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 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 61, 2017 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Petar S Varbanov, Rongxin Su, Hon Loong Lam, Xia Liu, Jiří J Klemeš 
Copyright © 2017, AIDIC Servizi S.r.l. 

ISBN 978-88-95608-51-8; ISSN 2283-9216 

The Kinetics for Mathematical Modelling on the Anaerobic 

Digestion of Organic Waste- A Review  

Cassendra P.C. Bonga, Li Yee Lima, Chew Tin Leea,*, Wai Shin Hoa, Jiři J. Klemešb 

aFaculty of Chemical and Energy Engineering, Universiti Teknologi Malaysia (UTM), 81310 Johor Bharu, Malaysia. 
bSustainable Process Integration Laboratory- SPIL, NETME Centre, Faculty of Mechanical Engineering, Brno University of  

 Technology- VUT Brno, Technická 2896/2, 61600 Brno, Czech Republic. 

 ctlee@utm.my 

Anaerobic digestion is a biological process that could provide high valorisation value of waste by producing 

biogas as a form of renewable energy and the digesate as biofertiliser. A mathematical model is an essential 

tool to observe, predict, simulate and optimise the system’s behaviour at different conditions. The modelling 

approach for wet digestion and dry digestion is different due to the presence of solid content. The selection of 

the limiting step for modelling can be influenced by both substrate and microbial loadings. The complex substrate 

is heterogenous by nature and consists fractions of organic matter with different biodegradability. The 

formulation of the model is affected by the type of substrates and its characteristic, which exhibits different 

substrate degradation kinetics, intermediate production and consumption behaviour and the biogas production. 

This review aims to look at the rationalisation and findings that lead to the different formulation of the models, 

from the point of view of the variation of the organic waste, including municipal solid waste, food waste and 

animal manure.  Recent research results for the anaerobic digestion of organic waste is also discussed, which 

serves to provide better insight for future generalisation of a model.  

1. Introduction 

Anaerobic digestion (AD) is a biological treatment that could realise the waste-to-energy concept. The organic 

waste undergoes several parallel and sequential processes to produce the end-product, biogas, which can be 

transformed to renewable energy. The main steps in AD include hydrolysis, acidogenesis, acetogenesis and 

methanogenesis. The use of AD for treating wastewater has been well-established as compared to the AD of 

complex substrates (Momoh et al., 2013), such as municipal solid waste (MSW), which is heterogenous in 

nature and varied along season and geographical location.  

Mathematical model is a set of mathematical equations that can represent the main aspects of a biological 

system, formulate and predict the system’s behaviour under different conditions (Donoso-Bravo et al., 2011), 

quantify the empirical behaviour and simulate the system (Eposito et al., 2008), predict the most influential 

process variable in the system (Fdez-Güelfo and Álvarez-Gallego, 2011) and to identify elements for 

optimisation (Lauwers et al., 2013). The current mathematical models can be divided into several generations 

as stated by Donoso-Bravo et al. (2011). 

The first-generation models were modelled based on either hydrolysis or methanogenesis as the limiting step. 

The second-generation stressed on the intermediate products and their influences over the AD process. The 

third-generation of models incorporated inhibition species on the microbial activities and the consideration of 

different substrates used in AD. Some of the popular models found in literature incudes the anaerobic digestion 

model 1 (ADM1), first-order kinetic model, Gompertz model and their modifications. For example, the use of the 

first-order model for degradation kinetics and the use of modified Gompertz equation for simulating biogas 

accumulation (Nielfa et al., 2015). There are several new approaches on the modelling for complex substrates, 

where focus have been paid on the role and observation of microbial adaptation, the effect of different substrate 

to microbe (S/M) ratio, total solid (TS) content, two-phase or three-phase substrate interaction and more.  

The mathematical modelling for AD of a complex substrate is challenging due to the complicated nature of the 

technology itself and the substrate. In complex substrates, the particulate substrate is first disintegrated into 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1761276

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Bong C.P.C., Lim L.Y., Lee C.T., Ho W.S., Klemeš J.J., 2017, The kinetics for mathematical modelling on the 
anaerobic digestion of organic waste- a review, Chemical Engineering Transactions, 61, 1669-1674  DOI:10.3303/CET1761276  

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macromolecules, such as carbohydrates, proteins and lipids. These will then hydrolyse to simple sugars, amino 

acids, and long-chain fatty acids (LCFA) which are then converted to fatty acid, acetate, H2 and CO2 via 

acidogenesis. The acetate from acetogenesis will then be converted to CH4 by methanogenesis (Bollon et al., 

2011). The intermediary reaction mechanisms for biogas production requires more understanding but 

experimental studies are complicated as the biogas production is affected by many process parameters 

(Rajendran et al., 2014). These parameters include the type of substrate, type of microbial inoculant (MI), the 

substrate to MI (S/M), total solid (TS) content, organic loading rate (OLR), pH and more.   

Lauwers et al. (2013) performed a review on the mathematical modelling of AD of biomass and waste. The 

paper focuses on discussing the power and limitation of a mathematical model, including ADM1, and concluded 

microbial community data and black-box model can be used to further improve the model performance. 

Kythreotou et al. (2014) performed a review of simple to scientific models for AD, stating exclusion of critical 

parameters from the model limits the model’s accuracy. The paper concluded that identifying rate-limiting step 

and the intermediate fermentation products is crucial but challenging for the AD of complex substrates. Both 

reviews highlighted the important role of the inclusion of microbial interactions for further improvement on 

model’s performance. In this paper, the mathematical models focusing on complex organic waste, including 

municipal solid waste (MSW), food waste (FW) and manures, is being reviewed and discussed. Recent findings 

related to the substrate, intermediates and products were presented which can be used to improve the model. 

One of the highlight of the paper is that the point of discussion is based on the categorisation of waste (MSW, 

FW and manure) rather than based on the characterisation of mathematical model (for example, mechanistic, 

empirical, first-order, ADM1 and more), which allows a better insight on the suitability of different models in 

respective to the waste type. 

2. Methodology 

This review is based on a literature search using ScienceDirect with the focus on several keywords: 

mathematical model for anaerobic digestion on organic waste, mathematical model for anaerobic digestion on 

solid waste, mathematical model for anaerobic digestion on food waste, kinetic on anaerobic digestion of organic 

waste, kinetic on anaerobic digestion on solid waste and kinetic on anaerobic digestion on food waste. 

3.  Mathematical modelling for AD of complex substrate 

Mathematical modelling for AD of complex substrate is different from wastewater due to several reasons as 

identified by Momoh et al. (2013), which included:  

(1) The unsuitability of modelling approach based on maximum specific growth rate coupled with a short 

hydraulic retention time (HRT) for complex substrate. 

(2) Challenges in differentiating between bacteria volatile suspended solid (VSS) and complex biomass VS.  

(3) A non-soluble form of complex substrate  

(4) Presence of recalcitrant fraction 

(5) Heterogenous nature of the complex substrate. 

3.1 The limiting steps 

Anaerobic digestion (AD) can be divided into two categories, based on the total solid (TS) content of the medium: 

the wet digestion with less than 15 % TS and the dry digestion with more than 15 % TS (Mata-Alvarez et al., 

2000). The kinetic model between wet digestion and dry digestion is different due to the presence of particulate 

matter and high solid content, which affect the rheological behaviour, chemical equilibria and mass transfer 

(Bollon et al., 2011). 

Hydrolysis is considered as the main limiting step in many studies. Hydrolysis has been traditionally modelled 

using the first-order kinetics. Fractionisation of an organic pool is gaining interest in the modelling for a complex 

substrate, which includes fractions with different biodegradability, including hardly degradable material (Vavilin 

et al., 2008). The disintegration step and hydrolysis step are often modelled by first-order kinetic (Bollon et al., 

2011). However, due to the difference in experimental conditions, hydrolytic biomass to substrate ratio and the 

lumped effect of disintegration and hydrolysis, different kinetic coefficients for the first-order hydrolysis rate were 

reported (Vavilin et al., 2008). Substrate characterisation should be divided into carbohydrates, proteins, lipids 

and the inert fraction with the disintegration or hydrolysis as the limiting step (Galí et al., 2009). In terms of the 

degradation kinetics of the macromolecules, carbohydrates are degraded first as the preferable substrate for 

the microbes. Protein and lipid had slower degradation kinetics and occurred after microbial acclimatisation has 

taken place (Nielfa et al., 2015).  

Myint et al. (2007) studied on the modelling during the dry digestion of cow manure residues in a leach-bed 

reactor. The model displayed two distinct segments which were due to the presence of two components in the 

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substrate: a readily degradable fraction, the hemicellulose and the slowly degradable fraction, the cellulose. The 

rate of hydrolysis is dependent on the surface area of the particles occupied by the organisms, which included 

both the native microbes from the manure and the introduced microbes as an inoculant. Hydrolysis efficiency in 

AD should consider the complex interaction between the different biomass species and intermediate products, 

with their relative spatial distribution (Vavilin et al., 2008). 

Acidogenesis is modelled based on the assumption that the acidogenic biomass grows on the soluble products 

of hydrolysis. The growth of the biomass is modelled as a single biomass, which is the acidogens, that feeds on 

the hemicellulose and cellulose at a different kinetic constant (Myint et al., 2007). The acidogenesis is governed 

by the respective microorganisms that convert the hydrolysates to several acids. The acidogenenic biomass 

can be integrated into one which the growth is represented by the Monod equations (Bollon et al., 2011). 

Methanogenesis may be more accessible from the perspective of monitoring as it has a slower happening rate 

than acidogenesis and acetogenesis. The sum of all gas compounds produced during the process and pH are 

the most widely used parameters for modelling application. Accumulation of CH4 can be modelled by Monod 

model when acetoclastic methanogenesis is the rate-limiting step, provided that there is no inhibition at high 

OLR or high organic solids to inoculum ratio (Vavilin et al., 2008). Figure 1 shows the possible consideration to 

be included in the mathematical model and kinetic study when there is variation in the digestion type (wet or 

dry), substrate type (mixed, segregated, heterogenous, homogenous), the production and possible inhibition of 

intermediates products and lastly, the production and accumulation of biogas or CH4. 

 

 

Figure 1: Considerations affecting the formulation of mathematical model following changes in digestion system, 

substrate, intermediates and gas production. 

3.2 Type of organic waste, degradation kinetics and mathematical model 

3.2.1 Municipal solid waste 

The organic fraction of municipal solid waste (OFMSW) is heterogenous and composed of different waste with 

different biodegradability (Nielfa et al., 2015). The hydrolysis is the limiting factor in a high-solid system. In the 

case of OFMSW, the hydrolysis rates differed among different particulate components and microbes prefer one 

substrate over the other.  

Sun et al. (2013) modelled on the co-digestion of municipal biowaste (MBW) and waste activated sludge (WAS). 

The model included disintegration process, kinetic rate equations for soluble components, hydrolysis process, 

acid production process, H2 production process and CH4 production process. At high OLR, methanogenesis is 

the rate-limiting step due to the decrease in pH and accumulation of VFA, which inhibit methanogenic activity. 

At low OLR, hydrolysis is the main limiting step which is dependent on the substrate concentration and followed 

a first-order reaction. 

3.2.2 Food waste 

Food waste (FW) is one of the best substrate for anaerobic digestion (AD) due to its CH4 production potential 

(Zhang et al., 2011). FW has several characteristics: high volatile solid (VS), high moisture, low pH and high in 

carbohydrate, protein and lipid. The mono-digestion of FW often encounters restriction at high loading due to 

rapid hydrolysis and subsequent acid accumulation. Liu et al. (2009) identified the food to microorganism (F/M) 

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ratio as the most critical feature in the degradation of VS of organic solid particles. The choice of type of MI and 

the S/I ratio influences the substrate degradation kinetics and VFA production (Bong et al., 2017). Durruty et al 

(2013) proposed a model to evaluate the effect of organic fractions on biogas generation from potato residues. 

The organic fractions are divided into inert and biodegradable fractions, where each fraction can be further 

divided into soluble and particulate fractions.  

Dhamodharan et al. (2015) did a kinetic study on the AD of FW using seven animals dungs. The study illustrated 

the effect on the choice of MI and the food to microbes (F/M) ratio on CH4 production. Among the seven animal 

dungs, pig dung and cow dung exhibited highest CH4 production and VS destruction. The use of high F/M ratio 

in high solid AD allows a successful digestion without the need of pH adjustment due to the presence of a large 

amount of inoculant. But in the case of FW, low F/M ratio is preferable as FW is easily hydrolysed. The presence 

of a large amount of inoculant will lead to rapid degradation and subsequent acid accumulation, which inhibit 

the methanogenesis. This is similar to Motte et al. (2013) who stated the difference in modelling approach based 

on solid to microbial biomass (S/X) ratio. For lab work, high S/X ratio is preferable to observe the changes in a 

microbial community. For industrial, low S/X ratio is preferable to prevent process failure as high S/X ratio 

requires relatively longer adaptation and monitoring period.  

Wang et al. (2014) studied on the kinetics of FW under acidic conditions with two types of MI: aerobic activated 

sludge and anaerobic activated sludge. Hydrolysis of FW increased when anaerobic activated sludge was used 

at any pH. Among the macromolecules, the microorganisms preferred carbohydrates and it was first degraded 

to glucose than to pyruvate (Chen et al., 2013). Following next is the degradation of the protein, which can be 

categorised into two stages: the dissolution of protein at an early stage and then transformation at a later stage. 

The VFA profile is observed to be different. Acetate and butyrate dominated during hydrolysis of carbohydrates 

where acetate, propionate and butyrate dominated from protein and lipid hydrolysis.  

In the study by García-Gen et al (2015), they improved the ADM1 model by incorporating an extra disintegration-

hydrolysis step during the AD of individual fruits and vegetable wastes (FVW). The model considered the 

disintegration kinetics for readily and slowly biodegradable fractions. The disintegration-hydrolysis involved the 

breakdown of the OM with a conglomerate of macromolecules into the soluble macromolecules that can then 

be hydrolysed by the microbial enzymatic activities. This step has known to be the limiting step for solid AD. 

Poggio et al (2016) proposed a novel model based on the ADM1 model for describing the AD of FW and GW. 

The model assumed that the OM is made of a mixture of particulate and soluble fractions and subdivided them 

into four substrate fractionation models. The study concluded that model with one particulate and one soluble 

fraction is the best suitable for FW whereas the model with two particulate fractions is best fitted for FW. 

Particulate fractions have hydrolysis as limiting steps thus is described by the first order kinetics. Particulate 

fraction is further divided into readily and slowly degradable fractions. Soluble fractions are directly assimilated 

by microbes thus its kinetic is influenced by biomass concentration and their respective uptake rates.  

During the AcoD of FW and pig manure (PM) by Dennehy et al (2016), the authors proposed a dual pool first 

order model and compared with a conventional first order model and a Gompertz model. The new model 

incorporates a new parameter which is the ratio of rapidly degradable substrate to a total degradable substrate 

to capture the kinetics resulting from the labile fraction of FW. It showed that the newly developed model 

provided the best fit for the study of codigestion of FW and pig manure (PM) under different mixing ratio, which 

is 1/0, 4/1, 3/2, 2/3, 1/4 and 0/1. The dual pool model takes into the consideration of two hydrolytic constants, 

which is for the fast degradable and a slowly degradable substrate. The test of fitness showed that the first-

order model is suitable to determine the rate of hydrolysis provided there is no inhibition of VFAs. The first order 

model failed to capture the CH4 yield as most the CH4 was generated in the first five days. The Gompertz model 

performed slightly better but the accuracy greatly decreases when the mixing ratio of PM: FW is 0:1 and 1:4, 

when the PM proportion is high.  

3.2.3 Manure and digestate  

The scale of the digester can influence on the effect of parameters. Hulle et al. (2014) studied the kinetic on AD 

of cow manure between lab scale (3.78 L) and pilot scale (120 L). The study concluded that mixing and VFA 

accumulation have a great effect on the formulation of the model. The experiment work showed a significant 

increase of VFA concentration during the non-mixing period. Modelling of pilot scale system should include the 

effect of mixing to prevent overestimation, account for VFA accumulation and reduce the need for model 

calibration.  

Some models focus on the loss of material, for example, NH3, during the storage of the digestate. Whelan et al. 

(2010) studied on the non- steady state partitioning model for describing NH3 volatilisation during storage of the 

digestate. The model was able to describe the behaviour of NH3 well in the digestate, where NH3 loses is linear 

with time and open storage could result in significant losses of NH3 as compared with covered facilities, 

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especially those with a high depth to surface area ratio. There is a need to consider bacterial adaptation for 

modelling of the AD of complex biomass (Momoh et al., 2013).  

Figure 2 illustrates a summary which maps the possible selection of limiting step and the parameters following 

different choices of organic waste and input. 

 

 

Figure 2. The relationship among the type of organic waste, limiting step and parameters for modelling the 

kinetics of complex substrate. 

4. Conclusion 

Mathematical modelling and kinetic study for the anaerobic digestion (AD) of complex organic waste would 

require further insights. The main kinetic model includes ADM1, first order and Gompertz model whereas Monod 

model remains the main stream for modelling AD biomass. This paper reviewed several modelling works on the 

AD of complex organic waste and the effect on microbial loadings, which can influence the suitability of the 

model in describing and predicting the system behaviour.  For the modelling of complex organic waste, including 

municipal solid waste (MSW), food waste (FW) and manure, the following should be integrated to the model: 

(1) The parameter “fractionisation” should be included due to the presence of organic pool with different 

biodegradation kinetics, including a separate pool for inert materials. It is useful to apply on MSW where there 

is a mixed of waste, including food waste with labile OM and green waste with high cellulosic content. 

(2) Hydrolysis as the rate limiting step is influenced by the surface-interaction kinetics and the parameter 

“disintegration” should be added prior to the hydrolysis step. 

(3) Methanogenesis could serve as the rate limiting step at low loading where there is no acid inhibition on CH4 

production due to the rapid degradation of labile organic matter, such as when FW is used as the sole feedstock. 

(4) The role of microbial inoculant, its loading amount and its adaptation capability would require better study to 

allow a better integration with the currently available generalised kinetic models, such as first-order and 

Gompertz.  

(5) The difference between the modelling assumption between wet and dry digestion, bench scale and pilot 

scale should be considered, due to the presence of high solid content, non-ideal mixing and possible process 

failure due to VFA accumulation at non-ideal mixed AD plant at high loadings. 

Acknowledgments  

The authors would like to acknowledge the research grants from the Ministry of Higher Education (MOHE) 

Malaysia with grant no. 7301.4B145 and from Universiti Teknologi Malaysia (UTM) with the grant no. 

2546.14H65, 2546.12H89 and 2501.10H28. The authors would also like to thank UTM for providing the Ainuddin 

Wahid Scholarship for supporting the postgraduate study. The authors also acknowledged the funding and 

support from JICA-JST-SATREPS entitled “Development of Low Carbon Scenarios in Asian Region” and by the 

EU project Sustainable Process Integration Laboratory – SPIL, funded as project No. 

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CZ.02.1.01/0.0/0.0/15_003/0000456, by Czech Republic Operational Programme Research, Development and 

Education, Priority 1: Strengthening capacity for quality research under collaboration agreement with UTM. 

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