Contribution to the comprehension of the calcium ions transfer phenomena through a nanofiltration spiral wound membrane

A mathematical model is proposed to predict the transfer mechanism of calcium salts through a nanofiltration membrane. The model is a combination of Nernst-Planck and film theory equations and it is characterized by three transport parameters: solute permeability Ps, reflection coefficient σ and film thickness δ. In the present work, the influence of including concentration polarization phenomena on salt rejection is studied and the thickness of the boundary layer formed at the feed side adjacent to the membrane, δ, is estimated. Model results were shown to be in good agreement with experimental data at various operating conditions of pressure, temperature and initial concentration. The results show that including concentration, polarization has not a great effect on salt rejection for the present operating conditions. Estimated thickness of boundary layer formed near the membrane surface was too small and ranged between 10−13 and 10−12 m. In later work, we plan to perform some experiments at different operating conditions as higher initial concentration of calcium salts or lower tangential flow to bring the system in polarization concentration and then to apply, verify and validate the model proposed.

A coupled model, based on Nernst-Planck and film theory equations, was proposed to predict the transfer mechanism of divalent ions through nanofiltration membranes. The model is characterized by three transport parameters: solute permeability Ps, reflection coefficient σ and film thickness δ. The influence of concentration polarization phenomena on calcium salt rejection was studied. Model results were shown to be in good agreement with experimental data and show that including concentration polariza- C pi — Ion concentration in the permeate, C *i p mol. L−1 — Ion concentration in the membrane Di F Ji Jv — — — — at the permeate interface, mol. L−1 Diffusion coefficient of ion i, m2.s−1 Faraday constant, C. mol−1 Ion i flux, mol. m−2.s−1 Solvent flux, m.s−1 Pi* Ps — — — — — — — — Ion permeability, m2.s−1 Local solute permeability, m2.s−1 Universal gas constant, J, K−1.mol−1 Ion i rejection, % Absolute temperature, K Ion charge number, dimensionless Ion valence number, dimensionless Distance variable, m ℜ Fig. 7. Theoretical and experimental curves almost superimposed for all salts. Table 3 Estimated boundary layer thickness, δ, at 1.5 mmol/L Salt Concentration mmol. L−1 δ. 10−13 m Ca(NO3)2 Ca(CH3COO)2 CaCl2 1.5 1.5 1.5 5.5 Ri T Zi zi x Greek letters δ ∆ΨD ε σ σi tion has not a great effect on salt rejection for the present operating conditions. Estimated thickness of boundary layer formed near the membrane surface was too small and ranged between 10−13 and 10−11 m.