Bioresource Technology Reports 7 (2019) 100201
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Determination of biogas process efficiency - a practice-oriented alternative to the biomethane potential test
T
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René Casarettoa,b, , Fritz Thomsena, Jens Borna, Jens Bo Holm-Nielsenb a b
Flensburg University of Applied Sciences, Germany Aalborg University Esbjerg, Energy Technology, Denmark
A R T I C LE I N FO
A B S T R A C T
Keywords: Efficiency Lignin Biogas Measurement
The energy efficiency of biogas plants is fundamentally based on the biochemical degradation of input materials. This paper introduces a novel method of efficiency determination based on the gross calorific value (GCV), a very common method in conventional power generation. GCV investigation of five commercial biogas plants in northern Schleswig Holstein were conducted by a one-year time series analysis with weekly sample taking. Although initially a simple model was used to estimate the lignin content, the plausible results indicate the suitability of the proposed method. For comparison, methods like the FoDM and the classical biomethane potential test are highlighted to point out their traditional usage and their adaption for the biogas sector. Also the laboratory efforts they cause is taken into account.
1. Introduction The energy efficiency of commercial scale biogas plants is usually determined by the energy content of the fermentation residues. In a batch fermentation test, a specific process biology of the respective plant is used to determine the residual gas yield. Several commercial biogas plants use different substrates and thus, will have differing process biologies. Therefore, in order to compare the different plants and substrates, a suitable alternative measuring method should be developed. The established procedure to determine gross calorific value (GCV) is a suitable approach to adapt to the biogas process, with low laboratory demands. Considering, not all energy sources determined with the GCV method can be used anaerobically in the biogas process, hence, when evaluating efficiency this is necessary to consider. As a result the energy content of the inert components for the biogas process has to be excluded from the energy balance. Lignin has a significant impact on energy efficiency due to its relatively high GCV and fraction within the substrates (e.g. Maize-Silage has a lignin content of about 8% (Epie et al., 2018)) and on the other hand because of its relatively high GCV, the biomass component with the greatest impact on energy efficiency is lignin. An insight into lignin formation is necessary to precisely determine its content. Lignin is a cell wall polymer formed from three monomers: Sinapyl, Guiacyl and Coniferyl. These monomers vary in compositions
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within respective plant (e.g. grasses, woods). The release of phenolic compounds in lignin, which are toxic to fermentation microbiology, can inhibit the fermentation process. Additionally, lignin limits the biological degradation of the plants due to is high complexity and its characteristics (i.e. non-soluble in water; high degree of polymerization (Benner et al., 1984; Colberg and Young, 1985; Gomes et al., 2011)). In relation to these characteristics, the microbiology is incapable of hydrolyzing lignin under anaerobic conditions. Pertaining to renewable resources, the degree of lignification within an anaerobic digestion system is of increasing interest for plant owners. This degree allows estimating the non-degradable part of the organics within the fermentation system, based on the findings that lignin is under anaerobic conditions non-degradable (Amthor, 2003). The key objective is to establish a feasible method to measure lignin in an accurate and reliable way, with emphasis on the analytical procedures and the extent of their reliability (Hatfield and Fukushima, 2005). In most cases commercial scale biogas plants use annual plants and residues from the agricultural sector, excluding wood and forages, as input materials, for which a variety of analytical methods have been developed. In principle, there are two common ways to determine lignin contents: gravimetric methods (e.g. Klason-Lignin, Acid-Detergent-Lignin) and spectrophotometric methods (e.g. Acetyl-Bromide-Lignin) that are based on the absorbance of the soluble fractions with UV (Stephen and Dence, 1992).
Corresponding author at: Flensburg University of Applied Sciences, Germany. E-mail address:
[email protected] (R. Casaretto).
https://doi.org/10.1016/j.biteb.2019.100201 Received 7 March 2019; Received in revised form 16 April 2019; Accepted 17 April 2019 Available online 18 April 2019 2589-014X/ © 2019 Elsevier Ltd. All rights reserved.
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produced is measured. The fermentation test is considered complete, when the gas production is lower than 0.5% of the accumulated gas production. Batch systems are simple, easy to set up and allows for simplistic monitoring and evaluating the gas/methane production of several different materials. A determination of the degradation efficiency is possible with batch test, if the fermentation residue is also tested. But the maximum methane production depends on several influencing factors like grain size, trace element supply, possible inhibitors (e.g. Nitrogen) so that a general statement about the degree of conversion is not given.
Former methods for determining the lignin content, used in forage analytics, are unspecific and not useful for woods. Currently additional methods like Acetyl-Bromide-Lignin (AB-Lignin) are not evaluated in a high level until now (Hatfield and Fukushima, 2005). Klason-Lignin is the most common method for determining lignin in Wood species and forages (Hatfield and Fukushima, 2005). An error, with respect to accuracy of this method, is perhaps the formation of “pseudo Lignin” that can cause an over-estimation of the lignin content within the analyzed samples. In previous analysis, it has been shown, that the Klason-Analysis gives lignin concentrations in forages, which are 2–3 times higher than with ADL-Analyzing (Hatfield et al., 1994). The lignin content using the Acetyl-Bromide-Lignin compared to Klason Method is relatively similar. However, the thioglycolic method results in low lignin contents. (Moreira-Vilar et al., 2014) Whether the methods are also suitable for the determination of the lignin content from fermentation residues has not been sufficiently investigated to the knowledge of the authors. The authors objective is to show the different lignin, forage analytic, waste water analyzing methods to determine the residual energy potential of fermentation residues. This research will highlight techniques of lignin extraction and related standard methods to determine efficiency and its advantages/disadvantages. Special focus is set on the practicability for commercial biogas plants. Lignin is normally anaerobically non-degradable and contains an energy potential, therefore, correctly analyzing lignin content and determining its GCV, is necessary for describing the residual energy potential in the digestates. In comparison with the input energy a benchmarking system for the energy efficiency would be possible. In Fig. 1, the simplified model of a mass and energy balance based on lignin for a biogas plant is shown.
2.1.2. DM – VS calculation The Dry Material and Volatile Solids (DIN Deutsches Institut für Normung e.V., 2001) is a basic parameter for describing the water content and the organic material content of a sample. With these parameters a description of the degradation of organic material (i.e. material which is gasified) can be done. Taking into account that a small fraction of the organic material is used for growing the bacteria (Kaltschmitt et al., 2009), whereas the substantial portion is converted into carbon dioxide and methane, an efficiency can be calculated. Utilizing standard tables for calculating the methane content, while digesting the samples (Döhler, 2013; Fachagentur Nachwachsende Rohstoffe, 2013), a description of the energy content is possible. Moreover, the formation of a mass balance is possible by comparing the organic dry material content of the in- and output. 2.1.3. FoDM – calculation The fermentable organic dry material (FoDM) calculation of Weissbach (Weißbach, 2009a, 2009b) is an extended VS method for describing the fermentable organic material. This method allows for an estimation of the gas production of several different input materials for biogas plants. Furthermore, the correction of the volatile fatty acids in the organic dry material is necessary, as these components are partially vaporized during the oven drying at 105 ± 1 °C (Weißbach and Strubelt, 2008a, 2008b). However, with this estimation equation it is possible to predict the biogas production in an easy way without considering the biological parameters within the fermentation.
2. Common analytical methods 2.1. Standard and biological methods 2.1.1. Batch-fermentation test According to VDI 4630 (Ingenieure, 2006), VDLUFA (Verband Deutscher Landwirtschaftlicher Untersuchungsund Forschungsanstalten, 2013) and other publications (Bayerische Landesanstalt für Landwirtschaft, 2009; Fachagentur Nachwachsende Rohstoffe, 2009) methods for estimating the gas production from digestible materials described in the above named method collection. These methods are mainly based on biological degradation of the materials with an inoculum as starter culture to perform fermentation. In most cases, the starter culture is sewage sludge from the municipal waste water treatment plant. According to VDI 4630, the inoculum and the sample is mixed (VS based 50:50), placed into a water bath at 38 ± 1 °C and the gas
2.2. Lignin analysis methods In the following, different methods for determining the lignin content of annual and perennial plants are presented. This block is divided up into two parts: spectrophotometric and gravimetric methods. A special focus is set on the accuracy of these methods and how they could be used for determining the energy efficiency of commercial scale biogas plants.
Biogas
Input Material (Lignin / Carbohydr ates)
Heat and entropie Biogas Plant Degradable (Carbohydrates) Residue Non-degradable (Lignin)
Fig. 1. Simplified flow chart biogas plant. 2
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2.2.5. ADL-lignin The Acid-Detergent-Lignin method from Van Soest is based on the Weender Method, which is used in analyzing of forages. The extended Weender-Analysis allows for a deeper view into the scaffold substances (Goering and van Soest, 1970). The ADL-Lignin method is based on the extraction of the lignin with a multi-stage detergent system, NeutralDetergent-Fiber (NDF) (van Soest et al., 1991) and Acid-Detergent-Fiber (ADF) extraction before Acid-Detergent-Lignin (ADL) analysis. Prior to analyzing the ADL in the samples, the NDF and ADF must be determined according to VDLUFA 6.5.1 and VDLUFA 6.5.2 (Verband Deutscher Landwirtschaftlicher Untersuchungsund Forschungsanstalten, 2013). In the first step, the samples are dried, screened and added to filter bags. Pre-extraction of the samples including higher fat yields (> 5%) is preferred. In the next step, the filter bags are put into H2SO4. Afterwards, the samples are rinsed and oven dried. The dried sample is ashed in a muffle oven. The lignin content is calculated as described in (ANKOM Technology, 2009):
2.2.1. Spectrophotometric One procedure for determining the lignin content of the samples is with spectrophotometric methods. These methods are mainly based on a complete solubilization of the lignin with acids within the samples. After solubilization of the lignin, the content is determined with a spectrophotometer. Therefore, it is necessary to calibrate the spectrophotometer for pure lignin to differentiate the contents. 2.2.2. Acetyl-bromide-lignin The most commonly used Acetyl-Bromide-Lignin method is proposed by Morrison (Stephen and Dence, 1992), which is applicable for several different adaptions to increase the performance of extraction (Iiyama and Wallis, 1990; Iiyama and Wallis, 1988). To prepare for this method, the sample is grinded and the non-cell wall materials are removed by pre-washing with ethanol and chloroform. The lignin content is measured in an UV-Spectrum at 280 nm using the equation of Morrison (Stephen and Dence, 1992) (Iiyama and Wallis, 1990).
g %Lignin = 3.37 ∗ absorbance/sample conc. ⎡ ⎤ − 1.05 ⎣ l ⎦
ADL (DM ) =
for legumes:
(W 3 − (W 1 ∗ C1)) ∗ 100 W 2 ∗ DM
with: DM = Dry Matter. W1 = Bag tare weight. W2 = Sample weight. W3 = Weight after extraction process. C1 = Blank bag correction.
g %Lignin = 5.12 ∗ absorbance/sample conc. ⎡ ⎤ –0.74 ⎣ l ⎦ Iiyama et al. developed a method for the determination of AB-Lignin in herbaceous plants (Iiyama and Wallis, 1990). Additionally, Iiyama et al. developed an improved method for determination the AB-Lignin content in woods and wood pulps (Iiyama and Wallis, 1988).
2.2.6. Klason-lignin The Klason-Lignin determination is based on the complete hydrolysis and solubilization of the carbohydrate components in organic biomass. The residue, which in this case is determined gravimetrically, is identified as lignin. The Klason analysis consists of a strong acid treatment followed by a dilution to a mild acid and boiled, to complete the hydrolysis. To prepare, prior to acid treatment the removal of fat, waxes, lipids are necessary. This is done by extracting with ethanol-benzene or similar (Stephen and Dence, 1992). The Klason method is adapted several times for investigation of annual plants, developing wood and forage, smaller sample amounts, different acid concentrations and also for determining the acid soluble lignin content of the samples (Bagby et al., 1971; Effland, 1977; Stephen and Dence, 1992; Whiting and Goring, 1982).
2.2.3. Thioglycolic acid The thioglycolic acid lignin method is based on the complete solubilization of lignin and the lignin content is measured in the solvent. There are several different procedures available for determining the lignin content with thioglycolic acid (Hatfield and Fukushima, 2005; Suzuki et al., 2009). The original methods used for determining the lignin content in woody species have been modified for herbaceous species (Moreira-Vilar et al., 2014; Suzuki et al., 2009). The original thioglycolic lignin method used an extracted cell wall sample mixed with thioglycolic acid and HCl (Browning). The resulting lignin must be purified by dissolving in dioxane and precipitated from this solution by dilution with ether. The filtrated residue - insoluble lignin - is washed with ether and dried for gravimetrical measurement of the lignin content. This method has been modified for smaller sample amounts by (Bonello et al., 1993), (Bruce and West, 1989), (Lange et al., 1995). In the adapted method, the samples were treated with HCL and thioglycolate. Insoluble residues are recovered by centrifugation and are washed with water. The thioglycolate lignin is dissolved in NaOH. By centrifugation, the non-lignin materials are pelletized and removed. Acidification with HCl, the lignin is recovered from the solution and pelletized by centrifugation. After drying, the insoluble residue is dissolved in NaOH and diluted before reading the UV-Spectra at 280 nm. For this direct measurement it is necessary to add a lignin standard to reach information about mass changes due to derivatization of the lignin.
2.3. Gross calorific value The main method for determining the efficiency is comparing the energetical potential of In- and Output materials, by using a reliable and independent setup. These requirements and low laboratory demands are covered by the GCV. In the following the method is presented in detail. The GCV has been used for decades in energertical rating of input materials for commercial coal fired power plants and it is also used in every part of the traditional heating technology and oil refineries. However, the GCV is an independent measuring method, based on the complete oxidation of the sample, for determining the energy content of the sample (Boie, 1957; DIN Deutsches Institut für Normung e.V., 2017; Ertelt, 1976). The GCV is measured from the dry material of the sample using a bomb calorimeter. The sample is dried, grinded and pelletized for measuring. The bomb is filled with 30 ± 1 bar and placed in a water bath. After ignition and complete oxidation of the sample, the heat difference of the water bath is measured. This heat difference is equal to the energy content. Calibration of the system is mostly done by Benzoic-Acid, which has a well-known GCV (26.46 kJ/g − IKA C 723 Benzoic-acid pellets).
2.2.4. Gravimetric methods A different method for determining the lignin content in woody materials is the gravimetric methods. These methods are based on a complete hydrolysis and solubilization of the carbohydrates. In this case, the lignin is recovered in a relatively solid fraction and measured directly with gravimetric methods. However, as compared to AB-Lignin and Thioglycolic Lignin methods, the lignin content is indirectly measured in the liquid phase. The benefit is that no standards for calibration is needed in the gravimetric methods (Stephen and Dence, 1992). 3
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GCV [KJ/G]
R. Casaretto, et al.
30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15
Total energec efficiency 100.00% 95.00% 90.00% 85.00%
85.47%
83.50%
82.99%
80.73%
80.00% 75.00% 70.00%
0
10
20
30
40
50
60
70
80
90
100
90
80
70
60
50
40
30
20
10
65.00%
100 % LIGNIN
57.71%
60.00%
0 % CARBOHYD.
55.00%
Fig. 2. Correlation GCV lignin - cellulose.
50.00% Plant 1
3. Lignin corrected efficiency
Plant 2
Plant 3
Plant 4
Plant 5
Fig. 3. Total energetic efficiency.
The following approach is based on using GCV for determining efficiency. The GCV method can be used as an efficiency indicator for commercial scale biogas plants based on the following assumptions:
Total digestable efficiency 100.00% 95.00%
– The fermentation residue is a two-component mixture of lignin and cellulose – Lignin in fermentation residues has an GCV of 28.6 kJ/g – Carbohydrates (Cellulose) has an GCV of 17.4 kJ/g
93.32%
93.25%
93.17%
92.73%
Plant 4
Plant 5
90.00% 85.00% 80.00% 72.74%
75.00% 70.00%
This results in Fig. 2: The significant difference in the GCV allows conclusions to be drawn about the composition of the binary system using simple equations:
65.00% 60.00% 55.00% 50.00%
x C + xL = 1
Plant 1
x C ∗ GCVC + xL ∗ GCVL = GCVSubstrate xL =
Plant 3
Fig. 4. Total digestible efficiency.
GCVSubstrate − GCVC GCVL − GCVC
operating under mesophilic conditions (38 ± 2 °C). Samples were taken from the last gas-tight covered tank reactor from the respective biogas plant. The samples were taken before separating into a liquid and solid phase in the residual tank. In order to reflect the total retention time of the plant and the harvest of new materials, the samples were taken weekly from the input materials and the fermentation residue. Table 1 outlines an overview about the fermentation volume, retention time and the origin of the input materials. However, the total energetic efficiency without lignin correction of these plants were between 57.7% and 85.5%. With lignin correction, the efficiency was between 72.7% and 93.3%. The results are shown in Fig. 3 and Fig. 4. Error bars are the standard deviations.
with:
x C = Mass portion of Cellulose xL = Mass portion of Lignin The approximately determined portion of lignin, based on simple investigations of the substrate, allows to describe the unused but anaerobically digestible part of the lignin portion. The lignin corrected efficiency is determined as:
ηL = 1 −
Plant 2
GCVRes ∗ DMRes ∗ x C, Res GCVSubstrate ∗ DMSubstrate ∗ x C, Substrate
4. Results 5. Discussion The total energetic efficiency was determined using a one-year time series analysis based on weekly sampling of five commercial biogas plants in northern Germany – Schleswig Holstein. These biogas plants use mainly maize silage, rye silage, cattle dung/ manure and pig manure as input substrates. These biogas plants are
The presented method can provide plausible and with good input parameters comparable and correct results. While the range of calorific values available in the literature for cellulose is narrow, the correct value to be applied for lignin is between 17.3 kJ/g and 29.2 kJ/g.
Table 1 Overview biogas plants. Investigated biogas plants Fermentation volume
Retention time Materials
Plant 1
Plant 2
Plant 3
Plant 4
Plant 5
8000 m3
5800 m3
4400 m3
14,200 m3
11,300 m3
96 d Renewable Resources/manure
70 d Renewable resources/manure
118 d Renewable resources/manure
184 d Renewable resources/manure
4
134 d Renewable resources
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Table 2 GCV of lignin and cellulose. Gross calorific value Origin
Lignin [kJ/g]
Lignin Lignin Lignin Lignin Lignin residue from soft wood Lignin residue from hard wood Lignin from spruce wood (Organosolv) Lignosulfonate 1 Lignosulfonate 2 Lignin Lignin - hydrolitic extraction Birch lignin - NaOH extraction
26.66 26.67 23.26 25.58 23.5 21.45 25.9 17 28.6 28.6 27.4 29.2
Gross calorific value Origin
Cellulose [kJ/g]
Cellulose Cellulose Cellulose Cellulose Birch/Spruce Wood 3:1 Cellulose microcrystalline
17.33 17.3 18.6 17.4 17.3
Table 3 GCV calculation models by extraction methods. Gross calorific value calculation model
Boie Dulong Lloyd Michel Mott Scheurer Steuer Wilson
Applied for
Coal Coal Fossil fuels Coal Coal Municipal waste Coal Municipal waste
Year
1957 1837 1980 1938 1940 1996 1937 1972
Kraft
Soda-anthraquinone
Lignosulfonate
Organosolv
Ethanol-process
GCV [kJ/g]
GCV [kJ/g]
GCV [kJ/g]
GCV [kJ/g]
GCV [kJ/g]
25.99 23.63 25.97 26.32 29.63 26.56 26.37 25.86
26.65 24.30 26.44 26.93 30.52 27.27 27.22 26.82
17.43 14.07 17.69 18.44 22.92 18.65 17.61 16.85
25.21 22.70 25.20 25.48 29.21 25.91 25.60 25.07
23.65 20.88 23.41 24.03 28.11 24.45 24.06 23.56
methods use inoculum (Boulanger et al., 2012) as a starting culture for the biocenosis and do not represent the considered biogas plant (Ingenieure, 2006). A biomethane potential test only represents batch fermentation and not a continuous process. The method of Weissbach et al.(Weissbach, 2009) is another method to determine gas potential based on the FoDM of the used materials. Still, this method does not consider the process parameters of the respective biogas plants. The method of the German Biomass Research Center (DBFZ) is based on the FoDM and the GCV as energy benchmarking (Fischer et al., 2016; Fischer et al., 2015). However, this method does not consider the residual biologically degradable energy content in a reliable way, on the account that FoDM is based on estimations and not on reproducible measurements. All methods have several advantages but also disadvantages, like high laboratory demand or long retention times (e.g. gas potential test). Establishing a benchmarking system for the energetical use of biomass will allow the total and non-degradable energy content of biomass to be measured in a quick, fast, cheap way. This would have several different benefits:
Furthermore, these values also depend on the origin of the lignin and the method used to isolate it. Table 2 shows the GCV of lignin and cellulose which can be found in the literature (Demirbaş, 2001; Jenkins et al., 1998; Schug and Karin, 2005). A more laborious method, but possibly more accurate, is the direct determination of the lignin content. However, there is still research work to be done. The named methods for determining the lignin content of the samples are mainly based on the extraction of cellulose and other extractives. The residue of the extraction is determined as lignin. Nonetheless, there is some extent of uncertainty that the measured lignin content may not be realistic and can be interfered with by products like pseudo-lignin from acid hydrolysis in the Klason-Procedure (Sannigrahi et al., 2011). All methods were developed within the context of Wood, paper industries and animal-feeding. None of the methods was developed in the context of extraction of lignin for determining the energy content from a mixture of materials after anaerobic treatment in a commercial scale biogas plant. They were developed for energetical use of the biomass and not for biological degradation. Different equations to estimate the energy content from principal analysis of coal, gas, oil, municipal waste, wood are available (Friedl et al., 2005; Raveendran and Ganesh, 1996; Sheng and Azevedo, 2005), but none is applicable for fermentation residues of AD plants. (See Table 3) On the other hand, common methods for determining the biogas yield from biomethane potential tests are available. However, these
– – – –
5
Reliable energy potential measurements Direct comparison of biogas plants Detailed information of efficiency Investment decisions, for example refurbishing a plant, based on measurements and not estimations
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– Professionalism of operator and maintenance could be considered directly
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It seems that the lignin corrected GCV allows for a quick and reliable method for determining the energy potential of fermentation residues. Several advantages emphasize this notion: – Easy and quick measurement according to DIN 18125 (DIN Deutsches Institut für Normung e.V., 2017) – State of the art in all commercial power plants – Correction factors for loss of volatile fatty acids can be done by Weißbach method (Weißbach and Strubelt, 2008a, 2008b) On the other hand, there are some research questions that should be considered: – How can the content of lignin measured/calculated correct? – How can pure lignin be extracted and what is the GCV? – Direct influence on the 100% benchmark – Is the assumption of the fermentation residue being a binary mixture permissible at all or is the GCV distorted by other materials (e.g. waxes) As a conclusion the following statement is possible: Every commercial scale biogas plant has a keen interest in energetic efficiency, based on the business-case. Therefore, with an easy and independent method, like the GCV, a direct comparison and benchmarking of the respective plant is possible. 6. Conclusions Related to the base-load energy production and the fixed selling price of the energy, efficiency is of main interest for existing plants, based on the effort that the operation cost are increasing every year. The GCV method with lignin correction seems highly feasible to reflect the residual energetic potential. These results also show the potential, which could be lifted with repowering initiatives to increase the economic efficiency of the respective plant. With increasing precision, the method presented is not only an alternative to the BMP tests, but possibly an improvement and leads to important findings based on low laboratory effort. Acknowledgments This work is funded by the Large-Scale Bioenergy Lab. 2 of Interreg 5a Project. Declarations of interest Declarations of interest: none. References Amthor, J.S., 2003. Efficiency of Lignin Biosynthesis: a Quantitative Analysis. Ann. Bot. 91 (6), 673–695. https://doi.org/10.1093/aob/mcg073. ANKOM Technology, 2009. Method 8 - Determining Acid Detergent Lignin in Beakers. pp. 1–2. Bagby, M.O., Nelson, G.H., Helman, E.G., Clark, T.F., 1971. Determination of lignin in non-wood plant fiber sources. In: Technical Association of the Pulp and Paper Industry. 54. pp. 1876–1878. Bayerische Landesanstalt für Landwirtschaft (Ed.), 2009. Internationale Wissenschaftstagung Biogas Science 2009. LfL-Schriftenreihe Band 15/2009, Freising-Weihenstephan, (251 pp). Benner, R., Maccubbin, A.E., Hodson, R.E., 1984. Anaerobic biodegradation of the lignin and polysaccharide components of lignocellulose and synthetic lignin by sediment microflora. Appl. Environ. Microbiol. 47 (5), 998–1004. Boie, W., 1957. Vom Brennstoff Zum Rauchgas: Feuerungstechnisches Rechnen Mit Brennstoffkenngrößen Und Seine Vereinfachung Mit Mitteln der Statistik. vol. IV Teubner, Leipzig (101 S).
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