Performance and Kinetics of Anaerobic Digestion of Sewage Sludge Amended with Zero-Valent Iron Nanoparticles, Analyzed Using Sigmoidal Models

Abstract

Sewage sludge was treated with nanoscale zero-valent iron (nZVI) to enhance biogas and methane (CH₄) production, and the influence of key parameters on the material’s anaerobic digestion (AD) efficiency was analyzed using sigmoidal mathematical models. In this study, three dosages of nZVI (0.5%, 1.5% and 3%) were added to the anaerobic sludge digestion system to enhance and accelerate the sludge decomposition process. The results showed that cumulative biogas yield after 41 days of digestion increased by 23.9% in the reactor with a nZVI dosage of 1.5%. Correspondingly, the highest CH₄ production enhancement by 21.5% was achieved with a nZVI dosage of 1.5% compared to the control. The results indicated that this nZVI dosage was optimal for the AD system, as it governed the highest biogas and CH₄ yields and maximum removal of total and volatile solids. Additionally, to predict biogas and CH₄ yields and evaluate kinetic parameters, eight kinetic models were applied. According to the results of the modified Gompertz, Richards and logistic models, the nZVI dosage of 1.5% shortened the biogas lag phase from 11 to 5 days compared to the control. The Schnute model provided the best fit to the experimental biogas and CH₄ data due to highest coefficients of determination (R²: 0.9997–0.9999 at 1.5% and 3% nZVI dosages), as well as the lowest Akaike’s Information Criterion values and errors. This demonstrated its superior performance compared to other models.

1. Introduction

In municipal wastewater treatment plants (WWTPs), large amounts of sludge are generated during the wastewater treatment process, which accumulates in primary and secondary settling tanks [1,2]. Larger solid particles settle in the primary settling tanks, while biologically degradable materials accumulate in the secondary ones [3,4]. Sludge management accounts for up to 50% of all operational costs of WWTPs, making efficient sludge handling essential to reduce environmental pollution and optimize operational expenses [5].

Sludge management methods include disposal and secondary utilization. Disposal involves sludge incineration for energy production or landfill disposal; however, these methods can have a negative environmental impact [6]. Secondary utilization, such as using sludge in agriculture as fertilizer, requires strict control due to potential contaminants [7]. Therefore, anaerobic sludge digestion is becoming an attractive alternative, enabling a reduction in sludge volume and the production of biogas, which can be used for energy generation [8,9]. Anaerobic sludge digestion is a process where microorganisms break down organic matter present in wastewater sludge without oxygen, producing biogas, the main component of which is methane (CH₄) [10,11]. This method reduces sludge volume, stabilizes its composition, and generates renewable energy [12,13]. However, traditional anaerobic digestion (AD) processes have certain drawbacks, such as long processing times (20–30 days) and low organic matter degradation efficiency (30–40%) [9,14].

Despite its advantages, AD faces challenges such as the slow degradation rate of excess activated sludge and insufficient CH₄ yield [15,16,17]. Additionally, contaminants present in the sludge can inhibit microbial activity, and hydrogen sulfide (H₂S) formed in the biogas can cause corrosion and require additional cleaning. Therefore, it is necessary to explore ways to improve process efficiency and biogas quality [18,19,20].

To increase CH₄ yield and improve biogas quality, various methods for optimizing the AD process are being investigated. These include sludge pretreatment, such as ultrasonic disintegration, thermal treatment, chemical additives, optimization of microbial communities and application of a static magnetic field (SMF) [21,22,23]. Applying an SMF is an effective, innovative, and sustainable approach to enhance the biodegradation of organic matter and biogas production [23]. Additionally, it can serve as a pretreatment for sewage sludge to facilitate precipitation of compounds with fertilizing properties, such as struvite [24]. It is also important to control process parameters, such as temperature, pH, mixing intensity, and hydraulic retention time, to ensure optimal conditions for methanogenic bacteria [25,26,27]. One effective way to enhance the AD process is the addition of specific supplements, such as iron compounds [28]. Iron additives can reduce hydrogen sulfide concentrations in biogas, improve microbial activity, and increase CH₄ concentration [29].

Iron is a vital micronutrient, playing a crucial role in enhancing microbial growth [30]. Several studies showed that H₂ generated from the corrosion of nZVI stimulates hydrogenotrophic methanogenesis and reduces the oxidation–reduction potential to provide a better anaerobic environment for methanogens [31,32]. Wang et al. [33] also indicated that nZVI addition was beneficial for hydrogenotrophic methanogens rather than acetoclastic methanogens as the relative abundances of the genera Methanobacterium, Methanospirillum and Methanolinea were positively correlated with the nZVI dosages. However, elevated nZVI dosages can have a negative impact on bacterial activity. For example, a study performed by Yang et al. [34] indicated that Fe²⁺ or Fe(OH)⁺ released from nZVI can react with S²⁻ to form FeS or bind with PO₄³⁻ to create stable complexes, thus hindering phosphorus uptake by methanogens.

Different studies have shown that the introduction of iron additives enhances the startup phase and the rate at which total solids decompose as the content of volatile solids is directly linked to biogas production. It was shown that sludge treatment with nanoscale zero-valent iron (nZVI) improved the hydrolysis stage and methanogenic activity. For example, research performed by Feng et al. [35] observed an increase in CH₄ production up to 43.5% and an increase in activities of several hydrolysis/acidification enzymes up to 0.6/1 times in the presence of zero-valent iron. Other research indicated that iron oxide-based additives can increase CH₄ concentration in biogas by up to 8% [36]. Although the number of studies addressing the improvement in biogas quality has been growing recently, there is still a significant lack of recommendations to ensure this process in practice. The AD of wastewater sludge is not fully controlled, and there is a shortage of practically applicable measures to optimize the operation of wastewater treatment plants.

Mathematical models are important tools for understanding and predicting the dynamics of the AD process. They help analyze the impact of various parameters on process efficiency, optimize operating conditions, and predict biogas and CH₄ yields [37,38]. For example, modeling and simulation studies indicate that properly selecting process parameters can improve biogas production in plug–flow reactors [39]. Sigmoidal kinetic models have gained significant attention in the study of AD of sewage sludge due to their ability to accurately describe the characteristic S-shaped behavior of cumulative biogas and CH₄ production over time. These models, such as the modified Gompertz, Richards and logistic function equations, effectively capture the distinct phases of the AD process, including the initial lag phase, biogas production potential and biogas production rate. Unlike simple first-order or Monod-type models, sigmoidal models account for the adaptation period of microbial communities, substrate availability, and potential inhibitory effects, providing a more realistic representation of the dynamic nature of anaerobic systems. Studies have demonstrated the superior performance of sigmoidal models in fitting experimental biogas production data, with the modified Gompertz model showing high accuracy across various substrates and operational conditions [40,41]. Their flexibility and ease of parameter estimation make them valuable tools for predicting biogas and CH₄ yields, assessing process performance, and optimizing operational conditions in sewage sludge digestion processes.

Currently, most studies that have treated sludge with nZVI have primarily used the modified Gompertz model to assess the kinetics of the anaerobic digestion process [42,43,44]. However, there is a notable lack of comprehensive evaluations of AD of sewage sludge treated with nZVI using a variety of other sigmoidal kinetic models. Therefore, the aim of this study was to investigate the sludge digestion process using nZVI additive to enhance biogas yield, CH₄ concentration and the overall performance of AD. Eight different sigmoidal models were employed to analyze the kinetic parameters of AD, identify the best-fitting model for biogas and CH₄ production data, and determine the optimal dosage of nZVI.

2. Materials and Methods

2.1. Materials

Substrate (a mixture of primary and secondary sludge, SS) and inoculum (anaerobically digested sludge, Inoc.) were collected from a wastewater treatment plant located in Vilnius, Lithuania. The plant treats 120,000 m³ of wastewater daily. The collected substrate and inoculum were stored in stainless steel tanks before the start of the experiments. The characteristics of substrate, inoculum and their mixture are presented in

Table 1. Characteristics of substrate (SS), inoculum (Inoc.) and a mixture of SS and Inoc. (mean ± SD), n = 3.

Parameters	Substrate	Inoculum	Mixed Sludge (SS + Inoc.)
Total solids (TS, %)	8.52 ± 0.03	4.29 ± 0.03	7.33 ± 0.06
Volatile solids (VS, %)	5.14 ± 0.02	2.53 ± 0.02	4.39 ± 0.04
VS/TS (%)	60.27	58.96	59.86
pH	5.41 ± 0.11	7.69 ± 0.15	6.96 ± 0.003
Chemical oxygen demand (COD, g/L)	82.967 ± 0.586	35.533 ± 0.208	70.967 ± 0.252

. The substrate showed high concentrations of total solids (TS), volatile solids (VS) and chemical oxygen demand (COD) (8.52%, 5.14% and 82.9 g/L, respectively), and this proved its high potential for biogas production. Zero-valent iron nanoparticles (NANOFER STAR) in the form of agglomerates were purchased from Nano Iron s.r.o. (Židlochovice, Czech Republic). The nZVI particles were activated by mixing these with deionized water.

2.2. Experimental Setup

Figure 1 displays the schematic diagram of the experimental AD system used. The experiments were performed in batch mode and mesophilic conditions. The AD system consisted of four reactors, which were constructed using digestion tanks wrapped in heating mats and biogas collection tanks. All reactors were equipped with mechanical mixers with speed control (120 rpm). Additionally, the scheme is supplemented with visual representation of wastewater treatment plant, which was chosen to collect a mixture of primary and secondary sludge, and anaerobically digested sludge material (substrate and inoculum) for the AD experiment.

Each reactor had a total capacity of 17 L, with 14 L as the working volume. To each reactor, 10.5 L of sewage sludge and 3.5 L of inoculum were added to create 3 L of head space. Aliquots of nZVI particles were then added into three reactors (B2, B3 and B4) to reach final concentrations of 0.5%, 1.5% and 3% (based on TS) in the AD systems. The control (blank group without nZVI particles, B1) was subjected to the same procedures. The experimental design is presented in

Table 2. Overview of the anaerobic batch experiments.

Reactor	Added Substrate Amount (kg)	Added Inoculum Amount (kg)	Added nZVI Amount (g)
B1	10.5	3.5	0
B2			5
B3			15.1
B4			30.2

. After filling the reactors with the respective combination of substrate, inoculum and aliquots of nZVI, the reactors were tightly sealed. Once it was ensured that the reactors were properly sealed, they were placed in a temperature-controlled environment with heating mats and a temperature-controlling device, which was set to a mesophilic temperature of 37 °C ± 1 °C [45]. The amount of produced biogas was measured using the water displacement technique each day for 41 days. The biogas collection system was made up of graduated cylinders positioned upside down inside water-filled containers. This study was carried out until biogas production was completed. The whole experiment was performed in triplicate.

2.3. Analytical Methods and Performance Parameters

Biogas composition (CH₄, carbon dioxide and residual other gases in percentage) was analyzed using a calibrated and portable Gas Analyzer (GFM 406, Gas Data Ltd., Coventry, UK). pH was measured with a digital pH meter (SevenMulti^TM, Mettler Toledo GmbH, Schwerzenbach, Switzerland). TS and VS were measured by the standard methods and determined by differential weighing after drying at 105 °C and 550 °C, respectively [46,47]. COD was analyzed using a chemical oxygen demand colorimeter (Lovibond MD100, Tintometer GmbH, Dortmund, Germany). Specific biogas/CH₄ production in AD is typically represented based on the mass of added VS, serving as a measure of assay’s precision. Specific biogas/CH₄ production and the performance of the digestion were calculated according to Equations (1) and (2), respectively [48,49]:

$$\mathrm{Specific} \mathrm{biogas} \mathrm{or} \mathrm{C}{\mathrm{H}}_4\mathrm{production} ({\displaystyle \frac{\mathrm{m}\mathrm{l}}{\mathrm{g}-\mathrm{V}{\mathrm{S}}_{\mathrm{a}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{d}}}})={\displaystyle \frac{\mathrm{Volume} \mathrm{of} \mathrm{biogas}/\mathrm{C}{\mathrm{H}}_4}{\mathrm{Mass} \mathrm{of} \mathrm{added} \mathrm{VS} \mathrm{content}}},$$

(1)

$$\mathrm{Biogas} \mathrm{or} \mathrm{C}{\mathrm{H}}_4 (\%)={\displaystyle \frac{{(\mathrm{Biogas}/\mathrm{C}{\mathrm{H}}_4)}_{\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}}-{(\mathrm{B}\mathrm{i}\mathrm{o}\mathrm{g}\mathrm{a}\mathrm{s}/\mathrm{C}{\mathrm{H}}_4)}_{\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}}}{{(\mathrm{B}\mathrm{i}\mathrm{o}\mathrm{g}\mathrm{a}\mathrm{s}/\mathrm{C}{\mathrm{H}}_4)}_{\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}}}}\times 100,$$

(2)

where (CH₄)_treated—CH₄ production value obtained for the reactor with nZVI particles; (CH₄)_untreated—CH₄ production value obtained from the control reactor. COD, TS and VS reduction is also usually used to evaluate the reactor performance and stability of the digestate; therefore, the removal efficiencies (%) of COD, TS and VS were calculated according to Equation (3) [50,51]:

$$\mathrm{C}{\mathrm{O}\mathrm{D}}_{\mathrm{R}\mathrm{E}}\mathrm{or} \mathrm{T}{\mathrm{S}}_{\mathrm{R}\mathrm{E}} \mathrm{or} \mathrm{V}{\mathrm{S}}_{\mathrm{R}\mathrm{E}} (\%)={\displaystyle \frac{{\mathrm{C}}_0-{\mathrm{C}}_{\mathrm{t}}}{{\mathrm{C}}_0}}\times 100,$$

(3)

where C₀—initial COD/TS/VS concentration (g/L); C_t—final COD/TS/VS concentration (g/L).

2.4. Kinetic Simulation

Eight kinetic models, i.e., the Gompertz model (GM), modified Gompertz model (MGM), Richards model (RM), modified Richards model (MRM), logistic model (LM), modified logistic model (MLM), Cone model (CM) and Schnute model (SM), were used to describe the cumulative biogas/CH₄ potential and kinetic parameters. A description of these models, their equations and kinetic parameters is presented in Table 3. The kinetic parameters were determined using a cost function aimed at minimizing the residual sum of squares (RSS) between the model-predicted results and the experimentally obtained values. For this calculation, the generalized reduced gradient (GRG) non-linear solver in Microsoft Excel was utilized [52]. The mathematical formula for RSS is provided in Equation (4):

$$\mathrm{R}\mathrm{S}\mathrm{S}={\sum }_{\mathrm{i}=1}^{\mathrm{n}}{({\mathrm{Y}}_{\mathrm{i},\mathrm{e}\mathrm{x}\mathrm{p}}-{\mathrm{Y}}_{\mathrm{i},\mathrm{c}\mathrm{a}\mathrm{l}})}^2,$$

(4)

Generally speaking, all cumulative models can be classified into two types: exponential models (such as first-order, CM) and sigmoidal models (RM, GM, LM in their original or modified form) [53]. Sigmoidal models can be recognized by the inclusion of a lag factor (λ) in their equations, which accounts for the initial delay in growth before the exponential phase begins. The MGM, one of the most commonly used sigmoidal equations, effectively describes the decomposition of simple organic feedstock by utilizing a reverse L-shaped curve and the decomposition of more complex substrates using an S-shaped curve [54]. The MGM is an empirical non-linear regression approach used to characterize cell density during the growth phases of methanogenic bacteria, accounting for both exponential growth rates and the duration of the λ. Previous studies have demonstrated that the MGM equation is a widely used model for predicting biogas and CH₄ yields during the degradation of simple organic substrates. The RM incorporates a fourth parameter (the shape factor, v), which enhances its ability to align more closely with the experimental data [55]. RM is almost the same as MRM, which includes the fifth additional parameter (initial biogas/CH₄ production, P₀). Substituting the growth rate parameter (α) and scaling parameter (β) in the Schnute equation leads to the MRM [56]. Some studies have shown that CM demonstrates the highest performance in accurately modeling CH₄ production from co-digestion [40,57]. RM, CM and LM successfully represent S-shape curves, which are typical for complex substrates like those containing fats and lipids [54]. One kinetic parameter, which was described in this study more detailed, is λ, which is a crucial parameter that indicates the effectiveness of AD and can be calculated using different kinetic models (MGM, MLM, RM and MRM).

Table 3. Various kinetic sigmoidal models, their equations and parameters.

Model	Equation Form and Parameter Definition	Reference
Gompertz	$P\left(t\right)=P_{max}\times exp\left[-{\displaystyle \frac{r_0}{\alpha }}\times \mathrm{e}\mathrm{x}\mathrm{p}\times \left(-\alpha t\right)\right]$, where α: decay constant or growth rate parameter (1/d); r0: initial rate of biogas production (mL/g-VSadded/d).	[55]
Modified Gompertz	$P(t)=P_{max}\times exp\left[-exp\left({\displaystyle \frac{R_{max}\times e}{P_{max}}}\times \left(\lambda -t\right)+1\right)\right]$, where P(t): cumulative biogas/CH4 production at time t (mL/g-VSadded); Pmax: maximum production potential of biogas/CH4 (mL/g-VSadded); Rmax: maximum biogas/CH4 production rate (ml/g-VSadded/d); λ: duration of the lag phase (d); t: time of anaerobic digestion (d); e: Euler’s constant, which is equal to 2.7183.	[58]
Richards	$P\left(t\right)=P_{max}\times {\left[1+v\times exp(1+v)\times exp\left({\displaystyle \frac{R_{max}}{P_{max}}}\times \left(1+v\right)\times \left(1+{\displaystyle \frac{1}{v}}\right)\times \left(\lambda -t\right)\right)\right]}^{-{\displaystyle \frac{1}{v}}}$, where ν: shape factor (dimensionless).	[59]
Modified Richards	$P(t)=P_0+P_{max}\times {\left[1+v\times \mathrm{e}\mathrm{x}\mathrm{p}(1+v)\times exp\left({\displaystyle \frac{R_{max}}{P_{max}}}\times (1+v)\times (1+{\displaystyle \frac{1}{v}})\times (\lambda -t)\right)\right]}^{-{\displaystyle \frac{1}{v}}}$, where P0: initial biogas/CH4 production (mL/g-VSadded).	[56,60]
Logistic	$P(t)={\displaystyle \frac{P_{max}}{1+e^{-k(t-t_0)}}}$, where k: specific growth rate constant or reaction rate coefficient (1/d); t0: time at which production rate is the highest (d).	[61]
Modified logistic	$P(t)={\displaystyle \frac{P_{max}}{1+exp\left(\frac{4\times R_{max}\times (\lambda -t)}{P_{max}}+2\right)}}$	[59]
Cone	$P\left(t\right)={\displaystyle \frac{P_{max}}{1+{\left(k\times t\right)}^{-n}}}$, where n: shape factor (dimensionless); k: hydrolysis rate constant (1/d).	[62]
Schnute	$P\left(t\right)=P_{max}\times exp\left({\displaystyle \frac{-\alpha t}{\beta }}\right)\left(\exp \left(\alpha t\right)-{\displaystyle \frac{\beta r_0}{\alpha +\beta r_0}}\right)$, where β: scaling parameter controlling the curve shape (dimensionless).	[63]

Table 3. Various kinetic sigmoidal models, their equations and parameters.

Model	Equation Form and Parameter Definition	Reference
Gompertz	$P\left(t\right)=P_{max}\times exp\left[-{\displaystyle \frac{r_0}{\alpha }}\times \mathrm{e}\mathrm{x}\mathrm{p}\times \left(-\alpha t\right)\right]$, where α: decay constant or growth rate parameter (1/d); r0: initial rate of biogas production (mL/g-VSadded/d).	[55]
Modified Gompertz	$P(t)=P_{max}\times exp\left[-exp\left({\displaystyle \frac{R_{max}\times e}{P_{max}}}\times \left(\lambda -t\right)+1\right)\right]$, where P(t): cumulative biogas/CH4 production at time t (mL/g-VSadded); Pmax: maximum production potential of biogas/CH4 (mL/g-VSadded); Rmax: maximum biogas/CH4 production rate (ml/g-VSadded/d); λ: duration of the lag phase (d); t: time of anaerobic digestion (d); e: Euler’s constant, which is equal to 2.7183.	[58]
Richards	$P\left(t\right)=P_{max}\times {\left[1+v\times exp(1+v)\times exp\left({\displaystyle \frac{R_{max}}{P_{max}}}\times \left(1+v\right)\times \left(1+{\displaystyle \frac{1}{v}}\right)\times \left(\lambda -t\right)\right)\right]}^{-{\displaystyle \frac{1}{v}}}$, where ν: shape factor (dimensionless).	[59]
Modified Richards	$P(t)=P_0+P_{max}\times {\left[1+v\times \mathrm{e}\mathrm{x}\mathrm{p}(1+v)\times exp\left({\displaystyle \frac{R_{max}}{P_{max}}}\times (1+v)\times (1+{\displaystyle \frac{1}{v}})\times (\lambda -t)\right)\right]}^{-{\displaystyle \frac{1}{v}}}$, where P0: initial biogas/CH4 production (mL/g-VSadded).	[56,60]
Logistic	$P(t)={\displaystyle \frac{P_{max}}{1+e^{-k(t-t_0)}}}$, where k: specific growth rate constant or reaction rate coefficient (1/d); t0: time at which production rate is the highest (d).	[61]
Modified logistic	$P(t)={\displaystyle \frac{P_{max}}{1+exp\left(\frac{4\times R_{max}\times (\lambda -t)}{P_{max}}+2\right)}}$	[59]
Cone	$P\left(t\right)={\displaystyle \frac{P_{max}}{1+{\left(k\times t\right)}^{-n}}}$, where n: shape factor (dimensionless); k: hydrolysis rate constant (1/d).	[62]
Schnute	$P\left(t\right)=P_{max}\times exp\left({\displaystyle \frac{-\alpha t}{\beta }}\right)\left(\exp \left(\alpha t\right)-{\displaystyle \frac{\beta r_0}{\alpha +\beta r_0}}\right)$, where β: scaling parameter controlling the curve shape (dimensionless).	[63]

2.5. Kinetic Model Accuracy Evaluation

Additionally, the coefficient of determination (R²), which is known as a fitness index of the models, root mean square error (RMSE), normalized root mean square error (NRMSE, %), Akaike’s Information Criterion (AIC) and Akaike’s weight were used to statistically compare the data fits obtained by the previously mentioned kinetic models [58]. The R² and RMSE are two standard criteria, which are used to report the final model performance. The RMSE is commonly used as a standard statistical metric to evaluate the performance of kinetic models. The RMSE measures how much the predicted values deviate from the actual measured values, expressed in the same units as the variable. By normalizing the RMSE (NRMSE), readers can more effectively evaluate the accuracy of the model’s predictions. The AIC is a widely used tool in statistical modeling and is broadly accepted as a criterion for model selection, including selection of kinetic models applied to the biogas production in an AD system [52]. Lower RMSE, NRMSE, AIC values and higher R² and Akaike’s weight indicate that the model has a better fit. For each model, the R², RMSE, NRMSE, AIC values and Akaike’s weight were determined using the following equations, respectively [60,64,65,66,67]:

$$R^2=1-{\displaystyle \frac{\sum {\left(Y_{i,exp}-Y_{i,cal}\right)}^2}{\sum {\left(Y_{i,exp}-\overline{Y}\right)}^2}},$$

(5)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\left(Y_{i,exp}-Y_{i,cal}\right)}_i^2},$$

(6)

$$NRMSE=\left[{\displaystyle \frac{RMSE}{Y_{max}-Y_{min}}}\right]\times 100,$$

(7)

$$AIC=Nln{\displaystyle \frac{RSS}{N}}+2K,$$

(8)

$$Akaik{e}^{\prime }s weight={\displaystyle \frac{\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{\Delta AIC}{2})}{1+\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{\Delta AIC}{2})}},$$

(9)

where Y_i,exp—observed biogas/CH₄ production data point (mL/g-VS_added); Y_i,cal—predicted biogas/CH₄ production data point (mL/g-VS_added); Y_max—maximum experimental value of biogas/CH₄ production (mL/g-VS_added); Y_min—minimum experimental value of biogas/CH₄ production (mL/g-VS_added); $\overline{Y}$—mean of the observed biogas/CH₄ production data (mL/g-VS_added); N—number of data points; RSS—residual sum of squares; K—number of model parameters; ΔAIC—the relative difference between two AIC values.

2.6. Statistical Analysis

All physical and chemical analyses were conducted three times, and the average values along with the standard errors (SD) were reported. A one-way ANOVA was performed to determine statistical significance between groups, with a p value of less than 0.05 indicating significance [68]. All calculations and graphs were prepared using Microsoft Excel.

3. Results

3.1. Effect of nZVI on Cumulative Biogas/Methane Generation and AD Performance

The experimental Cumulative Specific Biogas Yield (CSBY), CH₄ content, Cumulative Specific Methane Yield (CSMY), kinetic parameters, TS, VS and COD removal efficiency results can be seen in Figure 2. The cumulative biogas and CH₄ production in each reactor kept rising until it stabilized on the 41st day, indicating that digestion was complete. The maximum CSBY in the control reactor B1 was 336.1 mL/g-VS_added and much higher than compared to other AD studies with sewage sludge (132 mL/g-VS, 167 mL/g-VS) [42,69]. As can be seen from Figure 2a and

Table 4. Efficiency (mean) of anaerobic digestion process in four reactors without (B1) and with nZVI particles (B2–B4).

Parameter	B1	B2	B3	B4
Max. CH4 content (%)	76.6 (day 28th)	75.3 (day 25th)	73.5 (day 23rd)	74.05 (day 20th)
Avg. CH4 content during steady period (%)	69.91	71.48	71.75	71.57
Cum. biogas yield (mL)	206,317	211,493	266,412	256,428
Cum. CH4 yield (mL)	133,106	134,204	168,614	169,579
Cum. specific biogas yield (mL/g-VSadded)	336.08	344.51	416.42	391.24
Cum. specific CH4 yield (mL/g-VSadded)	217.19	218.97	263.99	258.73
Biogas increase respect to the control (%)	-	2.5083	23.905	16.413

, the CSBY for digester B2 was similar to B1 and reached 344.5 mL/g-VS_added after 41 d. However, with the addition of higher nZVI dosages, the biogas yields improved remarkedly. CSBY increased to 416.4 mL/g-VS_added (B3) and to 391.2 mL/g-VS_added (B4) (p < 0.05). The reactor B3 had the highest CSBY, which increased by 23.9% compared to the control. This falls in a range found in other studies, which showed that a nZVI dosage of 1 g/L resulted in increased biogas production up to 18.11% and 29.55% [70,71]. This shows that CSBY improved significantly at the higher doses of nZVI; however, it was slightly inhibited at a dose of 3%. Similarly, Jia et al. [70] found that a nZVI dosage of 1 g/L (what is equal to 1.5% nZVI concentration in this study) was optimal for biogas generation due to largest cumulative production, while 2 g/L nZVI concentration decreased biogas production up to 46.45% due to an inhibitory effect on methanogen activity. Therefore, it can be stated that high concentrations of nZVI particles can have a toxic effect on the AD of sewage sludge, as previously reported [45]. However, it can be noted that 3% of nZVI in this study did not indicate serious process inhibition, as the biogas/CH₄ and VS removal values obtained were similar to a nZVI concentration of 1.5%.

Biogas enhancement, specifically the increase in CH₄ concentration in biogas, was also observed in sewage sludge treated with nZVI particles during AD. At the beginning of the AD process (day 3rd–6th), the CH₄ concentration was very low in all groups (18–32%). It was mainly because microbial activity was not high enough and was still undergoing an adaptation period [70]. From the 2nd day to the 13th day, the CH₄ concentration increased faster due to a rapid increase in methanogens in the early stage [71]. The concentration of CH₄ gradually increased from 5.9% to 57.7% in the control, but from 9.6% to 69.4% in the B3 reactor. This accounts for an 11.7% increase in the CH₄ concentration in the presence of 1.5% of nZVI compared to the control. This is in agreement with another study, which showed that the presence of nZVI accounts for a 5.1$–$13.2% increase in CH₄ content [72]. This could be explained by several mechanisms. Firstly, the dissolution of nZVI leads to the generation of H₂ gas, which, in turn, is important for stimulating the activity of homoacetogenesis bacteria and promoting hydrogenotrophic methanogenesis. Furthermore, the H₂ produced enhances the CH₄ content in the biogas by facilitating the conversion of CO₂ into CH₄ (CO₂ + 4H₂ → CH₄ + 2H₂O) [73]. The conversion of CO₂ to CH₄ is one of the main pathways for methane production, which is driven by hydrogenotrophic methanogens [74]. During the steady period, the average CH₄ concentration of the biogas in four reactors was 69.91% (B1), 71.48% (B2), 71.75% (B3) and 71.57% (B4) (Table 4).

At a nZVI dosage of 0%, 0.5%, 1.5% and 3%, CSMY after 41 d was 217.2 mL/g-VS_added, 218.9 mL/g-VS_added, 263.9 mL/g-VS_added and 258.7 mL/g-VS_added, respectively. The maximum CSMY in the B1 reactor was 217.2 mL/g-VS_added, which falls within the range of 191–282 mL/g-VS_added found in other studies [75,76]. Compared with the control, CSMY significantly increased up to 21.5% and 19.13% (at 1.5% and 3% nZVI dosages, respectively, p < 0.05). Similarly, Suanon et al. [77] showed that a 0.1% nZVI dose increased methane yield by 25.2% during the AD of sludge. Figure 2b shows CSMY during the AD of sludge with nZVI particles. The increase in CH₄ production in the presence of nZVI could be explained by the generation of electrons (Fe⁰ → Fe²⁺ + 2^e−), which enriched the AD process, enhanced the growth of hydrogenotrophic methanogens, and resulted in increased CH₄ yields.

Kinetic parameters were calculated according to the RM to provide an in-depth investigation of the effect of nZVI dosages on AD (Figure 2c). A positive and moderately strong (R² = 0.75) relationship existed between the maximum CH₄ production potential (P_max) and the nZVI dosage, while a stronger but negative relationship (R² = 0.87) was observed between the lag time (λ) and the nZVI dosage. However, it should be noted that P_max was highest and λ was lowest in reactor B3 (272.9 mL/g-VS_added and 6.58 d, respectively), suggesting that a 1.5% nZVI dosage had a greater effect on methanogenic activity than 0.5% or 3% dosages.

The removal efficiency (RE, %) plays a key role in evaluating the effectiveness of the AD process. The effect of different dosages of nZVI on sewage sludge reduction in the AD was characterized by changes in COD, TS and VS. Figure 2d shows the COD, TS and VS degradability assay of sludge under different nZVI dosages. COD removal can be defined as the amount of COD that is degraded by bacteria [78]. Overall, COD exhibited a decreasing trend across all reactors throughout the AD process, when COD decreased from 70.97 ± 0.03 g/L in the beginning of the experiment to 48.4 ± 1.57 g/L (B1) to 47 ± 0.87 g/L (B3) at the end. The average COD removal efficiencies in all treatment groups were similar and ranged from 31.79 ± 2.2% in the control reactor to 33.78 ± 1.2% in the B3 reactor. However, there was no significant difference between these two groups (p > 0.05) and the addition of 1.5% nZVI improved the COD degradation rate by only 2%. However, COD removal was higher compared to another study which showed 21% COD removal during the AD of waste activated sludge [79]. COD reduction occurs due to microbial activity, when microbes consume and break down organic matter [77]. After 41 d from the start of AD, a decrease in TS was also observed. VS reduction and TS reduction are commonly used as indicators to assess the efficiency of anaerobic sludge digestion. It can be summarized that a higher VS removal was observed in reactors B3 and B4 along with higher biogas and CH₄ production (Table 4). The decrease in VS suggests that organic matter was transformed into intermediate volatile fatty acids for CH₄ production [80]. VS destruction in the control reached 36.4 ± 1.8%, which was more than 10% less than the highest value obtained in the reactor B3 (48.05 ± 0.2%). In this study, VS reduction with 1.5% nZVI was higher than many other references, including ozone or microwave pretreatment (36% and 23.2% VS reduction, respectively) [81,82]. A similar trend was observed in the case of TS removal results, with a higher nZVI dose resulting in better TS destruction. The highest TS removal value was obtained in reactor B3 (38.54 ± 0.1%). The TS and VS removal findings can be attributed to the enhanced degradation of organic matter due to the addition of nZVI [71].

3.2. Biogas Production and Kinetics

Modeling the biogas data from sewage sludge amended with nZVI particles involved estimating kinetic parameters by fitting the experimental data from AD tests to various models, including the Gompertz (GM), modified Gompertz (MGM), Richards (RM), modified Richards (MRM), logistic (LM), modified logistic (MLM), Schnute (SM), and Cone (CM) models. As shown in

Table 5. Summary of kinetic analysis based on cumulative biogas data using eight sigmoidal models for the anaerobic digestion of sewage sludge amended with varying nZVI dosages.

Model	Parameter	Treatment
		Control	0.5% nZVI	1.5% nZVI	3% nZVI
Gompertz	Pmax	345.83	353.27	430.13	401.67
	r0	3.0782	2.0577	0.6374	1.0339
	α	0.1746	0.1676	0.1239	0.1454
	R2	0.9979	0.9986	0.9991	0.9995
	RMSE	7.278	5.6728	5.6195	4.1385
	NRMSE	2.1656	1.6467	1.3495	1.0578
	AIC	172.73	151.79	151	125.31
	Akaike’s weight	5.5752 × 10−25	1.0729 × 10−16	7.59875 × 10−9	0.0001
Modified Gompertz	Pmax	345.83	353.26	430.128	401.67
	Rmax	22.22	21.784	19.602	21.484
	λ	10.704	8.9965	5.1534	6.6156
	R2	0.9979	0.9986	0.9991	0.9995
	RMSE	7.278	5.6728	5.6195	4.1385
	NRMSE	2.1656	1.6467	1.3495	1.0578
	AIC	172.73	151.79	151	125.31
	Akaike’s weight	5.5752 × 10−25	1.0729 × 10−16	7.59875 × 10−9	0.0001
Richards	Pmax	344.17	334.82	430.09	401.56
	Rmax	1.831	18.688	0.0674	0.0625
	v	0.0311	0.6511	0.0013	0.0011
	λ	10.859	9.3603	5.1548	6.6159
	R2	0.9979	0.9999	0.9991	0.9995
	RMSE	7.0951	2.3556	5.6317	4.1456
	NRMSE	2.1112	0.6838	1.3524	1.0596
	AIC	172.59	79.972	153.19	127.45
	Akaike’s weight	5.9794 × 10−25	0.4223	2.54208 × 10−9	4.40391 × 10−5
Modified Richards	Pmax	334.57	334.82	431.33	401.53
	P0	20.198	16.197	0.4236	0.3667
	Rmax	1.0109	0.6509	0.0044	0.0041
	ν	3.1259	3.3468	0.1027	0.0916
	λ	11.296	9.3602	5.1192	6.6182
	R2	0.9999	0.9999	0.9989	0.9995
	RMSE	1.9359	2.3556	5.6777	4.1649
	NRMSE	0.5854	0.6838	1.3634	1.0645
	AIC	66.84	81.972	155.87	129.84
	Akaike’s weight	0.0549	0.1554	6.65634 × 10−10	1.33308 × 10−5
Logistic	Pmax	334.65	341.81	409.4	386.44
	k	0.2714	0.2611	0.1973	0.2289
	t0	18.656	17.278	16.116	16.119
	R2	0.9999	0.9999	0.9942	0.9955
	RMSE	1.9683	3.2779	13.342	11.117
	NRMSE	0.5857	0.9515	3.2039	2.8416
	AIC	62.881	105.73	223.64	208.32
	Akaike’s weight	0.3978	1.07736 × 10−6	1.27987 × 10−24	1.21099 × 10−22
Modified logistic	Pmax	334.65	341.81	409.4	386.44
	Rmax	22.709	22.312	20.196	22.113
	λ	11.288	9.6187	5.9804	7.3813
	R2	0.9999	0.9999	0.9942	0.9955
	RMSE	1.9683	3.2779	13.342	11.117
	NRMSE	0.5857	0.9515	3.2039	2.8416
	AIC	62.881	105.73	223.64	208.32
	Akaike’s weight	0.3978	1.07736 × 10−6	1.27987 × 10−24	1.21099 × 10−22
Cone	Pmax	349.55	359.909	473.5	422.93
	k	0.0536	0.0578	0.0579	0.0608
	n	4.6672	4.0764	2.4466	3.0501
	R2	0.9983	0.9987	0.9996	0.9997
	RMSE	6.3389	5.3697	3.9316	4.1621
	NRMSE	1.8862	1.5587	0.9441	1.0638
	AIC	161.12	147.18	120.99	125.79
	Akaike’s weight	1.8507 × 10−22	1.07549 × 10−15	0.0249	0.0001
Schnute	Pmax	346.14	354.08	442.83	406.83
	r0	253.84	157.39	256.25	367.07
	α	0.1713	0.1619	0.1022	0.1302
	β	0.0612	0.0923	0.298	0.1902
	R2	0.9977	0.9981	0.9997	0.9999
	RMSE	7.7492	6.4129	3.5178	3.2649
	NRMSE	2.3058	1.8615	0.8448	0.8345
	AIC	179.99	164.09	113.66	107.39
	Akaike’s weight	1.4783 × 10−26	2.28901 × 10−19	0.975035017	0.9996

Abbreviations: RMSE: root mean square error, NRMSE: normalized root mean square error, AIC: Akaike’s Information Criterion.

, these kinetic models were validated using error functions such as R², RMSE, NRMSE, AIC and Akaike’s weight. The selected statistical parameters helped to reduce the differences between the experimental results and the model predictions.

The experimental values obtained from all the reactors were simulated with the modified MGM and MLM, both of which provided the values of maximum biogas yield potential (P_max), biogas production rate (R_max) and lag phase time (λ) (Table 5). According to the MGM, the P_max value was the highest (430.1 mL/g-VS_added) in the reactor B3, which was predicted with lower errors (R² = 0.9991, RMSE = 5.6195, NRMSE = 1.3495%). This was followed by the B4 reactor with a P_max value of 401.7 mL/g-VS_added. Additionally, it can be seen from the MGM results that a 1.5% nZVI dosage resulted in the lowest λ (5.15 d), indicating the early hydrolysis and AD startup. The λ value represents the adaptation period of microorganisms to the changes in the reactor’s environment [58]. P_max increased rapidly with the increase in nZVI dosage up to 1.5%, while R_max and λ gradually decreased. According to the MGM results, P_max increased up to 22.34%, while R_max and λ decreased up to 11.78% and 51.86% compared to the control, respectively. These findings suggest that the addition of a 1.5% nZVI dosage gradually enhances biogas production potential and reduces the start-up time during the AD while simultaneously slowing down the biogas production rate. The same trends were observed based on the MLM’s results; however, comparing all models, MLM provided the best fit to P_max values in the reactor B1 due to the highest R² (0.9999)/Akaike’s weight (0.3978) and lowest AIC (62.881)/RMSE (1.9683)/NRMSE (0.5857). The LM produced identical simulation results for the B1 reactor (Figure 3a). The predicted results for reactor B1 using MLM and LM were accurate due to NRMSE values being below 1%. Compared to other kinetic studies, Deepanraj et al. [83] used three kinetic models (GM, MGM and logistic) to predict cumulative biogas production from AD of food waste. The best fit was obtained using the MGM due to the highest R², which was above 0.997 in all cases. Shitophyta et al. [84] compared four kinetic models for the modeling of biogas production from corn stover. The best fit and most accurate model was found to be logistic due to the smallest RMSE and lowest deviation. The logistic model was also determined to be the best fit for the prediction of cumulative biogas production during the anaerobic co-digestion of solid waste of vegetables, fruits and sewage sludge [52].

Biogas production from all reactors was simulated using other two similar S-shaped models (the Richards model (RM) and modified Richards model (MRM)), which provided the values of P_max, R_max, λ and fourth parameter, and shape factor (v). The MRM additionally included a fifth parameter, initial biogas/methane production (P₀). The kinetic constants obtained are presented in Table 5. The R_max in the RM was recorded in the ranges from 0.0625 mL/g-VS_added/d to 18.688 mL/g-VS_added/d. The theoretical cumulative biogas production ranged between 334.82 mL/g-VS_added and 430.09 mL/g-VS_added according to the RM and was similar or slightly lower compared to the MRM (334.57 and 431.33 mL/g-VS_added). The shape factor values calculated by the RM were close to zero, which ranged from 0.0011 to 0.6511. Comparing all models, the RMSE, NRMSE and AIC values according to the RM were observed to be lowest in the B2 reactor. Therefore, the RM provided the best fit to the P_max values in the B2 reactor (0.5% nZVI) due to the highest R² (0.9999), Akaike’s weight (0.4223) and lowest AIC (79.972) and errors (RMSE = 2.3556, NRMSE = 0.6838%) (Figure 3b).

Table 5 shows the kinetic values of the Gompertz model (GM) and Schnute model (SM), as well as their statistical parameters. Both models provided the values of P_max, initial rate of biogas production (r₀) and growth rate parameter (α). The SM additionally included a fourth parameter, scaling or Schnute parameter (β). The SM is a more generalized extension of the GM that incorporates a time-derivative approach [47]. The α value ranged from 0.1239 to 0.1746 in the case of the GM and from 0.1022 to 0.1713 in the case of the SM. Both models showed a good fit to the experimental biogas production results due to NRMSE values lower than 5% and R² values higher than 0.99. Comparing all models, the SM provided the best fit to the P_max values in the B3 and B4 reactors due to the highest R² (0.9997 and 0.9999, respectively)/Akaike’s weights (0.975 and 0.9996, respectively), lowest errors (RMSE = 3.5178, NRMSE = 0.8448 and RMSE = 3.2649, NRMSE = 0.8345, respectively) and AIC (113.66 and 107.39, respectively) (Figure 3c,d). Comparing the α values obtained by the GM and different studies, it is evident that an environment such as sludge with 1.5% nZVI particles was more favorable due to the higher microorganism growth rate parameter, which was 1.4 times faster compared to the value obtained for the AD of waste activated sludge [79].

The other two models, the Cone model (CM) and the logistic model (LM), were used to fit measured biogas data, both providing the kinetic parameters of P_max and reaction or hydrolysis rate coefficient (k). The difference between these two models is that the CM includes a third parameter, the shape factor (n), while the third parameter of LM is t₀ (time at which the production rate is highest). The calculated kinetic parameters for the two fitted models can be found in Table 5. A comparison of the k values obtained using the CM showed that higher nZVI dosages (1.5% and 3% nZVI) resulted in slightly higher degradation rates (0.0579 d⁻¹ and 0.0608 d⁻¹, respectively) compared to the control. It is known that a larger k value represents a higher degradation rate [40]. However, since the differences in k values among all treatment groups were minimal, it can be concluded that varying nZVI dosages did not significantly enhance the hydrolysis rate. A comparison of n values derived from CM showed that in all nZVI treatment cases, it was higher than 1, indicating the presence of a lag phase. In general, these models showed a good fit to experimental biogas data obtained from reactor B1 due to a high coefficient of determination (R² > 0.99). However, the LM provided a better fit to B1 data due to lower error values, which were interestingly identical to the statistical parameters calculated using MLM data. This indicates that the LM and MLM can be used equally well to accurately predict biogas production from the control sewage sludge (Figure 3a). The R² and RMSE values obtained by the CM were similar to the SM results in the case of AD of sewage sludge amended with higher nZVI dosages (1.5% and 3%); therefore, it could be argued that the CN was the second best after the SM for biogas production in reactors B3 and B4. However, when comparing RMSE, NRMSE, AIC values and Akaike’s weights, the SM showed the best fit to the experimental biogas data from reactors B3 and B4 among all models (Table 5). Correspondingly, a study conducted by Zhang et al. [85] found that the CM was the most suitable for modeling biogas production kinetics due to the highest R² (>0.999), as well as the lowest RMSE (0.44) and AIC (−89.59) values. It was suggested that the CM could be used to evaluate the biogas kinetics of AD of waste sludge amended with nZVI particles more reasonably.

The modeled results using all models were plotted together with cumulative biogas production and are shown in Figure 3. The predicted performance of biogas in reactor B1 can be ranked as follows: MLM/LM > MRM > CM > RM > MGM/GM > SM. Meanwhile, the predicted performance of biogas in B2 has the following order: RM > MRM > MLM/LM > CM > MGM/GM > SM. The opposite trend was determined in reactors B3 and B4, when the Schnute model was superior: SM > CM > MGM/GM > RM > MRM > MLM/LM. It can be seen that all models provided reasonably good fits for the experimental biogas data. The sigmoidal models used accurately reproduced experimental S-shape curves for the pretreated sewage sludge with nZVI particles. All models demonstrated a high goodness of fit (R² > 0.99) for all nZVI dosages. The lag phase time calculated using MGM, MLM, RM and MRM gave similar results, ranging from 11 for the control group to 5$–$6 days for the group with a 1.5% nZVI dosage. Information about the lag phase time can help to determine the sludge retention time for the methanogenesis stage, ensuring optimal interaction between the feedstock and the bacterial biomass [50].

3.3. Methane Production and Kinetics

Methane yield is an important performance index of reactor efficiency. Therefore, the experimental CH₄ results were fitted with eight sigmoidal models to determine a more suitable nZVI dosage for sludge AD and methane production (

Table 6. Summary of kinetic analysis based on cumulative methane data using eight sigmoidal models for the anaerobic digestion of sewage sludge amended with varying nZVI dosages.

Model	Parameter	Treatment
		Control	0.5% nZVI	1.5% nZVI	3% nZVI
Gompertz	Pmax	221.58	222.49	272.4	264.03
	r0	5.2887	4.0257	0.8842	1.3165
	α	0.1918	0.1893	0.1341	0.1585
	R2	0.9999	0.9999	0.9997	0.9998
	RMSE	2.8299	2.2328	3.2303	2.5219
	NRMSE	1.3029	1.0197	1.2236	0.9747
	AIC	93.382	73.474	104.49	83.701
	Akaike’s weight	4.51867 × 10−14	5.90669 × 10−6	1.16604 × 10−5	0.0035
Modified Gompertz	Pmax	221.58	222.49	272.39	264.03
	Rmax	15.64	15.495	13.439	15.401
	λ	12.077	10.867	6.6088	7.0439
	R2	0.9999	0.9999	0.9997	0.9998
	RMSE	2.8299	2.2328	3.2303	2.5219
	NRMSE	1.3029	1.0197	1.2236	0.9747
	AIC	93.382	73.474	104.49	83.701
	Akaike’s weight	4.51867 × 10−14	5.90669 × 10−6	1.16604 × 10−5	0.0035
Richards	Pmax	217.64	219.88	272.85	264.1
	Rmax	11.831	9.1118	0.3392	0.0488
	v	0.5096	0.3312	0.0094	0.0012
	λ	12.305	11.01	6.5779	7.0355
	R2	0.9999	0.9999	0.9996	0.9998
	RMSE	1.3621	1.6781	3.3011	2.5263
	NRMSE	0.6271	0.7664	1.2504	0.9764
	AIC	33.959	51.484	108.32	85.848
	Akaike’s weight	0.3619	0.3519	1.71807 × 10−6	0.0012
Modified Richards	Pmax	217.41	220.13	272.29	263.92
	P0	11.548	7.5148	0.0835	0.0394
	Rmax	0.5388	0.2914	0.0008	0.0004
	ν	2.9558	3.3062	0.0889	0.0841
	λ	12.344	10.975	6.6143	7.0233
	R2	0.9999	0.9999	0.9997	0.9998
	RMSE	1.3539	1.6665	3.2348	2.5262
	NRMSE	0.6234	0.7611	1.2253	0.9764
	AIC	35.451	52.9	108.61	87.843
	Akaike’s weight	0.1716	0.1734	1.48617 × 10−6	0.0004
Logistic	Pmax	215.01	216.19	260.15	255.48
	k	0.2979	0.2918	0.2137	0.2473
	t0	19.364	18.255	16.799	15.835
	R2	0.9999	0.9999	0.9979	0.9982
	RMSE	2.2344	3.4573	7.9469	6.9838
	NRMSE	1.0288	1.5789	3.0103	2.6992
	AIC	73.534	110.2	180.11	169.26
	Akaike’s weight	9.22464 × 10−10	6.25741 × 10−14	4.4263 × 10−22	9.10726 × 10−22
Modified logistic	Pmax	215.01	216.19	260.15	255.48
	Rmax	16.015	15.771	13.897	15.793
	λ	12.651	11.401	7.4396	7.7469
	R2	0.9999	0.9999	0.9979	0.9982
	RMSE	2.2344	3.4573	7.9469	6.9838
	NRMSE	1.0288	1.5789	3.0103	2.6992
	AIC	73.534	110.2	180.11	169.26
	Akaike’s weight	9.22464 × 10−10	6.25741 × 10−14	4.4263 × 10−22	9.10726 × 10−22
Cone	Pmax	222.87	224.7	290.63	274.33
	k	0.0517	0.0549	0.0574	0.0627
	n	5.3314	4.896	2.888	3.334
	R2	0.9999	0.9999	0.9998	0.9999
	RMSE	2.1576	1.9209	2.9226	2.8108
	NRMSE	0.9934	0.8772	1.1071	1.0864
	AIC	70.597	60.833	96.087	92.81
	Akaike’s weight	4.006 × 10−9	0.0032	0.0008	3.63324 × 10−5
Schnute	Pmax	221.79	222.88	277.34	266.27
	r0	359.42	282.69	288.49	367.28
	α	0.1889	0.1849	0.1176	0.1456
	β	0.0384	0.0513	0.2121	0.1552
	R2	0.9999	0.9999	0.9999	0.9999
	RMSE	2.9999	2.4217	2.4067	2.1522
	NRMSE	1.3813	1.1059	0.9117	0.8318
	AIC	100.28	82.294	81.773	72.384
	Akaike’s weight	1.43592 × 10−15	7.17968 × 10−8	0.9992	0.9902

Abbreviations: RMSE: root mean square error, NRMSE: normalized root mean square error, AIC: Akaike’s Information Criterion.

). The MGM and MLM theoretically predicted a relatively long lag phase, ranging from 7 days in the 1.5% nZVI group to 12–13 days in the control group. The 0.5% nZVI group exhibited a similar lag time (11 d) to that of the control. Therefore, it can be stated that conditions without nZVI particles or with a low dosage require more time to reach the maximum CH₄ production potential. This could indicate that methanogens may slowly adapt to inhibitory compounds released during the thermal hydrolysis process of thickened waste activated sludge [86]. According to both models, the R_max values obtained with 0.5% (15–16 mL/g-VS_added/d) and 3% (15–16 mL/g-VS_added/d) nZVI particles were similar to those of the control group (16 mL/g-VS_added/d), while the 1.5% nZVI group showed a slightly lower maximum methane production rate (13–14 mL/g-VS_added/d). Therefore, it can be concluded that the reduced lag phase and maximum methane production rate contributed to the higher methane yield. The R² value was highly significant (>0.99) in all sludge treatment cases according to the MGM and MLM. However, the MGM provided a better fit for the experimental CH₄ results in reactors B2, B3 and B4 due to lower RMSE, NRMSE (<1.5%) and AIC values, as well as higher Akaike’s weights, compared to the MLM. In the case of the control group, the MLM provided a better fit for the same reasons. Another study [87] showed that the MGM and MLM had a good fit to the experimental CH₄ production results from the AD of dense primary sludge (R² = 0.976 and error = −0.35; R² = 0.964 and error = −0.35%, respectively). Bakraoui et al. [88] compared four sigmoidal models (RM, LM, MGM and CM) and demonstrated that the MGM was the most suitable for predicting CH₄ production from the AD of sludge and wastewater recycled pulp and paper. The duration of lag phase obtained in this study using the MGM (12.08–10.87 days for reactors B1 and B2, respectively) was similar to the values of 10.17–14.60 days observed during anaerobic co-digestion of liquid manure with winemaking waste, food waste and biowaste [89]. The maximum methane production rate of 13.89–16.02 mL/g-VS_added/d obtained in this study was comparable to values reported in other AD studies. For instance, when sewage sludge was amended with silver nanoparticles, researchers observed R_max values of 16.62 mL/g-VS_added/d and 16.42 mL/g-VS_added/d using modified Gompertz and logistic function models, respectively [90].

Two other models (RM and MRM) were used to fit experimental CH₄ production data and were compared using appropriate statistical parameters. For CH₄ production modeled with the RM and MRM in reactors B1 and B2, the R² values were 0.9999 and very close to 1. It can be seen that the RMSE and NRMSE values were nearly identical; however, the RM demonstrated a better fit and higher accuracy for reactors B1 and B2 due to lower AIC values (33.959 and 51.484, respectively) and higher Akaike’s weights (0.3619 and 0.3519, respectively). Therefore, it can be concluded that the RM provided the best fit for CH₄ production data in reactors B1 and B2. Compared to other studies, Farghali et al. [87] showed that the RM was capable of accurately simulating the kinetic patterns of CH₄ generation due to its high R² value (0.987) and low error (−0.35%) in the AD of dense primary sludge. Meanwhile, Ali et al. [55] observed a similar model performance for the RM in CH₄ generation from cow and sheep manure (R² = 0.975 and R² = 0.981, respectively).

The cumulative specific methane yields obtained from the AD of four reactors were fitted using two other models, the GM and SM. The SM results showed that the R² values were identical (0.9999) across all treatment groups, confirming that the SM provided an almost perfect fit for representing the CH₄ accumulation process. The GM demonstrated a similarly strong fit (R² > 0.999). Among all examined models, the SM was the most accurate in predicting CH₄ data for the 1.5% and 3% nZVI groups due to the lowest RMSE, NRMSE (<1%) and AIC values, and highest Akaike’s weights (>0.99). According to the SM results, nZVI dosages of 1.5% and 3% positively affected the maximum CH₄ production potential of primary sludge, enhancing it by 25.05% and 20.06%, respectively, compared to the control. Notably, the highest increase in P_max was achieved at a nZVI dosage of 1.5%. This finding is consistent with another study, which showed that both the GM and SM effectively fitted the experimental methane production data due to high R² values (>0.99) [63]. The referenced study evaluated the AD of Thai rice noodle wastewater amended with varying amounts of chicken manure. However, it was noted that the SM provided greater flexibility and should be preferred for general use.

Based on the CM results, the methane yield potential of sludge pre-treated with 0.5%, 1.5% and 3% nZVI dosages was 224.7 mL/g-VS_added, 290.63 mL/g-VS_added and 274.33 mL/g-VS_added, respectively. The same trend was observed in the LM results (216.19 mL/g-VS_added, 260.15 mL/g-VS_added and 255.48 mL/g-VS_added, respectively). Both models suggested that the methane yield potential in the 1.5% nZVI group was slightly higher than that of the 3% nZVI group. The P_max values predicted were in good agreement with the experimental yields, as indicated by high R² values (>0.999) and low RMSE/NRMSE values for the CM. Other studies have also established a good fit of the CM to experimental methane data. For example, Pan et al. [91] compared several kinetic models for methane production from anaerobic co-digestion of food waste with sewage sludge and demonstrated that the CM yielded better performance parameters (AIC: 54–181; R²: 0.98–0.99).

Figure 4 illustrates the best-fit model, determined by plotting the observed CSMYs against the predicted values from eight cumulative sigmoidal models. All models resembled an elongated S-shape curve and closely fitted the experimental CSMY data (R² > 0.99). The CH₄ production data predicted for the B1 (control) reactor followed this order: RM > MRM > CM > MLM/LM > MGM/GM > SM. The same trend was observed for the B2 (0.5% nZVI) reactor, when the RM provided the best fit and the SM the worst, which is consistent with biogas modeling data. Among all models, the SM was the most accurate for reactors B3 (1.5% nZVI) and B4 (3% nZVI), as was previously determined for biogas data. This suggests that the AD process of sludge with nZVI additives could be effectively predicted using the Schnute model, a more generalized and flexible growth model capable of approximating various growth patterns, including exponential, linear and sigmoidal growth under certain conditions.

4. Discussion

Several studies have shown that incorporating nanoparticles into the sewage sludge during the AD process enhances AD efficiency and reduces the lag phase. For example, Grosser et al. [90] reported that the addition of silver nanoparticles to sewage sludge significantly increased biogas and CH₄ yields by 15% and 25%, respectively, compared to the control. Additionally, it was reported that silver nanoparticles had a positive effect on kinetic parameters, such as a shorter lag time and a higher CH₄ production rate. The duration of the lag phase in the aforementioned study was shorter (3 days according to the MGM and MLM) than in our study with a 1.5% nZVI dosage (7 days according to the MGM, MLM, RM and MRM). It is worth noting that the maximum CH₄ production rate in our study (13–14 mL/g-VS_added/d) was lower compared to the value (16 mL/g-VS_added/d) obtained by Grosser et al. [90] using the MGM and MLM. In another study, Bakari et al. [92] studied the effect of micro-scale ZVI (in the form of iron scrap and steel wool) on the AD of domestic wastewater. The authors found that iron scrap and a steel wool concentration of 10 g/L increased CH₄ production by 38.3% and 30.7%, respectively. In addition, it was reported that both types of ZVI shortened the lag phase (according to the MGM, LM, and RM) from 3 days to 2 days. The maximum CH₄ production rate achieved with ZVI ranged from 56.6 to 70.7 mL/g-VS_added/d, depending on the kinetic model used. Therefore, a correlation can be observed between a higher maximum CH₄ production rate and a shorter lag time. This relationship is largely influenced by microbial adaptation and activity. Faster adaptation of methanogens results in their accelerated growth, leading to shorter lag times and a higher CH₄ production rates. The length of the lag phase is influenced by various factors, including the initial cell concentration, the time needed for cells to recover from physical damage or stress, the time required to produce essential coenzymes or division factors, and the time necessary to synthesize new enzymes for metabolizing available substrates in the medium. During this phase, microbial growth is nearly zero [60], which was also evident in our study.

Higher doses of Fe additives can prolong the lag phase, as observed in our study. This study showed that the addition of 1.5% (equivalent to 1 g/L) significantly shortened the lag phase from 11 days to 5–6 days for biogas production and from 12–13 days to 7 days for CH₄ production, according to different models. However, increasing the nZVI dose to 3.0% (in our study) increased the lag phase to approximately 7 days. Similarly, Lizama et al. [42] reported that the highest dose of nZVI (9 mg/g-VS) caused a greater lag phase compared to lower nZVI doses. This was explained by the fact that a higher amount of solubilization could have caused negative effects on the anaerobic microorganisms.

A study by Wang et al. [33] found that the optimal nZVI dosage for methanogens was 1 g/L, while higher concentrations of iron nanoparticles inhibited the process due to long-term accumulation of byproducts from nZVI particle corrosion. The adverse impact of elevated nZVI concentrations is primarily attributed to the buildup of these corrosion byproducts in anaerobic digesters, which inhibits bacterial growth and reduces biogas production. The byproducts of nZVI particle corrosion include ferrous iron (Fe²⁺), ferric iron (Fe³⁺), hydrogen gas (H₂), electrons (e⁻), and reactive oxygen species [73]. Dong et al. [93] investigated microbial community changes in the anaerobic seed prepared by cultivating digested sludge under varying nZVI concentrations. Their study found that the highest nZVI dose (25 g/L) drastically reduced the DNA concentration of anaerobes. Moreover, the relative abundance of homoacetogens (Clostridium) increased, while the relative abundance of methanogens (Methanosarcina and Methanobacteriales) decreased. High doses of nZVI may cause DNA damage and disrupt metabolic activities of bacteria due to the strong reducing capacity of nZVI. This could also explain the slightly slower and lower biogas and methane production in our study with a 3% nZVI addition.

The operational characteristics of reactors, such as the intensity of mixing, can significantly influence their biogas and methane production efficiency [94]. The slow mixing rate (120 rpm) in this study is consistent with several other studies, which showed that lower mixing intensities (<200 rpm) can enhance biogas production and the CH₄ content. Stafford [95] observed lower gas production from primary sewage sludge at higher stirring rates (>700 rpm) due to shear forces separating the hydrolytic bacteria from their polymer substrates, while a low mixing speed at 150 rpm maximized biogas production. Intense mixing generates higher shear stress, which can disrupt and break apart flocculent formations, ultimately leading to a decrease in biogas production. Ghanimesh et al. [96] showed that digester with continuous mixing at 100 rpm produced a higher CH₄ content, which was 0.6 L/g-VS. In summary, anaerobic digestion studies should prioritize a lower mixing intensity. An appropriate level of mixing ensures an enhancement in biogas production, uniform substrate distribution, prevention of solid settlement and a consistent environment for anaerobic bacteria, which is a key factor in achieving optimal digestion [97]. Additionally, the optimization of mixing during anaerobic digestion is crucial for decreasing energy demand, as it can vary from 14% to 54% of the total energy demand in a plant [98]. Hydrodynamics and the energy required for mixing can be evaluated through experiments in operational plants, laboratory tests, or numerical simulations (e.g., computational fluid dynamics models) [99].

Temperature is another key operational parameter that plays a crucial role in the efficiency of the AD process [100]. It significantly influences both physicochemical properties and the composition of microbial communities, which adapt to changes in temperature. Many researchers have compared the advantages and disadvantages of thermophilic (55–60 °C) and mesophilic (35–40 °C) modes of sludge AD. A study performed by Chen et al. [101] observed that cumulative CH₄ production during mesophilic (37 ± 1 °C) anaerobic digestion (MAD) was higher than in thermophilic (55 ± 1 °C) anaerobic digestion (TAD), and nZVI effectively improved the methanogenesis in both processes. Moreover, both digestion temperatures and nZVI addition influenced the composition of functional bacterial and archaeal communities. In MAD, nZVI promoted the dominance of Candidatus Microthrix and Methanothrix, while in TAD, it enhanced the presence of Coprothermobacter and Methanothermobacter. Song et al. [102] found a similar trend, as single-stage MAD (35 ± 2 °C) outperformed TAD (55 ± 1 °C) in terms of specific methane yield, effluent quality and process stability. Therefore, it could be speculated that the mesophilic temperature (37 ± 1 °C) used in our study may yield better results compared to the thermophilic temperature.

The results of this study showed that nZVI could be a promising material for improv-ing the AD of sewage sludge. This technology could be easily integrated into existing wastewater treatment plant operations without requiring significant modifications. nZVI could be directly dosed into sludge before entering the AD process. However, proper dosing and monitoring would be necessary as high doses of nZVI may negatively impact microbial communities in AD processes. Nevertheless, practical application may be challenging due to the high cost of nZVI (100–500 EUR/kg). The potential economic benefits of nZVI for industrial applications should be further investigated through a techno-economic analysis.

5. Conclusions

This study demonstrated that the addition of nZVI particles during the anaerobic digestion of sewage sludge significantly increased the cumulative biogas and methane yields. Specifically, the addition of 1.5% nZVI particles led to a significant increase in the cumulative specific biogas yield by 23.9% and cumulative specific methane yield by 21.5%. Eight sigmoidal models were analyzed to accurately simulate biogas and methane production, with the Schnute model being the most suitable for the prediction of anaerobic digestion of sludge amended with 1.5% and 3% of nZVI particles. This was attributed to the highest R² values (0.9999), the lowest AIC, RMSE and NRMSE values, and the highest Akaike’s weights. The modified Gompertz, Richards and logistic model results showed that the addition of 1.5% and 3% nZVI particles can shorten the lag phase of biogas and methane production during the anaerobic digestion of sewage sludge by 51.9% and 45.3%, respectively, compared to the control. Future research should be conducted to evaluate and compare various kinetic parameters of anaerobic digestion of sewage sludge treated with other iron additives, such as magnetite (Fe₃O₄) or micro-scale ZVI. Additionally, it is important to carry out the kinetic modeling of anaerobic digestion of sewage sludge amended with iron additives by changing the sludge mixing rate and temperature.