Experimental method
Segregated household GW and FW were collected at a local recycle centre (Todmorden, UK) and stored in the laboratory at a temperature of 5 °C. Within 24 h, the samples were examined and large pieces of bone, plastic, metal, wood were removed to avoid damage to the homogenisation equipment and reduce sampling errors during later analysis. The samples were then homogenised using a commercial food mincer and sampled for physio-chemical analysis. The remainder of the biomass samples was stored in a freezer with a temperature of around −18 °C and thawed before feeding to the digesters.
The semi-continuous study into the production of biogas from GW and FW was performed in two 2-l laboratory digesters. The temperature of the digester was maintained at 37 °C by immersion in a water bath and mixing was provided by a vertical stirrer operating at 60 RPM for 30 s every minute. The inoculum for the experiment was obtained from a homogenised sample of laboratory digestate from other digestion experiments, which originated from a mesophilic digester treating primary and secondary sludge at a wastewater treatment plant.
The two digesters were fed chemical oxygen demand (COD) equivalent pulses of GW and FW over a period of 112 and 176 days, respectively, with a gradual increase in organic loading until failure of the process occurred, as shown in Fig. 1. In the case of GW, the experiment was terminated early due to excessive foaming in the digester. The methane production of the digesters was monitored continuously and samples for offline analyses were taken intermittently during the feeding operations.
The pulsed and irregular feeding of the experimental system is usual in AD research, especially for solid waste and in small scale digesters in rural areas [15] and it is applied in the present study. This approach was chosen for two reasons; (1) the data produced is richer in kinetic information when compared with steady organic loading rate (OLR) experiments, and (2) the feeding profile is more representative of small scale systems which are manually operated and which was the focus of the larger research programme.
Analytical methods
Measurement of the methane production for the laboratory digesters was performed using an AMPTSII gas flowmeter (Bioprocess Control, Lund, Sweden). In this system the produced biogas was scrubbed into a 3 M NaOH alkaline solution to remove the carbon dioxide and hydrogen sulphide, and its volume was determined using a multichannel volumetric measurement device with a resolution of 10 ml. Methane production was then calculated assuming a scrubber efficiency of 98 %, taking into account the overestimation caused from the initial flush gas content (nitrogen), subtracting the concentration of water vapour and reporting the volumes at STP (0 °C and 1 atm), as per the manufacturer’s guidelines.
The total solids (TS) and volatile solids (VS) were measured as per standard methods [16]. The concentration of volatile fatty acids (VFA) was measured using an Agilent 7890A gas chromatograph, with a DB-FFAP column of high polarity designed for the analysis of VFA columns, as per the manufacturer’s guidelines. Elemental analysis was determined using an elemental analyser (Flash EA2000, CE Instruments) equipped with a flame photometric detector (Flash EA 1112 FPD, CE Instruments). The theoretical chemical oxygen demand (CODth) was calculated from the empirical formula obtained from elemental analysis, considering the organic matter to be fully oxidised to carbon dioxide and water, with nitrogen being reduced to ammonia and sulphur oxidised to sulphuric acid [17].
Model description
Three simplified models of AD have been considered in this paper. The models included a one reaction model (1R), a two reaction model (2R) and a three reaction model (3R) and were based on the work of Donoso-Bravo et al. [10], Bernard et al. [5] and Mairet et al. [9], respectively, with some minor modifications as discussed below. It should be noted that parameters of the model, unless calibrated as part of this work, were maintained as per the original citations and therefore there are some differences in units as described in the nomenclature section and appendices. In general the nomenclature was maintained as per Bernard et al. [5]. As part of the model screening in this work, hydrolysis was modelled using the first-order, Contois and Monod equations, and methanogenesis by Monod, Haldane, Tessier and Moser equations as described in Sect. “Kinetics of reaction”.
Using these simplified models to describe the complex AD process requires several assumptions;
-
The AD process can be simplified by a limited number of reactions. For the study we consider only the hydrolysis stage and the methanogenesis stage with the other stages in the process being incorporated into the above reactions.
-
The organic matter in the substrate can be represented by either a single lumped fraction in the case of the 1R and 2R models, or two fractions in the case of the 3R model.
-
Inhibition only occurs in the methanogenic stage.
-
The methane produced is immediately transferred into the gas phase without undergoing the liquid–gas transfer process, in contrast to ADM1 which calculation of the gas–liquid mass transfer rate and therefore includes both the dissolved and headspace gases as dynamic states.
-
The digester is completely mixed and the biomass concentration is homogeneous.
The use of a completely mixed model (i.e. no spatial variation) is common in AD modelling even where solid substrates are fed to the system [18–20], and in which cases there will undoubtedly be stratification of the solid components within the system. We used an intermittent mixing regime for technical reasons (as recommended by the manufacturer) and observation of the experimental setup confirmed that the digester contents were sufficiently mixed at all times.
One reaction model (1R)
The 1R model used in this paper is a generic mass balance involving a single substrate (S
1) that is converted to methane and carbon dioxide by the action of a single population of microorganisms (X
1). The dynamic model is fully described in the following equations:
$$ \frac{{{\text{d}}X_{1} }}{{{\text{d}}t}} = r_{1} - DX_{1} $$
(1)
$$ \frac{{{\text{d}}S_{1} }}{{{\text{d}}t}} = k_{1} r_{1} + d\left( {S_{{1,{\text{in}}}} - S_{1} } \right) $$
(2)
Methane flowrate:
$$ q_{m} = k_{3} r_{1} $$
(3)
The reaction stoichiometry is given by Eq. 4.
AD reaction:
$$ k_{1} S \to X + k_{2} {\text{CO}}_{2} + k_{3} {\text{CH}}_{ 4} $$
(4)
Two reaction model (2R)
The 2R model includes a single lumped fraction of particulate organic matter (S
1). The hydrolysis and acidogenesis/acetogenesis stages are considered together and the particulate organic matter is converted into VFA (S
2) by the action of the hydrolytic microorganisms (X
1):
Hydrolysis:
$$ k_{1} S_{1} \to X_{1} + k_{2} S_{1} + k_{4} {\text{CO}}_{2} + k_{1} k_{n} N $$
(5)
The methanogenic step involves the uptake of the VFA by the action of methanogenic microbes (X
2) to produce methane:
Methanogenesis:
$$ k_{3} S_{1} \to X_{2} + k_{5} {\text{CO}}_{ 2} + k_{6} {\text{CH}}_{4} $$
(6)
The rate of methane production is directly related to the rate of the methanogenesis reaction by the coefficient k6 (20.29 L g−1) which has been modified from the original work to give the total methane flow rate in L day−1: the matrix description of the dynamic model is shown in Eqs. (7) and (8) and all the stoichiometric parameters in Eq. (5) and (6) can be found in Bernard et al. [5].
$$ \frac{{{\text{d}}\xi }}{{{\text{d}}t}} = Kr\left( \xi \right) + D\left( {\xi^{in} - \xi } \right) $$
(7)
$$ q_{\text{m}} = k_{6} r_{2} \left( \xi \right) $$
(8)
where
\( K,r , \,\)
\( \xi \) and \( D \) are expressed as shown in Eq. (9):
$$ \xi = \left[ {\begin{array}{*{20}c} {X_{1} } \\ {X_{2} } \\ {S_{1} } \\ {S_{2} } \\ C \\ N \\ Z \\ \end{array} } \right],\;r\left( \xi \right) = \left[ {\begin{array}{*{20}c} {r_{1} \left( \xi \right)} \\ {r_{2} \left( \xi \right)} \\ \end{array} } \right],\;K = \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ { - k_{1} } & 0 \\ {k_{2} } & { - k_{3} } \\ {k_{4} } & {k_{5} } \\ {k_{n} k_{1} } & 0 \\ 0 & 0 \\ \end{array} } \right],\;D = I_{6} d $$
(9)
An important modification from the original model formulation is the inclusion of an additional dynamic state that represents the ammonia concentration in the digester (N). This was included to allow ammonia inhibition to be implemented the model screening process since this is an important phenomenon in AD of solid wastes [21]. The reaction stoichiometric coefficient for ammonia (k
n) was calculated from the elemental composition of the waste multiplied by an estimated degradability coefficient (0.5 and 0.7 for GW and FW, respectively) and calculated as 1.033 and 1.842 mmol/g VS for GW and FW, respectively. Note that in the original model formulation, a pH variable was included and calculated as a function of the alkalinity, carbon and VFA state variables (Z, C and S2). However, this calculated variable had no impact on any other aspect of the model in terms of feedback inhibition and therefore this was omitted from this implementation.
Three reaction model (3R)
The 3R model includes a fractionation of the particulate organic matter into carbohydrates/fats (S
1a) and proteins (S
1b) and the hydrolysis stage consists of two reactions, namely hydrolysis of carbohydrate/lipid (10) and hydrolysis of protein (11). Each reaction produces VFA (S
2) by the action of hydrolysis biomass (X
1a and X
1b):
Hydrolysis of carbohydrates/fats:
$$ k_{1} S_{{ 1 {\text{a}}}} + k_{2} N \to X_{{ 1 {\text{a}}}} + K_{3} S_{2} + k_{4} {\text{CO}}_{ 2} $$
(10)
Hydrolysis of proteins:
$$ k_{5} S_{{ 1 {\text{b}}}} \to X_{{ 1 {\text{b}}}} + k_{6} S_{2} + k_{7} N + k_{8} {\text{CO}}_{ 2} $$
(11)
The methanogenic stage involves the conversion of VFA by the methanogenic population (X
2) to methane as shown in Eq. (12):
Methanogenesis:
$$ k_{9} S_{2} + k_{10} N \to X_{2} + k_{11} CH_{4} + k_{12} {\text{CO}}_{ 2} $$
(12)
The methane flow rate is obtained using Eq. (13):
Methane flowrate
$$ q_{\text{m}} = k_{11} r_{2} $$
(13)
It should be noted that the Eqs. (10), (11), (12) and (13) have been adapted from [9]. As with the 2R model, Eq. (1) describes the general dynamics of the three reactions model with \( K,r, \)
\( \xi \) and \( D \) expressed as shown in Eq. (14):
$$ \xi = \left[ {\begin{array}{*{20}c} {X_{{1{\text{a}}}} } \\ {X_{{ 1 {\text{b}}}} } \\ {X_{2} } \\ {S_{{ 1 {\text{a}}}} } \\ {S_{{ 1 {\text{b}}}} } \\ {S_{2} } \\ C \\ N \\ Z \\ \end{array} } \right],\;r\left( \xi \right) = \left[ {\begin{array}{*{20}c} {r_{{ 1 {\text{a}}}} \left( \xi \right)} \\ {r_{{ 1 {\text{b}}}} \left( \xi \right)} \\ {r_{2} \left( \xi \right)} \\ \end{array} } \right],\;K = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ { - k_{1} } & 0 & 0 \\ 0 & { - k_{5} } & 0 \\ {k_{3} } & {k_{6} } & { - k_{9} } \\ {k_{4} } & {k_{8} } & {k_{12} } \\ { - k_{2} } & {k_{7} } & {k_{10} } \\ 0 & 0 & 0 \\ \end{array} } \right],\;D = I_{7} d $$
(14)
Furthermore, the methane production coefficient for the 3R model (k
11) has been modified to allow direct comparison with the experimental data of Sect. “Experimental method” (13.44 L g−1). For the stoichiometric parameters in Eqs. (10), (11) and (12), see Mairet et al. [9].
Kinetics of reaction
It has being reported in the literature that the first-order kinetic and Contois models are able to best describe the hydrolysis process [22], while the Monod kinetic equation has predominantly been used for soluble substrates with Haldane being frequently chosen to represent the methanogenesis reaction due to its sensitivity to VFA [5, 7]. In addition, the kinetic model for the growth of the microbial population by Tessier and Moser [14] is considered for the methanogenesis stage in the present investigation. Expression for biochemical conversion rates [2, 5, 7, 14] is shown in Eqs. 15–20 and the expressions for the ammonia and VFA inhibition factors (I
N and I
vfa), applied by the multiplication by the rate of methanogenesis (r
2), are shown in Eqs. 21–22 (modified from Batstone et al. [2]):
First-order:
$$ r = k_{\text{hyd}} S $$
(15)
Contois:
$$ r = \mu_{\hbox{max} } \frac{S}{{k_{s} + S}}X $$
(16)
Monod:
$$ r = \mu_{\hbox{max} } \frac{S}{{k_{s} + S}}X $$
(17)
Haldane:
$$ r = \mu_{\hbox{max} } \frac{S}{{k_{s} + S + \frac{{s^{2} }}{{k_{i} }}}}X $$
(18)
Moser:
$$ r = \mu_{\hbox{max} } \frac{{S^{\lambda } }}{{k_{s} + S^{\lambda } }}X $$
(19)
Tessier:
$$ r = \mu_{\hbox{max} } \left( {1 - e^{{ - \frac{S}{{k_{s} }}}} } \right)X $$
(20)
Ammonia inhibition:
$$ I_{N} = \frac{1}{{\frac{{k_{i,N} }}{N} + N}} $$
(21)
VFA inhibition:
$$ I_{\text{vfa}} = \frac{1}{{\frac{{k_{{i,{\text{vfa}}}} }}{{S_{2} }} + 1}} $$
(22)
Inorganic species
The non-organic compounds, including the inorganic carbon and nitrogen are included in the presented model. For a detailed description of the equilibrium expression for inorganic carbon, VFA and nitrogen, as well as the charge balance associated with the dissociation of the ions the reader should refer to Bernard et al. [5] and Mairet et al. [9]. Since the present work only considers the methane production rate for comparison with the experimental data the CO2 production and inorganic carbon state variable, as well as the alkalinity have not been reported since they have no mathematical influence on the methane production.
Model summary
In Sects. “One reaction model (1R), Two reaction model (2R), Three reaction model (3R), Kinetics of reaction, Inorganic species”, descriptions have been given for three AD model structures (1R, 2R, and 3R), a range of the kinetic rate equations that can be used to describe the reaction rates and two common inhibition functions. These model components can be combined to make a large number of different AD system models with varying complexity and ability to describe different phenomena. The ability of these models to reproduce the behaviour exhibited in the experimental results is tested to determine their suitability for modelling GW and FW digestion.
Modelling methodology
The equations describing the dynamic variables of each model structure, the reaction kinetics and the inhibition function were implemented in Simulink (Mathworks, MA, USA) and solved numerically by employing a fourth-order Runga-Kutta method using the stiff solver ode15 s with a maximum step size of 0.002 days. Feeding pulses were represented as trapezoids in the dilution rate (d) with a duration of 0.004 days (~6 min) and height such that the integral of the flowrate for each pulse was equal to the volume of substrate added during each loading event as shown in Fig. 1.
The initial condition for the simulations was obtained by a simple parameter estimation performed on a batch incubation of the inoculum. In this method the sum of the concentration of particulate organic matter (S
1) and hydrolytic (X
1) and methanogenic organisms (X
2) was assumed to be the measured VS of the sample (14.4 kg m−3). The methane production from the batch was then used to estimate the initial conditions and this method yielded the following conditions which were used in the semi-continuous simulations; S
1 = 0.17 kg m−3, X
1 = 7.75 kg m−3 and X
2 = 6.48 kg m−3. The initial ammonia concentration of 75 mmol L−1 was based on a measured concentration of 1.28 gNH3 L−1 in the inoculum. As mentioned both in Sect. “Inorganic species”, C and Z had no impact on the model outputs of interest and were therefore not simulated. The descriptions of the green and food wastes are shown for each model in Table 1 including a justification for their selection.
Table 1 Feedstock description in the 1–3 reaction models (*β1 and β2 are part of the parameter estimation method)
Parameter estimation and parameter uncertainty
The parameter estimation technique used the non-linear least square method as supplied with the optimisation toolbox in Matlab (Mathworks, MA, USA). A multi-start strategy was employed where several different initial parameter sets were used to avoid the minimisation algorithm reaching a local minimum [23]. Despite using simplified models, in all cases investigated, except the 1R model, the number of parameters prohibits the estimation of a full parameter set. Therefore, the focus of this paper has been on identifying and assessing the suitability of a model by varying the parameters describing the reaction kinetics and inhibition rather than stoichiometry.
The exclusion of stoichiometric parameters (k
n,
β) from the estimation method can be justified since they should not significantly impact on the nature of the feedstock or process conditions. The exception to this is parameter(s) that expresses the yield of VFA from the degradation of the feedstock (k
1 in the 2R model and β
1 and β
2 in the 3R model) since, for solid wastes, this can be highly variable due to two main factors; the concentration of non-biodegradable substances including water, and the biochemical makeup of the organic material (e.g. lignin, fats, carbohydrates, etc.). Therefore, in the present investigation these parameters were critical to allowing a good fit of the model. Further, it should be noted that previous authors did include these parameters in their identification procedure and therefore this could be seen as a shortfall of these works [1, 6, 8].
In summary, the parameters that were estimated were the biomass to VFA stoichiometric parameters (k
1, β
1, β
2), the kinetic parameters (k
hyd, k
x, k
s, µ
max, λ) and the inhibition parameters (k
i, k
i,vfa, k
i,N). This means that the parameters estimated for each model combination varied between 2 in the simplest case (1R with the first-order kinetics and no inhibition) and 11 in the most complex model (3R with Contois hydrolysis, Moser methanogenesis, and VFA and ammonia inhibition).
Parameter sets for the mechanistic model of AD systems are not generic and are developed for specific cases, makes the estimation of its parameters specific for the case under examination. The standard for the decision on which the model best describes the physical phenomena involves finding the optimal solution of the model parameters based on a cost function. In this case, the cost function is given by Eq. (23), this is simply the sum of the square between the model and the experimental data points, and it is commonly used for parameter estimation studies in this field [13, 23, 24]. Nevertheless, to measure the extent of the deviation of the model results from observed value obtained from experimental investigations, the relative root mean square error (rRMSE) is implemented since this allows comparison of the data obtained from different experiments and it is expressed as a percentage of the time-based mean of the measured methane flow rate (σ
qm,exp).
It should be noted that only the measurements for methane flowrate are used for parameter estimation, rather than including other offline measurements, e.g. VFA. This choice was made because the flowmeter provided continuous online measurement and therefore many thousands of data points for use in parameter estimation whereas offline data only provided a small number of data.
$$ j = \hbox{min} \sum\limits_{i = 1}^{n} {\left( {q_{\text{m,exp}} - q_{\text{m}} } \right)^{2} } $$
(23)
$$ {\text{rRMSE}}\,\left ( \% \right) = 100\frac{{\sqrt {\left( {\frac{j}{n}} \right)} }}{{\sigma_{{q_{{{\text{m}},\exp }} }} }} $$
(24)
The standard errors associated with the parameter estimation technique were calculated as the diagonal elements of the square root of the inverse of the Hessian matrix with respect to the cost function (Eq. 23).