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Development and validation of mathematical model for aerobic composting process

By Shannon Franklin,2014-05-05 22:02
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Development and validation of for aerobic composting process

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.

Abstract

    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, RungeKuttaFehlberg 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;

    Model validation

    Article Outline

    1. Introduction

    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. Solidliquid 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

    Acknowledgements

    References

    1. Introduction

    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 [1]. Mathematical

    reactor models can serve as an essential tool for faster and better process designs, system analysis,

    and operational guidance [2]. 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 [3]. The models [1], [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] and [29] 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 [30]. 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 [27]. The assumption about constant air humidity is valid if the substrates with initial moisture content

    between 60 and 65% are used [31]. Air that leaves the compost is saturated in the case of the

    moisture content above 50% [17], [31] and [32]. 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 [18].

    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 [1] and [8]. 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 [4] and [18].

    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 [9]. 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:

    (1)

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 [9]:

    (2)

    k=kTkO2kH2OkFAS

    For describing the effect of temperature on reaction rate constant, the equation developed in [33]

    was used as a basis. Using the experimental data from [34] and taking into account that 20 ?C is the referent temperature and 60 ?C is the optimal temperature, Haug [9] developed the equation in the following form:

    (3)

kd=kd20[1.066(T?20)?1.21(T?60)]

    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:

    (4)

    kT=a[b(T?20)?c(T?60)]

    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 [15]: (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 [9]: (6)

where Sm is solid content of the substrate ().

    For free air space correction function, the following equation is used [9]: (7)

    Free air space (FAS) is calculated using the following equations [9]: (8)

(9)

(10)

    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 (0.150.4).

    2.4. Stoichiometry

    According to the assumption about initial elementary composition of the substrate, organic matter

    degradation in the substrate can be presented by the following equation:

    (11)

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

    following equation:

    (12)

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 liquidgas for gas i

    (kg h?1).

    The generation rate Ri is calculated by the following equation:

    (13)

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

    following equation:

    (14)

    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 [35]:

    (15)

    HeO2=101325e(66.7354?(8747.55/T)?24.4526 ln (T/100))

(16)

    HeCO2=?11418.84+43.8658T

(17)

    HeNH3=105e(14.48?(4341/T))

    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 [36]:

    (18)

The pH value is defined as a function of the hydrogen ion as follows:

    (19)

    pH=?log10[H+]

    where [H+] is concentration of the hydrogen ion (mol L?1). It is calculated from the equlibrium relation:

    (20)

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 NH4N (mol L?1).

    The equilibrium constant K is calculated from the following equation [37]: (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:

    (22)

    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:

    (23)

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 liquidgas for water (kg h?1).

    The generation rate Rw is calculated by the following equation:

(24)

where Yw is stoichiometric coefficient of water.

    The generation rate is calculated by the following equation:

    (25)

    where kLaw is mass transfer coefficient liquidgas 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 [35]:

    (26)

    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

    following equation:

    (27)

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:

    (28)

(29)

(30)

    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:

    (31)

2.6. Heat balance

    2.6.1. Gas phase

    The temperature of the gas phase is calculated as:

    (32)

    where cpi is specific heat capacity of gas i (J kmol?1 K?1). The convective heat transfer from solidliquid phase to gas phase can be described by Newton's

    equation:

    (33)

    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 liquidgas 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. Solidliquid phase

    The temperature of the solidliquid phase is calculated as:

    (34)

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 solidliquid

    and gas phase, respectively.

    The specific heat capacities are calculated by the following equation [9] and [38]: (35)

    cp=1.48?0.64ash+4.18wc

    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:

    (36)

    Qcw=UA(Ta?T)

    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 [31]: (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:

    (38)

where Δh is the reaction enthalpy .

    The fourth term of the numerator in Eq. (34) takes into account the heat transfer liquidgas 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 solidliquid 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

    (Table 1).

Table 1.

    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)

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