Predicting the Hydraulic Conductivity of Metallic Iron Filters: Modeling Gone Astray

Abstract

Since its introduction about 25 years ago, metallic iron (Fe⁰) has shown its potential as the key component of reactive filtration systems for contaminant removal in polluted waters. Technical applications of such systems can be enhanced by numerical simulation of a filter design to improve, e.g., the service time or the minimum permeability of a prospected system to warrant the required output water quality. This communication discusses the relevant input quantities into such a simulation model, illustrates the possible simplifications and identifies the lack of relevant thermodynamic and kinetic data. As a result, necessary steps are outlined that may improve the numerical simulation and, consequently, the technical design of Fe⁰ filters. Following a general overview on the key reactions in a Fe⁰ system, the importance of iron corrosion kinetics is illustrated. Iron corrosion kinetics, expressed as a rate constant k_iron, determines both the removal rate of contaminants and the average permeability loss of the filter system. While the relevance of a reasonable estimate of k_iron is thus obvious, information is scarce. As a conclusion, systematic experiments for the determination of k_iron values are suggested to improve the database of this key input parameter to Fe⁰ filters.

1. Introduction

Due to their low-cost components, metallic iron-based filtration systems (Fe⁰ filters) have a broad range of applications from small-sized units for the production of safe drinking water in remote and/or low-income communities (e.g., Africa, Latin America and South-East Asia) to large-scale reactive walls for groundwater remediation, progress in numerical simulation of such systems may reasonably be expected to increase the application potential of this proven environmentally-friendly technology.

Predicting the service life of Fe⁰ filters is a fundamental aspect in discussing the suitability of this technology, both in terms of affordability and (long-term) efficiency [1,2,3,4]. The present state of the art has been shown to be unsatisfactory [3,5,6,7,8,9,10,11,12]. This situation is caused by several reasons. One reason is a misinterpretation of the essential reaction causing contaminant removal. A reduction reaction implying electrons transfer from Fe⁰ to the contaminants (e.g., RCl = chlorinated hydrocarbon, cf. Equation (1)) can be at best a side reaction under environmental conditions [13,14,15,16,17,18,19].

Fe⁰ + RCl + H⁺ ⇒ Fe²⁺ + RH + Cl⁻

(1)

Iron corrosion by water (cf. Equation (2)) is spontaneous and quantitative.

Fe⁰ + 2H⁺ ⇒ Fe²⁺ + H₂

(2)

Iron corrosion is well-known as a major nuisance reaction in the present iron-based world causing billions of Euro of damage per year [20]. This reaction is quantitative even in situations where water is present as moisture only [21]. It is therefore clear that iron corrosion by water has to be considered as a major reaction in a context (water treatment or Fe⁰/H₂O system) where water is the solvent ([H₂O] = ~55.5 mol L⁻¹). Even if more powerful oxidizing agents (e.g., Cr^VI, Cu²⁺, NO₃⁻, O₂, RCl) should be present as solutes (even at the mg L⁻¹ level), Equation (2) still needs to be considered a dominant reaction in Fe⁰/H₂O systems.

It has been unambiguously demonstrated that contaminant reduction in Fe⁰/H₂O systems is mediated by (i) Fe^II species (Equation (3)) and H₂ (Equation (4)), which are primary corrosion products and (ii) secondary and tertiary corrosion products, including Fe₃O₄ and green rusts [10,11,22,23,24,25,26].

2 Fe²⁺ + RCl + H⁺ ⇒ Fe³⁺ + RH + Cl⁻

(3)

H₂ + RCl ⇒ RH + HCl

(4)

The fact that the Fe⁰ surface is constantly shielded by a multi-layered oxide scale (see [11] and the references cited therein) implies that, even under oxic conditions (O₂ is available), Fe⁰ oxidizes according to Equation (2). This reaction is even accelerated if reaction products (e.g., Fe^II species) are consumed (Le Chatelier’s principle). Further details about this aspect of the iron corrosion process are summarized in [21].

The presentation above clearly shows that rate constants for modeling contaminant reductive transformations in Fe⁰/H₂O systems should be exclusively based on Equations (3) and (4). Alternative strategies have been discussed in [16,19,27,28,29,30]. These alternative strategies are based on Equation (1) and mentioned herein for the sake of completeness.

Electrochemical reduction alone (Equation (1)) cannot explain all reported observations as (i) the generation of iron hydroxides and oxides for As or U removal [31,32] or (ii) the porosity loss of Fe⁰ beds flushed with distilled water [33,34]. These obvious discrepancies between various modeling approaches for Fe⁰ filters, however, had been a motivation for the discussion herein.

The following four facts suggest that it may be sufficient to characterize the long-term behavior of Fe⁰ corrosion while considering the coupled removal efficiency of selected species added as contaminants to have a realistic picture of Fe⁰ filters [10,11,25]: (i) Fe⁰ corrosion by water is quantitative; (ii) contaminants are chemically reduced by some iron corrosion products; (iii) contaminants are present in trace amounts; and (iv) contaminants without redox properties are quantitatively removed. Thus, the most important parameter to design a Fe⁰ filter is the long-term corrosion rate (k_iron; cf. Section 2) of the material to be used.

The aim of the present communication is to demonstrate that no reliable model for Fe⁰ filters can be presented without relevant k_iron values. Domga et al. [35] recently summarized the results of mathematical modeling of the spatial description of the process of permeability loss in Fe⁰ filters. Herein, it is shown that related temporal changes depend on k_iron values that are yet to be determined. The significance of k_iron will be first presented, followed by the specificity of Fe⁰ materials as used in water treatment and environmental remediation.

2. The Rate Constant of Iron Corrosion (k_iron Value)

2.1. Descriptive Aspects

Kinetic studies on contaminant reductive transformation in Fe⁰/H₂O systems currently use relationships like the following based on reactions similar to Equation (5):

$$\frac{\mathrm{d}[\text{RCl}]}{\text{dt}}=-\mathrm{k}\times [\text{RCl}]\times [{\mathrm{H}}^{+}]$$

(5)

Following this approach, k values for individual contaminants (e.g., k_RCl) have been determined and partly tabulated for modeling purposes [16]; however, considering that Equation (1) is faulty as Equations (3) and (4) are the main sources of electrons for any contaminant in a Fe⁰/H₂O system (indirect reduction). Accordingly, properly modeling the kinetics of any reductive transformation in Fe⁰/H₂O systems depends on the accurate characterization of the kinetics of the production of Fe²⁺ and H₂ or the determination of the kinetics of Fe⁰ corrosion by water (Equation (6)). The relationship defining k_iron can be written as:

$$\frac{\mathrm{d}\left({\text{Fe}}^0\right)}{\text{dt}}=-k_{\text{iron}}\times {[{\mathrm{H}}^{+}]}^2$$

(6)

where k_iron is the rate of iron corrosion and [H⁺] the concentration of protons. Reardon [36,37] has positively correlated k_iron and the volume of H₂ generated in the Fe⁰/H₂O system. However, properly correlating both values depends on the description of individual systems and the characterization of possible H₂ sinks [38]. In other words, characterizing the kinetics of iron corrosion (at pH > 4.0) by determining the volume of generated H₂ is just an approximation. This approach is arguably nearer to reality than any reasoning based on Equation (5) [39].

The abundant literature on aqueous iron corrosion reveals that there are three basic kinetics laws that characterize the oxidation rates of Fe⁰: (i) the parabolic rate law; (ii) the logarithmic rate law; and (iii) the linear rate law. These laws are modeled respectively as follows [40,41]:

x² = Kp × t + x₀

(7)

x = Kp × log(c × t + b)

(8)

x = K_L × t

(9)

where x is the thickness of the oxide scale on Fe⁰; Kp is the parabolic rate constant for scale growth; K_L is the linear rate constant for scale growth; t is time; a and c are constants. As a rule, the x values (scale thickness) are correlated to the Fe⁰ weight loss (metal loss) without considering the presence of individual corrosive species (e.g., Cl⁻, CO₂, H₂O, H₂S, O₂). The parabolic rate law assumes that the concentrations of diffusing species at the oxide/metal and oxide/water interfaces are constant and that the oxide scale is uniform, continuous and of the single phase type [41]. This law is applicable to high temperature engineering problems and is recalled here only for the sake of completeness. The logarithmic and the linear rate laws are more likely to be applicable to Fe⁰ water filters and are both empirical relationships. It is essential to notice that the corrosion rate (k_iron) is not investigated as a function of the amount of individual corrosive species (e.g., Cl⁻, contaminants, H₂S), but rather as the extent of iron corrosion as influenced by these and all other operational parameters (flow velocity, system design, temperature). In other words, considering that contaminants are the main corroding agents for Fe⁰ is alien to corrosion science [10,13,14,25,40,41].

2.2. Corrosion Rate and Extent of Mass Loss

The extent of aqueous Fe⁰ corrosion under environmental conditions is generally expressed either in terms of [40,41] (i) the change in Fe⁰ weight (mass loss), (ii) the dimensions of the corroded crevices, (iii) the number and quantity of formed pits or (iv) the amount of corrosion products (e.g., the thickness of the oxide scale; Equations (7)–(9)). Equation (10) expresses the corrosion rate (k_iron) in terms of mass loss:

$$k_{\text{iron}}=K\times \frac{m\mathrm{b}-m\mathrm{a}}{A}\times \mathsf{\rho }\times t$$

(10)

where k_iron (mm/y) is the corrosion rate; m_b (g) is the mass before exposure; m_a (g) is the mass after exposure; A (mm²) is exposed surface area; t (years) is the exposure time; ρ (g/cm³) is density and K a constant. Expressing the time in years suggests that, to be relevant, experiments should ideally last for several years.

Another conventional way to access k_iron uses the Faraday law. Here, k_iron is quantified through the relationship between the corrosion current density and the extent of metal dissolution [41,42].

2.3. Oxide Scale Formation and Its Effects on k_iron: Modeling Aspects

Equation (10) shows that k_iron depends on Δm = m_b – m_a, A and t. Δm is related to the amount of corrosion products within the system or the thickness of the oxide scale (x value; Equations (7)–(9)). The time-dependent changes of Δm and A render corrosion processes very complex. Generally, one should consider the following processes [40,41,43]: (i) the Fe⁰ dissolution (anodic reaction) at the free metal surface; (ii) water reduction (cathodic reaction) at the Fe⁰ surface; (iii) the precipitation equilibria yielding the oxide scale on the Fe⁰ surface; (iv) the dissolution of the oxide scale (abiotic and biotic); (v) chemical reactions within the oxide scale (including Equations (3) and (4)); (vi) adsorption, co-precipitation and mass transfer across and within the oxide scale; and (vii) electro-migration (ionic transport) through the oxide-scale. The latter depends on the permeability and the surface charge of the precipitates building the scale [43].

A comprehensive model for the corrosion process should take into account all of these phenomena. Such a model would contain a large number of parameters. Relevant parameters could not be unequivocally determined on the basis of the limited amount of data that is available in the literature. This is an argument for new, holistic and systematic investigations. However, appropriate simplifications could be made. For example, Anderko and Young [44] considered that separate species form on the corroding Fe⁰ surface in order to derive a mathematical model that represents the effects of scale formation on corrosion rates. The time-dependent change of each Fe⁰ surface fraction is mathematical expressed. Similarly appropriate assumptions could be made on the reaction orders, the absorption behavior (e.g., Freundlich, Langmuir) and all other relevant aspects [44,45].

The approach conventionally used in the Fe⁰ water filtration literature [16,19,27,28,29,30] is not derived from Equation (10) and related simplifications. Moreover, the presentation above reveals that chemical reactions yielding contaminants reductive transformation in Fe⁰/H₂O systems (Equations (3) and (4)) are included in just one of seven processes making up the dynamics of oxide scale formation and transformation. For this reason, it would be difficult to find a model really describing an actual Fe⁰/H₂O system within the remediation literature.

3. Metallic Iron for Water Treatment

3.1. Major Characteristics of Fe⁰ Materials for Water Filters

Reactive Fe⁰ materials used for water filters are characterized mostly by their size/thickness of less than 5.0 mm [46,47,48,49]. A widely-used Fe⁰ sample from iPuTech (Rheinfelde, Germany) has a grain size varying from 0.3 to 2.0 mm [16,47]. This diameter represents less than 1/10 of the thickness of the buried cast iron pipes investigated by Mohebbi and Li [50]. The pipe presented pits having widths varying from 13 to 117 mm and depths from 1.6 to 9.1 mm after a service life of 90 years. From the study of Mohebbi and Li [50], an average rate constant (k_iron) for a cast iron pipe under service conditions can be derived (Equation (10)), but such k_iron values cannot be extrapolated to Fe⁰ materials for water filters, even when they result from the same cast iron. Many material and application-dependent k_iron values can be found in the broad literature [51,52] using Fe⁰, but none of them can be adopted for Fe⁰ filters. Relevant fields include metal recovery by the cementation process, which was established in 1890s [53,54]. In the cementation process, metal (e.g., Fe⁰, Zn⁰) particles are used in technical devices for some weeks or months and the major (controlled) operating conditions (e.g., high temperatures or low pH values) are not comparable to those of water filters. In particular, high metal ion concentrations warrant the formation of an electronic conductive layer of recovered metals at the Fe⁰ surface, while in the water filtration industry, a multi-layered, nonconductive oxide scale (“passive” film) is inevitably formed [55], rendering quantitative direct reduction (electrons from Fe⁰) simply impossible.

3.2. Coping with the Singularity of Fe⁰ Filters

In order to assess the service life of Fe⁰ filters, it is imperative to determine corrosion rates (k_iron) of relevant Fe⁰ materials over the long term and to develop models for Fe⁰ depletion to be correlated with the extent of contaminant removal under operational conditions. Although many studies have been performed to determine the rate constant of the degradation (chemical reduction) of individual contaminants (k_contaminant; based on reactions similar to Equation (1)) (Section 2), little research has been undertaken, which could be used to derive k_iron values over longer time scale than days, weeks or a few months. The intention herein is to pave the way to fill the gap regarding the long-term corrosion rate of materials for Fe⁰ filters in the absence of historical data. Clearly, long-term experiments for the determination of k_iron values are urgently needed to assist the development of reliable models to predict the service life of Fe⁰ filters and/or describe their operation.

It is understood that the k_iron value is not an intrinsic property of the material, but depends on several inter-related site-specific parameters. For example, the same material will exhibit different k_iron values in reactive zones containing the same amount of Fe⁰, but having different volumetric Fe⁰:sand ratios, e.g., 10:90, 25:75 and 50:50. The rationale is that the long-term kinetics of iron corrosion also depends on the pressure within the filter (P_H2 in Equation (6)). The initial k_iron value (k⁰_iron) could be identical in the three systems, but with increasing service life, systems with higher Fe⁰ ratios will exhibit lower k_iron values. Remember that in designing Fe⁰ filters, the goal is to work with the material, for which k_iron value implies efficient decontamination under the operational conditions (e.g., bed thickness, flow velocity, water chemistry). Clearly, selecting an appropriate material for a given site is not seeking for the most reactive one.

4. Modeling Permeability Loss of Fe⁰ Filters

4.1. Lessons from Monitored Aquifer Recharge

In many regions of the world, monitored aquifer recharge (MAR) is applied as a tool to (i) overcome the over-exploitation of groundwater or (ii) store excess storm water in the subsurface [56,57]. While injecting water into an aquifer, clogging does often occur, possibly rendering MAR inefficient in the long-term. There are three clogging types: (i) biological clogging due to growing biofilms; (ii) chemical clogging due mostly to precipitation reactions; and (iii) physical clogging caused by suspended solids. In many cases, it has been demonstrated that physical clogging accounts for about 70% of the permeability loss [58].

A particular case of water injection into the aquifer is that of the re-circulation of iron-rich groundwaters. Here, the precipitation of Fe^II/Fe^III mineral phases significantly reduces the porosity of the aquifer system after a certain operation duration. The removal of dissolved iron by several techniques was proven efficient in sustaining MAR operations [59]. In the context of Fe⁰ filters, dissolved iron generates and precipitates within the system (in situ). Accordingly, dissolved iron cannot be removed, but only properly considered. Thus, the two major lessons from the MAR technology are that: (i) suspended substances should be filtered out; and (ii) the generation, migration and transformation of Fe species should be carefully considered (mass balance, including volumetric expansion).

4.2. Lessons from the Merrill-Crowe Process

The century-old zinc cementation for gold recovery from a pregnant leach liquor (the Merrill-Crowe process) reveals that clarification and deaeration (O₂ removal) of the leach liquor are two decisive stages to optimize the efficiency of Fe⁰ filters. Clarification corresponds to the elimination of suspended substances (Section 4.1) to avoid physical clogging.

In its original version, the cementation process involved the filtration of a gold-bearing cyanide solution through a packed bed of zinc shavings. This operation was proven to have little efficiency because the reaction rate was very slow and the Zn⁰ surface was soon “passivated”. The problem of passivation was solved by the addition of a lead salt (e.g., Pb(NO₃)₂). This allowed a Zn⁰/Pb⁰ bimetallic system to form at the Zn⁰ surface (shavings), enabling continued gold cementation (deposition).

Two further improvements were realized later: (i) using zinc dust rather than shavings; and (ii) the deaeration of the gold-bearing solution to O₂ levels lower than 1.0 mg L⁻¹. Zn⁰ dust provided a much larger surface area (“A” in Equation (10)) for gold precipitation (Equation (11)), while the deaeration significantly reduced Zn⁰ consumption through O₂. Clearly, a concurrent reaction for Zn⁰ consumption is eliminated.

Zn⁰ + Au²⁺ ⇒ Zn²⁺ + Au⁰

(11)

Replacing Zn⁰ by Fe⁰, in the context of Fe⁰ filters, iron corrosion by water (Equation (2)) is the main reaction and corresponds to Equation (11).

It has been established that disturbance through dissolved O₂ results from the acceleration of Equation (2) by consumption of Fe^II species that are the effective reducing agents for O₂. Considering permeability loss, Fe⁰ corrosion in the presence of O₂ yields more voluminous Fe^III oxides (e.g., Fe₂O₃, FeOOH) and hydroxides (

Table 1. Theoretical ratios (η values) between the volume of expansive corrosion products (V_oxide) and the volume of iron consumed during the corrosion process (V_Fe). η values are compiled by Caré et al. [60]. The ratios of ρ values (densities) indicate that corrosion products are 1.5 to 2.2 times larger in volume than the parent Fe⁰.

Species	Formula	ρ (kg m−3)	ρiron/ρoxide	Voxide/Viron
Iron	Fe	7800	−	−
Hematite	Fe2O3	5260	1.5	2.12
Magnetite	Fe3O4	5180	1.5	2.08
Goethite	α-FeOOH	4260	1.8	2.91
Akageneite	β-FeOOH	3560	2.2	3.48
Lepidocrocite	γ-FeOOH	4090	1.9	3.03

). Table 1 summarizes the densities of iron corrosion products and the corresponding coefficient of volumetric expansion. If it is assumed that all oxidized Fe remains in the system (no transport of Fe^II/Fe^III), then it becomes evident that, for a constant volume (initial porosity of a Fe⁰ filter), the systems with more oxygen will clog early [7,60].

The logical consequence of the presentation above is that the use of O²⁻ poor leach liquors is a prerequisite for efficient Au recovery through Zn⁰ cementation in packed beds. This process is ultimately known as the Merrill-Crowe precipitation process. Coming back to Fe⁰ filters, it becomes evident that the analysis of the chemistry of the system would have enabled the proper consideration of the volumetric expansive nature of iron corrosion (Table 1) prompted at the start of the technology some 25 years ago.

The major lesson from the Merrill-Crowe precipitation process for Fe⁰ filters is that controlling the O₂ level is essential for the efficiency of a filter. O₂ accelerates corrosion and can be beneficial in some cases. Where necessary, even stronger oxidizing agents can be used [32,61]. However, it should be carefully considered that more corrosion products implies rapid clogging, such that it is certain that long-term permeable systems should operate at low O₂ levels.

4.3. Conventional Approach

Conventionally, numerical models are used to predict the long-term behavior of Fe⁰ filters (see [19] and the references cited therein). Data for such models should be derived from well-designed laboratory column experiments performed under operational conditions (e.g., experimental duration, flow velocity, temperature and water chemistry) mimicking as closely as possible those expected in the field. However, such studies currently provide at best information for understanding the early service life of Fe⁰ filters (some months). In fact, experiments lasting for more than one year are very scarce [19,62,63], while it is not established that k_iron determined under such conditions would describe the behavior of Fe⁰ filters for 10 or 15 years. Moreover, according to Alamilla et al. [51], it is even not likely that this will be the case because of the well-documented non-linearity of the kinetics of iron corrosion [64] (Section 2).

It has been unambiguously demonstrated that all tools used to accelerate lab-scale experiments collectively impair the reliability of the obtained test results [35,65,66,67]. Relevant tools include adding oxidizing agents to accelerate corrosion [58], increasing flow velocity [34] and using Fe⁰ of smaller particle sizes and/or rapid small-scale column tests [65]. For the early phase of column operation, methylene blue (MB) discoloration has been proven a powerful indicator of reactivity [68,69,70,71,72,73,74]. Actually, the longest experiment with MB in this context lasted only for 4 months [69]. Accordingly, the suitability of MB discoloration for characterizing the long-term behavior of Fe⁰ filters is yet to be investigated.

While the reliability of data currently used to model processes in Fe⁰ filters is questioned, there is evidence that used k_iron values are inaccurate. For example, Moraci et al. [19] critically reviewed available approaches and presented a new modeling tool to simulate the permeability loss in Fe⁰ filters. The presented model takes into account: (i) the volumetric expansive nature of iron corrosion; (ii) the precipitation of foreign species; and (iii) gas formation. However, a look at the used k_iron values suggests that there are still misunderstandings to be addressed before acceptable models are presented.

4.4. An Alternative Approach

The process of permeability loss should be investigated at the micro-scale [35]. The size of the Fe⁰ particles, the volume of the space between individual grains and their modifications with the service life of the filter should be simultaneously considered.

4.4.1. Descriptive Aspects

Spherical Fe⁰ particles are considered. Each particle has an initial diameter of d₀ (t₀ = 0). At t > t₀, the residual diameter of the particle (non-corroded Fe⁰) is d. The in situ generated amount of corrosion products corresponds to Δd = d₀ – d. The initial mass of each particle is m_p = ρ_iron × V₀ (V₀ = π × d₀³/6) A certain Fe⁰ mass (m₀) is made up of N₀ particles, such that m₀ = N₀ × ρ_iron × π × d₀³/6. Assuming uniform corrosion, all N₀ particles corrode with the same kinetics (k_iron). At each time (t), the mass of non-corroded Fe⁰ can be determined and the corresponding decrease in diameter (Δd) deduced. The average corrosion rate is given by Equation (12). The best scenario is the one in which Δd is directly measured (in situ).

k_iron = Δd/t

(12)

The surface area of a Fe⁰ particle (A_p) with a diameter d is given by Equation (13):

A_p =π × d²

(13)

The number of Fe atoms covering A_p can be deduced using Equation (14), while considering that each Fe atom (r_atom = 1.24 Å or d_atom = 2.48 Å) covers an effective area equal to a square (d_atom²).

N_A = A_p/d_atom²

(14)

This reasoning has intentionally ignored the evidence that isolated Fe⁰ atoms do not exist; instead, each unit cell of Fe⁰ (BCC) contains two (2) Fe atoms.

One major challenge for the Fe⁰ filtration technology has been to properly consider the contribution of (volumetric expansive nature of) iron corrosion to permeability loss [16,33,34,35,75]. That is coupling Δd to the diminution of the porosity (Challenge 1). Challenge 1 has been resolved using mass balance equations [7,8,35,76]. For example the porosity loss or the residual porosity can be evaluated for several d values, e.g., d₀/2, d₀/4, d₀/8, d₀/16, d₀/32, d₀/64…, d = 0 at t_∞. The open question (Challenge 2) is how to correlate this spatial porosity loss to the service life of Fe⁰ filters (temporal porosity loss). Thus, determining k_iron is the key to model the operation of Fe⁰ filters. In the context of Fe⁰ filters, “permeability loss” and “porosity loss” can be randomly interchanged as permeability loss is mostly mediated by the occupation of the inter-granular pore space (porosity) and, thus, to reduced interconnectivity of available pores (physical clogging). This is the reason why the entrance zone of a filter comparatively experiences increased porosity loss as a rule. This evidence alone makes the discussion of common column parameters like Reynold or Peclet numbers a complex task.

4.4.2. On the Significance of k_iron Values

The k_iron values used by Moraci et al. [19] as calibration parameters are expressed in mmol kg⁻¹ d⁻¹. According to the references cited by these authors, such expressions are common in the Fe⁰ literature. Each k_iron value should be given in mmol particle⁻¹ d⁻¹. The reason being that all Fe⁰ particles are corroded by water (H⁺ or H₂O) with the same kinetics. Admittedly, in the entrance zone, dissolved O₂ and/or other oxidizing agents, including contaminants (e.g., Cu^II, Cr^VI, NO₃₋) and co-solutes (e.g., Ca²⁺, Cl⁻, NO₃₋), will accelerate iron corrosion and, thus, the extent of permeability/porosity loss. However, permeability loss due to Equation (2) [7,9,33,34,35] is ideally uniform. For example, a k_iron value of 30 mmol kg⁻¹ d⁻¹ corresponding to 1680 mg kg⁻¹ d⁻¹ implies a daily dissolution of 1.57 × 10⁻² mmol or 0.88 mg of iron from individual spherical particles having a diameter of 1.0 mm (d value). The equations used for calculations are summarized in the previous section, and the results of all k_iron values used by Moraci et al. [19] are summarized in

Table 2. Values of k_iron used by Moraci et al. [19] and the corresponding t_∞ values. Calculations are made for d = 1.0 mm and m = 1 kg.

kiron (mmol kg−1 d−1)	kiron (mg kg−1 d−1)	kiron (μg particle−1 d−1)	t∞ (years)
0.7	39	0.02	69.8
1.2	67	0.04	40.7
1.5	84	0.04	32.6
4.0	224	0.12	12.2
7.0	392	0.21	7.0
10.0	560	0.29	4.9
12.0	672	0.35	4.1
15.0	840	0.44	3.3
21.0	1176	0.62	2.3
30.0	1680	0.88	1.6
170.0	9520	4.99	0.3
200.0	11,200	5.87	0.2
242.0	13,552	7.10	0.2
300.0	16,800	8.80	0.2

.

Table 2 also summarizes the time to complete material depletion or Fe⁰ exhaustion (t_∞) for the k_iron values used by Moraci et al. [19] assuming uniform corrosion for d = 1.0 mm. It is seen that t_∞ varies from 0.2 to 69.1 years. Keeping in mind that subsurface Fe⁰ reactive walls (Fe⁰ filters) should have a service life in the range of decades, Table 2 suggests that, for d = 1.0 mm, a material exhibiting a k_iron value averaging 4.0 mmol particle⁻¹ d⁻¹ should be used. Such a material would last for 12.22 years (t_∞ value). These calculations suggest that in a cross-disciplinary approach, a challenge for material scientists (Challenge 3) would be to manufacture a (possibly porous) spherical material with a 1.0 mm diameter exhibiting an average of 0.12 μg particle⁻¹ d⁻¹. The same reasoning suggests that a material exhausting within one year (t_∞) should have a k_iron value of 1.47 μg particle⁻¹ d⁻¹. Such a material could be useful for household water filters [77].

It is understood that solving Challenge 3 will just be the first step in designing Fe⁰ filters. In fact, the target k_iron value (e.g., 4.0 mmol particle⁻¹ d⁻¹) depends on a myriad of operational parameters, including the porosity of the Fe⁰ materials, the Fe⁰ ratio in the reactive layer, the nature of the admixing aggregates (e.g., MnO₂, pumice, sand, TiO₂), the water flow velocity (residence time), the water chemistry (the composition of the electrolyte) and the thickness of the reactive layer. The next section proposes an approach for a systematic investigation yielding suitable materials for Fe⁰ filters.

Table 3. Values of the number of particles (N₀) making up 1 kg of Fe⁰ as a function of the d values. The corresponding particle area (A_p) and the number of atoms at the surface (N_A) are specified.

d (mm)	N0	Ap (m²)	NA	NA (μmol)
0.05	1.53 × 1010	7.86 × 107	5.1 × 1013	8.48 × 10−5
0.08	4.53 × 109	1.77 × 10−6	1.1 × 1014	1.91 × 10−4
0.13	9.77 × 108	4.91 × 10−6	3.2 × 1014	5.30 × 10−4
0.25	1.22 × 108	1.96 × 10−5	1.3 × 1015	2.12 × 10−3
0.50	1.53 × 107	7.86 × 10−5	5.1 × 1015	8.48 × 10−3
0.75	4.53 × 106	1.77 × 10−4	1.1 × 1016	1.91 × 10−2
1.00	1.91 × 106	3.14 × 10−4	2.0 × 1015	3.39 × 10−2
1.25	9.77 × 105	4.91 × 10−4	3.2 × 1015	5.30 × 10−2
1.50	5.66 × 105	7.07 × 10−4	4.6 × 1015	7.64 × 10−2
1.75	3.56 × 105	9.63 × 10−4	6.3 × 1015	1.04 × 10−1
2.00	2.39 × 105	1.26 × 10−3	8.2 × 1015	1.36 × 10−1
2.25	1.68 × 105	1.59 × 10−3	1.0 × 1015	1.72 × 10−1
2.50	1.22 × 105	1.96 × 10−3	1.3 × 1015	2.12 × 10−1
2.75	9.18 × 104	2.38 × 10−3	1.5 × 1015	2.57 × 10−1
3.00	7.07 × 104	2.83 × 10−3	1.8 × 1015	3.05 × 10−1

and Figure 1 summarize the trend of the expected changes in efficiency as a function of d values. It is seen that the number of particles decreases exponentially with increasing d values, while the number of atoms at the surface increases. The objective of this communication is to outline that the law of efficiency change as d₀ changes to d in a Fe⁰ filter is not yet established.

5. Designing Long-Term, Efficient Fe⁰ Filters

5.1. The Problem

The main problem of the Fe⁰ filtration technology is the lack of appropriate standard tools to characterize the reactivity of used materials [47,48,49]. In other words, ill-defined materials have been used by various research groups and remediation companies for the past two decades. Accordingly, it is not surprising that only highly qualitative results have been presented. Considering this vacuity, the intention herein is to pave the way for the determination of k_iron values useful for modeling purposes. Clearly, while determining k_iron values useful for models, experiments should be designed to characterize as much parameters as possible in a systematic approach [77,78]. The major advantage of this approach is that the same set of experiments will generate data enabling the discussion of the appropriateness of current approaches, as well. For example, in a three-contaminant system employing arsenic, fluoride and uranium, the correlation of k_iron values to the time-dependent extent of the removal of each contaminant can be properly discussed. In a similar way, repeating the experiments of Luo et al. [34] for the long term will enable the determination of k_iron of the same sample in deionized water, groundwater and acid mine drainage (AMD).

In the absence of a reference material, an operational reference can be selected; for example, one of the materials commercialized by iPuTech mbH (Rheinfelde, Germany) and widely used in laboratory and field studies [47,79]. A material scientist solving Challenge 3 would manufacture Fe⁰ materials to be characterized (k_iron values) and compared to the operational reference. Following this approach, available Fe⁰ materials could be classified according to their k_iron values, which are only indirectly correlated to the intrinsic reactivity. It is understood that neither the reference nor one of the individual materials could be the material for all situations. The objective is to come out with various materials exhibiting various k_iron values, such that selection to meet site-specificity would be simplified. Section 4.3 suggested for Fe⁰ materials with a 1.0 mm diameter, k_iron = 0.12 μg particle⁻¹ d⁻¹ is suitable for subsurface-reactive Fe⁰ walls, whereas k_iron = 1.47 μg particle⁻¹ d⁻¹ is suitable for household water filters.

5.2. Ways to Reliable k_iron Values

A proven tool to determine k_iron values is to allow Fe⁰ materials (initial diameter d₀) to corrode under specific conditions and to determine the time-dependent variation of either (i) the residual diameter (d) or (ii) the amount of generated corrosion product (corresponding to Δd = d₀ – d). The latter can be also expressed by the thickness of the oxide scale (x in Equations (7) to (9)). In essence, a good estimation can be realized by just weighing the dried materials at the start (t₀) and the end (t_∞) of the experiments (mass balance) [80]. In this approach, Δm = m_∞ – m₀ represents (approximately) the mass of generated corrosion products when the column is flushed by deionized water. Depending on the operational conditions (oxic/anoxic), reliable assumptions can be made on the nature of corrosion products to enable modeling porosity loss. If the system is flushed by other solutions, the mass balance of individual species must be considered. Proper assumptions on all reactions likely to occur in the system and their contribution to changes in porosity must be made.

Recent data have proven that (i) the residual diameter (d), (ii) the amount of generated corrosion products and (iii) the nature of corrosion products can be determined by in situ measurements/observations of both d and changes in the pore diameters [34]. In their excellent work using X-ray computed tomography (RX-CT), Luo et al. [34] used three different model waters: (i) distilled water, (ii) synthetic groundwater and (iii) synthetic AMD. Luo et al. [34] also investigated the impacts of three different Fe⁰/sand ratios (w/w): 10:90, 50:50 and 100:0. Duplicating the experiments of Luo et al. [34] for longer experimental durations and more applicable experimental set-ups (for water treatment) will certainly allow the determination of reliable k_iron values for predictive models. In this effort, a pure Fe⁰ column (100 % Fe⁰) could be used just as a tool to shorten the experimental duration or as a negative reference. For long-term basic experiments, a volumetric Fe⁰:aggregate ratio of 25:75 should be used. Such experiments should not be stopped because contaminant breakthrough has occurred. In fact, the focus herein is on the efficiency of tested Fe⁰ materials [69,71,72] and possibly characterizing the time to Fe⁰ exhaustion (t_∞). There is certainly an infinity of possibilities of experimental designs; the present effort will be limited to two approaches: (i) the sampling-port approach (in one column); (ii) the column-in-series approach.

5.2.1. The Sampling-Port Approach

Experiments could be performed with a single column (ID = 5.0 cm, L > 50 cm) having a reactive zone (Fe⁰-bearing) of 50 cm and equipped with five (5) sampling ports S₁, S₂, S₃, S₄ and S₅ at 10, 20, 30, 40 and 50 cm from the inlet, respectively (Figure 2). The influent solution is pumped upwards from a reservoir and at pre-defined time intervals. Effluents from each sample port are analyzed for pH value, dissolved iron, dissolved oxygen and concentrations of contaminants and co-solutes (e.g., Ca²⁺, Cl⁻, HCO₃⁻, H₂PO₄⁻, SO₄²⁻). In addition to aqueous analysis, time-dependent changes of the morphology of Fe⁰ and the pore structure should be documented by RX-CT, for example at the following distances from the inlet: 5 (S'₁), 15 (S'₂), 25 (S'₃), 35 (S'₄) and 45 cm (S'₅) of the reactive zone.

This approach enables a reliable characterization of processes implying the well-documented increased porosity loss in the entrance zone of the Fe⁰ filters (e.g., at S₁ vs. higher S_i stations) and how it is influenced by the presence of oxidizing agents (e.g., Cr^VI, dissolved O₂) in the influent solution. Moreover, by individually varying the influent O₂ concentration and other oxidizing agents, while monitoring its profile in the whole column (S₁ through S₅), a better discussion of its impact on the porosity loss is realized. The last important feature from this experimental approach is that, before O₂ breakthrough at S₄, the porosity loss at S’₅ is attributed solely to corrosion through water (Equation (1)) under the experimental conditions (anoxic conditions). Accordingly, by purposefully varying the composition of influent solution, the impact of relevant operational parameters (e.g., Cl⁻, HCO₃⁻, SO₄²⁻) on the extent of porosity loss through water can be characterized.

In the absence of RX-CT, or complementary to this tool, other approaches for monitoring changes of the water flow velocity at individual sample ports can be used. The reduction of the pore size is then deduced from the Darcy equation. At the end of the experiments (at least two years), the column is dismantled after the last RX-CT analysis, and the non-corroded Fe⁰ materials and the corrosion products are characterized by various analytical tools, including selective extraction. This effort characterizes the changes of the following parameters as a function of the distance from the inlet: (i) the extent of Fe⁰ depletion (e.g., in %); (ii) the average size of non-corroded Fe⁰; (iii) the nature, the abundance and the reactivity of individual corrosion products; and (iv) the extent of the reduction of pore sizes.

5.2.2. The Column-In-Series Approach

The sample-port approach requires some sophisticated equipment, including the column and the pump. For less equipped laboratories (e.g., in the developing world), this could be prohibitively expensive. For these situations, small research units could be made up of five shorter columns (e.g., ID = 5.0 cm; L = 30 cm) in series (Figure 3). Individual columns are mostly filled with sand and contain a 10 cm-thick reactive layer. In this embodiment, each column corresponds roughly to 10 cm of the sample-port approach (Figure 2). The larger sand mass between the reactive layers enables the characterization of the impact of in situ-coated sand on the filter’s efficiency. For Columns 1 through 4, the effluent from one column is routed to the next through a tubing system comprising a sampling station/valve (Figure 3).

The results are exploited similarly as for the sample-port approach, the main objective being to determine the long-term kinetics of iron corrosion (k_iron). In this approach the system can be gravity fed. For example, it can be daily fed by 1000 mL of a model solution. For both approaches, the level of dissolved O₂ in the influent solution can be regulated by pre-filters: a relevant pre-filtration system is a slow sand filter (SSF) in which O₂ is consumed within a biofilm [81]. However, working with an SSF implies a low water flow velocity. Alternative approaches to decrease/lower the O₂ level include using a sacrificial Fe⁰ column containing up to 10 vol% Fe⁰ possibly of a smaller particle size. In fact, Fe⁰ is a well-documented O₂ scavenger that is used in food packaging [82,83].

5.3. The Significance of the Short-Term Kinetics of Fe⁰ Corrosion

The effort herein is to prepare the way for the determination of k_iron values of crucial importance for modeling scenarios to design long-term, efficient Fe⁰ filters. Before being efficient for the long term, however, Fe⁰ filters have to be pre-conditioned in a “maturation” phase. Data from Kowalski and Sogaard [84] suggest that in their case, this maturation phase has lasted for six months. In the subsurface context, this means that the filtration system must contain short-term, efficient materials to assure contaminant removal in the initial service life of Fe⁰ filters. Examples are adsorbents or more reactive Fe⁰ materials, possibly with smaller particle sizes. The presentation above demonstrates that the short-term rate constants of Fe⁰ corrosion (k⁰_iron) have been mistakenly used to interpret results and/or to predict the behavior of long-term experiments. According to Alamilla et al. [51], the function describing the instantaneous rate of iron corrosion is expressed as follows:

v(t) = k_iron + (k⁰_iron - k_iron) × exp(−q₀ × t)

(15)

where q₀ is a constant, k⁰_iron is the short-term and k_iron the long-term Fe⁰ average corrosion rate. It can be seen that the instantaneous rate drops exponentially from k⁰_iron to k_iron (k⁰_iron > k_iron).

The real significance of k⁰_iron values for subsurface Fe⁰ filters is that materials should be selected such that after the initial rapid corrosion, the permeability of the reactive zone is still superior to that of the surrounding areas. If this condition is not satisfied, preferential flow would render the filter useless before the long-term corrosion rate (k_iron) is established.

While the short-term corrosion rate (k⁰_iron value) is to be properly considered in designing subsurface Fe⁰ filters, it could be sufficient to design some filters for the treatment of wastewaters from various sources (agriculture, industry, mine, residential area). Here, filters may be designed to operate just for some days to weeks, and material “regeneration” consists of just letting “used” materials be “activated” by humidity and air O₂ for a certain time before being re-used. k⁰_iron values would be also sufficient to design several decontamination options using Fe⁰ particles in batch or fluidized column systems. In this context, the system efficiency can be purposefully enhanced by using more or less powerful oxidizing agents [10,61], because bed clogging is not an important issue. All that is needed is increased Fe⁰ corrosion to generate more contaminant scavengers [11,85,86]. k⁰_iron values are also sufficient to design cementation beds for metal ion recovery.

6. Discussion

6.1. Contaminant Removal in Fe⁰/H₂O Systems

The reliance of modelers on direct reduction by Fe⁰ (electrons from the core metal) is not only of concern for modeling the long-term permeability of Fe⁰ filters [3,87,88]. It strongly masks the real mechanisms of contaminant removal as that is a complex interplay between Fe⁰ particles, Fe^II released into solution and Fe^II/Fe^III precipitation [89,90,91,92,93,94,95,96]. In particular, the transformation of Fe⁰ to Fe oxides goes through colloids and hydroxides. The transformation “colloids to hydroxides” is accompanied by contaminants enmeshment (co-precipitation). In fact, whether they are reduced or not, contaminants are removed by adsorption, co-precipitation and size exclusion. On the other hand, regarding the abundance of (partly nascent) adsorbing sites and continuous generation of Fe^II species, contaminant chemical reduction by so-called structural Fe^II (adsorbed Fe^II) is favored [96]. The abundance of Fe^II/Fe^III hydroxides/oxides in Fe⁰/H₂O systems has been experimentally demonstrated by [92]. However, the authors have suggested adsorptive removal as an important removal mechanism accompanying direct reduction, which is in contradiction to the mechanisms presented in Section 2. Accordingly, coming models for Fe⁰/H₂O systems should be adjusted to address the main contaminant removal pathways.

6.2. The Shortcomings of Current Modeling Efforts

An overview of the Fe⁰ filtration literature reveals conflicting information regarding the causes of permeability loss [7,75]. There is always a discrepancy between the observed and the predicted porosity loss; the actual porosity loss being higher than that predicted [75]. Two examples for illustration: (i) Mackenzie et al. [82] measured a 5 to 10% porosity loss using tracer tests, while calculations based on mineral precipitation predicted only 1 %; and (ii) Kamolpornwijit and Liang [97] measured porosity losses of 25 to 30% based on tracer tests and attributed 1.3% to trapped gas. Mackenzie et al. [82] attributed the remaining porosity loss (up to 9%) to H₂ gas production. Henderson and Demond [75] realized that available models for Fe⁰ filters [98,99,100] do not account for the effects of H₂ on permeability loss. In a series of carefully-designed experiments, Henderson and Demond [75] could validate their working hypothesis that the majority of porosity loss is not attributable to precipitates, but to gas.

The problem with the data of Henderson and Demond [75] is that they have not considered that each individual Fe⁰ atom corrodes and produces both volumetric expansive precipitates and H₂ gas [35]. Moreover, the pore filling by Fe precipitates was evaluated using tabulated molar volume values (ν = 7.6 cm³ mol⁻¹ for Fe⁰), which are not necessarily relevant for Fe⁰ particles confined in a packed bed (Fe⁰ filter). For example, the coefficient of volumetric expansion (η = V_oxide/V_iron) for magnetite (Fe₃O₄; ν = 45.0 cm³ mol⁻¹) using ν values is 5.0, while the η value for confined system is close to 2.2 [60]. This combined with the lack of appropriate k_iron values has diminished the value of the models presented.

6.3. The True Value of Mathematical Modeling

Mathematical modeling constitutes the third pillar of science and engineering, achieving the fulfilment of the two more traditional disciplines: (i) theoretical analysis; and (ii) experimentation [101]. Mathematical models explore new solutions in a very short time period. This increases the speed of innovation cycles. However, this is only possible when theoretical analysis and experimentation are perfectly performed. The presentation herein has recalled that for Fe⁰ filters, both theoretical analysis [102,103] and experimentation [104] have been unsatisfactory. This suggests that there is a huge gap between calculations and understanding [105]. Admittedly, all modelers are not chemists nor electrochemists, but even in the absence of a real cross-disciplinary approach, self-critical scientists should have realized that something fundamental is missing. Moreover, consultations with material and corrosion engineers would have been necessary in the fitness of things to save millions of money (mostly from tax payer) spent in intensive research on a biased basis. The damage is huge, as the credibility of modeling could be questioned [105]. This communication advocates for saving the credibility of mathematical modeling through the generation of credible data for Fe⁰ filters.

The true value of mathematical models was recalled by Grauer [106] by the following wording: “theory and models cannot be validated in the strict sense but only refuted“. The presentation herein has collectively refuted all modeling efforts regarding the design and operation of Fe⁰ filters for the past two decades. The reason is the very poor theoretical analysis. The equation currently used for k_iron (corresponding to Equation (1)) is simply wrong, and the reductive transformation of contaminants is at best a side process (Equations (3) and (4)) (Section 2).

Better models are possible, they should be based on a systematic analysis of the Fe⁰/H₂O system. A sound analysis [102,103] has demonstrated that contaminants are removed by adsorption, co-precipitation and size exclusion. Accordingly, chemical reduction when it occurs is not responsible for (quantitative) contaminant removal and is rather a side reaction occurring within the oxide scale in the vicinity of Fe⁰ and not at its surface. The evidence that the surface of Fe⁰ and oxides changes with time and the Fe⁰ surface is not accessible to the contaminant question of the normalization of rate constants to the surface area [107]. Lastly, k_iron values are the pillar for modeling the temporal realization of permeability loss. These values are yet to be determined in well-designed long-term experiments.

6.4. Messages to Anonymous Collaborators

There is evidence in the Fe⁰ literature that the aspects challenged herein are difficult to accept [108,109,110,111,112]. However, this argumentation corresponds to mainstream science (chemical thermodynamics). The first message to an anonymous collaborator (a reader) is “no single reference is a quality guarantee”. Accordingly, sound scientific arguments should justify each reference. Moreover, the rationale for preferring a work to another should be known and named, if applicable.

The second message is for a reviewer either for a submitted manuscript or a grant proposal. An author/applicant is a colleague and needs collegial remarks to improve his/her manuscript/proposal. A minimal requirement is that reviewer comments focus on points that need improvements or are unacceptable [113,114,115].

The third message is to the attention of funding agencies. Each applicant should have the opportunity to write some comments on the reviewer's evaluation before the final decision is made. This is of particular importance when the majority of available comments is negative. Admittedly, this could/would lengthen the review process, but this is the way to greater fairness. This procedure is already adopted by some agencies, but should be universally adopted. The reason is that the best evaluator could not recognize an innovation at first glance.

The fourth and last message is for everyone. Research on water treatment and environmental remediation has been wrongly directed from the beginning onward. Warnings from chemists [116,117,118,119] were simply ignored. Whenever a mistake is identified, starting on a new basis is always a good decision. Individual researchers already went down this path [10,119]. As an example, the author of this communication has published on reductive precipitation of U^VI by Fe⁰ [120], then on reduction as a removal mechanism, until 2010 [14], only with [121] and hints from anonymous reviewers did he realize that at concentrations relevant for safe drinking water, reduction cannot be a stand-alone removal mechanism for any contaminant [78,122]. For environmental remediation, chemical reduction (degradation) may be sufficient when reaction products are readily degradable, for instance when nitrobenzene is quantitatively reduced to aniline. For safe drinking water provision, even reduction products (here, aniline) must be removed [123,124,125,126].

7. Concluding Remarks

Iron corrosion is a stochastic, probabilistic phenomenon that requires interdisciplinary concepts, including chemistry, electrochemistry, hydrodynamics, materials science, metallurgy, physics and surface science. From a pure chemical perspective, the thermodynamics and kinetics of the involved processes must be understood in order to effectively design and fabricate materials to be used for optimal (efficient and affordable) applications. This communication has recalled that the chemistry of the Fe⁰/H₂O system is being incompletely considered while using Fe⁰ as a remediation material. It is particularly disappointing/frustrating that the principles of corrosion as already understood and largely disseminated in the broad scientific literature have been simply ignored. The net result is that a fictive reaction (Equation (1)) has been used for dimensioning Fe⁰ filters for more than two decades (25 years).

This communication has also recalled that mathematical modeling is the third and only the third pillar of science and technology. The two first pillars are theoretical analysis (system analysis) and experimentation. The best model will not correct mistakes in system analysis or experimentation. It is clear that Fe⁰ filtration technology requires more accurate models that take into account the various mechanisms that underpin the process. The year-long experimentation on iron corrosion (prior to the advent of Fe⁰ filters) has demonstrated that uniform corrosion assumed by almost all models is an over-simplification. In other words, even considering reliable values of k_iron will only describe how Fe⁰ filters would behave considering the assumptions made (without biotic processes). This evidence confirms/suggests that (independent, where necessary) monitoring of the performance of Fe⁰ filters is mandatory.

To conclude, Rolf Grauer will be paraphrased as follows: It would make more sense to devote more time and energy to determine k_iron values than to develop sophisticated, but unreliable numerical models. It is clearly understood that no particular Fe⁰ material is a universal solution for all contaminations. Each and every case has to be considered in its totality before a decision is made on the proper material, characterized by its k_iron value. Therefore, a sort of database for k_iron values is needed. Establishing such a database is a scientific challenge and not just laborious work.