Determination of sorption isotherm based on the measurements of contaminant outflow

Figures & data

Figure 1. Sketch of the initial state of the experiment.

Figure 2. Some basis functions of Bessel splines for $M=100$.

Figure 3. Two different functions $\mathit{\psi }$ that in our scenario produced the equal outflow contaminant flux.

Figure 4. Red (dot): initial linear iteration ${\mathit{\psi }}_0$; black (dash-dot): Freundlich isotherm (56(56) $$\begin{array}{c}\hfill \mathit{\psi }(w)=w^{0,75},\end{array}$$(56) ); blue (solid): final iteration ${\mathit{\psi }}_{165}$.

Figure 5. Red (dot): initial linear iteration ${\mathit{\psi }}_0$; black (dash): Langmuir isotherm (57(57) $$\begin{array}{c}\hfill \mathit{\psi }(w)=\frac{2w}{w+1}.\end{array}$$(57) ); blue (solid): final iteration ${\mathit{\psi }}_{270}$.

Figure 6. (a) Red (dot): initial linear iteration ${\mathit{\psi }}_0$; black (dash): Freundlich isotherm (56(56) $$\begin{array}{c}\hfill \mathit{\psi }(w)=w^{0,75},\end{array}$$(56) ); blue (solid): final iteration ${\mathit{\psi }}_{177}$. (b) Red (dot): $q(d,t)w(d,t)$ corresponding to ${\mathit{\psi }}_0$; black (dash): $q(d,t)w(d,t)$ corresponding to Freundlich isotherm (56(56) $$\begin{array}{c}\hfill \mathit{\psi }(w)=w^{0,75},\end{array}$$(56) ); blue (solid): both $q(d,t)w(d,t)$ corresponding to final iteration ${\mathit{\psi }}_{177}$ and $\tilde{m}$ obtained by running simulation using Freundlich isotherm (56(56) $$\begin{array}{c}\hfill \mathit{\psi }(w)=w^{0,75},\end{array}$$(56) ) and subsequently altered by the perturbation (58(58) $$\begin{array}{c}\hfill \tilde{m}=m(1+c_{1}sin(ln(t+1))+c_{2}cos(ln(t+1))).\end{array}$$(58) ).

Figure 7. (a) Red (dot): initial linear iteration ${\mathit{\psi }}_0$; black(dash): Langmuir isotherm (57(57) $$\begin{array}{c}\hfill \mathit{\psi }(w)=\frac{2w}{w+1}.\end{array}$$(57) ); blue (solid): final iteration ${\mathit{\psi }}_{319}$. (b) Red (dot): $q(d,t)w(d,t)$ corresponding to ${\mathit{\psi }}_0$; black (dash): $q(d,t)w(d,t)$ corresponding to Langmuir isotherm (57(57) $$\begin{array}{c}\hfill \mathit{\psi }(w)=\frac{2w}{w+1}.\end{array}$$(57) ); blue (solid): both $q(d,t)w(d,t)$ corresponding to final iteration ${\mathit{\psi }}_{319}$ and $\tilde{m}$ obtained by running simulation using Langmuir isotherm (57(57) $$\begin{array}{c}\hfill \mathit{\psi }(w)=\frac{2w}{w+1}.\end{array}$$(57) ) and subsequently altered by the perturbation (58(58) $$\begin{array}{c}\hfill \tilde{m}=m(1+c_{1}sin(ln(t+1))+c_{2}cos(ln(t+1))).\end{array}$$(58) ).

Figure 8. (a) Red (dot): initial linear iteration ${\mathit{\psi }}_0$; black (dash): Freundlich isotherm (56(56) $$\begin{array}{c}\hfill \mathit{\psi }(w)=w^{0,75},\end{array}$$(56) ); blue (solid): final iteration ${\mathit{\psi }}_{96}$. (b) Red (dot): $q(d,t)w(d,t)$ corresponding to ${\mathit{\psi }}_0$; black (dash): $q(d,t)w(d,t)$ corresponding to Freundlich isotherm (56(56) $$\begin{array}{c}\hfill \mathit{\psi }(w)=w^{0,75},\end{array}$$(56) ); blue (solid): $q(d,t)w(d,t)$ corresponding to final iteration ${\mathit{\psi }}_{96}$; green (dash-dot): $\tilde{m}$ obtained by running simulation using Freundlich isotherm (56(56) $$\begin{array}{c}\hfill \mathit{\psi }(w)=w^{0,75},\end{array}$$(56) ) and subsequently altered by the random noise.