ORIGINAL PAPER

Exponential model describing methane production kineticsin batch anaerobic digestion: a tool for evaluationof biochemical methane potential assays

Mathieu Brule • Hans Oechsner • Thomas Jungbluth

Received: 15 October 2013 / Accepted: 4 February 2014

� Springer-Verlag Berlin Heidelberg 2014

Abstract Biochemical methane potential assays, usually

run in batch mode, are performed by numerous laboratories

to characterize the anaerobic degradability of biogas sub-

strates such as energy crops, agricultural residues, and

organic wastes. Unfortunately, the data obtained from these

assays lacks common, universal bases for comparison,

because standard protocols did not diffuse to the entire

scientific community. Results are usually provided as final

values of the methane yields of substrates. However,

methane production curves generated in these assays also

provide useful information about substrate degradation

kinetics, which is rarely exploited. A basic understanding

of the kinetics of the biogas process may be a first step

towards a convergence of the assay methodologies on an

international level. Following this assumption, a modeling

toolbox containing an exponential model adjusted with a

simple data-fitting method has been developed. This model

should allow (a) quality control of the assays according to

the goodness of fit of the model onto data series generated

from the digestion of standard substrates, (b) interpretation

of substrate degradation kinetics, and (c) estimate of the

ultimate methane yield at infinite time. The exponential

model is based on two assumptions: (a) the biogas process

is a two-step reaction yielding VFA as intermediate pro-

ducts, and methane as the final product, and (b) the

digestible substrate can be divided into a rapidly degrad-

able and a slowly degradable fraction.

Keywords Biogas � Anaerobic digestion � BMP assay �Batch � Model

Introduction

Optimizing biochemical methane potential assays

BMP assays, generally run in batch mode, are the most

valuable tool to evaluate and compare the methane pro-

duction of biogas substrates. Unfortunately, there is a lack

of widely accepted methodologies for these assays. Inter-

national standards have been reviewed by Rozzi and

Remigi [1] and Muller et al. [2]. These standards are not

widely applied because the preparation of complex mineral

mediums is not convenient. It is easier for scientists to

collect and use undiluted inoculums without addition of

minerals, which prove to be satisfactory in most cases.

There is a need to promote common rules, which would be

accepted as good practices by the scientific community [3].

The German directive VDI 4630 [4] is a first step towards a

simplification of the protocols for BMP assays. However,

this German standard may be too detailed and lacks

international awareness.

The difficulty in mastering BMP assays results from

multiple factors [1, 2, 4–9]: (a) assay start-up protocol,

i.e., weighing of substrate and inoculum, closing of

digesters; (b) origin, quality and homogeneity of the

inoculum; (c) pre-processing of the inoculum, e.g., pre-

fermentation, particle removal via sieving or decantation;

(d) substrate to inoculum ratio; (e) nutrient and micro-

nutrient content of the medium; (f) reactor mixing;

(g) temperature range and maintenance of constant

temperature; (h) measurement methods for gas volume

and methane content; (i) reference system (fresh weight,

total solids, volatile solids, COD); (j) calculation method

for the specific methane yield of substrates, e.g., method

for subtracting the inoculums’ own methane production.

Hence, researchers often face methodological issues they

M. Brule (&) � H. Oechsner � T. Jungbluth

State Institute of Agricultural Engineering and Bioenergy,

University of Hohenheim, Stuttgart, Germany

e-mail: mathieu.brule@uni-hohenheim.de

123

Bioprocess Biosyst Eng

DOI 10.1007/s00449-014-1150-4

are not aware of. For example, Schimpf and Valbuena

[10] investigated the effect of enzymatic pre-hydrolysis

of a crop substrate in water for 24 h prior to anaerobic

digestion in a BMP assay. In doing so, they noticed that

the amount of water added to the digestion medium can

have a tremendous influence on methane production rate.

Lemmer [11] suggested that the widespread use of

standard substrates of defined characteristics could build

a basis for comparison and validation of BMP assays.

Crystalline cellulose has been used in inter-laboratory

testing and might be a good candidate to become a

standard substrate on international level [12]. The refer-

ence system used while reporting the values should also

be standardized. Angelidaki and Sanders [6] suggested

that for this purpose the measurement of volatile solids

(VS) content should be preferred to COD determination.

The latter would suffer from chemical interferences

through non-biodegradable reducing agents and would be

inaccurate for solid substrates. Nevertheless, VS mea-

surement holds a major drawback in relation to the share

of organic matter that is volatile. Volatile compounds,

such as volatile fatty acids (VFA) and alcohols, are

partly lost upon drying, while contributing the organic

matter fraction [13–17]. On plant silages, this bias can

lead to an overestimation of methane yields as high as

15 % [18]. In the field of animal nutrition, methods have

been developed to overcome this bias, but they

vary according to substrate characteristics and are, thus,

difficult to implement on a wide variety of substrates

[19–24].

Modeling of biological processes

Models describing the kinetics of batch anaerobic

digestion have been reviewed by researchers dealing

with animal nutrition [25–27], biogas production [28–

36], and landfill gas production [37–40]. Models that

simulate bacterial growth and biochemical reactions,

such as ADM1 [41], or the Chen and Hashimoto model

[42], are quite complex. These models are generally

derived from Monod [41, 43, 44] or Contois [42, 45,

46] kinetics. Monod equation is similar to Michaelis–

Menten equation, which was developed in the field of

enzymology. Michaelis–Menten equation describes a

saturation effect in enzymatic reactions (represented by

a saturation constant), which is due to the temporary

formation of enzyme–substrate complexes [47]. Monod

kinetics assumes the growth rate of a bacterial culture

affected by a growth-limiting nutrient to undergo as

well a saturation effect [43]. Contois kinetics accounts

for an additional saturation effect related to higher

bacterial population densities, which further impedes

the growth rate [45].

Under certain conditions, saturation effects can be

neglected, and the Monod equation (or the Michaelis–

Menten equation) can be simplified to become first-order

kinetics, which is common in chemical reactions [34,

48], and can be used to design simpler models. In these

models, the variable followed would not necessarily be

the rate of bacterial growth, but rather the kinetics of

substrate degradation or product formation. Following

this approach, four models were described and imple-

mented to evaluate methane production kinetics in batch

BMP assays: first-order (Model A), two-step (Model B),

dual-pool first-order (Model C), and dual-pool two-step

(Model D).

Description of the models

In the field of chemistry, the first-order model assumes the

rate (R) or velocity of reactant utilization (here: substrate)

to be proportional to the amount of reactant available in the

medium:

R ¼ k � St ð1Þ

k first-order kinetics constant

St amount of undegraded substrate remaining at time

t (variable).

Integration along reaction time yields an exponential

equation, which gives the remaining (undegraded) sub-

strate at time t (St) [49]:

St ¼ S� e�kt ð2Þ

S total amount of degradable substrate

k first-order kinetics constant

t time after experiment start-up.

Applying the kinetics of product formation to batch

anaerobic digestion, the cumulated amount of methane

generated at time t, (Mt) can be expressed as follows

(Model A) [50–58]:

Mt ¼ S� ð1� e�ktÞ ð3Þ

From a biochemical point of view, anaerobic digestion

may be described as four subsequent steps: hydrolysis,

acidogenesis, acetogenesis and methanogenesis [29, 35,

59–61]. However, from a process-engineering point of

view, the biogas process can be divided into two steps: an

acidification step, comprising both hydrolysis and acido-

genesis, and a methane production step, comprising both

acetogenesis and methanogenesis, as shown in Fig. 1 [24,

60, 62]. Following this pattern, bacteria that carry out the

side reactions of acetate oxidation [63] and of homo-

acetogenesis [64] can be comprised into the methane pro-

duction step. The simplistic assumption of a two-step

Bioprocess Biosyst Eng

123

process does not account for the complexity of biochemical

and biological interactions at work in anaerobic digestion.

Nevertheless, this basic differentiation is often applied to

describe the overall balance of the biological system.

Following basic chemistry kinetics, these two steps may

appear as two consecutive first-order reactions: the first

step of acidification generates intermediate products, i.e.,

VFA, which are converted into the final product, i.e., bio-

gas, in the second step of methane production [60, 62, 65–

68].

According to Shin and Song [60], the series of two

consecutive first-order reactions resulting from a two-step

process can be aggregated into one single equation, which

can be applied to batch anaerobic digestion (Model B):

Mt ¼ S� 1þ kH � e�kVFAt � kVFA � e�kHt

kVFA � kH

� �ð4Þ

kH first-order kinetics constant of substrate degrada-

tion into VFA (first step)

kVFA first-order kinetics constant of VFA degrada-

tion into methane (second step).

VFA are soluble and reside in the liquid phase. VFA

concentrations should be followed closely because these

compounds inhibit bacterial development, especially at low

pH as they are in their protoned form [69]. VFA are not the

sole intermediate product, but H2 and CO2 are also gen-

erated in major amounts following the acidification step.

However, both H2 and CO2 are taken up very rapidly either

by methanogens [63] or by hydrogen-consuming homo-

acetogens [64]. Furthermore, CO2 can also migrate towards

the gas phase, or dissolve into the liquid phase, mainly as

carbonates (CO32-) and bicarbonates (HCO3

-) [70].

In biogas reactors fed with particulate substrates,

hydrolysis is often considered the rate-limiting step. If the

hydrolysis step performs much slower than the methane

production step, substrate conversion into methane follows

first-order kinetics as described in Eq. (1). However, the

chemical composition of particulate substrates is generally

heterogeneous. Particulate substrates may be divided into

several fractions, which are modeled as pools or com-

partments with different hydrolysis conversion velocities

[27, 71]. Based on this approach, Rao et al. [72], Luna del

Risco et al. [73] and Kusch et al. [74] described methane

production kinetics in batch anaerobic digestion with a

model assuming the substrate to be divided into two pools,

each following first-order kinetics (Model C):

Mt ¼ S� 1� a� e�kFt � ð1� aÞ � e�kLt� �

ð5Þ

a ratio of rapidly degradable substrate to total degradable

substrate

kF first-order kinetics constant for the degradation of

rapidly degradable substrate

kL first-order kinetics constant for the degradation of

slowly degradable substrate.

Based on previous models, a more complex model can

be developed, which considers both two different substrate

pools (dual-pool model), and two consecutive reaction

steps occurring in each compartment (two-step model).

The separation of the substrate into two pools can be

expressed as follows:

Mt ¼ MFt þMLt ð6Þ

MFt cumulated amount of methane from rapidly degrad-

able substrate pool at time t

MLt cumulated amount of methane from slowly degrad-

able substrate pool at time t.

The total amount of the rapidly degradable substrate

pool (SF) can be expressed as follows:

SF ¼ a� S ð7Þ

The total amount of the slowly degradable substrate pool

(SL) can be expressed as follows:

SL ¼ ð1� aÞ � S ð8Þ

Merging Eqs. (4)-(6)-(7)-(8), the dual-pool two-step

model can be expressed as follows (Model D):

Mt ¼S� a� 1þ kF � e�kVFAt � kVFA � e�kFt

kVFA � kF

� ��

þð1� aÞ � 1þ kL � e�kVFAt � kVFA � e�kLt

kVFA � kL

� ��ð9Þ

The process diagrams of all four models (A, B, C, D) are

summarized in Fig. 2.

Fig. 1 Reaction steps of anaerobic digestion from the points of view

of biochemistry and of process engineering (modified after Shin and

Song [60])

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123

Materials and methods

Biochemical methane potential assay

A test substrate (hay from permanent grassland) was

digested together with a bacterial inoculum for 87 days at

37 �C in a laboratory-scale BMP assay. Substrate origin

and composition were not determined. Hay was dried and

ground to 1-mm-fiber length. Inoculum was produced in a

laboratory reactor as described previously [9]. Dry matter

contents (Total Solids, TS) of hay and of the inoculum

were determined by drying samples at 105 �C for more

than 24 h. Subsequently, the organic matter content (vol-

atile solids, VS) was assessed by burning the samples in an

oven at 550 �C for more than 4 h. Dry matter and organic

matter contents of hay were 93.10 % (related to fresh

mass), and 92.26 % (related to dry mass), respectively. Dry

matter and organic matter contents of the inoculum were

3.99 % (related to fresh mass), and 61.41 % (related to dry

mass), respectively. 400-mg hay and 30-g inoculum,

respectively, were added into three digesters (replicates) at

the beginning of the experiment. Three other digesters were

fed with 50-g inoculum only and no substrate addition

(control variant). A higher amount of inoculum was used in

the control variant (50 g instead of 30 g for the substrate

variants) to increase the sensitivity of gas measurement,

since the inoculum had a low gas production.

Anaerobic digestion was performed according to the

Hohenheim Biogas Test (HBT) [75, 76], which is described

in the German directive VDI 4630 [4] and has been pat-

ented [77]. The HBT process was operated as described

previously [8, 75]. The cumulated methane yield of dry gas

under standard conditions (0 �C, 1,013.25 hPa) was cal-

culated according to the ideal gas law by applying the

formula described in VDI 4630 [4]. The average methane

production per gram of inoculum of the control variant fed

with inoculum only was calculated, multiplied by the

weight of inoculum in the substrate variant and subtracted

from the total methane yield of the substrate variant to

determine the methane yield of the test substrate (i.e.,

specific methane yield, m3/kg VS from the test substrate)

The values from the three reactors (replicates) were aver-

aged to build a data series (time; cumulated methane yield),

where time was expressed in days and cumulated methane

yield in m3/kg VS. After completion of the 87-day diges-

tion period, the average-specific methane yield amounted

to 0.338 ± 0.002 m3/kg VS, i.e., the standard deviation

amounted to 0.6 % of the average final value.

Data fitting to the models

The four models (A, B, C, D) previously described could

be fitted on the data series. For this purpose, command

lines were written and executed under the software Mat-

lab� version 7.4.0 (R2007a). A method was designed to

allow data fitting of model equations containing up to five

unknown constants. In this method, the simplest model was

used as building block to yield a better fit of solutions on

more complex models, according to the following steps:

1. Setting arbitrary initial conditions to unknown con-

stants of a simple model (Model A)

2. Executing software data-fitting function which runs

iterations starting from the initial conditions to

estimate values for unknown constants

3. Feeding the values obtained for model constants as

initial conditions for iterations of a more complex

model

4. Executing software data-fitting function on the more

complex model.

Fig. 2 Process diagrams of the

four models (A, B, C, D)

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The data-fitting function of Matlab� was set on

Levenberg–Marquardt algorithm [78, 79]. Equations and

constants for each model (A, B, C, D) have been defined

previously. Initial conditions for the iteration process

applied to each model are described in Table 1. Matlab�

command lines, which are compiled as a self-sustaining

modeling toolbox, are listed in the ‘‘Appendix’’ section.

Results and discussion

Convergence of the models to a solution

The Matlab� optimization function [lsqcurvefit] vas used

(‘‘Appendix A’’). [Optimset] was kept at default values

(‘‘Appendix B’’). The number of iterations was 6, 19, 50

and 29 for Models A, B, C, and D, respectively. Models A,

B, and D satisfied to the default optimization criteria:

directional derivative along search direction less than

[TolFun] and infinity-norm of gradient less than

[10*(TolFun ? TolX)], where the default value for both

[TolFun] and [TolX] was 1 9 10-6. On the opposite,

Model C failed to converge to a solution.

Analysis of the residuals

The output of software plotting commands on the data

series [time; cumulated methane yield (m3/kg VS)] repre-

senting hay digestion in the BMP assay is shown in Fig. 3.

For each of the four models, data are presented as dots, and

model output as solid lines, respectively. Plots of the

residuals are shown below the graphs of model outputs.

Residuals are defined as the difference between experi-

mentally measured and model values of cumulated meth-

ane yield for each data point of the time series. In a perfect

model, residuals would be randomly distributed along time,

meaning that the deviation of the model from reality would

be only related to measurement incertitude. This was not

the case here, with the residuals peaking to the beginning

and in the middle of the experimental period. The non-

random distribution of the residuals implies that the models

tested may not perfectly reflect the reality of the anaerobic

digestion process: these models remain an estimate of the

output of more complex biochemical processes. Models A,

B, and C yielded residuals in the range 2 9 10-2. The

dual-pool two-step model (Model D) yielded a much better

fit than other models, with residuals being much lower, i.e.,

in the range 4 9 10-3. This means that the more complex

design of the latter model was useful to reach a better fit on

the data. While being a useful parameter to demonstrate the

goodness of fit of the model, the residuals alone do not

verify if a model is applicable for evaluation purposes. This

issue is illustrated by Model C, which yielded low residuals

while failing to converge to a solution. Moreover, as dis-

cussed further, the estimated constants for Model C were

inaccurate.

Estimation of model constants

The values obtained for model constants are shown in

Table 2. From an experimental point of view, the most

critical value is S, which is the ultimate methane yield, i.e.,

the cumulated methane yield at t = ?. The models A, B,

and D yielded a reasonable estimate of S to be in the range

0.32–0.35 m3/kg VS; Model A has been the most widely

used by biogas researchers, presumably because it is simple

and known to provide realistic estimates of S [50–58]. In

Model B, the simulation of transitory VFA intermediate

formation and consumption according to a two-step reac-

tion did not affect S. In contrast, Model C yielded an

overestimation of S as amounting to 0.51 m3/kg VS, and

failed to converge to a solution. Model C may be unsuited

to the shift of the methane production curve resulting from

transitory accumulation of intermediates, i.e., VFA. This

issue was dealt with in Model D by incorporating the two-

step reaction principle into the dual-pool model. However,

the performance of Model C has already been revealed

under conditions where only little VFA accumulation

occurs in the course of the digestion process [72–74]. Such

conditions may include high inoculum to substrate ratio,

slowly degradable substrates, and post-methanation of

digestate samples. Under such circumstances, Model C

may be more accurate and even outperforms Model D due

to the estimation uncertainty which would result in low

VFA concentrations. Therefore, depending on substrate

characteristics and experimental conditions, the models

yielding the best estimates of S may differ. The modeling

toolbox should facilitate the selection of the most appro-

priate model under each circumstance depending on both

Table 1 Initial conditions for the iterations as defined in data-fitting

commands

Model Model type Constants

A First-order S0 k0

1 1

B First-order two-step S0 kH kVFA0

Sa ka ka 9 2

C Dual-pool first-order S0 a0 kF0 kL0

Sa 0.5 ka ka/2

D Dual-pool two-step S0 a0 kF0 kL0 kVFA

Sa 0.5 ka ka/2 kFa9 2

a Model constants obtained after completion of data fitting for

Model A

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the characteristics of the batch digestion process and the

quality of data being gathered.

Analysis of substrate degradation according to Model D

An interesting pattern of the dual-pool two-step model

(Model D) is that it provides some information about

substrate degradation. In Fig. 4, this model was divided

into the following components: cumulated methane pro-

duction from rapidly degradable substrate, cumulated

methane production from slowly degradable substrate,

concentration of intermediate products in the digestion

medium (assumed to be VFA, shown as equivalent to

methane production). In order to calculate the amount of

intermediate products generated along the digestion period,

the constants estimated for Model D were applied to

Model C which was then subtracted from Model D. The

measurement of VFA concentrations at different time

points during the digestion period was not performed,

though such a procedure would have been useful to eval-

uate the realism of the model. However, the transitory

accumulation of intermediates (VFA) to the beginning of

the process, which is observed in Fig. 4, is known to be a

typical pattern of batch anaerobic digestion [74, 80–82].

Validity range of Model C and Model D

Model D relies on many assumptions which can affect its

range of validity: (a) reaction rates are proportional to the

amount of substrate available; this implies that neither

Fig. 3 Output of software plotting commands

Fig. 4 Time course of the components of the dual-pool two-step

model (Model D)

Table 2 Model constants as estimated with data-fitting commands

Model Model type Constants

A First-order S K

0.3239 0.1243

B First-order

two-step

S kH kVFA

0.3212 0.1412 2.0410

C Dual-pool

first-order

S a kF kL

0.5108 0.5893 0.1372 0.0019

D Dual-pool

two-step

S a kF kL kVFA

0.3460 0.7202 0.2365 0.0258 0.9044

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saturation nor inhibition effects occur, (b) VFA are the sole

reaction intermediates produced, i.e., the role of H2 and

CO2 as intermediate products is neglected, (c) VFA initial

concentration is equal to zero, i.e., neither substrate nor

inoculum do contain VFA at the start of the process, and

(d) VFA are formed in the course of anaerobic digestion,

i.e., VFA degradation is not much faster than substrate

acidification, in the latter configuration Model C would be

best suited.

Furthermore, the application of Model D requires the

estimation of five model constants. Hence, data should be

of good quality, with accurate, quite regularly spaced data

points and the application of a long retention time to pro-

vide a correct estimation of the final methane yield

S. Hence, prior knowledge of S can be useful to widen the

scope of application of Model D.

Conclusion

The modeling toolbox may be applied for the quality

control of BMP assays, to overcome the number of

experimental factors affecting the results. As quality

parameters, both the value of the ultimate methane yield

(S) and the goodness of fit of the data (evaluated as

residuals distribution or as other statistical parameters) can

be chosen.

The limitations of the modeling toolbox remain to be

investigated by testing a wide range of different substrates

and conditions, as well as analyzing the validity range and

statistical robustness of the estimation method employed in

Matlab�. If this tool proves to be reliable, it could be run

routinely on data collected from the anaerobic digestion of

standard test substrates in different batches of BMP assays.

The interpretation of the results would assist in constant

improvement of the experimental procedure and in the

mitigation of experimental errors. Thus, modeling may be

an alternative or at least a complementary approach to the

implementation of standard experimental protocols. The

modeling toolbox provides a set of useful information, such

as the identification of slowly degradable substrates and of

weak inoculums. This information allows the researchers to

decide about the need to replace the inoculum or to extend

the digestion period to provide more reliable results.

The dual-pool two-step model (Model D) holds several

advantages: it ensures best data fitting and may yield an

estimation of transitory VFA accumulation in the course of

anaerobic digestion. This information can be useful,

because VFA concentrations should be kept at low levels to

ensure optimal performance of BMP assays. The modeling

toolbox may have numerous other applications. The

information gained from analyzing data of substrate, or of

digester effluent samples tested in post-methanation assays

[83, 84], may be interpreted for sizing or optimizing full-

scale biogas plants with the help of BMP assays. Further-

more, the toolbox may also be appropriate to evaluate

in vitro digestion assays in the field of animal nutrition.

However, the models developed in this toolbox remain a

simplification of the complex biochemical processes at

work in anaerobic digestion. They may be valid only under

certain conditions, which should be determined thoroughly

in further experiments.

Acknowledgments The authors would like to express their gratitude

to the Doctorate Program of the Faculty of Agricultural Sciences of the

University of Hohenheim for granting a Ph.D. scholarship to Mathieu

Brule. Thanks a lot to Dr. Simon Zielonka, scientific assistant at the

State Institute of Agricultural Engineering and Bioenergy at the Uni-

versity of Hohenheim, for commenting and correcting the first draft.

Appendix

A. Modeling toolbox

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B. Optimset values

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