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
A new residence-time distribution (RTD) function has been developed and applied to quantitative dye studies as an alternative to the traditional advection-dispersion equation (AdDE). The new method is based on a jointly combined four-parameter gamma probability density function (PDF). The gamma residence-time distribution (RTD) function and its first and second moments are derived from the individual two-parameter gamma distributions of randomly distributed variables, tracer travel distance, and linear velocity, which are based on their relationship with time. The gamma RTD function was used on a steady-state, nonideal system modeled as a plug-flow reactor (PFR) in the laboratory to validate the effectiveness of the model. The normalized forms of the gamma RTD and the advection-dispersion equation RTD were compared with the normalized tracer RTD. The normalized gamma RTD had a lower mean-absolute deviation (MAD) (0.16) than the normalized form of the advection-dispersion equation (0.26) when compared to the normalized tracer RTD. The gamma RTD function is tied back to the actual physical site due to its randomly distributed variables. The results validate using the gamma RTD as a suitable alternative to the advection-dispersion equation for quantitative tracer studies of non-ideal flow systems. 1. Introduction Researchers have used the distribution of residence times to examine the characteristics of a nonideal flow reactor or system. The residence-time distribution (RTD) was first proposed to analyze chemical reactor performance in a paper by MacMullin and Weber in 1935 [1–3]. Only after Danckwerts’ publication of “Continuous flow systems. Distribution of residence times,” in 1953, was the RTD theory organized in a more structured manner and most of the distributions were classified [2–5]. Many people still use Danckwerts’ work as their foundation for analysis of systems with the RTD model. The residence-time distribution of a system characterizes the mixing that happens in a system. The residence-time distribution function is quantified by the term . describes quantitatively the amount of time that different fluid particles have spent in the system. is also a probability density function (PDF) that defines the probability that a particle entering the system will remain there for a time (see [1–8] for a thorough explanation of the background theory to mixing and RTD). Equation (1) is generally used to determine the RTD function [2, 7] as where is the concentration of the tracer over time and the plot of concentration versus time is the tracer
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.
[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.
[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.
[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.
[10]
J. ？ermáková, 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[35]
V. Zahraeifard and Z. Deng, “Hydraulic residence time computation for constructed wetland design,” Ecological Engineering, vol. 37, no. 12, pp. 2087–2091, 2011.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[48]
S. Yue, “A bivariate gamma distribution for use in multivariate flood frequency analysis,” Hydrological Processes, vol. 15, no. 6, pp. 1033–1045, 2001.
[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.
[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.
[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.
[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.
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.
[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.
