In Figure 3, the typical data of the flux behaviors in downward UF of three kinds of nanoparticle suspensions are shown in the form of the reciprocal filtration rate (dθ/dv) versus the cumulative filtrate volume v per unit effective membrane area. This plot is well known as the Ruth plot [4] in the classical cake filtration. In conventional downward filtration, without the filter cake exfoliation, the filtration rate declines gradually due to the build-up of a particulate cake with progression of filtration. Therefore, the plots in Figure 3 appeared to be virtually linear throughout the course of filtration in accordance with the Ruth filtration rate equation described by [4] (1) d θ d ν = μ α av ρ s p ( 1 − m s ) ν + ( d θ d ν ) mwhere μ is the viscosity of the filtrate, α_av is the average specific filtration resistance, ρ is the density of the filtrate, s is the mass fraction of the particles in the suspension, p is the applied filtration pressure, m is the ratio of wet to dry cake mass, and (dθ/dν)_m is the reciprocal filtration rate at the start of filtration, equivalent to the flow resistance of the membrane. It is believed that these plots are still increasing linearly even if ν becomes larger beyond the range of the figure because the filter cake continues to grow. Also, the figure clearly demonstrates that the slope of the plot becomes steeper, i.e. the filtration rate becomes lower as the particle size becomes smaller. According to the Kozeny-Carman equation, the local specific filtration resistance α in each cross-section of compressive filter cake layer is expressed as [5] (2) α = 180 ( 1 − ɛ ) ρ p d s 2 ɛ 3where ε the local porosity of compressive filter cake, and ρ_p is the density of the particle. This equation means that α is inversely proportional to the square of the particle diameter d_s when ε is the same. Although this equation also means that α is strongly affected by ε, the dependence on d_s is considered to be more dominant in this study since nanoparticles with quite different sizes were used. Thus, the filter cake with higher specific filtration resistance is formed when the smaller particles are filtered, resulting in significant decrease of the filterability.

In Figure 4, the plots of dθ/dν versus ν in upward UF are shown for three kinds of nanoparticle suspensions. In upward UF, a decline of the filtration rate was suppressed because the filter cake was exfoliated continuously by the effect of gravity, and as a result the reciprocal filtration rate showed a tendency to approach a steady value (dθ/dν)_e. Since the filter cake consisting of the micron-size particles does not exhibit the flowability in general under such a mild force as gravity, this behavior in the upward mode is one of the distinct features in membrane filtration of the nanoparticles. This figure also distinctly shows that the filtration rate increased significantly with a decrease in particle size. This may be because the filter cake consisting of smaller particles can return to the bulk suspension more easily under the influence of gravity due to the flowability caused by extremely large specific surface area. It is of great interest to note that the downward and upward modes are opposite in the dependence of the filterability of nanoparticle suspensions on the particle size.

In order to evaluate the exfoliation properties of the nanoparticle cake in upward UF, the steady filtration rate q_e (= (dθ/dν)_e⁻¹) is used from the following section. Since some experimental data of filtration rate did not reach enough steady value during experiment, the values of q_e were determined as follows. The variable J is defined using the reciprocal filtration rates (dθ/dν)_d and (dθ/dν)_u in downward and upward modes and the reciprocal pure water flux (dθ/dν)_m of the membrane as [2] (3) J ≡ (d θ / d ν ) d − (d θ / d ν ) m (d θ / d ν ) u − (d θ / d ν ) m

In Figure 5, J calculated from the data of Figures 3 and 4 is plotted against the filtrate volume ν per unit membrane area. This figure indicates that the plot of J versus ν can be approximated by a straight line. Considering that (dθ/dν)_u becomes q_e⁻¹ when ν is infinity, q_e is represented as the following equation using the slope σ_d of a straight line in Figure 3 and the slope σ_j of a straight line in Figure 5.

The values of q_e obtained in this manner are depicted by the dashed lines in Figure 4. In addition, the solid lines in Figure 4 indicate the calculations based on the approximate straight lines in Figure 5.

The experimental data of steady filtration rate q_e in upward UF were obtained using single component suspensions of ST-XS, ST-20 and ST-ZL, and binary mixtures of ST-XS and ST-ZL, with various concentrations s (s_XS, s₂₀, s_ZL, or s_XS + s_ZL). The q_e data are plotted against the surface mean diameter d_s of particles in Figure 6 and against the particle concentration s in Figure 7, respectively. The surface mean diameter d_s of a binary mixture of small and large particles is calculated from the mean diameters d₁ and d₂ and the mass concentrations s₁ and s₂ of the small particle (subscript 1) and large one (subscript 2) by (5) d s = ( s 1 + s 2 ) d 1 d 2 s 1 d 2 + s 2 d 1

For ST-XS/ST-ZL mixtures in Figures 6 and 7, d_s is 52.3 nm since the concentration ratio s_ZL/s_XS was adjusted to 20.8. As shown in Figure 6, the plots of q_e monotonically decreased with an increase of d_s under the condition of constant total particle concentration. And also, the q_e values in Figure 7 tended to decrease with an increase of s when the same particles were filtered, and it was revealed that the higher the particle concentration is, the more difficult the filter cake exfoliation becomes. This may be because the cake exfoliation is induced by the density difference between the filter cake and the bulk suspension. The behavior of plots in Figure 7 indicates the presence of the critical concentration s* that q_e becomes zero due to no cake exfoliation. It is thought that the density difference is too small to induce the cake exfoliation under the condition of the critical concentration s*. Here, the value of s* is considered as 0.1 for every colloidal silica suspension from extrapolation lines.

As mentioned above, the steady filtration rate q_e is highly dependent on the particle diameter d_s and the concentration s. Now, it is assumed that the estimation equation to comprehensively describe these influences can be expressed by an exponential form q e = a d s b ( s * − s ) c. The multiple regression analysis was applied on the basis of the experimental data in Figures 6 and 7 to determine the constants a, b and c, and then the following estimation equation was obtained.

Figure 8 compares the calculated q_e values based on Equation (6) with the measured ones in Figures 6 and 7. This figure indicates that there is generally good correlation between both data, and therefore Equation (6) would be appropriate as an estimation equation of upward filtration rate observed for a broad range of particle diameters and concentrations.

In the previous section, either dependence on the particle diameter or the concentration has been investigated. In this section, moreover, the influence of these conditions on the steady filtration rate is explored in case both mean particle diameter and particle concentration are simultaneously altered. For example, it includes the case where small particles are added into a given concentration of large particle suspension. In this operation, the addition of small particles leads to both a decrease in mean particle diameter d_s and an increase in total particle concentration s. This fits, for example, the operation that the nanoparticles are added into the particulate suspension as a filter aid, because the filtration rate is markedly increased as the mean diameter of particles included in the suspension becomes smaller, as shown in Figure 6, and therefore the nanoparticles may be available as a filter aid for improvement of the filtration rate in upward or dynamic filtration.

Figure 9 shows dθ/dν versus ν in upward UF of the mixtures of a constant concentration of larger ST-ZL particle (s_ZL = 0.005) and various concentrations of smaller ST-XS particle (s_XS = 0 to 0.03). The filtration rate was markedly increased by the addition of ST-XS particles in the range of small ST-XS concentration of 0 to 0.005, while the filtration rate showed a tendency to decrease gradually with an increase in ST-XS concentration in s_XS range of more than 0.005. It is considered that the former was caused by the reduction in mean diameter, and the latter was due to the effect of the increase in total particle concentration. Figure 10 depicts the relationship between the steady filtration rate q_e estimated from the data in Figure 9 and ST-XS concentration s_XS. The plot of measured values shows a definite maximum around s_XS of 0.005. Thus, the upward filtration rate in this operation is determined by a balance between contributions of the mean particle diameter and the concentration to the filter cake exfoliation.

Next, the applicability of the estimation equation is focused on. In Figure 10, the calculated value based on Equation (6) is indicated by a solid line as well as the measured one. Although the estimation equation underestimated q_e value a little, it could describe the overall trend of measured values roughly, which demonstrates the effectiveness of this estimation equation. In addition, the predictions of q_e versus s_XS for other ST-ZL concentrations (s_ZL = 0.001, 0.01 and 0.02) are also indicated by the dashed lines in Figure 10. There is a definite maximum on any predicted line, which corresponds to the optimal dosage of nanoparticles where the nanoparticles are applied as a filter aid. The optimal dosages of ST-XS particles estimated in this manner are shown against the initial concentration of ST-ZL suspensions in Figure 11. Furthermore, Figure 12 shows the effect of dosage of nanoparticles with different diameters (d_s = 4.8, 10, 20 nm) for a given concentration of ST-ZL suspension (s_ZL = 0.005) on the steady filtration rate. Thus, this estimation equation is applicable to the prediction of the steady filtration rate for various conditions of feed suspensions, and also can determine the optimal dosage of nanoparticles.