Introduction
Methods
Column experiment
Mathematical models
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Linear Henry model (H model)where KH is the Henry distribution coefficient [L3 M−1]$$ s={K}_{\mathrm{H}}C $$(2)
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Nonlinear Freundlich model (F model)where KF is the Freundlich sorption coefficient [L3 M−1] and nF is the Freundlich sorption exponent [−].$$ s={K}_{\mathrm{F}}{C}^{n_{\mathrm{F}}} $$(3)
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Nonlinear Langmuir model (L model)where αL is the Langmuir constant [L3 M−1] and βL is the total sorption capacity of the solid phase [M M−1].$$ s=\frac{\alpha_{\mathrm{L}}{\beta}_{\mathrm{L}}C}{1+{\alpha}_{\mathrm{L}}C} $$(4)
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Linear irreversible sorption model (I model)where k1 is the irreversible sorption rate coefficient [L3 M−1 T−1]$$ \frac{\partial s}{\partial t}={k}_1C $$(5)
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Linear reversible sorption model (R model)where k2 is the first reversible sorption rate coefficient [L3 M−1 T−1] and k3 is the second reversible sorption rate coefficient [T−1].$$ \frac{\partial s}{\partial t}={k}_2C-{k}_3s $$(6)
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Henry and linear irreversible sorption model (H-I model)$$ \Big\{{\displaystyle \begin{array}{c}{s}_{\mathrm{e}}={K}_{\mathrm{H}}C\\ {}\frac{\partial {s}_{\mathrm{n}}}{\partial t}={k}_1C\end{array}} $$(8)
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Freundlich and linear irreversible sorption model (F-I model)$$ \Big\{{\displaystyle \begin{array}{c}{s}_{\mathrm{e}}={K}_{\mathrm{F}}{C}^{n_{\mathrm{F}}}\\ {}\frac{\partial {s}_{\mathrm{n}}}{\partial t}={k}_1C\end{array}} $$(9)
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Langmuir and linear irreversible sorption model (L-I model)$$ \Big\{{\displaystyle \begin{array}{c}{s}_{\mathrm{e}}=\frac{\alpha_{\mathrm{L}}{\beta}_{\mathrm{L}}C}{1+{\alpha}_{\mathrm{L}}C}\\ {}\frac{\partial {s}_{\mathrm{n}}}{\partial t}={k}_1C\end{array}} $$(10)
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Henry and linear reversible sorption model (H-R model)$$ \Big\{{\displaystyle \begin{array}{c}{s}_{\mathrm{e}}={K}_{\mathrm{H}}C\\ {}\frac{\partial {s}_{\mathrm{n}}}{\partial t}={k}_2C-{k}_3{s}_{\mathrm{n}}\end{array}} $$(11)
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Freundlich and linear reversible sorption model (F-R model)$$ \Big\{{\displaystyle \begin{array}{c}{s}_{\mathrm{e}}={K}_{\mathrm{F}}{C}^{n_{\mathrm{F}}}\\ {}\frac{\partial {s}_{\mathrm{n}}}{\partial t}={k}_2C-{k}_3{s}_{\mathrm{n}}\end{array}} $$(12)
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Langmuir and linear reversible sorption model (L-R model)$$ \Big\{{\displaystyle \begin{array}{c}{s}_{\mathrm{e}}=\frac{\alpha_{\mathrm{L}}{\beta}_{\mathrm{L}}C}{1+{\alpha}_{\mathrm{L}}C}\\ {}\frac{\partial {s}_{\mathrm{n}}}{\partial t}={k}_2C-{k}_3{s}_{\mathrm{n}}\end{array}} $$(13)
Descriptors
Transport parameters | Symbol | Unit | Parameter’s value | References | ||
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Lower | Basic | Higher | ||||
Filtration parameters | ||||||
Hydraulic conductivity | k | m s−1 | 5.0 × 10−5 | 1.0 × 10−4 | 5.0 × 10−4 | Appelo and Postma 1999 |
Effective porosity | ne | – | 0.27 | 0.30 | 0.33 | Domenico and Schwartz 1998 |
Diffusion and dispersion parameters | ||||||
Diffusion coefficient | DM | m2 s−1 | 1.0 × 10−9 | 5.0 × 10−9 | 1.0 × 10−8 | Maraqa 2007 |
Longitudinal dispersivity | α | m | 0.008 | 0.010 | 0.020 | |
Sorption parameters | ||||||
Henry distribution coefficient | KH | dm3 kg−1 | 0.005 | 0.010 | 0.020 | Fohrmann et al. 2001 Maraqa and Khashan 2014 |
Freundlich sorption coefficient | KF | dm3 kg−1 | 0.005 | 0.010 | 0.020 | Khan et al. 2010 |
Freundlich sorption exponent | nF | – | 0.6 | 0.8 | 1.0 | Dubus et al. 2003 Maraqa 2007 |
Langmuir constant | αL | dm3 mg−1 | 0.04 | 0.07 | 0.10 | Barrow 1978 |
Total sorption capacity of the solid phase | βL | mg kg−1 | 1.0 | 1.2 | 1.4 | Ahmed et al. 2011 |
Irreversible sorption rate coefficient | k1 | dm3 kg−1 s−1 | 1.0 × 10−6 | 5.0 × 10−6 | 1.0 × 10−5 | Wehehan et al. 2007 |
First reversible sorption rate coefficient | k2 | dm3 kg−1 s−1 | 1.0 × 10−6 | 5.0 × 10−6 | 1.0 × 10−5 | Liu et al. 1991 |
Second reversible sorption rate coefficient | k3 | s−1 | 1.0 × 10−4 | 2.0 × 10−4 | 4.0 × 10−4 | Wehrhan et al. 2007 Liu et al. 1991 |
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Relative tracer recovery ε [%]:where time t1 was determined from the breakthrough curve C(t) as the last moment at which C(t1) < 0.01·C0, whereas time t2 was determined from the breakthrough curve C(t) as the first moment at which C(t2) < 0.01·C0$$ \varepsilon =\frac{100}{C_0{t}_{\mathrm{in}}}\underset{t_1}{\overset{t_2}{\int }}C(t) dt $$(17)
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Time of maximum tracer concentration at the output tmax [min]:where Cmax is the maximum tracer concentration at the output$$ {\left.{t}_{\mathrm{max}}=t\right|}_{C={C}_{\mathrm{max}}} $$(18)
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Spread of breakthrough curve σ [min]:$$ \sigma =\sqrt{\frac{\int {\left(t-{t}_{\mathrm{max}}\right)}^2C(t) dt}{\int C(t) dt}} $$(19)
Results
Algorithm of sorption model preselection
Example of column experiment and algorithm application
Tracer | ε [%] | tmax [min] | σ [min] | δε [%] | δtmax [%] | δσ [%] | Model |
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Cl | 99.7 | 346.0 | 45.7 | 0 | 0 | 0 | A-D |
Li | 100.2 | 391.1 | 51.9 | 1 | 13 | 14 | H or F |
Cu | 22.3 | 391.1 | 106.8 | 78 | 13 | 134 | I or H-I or F-I |
Zn | 32.4 | 391.1 | 163.7 | 67 | 13 | 258 | I or H-I or F-I |
Model | Parameter | Chloride ions | |||
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Symbol | Unit | Value | RMSE | r | |
A-D | k | m s−1 | 4.07 × 10−4 | 0.241 | 1.00 |
ne | – | 0.31 | |||
DM | m2 s−1 | 0 | |||
α | m | 0.0027 |
Model | Symbol | Unit | Li | Cu | Zn | ||||||
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Value | RMSE [mg dm−3] | r [−] | Value | RMSE [mg dm−3] | r [−] | Value | RMSE [mg dm−3] | r [−] | |||
H | KH | dm3 kg−1 | 2.16 × 10−2 | 0.0668 | 0.990 | 3.40 × 10−2 | 0.1729 | 0.926 | 3.58 × 10−2 | 0.1467 | 0.927 |
F | KF | dm3 kg−1 | 4.23 × 10–3* | 0.0347* | 0.998* | 2.52 × 10−2 | 0.1730 | 0.937 | 2.85 × 10−2 | 0.1470 | 0.927 |
nF | – | 1.80* | 1.20 | 1.18 | |||||||
L | αL | dm3 mg−1 | 2.16 × 10−4 | 0.0669 | 0.990 | 4.63 × 10−1 | 0.1627 | 0.863 | 1.00 × 100 | 0.1173 | 0.831 |
βL | mg kg−1 | 99.90 | 2.28 × 10−1 | 2.06 × 10−1 | |||||||
I | k1 | dm3 kg−1 s−1 | 1.35 × 10−6 | 0.2733 | 0.830 | 1.85 × 10−5 | 0.0318 | 0.663 | 1.67 × 10−5 | 0.0434 | 0.567 |
R | k2 | dm3 kg−1 s−1 | 1.56 × 10−4 | 0.0647 | 0.991 | 2.08 × 10−5 | 0.0286 | 0.754 | 2.15 × 10−5 | 0.0350 | 0.753 |
k3 | s−1 | 7.25 × 10−3 | 3.10 × 10−5 | 7.00 × 10−5 | |||||||
H-I | KH | dm3 kg−1 | 2.16 × 10−2 | 0.0668 | 0.990 | 2.82 × 10−2 | 0.0119 | 0.958 | 3.22 × 10−2 | 0.0171 | 0.943 |
k1 | dm3 kg−1 s−1 | 2.33 × 10−14 | 1.58 × 10−5 | 1.30 × 10−5 | |||||||
F-I | KF | dm3 kg−1 | 4.23 × 10–3* | 0.0347* | 0.998* | 2.16 × 10–2* | 0.0111* | 0.964* | 2.85 × 10–2* | 0.0169* | 0.944* |
nF | – | 1.80* | 1.47* | 1.18* | |||||||
k1 | dm3 kg−1 s−1 | 2.22 × 10–14* | 1.57 × 10–5* | 1.30 × 10–5* | |||||||
L-I | αL | dm3 mg−1 | 2.22 × 10−4 | 0.0669 | 0.990 | 3.17 × 10−4 | 0.0119 | 0.958 | 3.44 × 10−4 | 0.0171 | 0.943 |
βL | mg kg−1 | 97.29 | 89.09 | 93.67 | |||||||
k1 | dm3 kg−1 s−1 | 7.38 × 10−13 | 1.58 × 10−5 | 1.30 × 10−5 | |||||||
H-R | KH | dm3 kg−1 | 1.81 × 10−2 | 0.0649 | 0.991 | 2.82 × 10−2 | 0.0119 | 0.958 | 3.22 × 10−2 | 0.0171 | 0.943 |
k2 | dm3 kg−1 s−1 | 1.47 × 10−4 | 1.58 × 10−5 | 1.30 × 10−5 | |||||||
k3 | s−1 | 4.22 × 10−2 | 2.22 × 10−14 | 4.10 × 10−8 | |||||||
F-R | KF | dm3 kg−1 | 4.23 × 10–3* | 0.0347* | 0.998* | 2.16 × 10–2* | 0.0111* | 0.964* | 2.86 × 10–2* | 0.0169* | 0.944* |
nF | – | 1.80* | 1.47* | 1.17* | |||||||
k2 | dm3 kg−1 s−1 | 2.22 × 10–14* | 1.57 × 10–5* | 1.30 × 10–5* | |||||||
k3 | s−1 | 1.00 × 10–4* | 2.10 × 10–7* | 2.39 × 10–7* | |||||||
L-R | αL | dm3 mg−1 | 1.04 × 10−4 | 0.0650 | 0.991 | 2.94 × 10−4 | 0.0119 | 0.958 | 3.44 × 10−4 | 0.0171 | 0.943 |
βL | mg kg−1 | 83.76 | 96.24 | 93.57 | |||||||
k2 | dm3 kg−1 s−1 | 6.28 × 10−5 | 1.58 × 10−5 | 1.30 × 10−5 | |||||||
k3 | s−1 | 4.88 × 10−3 | 4.37 × 10−14 | 3.40 × 10−8 |
Discussion
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For lithium (Li) ions the preselection algorithm proposed simple H and F models. The best fit between numerical and experimental breakthrough curves was obtained for the F, F-I and F-R models. The analysis of the identified parameter values shows that the differences between the breakthrough curves obtained from the F, F-I and F-R models are insignificant. In the hybrid models, the values of parameters k1, k2 and k3 are very low. This shows that equilibrium sorption is dominant in the process of lithium ion transport; therefore, the F model describes the process of lithium ions sorption with sufficient accuracy.
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For copper (Cu) ions and zinc (Zn) ions, the algorithm proposed a simple I model and hybrid H-I and F-I models. For these ions, both components of equilibrium and nonequilibrium sorption are significant. Goodness-of-fit indices, RMSE and r, prefer the F-I and F-R models. However, the dominance of irreversible sorption is indicated by descriptor ε, which suggests that the recovery rate of copper and zinc ions is 22.3 and 32.4%, respectively.