A new membrane fractionation process based on the combination of hybrid membrane cells and differential diffusion of two solutes

A new membrane fractionation process based on the combination of hybrid membrane cells and differential diffusion of two solutes was studied by numerical methods. The hybrid membrane cell comprises semipermeable and fully-permeable membrane sub-sections. While the semi-permeable membranes are permeable to the solvent and impermeable to both solutes, the fully-permeable membranes are permeable to all components of the solution, including the solvent. This cell is used to separate the solute with the lower diffusivity from the solute with the higher diffusivity since the former preferentially accumulates over the surface of the semipermeable membranes and it is preferentially removed in the fully-permeable sub-sections. The separation is demonstrated by numerical results. The effects of the solvent velocity (velocity of the flow permeating the semipermeable membranes), concentrate velocity (velocity of the flow permeating the fully-permeable membranes) and the number of sections are studied. In the concentrate stream, the purity of the solute with lower diffusivity increases with the solvent velocity, and decreases with the concentrate velocity. The purity increases slightly with the number of sections.

/ A new membrane fractionation process based on the combination of hybrid membrane cells and differential diffusion of two solutes was studied by numerical methods. The high purity of the solute with the lower diffusivity in the concentrate stream and the important recovery C c C in Cm cm Cp solute concentration normalized solute concentration solute concentration at the feed stream solute concentration at the membrane surface normalized solute concentration at the membrane surface permeate solute concentration cp / f CA / f CB / s CA / s CB / f ðCA Þi / f ðCB Þi f /C A f /C B f /ðC Þ A ref / / / / s ðCA Þi s ðCB Þi s ðCA Þm s ðCB Þm D H k kA kB Lout lout Lin lin Lm normalized permeate solute concentration solute concentration of A in the concentrate stream solute concentration of B in the concentrate stream solute concentration of A at the surface of the semi-permeable sub-sections solute concentration of B at the surface of the semi-permeable sub-sections mean solute concentration of A in the concentrate stream for section i mean solute concentration of B in the concentrate stream for section i mean solute concentrations of A in the concentrate stream for the entire cell mean solute concentration of B in the concentrate stream for the entire cell mean solute concentration of A in the concentrate stream for the entire cell and for the densest grid mean solute concentration of A at the surface of the semi-permeable membrane of section i mean solute concentrations of B at the surface of the semi-permeable membrane of section i mean solute concentration of A at the surface of all semi-permeable membranes mean solute concentration of B at the surface of all semi-permeable membranes molecular diffusivity distance between parallel plates current time step first virial coefficient for component A first virial coefficient for component B length of the outlet section normalized length of the outlet section length of the inlet section normalized length of the inlet section total length of the membrane sections lm L l Ls ls Lf lf / / / / / n nis nif nj pA=B ðpf =B Þi A / pf =B A / ðps =B Þi A / ps =B A / ps =B A / p=B A / / / / / pin=B A Rm RecA RA  V0 Vm vm s ðVm Þi s Vm VX total normalized length of the membrane sections length of the cell normalized length of the cell length of a semi-permeable membrane sub-section normalized length of a semi-permeable membrane sub-section length of a fully permeable membrane sub-section normalized length of a fully-permeable membrane sub-section number of sections number of longitudinal nodes in the semi-permeable section number of longitudinal nodes in the fully permeable section number of nodes in the vertical direction purity of A in relation to B mean purity of A for the fully-permeable section i mean purity of A for all fully-permeable sections in the cell mean purity of A for the semi-permeable section i mean purity of A for all semi-permeable sections in the cell mean purity of A over a semi-permeable membrane mean purity of A for a membrane with an infinite number of sections purity of the feed stream membrane resistance recovery of component A residual of the mass transport equation mean feed velocity permeate velocity normalized permeate velocity mean velocity of the solvent stream for sub-section i mean velocity of the solvent stream for the entire cell longitudinal component of the velocity VZ vx vertical component of the velocity normalized longitudinal component of the velocity normalized vertical component of the vz velocity X longitudinal coordinate x normalized longitudinal coordinate Z vertical coordinate z normalized vertical coordinate W width of the cell Sc Schmidt number Pe Peclet number DP À D0 =Rm V0Z Pv P0 P0 ¼ P0 =DPm PA0 , PB0 Eqs. (27) and (28)   Ps solvent velocity of a non-polarized v semi-permeable membrane f Pv concentrate velocity Re Reynolds number Sherwood number for an impermeable Shimp cell Shm logarithmic Sherwood number ln Shln Sherwood number for component A A based on the film equation ln ShB Sherwood number for component B based on the film equation / / / / / / / / / / / / / Greeks DPm static pressure difference across a semipermeable membrane o fraction of fully-permeable pores o Cf numerical error A o crit numerical error of the concentration at a critical location  osmotic pressure feed osmotic pressure p0 osmotic pressure difference across the Dp0 semi-permeable membrane c stream function v vorticity r density m viscosity f ln(c )