Permeate flux
The filtration equation can be expressed using cake filtration theory as follows 17,
\(J=\frac{P}{\mu R_{t}}=\frac{P}{\mu\left(w_{c}\alpha_{\text{av}}+R_{m}\right)}\)(2)
where J is the permeate flux, μ is viscosity, P is the operating pressure, Rt is the overall filtration resistance, and Rm is the membrane intrinsic resistance calculated using the Kozeny-Carman equation18
\(R_{m}=\frac{\left(1-\varepsilon_{m}\right)^{2}K_{p}}{\varepsilon_{m}^{3}}\)(3)
where εm is the membrane porosity, andKp is membrane specific constant which was determined experimentally to be 8 ×1012m-1.
The average specific filtration resistance of the cake layer (αav ) can be calculated by eq. (4)
\(\alpha_{\text{av}}=\frac{k_{o}S_{p}^{2}\left(1-\varepsilon\right)}{\rho_{S}\varepsilon^{3}}\)(4)
where ko is the Kozeny constant,ρs is the particle density, andSp is the specific surface area of particles. For spherical particles, the values of ko andSp are equal to 5.0 and6 /dp , respectively, wheredp is the particle diameter 19. Since enzyme and dextran coexist in the cake layer, the flow channel of the liquid is made up of the porous spaces formed by both enzyme and dextran. Therefore αav was assumed as the connection of each resistance in parallel 17, and can be expressed as
\(\alpha_{\text{av}}=\left[\frac{\omega}{\alpha_{e}}+\frac{1-\omega}{\alpha_{d}}\right]^{-1}\)(5)
where αe and αd are the average specific filtration resistances of enzyme and dextran, respectively, and ω is the effective volume fraction of enzyme in the cake layer.
The cake mass, wc , is obtained from
\(w_{c}=\rho_{s}\left(1-\varepsilon\right)L\) (6)
where L is the cake thickness, and ε is the average cake porosity which can be calculated from following equation
\(\varepsilon=1-(1-\varepsilon_{o})\left(\frac{d_{p}+2\delta}{d_{p}+D}\right)^{3}\)(7)
where δ is the stern layer thickness, andεo is the porosity and is assumed to be 0.420. The equilibrium distance between neighboring particles, D , can be evaluated based on the force balance at which the net interparticle force (Fn ) is equal to the solid compressive force (Fs ).Fs is generated from the fluid flowing through the cake, and can be calculated as
\(F_{s}=-2\pi\mu\frac{J\left(d_{p}+2\delta\right)\left(3+2r^{5}\right)}{2-3r+3r^{5}-2r^{6}}\)(8)
\(r=\sqrt[3]{1-\varepsilon}\) (9)
The Fn can be estimated by using (DLVO) theory, and mainly consists of van der Waals force (Fv ) and electrostatic force (Fe ). TheFv can be calculated as
\(F_{v}=-\frac{{d_{p}A}_{H}}{24D^{2}}\left(1-\frac{1}{1+\frac{\lambda_{p}}{\text{cD}}}\right)\)(10)
where λp is the characteristic wavelength of the particle and is assumed to be 100 nm, AH is the Hamaker constant, and the constant c is equal to 5.3220.
The Fe , calculated based on the Poisson-Boltzmann equation, is a function of an electric double layer thickness, and its reciprocal, κ , can be estimated by
\(\kappa=\left[\frac{e^{2}\sum{n_{i}z_{i}^{2}}}{e_{0}e_{r}k_{B}T_{e}}\right]^{1/2}\)(11)
where kB is the Boltzmann constant, e is the electrical charge (=1.6×10-19 C),eo is the absolute permittivity (=8.85×10−12 C2J−1 m−1), erthe dielectric constant, zi andni are the valence of ions and the number of ions per unit volume in the bulk solution, respectively, andTe is the absolute temperature. As the thickness of an electric double layer is less than particle radius, theFe can be calculated as
\(F_{e}=2\pi e_{0}e_{r}\zeta^{2}\kappa\left[\frac{1}{1-exp\left(-\kappa D\right)}-1\right]\)(12)
where ζ is the particle zeta potential.
Fe can be also simplified as:
\(F_{e}=\pi e_{0}e_{r}\frac{d_{p}^{2}}{\left(D+d_{p}\right)^{2}}\zeta^{2}\exp\left(-\kappa D\right)\left[1+\kappa\left(D+d_{p}\right)\right]\)(13)