1 Introduction

The combustion of fossil fuel, resulting in environmental challenges and the escalation of resource depletion, brought about the development of renewable and sustainable energy (Song & Zhang, 2015). Due to the inefficient and inconsistent disposal, management techniques are known as the problems generated from agriculture and agro-industries waste (Tripathi et al., 2019). On the other hand, various technologies and alternative processes, such as thermal and thermochemical conversion and biochemical processes, are employed to extract desired components from biomass (Gumisiriza et al., 2017; Selim et al., 2021). To ensure an alternative sustainable waste treatment strategy, anaerobic digestion of organic waste for bioenergy is a good greenhouse gas mitigation (Chen et al., 2014). The process of anaerobic digestion technology is defined as a multi-stage biochemical process involving anaerobic bacteria in converting organic waste into biogas (methane-rich gas). The process is an energy-efficient technology combining sustainable waste disposal and renewable energy production. Indeed, the technology takes place in four steps: hydrolysis, acetogenesis, acetogenesis, and methanogenesis (You et al., 2018). During this process, many factors contribute to the inefficiency of biogas production. Such factors, for instance, include the nature and high carbon-to-nitrogen (C/N) ratio of agricultural waste and the absence of low-cost technology to overcome substrate recalcitrance (Sambusiti et al., 2012; Zhurka et al., 2019).

As a result of this, pre-treatment is regarded as a suitable technique to address this problem. The pre-treatment technique increases the biodegradability of hemicellulose, thereby reducing lignin content to improve and accelerate biogas production and optimize cost and energy (Carrère et al., 2010; Hendriks & Zeeman, 2009). A typical example is the pre-treatment of agricultural residues, using chemical and biological pre-treatment methods to improve digestibility and reduce the digestion time during hydrolysis (Song et al., 2013). In another study, Aklilu and Waday (2021) reported that a practical pre-treatment step is necessary to reduce resistance by altering biomass's chemical and structural features. This pre-treatment application involves the physical, chemical, thermal, and biological processes. In a nutshell, thermal pre-treatment increases biogas production and decreases digestion time (Solé-Bundó et al., 2018). It is known to be the simplest and cheapest method when used in low-temperature heat applications. The methodological problem of this type of pre-treatment deals with the occurrence of Maillard reaction during the application of thermal pre-treatment at a high or low temperature with a long reaction time (Carrère et al., 2010). To address this problem, Carrere et al. (2010) recommend the combination of thermal pre-treatment of low temperature with chemical pre-treatment. Referring to chemical pre-treatment also enhances anaerobic digestion by disintegrating the lignocellulosic structure. The acidic (sulfuric acid), alkaline (sodium hydroxide), and oxidant (hydrogen peroxide) are the main chemical agents used for chemical pre-treatment.

In terms of scaling up and coupling with treating organic waste, anaerobic digestion offers great potential. For instance, the recent report released by the U.S. EPA in 2019 reported over 1200 anaerobic digesters coupled with water resource recovery facilities and another 248 digesters being operated on animal farms. However, these digesters are not performing based on the issue of unexpected perturbations in biogas production, which has resulted in the poor performance of the system as well as high operational costs (Gaida et al., 2015). The diversity and variability of the feed stream and the operation of anaerobic digesters contribute to this development (Wang et al., 2020). One way to address this is through a mathematical model. Mathematical models assist in improving and making significant contributions towards the performance prediction of anaerobic biogas digesters. Despite the advantages of the mathematical model, (Batstone et al., 2015) reported the challenges encountered by the mechanistic model because of a lack of detailed understanding of anaerobic digestion and its application, resulting in inaccurate prediction performance. Therefore, the present study reviews some novel methods capable of predicting digester performance accurately. These algorithms/methods include artificial neural networks (ANN), multilinear regression (MLR), response surface methodology (RSM), etc. One advantage of these models is that they provide the latent interaction between numerous input features and output results to achieve output prediction.

The optimization process parameter affects the biogas digester, thereby increasing the biogas production rate either in a batch or continuous system. For the batch system, the problems of anaerobic digestion, such as overloading or improper substrate ratio, are not stimulated, thereby causing disturbance and failure of the digester system. In contrast, the continuous system deals with the measurement of biogas production by a load of the substrate, which can be used to investigate the actual productivity of the biogas and the performance of the anaerobic digestion system in the long run (Duong & Lim, 2022a). According to Duong and Lim (2022b), the model equation involving temperature data in a continuous system and other factors is essential in estimating biogas production. It is interesting to state that the optimization process is relevant for biogas production from organic waste (Yılmaz & Sahan, 2020). Due to several experiments required and a significant amount of time and resources consumed, traditional methods are said to be ineffective for optimization studies.

Studies have been conducted on the pre-treatment and mathematical model as techniques for improving biogas yield. It is interesting to note that the majority of these studies were conducted experimentally in the laboratory. For instance, for pre-treatment techniques, studies were carried out by Baradar et al. (2016a), Abraham et al. (2020), Shitophyta et al. (2022), Lee et al. (2019); Wadchasit et al. (2020), Nugraha et al. (2018) and Karimu et al. (2019). On the other hand, some of the mathematical model studies include Wang et al. (2020); Obileke et al. (2021); Aklil and Waday (2021); Iweka et al. (2021); Liano et al. (2021); Beltramo et al. (2016); Taherzadeh et al. (2008); Asadi et al. (2020); Mougari et al. (2021); Xu et al. (2014); Rossi et al. (2022) and Diong and Lim (2022a). The comprehensive findings from these studies are discussed and presented in the present study. In these studies, it is established that pre-treatment increases the anaerobic biodegradability of the substrate, whereas the mathematical model predicts the optimum performance of biogas yield in anaerobic digestion. However, there is a need for the combination, incorporation, and integration of both pre-treatment and mathematical models as an optimization technique for biogas yield improvement. This seems lacking in the literature and needs to be well studied, creating a research gap. With that in mind, the present study fills a research gap in the existing literature and technologies regarding both techniques, which can now be accessed in a single paper. To our knowledge, this study is the first in which both techniques are combined and presented as a review paper. Therefore, this review aims at (1) comparing the pre-treatment and mathematical model of improving biogas yield and (2) discussing, analyzing, and drawing conclusions and recommendations, as the study offers effective tools for improving the anaerobic digestion of biogas yield.

1.1 Why optimisation technique (methodology of the study)

The optimization helps to support decision-making regarding sustainable development strategies that utilize eco-friendly technologies for efficient power generation. In this case, it usually predicts the model behaviour from biomass residues (Hatata et al., 2021), requiring many experimental datasets (Şenol, 2021). The optimization of process parameters in an anaerobic digestion process can accelerate the hydraulic retention time and assist in optimizing the specific parameters that are considered predictors of the desired response (Cremona et al., 2022). The employment of pre-treatment methods as an optimization process in the anaerobic digestion process has been widely confirmed to be efficient and reliable based on experimental studies (Varjani et al., 2022; Zhan et al., 2022). The exploitation of models in predicting and optimizing the performance of the anaerobic digestion process has led to the identification of challenges and a better understanding of the dynamic or kinetic processes associated with the overall behaviour of the plant (Enitan et al., 2017). Having established the need for the methodology used (optimisation technique), a conceptual review was employed for this study amongst the existing literature review types. From the objective of the study, which aimed to synthesize conceptual knowledge, thereby contributing to a better understanding of the subject topic, the conceptual review was chosen. The authors sought it wise and necessary to use the abstract evaluation because it provides the sensitive areas of the study on improving biogas yield via pre-treatment and mathematical model. This was deemed essential for the overall detailed, broad review and collection of data information, including Springer, ScienceDirect, SCOPUS, Web of Science, and Google Scholar (90 articles were reviewed). In addition, conference proceedings and scientific/technical reports were also used.

2 Pre-treatment techniques to improve biogas yield

There have been advancements toward the improvement of biogas yield by pre-treatment techniques. These techniques are physical/mechanical, chemical, and biological processes and will be briefly discussed in this section.

2.1 Physical and mechanical pre-treatment

The physical/mechanical pre-treatment improves the material's flow, porosity, and bulk density characteristics, forming a new surface area (Barakat et al., 2014). In this type of pre-treatment, the structure of the biomass is altered, and the size of the particles is reduced through the application of physical force. This process increases the surface area of the particles susceptible to microbial and enzymatic attack, thereby improving biogas production via the AD process (Karuppiah & Azariah, 2019; Zheng et al., 2014). Examples of physical and mechanical pre-treatment include milling, microwave irradiation, extrusion, etc.

2.1.1 Milling

The essence of milling pre-treatment is to reduce the biomass size, thereby opening the cellular structure and improving the bio-accessibility of the cell tissue. This is done by increasing the specific surface area of the biomass. A particular example where milling techniques can be carried out is on lignocellulose and algal biomass (Motte et al., 2014). The reduction of particle size is associated with the milling process, increasing the rate of enzymatic degradation and reducing viscosity in the biogas digester (Karuppiah & Azariah, 2019). A 1–2 mm particle size is recommended for an effective milling process (See Table 1). Table 1 shows examples of the effect of the milling process on biogas yields from the selected studies.

Table 1 Effects of the milling process on biogas yield
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2.1.2 Microwave irradiation (MI)

Microwave irradiation occurs as a result of microbial cell destruction by the action of the disruption of the chemical bond present in the cell walls and membranes. This is then polarized by macromolecule parts aligning with the poles of the electromagnetic field. This process is referred to as denaturation (Karuppiah & Azariah, 2019). However, microwave irradiation is characterized as having a wavelength between 1 mm and 1 m, and non-ionizing radiation can convey energy and separate substances. The MI is found on the electromagnetic spectrum between 300 and 300 000 MHz (IEO International Energy Outlook, 2009). It is seen that the dielectric characteristics of feedstock are one factor affecting microwave irradiation. The dielectric feature usually depicts the strength of the material to stock electromagnetic energy into heat. Table 2 presents examples of studies to show the effect of pre-treatment of feedstock on biogas yield.

Table 2 Effect of microwave irradiation pre-treatment on biogas yield
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2.1.3 Extrusion

The feedstock undergoes heat, compression, and shear force during the extrusion pre-treatment. This leads to the destruction, chemical modification, and alteration through the extruder process (Karuppiah & Azariah, 2019; Olatunji et al., 2021). The extrusion process operates through a single or twin screw that twists into a firm barrel with the presence of temperature control apparatus. As a result of this, there is an increase in pressure and temperature through the barrel due to the friction and energetic shearing experienced by the feedstock. Its effect on biogas yield occurs at the finishing ends when the feedstock releases pressure, which causes structural changes in the processed biomass, enabling easy anaerobic digestion (Zheng et al., 2014). Studies on the extrusion pre-treatment of feedstock and its effect on biogas yield are presented in Table 3

Table 3 Improvement of biogas yield by extrusion pre-treatment
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2.2 Chemical pre-treatment

According to Zhou et al. (2012), chemical pre-treatment offers a better advantage than physical and biological pre-treatment. This is because of its effectiveness and ability to biodegrade complex feedstock. Examples of chemical pre-treatment commonly used to improve biogas yield include acidic, alkaline, oxidative, and ozonation.

2.2.1 Acidic pre-treatment

The Acidic pre-treatment increases the rate and efficiency of the AD process through the release of intracellular organics as a result of the sludge disintegration and cell lysis. Various forces are responsible for the disruption during the acidic pre-treatment. These include the Van der force, hydrogen, and covalent bonds comprising biomass composition. The hemicellulose is broken down during this process, and cellulose is reduced (Li et al., 2010). Acidic pre-treatment is the most typical conventional pre-treatment for most biomass material treatment; however, some chemicals make the technique less appealing. Based on that, Saha et al. (2005) recommended that biogas digesters be constructed to withstand most acids' corrosive and toxic properties. Examples of such acids are the HCl and H2SO4 (See Table 4).

Table 4 Improvement of biogas yield by acidic pre-treatment
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2.2.2 Alkaline pre-treatment

This pre-treatment induces the swelling of particulate organic at high pH, which enables the biomass cellular substances to be more susceptible to the action of enzymes (Gumisiriza et al., 2017). The act of alkaline pre-treatment depends on the feedstock used for biogas production. For instance, the enhancement of hydrolysis of RNA, organic liquefaction of proteins, and saponification in macroalgae, as well as the swelling, delignification, and de-esterification of the intermolecular ester bond in lignocellulose biomass (Karuppiah & Azariah, 2019). Interestingly, the alkaline pre-treatment is said to be economical but very costly at the downstream processing. This is attributed to the large volume of water required to remove the salt from the feedstock. Hence, this process takes time. Studies have been conducted to show the effect of alkaline on the feedstock treatment to improve biogas yield, as seen in Table 5

Table 5 Effect of alkaline pre-treatment of feedstock on biogas yield
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2.2.3 Oxidative pre-treatment

These include the application of ozone, FeCl3, hydrogen peroxide, oxygen, and air as oxidizing agents towards the enhancement of hydrolysis of cellulose to solubilize the lignin and hemicellulose of feedstock (Monlau et al., 2013). On the other hand, it involves the complete contact degradation of organic compounds into carbon dioxide and water due to the enhanced contact between molecular oxygen and organic matter. Hence, high temperature and pressure are required (Strong et al., 2011). In addition, the high pressure is a function of the high-temperature conditions, increase in concentration, and oxidation rate. One major challenge of the oxidative pre-treatment technique is its effect and damage to hemicellulose, making it difficult for digestion (Lucas et al., 2012).

2.2.4 Ozonation/Ozonolysis pre-treatment

According to Olatunji et al. (2021), the ozonation technique operates under ambient temperature and pressure. This makes it different from other chemical pre-treatment techniques. In addition to that, it does not produce toxic material. It is environmentally benign, thereby having no effect on other processes, such as yeast fermentation and hydrolysis of enzymes after pre-treatment (Hu & Wen, 2008). Cardena et al. (2017) published that treating different types of feedstocks with the ozone method has generated and improved biogas yield by 66% (specifically pre-treatment of microalgae feedstock pre-treated with ozone). The ozonolysis process is considered uneconomical; hence, the process is not suitable for industrial or large-scale pre-treatment.

2.3 Biological pre-treatment

The biological treatment uses microorganisms to enhance the biodegradation of organic material to improve biogas yield. It offers more effective, environmentally friendly treatment and requires less energy consumption when compared with chemical and mechanical pre-treatment (Park et al., 2010). During the biological pre-treatment, the microorganism degrades the feedstock lignin content for biogas production. The lengthy incubation time is reported to be a significant factor affecting the biological pre-treatment. This is attributed to the degradation of cellulose and hemicellulose together with lignin (Millati et al., 2011) (Table 6).

Table 6 Examples of biological pre-treatment to improve biogas yield (Olatunji et al., 2021; Bremond et al., 2018; Carrere et al., 2016; Fountoulakis et al., 2002; Fu et al., 2015; Ghosh & Bhattacharyya, 1999; Hosseini et al., 2019; Tsapekos et al., 2017)
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3 Research on pre-treatment techniques for improving biogas yield

The choice of pre-treatment process to improve the biogas yield depends on the physicochemical properties and the structural arrangement of the substrate or feedstock. This enhances the formation of organic feedstock (Xu et al., 2014). According to Taherzadeh and Karimi (2008), the pre-treatment technique aims to avoid the degradation or loss of carbohydrates, reduce the possible impact on the environment, and act as an avenue that leads to the degradation of feedstock.

In addressing the long retention and low efficiency as a problem associated with anaerobic digestion, Baredar et al. (2016b) reviewed the enhancement of biogas yield by pre-treatment and adding additives. Part of the study's methodology includes considering different organic wastes and their biogas yield and reviewing various methods used to minimize the challenges of anaerobic digestion, such as the thermal, chemical, thermo-chemical, and ultrasonic pre-treatment techniques. On the other, the organic and inorganic additives were explored. From the review's findings, organic wastes generate a biogas yield of 0.1–0.9 m3kg−1VS, depending on its type. However, the pre-treatment technique and additives employed resulted in an increase of 60–150% and 80–150% methane production, respectively. Interestingly, adding additives to the pre-treatment provides novelty in their study. Usually, additives increase microbial activities, thereby increasing the digestion rate of the process, which results in a high biogas yield. Therefore, the authors recommend the combination of pre-treatment and additives for the improvement of biogas yield as well as for reducing the retention time of anaerobic digestion.

The problem with the lignocellulose substrate has to deal with the proper digestion of biomass during anaerobic digestion. Hence, this is attributed to feedstock's complex and recalcitrant nature (Patinvoh et al., 2017). In addressing this limitation, various processes (co-digestion, bioaugmentation, nutrient supplementation, and solid-state anaerobic digestion) have been deployed and used to degrade lignocellulosic feedstock. However, the pre-treatment of biomass remains the best option for improving methane production and rate of digestion. This was confirmed in a review by Abraham et al. (2020) on the pre-treatment strategies for enhanced biogas production from lignocellulosic biomass. In the study, the authors reviewed and presented types of pre-treatment methods and their effects on biogas production, stating their anaerobic conditions, which are still unclear in the literature. To this effect, it was revealed that the best and most common anaerobic condition for the pre-treatment technique is the batch at the temperature of 37 °C for 30–110 days. In an experimental study to establish the impact of pre-treatment, ionic liquid was used as a pre-treatment strategy for anaerobic digestion, and it was reported that it improved biogas production by 1200%. The authors propose future studies entailing the cost-effective and sustainable pre-treatment methods for biogas yield using different lignocellulosic substrates.

The study conducted by Shitophyta et al. (2022) was motivated by a gap in the literature on utilizing alkali and organic solvent pre-treatment in the anaerobic digestion of food waste. Based on that, the effect of alkali pre-treatment and organic solvent pre-treatment on biogas yield is determined. The authors also considered the impact of physical pre-treatment on biogas production. As part of the methodology, NaOH was used as the alkali reagent, C2H5OH as the organic solvent, and food wastes (rice, fruits, and vegetables) were the types of waste used in the study. Also, the cow rumen was used as the inoculum. The food waste was soaked at room temperature (20 °C to 25 °C) in a chemical reagent for 24 h during pre-treatment and was conducted in a 1L batch biogas digester. The result of the study confirmed that pre-treatment increased the biogas yield with respect to their concentrations. Based on the statistical result employed, which is not usually considered, the physical pre-treatment had a significant effect (p < 0.05) with the highest cumulative biogas yield of 58.2 mL/gVS. In contrast, alkalis and organic solvent pre-treatment proved no significant effect (p > 0.05) on biogas yield.

From the point of view of Lee et al. (2019), the issue of low C/N ratio and poor biodegradability of sewage sludge was said to have brought about the need for a pre-treatment technique. These factors affect significant pollution control and energy recovery offered by anaerobic digestion, especially concerning sewage sludge. The application of the pre-treatment technique improves solubilization during anaerobic digestion and co-digestion. Focusing on the study's methodology, microwave, ultrasonic, and heat pre-treatment was applied to the sewage containing waste-activated primary sludge and 20 g/L TS. Prior to the pre-treatment, the sewage sludge was sieved in a microwaved 500 mL bottle at 2.450 MHz/700 W for 6 min, while for ultrasonic, the sewage sludge was ultrasonicated in a 2L reactor at 20 + 30 kHz for 60 min, and thereafter stirred. The heat pre-treatment involved autoclaving sewage sludge in a 500 mL bottle at a temperature of 110C under 1.0 to 1.3 atmospheric pressure for 60 min. Afterward, it was subjected to cooling for about 2 h until it reached room temperature. It was revealed from the study that microwave irradiation at 700W for 6 min yielded the highest biodegradability of 62.0% concentration, a solubilization efficiency of 59.7%, and methane production of 329 mL/g VS. From the findings of the study, the choice of pre-treatment is essential as it plays a role in the efficient anaerobic digestion, thereby enhancing methane yield.

The production of biogas from raw EFB usually gives a low gas yield. Therefore, Wadchasit et al. (2020) employed a pre-treatment method to improve the yield, thereby considering its effects. This was done using chemical (NaOH solution), physical (size reduction), and biologically (activated sludge) pre-treatment at three different concentrations in a batch mode under high solid anaerobic digestion (10–15%TS). However, before the pre-treatment of the EFB, the total solid (TS) and volatile solid (VS) of the effluent were determined using the standard methods for the examination of water and wastewater (APHA). Thereafter, an inoculum of methanogenic bacteria was obtained from an anaerobic sludge to speed up biodegradation. The authors reported that after the pre-treatment process, the residue of the EFB was washed with water several times to obtain a neutral pH, which was used to analyze the composition of lignin, hemicellulose, and cellulose. The study was conducted at a temperature of 55 °C in 500 ml serum bottles covered with airtight caps used as digesters and was monitored for 35 days. As expected, all the pre-treatment techniques used in the study improved biogas yield significantly and differently, with the physical (size reduction) obtaining the highest biogas yield of 429.9 ml/gVS (> 90% increase). Notably, the highest biogas yield is because of the increase in the surface area, cellulose content, and reduced sugar concentration, which is said to be readily digested by microorganisms. This differs from the change in the composition of cellulose and lignin of the EFB after the pre-treatment process, which plays a vital role in biogas yield improvement.

To address the problem of resistance to biodegradation or hydrolysis of lignin during the production of biogas, the effect of acid pre-treatment using acetic acid and nitric acid in biogas production from rice husk during solid-state anaerobic digestion was conducted. Indeed, the role of pre-treatment is to assist in the degradation of lignin contained in agricultural waste. The pre-treatment method is suitable for increasing biomass hydrolysis efficiency with increasing accessibility enzymes. This is because of its effective hydrolysis, less degradation product, and more sugar produced, especially using organic acid pre-treatment. The Nugraha et al. (2018) study used acid pre-treatment of acetic and nitric acid in the 3% and 5% percentage variety levels, respectively. After 60 days of the retention period, the biogas production was measured using the water displacement method. The study showed that pre-treatment of 5% and 3% acetic acid on rice husk yielded 43.28 and 45.86 ml/gr.TS, whereas that of the 5% and 3% nitric acid generated 12.14 and 21.85 ml/grTS of biogas production, respectively. From the findings obtained, both pre-treatments of rice husks improved the biogas yield, with acetic acid performing better than nitric acid pre-treatment. This is attributed to the elimination of inhibitory compounds (lignin and cellulose) by acetic acid, which are more susceptible to the degradation of enzymes, thereby increasing the production of biogas.

Karimou's (Karımou, 2019) study conducted the effects of physical and chemical pre-treatments on biogas yield from maize straw and cattle manure. The essence of the study was to break down lignocellulosic biomass through pre-treatment and then determine their effect on biogas production. Usually, it is recalled that lignocellulose in biomasses decreases the degradation by anaerobic microorganisms and biomethane production potential. Therefore, the employment of pre-treatment considered in the study increases the solubility and degrade more efficiently using anaerobic bacteria. Prior to that, the authors carried out the analytical test for total solids (TS) and volatile solids (VS) for maize straw and cattle manure. From the test, both agricultural wastes obtained a TS of 41.2% and 86.1%, while the VS was 51.6% and 82.0%, respectively. In terms of the chemical pre-treatment (acidic and alkaline) used in the study, the alkaline pre-treatment was reported as the effective pre-treatment method for the maize straw, thereby reporting an increase of 163% in cumulative biogas. In contrast, the microwave pre-treatment produced the highest increase of biogas of about three folds. In the case of the cattle manure, the acid pre-treatment was effective, with a 103% increase in the cumulative biogas, while the microwave pre-treatment generated a 97% increase in biogas. Based on the study findings, maize straw has a higher biogas production potential than cattle manure. However, the handling associated with maize stream regarding mechanical breakdown requires good machinery and high energy input.

4 Mathematical models to improve biogas yield

This section presents the various mathematical models used to predict biogas yield. Such models include the ANN, MLR, RSM, AP, etc.

4.1 Aspen plus (AP)

Aspen Plus is a process simulation software used primarily on industries to stimulate thermochemical and chemical reactions regarding industrial processes. Also, it uses mathematical models to predict the performance of user-defined anaerobic digestion processes (Llano et al., 2021). Usually, the single-stage and two-stage anaerobic digestion uses AP v 10 to indicate the biogas yield, among other techno-economic and environmental parameters.

In Fig. 1, the single-stage model (a) deals with the one stoichiometric reactor, where the four stages of anaerobic digestion (hydrolysis, acidogenesis, acetogenesis, and methanogenesis) occur in the same unit based on the scheme and extent of reaction (Nduse & Oladiran, 2016). On the other hand, the two-stage model (b) focuses on converting carbohydrates, proteins, and fats into sugar, amino acids, and fatty acids using one stoichiometric reactor in the first reactor. The second reactor of the two-stage AD is a continuous stirred tank reactor (CSTR) for converting such compounds into biogas.

Fig. 1
figure 1

Schematic diagram of Aspen Plus model technique–Extracted from Llano et al. (2021)

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4.2 Artificial neural networks (ANN)

The ANN is called data-driven, approximating the nonlinear relationship between the independent input process variables and the dependent output variables. It describes the structured system with several interconnected neurons ordered in layers. Usually, the ANN deals with multi-layered perception that consists of input, hidden, and output layers (Beltramo et al., 2016). In comparing the ANN with the multilinear regression (MLR), principal components regression (PCR), and partial least square regression (PLS), Kessler (2007) stated that the ANN enables a more reliable and accurate approximation of the relation between the input variables and the predicted output variables (Gueguim Kana et al., 2012). In Hitzmann and Kullick (1994) and Hitzman et al. (1998) study, the ANN provides the nonlinear sigmoid function at the hidden layer, thereby giving the technique computational flexibility with respect to the linear regression methods. Also, the ANN provides accurate prediction performance of the desired variables. In evaluating and optimizing an anaerobic digester, the ANN is a powerful tool for this purpose. Hence, it has been used to optimize biogas production from waste biogas digesters and control the digester process (Abu Qdais et al., 2010; Holubar et al., 2000, 2002). In addition, the ANN has been employed to predict the trace compounds in biogas (Strik et al., 2005) and methane fraction from field-scale landfill bioreactors (Ozkaya et al., 2007).

4.2.1 Adaptive network-based fuzzy inference system (ANFIS)

The ANFIS is seen as the framework of the membership function and successful methodology for solving problems, modeling, data mining, and abating the intricacy of data (Jang, 1993). The system was developed based on the Takagi–Sugeno fuzzy inference system (FIS) (Güler & Übeyli, 2005). According to Heddam et al. (Heddam et al., 2012) and Salahi et al. (Salahi et al., 2015), the ANFIS benefits from the advantages of both the ANN and FIS with high training rates and efficient learning algorithms. Interestingly, using the ANFIS, both complex and nonlinear systems are said to be accurately modeled. In building the FIS structure, three methods encompassing grid partition (GP), thereby employing different kinds of membership functions and subtractive clustering (SC) as fuzzy c-means clustering (FCMC). These are clustered using the Gaussian type of membership function (Sivanandam et al., 2007). The three methods include genfis 1, genfis 2, and genfis 3, which describe the GP, SC, and FCMC. The genfis 1 (GP) produces a fuzzy system according to the membership function types and numbers, whereas the genfis 2 determines the number of rules and membership functions generated in FIS. For the genfis 3, the function produces the fuzzy system due to FCMC and extracts a set of rules that stimulate data behaviour.

4.2.2 Multiple linear regression (MLR)

Multiple linear regression (MLR) is also known as a simple regression model, which is essential in predicting biogas yield from feedstock characteristics and maximizing biogas yield through the management of anaerobic digestion. Further, it can be useful in performing preliminary energetic and economic evaluations (Duong & Lim, 2022a). The use of MLR for biogas prediction is attractive because of its simplicity and effectiveness. In addition, the technique can compare the accuracy of predicted and actual values based on the model equation. Despite the limited selection of input variables, types, and ranges associated with the MLR, the technique is much easier than the machine learning model (Duong & Lim, 2022a). Hence, the machine learning model is known to predict biogas production with high accuracy; however, it requires statistical skills and training because of its complexity. Primarily, the MLR deals with modeling a linear relationship between the independent and dependent variables. According to Abdipour et al. (2019), MLR is based on the linear and additive association of explanatory variables in focus to attempt the model. This focuses on the relationship between dependent and two or explanatory variables by assumption. Since the efficiency prediction of the MLR model depends on linear relationships of the input and output parameters, various studies have been conducted and published because of their simplicity.

5 Research on mathematical modeling for improving biogas yield

The process parameters used for optimization usually influence the biogas digester, thereby increasing the biogas yield. Based on this, Yilmaz and Sahan (2020) stated that this is significant, primarily for biogas production, especially from organic waste. However, the traditional optimization methods are said to be ineffective because of the involvement of several experiments. To address this problem, Aklilu and Wadey (2021) conducted a study aiming to maximize the biogas yield from anaerobic co-digestion of alkali-treated corn stover and poultry manure using the ANN and RSM. The study used independent process parameters such as temperature (25–45 °C), hydraulic retention time (4–20 days), pH (6–8), and (50–90%) percentage blending ratio (poultry manure to alkali treated CS ratio) on the biogas yield response. Part of the study's methodology is using a quadratic model built with Design Expert software version 11.12.06 for the RSM, while that of the RSM was done with the help of the MATLAB Neural Network Toolbox. More importantly, the study employs RSM for the experimental design and statistical analysis, whereas the ANN deals with the modeling and predicting biogas production. From the study findings, a maximum biogas yield of 745 ml/g TS was achieved at the following optimum process parameters: Temperature (13 °C), HRT (13 days), pH (7), and blending ratio (80%) with the desirability of 0.995. Interestingly, the study findings are attributed to the mixture of poultry and alkali-treated corn stover used in the study, which is a good substrate for enhancing biogas yield. Hence, the ANN model was said to be more efficient than the RSM model in terms of data fitting and prediction capabilities and accuracy from the determination coefficient (R2), root mean square error (RMSE), standard error of prediction (SEP), mean absolute error (MAE) and absolute average derivation (AAD).

One of the applications of optimization studies has to deal with the decision-making process. Obileke et al. (2022) study was motivated to address the global challenge of facilitating a decision-making process based on methane production using a predictive model. In addition, evaluating the performance of the biogas digester concerning operating conditions for optimization purposes forms the aim of the study. The study employed the constrained linear square to optimize the input parameter (pH, global irradiance, humidity, and temperature) concerning the desired response (biogas). Four hundred thirty measured datasets were collected in 18 days of the experiment, which were split into two, 286 (67%) and 144 (33%) as trained dataset (model) and test dataset (validation). An exciting aspect of the study is the novel design of the gas temperature and pressure measurement sensor (GTPMS) and data acquisition system (DAS), which aimed to monitor the performance and collection of data. The GTPMS consists of the gas sensor, pressure sensor, air pump, hydrophobic filter, and thermocouple modules, while the DAS contains the datalogger, power supply unit, circuit breaker, and converter. The result of the study demonstrated a strong validity as the determination coefficient (R2) between the developed model and the optimized output over the sample observation was reported as 0.968. This indicates that the modeled value of the methane fits well with the measured value for validation. Hence, an improvement of up to 26.5% in methane was obtained compared with the maximum volume from the optimization process, with the maximum methane volume (54.5%) produced in the biogas digester. Authors attributed this increment or improvement to the effect of temperature and other meteorological factors used for the developed model.

Another study based on the decision-making process of optimization was conducted by Iweka et al. (2021). The study focuses on optimizing biogas production from biomass waste through anaerobic digestion, using central composite design (CCD) of response surface methodology (RSM) and Phyton coding. Also, the authors considered the feasibility of using corn chaff inoculated with cow dung under mesophilic temperature conditions. The essence of the study ensures and helps facility operators dealing with biogas facilities to facilitate the process of decision-making with respect to the predictive model obtained. The study was carried out in a glass digester of a volume of 256 mL with rubber stoppers and was operated at a temperature of 25–33 °C. With the aid of the methodology, the corn chaff was inoculated with cow dung using different mixing ratios of 1:1, 1:1.55, and 1:3.5 for the hydraulic retention time (HRT) of 25, 31, and 37 days of monitoring. This was done using a central composite design to optimize and predict the optimal response. The result from the study revealed that the mixing ratio of 1:1.55 for 37 days produced the highest cumulative biogas yield of 6.19 L within a mixing ratio of 0.65. From the Python coding, input factors resulted in an optimal value of 4.71L, similar to the CCD result. The result confirmed that both CCD and Python coding are suitable and reliable for the optimization of the production of biogas. Hence, both predict the same optimal values and the highest cumulative biogas yield. However, using the GC–MS characterization, the biogas composition generated in the study was 68% methane and 22.76% carbon dioxide.

Biogas production is paramount in reducing carbon emissions and optimizing the recovery of resources from waste streams. Based on this, the optimization of biogas production through anaerobic digestion of municipal solid waste, food waste, and lignocellulosic biomass was conducted by Llano et al. (2021) using the Aspen Plus (AP) v ten software. The essence of the Aspen Plus in the study was to stimulate the different anaerobic digestion conditions and feedstock, aiming to optimize the biogas and methane production using 16 scenarios at thermophilic temperature conditions of 55 °C using the model approach for stimulation. These two model approaches are one digestion stage and two stages coupled in series, accounting for each of the scenarios in terms of inlet mass flow and varying feedstocks (municipal solid waste, food waste, and lignocellulosic biomass (LCB) as well as co-digestion of various feedstocks. During the analysis of each scenario, the following factors were considered, including the sizing, costing, and environmental aspects. From the simulation result, the single-stage approach obtained a biogas yield ranging from 305.5 to 406.4 mL/g VS, whereas the two-stage approach generated a biogas yield of 64.78 to 358.8 mL/g. Meanwhile, from the feedstock used in the study, the LCB was reported to have yielded the maximum methane, resulting in 106.0 mL/g VS.

Developing a tool capable of analyzing the rate of the biogas production process concerning its flow rate has been a challenge. As a result of this, the artificial neural network prediction of the biogas flow rate optimized with the ant colony algorithm (ACA) was conducted by Beltramo et al. (2016). The study is a methodology approach to analyze the biogas production process, interested in predicting the biogas flow rate via the ADM1 model and then comparing it with the ANN approach. Also, as part of the study's methodology, the simulation was performed for three months with a frequency of 20 simulation points per simulation day, and the substrate used was the co-digestion of cow manure and grass silage. The ANN model consists of input and output neuron layers in the study. Here, the input layer represents the independent process variables, whereas the output neuron is the dependent predicted process variable (biogas flow rate). Based on the data used for the study, this was split into three training (70% of the data), validation (15% of the data), and test (15% of the data) data set. These data are used to adjust the model (training dataset), measure the network generalization, and stop the training before overfitting (validation dataset) and prediction (test data- set). The study employed the ANN models using 11 input and 10 hidden neurons. In this case, the model with 11 input neurons and 10 hidden neurons obtained the best performance in terms of prediction (RMSE of 5% and R2 of 0.92). The ant colony algorithm used in the study as an optimization technique simplifies the model dimension and improves the performance of the prediction. Hence, this reduces the analytical time and cost of managing substrate composition. The study indicates and reveals the evaluation of feed composition and improving process output.

As mentioned earlier, finding the most suitable optimization tool for the performance of biogas production has been a challenge. Previously, the ant colony algorithm was used as an optimization technique by Beltramo et al. (2016); however, Hatata et al. (2021) study proposed the moth flame optimization (MFO) technique to identify the optimal structure of multilayer feedforward network (MFFNN) to predict the biogas production. The study used four novel two-dimensional mathematical models (TDMMs) and ANN to stimulate and predict biogas production via an anaerobic co-digestion process using waste-activated sludge and wheat straw. The authors aim to use various ratios of the substrate (waste-activated sludge and wheat straw) to improve the C/N ratio in enhancing the anaerobic biodegradability and obtaining the highest yield of biogas. This was done in a batch reactor (six) of volume 5L filled with 2.50 kg sludge for 130 days of monitoring. In the first reactor, only activated sludge was contained, while in the other reactor, the sludge was mixed with wheat straw at 3%, 4%, 5%, 6%, and 7%. For the properties of the substrate (activated sludge/wheat straw), the following characterization was done TS (1.20/90.00%), TVS (68.50/87.50%), COD (16.00 g/L), total nitrogen 4.85/0.59%), total carbon (24.50/48.50%), phosphorous (2.80/1.30%), potassium (0.35/2.13%), and pH (7.20). On the other hand, the input parameter used in the study includes the time of day and percentage mixture of the substrate. In contrast, the output parameter is the cumulative biogas production. The study findings revealed that a 7% ratio produced the highest yield of biogas and reduced the TS (58.06), TVS (66.55%), and COD (74.67%). The developed ANN and TDMMs also showed highly satisfactory conformity with the experimental data for biogas production. Thus, for the ANN result, the MSE values for training and testing the proposed MFFNN-MFO model were 3.361 × 10–2 and 14.8662, respectively. The integration of MFFNN with the MFO method is recommended to obtain an optimal neural network for the prediction accuracy of biogas yield.

Models have been developed to stimulate various anaerobic digestion processes, specifically biochemical, biological, and physicochemical (Najafi & Ardabili, 2018). Indeed, the ANN and adaptive network-based fuzzy inference system (ANFIS) approaches have been employed to predict biogas. Based on this, the large variability of the municipal wastewater treatment plant (MWTPs), the operation, geographical locations, and facility-specific models must be considered for the accurate capabilities of the prediction. Assessing the reliability of different structures of ANN and ANFIS model for biogas production from MWTPs digester in cold regions has not been explored. On the other hand, reducing the input data via principal component analysis (PCA) for the efficiencies of ANN and ANFIS models for the processes' input parameter variables is also yet to be studied. Based on this, Asadi et al. (2020) published an article on estimating biogas production using data-driven approaches for an anaerobic digester in cold region MWTPs. This was necessary to address the gap above. The study aimed to evaluate the biogas production rate from the anaerobic digester using ANN and ANFIS models. The dataset point of the study was 168 from 2014 to 2016. For the substrate physiochemical properties, the following parameters were considered: the total solid (TS), volatile solids (VS), fixed solids (FS), volatile fatty acid (VFA), and the pH, monitored according to the American Public Health Association (APHA), American Water Works Association (AWWA) and Water Environmental Federation (WEF) 2005. The ANN model of the study consists of the input, hidden, and output layers, whereas the ANFIS was built in terms of grid partition (GP), subtractive clustering (SC), and fuzzy c-means clustering (FCMC). The input parameter in the model equation includes the VFA, TS, FS, pH, and inflow rates, while the output parameter is biogas production. It is revealed from the study that the ANFIS-FCMC model estimation was found to be more accurate and resulted in the emission rate of methane (3.086 g/min), carbon dioxide (6.351 g/min), and hydrogen sulphide (41.46 g/min), respectively, with the observed data. Hence, the ANFIS resulted in an R2 of 0.92 and RMSE of 0.54, while the ANN recorded an R2 of 0.88 and RMSE of 0.23. The result is important for determining the economic feasibility of implementing combustion gas heat and electricity generation and increasing the biogas yield.

The practical model for predicting the methane yield in solid-state anaerobic digestion remains unclear. The employment of conventional methods (laboratory experiments) is time-consuming and labour-intensive. On the other hand, using results obtained from various laboratory experiments is often different because of discrepancies in materials and experimental procedures used in the study. To address this, Triolo et al. (2011) stated that data-driven models (statistical and ANN methods) tend to predict solid-state anaerobic digestion without extensive laboratory experiments. To establish this, Xu et al. (2014) study indicated the methane yield of lignocellulosic biomass in mesophilic solid-state anaerobic digestion. The study provided 50 dataset points from 10 publications for statistical analysis. Of these 50 dataset points, 20% were used for model validation from three independent studies by Liew et al. (2011), Xu and Li (2012), and Xu et al. (2013). On the other hand, 40 data set points from other published literature were used for model calibration for MLR and ANN. From the parameters or variables used in the study, it was observed that cellulose, lignin, extractives in feedstocks, and inoculation size (F/E ratio) were found to be important parameters in both ANN and MLR. Hence, the interaction between the F/E ratio and lignin was a significant factor in MLR. Both MLR and ANN models were calibrated and validated with a different set of data from the literature, and it was reported that both methods were able to satisfactorily predict the yield of methane of the solid-state anaerobic digestion. Although, the lowest standard error for prediction obtained in the study is the ANN model. The authors suggested that guidance and ideas for future feedstock evaluation and process optimization models should be developed for solid-state anaerobic digestion through the study.

Mostly, research on the optimization of biogas production deals with feedstock composition, process management (OLR, HRT, and mixing ratio), and operating conditions (pH and temperature), according to Rocamora et al. (2020), Panigrahi et al. (2019). OLRs higher than 5–6 g TVS/(L d) optimize the specific production of biogas while thermophilic conditions increase the microbial kinetics, thereby permitting the treatment of higher OLREs than mesophilic processes (Kothari et al., 2014). Despite all this, it has been reported that the intrinsic variability of experimental results and the unrealistic methodology used for the optimization studies remain challenging. This is specifically to achieve high accuracy of the predictive models. Based on such observation, Rossi et al. (2022) proposed to develop a predictive model (MLR) for biogas production from dry anaerobic digestion of organic fraction of municipal solid waste (OFMSW). The study used a data set from an experimental test on a pilot scale plug flow reactor (PFR), including 332 observations, to build the model. The biogas digester was conducted using PFR of a stainless-steel volume of 37L and a working volume of 28L and fed with OFMSW under the thermophilic temperature of 53 ± 2 °C. Methane and carbon dioxide concentration was measured and monitored using two infrared sensors. On the other hand, the inlet feedstock was analysed for TS, TVS, pH, and bulk density. Also, the parameters used to develop the model are the TVS, OLR, HRT, C/N ratio, lignin content, and VFAs. For proper analysis of the study, the Pearson correlation matrix and the principal component analysis were used to examine the relationships between variables. These parameters significantly correlate with specific methane production, especially the lignin content. Hence, a simple linear model based on the lignin content could not describe the data. This was possible through the MLR. However, the MLR, including all the factors and parameters used for the model as predictors, showed a good fitting of the experimental data and optimal fitting at R2 = 0.87, SEP = 18.53 NLCH4/kgTVS but prevented some perturbation at R2 = 0.78, specific methane production (SEP) = 40.99 NLCH4/kg TVS during the validation stage. Thus, this indicates a good compromise regarding fitting ability, predictive ability, and several predictors to be included in the model.

Determining the accurate prediction of biogas yield in different working conditions to increase the efficiency of anaerobic digestion's performance seems challenging. A positive response will ensure proper, effective decisions regarding optimizing the biogas yield. To address this, Duong and Lim (Hendriks & Zeeman, 2009) developed a regression model to estimate the production of biogas from the co-digestion of swine manure (SM) and waste kitchen oil (WKO). The aim is to provide and guide as a decision support tool to predict biogas production. The study used a glass jar of 1.9 L and a working volume of 1.4 L as a biogas digester in a mesophilic temperature throughout the 21 days of monitoring. A dataset of 247 observations, including one response variable (biogas production) and three key feature variables or predictors (number of pigs, manure production, and VS of manure and oil), was reported during the monitoring period. The study employed the R software v 4.2.0 to analyze the data (mean, standard deviation, linear, and polynomial models). It is established from the study that temperature has a significant impact on biogas production during co-digestion. Hence, a mesophilic temperature of 40 °C resulted in a high biogas yield (6000 mL/d) compared to lower-temperature regions (3000 and 5000 mL/d for 30 and 35 °C, respectively). The R2, one tool or factor to guide the improvement of biogas yield, was much higher in the polynomial regression (0.9656) than in the simple linear regression model (0.7167). From the findings of the study, it was reported that the estimation of biogas yield increased from 0.2 to 6.7% between the predicted and actual values. In conclusion, the study exposes that the improvement of biogas yield is possible through the model accuracy. In this case, calibrating the model by comparing the predicted data with the biogas yield from an anaerobic digester or other model.

5.1 Comparison study of improving biogas yield via optimization technique

The section examined the comparison studies of improving biogas yield via mathematical modelling and pre-treatment. Previously, improvement of biogas yield by pre-treatment and mathematical modelling has been discussed; however, it is necessary to briefly consider the performance comparison of these techniques from recent studies.

A combination of pre-treatment digestion techniques and a mathematical model for improving the biogas digester is possible. This was proven in the study conducted by Almomani (2020). In the study, the author compared the improvement of biogas yield using co-digestion (agricultural solid wastes and cow dung) by applying chemical pre-treatment (addition of NaHCO3) and ANN. The ANN was developed to model and optimise the cumulative biogas yield. It was obtained that the ANN predicted the cumulative biogas yield with 99.1% of data within ± 10% deviation of the mean experimental value. At the same time, the chemical pre-treatment improved the biodegradability of the substrate, thereby increasing the cumulative biogas yield by at least 43%. Both approaches tend to forecast the cumulative biogas yield as a function of temperature, substrate composition, and chemical dose applicable to scale up and cost analysis.

Gopal et al. (2021) compared the improvement of biogas production from flower waste through optimization and pre-treatment techniques. The response surface methodology (RSM) and artificial neural network were employed for the optimization method. At the same time, the pre-treatment was studied using physical (milling, microwave, and ultrasonic), chemical (acidic, alkaline, and solvent), hydrothermal (autoclave), and biological methods (Aspergillus fumigatus SL1). The authors used the substrate concentration, pH, temperature, and agitation time as the input parameters for the RSM and ANN. The study's outcome shows that the RSM and ANN were reported to be 0.9951 and 0.999, respectively, near the experimental value. However, the ANN prediction was found to perform better than the RSM. The chemical pre-treatment was reported to improve the biogas yield with higher biomethane kinetic and cumulative yield in the pre-treatment process. The study proved that mathematical modelling and pre-treatment optimization techniques significantly improved biogas yield.

One of the objectives of optimizing biogas yield from the anaerobic digestion of organic wastes focuses on maximizing energy recovery. Olatunji et al. (2021) used the pre-treatment of lignocellulose material and response surface methodology (RSM) to achieve this. The RSM was used to optimize biogas yield using temperature and hydraulic retention time (HRT) as the statistical predictive models, whereas the mechanical as a pre-treatment method of groundnut shells was used. From the study, the temperature, HRT, and particle size were said to have significant effects of p < 0.05, favours the organic dry matter of biogas (20.80 and 19.09 kg FM), optimum experimental and predicted yields (24.00 and 22.68 1NCH4ODM). The R2 obtained for the four yield components were 0.6268, 0.5875, 0.6109, and 0.5547, which are lower and show a sign of the average fit of the model. The study proved that statistical optimization and pre-treatment improve the biogas yield and groundnut shell.

A mathematical model comparison with pre-treatment (microwave irradiation) of mixed sludge was carried out by Elagroudy et al. (Elagroudy et al., 2020). The aim was to compare both processes as an optimization technique for improving the biogas yield. Pre-treatment enhances anaerobic digestion in terms of organic solubilization, sludge dewaterability, and biogas production. On the other hand, the mathematical model focuses on the model fit of the experimental data to predict a high regression coefficient for maximum biogas yield. Microwave irradiation was applied to the waste-activated sludge under seven different microwave intensities, measuring the temperature. For the mathematical model, Modified Gompertz (MG), Logistic function (LF), Reaction Curve (RC), and Exponential rise (ER) were optimized under seven different cases. The pre-treatment result revealed a significant difference in biogas production between the untreated and treated samples after 43 retention time. Hence, maximum accumulated biogas production was reported. Also, microwave irradiation increases the substrate's overall biodegradability, improving the biogas yield. On the part of the mathematical model, the LF and MG are said to be the best fit with experimental data than the RC and ER models. Hence, among the tested models used in the study, the MG proved to be the best model based on the percentage error of 4.5%, while the RC result with the highest error was 16.77%. The study showed that both techniques could improve biogas yield.

To summarise this section, Table 7 compares the pre-treatment and mathematical model as the primary optimisation technique to improve biogas yield.

Table 7 Comparison of the pre-treatment and mathematical model of the optimization technique
Full size table

5.2 Significance of findings from the study

From the research findings reviewed, it is necessary to highlight the importance of the study and its benefit to readers and audience.

  • The study's findings will assist in determining the best possible optimization technique for improving biogas yield.

  • It is established that optimizing biogas yield has significance on the process parameter or variables.

  • The study findings show that the longer the pre-treatment time, the more accessible the bacteria are on the cellulose, thereby increasing biogas yield.

  • Through ensuring nutrient balance, the findings from the study suggest that optimization techniques are a promising tool for improving biogas yield.

  • It is indicated from the findings that the study is imperative for the decision-making of the acceptance of the anaerobic digestion process.

  • Towards enhancing the biogas yield, the study provides systematic processes for overcoming the shortfall and challenges of anaerobic digestion.

5.3 Potential application of findings from the study

It is essential to summarise the application of the study's findings briefly. These include.

  • The total stability of the input parameter process used in the mathematical model gives an idea of the mode of operation being a risk to the sustainability of the process.

  • It helps to break down the structural material used for biomass for biogas production by reducing the lagging time of the anaerobic digestion and increasing biogas yield. Hence, it is economically feasible.

  • For the audience in sciences, as per the mathematical model, using a differential system equation, where necessary, is applicable in satisfactorily predicting the behaviour of a biogas digester.

  • The technique enhances biogas yield and organic waste management/treatment, especially in the agricultural field of research.

  • It is essential in facilitating the decision-making process for biogas production for energy engineers and economists.

  • Evaluating the feed composition and improving the process output, specifically for the microbiologist and chemical engineers.

  • To scale up anaerobic digestion in terms of techno-economic analysis (cost analysis) and finally to predict the system failure. This is needful for economists and engineers as well.

6 Conclusion

The review successfully looked at the pre-treatment and mathematical model approach to improving biogas yield. The authors have referred to these two approaches or methods as optimization techniques. As much as the pre-treatment focuses on the substrate or feedstock, the mathematical model deals with the model fit of the experimental data to obtain a good, quiet fit. In the study, the pre-treatment technologies consider the physical/mechanical, chemical, and biological, whereas the mathematical model includes the popular ANN, MLR, RSM, etc. Given the findings from the review, it is revealed that both methods have proved that they can be better used to improve biogas yield; the chemical and physical pre-treatment and the ANN perform better towards the biogas yield. Various factors are attributed to their performance, and these were presented in the study. The authors recommend more application of the chemical and physical pre-treatment and the ANN. This provides the challenges of each of these techniques for future studies. Having looked at the potential applications of the review, further research is necessary on the full-scale applicability of the study, as earlier mentioned, which deals with predicting biogas yield from the characteristics and biodegradation of the substrate (feedstock) and managing/controlling the digestion process. Also, further consideration, evaluation, and understanding of the reaction engineering parameter and its behaviour in the biogas digester are essential. All of these will enhance and improve the yield of biogas. However, the deep learning methodology was not used to obtain the results from the study. Therefore, the authors recommend using deep learning methodology in these further studies. Thereby providing improved performance, predictive modeling, and handling nonlinear relationships and large/complex data. This will also assist in adequately developing the kinetic model equation for designing biogas digesters and possible future research development.