Introduction

Kinetic analysis can be carried out in terms of dynamic models that consider time as a variable. Numerical modelling investigates both dynamic modelling and static modelling of the system without any practical experiments done [4, 8]. There are several kinetic expressions used for describing anaerobic digestion focusing on different variables [2, 3].

Anaerobic digestion offers significant advantages over aerobic process, like low energy consumption, reduced solids formation, low nutrient requirement and potential energy recovery from the methane produced [15]. Additionally, the use of the anaerobic digestion reduces the volume of waste, and the digestate could be used as soil amendment. This process is now widely used in many environmental applications, in different configurations and modes of operation [13].

In a study investigating anaerobic digestion of biomass, a biokinetic model was developed based on material balance of the substrate, biomass and biogas [8]. A balance sheet of material on biomass, the decomposition of the substrate and methane production accompanied by other gases (CO2, H2, NH3) were carried out based on the mass conservation law. Author undertook experiments to validate the model and compared it to two existing models and experimental data. Author concluded that the biokinetic model that was established was able to monitor the operation of the digestor over time including the monitoring of degradation of organic matter, evolution of biomass, biogas composition and methane production. Through these findings, the model presented a concordance result with other models.

In another study, a modified Gompertz model (MGM) and the first-order kinetic model (FOKM) were used to predict the yield of biogas production using okra as substrate [17]. The MGM assumed that the rate of biogas production corresponds to the bacterial growth rate in the digester like the FOKM. The above models focus on variables such as the potential methane production, the maximum methane production rate, the measured digestion time and the lag phase (number of days) which all make up the accumulated methane at time t. The substrate was treated with various treatments and co-digested with sheep slurries as the inoculum, and from the predictions which were made and compared to the obtained results, the author found out that the (FOKM) was the best model since it gave the least difference in the predicted results as compared to the obtained results.

In a different study, several variables have shown that biogas production is affected by pH levels, temperature regions and treatment of substrate [9]. Several mathematical models predict that the activity of microorganism’s growth is directly proportional to biogas production rate [9].

Kinetic models are mathematical representations of the fermentative processes, which enable the understanding of reactions taking place and allow for large-scale simulations [11]. The models widely used for predicting biogas yield are the modified Gompertz model (MGM) and the first-order kinetics model (FOKM). The MGM model is used to predict the biogas potential corresponding to the bacterial growth rate in the digester.

A study was carried out on the kinetic model of biogas yield from vinasse at various initial pH: comparison between the modified Gompertz model and first-order kinetic model [6]. The study showed that the pH levels affected the input parameters of both models, and the prediction of the biogas from both models showed that the modified Gompertz model had the least percentage variation from the measured biogas. In a different study, the modified Gompertz model and first-order kinetic model were used to compare and adjust experimental results [5]. The study was focused on the treatment of sludge from recycled mill paper through anaerobic digestion as an available treatment to produce methane and to minimize environmental problems. The researchers ran an experiment to produce methane, and the predicted results obtained from the models showed that the modified Gompertz model showed the least deviation of (1–13%) as compared to using the first-order kinetic model that had a deviation of (5–17%) [5].

In many studies for prediction of methane yield in an anaerobic digestion system, most authors utilize any of the kinetic models to undertake numerical studies. There has not been a study comparing the five kinetic models for prediction of methane yield for a particular food material. In some recent works, the focus of researchers has been in developing kinetic models to address process parameters and predict methane yield [12, 18]. The present work addresses this gap as it compares the five kinetic models for methane production using two waste materials—banana and orange peels. Therefore, the present work is aimed at comparing predicted biogas yield of five selected models using orange and banana peels as substrate and comparing the output to experimental digestion of the selected substrates. The results obtained from this study are very useful for selection of appropriate kinetic model for methane yield prediction in anaerobic digestion. It presents a documented performance of the five kinetic models in predicting biogas yield and presents a reference for theoretical researchers.

Kinetic Studies

A review of existing models for predicting methane production was carried out, and five models were selected (Table 1). The models’ input parameters are the maximum methane potential yield (A), the lag phase (\(\lambda )\), biogas rate constant (k), the shape factor coefficient (d), the incubation time (t) and the biogas production rate (\(\mu\)m) together with the limitations and formulas.

Table 1 Existing models and their limitations
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Input data for the various models were obtained from experimental data obtained from research (Table 2). The values of the input parameters vary depending on the substrate being studied.

Table 2 Biogas properties specific for each model
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The selected models were programmed on MATLAB with the required input parameters to obtain the graphs necessary for comparative analysis of the models. This was carried out by following the steps shown in Fig. 1:

Fig. 1
figure 1

Kinetic modelling approach adopted from [10]

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The models reviewed were all existing with clear input parameters; hence, steps one and two were skipped. The models were tested using arbitrary values as input parameters to see the behaviour of each model. The steps were then repeated with more arbitrary values to observe the model’s behaviour. The required data were obtained from research and the data were used to run the models on MATLAB software. Further simulations were done for all the substrates using the selected models to produce varying results. The results obtained from the models were then analysed for the purposes of comparative analysis. The obtained solutions were compared with the experimental results to check for consistency and variation. Once confirmed, the collected data from the substrates were then used to simulate the models, and all the results obtained were then compared to those of the experimental results. The procedure of testing these models is depicted in Fig. 2:

Fig. 2
figure 2

Procedure of comparative study between kinetic analysis and experimental

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Results and Discussion

Results of analysis of the models for banana peels and orange peels are presented in Figs. 3 and 4, respectively.

Fig. 3
figure 3

Cumulative methane yield of banana peels using all models

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Fig. 4
figure 4

Cumulative methane yield of orange peels using all models

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Table 3 presents a summary of methane yield from banana peels for the various models and the hydraulic retention time (HRT) for the different models. These data are derived from Fig. 3. The amount of methane produced from orange peels in one day and number of days to reach the maximum methane yield obtained from figure are presented in Table 4.

Table 3 Cumulative methane yield and HRT for banana peels
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Table 4 Cumulative methane yield and HRT for orange peels
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Figure 3 presents a comparative cumulative methane yield for the five models using banana peels as substrate and the experimental data. The result of Experiment 1 was obtained from the research from which the input parameters were obtained; hence, all the analytical results reach approximately the same results. Experimental results 2 to 4 were obtained from [1] in which Experiment 2 represents the cumulative methane yield of banana peels treated both mechanically and chemically having the highest cumulative methane yield of 366.2 ml. Experiments 3 and 4 show the cumulative methane yield of the banana peels treated chemically with a cumulative methane yield of 92.3 ml and mechanically with a cumulative methane yield of 77.8 ml, respectively.

Figures 3 is analysed to produce Table 3 which helps with comparing the results obtained from the analytical results to the experimental ones. All five models reach a maximum methane yield of 352.9 (ml) that is 99.23% accurate when compared to the experimental 1 result of 350.2 (ml), which was the main experiment used to obtain input parameters for the models. Although all the models reached the same methane yield, the time taken to reach the maximum methane yield differs. The Richards and transfert models reach the maximum yield in 15 days, which is a 20% deviation as compared to the actual result. The deviations obtained from logistic, Gompertz and first-order kinetic models are 60%, 116% and 220%, respectively. The first-order kinetic model has the highest deviation as it does not consider the lag phase and the methane production rate.

During the first day of methane production on the experiment, the yield achieved was 20.3 (ml). The Gompertz model had the least deviation from the actual yield with a deviation of 113.2% followed by the logistic, Richards, first-order and transfert models with deviations of 187.4%, 218.1%, 250.2% and 803.4%, respectively. Experimental results 2, 3 and 4 were obtained from [1], and the purpose for the comparison was to show that the effects of treatment to the substrate affect the cumulative methane yield produced. The percentage deviations between the substrates when comparing results for the models directly are very high, and this may have been caused by errors, which occurred when the experiments were conducted. The concentration of the substrates, ambient temperature and pressures inside the digesters during experiments may have affected the results, which is why most of the models’ deviations were high. There is no certainty that the input parameters were taken the day that the experiment was conducted or whether the collected substrates were not contaminated during handling, storing and running the experiments.

Figure 4 presents a comparative methane yield for the five models using orange peels as substrate and is analysed to produce Table 4 which helps with comparing the results obtained from the analytical results of Experiment 1. All five models reach a maximum methane yield of 464.6 (ml), which is 96.21% accurate when compared to the experimental result of 447 (ml). Experiments 2 and 3 show the experimental results and different pre-treatments of the substrate and mainly used to show how the treatment of the substrate affects the cumulative methane yield. Although all the models reached the same methane yield, the time taken to reach the maximum methane yield differs. The Richards and logistic models reach the maximum yield in 17 days, which is a 4.6% deviation as compared to the actual result. The deviations obtained from transfert, Gompertz and first-order kinetic models are 10.8%, 35.4% and 35.4%, respectively. The first-order kinetic model and Gompertz had the highest deviation as it does not consider the lag phase and the methane production rate.

During the first day of methane production on the experiment, the yield achieved was 50.6 (ml). The Gompertz and first-order kinetic models had the least deviation from the actual yield with a deviation of 26.1% followed by the logistic and Richards models both having a deviation of 76.4% and lastly the transfert model with a deviation of 355.1%.

Conclusion

A thorough study of Gompertz, logistic, first-order, Richards and transfert models was undertaken, and objective functions were identified. Similarly, all input parameters such as the maximum methane potential yield (A), the lag phase (\(\lambda )\), biogas rate constant (k), the shape factor coefficient (d), the incubation time (t) and the biogas production rate (\(\mu\)m) were identified for the analysis of the models.

All models showed an accuracy of predicting methane potential of over 95% on both the substrates that was focused on. When looking at the results obtained from Tables 1, 2, 3, 4, the model with the least deviations in terms of the total methane yield, the methane yield produced in one day and the time taken to reach maximum methane yield as compared to the experimental data was the Gompertz model with a cumulative deviation of 76.6%. The model, which was the least accurate, was the transfert model which had the cumulative deviation of 274.7%.

The study indicated that the models that were most accurate in terms of the total methane yield, the methane yield produced in one day and the time taken to reach maximum methane yield as compared to the experimental data for the orange substrate were the Gompertz and first-order kinetic models with a cumulative deviation of 30.67%. The model with the highest deviation was the transfert model with a cumulative deviation of 13–2%.