## 2 Advancements in Ultrafiltration for Protein and Peptide Separation
### Charged-Membrane  
UF is a pressure-driven membrane process that uses a pressure difference to drive mass transfer through a porous membrane for separating solutes and solvents in the feed solution. The rate of solvent permeation trough the semipermeable membrane, or permeate flux (L m\({}^{-2}\) h\({}^{-1}\)), is generally defined by  
\[J_{v}=L_{p}\;dP/dx \tag{1}\]  
where \(L_{p}\) solvent permeability (L m m\({}^{-2}\) h\({}^{-1}\) bar\({}^{-1}\)), \(dP\) is TMP (bar), and \(dx\) is the membrane thickness (m). The ability of the membrane to separate the solutes, or sieving coefficient \(S_{o}\), is expressed as [62]  
\[S_{o}=c_{p}/c_{r} \tag{2}\]  
Here, \(c_{p}\) and \(c_{r}\) are concentrations of solute in permeate and retentate, respectively. Solutes that are retained by the membrane build up on the membrane surface and create a new layer. The presence of a new layer leads to increased mass transfer resistance. Sometimes, this layer becomes a new filter that has a smaller aperture than the membrane itself. The concentration of solute in the membrane wall, \(c_{w}\) can be estimated by [62]  
\[c_{w}=c_{b}\;\{S_{o}+(1-S_{o})\;\text{exp}\;\{J_{v}/k\}\} \tag{3}\]  
Equation (3) shows that \(c_{w}\) depends on concentration of solute in the bulk solution (\(c_{b}\)), sieving coefficient (\(S_{o}\)), permeate flux (\(J_{v}\)), and mass transfer coefficient at the boundary layer (\(k\)). Charged-membrane UF is used to address the drawbacks of size-based sieving owing to the electrostatic repulsion or rule-of-thumb of at least a factor of 10 in molecular mass difference for complete separation [63]. It can be seen in the exemplified performances of the charged UF membrane in protein separations, tabulated in Table 4, that performance of the charged UF membrane improved by more than 50% of that of pristine-membrane UF. The advantage of using a charged membrane are implied in the flexible design of membrane and exemplified in the elimination of nearly complete protein in permeate to produce whey protein concentrate, employing the polarization index (\(\beta\)) constant, which is defined by [62]:  
\[\beta=\frac{J_{v}}{k}=\frac{J_{v}}{0.816\;\left(\frac{6\;Q_{R}}{\frac{Q_{R}}{ \frac{Q_{R}}{V_{HR}}}D^{2}}\right)^{1/3}} \tag{4}\]  
where \(Q_{R}\) = retention flow rate or average recirculation rate, \(h\) = spacer height; \(V_{HR}\) = retentate hold-up volume, and \(D\) = protein diffusion coefficient \(\approx\) 2 \(\times\) 10\({}^{-7}\) cm\({}^{2}\) s\({}^{-1}\). Since parameter \(\beta\) represents the membrane system design, scale-up with the same performance can be achieved with different materials, module geometry, and membrane area, provided that \(\beta\) is set constant (in order to maintain stable thickness of deposit layer). At the same time, controlling \(\beta\) will stabilize the flux, and hence increase membrane lifetime due to minimization of cleaning-in-place [64]. This stable and predictable target would be rather unlikely to achieve if TMP is held as reference as in normal practice in dairy industries [65; 66]. [PERSON] and [PERSON] [67] reported a study of the effect of membrane charge on permeate flux and protein transport through the UF membrane. The actual sieving coefficient, \(S_{a}\), for a charged UF membrane is estimated by the following equation  
\[S_{a}=(1-r_{s}/r_{p})^{2}\ K_{c}\ \text{exp}\ E_{s} \tag{5}\]  
where \(r_{s}\) and \(r_{p}\) are radii for solute and membrane pores, \(Kc\) is the convection hindrance factor, and \(Es\) is the dimensionless interaction electrostatic energy. \(E_{s}\) depends on ionic strength of the feed solution, membrane pore radius, and the particle size or radius of solutes. Equation (5) shows that the transmission of protein decreases with the increasing membrane potential, which is in agreement with the experimental results. Charge on the membrane surface creates electrostatic interaction with protein improving the sieving coefficient.