Results and Discussion
4.3 Analysis of Membrane Fouling of UF Membrane for Oil-in-Water Emulsion
4.4.1 Concentration polarization and gel concentration
During ultrafiltration of pure water, the permeate flux is directly proportional to the transmembrane pressure, it can been expressed as:
m o
w R
J P η
= ∆
0 (4.4.1)
where Jwo is the permeate flux of pure water (l/m2h); ∆P is transmembrane pressure (bar); Rm the intrinsic resistance of the clean membrane (1/m); η0 is water viscosity (N s/m2).
The permeate flux is directly proportional to the operating pressure only if the concentration and pressure are below a certain limit. However, its membrane resistance is greater than that of pure water.
61
Chapter 4.4. Characterization of Gel Concentration
During ultrafiltration the solutes are carried and accumulated at the membrane surface, and formed a concentration difference between the membrane surface and bulk solution. It results that the solutes diffuse into the bulk solution backward till a balance situation of concentration is attained (see Figure 4.4.1).
Membrane
Cm
Cb
Boundary layer JwC
dx Ddc
δ
Js
x = 0 Bulk
solution
Figure 4.4.1 Concentration profile in the boundary layer of UF
The following is the differential equation of mass transfer for steady state ultrafiltration:
2 0
2 =
− dx C Dd dx
Jw dC (4.4.2)
where D is the diffusion coefficient of solute (m2/s). By integrating equation (4.4.2) it can give the following equation:
C1
dx DdC C
Jw − = (4.4.3)
62
Chapter 4.4. Characterization of Gel Concentration
where JwC is the solute flux on to membrane;
dx
DdC is the solute diffusion flux in the backward direction. The difference is equal to the solute permeate flux, which is a constant at a stable situation. Hence, the integral constant C1 can been replaced with Js. Then, concentration in the permeate (vol%). According to the boundary conditions: x = 0, C
= Cb; x = δ, C = Cm. By integrating equation (4.4.4) we can obtain the following solute concentration at the membrane surface (vol.%); δ is the thickness of the boundary layer (polarization layer) (m).
If the retention of ultrafiltration membrane is perfect, there is no any solute in the permeate, Cf can be ignored. Thus equation (4.4.5) can be simplified as:
b
K = D , mass transfer coefficient.
Although equation (4.4.6) does not present the relation between the pressure and other factors, an increasing pressure can improve permeate flux of water, and the solute concentration at the membrane surface also increases. The concentration polarization becomes more severe, which causes the flux of the solute diffusion backward to be increased. As an UF process becomes steady state at a certain pressure, the logarithm functional relation between Jw and Cm fulfils equation (4.4.6).
In addition, the thickness of boundary layer in equation (4.4.6), δ, depends on the hydrodynamic conditions, such as, the flow velocity is parallel to the membrane surface. The diffusion coefficient D is related with the solute property and feed
63
Chapter 4.4. Characterization of Gel Concentration
temperature. If the treated object is a macromolecular solution, the solute concentration at the membrane surface, Cm, increased greatly because of the smaller D, and the backward-diffusion flux of solute is lower as well. It causes an increase in the ratio of Cm/Cb. If Cm is increased to yield the gel layer under a certain pressure, the pressure at that moment is called critical pressure. The solute concentration at the membrane surface is named of gel concentration (Cg). Therefore equation (4.4.6) can be changed into:
f g
w C
D C
J ln
= δ (4.4.7)
For a selected solute, the gel concentration can be regarded as a stable value under certain conditions. The gel concentration is related with the solubility of the solute in water. Thus, Jw can also be considered as a determined value. If the transmembrane pressure increases continually, the backward-diffusion flux of the solute can not be enhanced. In a short time the permeate flux may be increased, but the pressure increased is balanced by the gel layer resistance quickly with increasing the thickness of the gel layer. Thus, the permeate flux of water returns to the previous level.
According to equation (4.4.7) the following conclusions can be seen: (1) When the gel layer is formed the permeate flux of water does not increase with the pressure. (2) The permeate flux decreases linearly with the logarithm relation of the solute concentration, Cb. (3) The permeate flux still depends on the hydrodynamic conditions which defined the thickness of the boundary layer.
In a word, the relation of Jw and ∆P can be summarized, as shown in Figure 4.4.2 for the UF process and macromolecular solution. The relation between Jw and ∆P can be considered within three regions:
The first is a direct line, which stands for a direct proportional relation like in equation (4.4.1).
The second region shows that the Jw is a functional relation with ∆P, and the relation of Jw and Cm can be expressed by equation (4.4.6).
The third region is nearly a parallel line, which shows that the Jw has no relationship with ∆P, Cm is equal to Cg. Jw can be calculated based on equation (4.4.7).
64
Chapter 4.4. Characterization of Gel Concentration
Figure 4.4.2 Relationship between permeate flux and transmembrane pressure
On the other hand, there is a polarization layer resistance besides the membrane resistance if the polarization layer can not be ignored. According to the additivity of resistance, the permeate flux can be expressed as:
)
where Rp is the resistance of polarization layer (1/m); η is the permeate viscosity (N s/m2)
As the gel layer is formed, the resistance of ultrafiltration includes still the resistance of gel layer (acts as main action). The permeate flux is governed by the so-called general filtration equation given as:
)
Chapter 4.4. Characterization of Gel Concentration
where Rg is the resistance of gel layer (1/m).
From equation (4.4.9), it can be seen:
(1) Because Rg >> Rp, Rp can be ignored. Thus equation (4.4.9) can be simplified as:
(2) If the pressure is variable, an increase in the pressure can enhance the permeate flux in a shorter time and forces more solute to the membrane surface, the thickness of gel layer and the resistance of gel layer increases. Thus it seems that Rg ∝ ∆P at that time, equation (4.4.10) can be modified as:
)
(3) Comparing equation (4.4.11) with equation (4.4.6), it can be seen that equation (4.4.11) can not reflect the influences of flow velocity of bulk solution and feed oil concentration. Moreover equation (4.4.6) can not show directly the effects of the pressure and resistance. However there is a common fact between the equations above, which shows the relations of Jw — ∆P and Jw — Cm under the concentration polarization and gel layer respectively. Substituting equation (4.4.6) into equation (4.4.11) and rearranging it the results is:
b
Subsequently the following equation can be attained
Chapter 4.4. Characterization of Gel Concentration
With respect to the equation above, it can calculate approximately the solute concentration within concentration polarization region under different pressures and gel concentration under critical pressure at the membrane surface, respectively.