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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.

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 [

It is important to note that all molecules will eventually leave the system (this is also a method used to normalize the distribution) [

Generally, the main model used to describe the residence-time distribution of a system has been the one-parameter advection-dispersion equation or model (AdDE) [

This jointly combined four-parameter gamma model allows for more flexibility to account for the nonlinear aspects [

Regarding non-ideal flow systems, we are assuming the system is isothermal and homogeneous and that the volume changes during the tracer study are assumed to be negligible [

We are assuming that the residence time of tracer particles is similar to travel times of discrete water molecules in a non-ideal flow system along flow paths. The flow paths for discrete water particles vary in length, local hydraulic gradient, and cross-section. Tracer sample concentration as a measure of the tracer flux at a given time is randomly distributed, but the approach developed in this paper does not apply a residence-time distribution directly to the concentration data. Instead, the arrival of molecules at the sampling point at a particular time is seen as a random event dependent on the distance traveled and speed of travel. Thus, the relation between travel distance and velocity reflected in the space time (

For modeling non-ideal flow systems, addressing the interaction of

The literature suggests that the gamma distribution does well in describing tracer breakthrough curves for non-ideal flow systems [

The following mathematical discussion and (

Assistance in deriving the intermediate steps between equations (

The one-parameter advection-dispersion equation RTD model is obtained from the dimensionless effluent tracer concentration in (

The Péclet number (Pe) in (

A steady-state, non-ideal reactor of glass tubes was set up in the laboratory to simulate a plug-flow reactor (PFR). The glass tubing was borosilicate glass with an inner diameter of 0.4 cm. The flow path model consisted of 1.4 m straight segments of glass tubing. The straight segments of glass tubing were connected by 180° elbows made of Teflon tubing. The radius of each elbow was 0.079 m and the inner diameter of the Teflon tubing was 0.4 cm. The linear length of the system was 32 m. The system was calibrated such that flow rate in the system was maintained at 2.0-mL/min. The injection mechanism to introduce the conservative tracer was a syringe delivering a volume of 5 mL for each trial. Two trials were conducted using 10 ppm of the tracer dye rhodamine WT-20 and 10 ppm Zn, zinc chloride (ZnCl2). Discharge samples of the simulated PFR were collected at 20-minute intervals and analyzed using fluorometry and inductively coupled plasma optical emission spectrometry (ICP-OES) for rhodamine and zinc chloride, respectively. The rhodamine WT-20 tracer data was applied to the gamma and AdDE RTD models.

The results of the tracer study were used to develop the residence-time distribution (RTD) function. The RTD function (

Equation (

The normalized forms of the RTD for the tracer, the AdDE model, and the gamma model were computed in LibreOffice Calc. In order to determine the better RTD model, either the AdDE or the gamma, we had to calculate the mean-absolute deviation (MAD) [

The MAD associated with the gamma RTD model was approximately 0.16 while the MAD for the AdDE RTD model was approximately 0.26.

We used the DEPS algorithm to determine the four parameters (

Tracer concentration versus time for the laboratory experiment. These data were used to determine the mean, variance, and Péclet number for the plug-flow reactor.

Comparing the normalized forms of the advection-dispersion equation RTD and the gamma RTD models to the normalized tracer RTD model for the laboratory tracer study.

Figure

The results for the laboratory plug-flow reactor rhodamine dye study conducted are as follows The mean residence time (^{2}) is ^{2} which is from (

To compare the 3 RTD models to each other (gamma from (

Figure

The results for the normalized gamma RTD model’s interpretation of the laboratory plug-flow reactor rhodamine dye study conducted using

The dimensionless mean residence time or mean time in the system (^{2} which is from (

In an ideal plug-flow reactor (PFR) the apparent reactor velocity should be strongly correlated to the velocity of the peak of the tracer curve. This strong correlation is based on the shape of the velocity profile. In our study, the apparent reactor velocity (volumetric flow rate/area of the tube) is

The normalized form of the gamma RTD function had a better fit with the tracer RTD function than the advection-dispersion equation RTD function. The mean-absolute deviation (MAD) from the normalized tracer RTD function for the normalized gamma RTD function was

As previously discussed in Section

In addition, the normalized gamma RTD function allows for the calculation of the mean travel linear velocity and the mean travel distance which are obtained from the

For those reasons, we conclude that the jointly combined four-parameter gamma distribution RTD function better models the non-ideal flow present in the laboratory plug-flow reactor than the one-parameter advection-dispersion equation RTD function. Thus, the gamma RTD function is a suitable alternative to the advection-dispersion equation RTD function for quantitative tracer studies of other non-ideal flow systems.

Travel distance

Velocity

Time

Mean residence time

Distribution variance

Dimensionless time

Concentration over time

Concentration at time

Concentration at time

Reactor or system volume

Equivalent volume

Equivalent area

Equivalent

Volumetric flow rate

Space time

Independent random variable

Residence-time distribution

Residence-time distribution function

Transit time distribution

Detention time distribution

Hydraulic residence-time distribution

Advection-dispersion equation

Axial dispersion equation

Axial dispersion model

Convection dispersion equation

Probability density function

Plug-flow reactor

Péclet number

Mean-absolute deviation

Differential evolution and particle swarm optimization algorithm.

We will compare the four-parameter gamma RTD function to the advection-dispersion equation RTD function for a quantitative dye study that was performed at Mammoth Cave National Park, Ky, USA.

The authors would like to acknowledge financial support from the United States Department of Education Title 3; United States Department of Energy (DOE) National Nuclear Security Administration (NNSA); United States Geological Survey (Tennessee Water Science Center); and Tennessee State University (TSU) College of Engineering and the Department of Civil and Environmental Engineering. The authors also would like to acknowledge the assistance provided by other software not cited in this paper: LibreOffice Writer for their document processing; the JabRef reference manager (