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).