Development and validation of mathematical model for aerobic composting process
I. Petrica, , and V. Selimbašićb
aDepartment of Process Engineering, Faculty of Technology, University of Tuzla. Univerzitetska 8,
75000 Tuzla, Bosnia and Herzegovina
bDepartment of Environmental Protection, Faculty of Technology, University of Tuzla.
Univerzitetska 8, 75000 Tuzla, Bosnia and Herzegovina
Received 6 November 2006;
revised 30 July 2007;
accepted 3 August 2007.
Available online 17 August 2007.
By integrating the reaction kinetics with the mass and heat transfer between the three phases of the
system, a new dynamic structured model for aerobic composting process was developed in this
work. In order to evaluate kinetic parameters in mathematical model and to validate the model,
experiments were performed with the reactor of volume 32 L, in controlled laboratory conditions.
Different ratios of poultry manure to wheat straw were mixed and used as a substrate. Rosenbrock
optimization method was used for parameter estimation. In order to solve the system of 12
non-linear differential (and corresponding algebraic) equations, Runge–Kutta–Fehlberg method
was used, with approximation of fourth and fifth order and adjustment of step size. Both
algorithms were implemented in FORTRAN programming language. In order to achieve as
accurate description of the process dynamics as possible, the developed mathematical model was
validated by the results of several experimentally measured dynamic state variables. Comparisons
of experimental and simulation results for temperature of substrate, organic matter conversion,
carbon dioxide concentration and oxygen concentration, in general showed good agreement during
the whole duration of the process in a reactor. In the case of ammonia, an agreement was achieved
for the first 4 days and for the last 3 days of the process. A sensitivity analysis was performed to
determine the key parameters of the model. Analysis showed that two parameters had a great
influence on the main characteristics of the process. With validated model for aerobic composting
of mixture of poultry manure and wheat straw, optimal values were determined: initial moisture
content (70%) and airflow .
Keywords: Mathematical model; Aerobic composting; Reactor; Model parameters; Simulation;
2. Materials and methods 2.1. Description of model 2.2. Model assumptions and simplifications
2.3. Process kinetics 2.4. Stoichiometry 2.5. Mass balance
2.5.1. Dissolved gases in interstitial water (O2, CO2, NH3)
2.5.2. Water in composting material
2.5.3. Gases in gas phase (O2, CO2, NH3, N2, H2O)
2.6. Heat balance
2.6.1. Gas phase
2.6.2. Solid–liquid phase 2.7. Structure of mathematical model
2.8. Model inputs
2.9. Numerical methods 2.10. Experimental materials 2.11. Composting apparatus 2.12. Experimental design and analysis
3. Results and discussion 3.1. Model evaluation
3.2. Sensitivity analysis 4. Conclusions
The objectives of modelling are the development of mathematical tool to allow an integration of
knowledge on the considered phenomena, to orientate experimental design, to evaluate
experimental results, to test hypothesis, to reveal relations among variables, to predict the
evolution of a system and, finally, to design optimal process and management strategies.
Composting is a complex bioprocess that involves many coupled physical and biological
mechanisms. These coupled, and often nonlinear, mechanisms yield a broad spectrum of process
behaviours that are challenging to analyze both empirically and theoretically. Mathematical
modelling provides one approach for understanding the dynamical interactions between these
coupled mechanisms, and provides a framework for rational process design . Mathematical
reactor models can serve as an essential tool for faster and better process designs, system analysis,
and operational guidance . Increased computational power has made it feasible to use mathematical models of the
composting process, which can improve understanding and reduce the need for costly
experimentation. Mathematical models of the composting process have appeared in the literature
since 1976 . The models , , , , , , , , , , , , , ,
, , , , , , , , , , ,  and  showed more or less
success in predicting the profiles of: temperature, moisture, solids, oxygen and carbon dioxide.
Each of these models had some advantages and disadvantages, but there are some general lacks in
these models. Firstly, the above mentioned models, partly or not at all, described mass and heat
transfer between three phases of the system, and most of them did not describe dynamics of the
gas phase and the dissolved gases in water at composting material. Secondly, most of these models
did not use the original parameter values but they used those from existing literature. Thirdly,
model validations were carried out either only with one or two experimentally measured dynamic
state variables, or were not carried out at all.
The aims of this work were the following: (1) to develop the new dynamic and structured model
for aerobic composting process by connecting the reaction kinetics with mass and heat transfer
between three phases of the system, (2) to evaluate the kinetic parameters in suggested kinetics of the model by using the experimental results from laboratory reactor, (3) to validate the model with several experimentally measured dynamic state variables, (4) to show the efficiency of the validated model through the determination of the effects of the main process factors on the degradation of organic waste and evaluations of their optimum values.
2. Materials and methods
2.1. Description of model
The model describes the three-phase system and it is based on basic principles of chemical reaction engineering: kinetics, stoichiometry, mass and heat balances. At the beginning of the process, the substrate consists of organic part, inorganic part and water. Organic part of the substrate is degraded by biochemical reaction, with consumption of oxygen and generation of carbon dioxide, water and ammonia. Because of exothermic reaction, the heat is released. Considering substrate as a reactant, the model of batch reactor can be assumed. Air of constant composition is introduced into reactor, and gas phase composition is changed at reactor outlet. The role of air is to ensure sufficient concentration of oxygen for oxidation of organic matter and to take away the excess of moisture from the substrate. The complete mixing of material is assumed and it is achieved by agitation or efficient aeration. Reactor model can be approximated by the model of continuous stirred tank reactor (CSTR) at unsteady state with respect to present gases.
One part of the reactor is filled with substrate (represented by solid and liquid phases), while the rest of the volume is occupied by gaseous mixture (oxygen, nitrogen, carbon dioxide, water vapour, ammonia) (Fig. 1). According to biochemical reaction, solid and liquid phases are
responsible for the release of heat. Mass transfer of dissolved gasses and evaporated water occurs on the boundary between liquid and gas phases. The considered heat transfers are: heat transfer released from biochemical reaction, heat transfer from reactor to surroundings, convective heat transfer between phases, heat transfer of evaporated/condensed water.
Full-size image (26K)
Fig. 1. Mass and heat transfer phenomena included in the model.
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2.2. Model assumptions and simplifications
The following assumptions were taken into account while developing the model:
- The part of reactor volume with gas mixture has a constant value.
- The system maintains a constant pressure.
- Gas mixture is saturated with water vapour.
- Mass flows of the air at reactor inlet and outlet are equal (air has a constant flow).
- Liquid and solid phases have uniform temperatures.
- The substrate is a homogeneous mixture of uniform composition.
- Elementary composition of organic matter in the substrate (carbon, hydrogen, oxygen, nitrogen)
is known at the beginning of the process.
- The composting rate is expressed as the rate of organic matter degradation.
The assumption about constant volume of the gas phase above the composting material is based
on considerations of anaerobic digestion model . In reality, the volume of gas phase depends on variations in water content, degradation of organic matter and compaction of the material.
The assumption about maintenance of the constant pressure is justified because variations of total
pressure of the gas phase are small comparing to pressure of the surroundings . The assumption about constant air humidity is valid if the substrates with initial moisture content
between 60 and 65% are used . Air that leaves the compost is saturated in the case of the
moisture content above 50% ,  and . Typical composting process maintains the moisture content above 50%.
The assumption about constant airflow ensures that oxygen is distributed uniformly in the voids,
eliminating anaerobic conditions .
The assumption about uniform temperature comes from the fact that there is little or no resistance
to heat transfer from the compost matrix to the air in the reactor  and . The assumption about uniform substrate allows the model to neglect the statistical and spatial
variations in substrate composition and density which are known to be usually present. The model
predictions will be average values around which statistical variations occur  and .
With known mass fraction of carbon, hydrogen, oxygen and nitrogen in the substrate, the
stoichiometric coefficients for oxygen, carbon dioxide, water and ammonia can be calculated from
the stoichiometry of the oxidation reaction of the substrate's organic part . Inorganic matter does not participate in the reaction.
The simplifications made in the model development were:
- All heat capacities are constant.
- All enthalpies are independent from the pressure.
- Gas phase consists of ideal gases.
2.3. Process kinetics
In description of kinetics for the substrate degradation, the following equation is suggested:
where mOT is the mass of organic matter in the substrate (kg), t is time (h), k is reaction rate
constant (kg1?n h?1) and n is reaction order (–).
Reaction rate constant is the function of temperature, oxygen, moisture and free air space :
For describing the effect of temperature on reaction rate constant, the equation developed in 
was used as a basis. Using the experimental data from  and taking into account that 20 ?C is the referent temperature and 60 ?C is the optimal temperature, Haug  developed the equation in the following form:
where kd20 is reaction rate constant at temperature 20 ?C (h?1) and T is substrate temperature (?C).
In the model, modification of the Eq. (3) is suggested as:
where a, b and c are constants that need to be determined as well as the reaction order n in Eq. (1).
For oxygen correction function, the following equation is used : (5)
where O2 is oxygen concentration (kg O2 m?3) and KO2 is oxygen saturation constant (kg O2 m?3).
For moisture correction function, the following equation is used : (6)
where Sm is solid content of the substrate (–).
For free air space correction function, the following equation is used : (7)
Free air space (FAS) is calculated using the following equations : (8)
where δm and δw are density of composting material and water (kg m?3); Gs, Gv and Gf are specific gravity of solids, specific gravity of volatile fraction of the solids (=1) and specific gravity
of the fixed fraction of the solids (=2.5); Vs is volatile fraction of the solids (–); C is bulk weight coefficient for the substrate (0.15–0.4).
According to the assumption about initial elementary composition of the substrate, organic matter
degradation in the substrate can be presented by the following equation:
where a, b, c and d are indexes which describe the molar fraction of carbon, hydrogen, oxygen and
nitrogen in the organic part of the substrate. The stoichiometric coefficients for oxygen, carbon
dioxide, water vapour and ammonia can be calculated using the defined molecular formula of
organic part of the substrate and the Eq. (11).
2.5. Mass balance
2.5.1. Dissolved gases in interstitial water (O2, CO2, NH3)
The general mass balance for dissolved gases in the water within the substrate is given by the
where mi is mass of dissolved gas i in solution (kg), Ri is generation rate of gas i toward
biochemical reaction in liquid phase (kg h?1) and is rate of mass transfer liquid–gas for gas i
The generation rate Ri is calculated by the following equation:
where Yi is stoichiometric coefficient of gas i. The sign (+) in Eq. (13) is valid for carbon dioxide
and ammonia, and sign (?) is valid for oxygen.The generation rate is calculated by the
where kLai is mass transfer coefficient for gas i (kg h?1 Pa?1), Hei is Henry's constant for gas i (Pa), fi is dissociation factor for gas i in the solution (–), Xi is molar fraction of gas i dissolved in the solution (–) and pi is partial pressure of gas i in gas phase (Pa).
The Henry's constants for oxygen, carbon dioxide and ammonia are fitted by literature data :
HeO2=101325e(66.7354?(8747.55/T)?24.4526 ln (T/100))
where T is temperature of composting material (K).
It was assumed that dissociation factors for oxygen and carbon dioxide in water solution equal to 1.
The following dissociation factor for ammonia is used :
The pH value is defined as a function of the hydrogen ion as follows:
where [H+] is concentration of the hydrogen ion (mol L?1). It is calculated from the equlibrium relation:
where K is the equilibrium constant (mol L?1), [NH3] is the concentration of free NH3 dissolved
in the water phase (mol L?1), [NH4+] is the concentration of water-soluble NH4–N (mol L?1).
The equilibrium constant K is calculated from the following equation : (21)
where K298 is the equilibrium constant at temperature 298 K (K298 = 10?9.24 mol L?1), ΔH0 is the change of reaction enthalpy (ΔH0 = 86400 J mol?1), R—universal gas constant (R = 8.314 J mol?1 K?1).
The concentration of NH3 in the gas phase is assumed to be in equilibrium with free NH3
dissolved in the water phase of compost.
The partial pressure of gas in the gas phase pi can be described by equation of ideal gas state:
where ni is number of mole of gas i (kmol), R is universal gas constant (J kmol?1 K?1), ψ is temperature of gas phase (K), Vg is volume of gas phase (m3) (it is calculated as a difference
between reactor volume and volume occupied by composting material).
2.5.2. Water in composting material
The general mass balance for water in composting material is given by the following equation:
where mw is mass of water in composting material (kg), Rw is generation rate of water toward
biochemical reaction (kg h?1) and is rate of mass transfer liquid–gas for water (kg h?1).
The generation rate Rw is calculated by the following equation:
where Yw is stoichiometric coefficient of water.
The generation rate is calculated by the following equation:
where kLaw is mass transfer coefficient liquid–gas for water (kg h?1 Pa?1), Pv is pressure of water vapour in gas phase (Eq. (19)) (Pa) and Ps is pressure of water vapour saturated at
temperature of gas phase (Pa).
The pressure Ps is fitted by the literature data :
Ps=10(22,443?(2795/ψ)?1,6798 ln ψ)
2.5.3. Gases in gas phase (O2, CO2, NH3, N2, H2O)
In general, the mass balance equation for the components in the gas phase can be described by the
where Fi,0 and Fi,f are molar flows at inlet and outlet for component i (kmol h?1) and is
generation rate for component i at outlet from liquid phase (kmol h?1). It was assumed that there was no carbon dioxide in inlet air. The molar flows of oxygen, nitrogen
and water vapour are calculated by the following equations:
where PT is total pressure (Pa) and Q is volumetric airflow (m3 h?1). The pressure of water vapour Ps is calculated by the Eq. (23), but at initial temperature of gas phase ψ0.
The outflows are calculated by the following equation:
2.6. Heat balance
2.6.1. Gas phase
The temperature of the gas phase is calculated as:
where cpi is specific heat capacity of gas i (J kmol?1 K?1). The convective heat transfer from solid–liquid phase to gas phase can be described by Newton's
where hc is convective heat transfer coefficient between two phases (J h?1 K?1). The third term of the numerator in Eq. (32) takes into account the heat transfer liquid–gas in the
following way (max is elemental intrinsic function which returns the maximum value in an
argument list): (1) if then follows that the difference between temperature at phase
border and gas phase are equal to (T ? ψ), (2) if then follows that the difference between temperature at phase border and gas phase are equal to 0.
2.6.2. Solid–liquid phase
The temperature of the solid–liquid phase is calculated as:
where cpw, cpOM and cpIM are specific heat capacities of water, organic matter and inorganic
matter, respectively; and are molar enthalpies of gas at temperature of solid–liquid
and gas phase, respectively.
The specific heat capacities are calculated by the following equation  and : (35)
where ash is the ash or mineral content of the material (?) and wc is the dry-basis moisture content (?).
The heat transfer through the reactor walls Qcw is calculated as:
where U is overall heat transfer coefficient (J h?1 m?2 K?1), A is area of heat exchange (m2) and Ta is ambient temperature (K).
The overall heat transfer coefficient U is calculated by the following equation : (37)
where λf is thermal conductivity of insulator (J h?1 m?1 K?1), Alm is logarithmic mean of
surface area of insulator surrounding the reactor (m2) and L is mean thickness of insulator (m).
The biochemical heat generation QG is calculated as:
where Δh is the reaction enthalpy .
The fourth term of the numerator in Eq. (34) takes into account the heat transfer liquid–gas in the following way (max and min are elemental intrinsic functions which return maximum and
minimum value in an argument list): (1) if then follows that molar enthalpy of gas is at temperature of solid–liquid phase T, (2) if then follows that molar enthalpy of gas
is at temperature of solid–liquid phase ψ.
2.7. Structure of mathematical model
Mathematical model consists of 12 nonlinear differential equations (Eqs. (1), (12), (23), (27), (32) and (34)) with corresponding algebraic equations (Eqs. (2), (4), (5), (6), (7), (8), (9), (10), (13), (14), (15), (16), (17), (18), (19), (20), (21), (22), (24), (25), (26), (28), (29), (30), (31), (33), (35), (36), (37) and (38)). Therefore, the system of equations is described by 12 dynamic state variables
Dynamic state variables in the mathematical model
No. Dynamic state variable Symbol Unit Equation
1 Mass of organic matter mOT kg (1)
2 Mass of dissolved O2 mO2 kg (12)
3 Mass of dissolved CO2 mCO2 kg (12)
4 Mass of dissolved NH3 mNH3 kg (12)
5 Mass of water in the substrate mw kg (23)
6 Molar amount of O2 (gas phase) nO2 kmol (27)
7 Molar amount of CO2 (gas phase) nCO2 kmol (27)