Membranes concentration profile

In addition to the ion-clustered gel morphology and microcrystallinity, other structural features includes pore-size distribution, void type, compaction and hydrolysis resistance, capacity and charge density. The functional parameters of interest in this instance include permeability, diffusion coefficients, temperature-time, pressure, phase boundary solute concentrations, cell resistance, ionic fluxes, concentration profiles, membrane potentials, transference numbers, electroosmotic volume transfer and finally current efficiency. 

This theoretical study is focused on the process combination of a distillation column and a pervaporation unit located in the side stream of the column. This hybrid membrane process can be applied for the separation of azeotropic mixtures such as acetone, isopropanol and water. Water is removed from the side stream of the column by pervaporation, while pure acetone and isopropanol are obtained at the top and bottom of the column. Detailed simulation studies show the influence of decisive structural parameters like side stream rate and recycle position as well as operational parameters like reflux ratio and mass flow on concentration profiles, membrane area and product compositions. 

FIG. 22-58 Concentration profile of electrolyte across an operating ED cell. Ion passage through the membrane is much faster than in solution, so ions are enriched or depleted at the cell-solution interface, d is the concentration boundary layer. The cell gap, A should he small. The ion concentration in the membrane proper will he much higher than shown. (Couttesij Elsevier.)... 

FIG. 22-81 Permeant -concentration profile in a pervaporation membrane. 1— Upstream side (swollen). 2—Convex curvature due to concentration-dependent permeant diffiisivity. 3—Downstream concentration gradient. 4—Exit surface of membrane, depleted of permeant, thus unswollen. (Couttesy Elseoier )... 

Figure 10.10 Partial pressure and concentration profiles across membranes. 

For microporous membranes, the partial pressure profiles, in the case of gas (vapor) systems, and concentration profiles are continuous from the bulk feed to the bulk permeate, as illustrated in Figure 10.10a. Resistance to mass transfer by films adjacent to the upstream and downstream membrane interfaces create partial pressure and concentration differences between the bulk concentration and the concentration adjacent to the membrane interface. Permeability for microporous membranes is high but selectivity is low for small molecules. 

In Figure 10.10a, it can be seen that for porous membranes, the partial pressure and concentration profiles vary continuously from the bulk feed to the bulk permeate. This is not the case with nonporous dense membranes, as illustrated in Figure 10.10b. Partial pressure or concentration of the feed liquid just adjacent to the upstream membrane interface is higher than the partial pressure or concentration at the upstream interface. Also, the partial pressure or concentration is higher just downstream of the membrane interface than in the permeate at the interface. The concentrations at the membrane interface and just adjacent to the membrane interface can be related according to an equilibrium partition coefficient KM i. This can be defined as (see Figure 10.10b) ... 

Fig. 18. Calculated concentration profiles within membrane, x/d = —0.5 at cathode interface +0.5 at anode interface.

Figure 3 Diffusion across a membrane. The solute molecules diffuse from the well-mixed higher concentration cY to the well-mixed lower concentration c2. Equilibrium is assumed at the interfaces of membrane and solutions. The concentrations on both sides of the membrane are kept constant. At steady state, the concentrations cm remain constant at all points in the membrane. The concentration profile inside the membrane is linear, and the flux is constant.

Similarly, applying the concentration profile of Eq. (38) to membrane diffusion gives... 

Figure 5 shows the diffusion of a solute into such an impermeable membrane. The membrane initially contains no solute. At time zero, the concentration of the solute at z = 0 is suddenly increased to c, and maintained at this level. Equilibrium is assumed at the interface of the solution and the membrane. Therefore, the corresponding membrane concentration at z = 0 is Kc1. Since the membrane is impermeable, the concentration on the other side will not be affected by the change at z = 0 and will still be free of solute. This abrupt increase produces a time-dependent concentration profile as the solute penetrates into the membrane. If the solution is assumed to be dilute, Fick s second law Eq. (9) is applicable ... 

Figure 5 Diffusion into a semi-infinite membrane. The membrane initially contains no solute. At time zero, the concentration of the solution at z = 0 is suddenly increased to and maintained at cx. This abrupt increase produces time-dependent concentration profiles as the solute penetrates into the membrane.

Figure 6 Unsteady diffusion across a membrane. The membrane is initially free of solute. At time zero, the concentrations on the two sides of the membrane are increased to and maintained at c, and c2. The solute penetrates into the membrane from both sides, resulting in time-dependent concentration profiles within the membrane.

Equation (128) with the initial and boundary conditions of Eqs. (129)—(131) is very similar to the model equations for diffusion in a semi-infinite membrane [Eqs. (64)-(67)]. The concentration profile is... 

In practice, estimation of Laq requires information on the rate of solute removal at the membrane since aqueous resistance is calculated from experimental data defining the solute concentration profile across this barrier [7], Mean /.aq values calculated from the product of aqueous diffusivity (at body temperature) and aqueous resistance obtained from human and animal intestinal perfusion experiments in situ are in the range of 100-900 pm, compared to lumenal radii of 0.2 cm (rat) and 1 cm (human). These estimates will necessarily be a function of perfusion flow rate and choice of solute. The lower Laq estimated in vivo is rationalized by better mixing within the lumen in the vicinity of the mucosal membrane [6],... 

Fig. 10.2 (A) Cross-section SEM micrograph of the hybrid membrane containing the receptor 1, (B) membrane transport concentration profiles and (C) molecular recognition principles of acidic I and zwitterionic II L-phenylalanine in the heteropoly-siloxane material membrane (1-hydrogen bonding, 2-charge interaction, 3-Van der Waals forces) [29].

Fig. 14. Concentration profiles during the continuous indirect electrochemical oxidation of 4-ethylphenol catalyzed by the enzyme EPMH in the electrochemical enzyme membrane reactor...

The major significance of this work is that yinth-ini c. compaction for one solute and aqucoui iotwtion c i )Cct6 for different solutes are measured for a commercial hyperfiltration membrane as a function of applied differential pressure. The results are obtained via simulation of the steady state concentration profile adjacent to the planar surface of the membrane for... 

Figure 12-5 Concentration profiles of a reactant that migrates into one phase through a membrane and into a second phase for reaction.

Fig. 5. Concentration profile of cephalosporin in an emulsion liquid membrane...

The typical concentration profile of solute in an SLM system with quaternary ammonium salt as carrier is schematically shown in Fig. 6. To model the facilitated transport within a supported liquid membrane [58,59], the following assumptions are usually made ... 

Fig. 6. Ty pical concentration profile in a supported liquid membrane system...

It must be borne in mind, however, that the implied condition is made that the Lik s have the same values for the different measurements. As, however, the Lih s are concentration dependent and the concentration-profiles in the membrane change their shape dependent on the kind of experiment, this condition is not satisfied in general. 

For the calculation of membrane phenomena as diffusion through membranes, membrane potentials, electrical resistance, transference numbers during electrodialysis, concentration profiles in the membrane under different circumstances, the flux equations have to be solved with the appropriate boundary-conditions. 

Besides the transference numbers Schlogl and Schodel were able to calculate the concentration profiles in the membrane. If the solution concentrations are different on both sides of the membrane, these profiles change if the direction of the current is reversed. By consequence, the electrical resistance depends on the direction of the current. This is called the rectifier effect. 

I.2. Interdiffusion. F. Helfferich and H. D. Ocher (54) studied the interdiffusion of counterions through an ion-exchange membrane. Bases for their calculations were the Nemst-Planck flux equations combined with the M.S.T. model. Ion-fluxes and concentration profiles in the membrane were calculated. 

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