Analogous study of cumulative biogas production by anaerobic digestion of sewage treatment plant sludge, the proposal of universal dimensionless models

The current global concern is how to replace polluting fossil fuels with sustainable green fuels. On this basis, in this study, a characterization protocol for sewage sludge in Algeria was carried out. Their objective was to study the process of anaerobic digestion for biogas production by analogy to experiments that have already been made in the literature. The performance of the proposed models has been validated with experimental data from the scientific literature. Biodegradable organic matter levels are high (160–500 mg L−1) giving an acceptable yield for biogas production. The modeling of cumulative biogas production was carried out by Gompertz and models were proposed for different products. It has been observed that there is a better agreement of the models proposed with the experimental data with a maximum value of r2 = 0.9996. The modeling of the degradation of organic matter was carried out by the first order‐model, and dimensionless models were proposed. The latter gave a good agreement with the experimental data better than the model in the literature, with a maximum value of r2 = 0.9985. The proposed models to study the anaerobic digestion process of organic matter, as well as those related to the process of cumulative biogas production, are universal (without units of measurement), which will help producers in this field to improve production, as well as reduce the investment cost through preliminary simulations.

waste, which constitutes a huge renewable reservoir of nonfossil green gas. Increasing concern about the safe disposal of different wastes generated in society necessitates the scientific community to collect and treat the waste effectively. Significant developments have taken place in treating the waste-waters during the last two decades. This has resulted in increasing sludge production, which is consuming 50% of the current operating costs of waste-water treatment plants. 1 This sludge is a by product obtained from different operation units of wastewater treatment plants during different physical, chemical, and biological processes, which include: a clarifier, a biological reactor, a centrifuge, and so forth. 2 The problem of disposal of sludge from sewage treatment plants is more delicate; it has increased production of waste-water and regulations that are becoming more demanding. 3 The use of sewage sludge in agriculture has been widely practiced in most developed countries. 4 Sludge processing pathways have always had goals mainly related to the reduction of volume and fermentability, more particularly their stabilization. 5 Different strategies for the treatment and the final disposal are possible, but the general opinion is that sewage sludge is a valuable source of energy and materials. 6 Sewage sludge treatment has relied on three main processes: Dewatering, drying, and stabilization by anaerobic digestion and thermal treatment. 7 "Anaerobic digestion is the preferred stabilization method that produces biogas, a valuable energy source." 8 In fact, cumulative biogas production is influenced by the amount of added organic material, the pH of the digester, any toxic substances, and the C/N ratio in anaerobic condition. Additionally, biogas production is affected by the proximate composition and characteristics of the initial organic material, which have significant effects on the decomposition efficiency of the anaerobic organic material and methane production. 9,10 Min-Jee et al. 11 have studied the effects of proximate composition ratios on biogas production. Their findings reveal that biogas production can be improved by mixing agricultural byproducts that have a low biogas production rate to achieve the same proximate composition as an agricultural byproduct that has a high biogas production rate. Laskri et al. 12 have found that the cumulative biogas production during the digestion of sludge from sewage treatment plants (5000 mL) is 10 times greater than when digesting landfill waste organic matter (500 mL) during 30 days of digestion. A number of studies have dealt with the effects of sludge thermal treatments on biogas production enhancement during anaerobic digestion. [13][14][15][16] On the other hand, Pérez Garcia et al. 17 studied the agronomic value of sewage sludge from Tenerife. The influence of drying methods on thermodynamic parameters and a comparative approach to the performance of sludge drying was studied by Ameri et al. 3,18 After citing the research carried out in this field, it appears that numerous works have studied the cumulative biogas production from the sludge of waste-water treatment plants by anaerobic digestion, but a limited number of studies have addressed the simulation of this process. For example, anaerobic digestion and biogas potential: simulations of industrial and laboratory processes have been studied by Hamawand et al. 19 This simulation showed the ability to overcome the uncertainty and discrepancy of measured biogas from an industrial digester. In the case of the lagoon digester, it was shown that the discrepancy in the measured biogas was around 250%. The measured biogas was higher by 2.5-fold than that predicted by simulation. Successful simulation studies were carried out on the anaerobic codigestion process to find the optimal ratios of substrates in the suspension introduced into the digester by Inayat et al. 20 The optimal substrate composition found is 50% waste-water, 25% livestock manure, and 25% biomass. The development of a process modeling simulation by Aspen Plus, for the anaerobic digestion process, was investigated by Al-Rubaye. 21 The AD model developed at Aspen aims to find the optimum temperature, reactor size, and substrate flow rate to achieve an optimum process. The percentage of methane gas has dropped from about 70% to about 90%. 22 Concerning biogas from anaerobic codigestion of food waste and primary sludge, the system efficiency using the heat generated increased to 75.25% and 78.43% for thermophilic and mesophilic conditions, respectively. However, despite the high level of biogas production at thermophilic conditions (137.4 m 3 h −1 ) compared to a mesophilic scenario (67.74 m 3 h −1 ). 23 The proposed approach overcomes various limitations in the reviewed works, which is the impetus of the proposed work. 24 Most of the previous articles have partially addressed effectiveness evaluation with simple models, such as the response surface methodology (RSM) 20 and the AD model developed in Aspen. 22 However, proposed empirical models, capable of addressing most of the problems addressed and serving as a tool to meet future requirements for biogas production, are limited in the literature. Validation of the research effort in terms of novelty and originality, presented as follows: 2. The results obtained from this analysis encourage us to search the literature for sludge with the same properties as our own sludge and that was already used to produce biogas, to submit it to a similarity study with our own sludge, and it gives the work a unique value. 3. This article proposes the development of new universal empirical models that do not depend on units of measurement, that was limited in previous works, that is, more flexibility to use them with more efficiency and precision, and so on. 4. The proposed models aim to improve the digestion process by monitoring the kinetics of biogas production and the degradation of organic matter. 5. Validate the effectiveness of the proposed models by testing them on the different raw materials used in the literature for the production of biogas, as well as by comparing them with the different models available in this field. Whether with regard to the models that follow the kinetics of biogas production or those related to the monitoring of the digestion of organic matter.
The questioner can wonder about the usefulness of these models, more than that, what is the interest of modeling simulation. However, in fact, the benefit they provide is qualitative. The proposed models are universal and are not related to measurement units, which means that their application is easy and includes all products. not only methane but all gases such as hydrogen, oxygen, and others. In addition, they give us predictive information about the value of a process for producing such gases, based on preliminary data already obtained by analyzing the raw material from which the gas must be extracted. This gives us a qualitative advantage represented by a saving of time, effort, and money. It is extremely prudent for everyone to be convinced that the environment relationship can be harmonized by shifting our energy reliance from fossil fuels to greener, sustainable, and renewable energy vectors like biogas (methane, hydrogen, etc.). On this basis, the main objective of this study was to characterize the sludge. Based on this characterization, the second objective is to use the results of cumulative biogas production from a similar sludge to our own product for the validation of five new universal and dimensionless models with the experimental production data of biogas and the degradation of organic matter. As for the third objective, the aforementioned models were evaluated and compared to different models from the scientific literature to select the one that best suited the experimental data on cumulative biogas production and organic matter degradation. The latter was evaluated with other methane and hydrogen production data from existing studies in the scientific literature.

| Methanization
Methanization or anaerobic digestion is the decomposition of organic material by micro-organisms in the absence of oxygen. This process involves several bacterial species that simultaneously transform organic waste into biogas. Anaerobic fermentation can take place in three temperature ranges: Psychrophiles: 15-25°C, Mesophiles: 25-45°C, and Themophiles: 55-65°C. 25 Results of anaerobic digestion include both biogas and digestate, the latter requires solar drying for its stabilization, as what Ameri et al. 3 ) did on mud, Badaoui et al. 26 on tomato waste, Djebli et al. 27 on potatoes, and vacuum drying as Keskes et al. 28 did on pharmaceutical powders.

| Anaerobic digestion of sludge from sewage treatment plants
To study this step, a sample of sludge was taken from the wastewater treatment plant (ONA) of Wilaya de Boumerdes (Algeria), where detailed analyses were carried out on it, to determine its susceptibility to anaerobic digestion. This sample is placed in the digester with a dilution rate of 80%. Tables 1 and 2 give the physiochemical characteristics of our sludge and the sludge that has undergone anaerobic digestion. The factors that influence the production of biogas were mainly based on the operating conditions, as well as the type of feed to the digester. Operating conditions such as pH and temperature directly influence microorganisms. The composition and the concentration of sludge are also important, without forgetting the toxic compounds and the inhibitors of the methanogenic phase. Sometimes the toxic compounds are not initially present in the feed, but they are produced inside the reactor from the degradation of the substrate, for example, volatile fatty acids (VFAs) and ammonia.
Due to the impossibility of conducting an experimental study of methanization, at the time of the emergence of the coronavirus (COVID- 19), the results of a study carried out on sludge with the same properties as our own product were taken, as they show in Tables 1  and 2. Based on this analysis, it was noted that there is a great convergence of results, which indicates that our sludge will produce a quantity of gas close to the results of Table 3.
Tables 1 and 2 show that the two substrates are rich in organic matter, and therefore they could easily promote anaerobic digestion. The results are given in Table 3.
The biodegradability of the sludge from the sewage made it possible to recover flammable biogas after 24 h of digestion, and the production of the biogas reached almost 5 L for an initial concentration of 700 mg L −1 . The reduction in COD is very significant as is the purification efficiency, which reaches 80% ( Figure 1).
According to Figure 1, it is remarkable that biogas becomes flammable from the 3rd day of digestion, has a neutral pH of around 7, and that the degradation of organic matter is more visible because the curve corresponding to the evolution of the COD begins to decrease. There appears to be a slight decrease in the pH from Day 0 until the 3rd day of digestion, when the pH was lowered from 7.17 to 6.75 as shown in Table 3. When the pH increases on the 6th day, this indicates that the acidogenic phase has been exceeded and the degradation of organic matter during methanogenesis has occurred resulting in the production of biogas. The temperature during digestion was maintained at around 36°C.

| Modeling of cumulative biogas production
The biogas production curves were modeled using the modified Gompertz equation. 30 The below equation was used to analyze the cumulative production of biogas.
T A B L E 3 Anaerobic digestion of WWTP sludge. 12 Residence time (day) Nature of gas formed Cumulative biogas production (mL) where X B (t) represents the cumulative biogas production (mL) as a function of time t (h); t L is the latency time (h); X P is the ultimate potential of biogas (mL); R max is the maximum speed of biogas production (mL h −1 ). Roeland 31 cited in his article a set of Gompertz models in Growth Analyzes and New Approach Gompertz Model: An addition to the Unified Richards Family, some of the parameterizations of the Gompertz model found in the literature are more useful than others because they have parameters that are easy to interpret. A valuable and commonly found reset is: Such that K G is a growth rate coefficient (which affects the slope), and t i represents the inflection time.
In growth curve analyses of bacterial (or microbial) counts, in particular the adaptation of a four-parameter Gompertz model, as suggested by Gibson et al., 32 the model becomes: -The modification of Zwietering The reparametrization proposed by Zwietering et al. 33 is often referred to as "modified Gompertz" which is generally applied to bacterial growth data, particularly in food. It can be given as: where K Z is the absolute growth rate (i.e., tangent to the curve) at T Lag time, called the "lag time." -The resetting of Zweifel and Lasker The parameterization of Zweifel and Lasker 34 was copied by Ricker 35 in his book and found its place in the study of fish growth. Today, we often talk about the Ricker model. This model is mainly used for the growth of fish, but it also fits the growth data of other animals, for example, crustaceans. It can be expressed as follows: where K G is the value of the growth coefficient, W 0 is specified as the initial value (number, density, mass, length, etc.). It gives the starting point of the growth curve.
-The Gompertz-Laird Another very common type II parametrization is the version of the Gompertz model originally proposed by Laird 36 to describe the growth in tumor size, but it is often adapted to the growth in number (populations) of cells and microbes. With Aggrey's 37 notation (often encountered in many studies in domestic animals), the Gompertz-Laird model becomes: It is possible to consider this model as a variant of the model (Equation 5) (or vice versa), but in reality, their parameters behave very differently. The W 0 parameter is comparable to that of the model (Equation 5), but the other parameters are not. Interpretations of the K and L parameters vary in the literature and are often ambiguous or poorly explained.
Another reparameterization is suggested by Norton. 38 It is sometimes mistakenly considered a Gomperz-Laird model and looks like this: This model has very different parameters than Laird's. It has the same significant coefficient ratio and the same parameter for the initial value (or starting point) as the model (5).
-Absolute growth rate Previous authors have also noted, more or less explicitly, that it is possible to reparametrize the Gompertz model so that the growth parameter returns a relative or absolute growth rate.
The traditional three-parameter Gompertz model, as in the version shown in Equation (1), is a special case of the four-parameter Richards model, for example, where k R is the growth constant specific to the model controlling the maximum growth rate and the parameter determining the inflection value. This model (Equation 9) presents the same problem as the traditional Gompertz models. Specifically, the growth parameter (k G ) is not comparable to the growth coefficients of versions of other traditional models. In addition, these growth parameters (or growth coefficients) are more difficult to interpret because they do not constitute the absolute or relative growth rate. Therefore, we recommended two forms of the Richards model, called the Unified-Richards (or U-Richards). The first of them, the t i shape of U-Richards, 39 is given by: where d is the fourth parameter (shift of the inflection value). The second, the W 0 form of U-Richards 40 then becomes: We find two general expressions that group together almost all models in the literature.
The first (concerning the equations from Equation 1 to 8) is made up of two exponential contributions grouping nine parameters (Equation 12).
The second expression (concerns the equations from Equation 9 to 11) is made up of a single exponential contribution grouping nine parameters (Equation 13): The parametric matrix of the two models (Equations 12 and 13) is given in Table 4.
To avoid the dimensional problem of the variables of the equations, the following steps must be followed:  Equation (1) becomes: Knowing that: , , and B,r P,r are dimensionless.
In the same way, one does with the other equations according to the two general adimensional models proposed (Equations 12 and 13). Table 5 is a recapitulation of the 13 new dimensionless models. The experimental data were fit to the models using Origin software. Models are empirical and their constants have no physical meaning.
T A B L E 4 Parametric matrix of the two models.

Model parameters
Number of parameters T A B L E 5 Kinetic parameters of the proposed models and the models in the literature by analogy.  Equations (models) were used to estimate the cumulative biogas production, as shown in Table 5. Eleven modified Gompertz models and two proposed models were used to describe the cumulative biogas production curve. The values of the coefficient of determination (r 2 ), and the mean systematic error RMSE vary from 0.9844 to 0.9892 and 1.27 × 10 −2 to 1.84 × 10 −2 , respectively. The two models proposed (Equations 12 and 13) give the highest values of the coefficient of determination and the lowest mean systematic error values RMSE.
The two models offered have the following advantages: -Gives the best smoothness for the cumulative gas production curves. -By analogy facilitate the use of equations existing in the literature.
Considering the efficiency of the two proposed models, we will try to improve them by proposing three models represented on the following equations: The models' predictions were compared to our own experimental results, which were previously described. The models proposed (Equations 15, 16, and 17) gave better smoothing of the cumulative gas production curves when compared with Equations (1), (12), and (13). The proposed model (Equation 16) with a single exponential and nine parameters gave the highest coefficient of determination value with the lowest mean systematic error value RMSE. Figures 2 and 3 represent the smoothing results of the cumulative gas production curves.
The RMSE for the proposed models is: for Equation (15)   Then a model combining the two contributions is proposed (Equations 16 and 17), and the expression for the proposed model becomes: If γ = 1, Equation (16) is obtained. If γ = 0, Equation (17) is obtained.
In the simulation, the first comparison is made between the responses of the models and the experimental data. The agreement between model predictions and experimental values was quite good, as shown in Figure 4.
The second comparison was made between the proposed models (Equations 16, 17, and 18) and the best model found for each product selected in the literature; BPK model for Sludge and plant waste (Methane: CH 4 ), 41 Gompertz equation for Plant crop residues (Methane: Comparison of the rate of experimental biogas with the volume predicted by the models: (A) Gompertz model Equation (1); (B) Proposed model Equation (16).
T A B L E 6 Kinetic parameters of the imp[roved proposed models.

Models
Parameters   45 using the coefficient of determination and the mean square error are indicated in Table 7.
The selected products have, naturally, different chemical compositions, structures, and porosities compared to our mud. As shown in Figure 4, the proposed models proved to be in good agreement with the experimental data. The new models have high values of (r 2 ) with a max of 0.9996 and low values of RMSE with a min of 6.34 × 10 −4 obtained by the proposed model Equation (18), compared to those obtained with the models of modified Gompertz, BPK, logistic function, transfer function, and exponential for both cases (production of Methane and Hydrogen), as shown in Figure 5.
The model proposed (Equation 18) with 11 constants gave good results compared to the two other models proposed (Equations 16 and 17) which include 10 and 9 constants, respectively. This may mean that the high number of constants gives flexibility to the model, making the best possible predictions. Only the constraints are the initial values input into the software to fit the model with the experimental data.
The data obtained, presented in Table 7, confirms that the new models are universal and can characterize perfectly the production curves of methane and other gases such as hydrogen of different products.

| Biodegradation kinetics
According to Chynoweth et al. 9 and Nikolaeva et al., 46 the anaerobic digestion of most substrates follows firstorder degradation kinetics. The cumulative biogas production is proportional to the amount of COD consumed during the fermentation process. The rate of substrate removal is described by Equation (19).
By integration of Equation (19) and with Y t = L u at t = 0 where Y t is the organic matter content at time t; L u is the initial amount of biodegradable organic matter.
By analogy to Equation (19), the dimensionless model is proposed in this form: We define degradation kinetics of order (n) The solution of Equation (21), gives the kinetics of order (n) where α is the fractional coefficient of the model In Figure 6, the simulated and experimental biodegradation rates are shown. The COD simulated by the literature model (Equation 20) had a slightly slower degradation from the 1st day until the 5th day of degradation, compared to the experimental measurements. Then it became very rapid from the 8th day and stopped on the 17th day, which gives a poor monitoring of the degradation process. On the other hand, the COD simulated by the new models showed a degradation very consistent with the experimental values, and this from the first point to the last point, which gives a good follow-up of the process.
The proposed model (Equation 23) gives the value of the highest coefficient of determination with the lowest mean systematic error value RMSE. Therefore, the previously mentioned best describes the behavior of COD degradation during anaerobic digestion ( Figure 6).
The data predicted by the models were plotted against experimentally measured biodegradation rate values (see Figure 7), and this helped to confirm the accuracy of the models. A good agreement between the experimental and predicted biodegradation rate values has been noted for the two suggested models. All data from the two T A B L E 7 Results obtained from the models proposed for the cumulative biogas production from products in the literature. F I G U R E 5 Evaluation of the coefficient of determination for the five products is represented in Table 7 suggested models are closer to the straight line indicating the best fit.

Reference
To ensure the performance of the proposed models, they were validated using experimental data from the literature.
According to Figure 8, the proposed models (Equations 22 and 23) give a good accord between the experimental and predicted COD biodegradation values.
The proposed models (Equations 22 and 23) validation is checked by comparing the coefficient of determination with Gelatin product used in Schneider 47 study, the residual organic matter product used in Arras 48 study and Sludge product used in Liu et al. 49 study as given in Figure 7. The proposed models (Equations 22 and 23) have the highest coefficient of determination.
A comparison was made between the proposed models (Equations 22 and 23) and the best model found for each product selected in the literature; Equation (20) model for Gelatin, Residual organic matter, and for sludge using the coefficient of determination, and root mean square error is indicated in Table 9. As a result, the proposed models (Equations 22 and 23) have higher values of r 2 and lower values of RMSE than those obtained using Equation (20) from the scientific literature.
The data obtained, presented in Table 9 and Figure 8, confirm that the new models are universal and make it possible to describe the COD biodegradation curves of different products.

| CONCLUSION
A comparative study of cumulative biogas production by anaerobic digestion of sewage sludge was conducted by analogy with similar studies. The first objective of this study is to give an energy value to the sludge by identifying the following points: • Experimental characterization of the chemical composition of the targeted sludge was carried out on site. • A theoretical study in the scientific literature to find a sludge that has the same chemical composition as our own sludge, thus identifying the models that have already been used to follow the kinetics of cumulative biogas production and the degradation of biodegradable organic matter • Five models have been proposed: three to follow the kinetics of cumulative biogas production and two for the degradation of biodegradable organic matter The modeling of the cumulative biogas production was carried out by that of Gompertz and the models proposed (Equations 16, 17, and 19). These models were compared according to the root mean square error RMSE, the coefficient of determination (r 2 ). A good agreement was observed between the proposed models and the experimental data.
The modeling of the degradation of organic matter was carried out by the first-order model (Equation 20), and with two proposed models (Equations 22 and 23). The latter gave a better agreement with the experimental data than the model of the scientific literature.
The proposed models are universal and do not depend on measurement units, which enables researchers to apply them to a wide range of products.

BOD
biochemical oxygen demand COD chemical oxygen demand ONA national sanitation office r 2 coefficient of determination R max peak biogas production rate (mL h −1 ) RMSE root mean square error T air temperature (°C) t L latency time (h) WS wastewater sludge X B cumulative biogas production (mL) as a function of time t (h) X B,r cumulative biogas production X P ultimate potential of biogas (mL) Y t organic matter content at time t χ 2 reduced chi-square