Natural Zeolites for the Sorption of Ammonium: Breakthrough Curve Evaluation and Modeling

The excessive use of ammonium fertilizer and its associated leakage threatens aquatic environments around the world. With a focus on the treatment of drinking water, the scope of this study was to evaluate and model the breakthrough curves for NH4+ in zeolite-filled, fixed-bed columns. Breakthrough experiments were performed in single- and multi-sorbate systems with the initial K+ and NH4+ concentrations set to 0.7 mmol/L. Breakthrough curves were successfully modeled by applying the linear driving force (LDF) and Thomas models. Batch experiments revealed that a good description of NH4+ sorption was provided by the Freundlich sorption model (R2 = 0.99), while unfavorable sorption was determined for K+ (nF = 2.19). Intraparticle diffusion was identified as the rate limiting step for NH4+ and K+ during breakthrough. Compared to ultrapure water, the use of tap, river, and groundwater matrices decreased the treated bed volumes by between 25% and 69%—as measured at a NH4+ breakthrough level of 50%. The concentrations of K+ and of dissolved organic carbon (DOC) were identified as the main parameters that determine NH4+ sorption in zeolite-filled, fixed-bed columns. Based on our results, the LDF and Thomas models are promising tools to predict the breakthrough curves of NH4+ in zeolite-filled, fixed-bed columns.


Introduction
To meet worldwide food demands, the use of nitrogen (N) fertilizer in agricultural applications has increased by 985% between 1961 and 2020 [1]. The use of N fertilizer (mainly in the form of ammonium-NH 4 + ) has several advantages: increased crop qualities; increased yields; and increased resistance to environmental stresses (e.g., limited water availability) [2,3].
Farmers in threshold and developing countries often use excessive amounts of N fertilizer because of difficulties in predicting the requirements of each specific plant and soil condition. This leads to N leakage of between 40% and 50% on average globally and has major impacts on both soil and water environments (e.g., increased oxygen demand and simulation of eutrophication) [4,5]. Under anaerobic groundwater conditions, NH 4 + concentrations as high as 390 mg/L have been recorded in a coastal aquifer-aquitard system in the Pearl River Delta, China [6]. Considering that global food demands continue to increase, it is reasonable to assume that NH 4 + levels will also increase across various water matrices and environmental niches.
Compliance with local threshold values is the determining factor when selecting a suitable process for the removal of NH 4 + from drinking water. NH 4 + is only considered harmful to humans at levels above 100 mg/kg body weight per day [7]. During drinking water treatment processes, the formation of toxicologically relevant NO 2 − species (via nitrification) and harmful chlorination side products must also be considered. In addition, NO 3 − can be reduced to NO 2 − in the stomach, which causes methemoglobinemia in infants. Finally, NO 3 − is a precursor of a variety of carcinogenic and teratogenic N-nitroso compounds [8].
To overcome these threats to human health, NH 4 + contaminated water can be treated using a range of technologies and processes. Such processes include: biological treatment processes (activated sludge and biofilm); air-stripping; membrane processes; and ion exchange [9,10]. When considering the treatment processes listed above, high energy expenditures can be avoided by excluding aeration processes. The use of additional chemicals and high pressures can be bypassed by using natural zeolites as sorbents.
Natural zeolites, particularly clinoptilolite, are promising sorbents for use in the processing of drinking water. Natural zeolites are abundantly available and cheap [11][12][13]; they can be subsequently used as a sustainable, slow-release fertilizer in agriculture [14]; they can be regenerated in a cost-effective manner through a simple brine treatment [15][16][17]; they have a broad range of applications; and most importantly, they demonstrate outstanding NH 4 + removal efficiencies [18]. Natural zeolites consist of three-dimensional AlO 4 − -and SiO 4 − -tetrahedrons covalently connected through oxygen atoms, which feature a negative surface charge. This surface charge can be selectively compensated by NH 4 + , which exchanges with Na + , K + , Ca 2+ , and Mg 2+ ions [19].
Previous studies have thoroughly investigated the use of natural zeolites with respect to the removal of heavy metals [20][21][22], organic contaminants [23], and high NH 4 + loads during wastewater treatment [23]. It has been reported that K + is the main competitor to NH 4 + due to their similar hydrated ionic radii and charge densities [24]. Recently, we investigated granular natural zeolites in fixed-bed columns for the removal of NH 4 + from drinking water sources [18,25]. Here, we investigate the NH 4 + breakthrough curves of finer zeolite particles in order to reduce the dimensions and run time of the zeolite columns.
Modeling of the NH 4 + breakthrough curves promises to further reduce time-consuming laboratory investigations. To date, different models have been used to calculate breakthrough curves. These models include: the LDF model, the Thomas model, the Yoon-Nelson model and the dose response model. With the exception of the LDF model, all of these models were previously applied to the modeling of NH 4 + breakthrough curves in zeolite-filled, fixed-bed columns. These previous studies investigated the influence of model water and wastewater matrices and, in doing so, enabled the extrapolation of zeolite sorption performance for a variety of NH 4 + removal scenarios [26,27]. The weakness of these models is that the description of the sorption mechanisms is reduced to simplified mathematical relationships. The LDF model also makes simplified assumptions, but it also includes a detailed evaluation of equilibrium and kinetic behavior. To the best of our knowledge, applying the LDF model for the modeling of NH 4 + breakthrough curves in zeolite-filled, fixed-bed columns has not previously been reported in the literature. To date, the LDF model is best known for modeling the sorption performance of activated carbon [28].
The main objective of this study was to investigate the influence of single-and multisorbate systems and of natural water matrices on the NH 4 + breakthrough curves in natural zeolite-filled, fixed-bed columns. In a departure from previous studies, the LDF and Thomas models were used to model the breakthrough curves for NH 4 + and to investigate the influence of its main competitor-K + . To achieve this, batch experiments and kinetic studies were conducted to determine the parameters necessary to model breakthrough curves. Furthermore, the influence of various water matrices (single-and multi-sorbate systems in ultrapure water, tap-, surface-, river-and groundwater) on the breakthrough curve of NH 4 + was investigated. Here, we demonstrate a promising tool with which to predict the NH 4 + breakthrough curves and to improve drinking water quality, using cost-effective techniques. Improving the quality of drinking water is increasingly relevant, especially in threshold and developing countries.

Modeling and Experimental Validation
Initially, the parameters of the Freundlich sorption model and for intraparticle diffusion were determined (see Table 1). The experimental isotherm and kinetic data can be found in Figures S1 and S2. . V: volume rate; v F : filter velocity; EBCT: empty bed contact time; D L : diffusion coefficient in the aqueous phase; ρ B : bed density; ε B : bed porosity; a v : specific particle surface; ν: kinematic viscosity; Re: Reynolds number; Sc: Schmidt number.
By comparing the correlation coefficients (R 2 , Table 1) for each water matrix, it can be seen that NH 4 + sorption is well described by the Freundlich sorption model. Several studies propose the Langmuir sorption model for a better correlation [15,[29][30][31], while others describe the Freundlich sorption model as more convenient [14,23,32,33]. As Lin et al. and Chen et al. described, it is most likely that ion exchange is the predominant sorption mechanism at NH 4 + concentrations below 1000 mgN/L for clinoptilolite-like zeolites [29,34]. In a previous study, we added support to this assumption by balancing the initial and end point ion equivalents for the same NH 4 + concentration (0.7 mmol/L) used in this study [18]. The Langmuir sorption model was not investigated here because the LDF model is based on Freundlich parameters.
It is notable that the Freundlich parameters, K F and n F, differ greatly between the isotherms investigated. Comparatively low K F values indicate a weak sorption of the sorbate to the sorbent, which is the case for K + (sorbent loading is low at low concentrations). Jaskūnas et al. determined similar values for K F (1.73 × 10 −5 L 0.23 * mg 0.77 /g) and n F (4.28) by investigating the sorption of K + to a natural zeolite (clinoptilolite) [35]. The low ability of K + to exchange charge balancing cations in the natural zeolite's framework can be related to the following characteristics: (1) after Ca 2+ , K + represents the second highest concentration ( Table 2); and (2) K + is able to exchange Ca 2+ , Mg 2+ and Na + , but due to its comparatively large hydrated radii, Ca 2+ (highest concentration in the zeolite's framework) and Mg 2+ cannot be easily exchanged. In contrast, NH 4 + is able to exchange all of the charge-balancing cations (Ca 2+ , Mg 2+ , Na + , K + ) in the zeolite's framework, which also results in higher K F values. Comparing the NH 4 + isotherm without the influence of competing K + ions (NH 4 + : K F = 1.02 L 1.02 * mg 0.07 /g) with the isotherm where K + is present (NH 4 + , (+K + ): K F = 1.21 L 1.21 * mg 0.37 /g) suggests that the presence of K + strengthens the sorption of NH 4 + . The n F values (n F (NH 4 + ) = 0.93 > n F (NH 4 + , (+K + )) = 0.63) imply that a faster breakthrough of NH 4 + occurs in the presence of competing K + ions-see Figure 1a,b. The isotherm for NH 4 + in the absence of K + displayed almost linear behavior and was reflected in the very similar values for K F and n F . These results support the findings of our previous study, using the same zeolite type (8-16 mm grain size) and over a range of concentrations we deemed relevant to drinking water (c 0 (NH 4 + ) = 1-110 mgN/L), a linear isotherm shape was observed [18].
In the next step, intraparticle diffusion kinetics were investigated in a differential circuit batch reactor. The impact of film diffusion was negated by operating at a filter velocity of 29 m/h. Without the influence of the competing K + , the fastest intraparticle diffusion rate was determined for NH 4 + (k s a v = 1.1 × 10 −4 1/s). As the intraparticle diffusion is dependent on the molecular weight of the sorbate and the temperature, it is not surprising that the diffusion of K + is comparatively slower (k s a v = 3.6 × 10 −5 1/s). This can be attributed to the (>2 x) greater molecular weight of K + (39 g/mol) compared to NH 4 + (18 g/mol). The molecular weight of K + results in a hindered K + transport-as a result of confinement and interaction of K + with the pore. The use of a constant temperature in each experiment meant that this parameter did not influence intraparticle diffusion. The slowest intraparticle diffusion (k s a v = 9.8 × 10 −5 1/s) was determined for NH 4 + in the presence of K + . It is assumed that K + not only competes for the same sorption sites, but also interacts with and reduces the rate of intraparticle diffusion for NH 4 + . Furthermore, a dependence on sorbate concentration or sorbent loading may occur as the result of a combined mechanism or as the result of a dependence of loading on the surface diffusion coefficient (D S ) [28].
It is also not surprising that the mass transfer coefficients k F a v for the film diffusion are much faster than those of the intraparticle diffusion (k s a v ). The difference in k F a v between NH 4 + and K + is related to their diffusion coefficients in the aqueous phase (D L ). As the molecular weight of K + is higher than that of NH 4 + , D L is comparatively lower for K + , and this results in a reduced rate of film diffusion.
After determining all necessary parameters, the experimental data were compared to the LDF and Thomas models. All of the parameters displayed in Table 1 were implemented in the program LDF 2.6© to obtain the modeled breakthrough curves for NH 4 + and K + . The nonlinear Thomas model curve fitting was performed using the program OriginPro 2023© (see also Table S1). Figure 1a,b show comparisons of the experimental data with curves predicted by the LDF and Thomas models, respectively. After 80 h, each breakthrough curve showed a complete saturation of the zeolites used. The breakthrough curve for NH 4 + was represented by a typical S-shape both with and without the influence of K + . In contrast, K + achieved a breakthrough level of 23% after only 30 min. The offset of approximately 23% and the much faster breakthrough of K + can be related to its weak sorption onto zeolites (K F = 8.39 × 10 −3 L 2.19 * mg −1.19 /g), high abundance in the zeolite's framework, partial washout from the zeolite used, and its unfavorable sorption onto the natural zeolites used (n F > 1). In general, K + concentrations up to 0.7 mmol/L (equivalent to 27 mg/L) are quite unusual in drinking water applicationstypically K + concentrations are found to be between 0.02 mmol/L and 0.1 mmol/L [18]. A relatively high K + concentration was chosen to simulate a worst case scenario and to investigate the influence of K + and NH 4 + at the same concentrations (i.e., in a 1:1 ratio). It was assumed that the NH 4 + breakthrough curve obtained in the presence of concentrations of K + that were relevant to drinking water would be very similar to the NH 4 + breakthrough curve obtained in the absence of K + .
A steady saturation of the zeolite-filled, fixed-bed column was observed in all cases, particularly pronounced for NH 4 + in the absence of K + . The relatively delayed breakthrough in all cases indicate that intraparticle diffusion is the diffusion limited step. This outcome is also in accordance with the findings of Damodara Kannan and Parameswaran, who investigated the NH 4 + uptake by clinoptilolite from swine wastewater permeate [26]. Even though the intraparticle diffusion rate was 1.12 times higher for NH 4 + (without K + ), and the sorption was more favorable for NH 4 + in the presence of K + (n F = 0.63), the breakthrough of NH 4 + in the absence of K + was reduced. This is strong evidence for the competing effect of K + on the sorption of NH 4 + onto zeolites. Because NH 4 + and K + compete for many of the same sorption sites, a much faster NH 4 + breakthrough occurred in the presence of K + .
With regards to the breakthrough curves for NH 4 + , both the LDF and Thomas models were successfully validated-the modeled breakthrough curves of NH 4 + were in close agreement with the experimentally obtained breakthrough curves. For K + , the modeling was not as accurate. The LDF model did not adequately describe the breakthrough curve for K + during the first 16 h, whereas the Thomas model was inadequate to describe behavior after 24 h. Furthermore, it was not possible to maintain a typical S-shape breakthrough curve for K + using the LDF model. The nonlinear Thomas model fitting provided a much better S-shape breakthrough curve for K + . To obtain a better fitting of the Thomas model, mainly in the first 6 h, the empty bed contact time (EBCT) should be increased in future studies-2-5 min EBCT are recommended for drinking water treatment applications [28].
To the best of our knowledge, this study is the first report of the LDF model being applied to NH 4 + breakthrough in the presence of K + in zeolite-filled, fixed-bed columns. The LDF model described the experimental isothermal and kinetic data sufficiently well. Furthermore, the experimental data were successfully fitted using the nonlinear Thomas model.
From a practical point of view, time-consuming and extensive field investigations could be reduced by switching to time-saving laboratory experiments. In turn, these laboratory experiments, which are still time-consuming, can be made easier still-by using simple breakthrough investigations through application of the Thomas model. Thus, accurate predictions of the dimensions of decentralized zeolite filters and their breakthrough characteristics are highly promising areas.

Ammonium Breakthrough in Natural Water Matrices
In addition to modeling NH 4 + breakthrough in zeolite-filled, fixed-bed columns in single-and multi-sorbate systems, the influence of natural water matrices on NH 4 + breakthrough characteristics is also of great interest. Therefore, the effects of different alkali metals, earth alkali metals and dissolved organic carbon (DOC) were investigated, at concentrations typically found in natural water matrices, in order to evaluate their influence on NH 4 + breakthrough. Here, tap water, a 1:10 mixture of Elbe river water and tap water, Elbe river water, groundwater, and ultrapure water as a control were used as matrices. Table 3 and Table S2 summarize the initial cation and DOC concentrations as well as providing general information on the different matrices investigated.  Table 3 represent typical matrix-dependent concentrations. In a previous study, the sorption of NH 4 + onto the same natural zeolite was most affected by the presence of K + , followed by Na + , Mg 2+ , and Ca 2+ [18]. This is generally in agreement with other reported studies (although these can differ with respect to Mg 2+ and Ca 2+ [36,37]). Furthermore, zeolites can be used to remove DOC in a targeted manner [33] and in more complex applications [38]. It is suspected that macromolecular humic compounds function to block zeolite pores, or function as an additional molecular sieve (providing diffusion resistance) for NH 4 + , resulting in minor equilibrium loadings in batch investigations [18,28]. This is why the parameters that influence NH 4 + sorption onto natural zeolites, mainly K + and DOC concentrations, are of particular interest. The elevated cation concentrations found in groundwater and the higher DOC concentration found in the Elbe river water (see Table 3) can influence the breakthrough of NH 4 + . The effects of different, natural water matrices on NH 4 + breakthrough curves are shown in Figure 2. Compared to the control with ultrapure water, an accelerated NH 4 + breakthrough was observed for all natural water matrices. Tap water and Elbe-/tap water exhibit a very similar breakthrough behavior. As the treated bed volume (BV) increases further, the Elbe-/tap water mixture exhibits a higher rate of breakthrough than tap water. The breakthrough of groundwater and Elbe river water is almost the same over the range of treated BV. To achieve a breakthrough level of 50%, the following matrix-dependent BVs were treated: 435 BV for ultrapure water, 325 BV for tap water, 268 BV for the Elbe-/tap water mixture, 176 BV for groundwater, and 135 BV for Elbe river water. Compared to the control, it is remarkable that the BV that was treated before a breakthrough of 50% was achieved decreased by 25% for tap water, by 38% for the Elbe-/tap water mixture, by 59% for groundwater and by 69% for Elbe river water. The typical S-shapes, observed for the breakthrough curves in single-and multi-sorbate systems (Section 2.1), could barely be seen for natural water matrices and supported the idea of a faster NH 4 + breakthrough. An increase in the EBCT, from 2.16 min up to 5 min, should be employed in future studies and especially in field applications.

The cation and DOC concentrations shown in
It is most likely that the concentrations of K + and DOC were the determining factors in the shift of the breakthrough curves to the left (breakthrough occurring at earlier time points and lower treated BVs in the presence of these materials). The competing effect of K + on NH 4 + breakthrough can be interpreted by considering the physical characteristics of both ions as well as the dimensions of the framework of natural zeolites-shown in Tables 4 and 5.   [42], (b) [43], (c) [44].
Zeolites are three-dimensional solids with well-defined structures. Natural zeolites are built around three ring types, each with different channel dimensions. Different ring types provide different sorption sites for hydrated cations. Ca 2+ and Mg 2+ possess relatively large hydrated radii compared to the other cations investigated. Due to the larger hydrated radii, as well as the molecular sieving effects that occur within natural zeolites, Ca 2+ and Mg 2+ cannot diffuse to all possible sorption sites (only to the center of horizontal ten-membered rings and horizontal eight-membered rings). Furthermore, it is thought that electrostatic repulsion of Ca 2+ and Mg 2+ can occur due to their relatively high charge densities. Similarly to Ca 2+ and Mg 2+ , Na + can only occupy certain sorption sites [40]. In contrast, NH 4 + and K + , with identical hydrated radii and charge densities, can diffuse into and be exchanged within all ring types of natural zeolites [40]. Therefore, K + is the main competitor to NH 4 + and the most important cation to consider when discussing NH 4 + breakthrough in zeolite-filled, fixed-bed columns.
The influence of different anions, those that typically occur in natural water matrices, can be excluded. Even at high anion concentrations, far outside what is usually found in drinking water (3.5 mmol/L for NO 2 − , NO 3 − , HPO 4 2− /H 2 PO 4 − and SO 4 2− ), no significant effect on the loading of NH 4 + onto the same natural zeolite was observed in batch experiments [18].
The point of zero charge (pH PZC ) of the same natural zeolite was observed to be in the range between pH PZC 6.24 and 6.47 [18], indicating that the surface is mostly positively charged at pH < pH PZC . Since the pH values used in our natural water matrices were higher than the pH PZC , it can be assumed that the zeolites' surfaces are mostly negatively charged. Thus, the surface will repel the negative zeta potential of humic compounds in the DOC at pHs > 1.6. Nevertheless, it is suspected that the comparatively high DOC concentration in the Elbe river water (due to macromolecular humic compounds) functioned to block pores, or as an additional molecular sieve for NH 4 + , resulting in faster NH 4 + breakthrough [18,28].
Influence of Potassium and the DOC on the Ammonium Breakthrough Curve DOC and K + concentrations in natural water matrices are thought to increase the rate of NH 4 + breakthrough in zeolite-filled, fixed-bed columns. Figure 3a,b shows how DOC and K + influence the Thomas model rate constant k Th , respectively. All Thomas model constant determined and modeled NH 4 + breakthrough curves can be found in Table S1 and Figure S3. The dependency of k Th on the DOC (R 2 = 0.98) and K + (R 2 = 0.94) concentration in natural water matrices is well described by a linear fit. According to Figure 3, a higher DOC and K + concentration results in a higher k Th and an accelerated breakthrough of NH 4 + . So, in addition to K + (see Section 2.2), the DOC concentration also has a strong influence on the breakthrough of NH 4 + in zeolite-filled, fixed-bed columns. In order to assess whether K + or the DOC has a stronger influence on k Th , a two-factor variance analysis was carried out by using the program OriginPro 2023© in Figure 4.  Figure 4 shows that the DOC has a stronger influence on k Th than potassium, especially at concentrations > 6 mg/L. There was a significant dependency between k Th and the DOC concentration, which is an important finding for field applications.
To further confirm the dependency between k Th and K + and the DOC, more natural water matrices should be investigated.
Nevertheless, due to the complex composition of DOC, it is challenging to describe a precise mechanism for the influence of DOC on k Th and the breakthrough of NH 4 + in zeolite-filled, fixed-bed columns. DOC consists of about 50% humic compounds, with relatively large molecular structures (humic and fulvic acids). The remainder is made up of low-molecular acidic and neutral substances and other compounds [28]. Because of the variety of molecular structures and weights, no individual species can be identified and analytically determined.
In summary, breakthrough curves of NH 4 + in zeolite-filled, fixed-bed columns can be better predicted, and location-dependent estimations of k Th can be made by: (1) measuring K + and DOC concentrations; (2) applying the linear equations of Figure 3; and (3) applying a two-factor variance analysis.

Zeolite Characteristics
CLP85+, the natural zeolite (clinoptilolite) under investigation, was supplied by Zeolith Umwelttechnik Berlin GmbH, Berlin, Germany. In Table 2, the chemical composition and general characteristics of the zeolite are shown. In order to remove particulate matter, salts, and excess adsorbed cations, the zeolites were washed with ultrapure water until the electrical conductivity was less than 10 µS/cm (and the water appeared clear by visual inspection). The zeolites were subsequently dried at 80 • C for 24 h before beginning the experiment.

Column and Batch Set-Up
The experimental determination of breakthrough curves was carried out in continuous flow mode through fixed-bed columns (inner diameter d i : 0.9 cm; bed height H B : 6.2 cm; zeolite mass m Z : 3.6 g- Figure 5a). Before each experiment, the zeolites were contained, using a plastic mesh and glass wool, in the adsorber compartment. A bed of glass spheres (d: 0.2 mm; H B : 1.3 cm) was placed under the plastic mesh to achieve a uniform incoming flow to the zeolite bed. The feed was pumped (Piston pump: REGLO-CPF Digital; Cole-Parmer GmbH, Wertheim, Germany) in an upwards direction (flow rate . V: 1.84 mL/min; filter velocity v F : 1.75 m/h; empty bed contact time EBCT: 2.16 min). To analyze breakthrough of the sorbate in the effluent, samples were taken from 10 min to 4800 min (3.3 days). A differential circuit batch reactor (Figure 5b) was used to investigate intraparticle diffusion (d i : 1.8 cm; H B : 3 mm; m Z : 0.7 g; v F : 29 m/h; solution volume: 1 L). A comparatively high filter velocity (29 m/h) was used to negate the impact of film diffusion. Glass spheres (d: 0.2 mm; H B : 9.5 cm) were placed under the plastic mesh, and glass wool was used to fix the zeolite bed in place. As for the fixed-bed column, the feed here was also pumped in an upward direction. Zeolite loading was investigated by time-dependent sampling from 30 min to 2880 min (2 days).
Batch experiments were performed using the bottle point method and were carried out in borosilicate flasks (Borosilicate 3.3 glass; VWR International GmbH; Dresden, Germany) with 100 g of zeolites per liter of water [28]. Flasks were stirred at 100 rpm (SM-30 orbital shaker; Edmund Bühler GmbH; Bodelshausen, Germany), and sampling was performed until equilibrium was reached.
The initial NH 4 + and K + concentrations of 0.7 mmol/L were obtained by diluting 0.3 mol/L NH 4 Cl and 0.7 mol/L KCl (analytical grade) stock solutions, respectively. To avoid microbial contamination, the water matrices used, as well as the column and batch set-ups, were autoclaved prior to experiments. Collected samples were filtered through a 0.45 µm PET filter (CHROMAFIL ® Xtra PET-45/25; Macherey-Nagel GmbH and Co. KG; Düren, Germany) before determining the amounts of NH 4 + , cations and DOC. By using an ultrapure water control, a mostly non-competitive water matrix was provided. All experiments were carried out in duplicate and at room temperature (22 • C). Before an experiment, the pH, electrical conductivity, and water temperature were recorded.

Breakthrough Curve Modeling
In order to reduce the mathematical effort involved in mass transfer models, the linear driving force (LDF) model was used for breakthrough curve modeling. The LDF model represents a simplified mathematical structure of intraparticle diffusion; it is widely used for the modeling of sorption processes in fixed-bed columns. Instead of Fick's law, a mass transfer equation with LDF is used. Under the assumptions of negligible dispersion (because of high filter velocities v F ) and validity of the Freundlich isotherm, the differential mass balance can be formulated as in Equation (1): Equation (1) consists of three terms: (1) describes the advection with the linear v F (m/h), the sorptive concentration c (mg/L), and the way z (m); (2) includes the bed porosity ε B (−) and time t (h); and (3) describes the sorption for a given bed density ρ B (g/cm 3 ) and the average loading of the sorbent q (mg/g).
Sorption processes can be described as consisting of four sub processes: (1) advection of the sorbate within the solution near the sorbent particle; (2) film diffusion; (3) surface diffusion; and (4) pore diffusion into the sorbent particle and further sorption of the sorbate. It is generally known that the rate-limiting steps during sorption are the film and intraparticle diffusion processes.
Firstly, the film diffusion, with a linear decrease in concentration (c) to the sorbent surface, is supposedly due to a steady increase in the equilibrium concentration c eq (Equation (2)). Equations (3)-(11) describe the coefficients necessary to calculate the time-dependent (q) in Equation (2).
Sh T = 0.037 * Re 0.8 * Sc Under the simplified assumption of spherical particles with a radius r P (m), the specific particle surface a v of the fixed bed can be calculated. ε B is a function of ρ B and the apparent particle density ρ P (density of water-filled particles) (g/cm 3 ). The mass transport coefficient k F (m/s) can be calculated by the Sherwood number Sh (−), the diffusion coefficient of the sorbate in water D L (m 2 /s) [28], and the particle diameter d P (m). There are several correlations to calculate Sh. In this study, the approach by Gnielinski [45] was used because the validity range of the Schmidt number Sc (Sc < 12,000) (−) was given. The Reynolds number Re (−) is calculated by v F , d P , ε B , and the kinematic viscosity ν (m 2 /s). D L is a function of the temperature T (K), the dynamic viscosity η (Pa*s), and the molecular weight of the sorbent (g/mol).
Secondly, the particle diffusion is composed of surface and pore diffusion. Under the assumption of LDF in the direction of the sorbent particle, the mass transport Equation (12) is used. Equations (13) and (14) describe the needed coefficients for the time-dependent loading q of the sorbent. ∂q ∂t = k s * (q eq − q) q eq,F = K F * c eq 1/n F (14) k s (1/s) describes the effective volumetric mass transport coefficient of the surface diffusion and is often dependent on q, and the empirical parameter w (g/mg) describes the strength of the influence of q on the intra-particle mass transfer. k s (0) (1/s) is the intrinsic mass transfer coefficient. The equilibrium loading q eq (mg/g) can be calculated by the Freundlich sorption model. K F (L n F * mg 1−n F /g) characterizes the sorption strength, and n F (−) determines the curve of the sorption isotherm. k s (0) and w can be experimentally determined with the previously described differential circuit batch reactor. Furthermore, Equation (15) is used to calculate the experimental q eq,exp (mg/g) of the described batch set-ups: c 0 and c eq are the initial and equilibrium concentrations (mg/L) of the sorbent in a defined volume of water V (L). m Z is the sorbent mass of natural zeolites (g).
The following experimental steps and programs were used for breakthrough modeling: The following boundary parameters were needed for the calculation: c 0 , K F , n F , k F a v , k s a v , m Z , v F and ρ B . Furthermore, experimental data of breakthrough curves can be loaded into LDF 3.0b and compared with modeled curves. Three different experimental breakthrough curves with the same initial concentrations of step (2) in ultrapure water were recorded.
The experimental breakthrough data and the LDF model were compared to the Thomas model (Equation (16)). The Thomas model is used to describe the ion exchange of zeolites in fixed-bed columns and to estimate the maximum sorbate sorption capacity and sorption rate in the column [46]. The nonlinear Thomas model fitting was performed using the program OriginPro 2023© and by simplifying Equation (16) to Equation (17) with Equations (18)-(20):

Investigation of Natural Water Matrices
Alkali metals and alkaline earth metals are ubiquitous cations in natural water matrices. For the investigation of natural water matrices, tap water, Elbe river water, a 1:10 mix of Elbe river and tap water, and groundwater were analyzed. The NH 4 + concentrations of the autoclaved water samples were measured prior to each experiment, and equal NH 4 + levels were achieved by spiking with a 0.03 mol/L NH 4 Cl stock solution. To estimate the effect of cations and the DOC content on the breakthrough curves, the levels of both were measured before each experiment.

Analytical Methods
The standard parameters of pH, electrical conductivity, and temperature were recorded using a multimeter and sensors (Sentix ® 41 pH electrode, TetraCon ® 325 conductivity cell, Multi 340i multimeter; Xylem Analytics Germany Sales GmbH and Co. KG; Weilheim, Germany). NH 4 + and cations were measured by a chromatography device (930 Compact IC; Methrom AG; Herisau, Switzerland). DOC was determined by an analyzer (TOC-VCPN Analyzer; Shimadzu Corporation, Kyoto, Japan).

Conclusions
This study is the first to apply the LDF model to the modeling of NH 4 + breakthrough curves in zeolite-filled, fixed-bed columns. Investigating the influence of natural water matrices further strengthened our understanding of NH 4 + breakthrough for field applications. The conclusions of our study can be summarized as follows: • The sorption of NH 4 + onto natural zeolites was more favorable than the sorption of K + . For this reason, natural zeolites are highly promising sorbents with which to remove NH 4 + in drinking water applications. • Intraparticle diffusion is the rate limiting step during NH 4 + breakthrough-as determined by comparison of k F a v and k s a v values. The presence of K + decreased the intraparticle diffusion rate of NH 4 + .

•
The LDF and Thomas models were successfully validated by experimental NH 4 + breakthrough data and can reduce time-consuming field investigations. The Thomas model can further reduce the complexity of experimental investigations because only breakthrough data are necessary (this is in contrast to the LDF model that is based on isothermal and kinetic data). • Natural water matrices strongly decreased the treated bed volumes during NH 4 + breakthrough. In particular, K + and DOC concentrations have to be considered when applying natural zeolites to the treatment of drinking water.
In summary, the time-consuming and cost-intensive investigations of field experiments can be reduced by using predictions of NH 4 + breakthrough when applying the LDF and Thomas models. The Thomas model is a promising tool to reduce planning and investigation costs for decentralized applications, especially in threshold and developing countries.
In future studies, we will further validate the results reported here while also studying the influence of additional parameters (e.g., competing cations, pH, organic compounds, and zeolite regeneration using brine solutions) on the breakthrough of NH 4 + in zeolitefilled, fixed-bed columns.
The main findings of this study are that the LDF and Thomas models sufficiently predict the breakthrough of NH 4 + in fixed-bed columns and that natural zeolites are highly promising sorbents for the removal of NH 4 + in drinking water applications.

Informed Consent Statement: Not applicable.
Data Availability Statement: Data are available upon request.