2.2 Membrane based self-seeding process.
In the membrane assisted crystallization, beyond the heterogeneous
interface to accelerate heterogeneous nucleation, the hollow fiber
membrane possesses the additional function. The crystals appeared on the
membrane surface will then fall off with the flow of the fluid, entering
the crystallizer. The micro force field model on the crystal
particle-membrane surface to analyze this process is illustrated as
followed.
As defined Figure 1b, F1, F2 and
F3 represent the intermolecular forces (mainly van der
Waals forces), hydrostatic pressure force and hydraulic forces,
respectively; F5 and F6 are gravity and
static friction; F7 is the reaction force of
F1 and F2. F4 is
buoyancy. F1~F5 can be
expressed by the following
formula40-43,
\(F_{1}=\frac{A}{6}\left(\frac{rr_{c}}{Z_{0}^{2}\left(r_{c}+r\right)}+\frac{r_{c}}{\left(Z_{0}+r\right)^{2}}\right)\)(7)
\(F_{2}=1.7\times 6\pi\mu r_{c}v_{r_{c}}\) (8)
\(v_{r_{c}}=\frac{3r_{c}}{h}U_{m}\) (9)
\(F_{3}=\rho_{l}\text{gH}L^{2}\) (10)
\(F_{4}=\rho_{l}gL^{3}\) (11)
\(F_{5}=\rho_{c}gL^{3}\) (12)
Where L and rc are the crystal size and the spherical
equivalent radius of the crystal, respectively. A is the Hamaker
constant (the general order is 10-20J). r is the
roughness of the membrane surface; Z0 is the distance
between the crystal and the membrane surface; ρ 1and ρ c are the density of the fluid and the
crystal, respectively; g is the acceleration of gravity andH is the position of the crystal in the fluid; \(\mu\) is the
fluid viscosity; \(v_{r_{c}}\) represents the velocity of the fluid at
the center of the crystal; h represents the thickness of the
fluid layer; \(U_{m}\) is the average velocity of the fluid.
According to the principle of force balance, the forces in the
horizontal and vertical directions of the crystal have the following
relationship43,
\(F_{7}=F_{1}+F_{2}\) (13)
\(F_{6}=F_{3}+F_{4}-{F_{5}}\) (14)
In MACC, when the crystal is inclined to detach from the surface of the
separation membrane, the overall force acting on the crystal at the
membrane surface must satisfy the following
formulas43,
\(F_{3}+F_{4}-F_{5}\geq F_{\max}\) (15)
\(F_{\max}=K\left(F_{1}+F_{2}\right)\) (16)
Where F max the maximum static friction force of
certain crystal, and K is the maximum static friction
coefficient. Based on the equation, the criterion for the crystal to
slide or detach from the membrane surface can be expressed
as43,
\(\frac{1.7\times 6\pi\mu r_{c}\frac{3r_{c}}{h}U_{m}+(\rho_{l}-\rho_{c})gL^{3}}{\rho_{l}\text{gH}L^{2}+\frac{A}{6}\left(\frac{rr_{c}}{Z_{0}^{2}\left(r_{c}+r\right)}+\frac{r_{c}}{\left(Z_{0}+r\right)^{2}}\right)}\geq K\)(17)
The relationship of the different force acting on the crystal with
different size was shown in Figure 1c. As discussed in authors’ previous
work, the value of the crystal detach criterion can be increased first
and then decreased to zero as the crystal size
increases19. This means
that the tiny seeds detached from the surface of the separation membrane
have a relatively concentrated particle size distribution due to the
auto-selection mechanism based on the micro force field of the membrane
surface (also illustrated in Figure 1b). Only the crystals within a
certain interval can detach from the membrane surface. According to Eq.
(7) to (12), the peak of the crystal detachment curve or the detachment
space varies with the fluid flow rate, viscosity and roughness of the
separation membrane. Thus, the overall MACC process is shown in Figure
1d. The nucleus first produces crystals on the membrane, then grows to a
certain extent and detaches from the membrane surface into the membrane
module (achieving the ‘auto-seeding’ process). Thereafter, the crystal
is transported to the crystallizer to complete the following growth
process. According to Figure 1c, the detachment motion and the
auto-seeding process are highly size-dependent. Thus, different from the
artificial seeding operation (Figure 1e), the seeding operation in MACC
became automatically and controllable. These crystal particles as seeds
are uniformly transported through the membrane module to the
crystallizer and continue to grow in the low supersaturation of the bulk
solution. By provided abundant nucleus with uniform initial size, MACC
can avoid the explosive nucleation and exceeding secondary nucleation
that easily occurred in conventional cooling crystallization (especially
the spontaneous nucleation operation, shown in Figure 1f).