**4. Results and Discussion**

Generally, it is well known that the following key parameters primarily affect the rejection of organic solutes during NF separations and are related to:


It has to be clearly stated that the negatively hydrophilic solutes can be rejected by electrostatic repulsion through negatively charged membrane surfaces. Electrostatic interactions between charged solutes and a porous membrane have been frequently reported to be an important rejection mechanism. Ions such as Na<sup>+</sup>, K<sup>+</sup>, Ca2+, and Mg<sup>2</sup>+ in feed water reduced the negative zeta potential of a membrane.

The experimentally obtained retention rates in the NF process for all model solutions were presented in Figure 3. The obtained retentions in process time slightly changed only initially and reached constant values after 40 min for whole processes which lasted up to 240 min. It has to be noticed, that two model solutions containing 3.6 g/<sup>L</sup> of succinic acid have displayed totally different values of retention. When sodium hydroxide was used as a pH regulator, the retention rate varied between 67% and 77% (see MS1 in Figure 3). When magnesium hydroxycarbonate was used, the retention decreased from significantly reaching a range of 20–22% (see MS3 in Figure 3). On the other hand, when the concentration of succinic acid increased from 3.6 g/<sup>L</sup> (MS1) to 36 g/<sup>L</sup> (MS2), the retention decreased form 67–77% to 3–16%. When retention profiles of MS1 and MS2 were analyzed separately, the initial increase of retention rate could be related to the fouling caused by concentration polarization, internal pore blocking or cake formation as it was described in [61]. However, since the MS2 has 10 times higher concentration than MS1, then it should be expected that retention rate would be kept at the same level, or that it should increase if there fouling would occur; but this was not the case. Therefore, it can be assumed that there is no typical fouling but a type of electrokinetic saturation of surface charges, the mathematical description of which is unknown to authors and requires further detailed studies.

The concentrations of sodium in the feed and permeate have changed significantly in the end of MS2 separation experiments, which is not the case in MS1 (Figure 4). That observation indicates that there is a mechanism of ion binding by the pore wall at higher concentration of sodium. Based on the obtained results, the ion binding for lower concentrations of succinic acid and pH-regulators (MS1 and MS3) can be excluded.

#### *4.1. Impact of Dynamic Viscosity on Modeled Permeate Flux*

The obtained dynamic viscosity values for all model solutions were presented in relation to the temperature in Figure 2. In modeling, it is generally assumed that viscosities of aqueous solutions with low solute concentrations are equal to water viscosity. For the MS1 at 303 K, the obtained viscosity value differed by approximately 10% in comparison with reference water viscosity. Such difference can influence the modeling results, which is highlighted by comparison of volumetric fluxes calculated with experimentally obtained viscosities and with viscosity being equal to water at certain temperatures which was presented in Figure 5. In general, the experimental viscosities in calculations of volumetric

fluxes for investigated solutions resulted in a decrease of calculated volumetric fluxes from 7.5% for MS1 and up to 66.4% for MS3 to those calculated with pure water viscosities. It has to be stated that including the experimentally obtained viscosities resulted in better description of the experimental volume flux for MS2. However, for MS1 and MS3 cases the discrepancies between utilization of water and experimental viscosities were rather far from the experimental, but still of the same order. Generally, for the investigated solutions, the use of experimental viscosities with combination of the Hagen–Poiseuille equation resulted in an underestimation of the calculated volume flux.

**Figure 3.** Experimental retention rates achieved for model solutions: MS1 (3.6 g/<sup>L</sup> succinic acid at pH = 9.7 regulated with granulate NaOH), MS2 (36.0 g/<sup>L</sup> succinic acid at pH = 8.8 regulated with granulate NaOH) and MS3 (3.6 g/<sup>L</sup> succinic acid at pH = 8.7 regulated with 4MgCO3 × Mg(OH)2 × 5H2O) in relation to duration of the NF process.

**Figure 4.** Concentration of sodium and magnesium cations in permeate and feed sides after 15 and 240 min of separation.

**Figure 5.** Comparison of calculated volume fluxes of model solutions (MS1, MS2, MS3) (10−<sup>5</sup> <sup>m</sup><sup>3</sup>/(m<sup>2</sup>·s)) with experimental viscosities (SV) and pure water viscosities (WV) at different temperatures.

#### *4.2. Comparison of the ddDSPM Model with the Standard Approach*

The standard approach considers only concentrations of solutes and ions coming from dissolved components, in the presented cases from succinic acid. The ddDSPM model takes into account all solutes, ions, and solvent, which include succinic acid, pH regulating solutions and water. In the standard approach, the model consists of 29 equations and 46 variables (*NC* = 2), while in the ddDSPM there are 62 equations with 88 variables (*NC* = 5), which were solved with the aim to estimate *Xd*. Additionally, estimation of *Xd* was also conducted under direct consideration of values of experimental fluxes (*Vexp*) instead of volume fluxes calculated with the Hagen–Poiseuille equation. All estimation results were compared in Figure 6. In the standard approach (i.e., DSPM model, ions coming from pH regulator not included), *Xd* is changing between −35.59 and +278.09 mol/m3, while in the detailed approach (i.e., ddDSPM model, ions coming from pH regulator included) the changes ranged between −35.73 and +875.69 mol/m3. The use of the water viscosity in the ddDSPM resulted in *Xd* values ranging between −34.98 and +939.67 mol/m<sup>3</sup> and use of experimental values of volume fluxes (*Vexp*) in ddDSPM (i.e., ddDSPM model, ions coming from pH regulator included) resulted in *Xd* values ranging between −19.57 and +871.74 mol/m3. It is important to highlight that the retention obtained with the models overlaid with the experimental values, regardless of which model was used.

Based on the obtained results, it is difficult to postulate which modeling approach is the best, since it is impossible to experimentally obtain the overall membrane charge density *Xd*. However, it is evident that the obtained values of *Xd* for higher concentration of solutes (MS2) differ significantly depending on the applied modeling approach. At first glance, it seems that there is no clear reason to use the ddDSPM model for diluted solution such as MS1; however, when feed contains a higher number of components or polyanions like in MS3, the ddDSPM would be recommended. The presented results clearly show that use of experimentally obtained volumetric flux in comparison to the ddDSPM with experimentally obtained viscosities does not significantly influence the *Xd* when computed volumetric flux is in good comparison to the experimental one (MS2). Otherwise, use of experimental volumetric flux in estimation of *Xd* is recommended.

#### *4.3. Variation of the Overall Volume Charge Densities in Relation to Used pH Regulator*

Comparison of the estimated values of the total volume membrane charge densities with standard and detailed models was presented in Figure 6 along with ionic strength and obtained experimental rejections for each investigated solution. The ionic strength *I* for each model solution was calculated according to Equation (22).

> *I*

$$=\frac{1}{2}\sum\_{i=1}^{NC} \left(c\_i \cdot z\_i^2\right) \tag{22}$$

**Figure 6.** Comparison of estimated total volume membrane charge densities (*Xd*) with different modeling approaches with relation to ionic strength (*I*) and experimentally obtained retentions (*R*) (ddDSPM—detailed described Donnan–Steric Partitioning Model; DSPM—Donnan–Steric Partitioning Model with standard approach; *Vexp*—experimental volume flux).

As presented in Figure 6, the estimated values of *Xd* for each model solutions were clearly different, but these values were adequate to retention rates obtained in the NF processes. In general, the higher ionic strength of the separated solutions resulted in a higher value of the total volumetric charge of the membrane. It should be noted that *Xd* values for all model solutions were values obtained for the experiments which reached the steady state and these values should not be confused with the fixed membrane charge because the *Xd* relates all electrokinetic phenomena present in the vicinity of membrane surface and not only those related to the membrane surface as it is in the case of fixed membrane charge [62–64]. The achieved *Xd* value for MS1 (3.6 g/<sup>L</sup> succinic acid of pH = 9.7 regulated with granulate NaOH) equal to −35.73 mol/m<sup>3</sup> may indicate the presence of strong electrostatic repulsion between succinate anions present in aqueous solution and membrane surface functional groups, hence the *Xd* reached a negative value and retention achieved 78%. For two other cases, the *Xd* values were positive. Such behavior could be related to the amphoteric characteristics of TiO2 active membrane layer, although it is generally explained by the charge change at isoelectric point at specific pH [46].

$$-Ti-OH + H\_3O^+ \rightarrow -Ti-OH\_2^+ + H\_2O \quad \text{at pH} < IPP \tag{23}$$

$$-Ti-OH + OH^- \rightarrow -Ti-O^- + H\_2O \quad \text{at pH} > IEP \tag{24}$$

In the studied cases, when pH was equal to approximately 9, it was expected that the active layer would possess negatively charged groups. The succinate anion has the lowest di ffusion coe fficient among other ions present in the system (Table 4), therefore succinate anions should grind and penetrate the membrane as the last anions. Although such strongly positive *Xd* value obtained for MS2 (36.0 g/dm<sup>3</sup> succinic acid at pH = 8.8 regulated with granulate NaOH) may reflect high concentration of sodium cations, their selective adsorption on membrane surface, which results in formation of additional surface layer and appearance of the electrostatic attraction of the succinic anions, resulted in very low retention (16%). In cases of basic salts, it is very di fficult to determine which hypothetical mechanisms might influence the separation and could result in such charge values. Although the estimated value for the MS3 (3.6 g/dm<sup>3</sup> of succinic acid with pH = 8.7 regulated with 4MgCO3 × Mg(OH)2 × 5H2O) was equal to +163.27 mol/m3, which suggests the selective adsorption of magnesium cations or other cations on the membrane active layer surface, also within the pores.

In general, the increase of the membrane charge densities with pH increase might be caused by the selective adsorption of ions at the membrane-separated mixture interface and additional adsorption in membranes' pores in active layer [65]. In this work, authors consider the total volume membrane charge density as the sum of the fixed membrane charge density and the number of adsorbed ions in the whole active membrane volume and its close vicinity at the feed side and membrane pores. The possible mechanism for the formation of membrane charge assumes that ions are partitioned from the bulk solution into the membrane pore under the influence of the Donnan potential. Among the partitioned ions in the membrane pores, either cations or anions are adsorbed selectively by the pore walls. Next, the adsorbed ions are bound on the pore wall and provide the electric charge to the membrane, which is specific to the investigated cases. In view of all that was stated above, the values of total volume membrane charge densities *Xd* will always be di fferent depending on the type of solution subjected to the NF process, although the pH values are the same. Despite that pH for all studied solutions was equal to approximately 9, the *Xd* values varied between −35.73 and +875.69 mol/m<sup>3</sup> (Figure 6). Therefore, it can be assumed that, the mechanism of selective ion adsorption plays a significant role despite similar pH values, the schematic explanation of which was presented in Figure 7. The overall volume membrane charge density in MS1 is negative because not all negative membrane surface group are associated with cations present in the solution, therefore succinic ions are repulsed by the membrane interface. The MS3 contains a similar amount of added magnesium hydroxide as sodium hydroxide in MS1. Since the magnesium ion possesses two positive charges, it can interact with two negative membrane sites and more negative membrane sites can be associated. Therefore, in MS3, the succinic ions can permeate more easily than in case of MS1. The lowest retention observed for MS2 could be related to the highest overall volume membrane charge due to high concentration of dissociated succinic acid and the highest amount of added sodium hydroxide, which in that case could compensate all present negative membrane charges.

**Figure 7.** Schematic representation of separation mechanisms in nanofiltration of aqueous solutions of succinic acid with different pH regulators (for details about MS1, MS2, and MS3 see Table 4, *Xd*—total volume membrane charge density; *Csuc2*−—molar concentration of succinate anion; IS*f-m*—feed-membrane interface; IS*f-m*—permeate-membrane interface).
