Study of Dynamic Concentration Gradient on Mass Transfer Coefficient: New Approach to Mobile–Immobile Modeling
Publication: Journal of Hazardous, Toxic, and Radioactive Waste
Volume 24, Issue 4
Abstract
The theory of mobile–immobile partitioning to capture a medium’s heterogeneity for simulating the interaction of contaminant mass between these two regions is still limited to the lump value of mass transfer coefficient (MTC) that fails to capture the long tails of breakthrough curves (BTCs). For a time-dependent solute source, BTCs consists of two parts, for example, rising and falling limbs. During the rising part, the concentration in the mobile region is higher and mass transfer occurs from the mobile to immobile region. However, during falling limb concentration in the immobile region have higher values, resulting in the reverse diffusive mass transfer process. This study focuses on overcoming the reported limitations of the mobile–immobile model (MIM) in the prediction of long tails of BTC during the falling limb. To achieve this objective, we propose an approach that is based on the dynamics of time resident concentration and its gradient between hydraulically coupled mobile and immobile regions. In this modified MIM, we estimated two distinct diffusive MTCs for rising and falling limbs (RFMT) of BTCs using a nonlinear least square optimization algorithm. Two experimental data sets available in the literature were simulated using a numerical solution of the proposed model and asymptotic time-dependent dispersion function. The estimated parameters supported the hypothesis that for pulse type input, liquid phase transport during the rising limb of BTCs is governed by advection and dispersion, whereas during the falling limb it is majorly diffusive dominated that can be represented by the new MTCs. Simulated results of RFMT are then compared with continuous-time random walk (CTRW) and constant mass transfer (CMT) approaches to compare the quality of the simulation. A better overall simulation of experimental BTCs was obtained using RFMT in comparison with other models. Sensitivity analysis is also carried out to evaluate the capabilities of RFMT over the rising and falling portions of BTCs. This theory finds its application in quantifying persistent chemical residuals in the immobile region that acts as a source when purging and subsequently helps when designing appropriate cleansing operations.
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Data Availability Statement
Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request (Extracted graphical data, Numerical code, optimization algorithm).
Acknowledgments
We sincerely acknowledge the funding support from DST project number ECR/2017/000392.
Notation
The following symbols are used in this paper:
- C
- volume-averaged dissolved concentration in liquid phase;
- Dal
- Damkohler’s number;
- D(t)
- macrodispersion function of time;
- f
- fraction of sorption site available for mobile region;
- Kd
- sorption coefficient;
- L
- length of any location from input;
- q
- Darcy’s velocity;
- tmax
- maximum experimental duration;
- tp
- duration of pulse type input;
- vm
- seepage velocity or average pore water velocity;
- θ
- porosity of medium;
- μ
- first order sorption coefficient;
- ρ
- bulk density of porous media;
- ω(t)
- MTC as a function of time;
- ω1
- MTC for rising limb;
- ω2
- MTC for falling limb;
- Subscript “m”
- parameters of liquid mobile phase; and
- Subscript “im”
- parameters of liquid immobile phase.
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Journal of Hazardous, Toxic, and Radioactive Waste
Volume 24 • Issue 4 • October 2020
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© 2020 American Society of Civil Engineers.
History
Received: Oct 6, 2019
Accepted: Feb 7, 2020
Published online: Jun 4, 2020
Published in print: Oct 1, 2020
Discussion open until: Nov 4, 2020
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