**1. Introduction**

Diminishing resources of fossil fuels, vulnerability of their prices, global warming, and environmental pollution result in a large interest in renewable and unconventional energy sources. Hence, one of these resources is biodiesel, which is currently under rigorous investigation [1]. Biodiesel is composed from renewable biological sources, such as vegetable oils and animal fats. It is biodegradable and nontoxic, and is characterized by low emission profiles and therefore is environmentally benign [2,3], unfortunately significant amounts of waste are generated during its production, which mainly consists of the glycerol phase. According to data presented by BP [4], the global biofuel production increased on the average by 5.13% (+4.1 Mtoe) in 2017 compared to 2014. In 2017, in the Europe region, biodiesel production reached approximately 14.167 million tons of oil equivalent (toe) whereas in North America nearly 38.190 Mtoe. Additionally, global primary energy consumption was projected to grow by 1.5 % per year between 2012 and 2035, whereas the energy from renewable resources, including biofuels, was expected to grow by 6.4% per year [5]. The biodiesel industry produces approximately 600 million tons of crude glycerol [6]. Therefore, due to the increase of glycerol waste, the technologies which allow for its beneficial reuse are of grea<sup>t</sup> interest. Many researchers focused on the biotechnological processing of glycerol by microbial bioconversion, delivering additional benefit to the environment in view of lower pollution. Through appropriate relation of microorganism clones, they postulate that it is possible to obtain low molecular weight organic compounds which are economically desirable, such as succinic acid [6–8]. In recent years, the interest in succinic acid production is growing because of the possibility to use it in the production of polymers and biodegradable plastics, surfactants, detergents, electrolytic coatings, and pharmaceutical active agents [9,10]. Generally, the technology of succinic acid biosynthesis ends at the stage of obtaining a post-culture liquid called post-fermentation broth, in which succinic acid produced by bacteria is dissolved [11–14]. Unfortunately, due to complex fermentation process, the main product is contaminated with various metabolites, especially with organic acids: acetic, formic, and others at low concentrations. The post-fermentation broth also contains wastes from the fermentation medium: residual glycerol, mineral salts, and minor amounts of polyols and proteins. Therefore, the success of bioconversion is determined by the process of purification [15–19]. Di fferent traditional methods are used to separate organic compounds from the post-fermentation broth e.g., precipitation, distillation, liquid–liquid extraction, ion-exchange, adsorption, crystallization or esterification, however there are alternatives to the traditional purification methods, such as the membrane techniques, for example nanofiltration (NF) [15,20,21], which is one of the newer membrane pressure methods for separation of liquid mixtures. The NF membrane can be either organic in nature (i.e., polymeric) or inorganic. Inorganic membranes, especially the ceramic ones, are more suitable for use with organic solvents due to their excellent chemical, mechanical, and thermal stability [22]. It is worth to notice that there is a commercially available ceramic membrane with a hydrophobized surface for organic solvent nanofiltration (OSN). In general, transport characteristics are much less investigated compared to polymeric membranes. Additionally, due to the multifaceted nature of interactions between membrane, solvent, and solute, the prediction of transport mechanism in OSN is much more complicated compared to aqueous applications [23]. Nevertheless, mathematical description of NF separations, whether it is organic or an aqueous solution containing organic compounds, is of grea<sup>t</sup> interest for many researchers [24–34] and it is far from truly being predictive.

In general, NF combines the removal of uncharged components at nanoscale with charge e ffects between the solution and the surface of the membrane [24]. Acids possess very low molecular weight (MW) (e.g., the MW of succinic acid is equal to 118.09 g/mol) in comparison to cut-o ff of ceramic membranes, therefore the so-called sieving e ffect cannot play the main role. In such case, the retention of acid is very low and process e fficiency is unsatisfactory. In order to enhance the separation of acid, the pH value is adjusted to obtain a dissociated form of acids above the p *Ka* (for succinic acid p*Ka*1 = 4.22 [35]). When the dissociated form of the acid is separated in NF, the main separation mechanism changes to the electrostatic repulsion. Nevertheless, for a charged compound, both steric hindrance and electrostatic interactions are responsible for e fficient separation—i.e., retention rate. Another important parameter in the transport and interpretation of retention is the membrane charge present along the surface of a membrane and also through the pores [32]. A strong charge present at the membrane surface has a crucial e ffect on the ion retention by the membrane [36] and therefore, the knowledge regarding the electrochemical interactions which occur at the membrane surface should o ffer the possibility to influence and describe the permeate flux, fouling tendency, retention of components, and cleaning conditions of the NF membranes more comprehensively. Understanding the ion-transport mechanism through a ceramic NF membrane is challenging and essential for further optimization of the NF processes.

The only available method adequate for characterizing the inherent membrane charge is the zeta potential of membrane. In the standard streaming potential measurements, the zeta potential is determined in order to link it to the streaming potential by the classical Helmholtz–Smoluchowski equation [37]. Streaming potential measurement gives the opportunity to obtain reliable information regarding the surface properties of ceramic membranes. Unfortunately, such measurement method requires sample in the flat, powder or eventually in fiber forms which requires destruction of a tubular membrane. Therefore, authors of this study postulate using the mathematical model to determine the total volume membrane charge density through parameter estimation based on the retention experiments.

Modeling of a nanofiltration membrane performance comprises two aspects: flux and rejection predictions [27]. Generally, the NF transport description models should consider the interaction between the charged membrane and ionic solutes [38]. Many charged membrane transport theories have been proposed which account for electrostatic e ffects as well as di ffusive and convective flow to describe the solute separation [39], such as the steric-hindrance pore model (SHP), electric-steric-hindrance pore model (ESHP) [40], Teorell–Mayer–Sievers model [41], frictional model or space-charge model [42]. However, the most popular and widely adopted mathematical model for NF process is based on the extended Nernst–Planck (eN-P) equation [39]. Bowen and co-authors [25,43–45] proposed the Donnan-steric-partitioning model (DSPM) which arose from the eN-P equation. The DSPM has been also used by many authors [32,46–49] with fairy good results. However, prediction of separation or membrane charge is very limited, and modeling is used to describe and correlate rejection results.

In order to predict the separation performance, it is important to evaluate the membrane charge density in well-defined solutions. Therefore, the authors aim to consider each ion and water in the modeling, and therefore to obtain values of the total volume membrane charge densities through mathematical modeling. As a case study, the nanofiltration of aqueous solutions of succinate sodium and magnesium salts in pH equal to 9 and in two di fferent concentrations (3.6 and 36.0 g/L) were studied experimentally and modeled with a comprehensively described DSPM model. Aside from sodium hydroxide, magnesium hydroxycarbonate (4MgCO3 × Mg(OH)2 × 5H2O) was also selected for pH adjustment of the feeds. It was because the magnesium hydroxycarbonate was used in the fermentation process reported in [50] to keep pH at approximately 8–9 in the bioreactor and at around 8.5 in the post-fermentation broth.
