develop a reliable mathematical model for the analysis of thermochemical conversion of bio- mass. This paper .... calculated based on the following equations, ...
A COMPREHENSIVE MATHEMATICAL MODEL FOR BIOMASS COMBUSTION
H. Fatehi and X. S. Bai
Division of Fluid Mechanics, Dept. of Energy Sciences,
This paper reports on an investigation of a comprehensive mathematical model for biomass combustion within the one-dimensional model framework. The model takes into account different thermochemical processes, e.g., moisture evaporation, pyrolysis, heterogeneous char reactions, intra-particle heat and mass transfer, and changes in thermo-physical properties. Different approaches to model the various processes involved in the thermochemical conversion of biomass are discussed, and a sensitivity study is carried out to investigate the performance of sub-models for the drying process. The comprehensive model is used to investigate the effect of moisture diffusion and vapour condensation inside the particle pores. The model is evaluated under different conditions and satisfactory comparison of the model results with experimental data and model results from other researchers is observed.
The increasing demand for energy and concern about global warming have led to an increased interest in using biomass and waste as an alternative to fossil fuels. Reduction of CO emission is one of the recent attempts in the development of combustion system. To assist the analysis and design of combustion systems utilizing biomass energy it is important to develop a reliable mathematical model for the analysis of thermochemical conversion of biomass. This paper presents an evaluation of a comprehensive mathematical model for large biomass particles in which there is a significant temperature gradient.
Different processes are involved in the thermochemical conversion of biomass; e.g., drying, primary pyrolysis, cracking and polymerization of tar or secondary pyrolysis, heterogeneous combustion and gasification of char, homogeneous combustion of the volatile, formation of ash and etc. Due to the high moisture content of biomass, the drying process is generally an important part of the thermochemical conversion of biomass. Alves and Figueiredo (1989), Peters and Bruch (2003) and Di Blasi et al. (2003) have performed measurements on drying of biomass particles. Di Blasi (1998) developed a detailed model investigating the moisture evaporation. The model takes into account the water vapour convection and diffusion, capillary water convection due to pressure gradient and the bound water diffusion in the pores of the particle. Depending on the moisture content, the particle properties and size, and the heating conditions, etc., two regimes of drying have been identified in the study, the regime of fast drying under high heating rate environment, and the regime of slow drying under slow convective heating.
Pyrolysis is another important process during thermochemical conversion of biomass particle since up to 90% of the mass loss of the dry particle can happen during pyrolysis. Extensive studies have been carried out on pyrolysis to understand and to categorize this pro-
2
cess. Gronli (1996) presented a joint modelling and experimental study of biomass pyrolysis, where a survey of various mathematical models up to 1996 can be found. A more recent review on this field is given by Di Blasi (2008). During pyrolysis the biomass particles undergo heat and mass transfer inside and around the particles, and complex chemical reactions occur during the structural change in the biomass particle. Hagge and Bryden (2002) studied the effect of shrinkage on the pyrolysis behaviour of a large range of particle sizes. They used the kinetic parameters of Thurner and Mann (1981) and considered the effect of tar cracking and polymerization. They found that shrinkage could affect both the pyrolysis time and products in the thermal wave regime. For particles in thermally thin and thick regimes this effect was found to be negligible. Di Blasi (1996) studied the effect of shrinkage on a slab particle with half thickness of 2.5
and reported the same qualitative behaviour of the temperature field,
with or without including shrinkage in the model.
Several authors have considered modelling of the complete combustion process of biomass. Yang et al. (2007c) employed two-dimensional conservation equations coupled with a global one-step pyrolysis kinetic rate for the simulation of a shrinking particle ranging in size from micrometre to several millimetres. They pointed out the violation of uniform temperature assumption for particle sizes larger than 250
. Lu et al. (2008) developed a one di-
mensional model of a single biomass particle with different sizes. They used the kinetic scheme of Shafizadeh and Chin (1977) to predict the decomposition of virgin fuel to gas, tar and char. They included a thermal and a mass boundary layer to the model to investigate the effect of the surrounding flame on the particle. They reported that the effect of boundary layer on the conversion behaviour was not significant. Haseli et al. (2011) presented a one dimensional model for combustion of biomass particle emphasizing on the role of pyrolysis and gaseous combustion during the biomass conversion. They adopted the kinetic scheme of Shafizadeh and Chin (1977) with the kinetic constants of Thurner and Mann (1981) for low
3
temperature and the rate constants of Di Blasi and Branca (2001) for high temperature pyrolysis. Mehrabian et al. (2012) developed a model for the thermochemical conversion of thermally thick particle, suitable for implementation in CFD calculations of a packed bed. The model was based on the layer model proposed by Thunman et al. (2002) and Porteiro et al. (2006). The results showed a good agreement with the experimental data.
Despite the many detailed studies as exemplified above, a systematic evaluation of the various models developed for biomass combustion is not available. Since a useful model for engineering simulations of biomass combustion should be highly efficient and accurate, it is desirable to introduce reasonable assumptions to simplify the model. To do so it is important to know the validity of the various model assumptions. In the present work, a comprehensive general model for biomass combustion is presented within the one-dimensional model framework. Intra-particle mass and heat transfer are coupled with kinetic data of different processes inside the particle. The model is validated using three sets of experimental data to examine the performance of the comprehensive model. Then, various approaches for the drying of biomass particles are evaluated to explore the limitations and strength of models.
A COMPREHENSIVE MATHEMATICAL MODEL FOR BIOMASS COMBUSTION
General Consideration
We consider a general model for biomass particles of arbitrary size, in which chemical reactions, convective, conductive and radiative heat transfer inside the particle as well as heat transfer from and to the particle with the surrounding gas, are taken into account. The particle
4
is modelled as one dimensional, to maintain a reasonably low workload for the computation. The model can be readily extended to two or three dimensions. The following assumptions are employed in the model:
The biomass particle is made up of multiple components, including solid, gas and liquid;
All these components are in thermal equilibrium;
The particle geometry can be spherical, cylindrical or slab (parallelepiped plate) and it is represented in a one dimensional frame work;
Gases inside the porous structure of particle obey the ideal gas law;
The momentum transport in the particle is governed by Darcy law;
Homogeneous gas reactions inside the particle are neglected.
Governing Equations
At time mass,
and at the spatial coordinate
of a biomass particle the density of the bio-
( , ), changes with time during thermochemical conversion. The change of the
density of the particle is described by follow equation, ( , )
where ̇ rate of
=−
−
−
= ̇
=−
−
−
(1)
is the net mass loss rate of the particle due to drying (at the
), volatile release (at the rate of
), and char formation (at the rate of
particle the mass of moisture (liquid) per unit volume of the particle is denoted as
). In the ( , ),
which changes during the drying process. If the convection and diffusion of moisture in the particle are neglected, the change of
( , ) in time can be modelled as,
5
( , )
=−
(2)
The pyrolysis rate is determined from the pyrolysis mechanism to be discussed later. The change of char mass in the particle is described in Eq. (3), where the source term ̇ accounts for the char production due to pyrolysis and the char consumption due to oxidation and gasification. ( , )
= ̇ =
−
−
−
(3)
The heterogeneous char reaction rates are presented in the following form,
=
where
−
,
is the reaction number (
) and = O , CO , H O.
is the specific internal surface area (
chiometric coefficient and structure.
, and
(4)
/
is the stoi-
) of char porous
is molecular weight.
The conservation equation of gaseous species can be written as,
+
1 ( ) =
where
is the porosity;
pores of the particle;
( ) ,
,
1 ( )
= ,
,
,
( ) ,
+ ̇
(5)
is the density of gas in the particle;
is the velocity of gas in the
is the mass fraction of the gas phase species . The effective gas diffu-
sivity in the particle pores is calculated as follows,
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=
=
where τ is tortuosity, which can be estimated as
.
(6)
= 1/ (Mehrabian et al., 2012). The for-
mation rate of gaseous species i, ̇ , is determined from the sub-models for drying, pyrolysis and char combustion/gasification. In the transport equations of species mass fractions, Eq. (5), ( ) is the Lamé coefficient accounting for non-Cartesian coordinate system. For the spherical, cylindrical, and large slab plate shaped particles, ( ) has the following form,
The gas phase mass fraction obeys the following equation,
+
1 ( )
( )
= ̇ =
̇ .
(8)
where N is the total number of the species in the gas.
The energy conservation equation is based on the assumption of local thermal equilibrium for gas, liquid and solid of the particle,
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+
+
+ (9)
+
where
1 ( )
( )
1 ( )
=
( )
+ ̇
is the temperature of the particle components at ( , ), ̇
includes the endother-
mic heat of evaporation, heat of pyrolysis and heat of gasification of char as well as the highly exothermic heat of char oxidation and the change in sensible enthalpy, and
The effective thermal conduction coefficient ( and conductive (
is heat capacity.
) is a sum of the radiative (
)
) heat transfer coefficients. The conductive heat transfer coefficient is a
mass weighted sum of moisture (
), char ( ), biomass (
), and the gas (
) thermal con-
ductivity. The effective conduction coefficient inside the particle is calculated as follows, =
=
+ (1 − )[(
+
+
+
)/(
+
+
)]
(10)
=
where
is Stefan-Boltzmann constant,
is emissivity, and
is the local pore diameter
inside the particle.
The specific heat of biomass, char (Gupta et al., 2003) and tar (Tinaut et al., 2008) are calculated based on the following equations,
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(−0.004 )
= 2400 − 2500 = 1430 + 0.355 −
7.32 × 10
= 506 + 1.16 − 1.16 × 10
−
(11) (12)
3.33 × 10
(13)
From Eq. (9) the temperature of the particle can be determined. From equation of state and the continuity equation, Eq. (8), and Darcy law, Eq. (14), the gas pressure and velocity at ( , ) are determined. Darcy law relates the gas velocity to the pressure gradient of gas in the particle,
=−
where
is the viscosity of the gas and
(14)
is permeability which is calculated based on the em-
pirical equation (Vafai and Sozen, 1990),
=
150(1 − )
.
(15)
Shrinkage Model
During thermochemical conversion it is often assumed that the change of particle volume is a linear function of the mass loss. A shrinkage factor change of volume due to shrinkage, which is modelled as
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is introduced to account for the
= 1 + (1 −
)(
The shrinkage factor tial volume.
,
− 1) + (
−
)(
− 1) + (
−
)(
− 1)
(16)
is defined as the ratio between the current volume of particle to its ini-
and
are empirical parameters between 0 and 1, representing the extent
of shrinkage due to moisture evaporation, volatile release and char gasification/combustion, respectively. Value 0 represents constant density and 1 represents constant volume during thermochemical conversion. Subscript 0 denotes the initial state of the particle.
The assumption here is that shrinkage happens across the grain only. Shrinkage not only happens due to the loss of mass from the particle but also due to the rearrangement of the chemical bond and physical structure of the virgin biomass and char. A value of
= 0.9
= 0.75 for the pyrolysis process. If char oxi-
was chosen for the evaporation process, and
dation is controlled by external diffusion, it can be assumed that during char reaction the density remains constant. This yields that
= 0.
Pyrolysis Model
Biomass pyrolysis is a multi-scale chemical and physical process, which involves a length scale of the particles ranging from 10 ticles of 10
and liquid droplets of 10
to 10 , cavity and cell walls in the paror less (Kersten and Garcia-Perez, 2013). A
fundamental modelling approach should be built up from the smallest relevant scales, taking into account the detailed physical and chemical processes. This approach may be possible for modelling of celluloses, hemicelluloses or lignin; however, for a practical biomass particle a common approach is based on global phenomenological models that neglect the fine details in the particles, e.g., in CFD modelling of biomass combustion process in furnaces firing straw
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(Kær, 2004, Yang et al., 2007a) or wood pellets (Klason and Bai, 2007) and wood powders (Elfasakhany et al., 2013).
For fast pyrolysis of pulverized wood particles of the size below millimetres in a combustion condition, Elfasakhany and Bai (2006) compared six different pyrolysis models, including a single step global model (Gronli, 1996) with prescribed pyrolysis product yield, a functional group model (Elfasakhany et al., 2008), and the Broido–Shafizadeh model (Varhegyi et al., 1994). It was shown that for low moisture (< 10 wt-%) the structures of the pulverized wood flame, the temperature field, and the major species (such as CO2) are not sensitive to the use of different models, provided that the kinetic rate and the other model constants are calibrated against the same set of experimental data. In the present model framework similar sensitivity study can be carried out for large particles under slower heating rate. Furthermore, other key model components such as drying, shrinkage, intra-particle heat and mass transfer can also be investigated. In this paper we focus on the sensitivity study of drying models since drying is a dominating process in large particles with high moisture content. For this purpose, the Broido–Shafizadeh mechanism (Varhegyi et al., 1994) is used to model the pyrolysis chemistry, with chemical kinetic constant of Di Blasi and Branca (2001). The pyrolysis model and char combustion/gasification reactions and the respective kinetic rates are given in Table 1.
A systematic approach is employed to compute the yield of different gas species and compositions of tar (Fatehi and Bai, 2012). In this approach the overall elemental mass conservation and energy conservation are used to determine the gas and tar composition. To close the system of equations, the empirical correlations of Neves et al. (2011) were used. Depending on the reactor condition, the amount and composition of char residue can be different. Typically, char consists of carbon and a smaller amount of hydrogen and oxygen. As the heat-
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ing rate to the particle increases, the hydrogen and oxygen content of remaining char is lower (Neves et al., 2011). This means that for operating condition similar to those of industrial furnaces with high heating rate environment, the assumption of pure carbon for char is reasonable.
Drying Models
Drying is an important step during conversion of solid fuel either as a pre-treatment process or happening simultaneously or before pyrolysis. High initial moisture can reduce the particle heating rate by a factor of 3 to 5 (Di Blasi et al., 2003). Water inside the particle can be found in three forms; free water, bound water and capillary water. Capillary and free water exist in liquid form in voids and cells while bound water exists as water molecules physically or chemically bonded to the pores surface of the particle or hydrated species. The bound water can exist up to fibre saturation point (FSP), which for many types of wood can be up to 30% of dry weight (Di Blasi, 2008). The moisture content larger than FSP is in form of free water. It can be assumed that the free water and capillary liquid have the same potential energy state as they would have outside the particle.
Regarding modelling the evaporation process, there are several types of models studied in the literature including the heat flux model, the equilibrium model, and the chemical reaction model. In the heat flux model, which can be referred to as the most frequently used drying model in literature, the evaporation is controlled by heat transfer to the particle. The
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model is based on the assumption that evaporation takes place in an infinitely thin region and at normal boiling point of water. Some modifications have been proposed to include evaporation at temperatures lower than boiling point of water (Bruch et al., 2003) or to overcome the numerical instabilities (Yang et al., 2007c). To evaluate the evaporation rate in the heat flux model, the steady state form of energy conservation is used. If
represents the moisture
evaporation rate, the heat flux evaporation model can be written as
