%0 Journal Article
%T 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 Irucka Embry
%A Victor Roland
%A Oluropo Agbaje
%A Valetta Watson
%A Marquan Martin
%A Roger Painter
%A Tom Byl
%A Lonnie Sharpe
%J ISRN Chemical Engineering
%D 2013
%R 10.1155/2013/539209
%X 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¨C3]. Only after DanckwertsˇŻ publication of ˇ°Continuous flow systems. Distribution of residence times,ˇ± in 1953, was the RTD theory organized in a more structured manner and most of the distributions were classified [2¨C5]. Many people still use DanckwertsˇŻ 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¨C8] 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
%U http://www.hindawi.com/journals/isrn.chemical.engineering/2013/539209/