The force balance model for determining particle deposition
The force balance model is used to determine the deposition possibility of particles (Fig. 1 ). The tangential drag force,Ft , induced by the tangential velocity in a stirred cell can be estimated by the modified Stokes law
\(F_{t}=\frac{3}{4}\text{πμ}d_{p}^{2}\gamma C_{1}\) (14)
where C1 is a correction factor resulting from the membrane and the cake. The value of C1 is equal to 1.7009 19. γ is the shear rate at the membrane surface, γ=τc /μ . For a stirred cell, the shear stress (τc ) can be calculated by21
\(\left\{\par \begin{matrix}\end{matrix}\right.\ \) (15)
where ωo is the stirring angular velocity, andR is the radius of the stirred cell, δcrepresents the thickness of momentum boundary layer,\(\delta_{c}=\sqrt{\frac{\mu}{\rho\omega_{o}}}\), andRc is the critical radius and can be expressed as21
\(R_{c}=\frac{D_{i}}{2}1.23\left(0.57+0.35\frac{D_{i}}{2R}\right)\left(\frac{b}{2R}\right)^{0.036}n_{b}^{0.116}\frac{\text{Re}}{1000+1.43\text{Re}}\)(16)
where Di is the diameter of the impeller,h is the blade height, Re is the Reynolds number and is defined as \(\text{Re}=\frac{\rho\omega_{o}D_{i}^{2}}{4\mu}\), andnb is the number of blades. For the Millipore stirred cell (Amicon 8050) used in this study, Di= 3.8 cm, R = 2.1 cm, h = 0.9 cm, andnb = 2.
The normal drag force, Fp , induced by permeate flow can also be calculated by the modified Stokes law because of the very small Reynolds number:
\(F_{p}=3\text{πJμ}d_{p}C_{2}\) (17)
The correction factor, C2 , in the equation can be obtained by
\(C_{2}=0.36\left(R_{t}\text{Ld}_{p}^{2}/4\right)^{-2/5}\) (18)
Similarly, the modified Stokes law was used to calculate the inertial lift force Fl as follows,
\(F_{l}=3\pi v_{l}\mu d_{p}C_{2}\) (19)
where vl is the inertial lift velocity of particle and can be evaluated by the equation proposed by Vasseur and Cox 22
\(v_{l}=\left(\frac{61\gamma^{2}}{576\nu}\right)\left(\frac{d_{p}}{2}\right)^{3}\)(20)
where ν is the kinematic viscosity.
Finally, the net gravitational force, Fg , is obtained by
\(F_{g}=\frac{\pi}{6}g\left(\rho_{s}-\rho\right)d_{p}^{3}\) (21)
where ρ and ρs are the densities of the fluid and particle, respectively.
For a deposited particle, the external forces can be divided into normal and tangential directions as presented in Figure 1 . If the net force (Fp +Fg -Fl ) is attractive (positive sign) in the normal directions and the friction force (Ff ) is larger thanFt in the tangential directions, the particles are deposited on the membrane surface. If not, the particles will be swept back into the bulk solution. Thus the condition for the particles to stably stick on the membrane surface is given as
\(\left\{\par \begin{matrix}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ }F_{g}+F_{p}>F_{l}\\ {F_{f}=\mu}_{\max}\left(F_{g}+F_{p}-F_{l}\right)>F_{t}\\ \end{matrix}\right.\ \) (22)
where μmax is the maximum friction coefficient.