## 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

## Abstract

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.

- Authors:

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

- Publication Date:

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1198485

- Resource Type:
- Published Article

- Journal Name:
- ISRN Chemical Engineering

- Additional Journal Information:
- Journal Name: ISRN Chemical Engineering Journal Volume: 2013; Journal ID: ISSN 2090-861X

- Publisher:
- Hindawi Publishing Corporation

- Country of Publication:
- Country unknown/Code not available

- Language:
- English

### Citation Formats

```
Embry, Irucka, Roland, Victor, Agbaje, Oluropo, Watson, Valetta, Martin, Marquan, Painter, Roger, Byl, Tom, and Sharpe, Lonnie. 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. Country unknown/Code not available: N. p., 2013.
Web. doi:10.1155/2013/539209.
```

```
Embry, Irucka, Roland, Victor, Agbaje, Oluropo, Watson, Valetta, Martin, Marquan, Painter, Roger, Byl, Tom, & Sharpe, Lonnie. 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. Country unknown/Code not available. doi:10.1155/2013/539209.
```

```
Embry, Irucka, Roland, Victor, Agbaje, Oluropo, Watson, Valetta, Martin, Marquan, Painter, Roger, Byl, Tom, and Sharpe, Lonnie. Tue .
"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". Country unknown/Code not available. doi:10.1155/2013/539209.
```

```
@article{osti_1198485,
```

title = {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},

author = {Embry, Irucka and Roland, Victor and Agbaje, Oluropo and Watson, Valetta and Martin, Marquan and Painter, Roger and Byl, Tom and Sharpe, Lonnie},

abstractNote = {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.},

doi = {10.1155/2013/539209},

journal = {ISRN Chemical Engineering},

number = ,

volume = 2013,

place = {Country unknown/Code not available},

year = {2013},

month = {1}

}

DOI: 10.1155/2013/539209