An integrated biochemical and physical model for the composting process

Bioresource Technology 98 (2007) 3278–3293

An integrated biochemical and physical model for the composting process Francina Sole-Mauri a,*, Josep Illa a, Albert Magrı´ Francesc X. Prenafeta-Boldu´ b, Xavier Flotats b b

b

a University of Lleida, Avda Rovira Roure, 191, E-25198 Lleida, Spain GIRO Technological Centre, Rambla Pompeu Fabra, 1, E-08100 Mollet del Valle`s, Barcelona, Spain

Available online 1 September 2006

Abstract A dynamic model for the composting process has been developed, which integrates several biochemical and physical processes. Different microbial populations (mesophilic and thermophilic bacteria, actinomycetes and fungi) have been considered, each specialized in certain types of polymeric substrates (carbohydrates, proteins, lipids, hemicellulose, cellulose and lignin) and their hydrolysis products. Heat and mass transfer between the three phases of the system have been taken into account. The gas phase was considered to be composed by nitrogen, oxygen, carbon dioxide, ammonia and water vapour. Model computer simulations provided results that fitted satisfactory the experimental data. A sensitivity analysis was performed to determine the key parameters of the model. The partition of both the composting mass and the active biomass into different major groups of substrates and specialized microbial populations, as well as the factors affecting the gas–liquid equilibrium, were important for an accurate description of the composting process.  2006 Elsevier Ltd. All rights reserved. Keywords: Modelling; Composting; Sensitivity analysis; Stoichiometry; Kinetics

1. Introduction The use of numerical methods for the simulation of biodegradation processes represents an important tool for the design of waste treatment plants, the establishment of control and operating strategies, and the optimization of process engineering (Vanrolleghem and Keesman, 1996). The mathematical modelling of biodegradation mechanisms involved during aerobic and anaerobic wastewater treatment has experienced an important progress during the last decades, resulting in models that have been recognized as references by the scientific community. That is the case of activated sludge models ASMx (Henze et al., 2000) and anaerobic digestion model ADM1 (Batstone et al., 2002). Despite the fact that a number of models for the composting

*

Corresponding author. E-mail address: [email protected] (F. Sole-Mauri).

0960-8524/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.biortech.2006.07.012

process have also been published and reviewed (Mason and Milke, 2005a,b), a standard generally accepted model has still not been proposed. This could partly be explained by the fact that composting is characterized by a high degree of intricacy related to the interactions of several biochemical and physical factors in a heterogeneous matrix of gas, liquid and solid phases, which fluctuate considerably over time. Consequently, modelling the inherent complexity of the composting process requires the adoption of a number of simplifying assumptions. The first and most elemental models were developed primarily considering physical process, being biochemical processes like biodegradation simplified to a first order kinetics in relation to the substrate. The parameters obtained from this type of models are applicable only under the same conditions in which the kinetic constants have been determined (Bari and Koenig, 2000; Bari et al., 2000; Ekinci et al., 2004; Hamoda et al., 1998; Keener et al., 1993, 1996; Kulcu and Yaldiz, 2004; Mohee et al.,

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1998; Nakasaki et al., 1987, 2000). Such models have been referred as inductive by some authors (Hamelers, 2001). Haug (1993) developed the most complete model within this category, which considered temperature, oxygen gas, moisture, biodegradable volatile solids and free air space (FAS), as parameters affecting the degradation rate. First order kinetics in relation to the substrate, when applied to the overall composting process, presents the fundamental drawback of neglecting biological variables, such as the biomass content and population dynamics, or the effect of substrate limitation on the growth of microorganisms. Furthermore, the interactions (Richard, 1997) between biotic and abiotic factors occurring during composting cannot be described just by using empirically-based first order kinetics. Hence, the applicability of these models is compromised by the need of fitting the kinetic constant value for every kind of waste and operational condition. This parameter adaptation requires a considerable experimental effort in order to monitor all relevant factors affecting the process. The fact that some of these parameters can not easily be measured complicates even further the development, applicability and accuracy of models based on generalized first order kinetics (Hamelers, 2001). In the search for a more universal model, other authors have approached the modelling of composting from a mechanistic (or deductive) point of view, by integrating the basic principles of physics, chemistry and microbiology involved in the composting process. These models initially considered a single population of microorganisms growing on an ‘‘average’’ substrate. The growth rate has commonly been calculated by a Monod kinetics on the content of substrate in combination with other growth factors such as oxygen (Seki, 2000), and moisture (Liang et al., 2004; Stombaugh and Nokes, 1996). The requirement of a nitrogen source has, so far, been neglected, although Liang et al. (2004) included the microbial immobilization of nitrogen. A very important improvement on modelling the biological activity has been introduced by Kaiser (1996). This author divided the microbial biomass into different populations, each with selective capability to degrade major categories of substrates. However, the effect of nitrogen limitation on the biomass development was overlooked in this model. Moreover, an important weakness common to these mechanistic models is that substrates were assumed to be readily available for the microorganisms, being the hydrolysis of particulate matter not treated as a separate process. Hydrolysis has often ben described as the rate-limiting step in several biodegradative processes (Choi and Mathews, 1996; Insel et al., 2002; Veeken and Hamelers, 1999), including the composting of fibres (Tuomela et al., 2000). Despite the importance of hydrolysis, only a few composting models have included this process (Hamelers, 1993; LiwarskaBizukojc et al., 2002; Tremier et al., 2005). Concerning the death and lysis of microorganisms, there is a general agreement in the literature that this process follows a first order kinetics, but only some models recycled the decayed biomass back into the system (Hamelers, 1993).

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Simplifying assumptions have also been applied to the output variables of the composting system, metabolised substrates are ultimately converted into water, carbon dioxide, and ammonia (Finstein, 1980; Haug, 1993), but these end-products have often been excluded. This is especially relevant for the emission of ammonia, which contributes to odour nuisance. Concerning the simulation of a three-phase system, to our knowledge, only the model developed by Hamelers (1993) combines the three phases by considering the particle as the modelling unit. Despite the new insight into composting brought by this model, its practical application is limited by the particle scale focus. If the gas/liquid transfer is not considered, the effect of the incoming air as limiting factor for the growth of microorganisms can be underestimated (Epstein, 1997; Finstein et al., 1987a,b,c,d). This simplification also influences the flow rate and composition of the exhaust gases produced, which are critical for predicting ammonia emissions (Liang et al., 2004; Sole-Mauri et al., 2003, 2005; Sole-Mauri, 2006). The present study is intended to include critical processes, which have so far been considered partly and/or dispersely in previous composting models, into a new nominal model that constitutes a basic structure easily adaptable in order to include future improvements. This structure is based on the matrix notation, as the ASMx (Henze et al., 2000) and ADM1 (Batstone et al., 2002) models. Different populations of microorganisms were regarded, which have specific substrate specificities concerning the soluble compounds arising from the breakdown of the particulate matter. Since microorganisms, are not able to directly assimilate complex polymeric substrates, a first degradation step in the form of hydrolysis has been introduced. In relation to the physical processes, the gas/liquid mass transfer has been considered for the main components of the gas phase, including nitrogen, oxygen, carbon dioxide, ammonia and water vapour. A set of parameter values for the model are presented and the corresponding model predictions are compared with experimental data. Finally, a sensitivity analysis has been carried out to determine the key parameters of the model. These results will facilitate the generation of simplified versions of the model and the orientation of future experiments design for calibrating the model. 2. Methods 2.1. Model theoretical principles A volume unit (V) formed by a gaseous and a liquid phase was considered, in which the solid particles were immersed or covered by a water film (Fig. 1). Both phases were regarded as homogeneous and the particles size was supposed to be small enough so that inner concentration or temperature gradients could be neglected. The gas phase volume (Vg) was assumed to be constant and the airflow occurred through convection, changing its composition

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Fig. 1. The basic structure for the three phase unit, with the energy and mass flows.

due to the liquid–gas transfer ðMrT k Þ. The considered aerobic reactions are exothermic and, consequently, heat is generated (qG) in the solid–liquid phase. As consequence of temperature difference between the volume unit and its surroundings, a conductive–convective heat flux ðqcw Þ occurred. The gas phase is heated by convective heat transfer ðqT c Þ from the solid–liquid phase and, for this reason, uniform and different temperatures for the gas (#) and the solid–liquid phase (T) were assumed. A schematic representation of all the processes included in the model is shown in Fig. 2. The model contains 30 state variables corresponding to particulate (14), soluble (8) and gas (5) components, moisture, solid–liquid phase temperature, and gaseous phase temperature (Table 1).

Six microbial populations, each with a particular substrate specificity and optimum temperature for growth, were considered: mesophilic and thermophilic bacteria (XMB and XTB), mesophilic and thermophilic actinomycetes (XMA and XTA), and mesophilic and thermophilic fungi (XMF and XTF). The composting mass was fractionated into carbohydrates (XC), proteins (XP), lipids (XL), hemicelluloses (XH), celluloses (XCE), lignins (XLG), and inert materials (XI). The products of cellular lysis are considered particulate substrate in the form of inert material and protein, newly available for hydrolysis. Stoichiometric coefficients of the considered biochemical processes were calculated from general chemical formulae (Appendix 1). The biological processes included in the model were the hydrolysis of particulate substrates, and microbial growth, death and lysis (Fig. 2). Hydrolysis of particulate matter was considered to be selectively performed by bacteria (XC, XP and XL), actinomycetes (XH) and fungi (XCE, XLG). Concerning the microbial degradation of the hydrolysed soluble products, bacteria were assumed to grow on the products resulting from the degradation of carbohydrates, proteins and lipids (SC, SP, and SL). Besides these compounds, actinomycetes could also consume the breakdown products of hemicellulose (SH), while fungi could use all the soluble substrates, including the products of lignin degradation (SLG). The substrate degradation consumed dissolved oxygen ðS O2 Þ and produced carbon dioxide ðS CO2 Þ and water, while ammonia was released to the liquid phase, where its concentration

Complex particulate waste XC XCE XP

XL

XH XLG

XI SOLID PHASE

HYDROLYSIS

LIQUID PHASE SC

SP

SL

GAS PHASE

SH SLG

Readily degradable soluble substrate SC

SP

SL

SH SLG

MICROORGANISMS GROWTH

O2 MICROORGANISMS LYSIS Death biomass

O2

LIQUID-GAS MASS TRANSFER XMB XTB XMA XTA XMF XTF

CO2 (l)

CO2

H2O

H2O

NH3 (l)

NH3

Microorganisms

MICROORGANISMS DEATH

Fig. 2. Schematic representation of the considered processes.

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Table 1 Components of the model (state variables) i

Symbol

Units

Description

Generalised formula

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

XC XP XL XH XCE XLG XI XMB XTB XMA XTA XMF XTF XDB SC SP SL SH SLG S O2 S CO2 S NH3 –NH4 IW nO2 nCO2 nNH3 nH2 Ov nN2 # T

kg kg kg kg kg kg kg kg kg kg kg kg kg kg kg kg kg kg kg kg kg kg kg kmol kmol kmol kmol kmol K K

Carbohydrates Proteins Lipids Hemicellulose Cellulose Lignin Inert organic material Mesophilic bacteriaa Thermophilic bacteriaa Mesophilic actinomycetesa Thermophilic actinomycetesa Mesophilic fungia Thermophilic fungia Decayed biomass Carbohydrates and cellulose hydrolysis products Proteins hydrolysis products Lipids hydrolysis products Hemicellulose hydrolysis products Lignin hydrolysis products Dissolved oxygen Dissolved carbon dioxide Total dissolved ammonium Water Oxygen gasb Carbon dioxide gasb Ammonia gasb Water vapourb Nitrogen gasb Gas phase temperature Solid–liquid phase temperature

C6H12O6 C16H24O5N4 C25H45O3 C10H18O9 (C6H12O6)n C20H30O6  C5H7O2N C5H7O2N C5H7O2N C5H7O2N C10H17O6N C10H17O6N C5H7O2N C6H12O6 C16H24O5N4 C25H45O3 C10H18O9 C20H30O6 O2 CO2 NH3 H2O O2 CO2 NH3 H2O N2  

Generalised formulae from Haug (1993) and Henze et al. (2000). a Considered heterotrophs. b Considered ideal gases.

was governed by chemical equilibrium (S NH4 ¼ NH3 þ NHþ 4 ). The biotransformation of a particulate substrate into its soluble monomers was represented by the Contois kinetics. Degradation of the resulting hydrolysis products and ammonia consumption were modelled according to the Monod kinetics. Temperature growth-limiting functions were adopted from Rosso et al. (1993), and the water growth-limiting function was taken from Stombaugh and Nokes (1996). This selection was based on the analysis of temperature functions made by Rosso et al. (1993), which was extended to other temperature functions (data not shown). Death and lysis of microbial biomass was represented by a first order kinetics. Reaction rates of the previously described processes and growth limiting functions are shown in Appendix 2. Mass transfer of oxygen, carbon dioxide, ammonia and water vapour was modelled as part of the physical processes, applying the two-film theory (Tchobanoglous et al., 2003). Mass transfer of dinitrogen gas was not considered to influence significantly any process and, consequently, was neglected in the model. Concentration and partial pressure in both the liquid and gas phase were

assumed to be uniform. In the system ammonium– ammonia, concentration of free ammonia was calculated as a function of the total concentration of the two possible ionisation states, of the pH and of the temperature (Anthonisen et al., 1976). In the system water/water vapour, the mass flow was considered to be proportional to the difference between the saturated water vapour pressure at the temperature of the gas phase, and the water vapour pressure existing in that phase. The mass balance equation (Eq. (1)) for every particulate and soluble component i (i = 1, . . ., 23) is: dmi ¼ Mri  MrTi ; dt

ð1Þ

where M is the total mass of compost, ri is the reaction rate and rTi the rate of liquid to gas mass transfer for water vapour and dissolved gases (Appendix 2). For the components in the gas phase (Eq. (2)) takes the form: dnk ¼ nk0  nk1 þ MrT k ; dt

ð2Þ

where nk0, nk1 (k = 1, . . ., 5, corresponding to i = 24, . . ., 28) are the in and out molar convective flows, respectively.

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Constant inflows (Eq. (3)) are determined by the volumetric air flow and standard air composition, while outflows are computed assuming ideal gas behaviour and constant total pressure P, ! 5 R# X 1 d# nk1 ¼ nk ðnk0 þ MrT k Þ þ ; ð3Þ PV g k¼1 # dt where R is the perfect gas constant and nk the molar mass of the gas k (state variable). The energy balance for the gas phase and the solid–liquid phase provides the differential equations to be applied (see Appendix 3) for both temperatures. Space gradients were not considered here and the extension of the model to the space will be the topic of further works. Hence, experiments to evaluate the model were performed in a completely mixed medium, as described in Section 2.3. 2.2. Model inputs Four different categories of data are required in the model: initial values (substrate composition), stoichiometric coefficients, kinetic and operation parameters. For the evaluation experiment, the initial substrate composition was defined as a mixture of different wastes previously characterized in the laboratory (Table 2). The amount of dissolved gases was assumed to be initially in equilibrium with normal air composition. Stoichiometric coefficients were calculated from the reactions shown in Appendix 1. The accuracy of these general formulae was verified against organic matter, carbon and nitrogen fractions from three different measured mixtures of compostable wastes (Table 3). Experimentally, samples were fractionated into proteins, lipids and fibres, and these measured values were used to estimate the amount of C and N, using the formulae implemented in the model.

Default kinetic parameter values used in the model are listed in Table 4. The values have been obtained after a literature review, choosing values with physical and biological sense in the framework of the developed model. Previous estimates on the hydrolysis constant for the whole process ranged from 0.00324 to 0.1798 h1 (Hamelers, 1993; Laspidou and Rittmann, 2001; Tremier et al., 2005; Veeken and Hamelers, 1999). Hydrolysis constants were selected in the range of the reported values for the whole process considering the different biodegradabilities of each fraction. Available data on the maximum specific growth rate for aerobic heterotrophic microorganisms growing on single substrates are comprised between 0.055 and 0.2836 h1 (Hamelers, 1993; Liwarska-Bizukojc et al., 2002; Stombaugh and Nokes, 1996; Tremier et al., 2005). Liang et al. (2004) considered different substrates but a single population, with maximum specific growth rate in the range 0.025–0.60 h1. Under competitive conditions, when different populations grow on the same substrate, described growth rates varied between 0.0222 and 0.20 h1 for bacteria, 0.0214–0.10 h1 for actinomycetes and 0.05–0.25 h1 for fungi (Agamuthu et al., 2000; Kaiser, 1996). As there is no published data on the maximum specific growth rate for different populations on different substrates, kinetic values used in the model were derived from those reported for different populations and one substrate (Table 4). A general yield coefficient (Y) of 0.35 gmicroorg gsubs1 was taken from the literature and assigned to all microorganisms (Stombaugh and Nokes, 1996). Saturation constant values used for substrate and oxygen were previously described by Stombaugh and Nokes (1996). Saturation constant for ammonium has not been reported in the literature for the composting process, as ammonium was not considered as limiting for the growth of microorganisms. For the preliminary test of the model, an approximate value is taken, similar to Henze et al. (2000). 2.3. Experimental set-up

Table 2 Characterization of the waste and mixture used to check the model

Paper pulp Poultry manure Fruit pulp Mixture 1

TS

C/N

51.2 81.3 16.5

>1000 6.2 42.6 19.83

Theoretical stoichiometric coefficients were verified against samples taken from a full-scale composting plant (Compost Segria`, Lleida, Spain). Samples were taken from three different windrows composed by the mixtures described in Table 5. Samples of 4 kg each were taken at three different points of every windrow, at 25 and 100 cm depth. Carbon and nitrogen were analysed by elemental

Units: (%) weight.

Table 3 Measurements and model estimation of C and N for three different mixtures of wastes Measurements

1 2 3

Model estimation

Deviation

Protein

Lipid

Hemicel

Lignocel

C

N

C

N

C

N

86.875 60.625 28.333

103.533 25.700 15.567

100.467 96.300 125.667

232.867 303.000 281.533

329.53 332.03 284.77

13.90 9.70 4.53

324.08 313.39 266.45

14.33 9.88 4.75

1.7 5.7 6.4

3.1 1.9 4.7

Estimations and deviations between measurements expressed in %, the rest in mg g1 dry basis.

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Table 4 Default values for coefficients and parameters used in the sensitivity analysis Symbol k hX c ;1 k hX p ;2 k hL;X MB ; k hL;X TB k hX c ;4 k hX p ;5 khH khCE, khLG khS lMB lTB lMA lTA lMF, lTF kS k O2 k NH4 bMB bTB bMA bTA bMF bTF kdec klO2 klCO2 klNH3 klH2 Ov

Name Hydrolysis constant of carbohydrates by XMB Hydrolysis constant of proteins by XMB Hydrolysis constant of lipids by XMB and XTB Hydrolysis constant of carbohydrates by XTB Hydrolysis constant of proteins by XTB Hydrolysis constant for hemicellulose by XMA and XTA Hydrolysis constant for cellulose and lignin by XMF and XTF Saturation coefficient for all Contois kinetics Specific growth rate for mesophilic bacteria on SC, SP and SL Specific growth rate for thermophilic bacteria on SC, SP and SL Specific growth rate for mesophilic actinomycetes on SC, SP, SL and SH Specific growth rate for thermophilic actinomycetes on SC, SP, SL and SH Specific growth rate for fungi on substrate SC, SP, SL, SH and SLG Substrate saturation constant for Monod kinetics Oxygen saturation constant Ammonium saturation constant Death constant for mesophilic bacteria Death constant for thermophilic bacteria Death constant for mesophilic actinomycetes Death constant for thermophilic actinomycetes Death constant for mesophilic fungi Death constant for thermophilic fungi Microorganisms decomposition constant (lysis) Oxygen liquid–gas mass transfer constant CO2 liquid–gas mass transfer constant Ammonia liquid–gas mass transfer constant Water vapor liquid–gas mass transfer constant

Table 5 Mixtures used to check accuracy of theoretical formulae of substrates (in 103 kg)

Fruit pulp Paper pulp Cattle manure Municipal sewage sludge Agro-industrial sewage sludge Poultry manure Poultry egg wastes

Mixture 1

Mixture 2

Mixture 3

46.8 15.9 33.6 20.8

36.4 53.0 16.8 20.8 6 3.6

36.4 79.5 15.6 9.1 10.6

analyser equipment, proteins content were derived from organic nitrogen, considering a conversion factor of 6.25, lipids were measured by Soxhlet method (APHA et al., 1995) and fibres by Van Soest method (Van Soest et al., 1991). Composting experiments were performed in a closed parallelly-piped container (0.60 long · 0.66 width · 0.40 height m) with semi-cylindrical bottom (0.66 diameter · 0.60 width m). The SA:V ratio was 9.0 and the total capacity was 0.26 m3, although the reactor was operated with a load of about 0.15 m3. This container was constructed on wood coated internally with stainless steel plates. Between steel and wood an isolation layer of polyurethane foam (1 cm) was set. An external cover of polyurethane (5 cm) provided additional thermal isolation. A horizontal rotating axis with blades working at 0.3 rpm on a 15 min intermittent

Units

Value

1

0.04 0.02 0.01 0.02 0.04 0.009 0.007 1E4 0.20 0.18 0.10 0.12 0.1 62.0 0.007 0.05 0.03 0.02 0.010 0.015 0.010 0.010 0.0025 1E7 1E6 1E7 1E7

h h1 h1 h1 h1 h1 h1 g g1 h1 h1 h1 h1 h1 g m3 g m3 g m3 h1 h1 h1 h1 h1 h1 h1 kg h1 Pa1 kg TM1 kg h1 Pa1 kg TM1 kg h1 Pa1 kg TM1 kg h1 Pa1 kg TM1

schedule, placed at the bottom of the geometric axis, ensured the complete mixing of the composting mass. Air through the mass was blown intermittently (15 min ON/ 105 min OFF) through small openings at the bottom of the container. The inflow gas volume was measured by a rotating vane sensor FVA915S140 (Ahlborn, Germany), connected to a datalogger ALMEMO 2590-9 (Ahlborn, Germany). Two thermocouples were used to measure the internal and external air temperatures. A lean fixed iron bar held four additional thermocouples at different levels inside the composting mass. All six thermocouples were K type (Ahlborn, Germany) and the temperature was registered every 5 min with a datalogger acquisition unit HP-34970A (Hewlett-Packard, USA). Oxygen levels in the inner atmosphere of the reactor were measured by an electrochemical sensor FY9600-O2 (Ahlborn, Germany) and recorded every 5 min by the ALMEMO datalogger. The temperature and relative moisture inside the reactor could exceed the working range of the oxygen sensor. For this reason, it was placed in a small hermetic plastic box outside the reactor through which a tiny compressor (5 dm3 min1) continuously recirculated the air in a closed loop which included a condenser coil and a water trap. The whole composting installation was placed on a weighing platform (Gram Precision, Spain) with a 0–10 V output signal recorded every 5 min by the HP datalogger. During the 15 days of the experiment, total solids (TS), carbon and nitrogen were measured daily. TS were analysed by drying sample at 105 C during 24 C, and an

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3). Coefficients, parameters and initial values were taken from experimental and literature data (Table 4). A good agreement was found between predicted and experimental results on the evolution of temperature, oxygen in the exhaust gas, carbon and nitrogen content, and ammonium (Fig. 3(a)–(d)). A lag phase of approximately 48 h was observed before the rise of temperature into the mesophilic range (Fig. 3(a)). After the thermophilic peak, the temperature gradually decreased and the experiment was stopped after 15 days of operation. The day–night temperature fluctuations were responsible for the minor 24 h period oscillations of the measured temperature in comparison with the simulated values. The deviation between predicted and modelled data between 72 h and 96 h is due to the population succession of mesophilic–thermophilic microorganisms, predicted by the model, although the little decrease in temperature observed experimentally during this transition was simulated as constant in time. At the end of the experiment, the observed temperature decreased faster than predicted, attributable to lower observed thermal inert when compared to the predicted values. The oxygen evolution shows two main negative peaks due to the exponential growth of mesophilic and

elemental analyser (Leco TrusPec CN) was used for carbon and nitrogen determinations. The initial mixture was made of paper pulp (29.15 kg), cattle manure (11.8 kg) and fruit pulp (42.6 kg). The model was implemented in MATLAB (Mathworks, USA). The numerical method used to solve the set of ordinary differential equations was the ODE15s routine from MATLAB software package (Mathworks, USA). Simulations were performed on a Pentium IV computer, and 6 s were taken to simulate 15 days. 3. Results 3.1. Model evaluation

75

25

60

20

45

15

O 2 [%]

30

10

15

5 Experimental

Experimental

Model

350

35

0.30

300

30

0.25

250

25

200

20

150

15

100

10

36 0

33 6

28 8

31 2

26 4

24 0

21 6

Model 33 6

31 2

24 0

21 6

19 2

16 8

14 4

96

12 0

48

72

36 0

Time [h]

28 8

Experimental

38 4

0.00 36 0

N-NH4+ [kg]

0.05

0 38 4

33 6

31 2

28 8

26 4

21 6

24 0

19 2

16 8

14 4

12 0

96

72

48

0

5 N experimental

0 24

0.10

24

N model

0.15

0

C experimental

0.20

26 4

50 C model

19 2

16 8

14 4

96

12 0

72

24

0

36 0

33 6

28 8

31 2

26 4

24 0

19 2

16 8

14 4

96

12 0

72

48

24

21 6

Time [h]

N [g/kg dm]

C [g/kg dm]

Time [h]

Time [h]

0.3 XMB

0.016

XTB

XMA

0.014

0.25 Microorganisms [kg]

Microorganisms [kg]

Model

0

0

0

48

T [°C]

A mixture of paper pulp, cattle manure and fruit pulp was used as a model substrate. The main characteristics of each waste and of the mixture are summarized in Table 2. The accuracy of the selected generalized stoichiometric formulae was previously checked against the chemical analyses of mixtures from heterogeneous organic wastes. Deviation between predicted and measured amounts of carbon and nitrogen ranged between 1% and 6% (Table

0.2 0.15 0.1 0.05

XTA

XTF

0.012 0.01 0.008 0.006 0.004 0.002

0

0 0

24

48

72

96 120 144 168 192 216 240 264 288 312 336 360

0

24

48

72

Time [h]

Fig. 3. Model predictions and experimental data.

96

120 144 168 192 216 240 264 288 312 336 360 Time [h]

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to appear and grew until the increasingly high temperature became inhibitory. Subsequently, thermophilic bacteria became the dominant population. Though in a minor proportion, thermophilic actinomycetes and fungi also developed in this phase. Consistently with the high microbial activity present during the thermophilic phase, the simulated concentration of oxygen in the exhaust gases was significantly depleted (Fig. 3(b)). Different simulations were performed with the model in order to study the influence of moisture and the ratio C/N onto the temperature, the total microbial populations and the composition of the exhaust gases (Fig. 4). Two moisture values (50% and 65%) and three C/N ratios (30, 40, and 50) were tested, while the air inflow rate was kept intermittently (15 min ON/105 min OFF) at 1.5 m3 m3 h1. At C/N value of 30, biomass development was comparatively bigger and the thermophilic phase was longer. At a C/N ratio of 40, microbial growth and temperature maintained a sustained growth throughout the whole simulation. This situation was again reversed at the highest C/N value of 50, when nitrogen became the growth limiting factor and the variables related to microbial activity (biomass, temperature, and ammonia) decreased upon nitrogen exhaustion.

thermophilic microorganisms (Fig. 3(b)). Predicted percentages of oxygen at the exhaust gas follow qualitatively experimental results. The deviation found is explained by the opening of the reactor during sampling periods, which supposed a higher apport more oxygen to the composting mass than the simulated values (Fig. 3(b)), and it could be also attributable to the variation of the liquid–gas transfer rate as the material is dried. The content of carbon and nitrogen progressively decreased, indicating the progressive oxidation and volatilization of the organic matter, which was consistent with the ammonia found in outflow gases (Fig. 3(c)). Ammonium increased throughout the experiment, being the deviation found between predicted and experimental data explained by the consideration of a constant pH in the model (Fig. 3(d)). The biomass development of the different microbial populations considered in the model is shown in Fig. 3(e) and (f). The evolution of the total biomass followed a similar pattern to that seen with the temperature (Fig. 3(a)), with its maximum coinciding with the thermophilic range of temperatures. Biomass from individual populations, however, peaked at different times forming a microbial succession. Mesophilic bacteria were the first microorganisms

70

60

60

50

50

Temperature [°C]

Temperature [°C]

Moisture=50% 70

40 30 20

30 20 10

0

0 0

24

48

72

96

120

144 168 192 216

240 264 288 312

336 360

0.7

0.7

0.6

0.6

0.5

Microorganisms [%]

Microorganisms [%]

Moisture=65%

40

10

0.4 0.3 0.2 0.1

0

24

48

72

96

120

144

168

192

216

240

264

288

312

336

360

0

24

48

72

96

120

144

168

192

216

240

264

288

312

336

360

0

24

48

72

96

120

144

168

192

216

240

264

288

312

336

360

0.5 0.4 0.3 0.2 0.1

0

0 0

24

48

72

96

120 144 168 192 216 240 264 288 312 336 360

0.5

0.5

0.4

0.4

0.3

0.3

NH3 [%]

NH3 [%]

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0.2

0.2 0.1

0.1

0

0 0

24

48

72

96

120 144 168 192 216 240 264 288 312 336 360 Time [h]

C/N=30

C/N=40

C/N=50

Time [h] C/N=30

Fig. 4. Simulations at different moisture contents and C/N ratio.

C/N=40

C/N=50

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60

The influence of different airflow rates (0.06, 0.12, 0.24 and 0.48 m3 m3 h1) on the composting process was also examined in a second set of simulations (Fig. 5), were initial C/N ratio was set at 20 and moisture at 60%. A low aeration rate resulted in oxygen limitation for the microbial activity. The slower growth rate resulted in a delayed start of the thermophilic phase, which was then maintained for a longer time (Fig. 5(a)). General trend of C/N ratio is described to low down, but as consequence of a low microbial activity and the temperature effect, more ammonia was volatilized than carbon degraded (Fig. 5(c)), and C/N ratio consequently increased over time (Fig. 5(b)). NH3 is present in greater amount in the exhaust gas as airflow rate is lower. This is due to a longer thermophilic period and to a greater reduction of moisture content (Fig. 5(c)).

40

3.2. Sensitivity analysis

20

A sensitivity analysis was performed to assess the relative importance of individual parameters on the overall model performance. Each examined parameter (Table 4) was successively modified to the ±25% and ±75% of its default value. The effect of these individual changes was investigated on seventeen objective functions: particulate total mass degradation rate (PTDR); maximum total microorganisms mass (MTMM); particulate degradation rate for carbohydrates (XCDR), proteins (XPDR), lipids (XLDR), hemicelluloses (XHDR), celluloses (XCEDR), and lignins (XLGDR); maximum mass for bacteria, actinomycetes, and fungi, mesophilic and thermophilic, (MBMM, TBMM, MAMM, TAMM, MFMM, TFMM); moisture elimination rate (MER); and the cumulative output of carbon dioxide and ammonia (CCO2, CNH3). Results from this analysis are summarized in the Electronic appendix. Interactions of the high number of parameters from the model make interpretations of the results not always simple. Changes in the hydrolysis constant (kh) of each substrate affect the degradation rate, but also the moisture elimination rate and the cumulative gases output. Hydrolysis constant for hemicellulose, cellulose and lignin had the highest sensitivity. The saturation constant of the Contois kinetics showed response with an increase on its value, but not with a decrease. In the microorganisms growth process, increasing the growth rate constant also increases the population density severely, while competition against the other populations decreases the total mass of microorganisms and, consequently, also total degradation. In agreement with the lower amount of microorganisms, moisture elimination rate and cumulative carbon dioxide and ammonia also decrease. Conversely, an increase in b results in an important reduction of the affected population biomass and a slightly opposite effect in other populations, being total maximum biomass lower than in the standard conditions. The effect of variation of the yield value and on several objective functions is depicted in Fig. 6. Higher values of substrate (ks), or oxygen ðk O2 Þ saturation constant, tended

The previously described patterns were similar for the two tested moisture values. At the water content of 65% the overall process was slowed down and more time was needed for the completion of the thermophilic phase volume (Fig. 4). The predicted biomass grown under these conditions was lower. Concerning the predicted evolution of ammonium and ammonia, a relatively low volatilization of ammonia occurred at the highest C/N ratio and moisture values, as more ammonia was retained by the higher amounts of biomass and liquid phase.

T (ºC)

80

0 0

24

48

72

96 120 144 168 192 216 240 264 288 312 336 360

0

24

48

72

96 120 144 168 192 216 240 264 288 312 336 360

0

24

48

72

96 120 144 168 192 216 240 264 288 312 336 360

0

24

48

72

96

C/N

30

20

10

310

C (mg/ g ms)

300

290

280

270

NH3 (‰)

120

80

40

0 120 144 168 192 216 240 264 288 312 336 360 Time (h) Q 0.06

Q 0.12

Q 0.24

Q 0.48

Fig. 5. Simulations at different aeration flow rates (m3 m3 h1).

F. Sole-Mauri et al. / Bioresource Technology 98 (2007) 3278–3293 150 100 50 0 -50 -100 -75

-25 PTDR

25

75

XPDR

XLDR

XHDR

XCEDR

-25 MTMM

MBMM

25 TBMM

TAMM

XLG

300

200

100

0

-100 -75

75

25

0

-25

-50 -75

-25

25 MER

CCO2

75

CCNH3

Fig. 6. Sensitivity analysis. Influence of the yield coefficient value on the objective functions response (tested variations: ±25% and ±75%).

to decrease all the objective functions with lower sensitivities. A summary of the results is shown in Table 6. 4. Discussion Modelling the complex interactions between relevant physical and biochemical processes during composting represents a considerable challenge. A compromise has to be found between the manageability of the resulting algorithms and their capacity to provide realistic predictions. This paper proposes a new model aimed to improve the limitations found in earlier works. Big differences can be found on the assumptions and the level of complexity implemented in previous models. Despite the fact that composting essentially results from the microbiological activity, some models completely disregarded biological aspects and treated the overall process as a first order kinetics respect to the substrate (Bari and Koenig, 2000; Bari et al., 2000; Ekinci et al., 2004; Hamoda et al., 1998; Keener et al., 1993, 1996; Kulcu and Yaldiz, 2004; Mohee et al., 1998; Nakasaki et al., 1987, 2000; Ndegwa et al., 2000; Robinzon et al., 1999; Smith and Eilers, 1980; Van Lier et al., 1994). Other models included biological aspects but simplified the whole process by considering it the result of just one

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substrate being readily available for one single type of microorganisms. Some authors in this group introduced a hydrolytic step, modelled as Contois (Tremier et al., 2005) or first order kinetics (Hamelers, 1993), which is in fact a particular case of the previous kinetics. This ‘‘single organism and substrate’’ assumption might simplify excessively the composting process, especially if one considers that the succession of very diverse microbial populations during composting was already described in Poincelot (1975) and has been confirmed by quantitative analytical techniques (Klamer and Baath, 1998). It was not until Kaiser (1996) that the active biomass was divided into different populations, each specialized in certain categories of substrates from the composting mixture. However, in this model, the hydrolysis was overlooked being all substrates considered to be readily available for microorganisms, and nitrogen forms were not considered, so prediction of the evolution of these forms was not possible. In an innovative approach, the present model combines the fractionation of the compostable wastes into different substrates, which are specific to particular groups of microorganisms, and the introduction of hydrolysis for each polymeric substrate modelled according to Contois kinetics. As in other works, biomass growth was modelled by multiple Monod kinetics for the organic substrate (Hamelers, 1993; Liang et al., 2004; Stombaugh and Nokes, 1996; Kaiser, 1996; Seki, 2000), and oxygen (Hamelers, 1993). Liang et al. (2004) included the immobilization of nitrogen due to microbial growth, as a function of the concentration of ammonium–N and the microbial C/N ratio, considering nitrogen to be degraded at the same rate than labile forms of carbon. Ammonium or aminoacids uptake and the limiting effect of nitrogen shortage have, to our knowledge, never been considered before. In the model here presented, growth limitation due to inadequate amounts of carbon or/and nitrogen source is implemented by including a Monod term for these elements in the general growth kinetic expressions. Lysis of microorganisms has commonly been represented by first order kinetics (Haug, 1993; Liang et al., 2004; Seki, 2000; Stombaugh and Nokes, 1996; Tremier et al., 2005) but the incorporation of this phenomenon into the modelling of the composting process has generally been ignored. Only Hamelers (1993) considered decayed microorganisms as polymeric substrate, newly available for hydrolysis. A similar criterion was taken in the proposed model but the death biomass was first converted to particulate substrates (proteins) by following a first order degrad-ation rate. The consideration of different qualitative and quantitative fractions of organic matter into the composting mixture facilitates the understanding of important changes during the composting process, i.e., variations in the availability of easily biodegradable matter, such as lipids or proteins, have a greater effect on the process than the non-easily degradable matter as fibres. Labile forms of organic matter are responsible for a fast growth of bacteria which is related to the intensity of thermophilic peak. In the last steps of degradation process, the

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Table 6 Qualitative results of the sensitivity analysis

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biodegradation of recalcitrant organic matter determines the stability of the final product. Consequently, labile and recalcitrant forms of organic matter, and the microbial populations responsible of their biodegradation, have to be taken into account in order to correctly predict biodegradation of the non-easily degradable organic compounds. The principles of mass and energy conservation represented another cornerstone of the model, and mass balances for carbon, hydrogen, oxygen, and nitrogen have been closed. This required the definition of the general stoichiometry for all the reactions included in the model. Two general approaches have previously been used for the determination of stoichiometric coefficients: the first is based on experimental measurements, as is the case of ASMx (Henze et al., 2000) and ADM1 (Batstone et al., 2002). The main limitation of this method is that empirically stabilised stoichiometric coefficients values may differ significantly for every type of waste. In these models mass is closed in terms of COD (chemical oxygen demand) and nitrogen among others, but this is not the case for carbon or hydrogen since they are not stabilised as state variables. Therefore, the empiricism of the stoichiometric coefficients reduces their universality conditioning their applicability. In order to find a more generalized composting model, a second approach has been taken, which considers generalised chemical formulae for every component. The resulting stoichiometric coefficients have a broader application and mass balances can be accurately closed for all elements. In this case, the assumption that a general formula describes a whole range of heterogeneous compound (i.e., lipids, proteins or fibres) could be questionable. In this work, the validity of this approach has been tested for three different mixtures sampled from a full-scale composting plant. Simulations generated with the new model fitted the experimental measurements on the time-course temperature, nitrogen and C/N ratio evolution (Fig. 3). Temperature was highly correlated to biomass activity and, hence, is a good indicator of the performance of the overall process. Furthermore, well known patterns from composting process which, like the microbial community dynamics, are sometimes difficult to follow experimentally were depicted in the simulated results. Evolution of biomass from different phylogenetic microbial groups corresponded to recent experimental evidence based on phospholipids fatty acid analysis (Klamer and Baath, 1998). Compared to other experimental studies, the model also provided realistic predictions on the evolution of complex waste composition (Madejo´n et al., 2001) and the exhaust gases. Simulations under non-optimum conditions, like high or low amounts of N, moisture or different flow rates, consequently resulted, as expected, in a relatively lower maximum temperature, biomass content, and duration of the thermophilic phase. The sensitivity analysis demonstrated that the microbial growth rate greatly influenced the evolution of composting. If substrate, oxygen or ammonium were present in adequate amounts, a low sensitivity was found for kS, k O2 or k NH4 , limiting the relative importance of Monod terms in the

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growth of microorganisms under standard (near to optimum) operating conditions. These results are consistent with those found by Stombaugh and Nokes (1996) or Hamelers (2001), and contribute to the understanding the interactions between the different substrates and microbiota, and their relative importance to the composting process. The objective functions showed to be sensitive to increases in the hydrolysis saturation constant but not to decreases. Two possible explanations are suggested to justify this asymmetric behaviour. If the saturation constant for hydrolysis is too low, the effect on objective function is only noticeable when it reaches a value which is high enough but, if this value is appropriate, the effect of this constant is only detected when it takes a too-large value. In this case hydrolysis could be modelled by first order kinetics. More experimental data about the hydrolysis process are needed in order to determine whether the simple first order is good enough to describe adequately the process, or the effect of microorganisms has to be taken into account through Contois kinetics, as suggested by Vavilin et al. (1996) for anaerobic digestion. Applicability of first order kinetics to simulate all the biochemical processes involved in the degradation has some limitations. For instance, Keener et al. (1992) revealed that first order degradation constant had to be updated after approximately three days when using data from chicken manure, or in a longer period of time for yard waste (Marugg et al., 1993). The need to implement different kinetic processes to avoid the limitations described using first order kinetic processes of degradation has also been demonstrated. The results from the sensitivity analysis are the basis for the future development of a simplified model, with a lower number of parameters, allowing calibration and validation processes. 5. Conclusions A novel mathematical model for the composting process has been developed which included most of the relevant biochemical reactions known to occur at the system interphases. Solid particulate substrate was divided in six groups, with different solubilities, and six different microorganism populations with different affinities for the different solubilized organic matter fractions were implemented in the model. Five components in the gas phase (nitrogen, oxygen, carbon dioxide, ammonia and water vapour) were considered with their corresponding liquid–gas mass transfer rates. Finally, solid–liquid and gas phase were characterized by different temperatures. The inclusion in the model of theoretical rather than empirical stoichiometric coefficients, assures a more universal application than previous works. Preliminary evaluation of the model was in general agreement with laboratory results and literature data. A sensitivity analysis revealed the parameters that have the greatest influence on the model, orientating the development of a simplified model and the execution of experiments designed for first calibrate and then validate the model. In summary, the model represents a valuable tool that will contribute to

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the understanding of the complex biological and physicochemical interactions of composting. Further research is directed to calibrate and validate the model, and to extend its applicability into a multidimensional space. Acknowledgements The authors thank Anna Picas and Sofia Forno´s for their help with the experimental works. Financial support from the Spanish Ministry of Education and Science (Project: REN2000-0049-P4-0) and the Spanish Ministry of Environment (Project: 111/2004/3) is gratefully acknowledged. Appendix 1. Equations used to calculate stoichiometric coefficients Growth of bacteria or actinomycetes on SC: C6 H12 O6 þ ð6  5aÞO2 þ aNH3 ! aC5 H7 O2 N þ ð6  5aÞCO2 þ ð6  2aÞH2 O Growth of bacteria or actinomycetes on SP: C16 H24 O5 N4 þ ð33=2  5bÞO2 ! bC5 H7 O2 N þ ð16  5bÞCO2 þ ð6  2bÞH2 O þ ð4  bÞNH3 Growth of bacteria or actinomycetes on SL: C25 H45 O3 þ ð134=4  5cÞO2 þ cNH3 ! cC5 H7 O2 N þ ð25  5cÞCO2 þ ð45=2  2cÞH2 O Growth of actinomycetes on SH: C10 H18 O9 þ ð10  5dÞO2 þ dNH3 ! dC5 H7 O2 N þ ð10  5dÞCO2 þ ð9  2dÞH2 O Growth of fungi on SC: C6 H12 O6 þ ð6  21=2eÞO2 þ eNH3 ! eC10 H17 O6 N þ ð6  10eÞCO2 þ ð6  7eÞH2 O Growth of fungi on SP: C16 H24 O5 N4 þ ð33=2  21=2f ÞO2 ! f C10 H17 O6 N þ ð16  10f ÞCO2 þ ð6  7f ÞH2 O þ ð4  f ÞNH3 Growth of fungi on SL: C25 H45 O3 þ ð139=4  21=2gÞO2 þ gNH3 ! gC10 H17 O6 N þ ð25  10gÞCO2 þ ð45=2  7gÞH2 O Growth of fungi on SH: C10 H18 O9 þ ð10  21=2hÞO2 þ hNH3 ! hC10 H17 O6 N þ ð10  10hÞCO2 þ ð9  7hÞH2 O

Growth of fungi on SLG: C20 H30 O6 þ ð49=2  21=2iÞO2 þ iNH3 ! iC10 H17 O6 N þ ð20  10iÞCO2 þ ð15  7iÞH2 O MW S , being YX/S the biowhere a; b; c; d; e; f ; g; h; i ¼ Y X=S  MW X mass yield (mass biomass produced per mass of substrate consumed). MWS and MWX are the molecular weights of substrate S and microorganism X, respectively.

Appendix 2. Process reaction rates (qj). Units: kgi h1 kgTM Process

Reaction rate

Hydrolysis (Processes j = 1, . . ., 12) Xi MB Carbohydrates, proteins and qj ¼ k hij khS XXMB þX i TM lipids mesophilic hydrolysis i = XC, j = 1; i = XP, j = 2; i = XL, j = 3 Carbohydrates, proteins and lipids thermophilic hydrolysis i = XC, j = 4, i = XP, j = 5, i = XL, j = 6

Xi TB qj ¼ k hij khS XXTB þX i TM

Hemicellulose thermophilic hydrolysis

XH q7 ¼ k hH khS XXTATAþX H TM

Cellulose and lignin thermophilic hydrolysis i = XCE, j = 8; i = XLG, j = 9

Xi TF qj ¼ k hi khS XXTF þX i TM

Hemicellulose mesophilic hydrolysis

XH MA q10 ¼ k hH khS XXMA þX H TM

Cellulose and lignin mesophilic hydrolysis i = XCE, j = 11; i = XLG, j = 12

Xi MF qj ¼ k hi khS XXMF þX i TM

Growth of microorganisms (Processes j = 13, . . ., 36) Uptake SC, SP, and SL by XMB qj = lMB Æ fT1 Æ fO2 i = SC, j = 13; i = SP, Æ fIW Æ fNH4 Æ fSi j = 14; i = SL, j = 15 Æ faB Æ XMB/TM Uptake SC, SP, and SL by XTB i = SC, j = 16; i = SP, j = 17; i = SL, j = 18

qj = lTB Æ fT2 Æ fO2 Æ fIW Æ fNH4 Æ fSi Æ faB Æ XTB/TM

Uptake SC, SP, SL and SH by XMA i = SC, j = 19; i = SP, j = 20; i = SL, j = 21; i = SH, j = 22

qj = lMA Æ fT1 Æ fO2 Æ fIW Æ fNH4 Æ fSi Æ faA Æ XMA/TM

Uptake SC, SP, SL and SH by XTA i = SC, j = 23; i = SP, j = 24; i = SL, j = 25; i = SH, j = 26

qj = lTA Æ fT2 Æ fO2 Æ fIW Æ fNH4 Æ fSi Æ faA Æ XTA/TM

F. Sole-Mauri et al. / Bioresource Technology 98 (2007) 3278–3293

Appendix 2 (continued)

Appendix 3. Energy balance

Process

Reaction rate

Uptake SC, SP, SL, SH and SLG by XMF i = 27, j = SC; i = 28, j = SP; i = 29, j = SL; i = 30, j = SH; i = 31, j = SLG

qj = lMF Æ fT1 Æ fO2 Æ fIW Æ fNH4 Æ fSi Æ faF Æ XMF/TM

Uptake SC, SP, SL, SH and SLG by XTF i = 32, j = SC; i = 33, j = SP; i = 34, j = SL; i = 35, j = SH; i = 36, j = SLG

qj = lTF Æ fT2 Æ fO2 Æ fIW Æ fNH4 Æ fSi Æ faF Æ XTF/TM

Lysis of microorganisms (Processes j = 37, . . ., 43) Death of microorganisms qj = bi Æ Xi/TM i = XMB, j = 37; i = XTB, j = 38; i = XMA, j = 39; i = XTA, j = 40; i = XMF, j = 41; i = XTF, j = 42 Microorganisms lysis q43 = kdec Æ XDB/TM Liquid–gas transfer (Processes j = 44, . . ., 47) Liquid–gas transfer qj = kli(Hei Æ xi i = O2, j = 44; i = CO2,  ni Æ R Æ #/Vg) j = 45; i = NH3, j = 46 Water evaporation– condensation

q47 = klH2 Ov (Psat(#)  nH2 Ov R Æ #/Vg)

TM (kg): total mass; Hei (Pa): Henry’s constant for dissolved gases; xi(): Molar fraction of gas i in solution; R (J kmol1 K1): ideal gas constant; Vg (m3): total volume of the gas phase.

Growth limiting functions S

i Substrate fSi ¼ kS þS 

Substrate availability i faB ¼ S CSþS ; L

i

Ammonia–ammonium fNH4 ¼

faB = 1 if i = Sp faA ¼ S C þSSLi þS H ; faA = 1 if i = Sp Si ; faF ¼ S C þS L þS H þS LG

; fNH4 = 1 if i = Sp

faF = 1 if i = Sp

Dissolved oxygen fO2 ¼ kO S NH

4

k NH4 þS NH

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S O 2

2

þS O

2

4

Moisture fIW ¼ 0 if m 6 m2 a fIW ¼ ðm  m2 Þ=ðm3  m2 Þ if m2 < m 6 m3 a fIW = 1 if m3 < m Temperature

The homogeneous gas phase is entered by the main gas stream composed of each molar gas flow nk0 (kmol h1) (k = 1, . . ., 5) at temperature #0 (K) and by the gas transfer rate MrT k (kmol h1) at the liquid phase temperature T (K). Each molar gas flow nk1 leaves it at the gas temperature #. Molar enthalpies of the gases hk (J kmol1) are assumed to be zero at the reference temperature (#ref = 273.15 K) and to have a linear dependence on temperature, being bk (J kmol1 K1) its slope. The convective heat transfer between solid–liquid and gas phase is assumed to be proportional to the total mass of compost M in the volume unit; and to the phases temperature difference qT c [J h1] = hSM(T  #). With such considerations and being nk (kmol) the state variable of molar mass of gas k in the gas phase, the energy balance in the gas phase provides the ordinary differential equation (ODE) for the gas phase temperature: d# qT c þ ð#0  #Þ ¼ dt

P

 P þ ðT  #Þ k bk maxf0;MrT k g P : k nk bk

k bk nk0

ð1:1Þ From the energetic point of view, the thermal capacity of the solid–liquid phase must be calculated from the constitutive parts: water (mwcpw), different organic components (micpi) and by part of the container (Ccon) being at the same temperature. The heat fluxes considered are: (a) heat transfer through the container’s wall qcw [J h1] = UA(Ta  T), proportional to the temperature difference between the surroundings (Ta) and the bulk mass (T); (b) convective heat transfer to the gas phase (qT c ); (c) enthalpy fluxes associated to the gas transfers, where the upwind schema from Patankar (1980) has been adopted. Thus, when the gas mass transfer is positive (flow from liquid to gas phase) the gas enthalpy (hk*) is evaluated at the solid–liquid temperature, and when the gas flows in the opposite direction the gas enthalpy is evaluated at the gas phase temperature; and d) biological heat generation (qG), which is assumed to be proportional to the oxygen consumption rate. The energy balance in the solid–liquid phase provides the ODE for the solid–liquid phase temperature:   P P14 dmw dmi dT qcw þ qG  qT c  k MrT k hk  ðT  #re Þ cpw dt þ i¼1 cpi dt ¼ : P dt Ccon þ cpw mw þ 14 i¼1 cpi mi

ð1:2Þ 2

T max ÞðT T min Þ fT b ¼ ðT opt T min ÞððT opt T minðTÞðT T opt ÞðT opt T max ÞðT opt þT min 2T ÞÞ

Supplementary data

a

m = IW/TM() : moisture content wb. (m2 = 0.40; m3 = 0.60); S i ðkgi m3 IWÞ : concentration of solubles in liquid phase. b Mesophilics fT1: Tmin = 5.1 C; Topt = 35.4 C; Tmax = 44 C. Thermophilics fT2: Tmin = 30.8 C; Topt = 57.2 C; Tmax = 65.5 C.

Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.biortech.2006. 07.012.

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