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.