. 101–107, 2006. View at Publisher · View at Google Scholar · View at Scopus
  11. C. Laquerbe, J. C. Laborde, S. Soares et al., “Computer aided synthesis of RTD models to simulate the air flow distribution in ventilated rooms,” Chemical Engineering Science, vol. 56, no. 20, pp. 5727–5738, 2001. View at Publisher · View at Google Scholar · View at Scopus
  12. S. J. Royaee and M. Sohrabi, “Comprehensive study on wastewater treatment using photo-impinging streams reactor: residence time distribution and reactor modeling,” Industrial & Engineering Chemistry Research, vol. 51, no. 11, pp. 4152–4160, 2012. View at Publisher · View at Google Scholar
  13. J. C. Williams and M. A. Rahman, “The continuous mixing of particulate solids,” Journal of the Society of Cosmetic Chemists, vol. 21, no. 1, pp. 3–36, 1970, http://journal.scconline.org/abstracts/cc1970/cc021n01/p00003-p00036.html. View at Google Scholar
  14. C. G. C. C. Gutierrez, E. F. T. S. Dias, and J. A. W. Gut, “Investigation of the residence time distribution in a plate heat exchanger with series and parallel arrangements using a non-ideal tracer detection technique,” Applied Thermal Engineering, vol. 31, no. 10, pp. 1725–1733, 2011. View at Publisher · View at Google Scholar · View at Scopus
  15. C. G. C. C. Gutierrez, E. F. T. S. Dias, and J. A. W. Gut, “Residence time distribution in holding tubes using generalized convection model and numerical convolution for non-ideal tracer detection,” Journal of Food Engineering, vol. 98, no. 2, pp. 248–256, 2010. View at Publisher · View at Google Scholar · View at Scopus
  16. A. P. Torres and F. A. R. Oliveira, “Residence time distribution studies in continuous thermal processing of liquid foods: a review,” Journal of Food Engineering, vol. 36, no. 1, pp. 1–30, 1998. View at Publisher · View at Google Scholar
  17. M. E. Rodrigues, A. R. Costa, M. Henriques, J. Azeredo, and R. Oliveira, “Wave characterization for mammalian cell culture: residence time distribution,” New Biotechnology, vol. 29, no. 3, pp. 402–408, 2012. View at Publisher · View at Google Scholar
  18. J. B. Bassingthwaighte, “Physiology and theory of tracer washout techniques for the estimation of myocardial blood flow: flow estimation from tracer washout,” Progress in Cardiovascular Diseases, vol. 20, no. 3, pp. 165–189, 1977. View at Publisher · View at Google Scholar · View at Scopus
  19. J. N. Carleton, Modeling approaches for treatment Wetlands [Dissertation], University of Maryland, College Park, Md, USA, 2009, http://hdl.handle.net/1903/9585.
  20. J. N. Carleton and H. J. Montas, “A modeling approach for mixing and reaction in wetlands with continuously varying flow,” Ecological Engineering, vol. 29, no. 1, pp. 33–44, 2007. View at Publisher · View at Google Scholar
  21. J. N. Carleton and H. J. Montas, “An analysis of performance models for free water surface wetlands,” Water Research, vol. 44, no. 12, pp. 3595–3606, 2010. View at Publisher · View at Google Scholar · View at Scopus
  22. Z. Q. Deng, H. S. Jung, and B. Ghimire, “Effect of channel size on solute residence time distributions in rivers,” Advances in Water Resources, vol. 33, no. 9, pp. 1118–1127, 2010. View at Publisher · View at Google Scholar
  23. R. A. Payn, M. N. Gooseff, D. A. Benson et al., “Comparison of instantaneous and constant-rate stream tracer experiments through non-parametric analysis of residence time distributions,” Water Resources Research, vol. 44, no. 6, Article ID W06404, 10 pages, 2008. View at Publisher · View at Google Scholar
  24. R. Painter, T. Byl, L. Sharpe, V. Watson, and T. Patterson, “A residence time distribution approach to biodegradation in fuel impacted karst aquifers,” Journal of Civil & Environmental Engineering, vol. 2, no. 5, article 121, 2012. View at Publisher · View at Google Scholar
  25. R. Painter, T. Byl, L. Sharpe, A. Kheder, and J. Harris, “The role of attached and free-living bacteria in biodegradation in karst aquifers,” Water, vol. 3, pp. 1139–1148, 2011. View at Publisher · View at Google Scholar
  26. T. D. Byl and R. Painter, “Microbial adaptations to karst aquifers with contaminants,” in Proceedings of the 19th Tennessee American Water Resources Association (AWRA), Tennessee Water Resources Symposium, pp. 2C-9–2C12, Burns, Tenn, USA, 2009.
  27. M. Martin and R. Painter, “Use of independent gamma distribution to describe tracer break-through curves,” in Proceedings of the 19th Tennessee American Water Resources Association (AWRA), Tennessee Water Resources Symposium, pp. P-10–PP10, Burns, Tenn, USA, 2009.
  28. I. Embry, V. Roland, R. Painter, R. Toomey, and L. Sharpe, “Quantitative dye tracing—development of a new interpretative method,” in Proceedings of the 22nd Tennessee American Water Resources Association (AWRA), Tennessee Water Resources Symposium, pp. 1C-6–1C-16, Burns, Tenn, USA, 2012.
  29. R. H. Kadlec, “Effects of pollutant speciation in treatment wetlands design,” Ecological Engineering, vol. 20, no. 1, pp. 1–16. View at Publisher · View at Google Scholar
  30. M. Hrachowitz, C. Soulsby, D. Tetzlaff, I. A. Malcolm, and G. Schoups, “Gamma distribution models for transit time estimation in catchments: physical interpretation of parameters and implications for time-variant transit time assessment,” Water Resources Research, vol. 46, no. 10, Article ID W10536, 15 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus
  31. J. W. Kirchner, X. Feng, and C. Neal, “Catchment-scale advection and dispersion as a mechanism for fractal scaling in stream tracer concentrations,” Journal of Hydrology, vol. 254, no. 1–4, pp. 82–101, 2001. View at Publisher · View at Google Scholar · View at Scopus
  32. J. W. Kirchner, D. Tetzlaff, and C. Soulsby, “Comparing chloride and water isotopes as hydrological tracers in two Scottish catchments,” Hydrological Processes, vol. 24, no. 12, pp. 1631–1645, 2010. View at Publisher · View at Google Scholar · View at Scopus
  33. J. W. Kirchner, X. Feng, and C. Neal, “Fractal stream chemistry and its implications for contaminant transport in catchments,” Nature, vol. 403, no. 6769, pp. 524–527, 2000. View at Publisher · View at Google Scholar
  34. A. Brovelli, O. Carranza-Diaz, L. Rossi, and D. A. Barry, “Design methodology accounting for the effects of porous medium heterogeneity on hydraulic residence time and biodegradation in horizontal subsurface flow constructed wetlands,” Ecological Engineering, vol. 37, no. 5, pp. 758–770, 2011. View at Publisher · View at Google Scholar · View at Scopus
  35. V. Zahraeifard and Z. Deng, “Hydraulic residence time computation for constructed wetland design,” Ecological Engineering, vol. 37, no. 12, pp. 2087–2091, 2011. View at Publisher · View at Google Scholar
  36. M. S. Field, “The QTRACER2 program for tracer break-through curve analysis for tracer tests in karst aquifers and other hydrologic systems,” U. S. Environmental Protection Agency, Office of Research and Development, EPA/600/R-02/001, 2002, http://oaspub.epa.gov/eims/eimscomm.getfile?p_download_id=36351.
  37. M. S. Field, “Efficient hydrologic tracer-test design for tracer-mass estimation and sample-collection frequency, 1. Method development,” Environmental Geology, vol. 42, no. 7, pp. 827–838, 2002. View at Publisher · View at Google Scholar · View at Scopus
  38. B. Zhou, Y. Jiang, Q. Wang, and M. Shao, “Chloride transport in undisturbed soil columns of the Loess Plateau,” African Journal of Agricultural Research, vol. 6, no. 20, pp. 4807–4815, 2011. View at Publisher · View at Google Scholar
  39. M. Vanclooster, D. Mallants, J. Diels, and J. Feyen, “Determining local-scale solute transport parameters using time domain reflectometry (TDR),” Journal of Hydrology, vol. 148, no. 1–4, pp. 93–107, 1993. View at Publisher · View at Google Scholar · View at Scopus
  40. R. D. Markovic, “Probability functions of best fit to distributions of annual precipitation and runoff,” Hydrology Papers no. 8, Colorado State University, Fort Collins, Colo, USA, 1965. View at Google Scholar
  41. I. Kotlarski, “On characterizing the gamma and the normal distribution,” Pacific Journal of Mathematics, vol. 20, no. 1, pp. 69–76, 1967, http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102992970. View at Google Scholar
  42. A. Stuart and K. Ord, Kendall’s Advanced Theory of Statistics: Volume 1: Distribution Theory, Oxford University Press, New York, NY, USA, 6th edition, 1994.
  43. K. V. Bury, Statistical Distributions in Engineering, Cambridge University Press, New York, NY, USA, 1999.
  44. N. T. Kottegoda and R. Rosso, Statistics, Probability, and Reliability for Civil and Environmental Engineers, McGraw-Hill, New York, NY, USA, 1997.
  45. B. Bobée and F. Ashkar, The Gamma Family and Derived Distributions Applied in Hydrology, Water Resources Publications, Littleton, Colo, USA, 1991.
  46. H. A. Loáiciga, “Residence time, groundwater age, and solute output in steady-state groundwater systems,” Advances in Water Resources, vol. 27, no. 7, pp. 681–688, 2004. View at Publisher · View at Google Scholar · View at Scopus
  47. S. K. Singh, “Simplified use of gamma-distribution/nash model for runoff modeling,” Journal of Hydrologic Engineering, vol. 9, no. 3, 240243 pages, 2004. View at Publisher · View at Google Scholar
  48. S. Yue, “A bivariate gamma distribution for use in multivariate flood frequency analysis,” Hydrological Processes, vol. 15, no. 6, pp. 1033–1045, 2001. View at Publisher · View at Google Scholar · View at Scopus
  49. H. Aksoy, “Use of gamma distribution in hydrological analysis,” Turkish Journal of Engineering and Environmental Sciences, vol. 24, no. 6, pp. 419–428, 2000. View at Google Scholar
  50. I. E. Amin and M. E. Campana, “A general lumped parameter model for the interpretation of tracer data and transit time calculation in hydrologic systems,” Journal of Hydrology, vol. 179, no. 1-4, pp. 1–21, 1996. View at Google Scholar · View at Scopus
  51. P. K. Bhunya, R. Berndtsson, C. S. P. Ojha, and S. K. Mishra, “Suitability of gamma, chi-square, weibull, and beta distributions as synthetic unit hydrographs,” Journal of Hydrology, vol. 334, no. 1-2, pp. 28–38, 2007. View at Publisher · View at Google Scholar · View at Scopus
  52. M. D. Springer, The Algebra of Random Variables, John Wiley & Sons, New York, NY, USA, 1979.
  53. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, McGraw-Hill, New York, NY, USA, 2nd edition, 1968.
  54. S. M. Selby, CRC Standard Mathematical Tables, The Chemical Rubber, Cleveland, Ohio, USA, 18th edition, 1970.
  55. A. Jeffrey, Handbook of Mathematical Formulas and Integrals, Elsevier/Academic Press, New York, NY, USA, 3rd edition, 2004.
  56. D. Zwillinger, S. G. Krantz, and K. H. Rosen, Eds., CRC Standard Mathematical Tables and Formulae, CRC Press, New York, NY, USA, 30th edition, 1995.
  57. “LibreOffice Calc (Version 3.5.4.2) [computer software],” The Document Foundation, 2012, http://www.libreoffice.org/.
  58. “Solver for nonlinear programming/NLPSolver (Version 0.9-beta-1) [computer software],” Sun Microsystems, Inc., 2009, http://extensions.services.openoffice.org/en/project/NLPSolver.
  59. Apache OpenOffice.org, “NLPSolver,” 2012, http://wiki.services.openoffice.org/wiki/NLPSolver.
  60. C. J. Willmott, K. Matsuura, and S. M. Robeson, “Ambiguities inherent in sums-of-squares-based error statistics,” Atmospheric Environment, vol. 43, no. 3, pp. 749–752, 2009. View at Publisher · View at Google Scholar · View at Scopus
  61. S. C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, McGraw-Hill, New York, NY, USA, 2nd edition, 2008.
  62. J. W. Eaton et al., GNU Octave (Version 3.6.1) [computer software], 2012, http://www.octave.org.
  63. B. Spitzak et al., FLTK (Version 1.3) [computer software], 2011, http://www.fltk.org/index.php.
  64. T. Williams, C. Kelley et al., gnuplot (Version 4.4.4) [computer software], 2011, http://www.gnuplot.info.