Maxwell–Stefan based modelling of ion exchange systems containing common species (Cd2+, Na+) and distinct sorbents (ETS-4, ETS-10)


Cadmium(II) is a toxic hazardous cation, whose presence in the environment causes great concern because of its bioaccumulation in organisms and bioamplification along food chain. Hence, the removal of cadmium compounds from industrial waters and wastewaters is particularly essential, which requires intensive experimental and modelling studies to deal with the problem. In this work, the ion exchange of Cd2+ ions from aqueous solution using microporous titanosilicates (ETS-4 and ETS-10) has been modelled using adapted Maxwell–Stefan equations for the ions transport inside the sorbent particles. The fundamentals of the Maxwell–Stefan equations along with correlations for the convective mass transfer coefficients have been used with advantage to reduce the number of model parameters. In the whole, the model was able to represent successfully the kinetic behaviour of 11 independent and very distinct curves of both studied systems (Cd2+/Na+/ETS-4 and Cd2+/Na+/ETS-10). The predictive capability of the model has been also shown, since several uptake curves were accurately predicted with parameters fitted previously to different sets of experimental data.

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Patrícia F. Lito wishes to thank Fundação para a Ciência e Tecnologia (Portugal) for the Grant (SFRH/BPD/63214/2009) provided for this study.

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Correspondence to C. M. Silva.



\(A\) :

External particle surface area (m)

\({\text{AARD}}\) :

Average absolute relative deviation

\(\left[ B \right]\) :

Matrix with MS diffusivities

\(C_{i}\) :

Molar concentration of \(i\) in bulk solution (mol/m3)

\(d_{\text{p}}\) :

Particle diameter (m)

\(D_{i}\) :

Self-diffusion coefficient of species \(i\) (m2/s)

\(D_{\text{Aw}}\) :

Diffusivity of the solute in solution (m2/s)

\(D_{ij}\) :

Interdiffusion coefficient of pair \(i{ - }j\) (m2/s)

Đ ij :

MS surface diffusivity of pair \(i{ - }j\) (m2/s)

Đ is :

MS surface diffusivity of pair \(i\)-fixed ionic charges (m2/s)

\({\text{F}}\) :

Faraday constant (C/mol)

\(k_{\text{f}}\) :

Film mass transfer coefficient (m/s)

\(K_{\text{LF}}\) :

Langmuir–Freundlich parameter

\([L]\) :

\(= [B]^{ - 1}\)

\(n\) :

Langmuir–Freundlich parameter

\(N_{j}\) :

Molar flux of counter ion j (mol/m2 s)

\(q_{j}\) :

Molar concentration of counter ion j in the particle, (mol/m3)

\(q_{\text{t}}\) :

Total concentration of ionic species in the particle (mol/m3)

\(\bar{q}_{j}\) :

Average concentration of j in the particle (mol/m3)

\(Q\) :

Ion exchanger molar capacity (mol/m3)

\(r\) :

Radial position in the particle (m)

\(Sc\) :

Schmidt number

\(Sh\) :

Sherwood number

\(\Re\) :

Gas constant (J/mol K)

\(R\) :

Particle radius (m)

\(Re\) :

Reynolds number

\(t\) :

Time, s (and h in the figures)

\(T\) :

Absolute temperature (K)

\(u_{i}\) :

Velocity of i relative to the solid, (m/s)

\(V_{\text{L}}\) :

Volume of fluid phase (m3)

\(V_{\text{s}}\) :

Volume of solid phase (m3)

\(x_{i}\) :

Molar fraction of i in bulk solution

\(y_{j}\) :

Molar fraction of counter ion j in the particle

\(z_{i}\) :

Charge of component \(i\)

Greek letters

\(\varepsilon\) :

Mixer power input per unit of fluid mass

\(\gamma_{{i,{\text{sol}}}}\) :

Activity coefficient of counter ion \(i\) in a solution in equilibrium with particle

\(\varphi\) :

Electrostatic potential (V)

\(\left[ \varGamma \right]\) :

Matrix of thermodynamic factors

\(\xi_{i}\) :

Related with the electrostatic potential gradient

\(\nu\) :

Kinematic viscosity

\(\mu_{i}\) :

Chemical potential of \(i\) (J/mol)



Counter ion initially present in bulk solution (Cd2+)


Counter ion initially present in particle (Na+)


Fixed charged groups of the particle

\(\infty\) :

Final equilibrium condition of experiment

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Lito, P.F., Aniceto, J.P.S. & Silva, C.M. Maxwell–Stefan based modelling of ion exchange systems containing common species (Cd2+, Na+) and distinct sorbents (ETS-4, ETS-10). Int. J. Environ. Sci. Technol. 12, 183–192 (2015).

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  • Cadmium(II)
  • ETS-4
  • ETS-10
  • Ion exchange
  • Maxwell–Stefan
  • Modelling