A Simplified Model Accounting for the Combustion of Pulverized Coal

Article Cite This: Energy Fuels 2017, 31, 11391-11403

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A Simplified Model Accounting for the Combustion of Pulverized Coal Char Particles in a Drop Tube Furnace Patrick Gilot, Alain Brillard,* Jean-François Brilhac, and Cornelius Schönnenbeck Laboratoire Gestion des Risques et Environnement EA2334, Université de Haute-Alsace, Institut J.-B. Donnet, 3bis rue Alfred Werner, Mulhouse F-68093, France ABSTRACT: A simple model is proposed, which predicts the evolutions versus time of the temperature and of the carbonaceous material conversion of a coal char particle during its combustion in a drop tube furnace under a very high heating rate of 1500 K/s. The values of the intrinsic reactivity parameters are obtained performing thermogravimetric analyses of the coal char particles in a thermobalance under a low heating rate of 10 K/min. In this simple model, the local evolutions of the particle porosity and density are not accounted for as only their mean values are considered at any time of the combustion process. The oxygen concentration gradient within the particle is accounted for, once a particle effectiveness factor related to the Thiele modulus is estimated. Comparisons between the experimental and simulated temperatures of the particle are performed for three regulation temperatures of the drop tube furnace, and the combustion regimes are analyzed in terms of the effectiveness factor of the particle.

1. INTRODUCTION Coal combustion is largely affected by the burning rate of the carbonaceous residue (char) obtained after the fast pyrolysis step. Modeling the coal combustion in pulverized coal-fired boilers and furnaces requires the determination of both the kinetic and the oxygen transport parameters. The determination of the kinetic parameters is a complicated challenge for the following reasons: (1) Char has a complex porous structure, which depends on its formation conditions and continuously changes during the combustion process. (2) Coal particle combustion involves a strong coupling between oxygen transport within and outside the particle and combustion of the carbonaceous material, which is highly affected by combustion conditions such as heating rate and oxidizing environment. (3) The kinetic parameters, advantageously determined under kinetically controlled combustion, must be accurate enough to be used for the simulation of a combustion process, especially when combustion is controlled by kinetics. In a thermobalance, the sample temperature is well controlled, and the mass loss is measured with high accuracy during the whole combustion process, as described in ref 1. Further, the active surface area is rate controlling,2 and combustion occurs in the zone I burning regime. The char particle burns in a uniform way, and the evolution of the porous structure during the combustion process may be represented by the random pore model proposed by Bhatia and Perlmutter.3 On the contrary, in industrial furnaces where high heating rates (up to 105 K/min) and temperatures (up to 2000 K) are at work, strong diffusional limitations appear. Both internal and external diffusional limitations must thus be taken into account in a model devoted to the description of the combustion of char particles in industrial furnaces. Further, a complex situation occurs during an intermediate burning regime, where combustion is nonuniform within the particle with local evolutions of the surface area, apparent density, pore © 2017 American Chemical Society

diameter, pore volume, etc. Thiele modulus and an effectiveness factor are currently used in the available models to simulate the combustion process that occurs in zone II; see ref 4 for example. In the present study, coal char particles have been produced from a South African bituminous coal through a pyrolysis process in a drop tube furnace. The combustion under an oxidative atmosphere of the collected coal char particles then has been performed in the drop tube furnace, which simulates the conditions occurring in industrial furnaces. The particle temperature has continuously been measured by a two-colors pyrometer located at the bottom of the drop tube furnace; see Figure 1b. The values of the kinetic parameters associated with the coal char combustion have been determined through a thermogravimetric analysis performed under a low heating rate of 10 K/min and under an oxidative atmosphere. The main purpose of the present work is to propose a simple model, which intends to simulate both the temperature and the carbonaceous material conversion of the coal char particle during the combustion process in a drop tube furnace. This simple model mainly consists of only two basic equations: an energy balance and an oxygen transport balance, which are written assuming uniform properties of structural parameters and temperature inside the particle. In this model, pore growth and coalescence are accounted for to derive a relation between the surface area and the carbonaceous material conversion, which we will simply call conversion throughout this work, when no confusion may occur. The porous structure of the particle is here characterized by a structural parameter. An evaluation of the Thiele modulus is performed. The oxygen concentration gradient within the particle is accounted for, once a particle effectiveness factor related to the Thiele Received: June 20, 2017 Revised: August 25, 2017 Published: August 29, 2017 11391

DOI: 10.1021/acs.energyfuels.7b01756 Energy Fuels 2017, 31, 11391−11403

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Energy & Fuels

Figure 1. A schematic representation of the drop tube furnace used for the present study for the coal char production (a) and for the char combustion (b).

coal combustion occurring in a large-scale furnace have been simulated13 through a CFD model. A coal combustion model has been presented,14 which is based on a kinetic model where the chemical reaction rate is considered as a function of coal intrinsic reactivity and fuel mass for a global reaction of order n. An oxy-fuel combustion has been simulated15 through a CFD model. The model that is described in the present study is much simpler than many of the models presented in the above indicated papers. It only requires the resolution of a system of two coupled balance equations accounting for the heat transfer (eq 3) and the oxygen transport (eq 4). To validate this simple model, comparisons between the simulated and observed temperatures of a particle introduced in the drop tube furnace are proposed at three regulation temperatures of the drop tube furnace (1100, 1200, and 1300 °C). The resolution of these equations is not time-consuming and does not require a dedicated software. The simulated and experimental particle temperatures do not quite well superimpose, although the differences between the simulated and experimental temperatures lie in the admitted uncertainties, which are confirmed in the present context through repeatability combustion experiments. It should indeed be noticed that pyrometry measurements of the particle temperature in a drop tube furnace often lack precision, mainly because of the radiations emitted by the reactor walls; see ref 16 that gives 50 °C as the uncertainty for particle temperatures measured in drop tube furnaces during coal pyrolysis and the results concerning the combustion of anthracite char coals in an isothermal plug flow reactor under different oxygen concentrations in ref 17. Nevertheless, the trends of the particle temperature curves are well predicted using this simple model when physical parameters such as the particle radius, oxygen concentration, etc., are modified. The simple model that is proposed may thus be considered for a quick and rough description of the particle behavior during the combustion process in a drop tube furnace.

modulus is estimated. Yet the local evolutions of the particle porosity and density are not accounted for in this model, as only their mean values are considered at any time of the combustion process. The combustion of coal particles and of coal char particles in furnaces or boilers has already been analyzed and modeled by many authors and for a long time. In ref 5, the author combines a Thiele analysis and a random pore model to describe the combustion of coke particles in an atmospheric fluidized bed combustor. A sophisticated model has been proposed by Mitchell,6 to simulate the physical changes of char particles occurring during a combustion process. Local values of the parameters have here been considered, which allow one to locally use the random pore model for the description of the local evolution of the surface area. A model accounting for coal combustion has been studied, 7 where the motion of representative coal particles is analyzed together with the evolution of the temperature. The oxy-fuel combustion of coarse size coal char in a fluidized bed has been modeled,8 through a generalized fully transient, one-dimensional, and nonisothermal oxy-fuel combustion model, using a volume reaction model. The devolatilization process of a single coal particle and the combustion of the residual char have been modeled.9 In these two last works, the authors introduced two sets of governing equations for the solid particle phase (component mass balance of gas species, total molar balance of gas mixture, and energy balance) and for the gas boundary layer (component mass balance, total molar balance, and energy balance). On the basis of five gas species and three main chemical reactions, this leads to a system of 8 + 7 equations, which describe the coupled evolutions of the temperatures (gas and particle), of the mass fractions, of the total molar flux of gas mixture, of the instantaneous mass concentration of carbon in solid char, and of the total concentration of gaseous mixture. For the resolution of this large system of evolution equations, the authors used the COMSOL Multiphysics software. In ref 10, the authors proposed a fully transient and coupled kinetic, heat-transfer model accounting for the pyrolysis of large coal particles. The combustion of a gasified semi-char in a drop tube furnace has been analyzed,11 using a pore development process and an effectiveness factor. A random pore model with intraparticle diffusion has been introduced,12 for the description of char particles combustion. Different operating conditions of

2. MATERIALS A South African bituminous coal has been used for the pyrolysis experiments in a drop tube furnace. The proximate and ultimate analyses of this coal and of the coal char, which is produced from this coal through a pyrolysis process in the drop 11392


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thermocouple was set at three given values: 1000, 1200, and 1300 °C, to investigate the gas temperature along the reactor vertical axis. Char particles were injected in the drop tube furnace in the configuration indicated in Figure 1b for combustion under an oxidative atmosphere (88% nitrogen, 12% oxygen). They were injected with a primary gas flow (40 L/h) through a water cooled injector and entrained by a preheated secondary nitrogen flow (360 L/h) preheated at 900 °C. During each injection sequence, an amount of 2 mg of char particles was introduced in the reactor. As for the coal particles, when entering in the injection probe, the coal char particles were obeying a transient velocity regime during a very short time length, after which they were carried through the reacting zone of the drop tube furnace at the gas velocity. This gas velocity is computed in terms of the flow and of the temperature. The falling time is then computed. Different falling heights have been tested. The particle temperature was continuously measured during the combustion process with a two-colors pyrometer IMPAC ISR 12-LO placed at the bottom of the drop tube furnace and pointing vertically toward the extremity of the injection probe; see Figure 1b. As indicated in its manual, the accuracy of this two-colors pyrometer is given as 0.4% of the measured value plus 1 °C, if the temperature is less than 1500 °C, which is about 6 °C. The devolatilization process occurring during the pyrolysis of the coal particles led to swelling phenomena. The diameters of the collected coal char particles have been observed using a granulometry laser procedure performed at room temperature, and the diameters of the dominant subpopulation were lying in the range 100−160 μm, to be compared to the diameter range of the parent coal particles, 36−72 μm. The collected coal char particles have then been smoothly sieved in this diameter range (mean value 130 μm) for the further combustion experiments. Such a swelling phenomenon has already been observed in ref 10 for coal particles submitted to a pyrolysis process, with pictures of the coal particles at different stages of the pyrolysis process. Successive identical coal char particle drops with temperature measurements were performed to analyze their repeatability. 3.2. Char Combustion in a Thermobalance. A preliminary thermogravimetric experiment was performed on roughly 100 mg of collected char particles. It was performed under nitrogen and under a temperature ramp equal to 10 °C/min. During the temperature increase until 300 °C (that is before the start of the combustion process), about 3% of the initial mass was lost. This mass loss was attributed to hydrocarbons and moisture, which may have previously condensed on the surface of the sample during its collection from the bottom of the drop tube furnace devoted to pyrolysis experiments. The hydrocarbons plus water content of the raw char material (rm) was thus estimated as (τHC,W)rm = 0.03 ± 0.01. The kinetics of char combustion were studied in a thermobalance (Setsys TG12 Setaram). About 8 mg of char collected from the drop tube furnace and sieved to the 100−160 μm granulometry was placed in a small cylindrical crucible (5.91 mm of internal diameter and 2.6 mm of internal height). Because of the thinness of the material layer, the internal diffusional limitations may be neglected, as was already described in ref 18. The surface of the powder material was leveled off at the mouth of the crucible to minimize the external diffusional limitations to oxygen transport toward the surface of the material layer. The oxidizing gas (90% nitrogen, 10% oxygen) was flowing at 166.7 cm3/min (normal conditions). The sample was heated from ambient temperature to 950 °C at 10 K/min. Runs were also performed with an empty crucible to assess the effects of drag and buoyancy on the weight measurements. Corrections associated with thermogravimetric analysis with an empty crucible were thus brought (subtracted) to the experimental results concerning the coal char particles, to obtain reliable thermograms. Because of the adsorbed water and hydrocarbons on the char sample and assuming that no combustion of the carbonaceous material occurs before 300 °C, see the preceding discussion, time 0 of the char combustion was taken when the temperature was reaching 300 °C. The sample mass measured at that temperature was taken as the initial

tube furnace, are indicated in Table 1, the measures being performed on the basis of the associated ISO standards. Table 1. Proximate and Ultimate Analyses of the Coal Considered in This Study (O Is Computed by Difference) and of the Coal Char Produced from the Coal by Pyrolysisa Coal proximate analysis (wt %, ar) volatile matter ash fixed carbon moisture

24.0 13.8 54.7 7.5

ultimate analysis (%, daf) C H N S O

67.64 3.77 1.81 0.76 26.02

Coal Char proximate analysis (wt %, ar) volatile matter ash fixed carbon moisture a

0.7 20.0 78.5 0.8

ultimate analysis (%, daf) C H N S O

76.63 0.89 1.26 0.53 20.69

ar, as-received; daf, dry and ash-free basis.

The high heating value of the coal used in the present study was measured at 27.3 MJ/kg, on dry basis. The density of the coal char material was measured as (ρbulk)rm = 296.6 kg/m3.

3. EXPERIMENTS AND METHODS 3.1. Coal Pyrolysis and Char Combustion at High Heating Rates in a Drop Tube Furnace. The vertical furnace that was used for the coal pyrolysis and then for the coal char combustion is heated by six lanthane chromite bars, allowing a power of 12 kW; see Figure 1. The maximal furnace temperature that can be obtained is 1600 °C, and it can be regulated adjusting electrical resistances, through a thermocouple located between the external wall of the reactor and the internal wall of the furnace. The reactor (60% alumina and 40% silica) has an internal diameter of 5 cm and a length of 140 cm. Before their introduction in the reactor, the coal particles were first crushed and sieved to the 36−72 μm size fraction, with a mean diameter of 50 μm. This granulometry perfectly corresponds to that of pulverized coal particles usually used in industrial boilers. The coal particles were fed into the drop tube furnace through an injection probe at a rate equal to 17 × 10−3 kg/h (Figure 1a). When entering in the injection probe, the coal particles were obeying a transient velocity regime during a very short time length, after which they were carried through the reacting zone of the drop tube furnace at the gas velocity. This gas velocity is computed in terms of the flow and of the temperature. The falling time is then computed. The reaction time for devolatilization of coal particles was about 1 s for a gas temperature of 1573 K, corresponding to a 1 m drop and a gas flow of 400 L/h. The devolatilization yield was determined at 0.39 (daf), using the ash tracer method. The uncertainties related to the devolatilization yields ranged from 0.045 to 0.110, according to the determined devolatilization yields corresponding to different drop heights in the drop tube furnace. These estimates have been obtained taking the uncertainties related to the ash contents measured in the raw material and in the char residue, as well as the water content in the raw material. The relative uncertainties associated with the value obtained for the devolatilization yield of the char prepared for combustion experiments (with 1 m drop) were taken equal to 0.074, leading to a devolatilization yield equal to 0.39 ± 0.03 (daf). The coal char particles were collected at the bottom of the drop tube furnace in the configuration indicated in Figure 1a. Without any injection of char particles in the drop tube furnace in the configuration of Figure 1b, the temperature of the regulation 11393


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Energy & Fuels sample mass. All of the parameters related to this initial sample have the superscript 0. The void volume fraction f void of the sample is taken equal to 0.5, and the apparent density (density of a char particle) of the initial material may be computed as ρ0p = (ρbulk)rm(1 − (τHC,W)rm)/(1 − f void) = 575.4 kg/m3. From the thermogram, the instantaneous conversion of the carbonaceous material in the thermobalance, defined as XC,th(t) = 1 − mC(t)/m0C, is computed as a function of time (or temperature) from the following expressions: m0C = m0(1 − τ0ash) and mC(t) = m(t) − m0τ0ash, where m0 is the mass of the initial material present in the crucible at time 0, and m(t) is that which is read on the thermogram at time t. The ash content τ0ash in the initial solid is taken equal to 0.245 ± 0.005, as the precision concerning τ0ash deduced from the thermogram is approximately equal to 2%. This value may slightly vary from one sample to another one, due to heterogeneities. The temperature ramp is described through T = at + T0, for the constant heating rate a (10 K/min) and where T0 is the initial temperature at time 0. 3.3. Measurement of the Specific Surface Area of the Coal Char Particles. The coal char surface area was measured with a Micromeritics device using CO2 adsorption, following the IUPAC recommendations indicated in ref 19. For microporous samples with a relatively low total porosity, like carbons or carbonaceous residue, adsorption with N2 requires low pressures at very low (cryogenic) temperatures. This may lead to degassing problems and lack of precision. The sample was degassed at 150 °C under vacuum during 1 day before CO2 adsorption. The CO2 adsorption was realized at 273 K under higher pressures. The Dubinin−Radushkevich equation described in ref 20 was used to obtain the volume of the micropores. The specific surface area of the micropores then was deduced. The specific surface area of the initial coal char material was estimated as A0 = 99 500 ± 1000 m2/kg, according to the precision (equal to 1%) indicated in the operating manual. For kinetics investigations, the evolution of the specific surface area of the coal char material during combustion was determined measuring the surface area of partially burnt samples obtained at different conversions; see Figure 2.

The presence of mineral matter leading to ash residue after the combustion process is accounted for to deduce the mass of the carbonaceous part of the sample. Yet a possible catalytic effect of these minerals is not considered. 4.1.2. Equations. Assuming a first-order reaction with respect to oxygen, the carbonaceous material consumption rate is written as RC,th = MCksACmC(CO2)∞, where the kinetic constant ks is given by an Arrhenius expression: ks = (ks)0 exp(−E/(RgT)), where (CO2)∞ is the inlet oxygen concentration and AC is the specific surface area based on the mass of carbonaceous material and whose increase is tentatively related to the carbonaceous material conversion X C,th in the thermobalance through Bhatia’s relation: A C = A C0 1 − Ψ ln(1 − XC,th)

(1)

according to refs 4 and 21. The calculated specific surface area A0C of the initial char, based on the carbonaceous part, is related to the measured specific area A0 through A0C = A0/(1 − τ0ash). For the initial coal char material, A0C is taken as equal to 131 800 m2/kg. The relative uncertainties concerning this value of A0C are evaluated through ΔA0C/A0C = ΔA0/A0 + Δτ0ash/(1 − τ0ash) ≃ 0.02, according to the previous indications. This leads to ΔA0C ≃ 6600 m2/kg. The variation of XC,th with respect to time is governed by the differential equation: dXC,th dt

R C,th

=

mC0

= (CO2 )∞ MCksA C0(1 − XC,th)

× 1 − Ψ ln(1 − XC,th)

(2)

For the determination of the kinetic parameters (ks)0 and E in the kinetic constant ks, a Scilab code has been built, which first solves eq 2 with initial values of these kinetic parameters. This code then includes an optimization procedure, which determines the optimal values of the kinetic parameters through the minimization of the “error” taken as J

error =

2

∑ ((XC,th)exp (t j) − (XC,th)sim (t j)) j=1

where (XC,th)exp(tj) and (XC,th)sim(tj) are the experimental and simulated conversions of the carbonaceous material in the thermobalance at time tj. Instead of taking all of the experimental measure times, only around 150 of them are selected, regularly distributed along the overall experiment duration tmax. This reduces in a significant way the computing times. 4.2. Modeling the Combustion of the Coal Char Particle during Its Fall in the Drop Tube Furnace. 4.2.1. Hypotheses of the Model. As was already indicated, the char particle is supposed to leave the injection probe at 323 K and to fall inside the reactor at the same velocity as the gas. The gas temperature profile along the reactor axis has been measured and set in the model. A boundary layer accounts for the diffusional limitations in the oxygen transport from the gas flow to the particle surface. The particle is supposed to be heated by radiation from the walls of the reactor and by the reaction enthalpy. A convective heat flux, represented by a coefficient h, is exchanged between the particle surface and the gas. The reaction enthalpy accounts for the formation of both CO and CO2. The molar ratio CO/CO2 is estimated as a function of temperature as in ref 22. No temperature gradient is

Figure 2. Experimentally measured (crosses) and computed (red □) specific surface area of partially burnt coal char particles.

4. CALCULATIONS 4.1. Determination of the Kinetic Parameters through a Thermogravimetric Analysis Performed under an Oxidative Atmosphere. 4.1.1. Hypotheses. The particle radius is supposed to be constant along the combustion process. The char combustion is supposed to be kinetically controlled by the CO2 surface area, which is tentatively correlated to the conversion XC,th. The evolution of the specific surface area during combustion, related to the pore structure evolution, is thus accounted for. The pore structure is represented in the random pore model described in ref 4 through a structural parameter Ψ, which has to be determined. 11394


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(CO2)∞), where (CO2)∞ is computed as (CO2)∞ = (XO2)∞P/ (RgTg), where (XO2)∞ is the inlet oxygen molar fraction. Because of the hypothesis of a quasi-steady-state combustion regime, the oxygen flux related to the carbonaceous material consumption rate is given as ΦO2 = RC/(ΛmC), with RC = ηpksmCAC(CO2)Rp/Λ and Λ = (τ + 1)/(τ/2 + 1). ΦO2 is thus given as ΦO2 = (CO2)∞/(Λ/(ηpksmCAC) + 1/(kd4πR2p)). The oxygen mass transfer coefficient is defined as kd = Sh(DO2)(F.bl)/ (2Rp) with (DO2)(F.bl) = 1.78 × 10−5(Tm/273)1.75. The Sherwood number Sh is taken equal to 2. The specific area AC of the particle is tentatively calculated according to the following expression, eq 1: A C = A C0 (1 − Ψ(1 − XC)) and the domain of validity of this expression will be deduced from experimental observations, see ref 16 and the discussion in section 5.1. ΦO2 is finally given through

considered within the particle, in agreement with the discussion presented16 for a coal particle. The presence of noncarbonaceous minerals is accounted for to estimate the heat capacity of the char particle. The combustion process is supposed to be controlled both by kinetics and by oxygen diffusion inside the particle and in a boundary layer. Estimating the particle effectiveness parameter ηp allows one to calculate the theoretical particle combustion rate. This parameter ηp is controlled by the porous structure, and the temperature is calculated at every time step. Both Fick and Knudsen diffusions are accounted for. A mean pore radius is estimated for the calculation of the Knudsen diffusion. The porous structure evolution is modeled by eq 1 with XC instead of XC,th, as was already proposed by Bhatia and Vartak.3,21 Fragmentation is not considered here, and a constant particle radius is thus supposed. The probability of particle fracture during the particle fall is indeed surely low because of the small residence time of the particles and of the absence of shocks between particles. SEM pictures of coal char particles injected in a drop tube furnace under different atmospheres are presented in ref 25. 4.2.2. Equations. Position of the Particle. The position of the particle during its fall is calculated as a function of time through xp = ∫ t0 ug dt, where ug is the gas velocity, which depends on the gas flow F0g and whose expression is known at 298 K: ug = (Tg/298)F0g/(π(dr/2)2). Heat Transfer. The energy balance during the char particle heating in the drop tube furnace is written as 0 (mC0(1 − XC)(Cp)C + mp0τash (Cp)ash )

∂Tp ∂t

3Λ ηpksρp0 4πR p3A 0(1 − XC) 1 − Ψ ln(1 − XC)

+

1 kd 4πR p2

Computation of the Particle Effectiveness Factor. Introducing x = r/Rp leads to the following dimensionless equation for the oxygen transport within the particle: ksA CρC 1 ∂ ⎛ 2 ∂XO2 ⎞ XO2 ⎜x ⎟ = R p2 2 ΛDOe2 x ∂x ⎝ ∂x ⎠

(4)

which leads to the following expression of the Thiele modulus:

= 4πR p2εpσ(TR4 − Tp4)

0 0 + 4πR p2h(Tg − Tp) + (τ |ΔR HCO | + |ΔR HCO |) 2

(CO2 )∞

ΦO2 =

ΦO2

Φ = Rp

1 + τ /2 (3)

The initial values of carbon and particle masses are calculated 4 as m0C = m0p(1 − τ0ash) and mp0 = ρp0 3 πR p3, respectively. The thermal parameters (Cp)C and (Cp)ash used in eq 3 are given by the following expressions:

= Rp

ksA CρC ΛDOe2 ksA 0ρp0 (1 − XC) 1 − Ψ ln(1 − XC) ΛDOe2

(5)

In eq 4, DeO2 is the effective oxygen diffusivity within the porous structure. This diffusivity, which accounts for both Fick and Knudsen diffusions, is given through: 1/DeO2 = 1/DeK + 1/ DeF, with DeK = DKθp/τp and DeF = DFθp/τp. The Knudsen diffusivity is related to the mean pore radius:

⎛ 1200 ⎞⎛ exp(1200/Tp) − 1 ⎞2 ⎟⎟⎜⎜ ⎟⎟ (Cp)C = exp⎜⎜ MC 1200/Tp ⎝ Tp ⎠⎝ ⎠ 3R g

(Cp)ash = 754 + 0.586(Tp − 273) rp =

see ref 23. The char emissivity εp is taken equal to 0.85. The convective coefficient h is calculated from (h)Tm = Nu(λg)Tm/ (2Rp), taking the Nusselt number Nu equal to 2. Tm is the mean temperature of the boundary layer: Tm = (Tp + Tg)/2. The gas thermal conductivity (λg)Tm is related to Tm through (λg)Tm = 0.0207(Tm/273)0.5(1 + 113/273)/(1 + 113/Tm). The reaction enthalpies for CO and CO2 formations are taken as

2rf θp A CρC

(6)

through DK = 4 rp 2R gTp/(πMO2) /3. The roughness factor rf is taken equal to 2.6 The particle porosity θp is a function of the particle density: θp = 1 − ρp/ρcr, where ρcr is the density of carbon crystallites, whose values are given as 1.4 and 1.82 g/ cm3 in ref 24 and 1.33 g/cm3 in ref 6. A mean value of 1.5 g/ cm3 is taken in the present model. The Fickian diffusivity DF is expressed as DF = 1.78 × 10−5(Tp/273)1.75. The density ρC of the carbonaceous part of the particle is related to the conversion XC through: ρC = ρ0p(1 − XC)(1 − τ0ash). The particle density ρp is expressed as ρp = ρ0p(1 − XC(1 − τ0ash)). The particle effectiveness factor ηp is related to the Thiele modulus Φ through

0 ΔR HCO = (−110.5 × 103) − 6.38(Tp − 298) 0 ΔR HCO = (−393.5 × 103) − 13.1(Tp − 298) 2

The CO/CO2 ratio τ is approximated as a function of the gas temperature through: τ = 102.5 exp(−25 000/(RgT)); see ref 22. Oxygen Flux. The oxygen flux ΦO2 entering the particle is defined through the expression: ΦO2 = kd4πR2p((CO2)Rp −

ηp = 11395

3 ⎛⎜ 1 1⎞ − ⎟ Φ ⎝ tanh Φ Φ⎠ DOI: 10.1021/acs.energyfuels.7b01756 Energy Fuels 2017, 31, 11391−11403

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For the determination of the optimal values of (ks)0 and E, which appear in the Arrhenius expression of the kinetic constant ks, the Scilab code, which has been described in section 4.1.2, has been applied in the temperature range 485− 635 °C, thus eliminating the domains of temperature corresponding to a very low conversion of the carbonaceous material and to conversions close to 1, for which large uncertainties may exist. The optimal kinetic constant is found equal to ks = 1050 exp(−153 200/(RgT)). The determination of the kinetic constant ks is affected by the knowledge of the initial specific surface area A0 and of its variations along the combustion process. Classical statistical tools lead to the following standard deviations of E: 2400 J/mol. The activation energy will thus be taken equal to 153 200 ± 2400 J/mol. Of course, the value of A0, computed from CO2 adsorption isotherms, remains questionable, especially when the complete description of the porosity is not performed. Regarding the variations of the specific surface area, modeled by eq 1 and using the structural parameter Ψ, this approach may be considered as a possible way to get a draft estimate of the porosity evolutions. A possible opening of closed pores during the combustion process is not considered in the random pore model; see ref 3. Despite the uncertainty about the porosity representation, it appears a priori important to account for the variations of the active surface area. Nevertheless, the value of 153 200 ± 2400 J/mol of this active surface area is in reasonable agreement with that of 134 000 J/mol, which is cited in ref 27 for a bituminous coal. 5.2. Char Combustion in the Drop Tube Furnace. 5.2.1. Experimental Gas Temperature. The gas temperature was measured with a thermocouple located at different points along the reactor axis for three different regulation temperatures of 1100, 1200, and 1300 °C without any char injection. The injected gas entered the reactor at the end of the injection probe at about 900 °C and then was heated by the reactor walls. The gas was reaching a temperature plateau at the temperature of the isothermal part of the walls; see Table 2, column 2.

In the experiments realized in the drop tube furnace, the initial time t = 0 is chosen when the char particle leaves the injection probe, which leads to the initial conditions: Tp = 323 K, XC = 0, and xp = 0 m. The problem under consideration in the present work thus mainly consists of two equations: an energy balance written for the particle (eq 3) and an oxygen mass transport balance within the particle (eq 4), assuming that the structural parameters (porosity, density, and pore radius) involved in this model are uniform inside the particle. The temperature is also supposed to be uniform in the particle. Equation 3 allows one to determine the particle temperature. Equation 4 allows one to calculate the oxygen concentration gradient within the particle and to define the Thiele modulus Φ through eq 5, which is then used for the computation of the effectiveness factor of the particle ηp, leading to the determination of ΦO2, which realizes the coupling between oxygen mass transport and energy balances. Numerical Resolution of the Problem Using Fortran Software. A Fortran code has been developed, which solves eq 3 through an explicit Euler scheme with respect to the time parameter. Starting from the above-indicated initial conditions, this code successively computes the values of the different quantities ug, m0c , ..., ηp, Φ, ..., ρp, as previously exposed. The application of the explicit Euler scheme leads to the value of Tp through eq 3, and then that of the conversion XC at time t + dt through XC(t + dt ) = XC(t ) + dXC(t ) = XC(t ) + dt(CO2 )∞ MCksA C0(1 − XC(t )) ×

1 − Ψ ln(1 − XC(t ))

The time step dt was taken equal to 2 × 10−4 s. The key point of this code is the resolution of eq 4, which is written in spherical coordinates. A simple and efficient second-order finite difference method has been elaborated for this resolution, which is described in ref 26.

Table 2. Gas Temperature Plateau Measured (Thermocouple) without Particle Injection and Initial Temperatures Recorded by the Pyrometer during Particle Combustion for Three Regulation Temperaturesa

5. RESULTS AND DISCUSSION 5.1. Determination of the Kinetic Constant ks through a Thermogravimetric Analysis of the Coal Char Particle. Measurements performed on the partially burnt coal char samples revealed that during the combustion process the specific surface area was increasing up to a maximal value equal to 300 000 m2/kg, corresponding to a carbonaceous material conversion equal to 11%, and then remained constant up to a total conversion; see Figure 2. The constant value of 300 000 m2/kg was thus imposed in the model for XC bigger than 0.11. For XC < 0.11, Bhatia’s relation (eq 1), with XC instead of XC,th was used to simulate the values of the specific surface area. When XC increases from 0 to 0.11, the temperature remains low enough during most of the corresponding interval 0−0.1 s, which means that the char combustion can be considered under kinetic control, validating the use of Bhatia’s relation. Taking Ψ = 37, eq 1 was found to reasonably represent the increasing evolution of the surface area. The following conditions were thus imposed: if XC ≤ 0.11, Ψ = 37.0, A0C = 131 800 m2/kg, and if XC > 0.11, Ψ = 0.0. As exposed later, this hypothesis will not lead to wrong simulations of the combustion process; see also the discussion in section 5.2.5.

regulation temperature

temperature plateau

initial pyrometer temperature

1100 1200 1300

1035 1132 1242

1048 1133 1236

a

All of the temperatures are measured in °C.

These uniform wall temperatures were measured between 58 and 68 °C smaller than the regulation temperature, depending on the regulation temperature, probably due to heat losses across the inlet and outlet sections of the reactor. After this plateau, the gas temperature decreased up to the outlet section of the reactor. 5.2.2. Experimental Particle Temperatures. For each regulation temperature, the particle temperature has been recorded by the two-colors pyrometer for at least nine successive injection sequences. For each regulation temperature, large variations of the particle temperatures have been observed either in the slopes of the increasing part of the curves or in the values of the maximal temperature. For the regulation 11396


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Figure 3. Mean temperature curves during combustion of char particles in a drop tube reactor for the three regulation temperatures of 1100, 1200, and 1300 °C.

temperature equal to 1200 °C, the maximal difference between the lowest and highest maximal temperatures is equal to 63.3 °C (between 1289.1 and 1352.4 °C, with a mean value of 1326.4 °C and a standard deviation equal to 25.0 °C). These maximal temperatures occur at times, which vary between 0.258 and 0.334 s, with a mean value equal to 0.289 s and a standard deviation equal to 0.027 s. Solomon gives16 50 °C as the uncertainty for particle temperatures measured in drop tube furnaces during coal pyrolysis. The overall duration of the combustion process remains the same in the different experiments for each regulation temperature. Quite similar observations can be done in the cases of combustions performed under 1100 and 1300 °C regulation temperatures. The precise causes of these variations are difficult to identify, but an important lack of repeatability of the particle injection process may occur. The initial pyrometer temperatures (see Table 2, third column) are close to the plateau temperature, whatever the regulation temperature. This means that the pyrometer also detects the wall temperature, which is the highest temperature in the reactor in the absence of combustion. When the particle enters the reactor, its temperature highly increases, and the pyrometer rapidly detects this temperature when it becomes higher than that of the walls. After a temperature peak, the particle temperature decreases and reaches the same temperature as that observed before the particle entrance. For each regulation temperature, a particular curve called the mean curve was selected among the set of experimental temperature curves, choosing the maximal pyrometer temperature as criterion. This mean curve exhibits a maximum that is roughly the mean of all of the maxima. On Figure 3, these mean curves are presented for the three regulation temperatures of 1100, 1200, and 1300 °C. In Figure 3, time 0 was arbitrarily chosen. As expected, the temperature maximum increases with the regulation temperature. Surprisingly, the slope of the increasing part of the curve corresponding to the 1300 °C regulation temperature seems abnormally low in comparison to the two other slopes.

5.2.3. Comparison between the Experimental and Simulated Particle Temperatures and Simulation of the Conversion. The model described in section 4.2 has been used to predict the particle temperature during its combustion in the drop tube furnace and the evolution of the conversion versus time. In the simulations, the initial time 0 was here set when the particle enters the reactor at a temperature of 50 °C. Figure 4a−c compares the experimental and simulated evolutions of the particle temperature for the three regulation temperatures. As was already discussed, for the experimental curve representing the temperature returned by the pyrometer, the time 0 at which the particle leaves the injection probe is not precisely known. The pyrometer signal starts increasing when the particle temperature reaches the flat small initial part of the pyrometer signal. The pyrometer signal and the simulated temperature curve are forced to cross at the point corresponding to the start of the pyrometer signal increase, through a simple translation of the pyrometer signal along the time axis. The pyrometer temperature signal and the simulated particle temperature exhibit a quite good agreement in Figure 4b, which corresponds to the 1200 °C regulation temperature. The maximal temperatures are quite close in the two curves (44 °C difference, corresponding to a percentage deviation approximately equal to 3%), and the combustion durations (about 0.5 s) are in good agreement. In the case of Figure 4a (1100 °C regulation temperature), the agreement is not so good, especially when considering the maximal temperature (73 °C difference, corresponding to a percentage deviation approximately equal to 6%). Yet remembering the differences between the lowest and highest observed temperatures of the particle inside the drop tube furnace (75 °C difference), the results of the present model may be considered as valid. Further, the combustion durations are in good agreement (about 0.8 s), still in this case of a regulation temperature of 1100 °C. In the case of Figure 4c (1300 °C regulation temperature), the pyrometer signal does not seem to be realistic because of a too low slope of the increasing part of the signal (136 °C difference, corresponding to a percentage deviation approx11397


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Figure 4. Comparison between model calculations and measurements of the particle temperature versus time during the particle drop in a drop tube reactor for a regulation temperature of (a) 1100 °C, (b) 1200 °C, and (c) 1300 °C. 11398


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Figure 5. Simulations of the carbonaceous material conversion as a function of the particle position, for the three regulation temperatures (1100, 1200, and 1300 °C).

Limitations to oxygen diffusion are at work during a large part of the combustion process. When the diffusional limitations for oxygen transport within the particle are high, combustion is very close to occur within a shell at the particle surface. This can be verified calculating the oxygen concentration versus the radial position (CO2)r = (CO2)Rp(Rp/r) sinh(Φr/Rp)/sinh(Φ). In Figure 7 corresponding to the 1200 °C regulation temperature, three oxygen concentration profiles within the particle are presented, which correspond to the times at which the three conversions, respectively equal to 0.096, 0.594, and 0.831, are reached. At the beginning of the combustion process, that is, at a time corresponding to 9.6% conversion, when only 9.62% of the carbonaceous material is consumed, the O2 concentration exhibits a very fast decrease close to the particle surface. Only the particle volume between r/Rp = 0.7 and the surface is concerned by combustion. When 59.4% of the carbonaceous material is consumed, the combustion front has only progressed toward the particle center up to r/Rp = 0.6. Finally, when 83.1% of the carbonaceous material is consumed, the particle center is also concerned by combustion. As expected, these oxygen concentration gradients are more pronounced when the regulation temperature increases (curves not presented here). The different combustion regimes depend on the Thiele modulus Φ given by eq 5. According to the hypotheses of the model, the value of the Thiele modulus is uniform in the whole particle in which the porous structure as well as the temperature are supposed to remain uniform during the whole combustion process. The values of the Thiele modulus corresponding to the three conversions 0.096, 0.594, and 0.831 are equal to 17.9, 10.6, and 3.8, respectively. 5.2.5. Influence of the Specific Surface Area. To investigate the effect of the specific surface area, another simulation of the combustion process has been realized assuming that the maximal value of the specific surface area is now equal to 400 000 m2/kg instead of 300 000 m2/kg, which represents a significant change. However, the simulation of the combustion with a 1200 °C regulation temperature with this higher value of

imately equal to 10%). The experimental temperature curve indicated in Figure 4c (and also in Figure 3) is yet the mean curve of repeated combustion experiments under the regulation temperature of 1300 °C. Nevertheless, the simulated temperature curve seems to be very realistic in the three cases, with a maximal temperature that increases and a combustion duration that decreases when the regulation temperature increases from 1100 to 1300 °C. The evolutions of the conversion versus the particle position along the reactor axis are presented in Figure 5 for the three regulation temperatures. As expected, a larger acceleration of the combustion process occurs when the regulation temperature increases. The combustion process is complete after a particle drop of about 30, 15, and 10 cm, for the three regulation temperatures of 1100, 1200, and 1300 °C, respectively. 5.2.4. Combustion Regime of the Particle. The value of the effectiveness factor ηp allows one to determine the combustion regime of the particle. Figure 6 presents the evolution of ηp versus time or versus the conversion XC for the three regulation temperatures of 1100, 1200, and 1300 °C. At the very beginning of the particle drop, the effectiveness factor remains almost equal to 1, as the combustion rate is here very small due to low particle temperatures. After this initial stage, a fast decrease of the effectiveness factor is observed, due to a fast particle heating. For the 1300 °C temperature regulation, the factor reaches a minimum value 0.1, which means that a mixed diffusional and kinetic regime occurs, with a very large diffusional contribution. For the 1200 °C temperature regulation, the minimum value is 0.15. This mixed diffusional and kinetic regime occurs during a short time (Figure 6a) and prevails during a large conversion range, roughly between 0.1 and 0.6 for the 1200 and 1300 °C regulation temperatures (see Figure 6b). The effectiveness factor ηp then increases up to 1. The diffusional limitations here decrease due to porosity increase and above a conversion between 0.6 and 0.9, depending on the regulation temperature. Combustion is here close to be controlled by kinetics. 11399


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Figure 6. Evolutions of the effectiveness factor ηp of the char particle during its drop in a drop tube reactor for three regulation temperatures (1100, 1200, and 1300 °C), as a function of (a) time and (b) conversion.

fall in the drop tube furnace, this simple model is also able to predict the evolution of the conversion during the combustion process and the impact of different structural parameters acting on the combustion process. The agreement between the simulated and measured particle temperature looks relatively poor, although the differences between the simulated and experimental particle temperatures lie in the uncertainties, which are admitted for such experiments and which are confirmed in the present study through repeatability combustion experiments. The resolution of the two equations involved in this model does not require time-consuming computations, and they can be solved using a Fortran software. This model is much simpler than the comprehensive model that has been introduced,9,10 for example, which is based on 8 + 7 equations. Of course, this simple model does not give as many details as the comprehensive one. The quite high differences between the experimental and simulated particle temperatures are surely due to both the

the specific surface area returned that the maximal combustion temperature was only increased by 4 °C and that the position of this maximum on the reactor axis was almost unchanged, as well as the corresponding conversion. This absence of sensitivity of the model toward the specific surface area is partially due to the fact that the combustion process is for a large part controlled by oxygen diffusion, limiting the influence of the kinetic parameters. Another reason is that the increase of the specific surface area decreases the mean pore radius rp, given in eq 6, which introduces an antagonistic effect on the combustion rate. The above-indicated approximate values of the specific surface area can thus be accepted.

6. CONCLUSIONS A simple model has been proposed, which simulates the evolution during a combustion process of the temperature of a coal char particle injected in a drop tube furnace. Especially devoted to the simulation of the particle temperature during its 11400


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Figure 7. Instantaneous oxygen concentration as a function of the radial position within the char particle at three times corresponding to three different conversions (0.096, 0.594, and 0.831) during the particle combustion in a drop tube reactor for a regulation temperature of 1200 °C.

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ACKNOWLEDGMENTS We thank Mrs. Fatiha Mechati, Mrs. Sandrine Lesieur, Mrs. Damaris Kerhli, and M. Olivier Allgaier for their fruitful contributions to the present study. We also thank Dr. L. Porcheron (EDF R&D), E. Thunin (EDF R&D), and Dr. P. Pilon (EDF R&D) for their support and helpful discussions along this study. We also thank the anonymous referees whose comments contributed to improve a previous version of the work.

difficulty to obtain pertinent experimental measurements of the particle temperature in such conditions (that is, using a twocolors pyrometer) and the poor knowledge of some physical properties of the char particle such as emissivity and thermal properties and their variations along the combustion process. Of course, the simplified particle structure, a unique pore model, which is here considered, may also lead to the observed relatively poor agreement. The particle structure drastically evolves locally during the combustion process, and parameters such as particle porosity, pore size, apparent carbon density, and specific surface area should be considered as local quantities, which evolve inside the particle along this combustion process. Because of diffusional limitations of oxygen transport, a combustion front progresses from the surface to the center of the particle. A porosity gradient thus appears within the particle. A unique pore model trivially does not account for this phenomenon. The only way to account for the evolution of the porous medium during the combustion process should be to consider local parameters depending on the radial position (carbon density, particle density, specific surface area, particle porosity, oxygen diffusivities, etc.) and to model local combustions occurring in successive elemental spherical annular volumes inside the particle. A more sophisticated model, using local quantities within the particle, is currently in progress in the lab. Nevertheless, the simplified model, which has been presented, could be useful for a quick and rough description of the particle behavior during the combustion process.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Alain Brillard: 0000-0003-0615-8880 Notes

The authors declare no competing financial interest. 11401

NOMENCLATURE a = constant heating rate for TGA (10 °C/min), K/s Ac = specific surface area of the carbonaceous part, m2/kg A0 = specific surface area of the initial material, m2/kg A0C = specific surface area of the carbonaceous part of the initial material, m2/kg (Cp)c = calorific capacity of carbon, J/kg·K (Cp)ash = calorific capacity of ash, J/kg·K CO2 = oxygen concentration within the char particle, mol/m3 (CO2)Rp = oxygen concentration at the particle surface, mol/ m3 (CO2)∞ = oxygen concentration far from the particle, mol/m3 dr = diameter of the reactor of the drop tube, m DF = Fick diffusivity of oxygen, m2/s DK = Knudsen diffusivity of oxygen, m2/s DeF = effective Fick diffusivity of oxygen, m2/s DeK = effective Knudsen diffusivity of oxygen, m2/s DeO2 = effective diffusivity of oxygen within the particle, m2/s (DO2)F,bl = diffusivity of oxygen in the boundary layer, m2/s E = activation energy, J/mol f void = void volume fraction in the bulk material F0g = gas flow, m3/s h = convective heat transfer coefficient, W/m2·K ks = kinetic constant, m/s kd = mass transfer coefficient of oxygen, m/s (ks)0 = frequency factor, m/s m = mass of the sample (TG), kg m0p = initial mass of the char particle, kg m0 = initial mass of the sample (TG) at 300 °C, kg DOI: 10.1021/acs.energyfuels.7b01756 Energy Fuels 2017, 31, 11391−11403

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(3) Bhatia, S. K.; Perlmutter, D. D. A random pore model for fluidsolid reactions: I. isothermal, kinetic control. AIChE J. 1980, 26, 379− 386. (4) Ma, L.; Mitchell, R. Modeling char oxidation behaviour under zone II burning conditions at elevated pressures. Combust. Flame 2009, 156, 37−50. (5) Hustad, J. E.; Vareide, D.; Sønju, O. K. The evolution of porosity and intrinsic reactivity of coke particles. Fuel Process. Technol. 1993, 36, 81−89. (6) Mitchell, R. E.; Ma, L.; Kim, B. On the burning behavior of pulverized coal chars. Combust. Flame 2007, 151, 426−436. (7) Hashimoto, N.; Watanabe, H. Numerical analysis on effect of furnace scale on heat transfer mechanism of coal particles in pulverized coal combustion field. Fuel Process. Technol. 2016, 145, 20−30. (8) Bhunia, S.; Sadhukhan, A. K.; Gupta, P. Modelling and experimental studies on oxy-fuel combustion of coarse size coal char. Fuel Process. Technol. 2017, 158, 73−84. (9) Sadhukhan, A. K.; Gupta, P.; Saha, R. K. Modeling and experimental studies on single particle coal devolatilization and residual char combustion in fluidized bed. Fuel 2011, 90, 2132−2141. (10) Sadhukhan, A. K.; Gupta, P.; Saha, R. K. Modeling and experimental investigations on the pyrolysis of large coal particles. Energy Fuels 2011, 25, 5573−5583. (11) Kelebopile, L.; Sun, R.; Wang, H.; Zhang, X.; Wu, S. Pore development and combustion behavior of gasified semi-char in a drop tube furnace. Fuel Process. Technol. 2013, 111, 42−54. (12) Everson, R. C.; Neomagus, H. W. J. P.; Kaitano, R. The random pore model with intraparticle diffusion for the description of combustion of char particles derived from mineral- and inertinite rich coal. Fuel 2011, 90, 2347−2352. (13) Al-Abbas, A. H.; Naser, J.; Hussein, E. K. Numerical simulation of brown coal combustion in a 550 MW tangentially-fired furnace under different operating conditions. Fuel 2013, 107, 688−698. (14) Barranco, R.; Rojas, A.; Barraza, J.; Lester, E. A new char combustion kinetic model 1. Formulation. Fuel 2009, 88, 2335−2339. (15) Bhuiyan, A. A.; Naser, J. Numerical modelling of oxy fuel combustion, the effect of radiative and convective heat transfer and burnout. Fuel 2015, 139 (0), 268−284. (16) Solomon, P. R.; Serio, M. A.; Suuberg, E. M. Coal pyrolysis: experiments, kinetic rates and mechanisms. Prog. Energy Combust. Sci. 1992, 18, 113−220. (17) Karlström, O.; Brink, A.; Biagini, E.; Hupa, M.; Tognotti, L. Comparing reaction orders of anthracite chars with bituminous coal chars at high temperature oxidation conditions. Proc. Combust. Inst. 2013, 34, 2427−2434. (18) Zouaoui, N.; Brilhac, J. F.; Mechati, F.; Jeguirim, M.; Djellouli, B.; Gilot, P. Study of experimental and theoretical procedures when using thermogravimetric analysis to determine kinetic parameters of carbon black oxidation. J. Therm. Anal. Calorim. 2010, 102, 837−849. (19) Thommes, M.; Kaneko, K.; Neimark, A. V.; Olivier, J. P.; Rodriguez-Reinoso, F.; Rouquerol, J.; Sing, K. S. W. Physisorption of gases, with special reference to the evaluation of surface area and pore size distribution. Pure Appl. Chem. 2015, 87 (9−10), 1051−1069. (20) Chen, S. G.; Yang, R. T. Theoretical basis for the potential theory adsorption isotherms. The Dubinin-Radushkevich and Dubinin-Astakhov equations. Langmuir 1994, 10, 4244−4249. (21) Bhatia, S. K.; Vartak, B. J. Reaction of microporous solids: the discrete random pore model. Carbon 1996, 34, 11:1383−1391. (22) Laurendeau, M. L. Heterogeneous kinetics of coal char gasification and combustion. Prog. Energy Combust. Sci. 1978, 4, 221−270. (23) Merrick, D. Mathematical models of the thermal decomposition of coal: 2. Specific heats and heats of reaction. Fuel 1983, 62, 540−546. (24) Su, J. L.; Perlmutter, D. D. Effect of pore structure on char oxidation kinetics. AIChE J. 1985, 31, 973−981. (25) Yi, B.; Zhang, L.; Yuan, Q.; Yan, S.; Zheng, C. The evolution of coal char structure under the oxy-fuel combustion containing high H2O. Fuel Process. Technol. 2016, 152, 294−302.

mC = mass of carbon within the char particle/mass of carbon within the char sample (TG), kg m0C = initial mass of carbon within the char particle/initial mass of carbon in the sample (TG) at 300 °C, kg MC = molar weight of carbon, kg/mol MO2 = molar weight of oxygen, kg/mol Nu = Nusselt number (here taken equal to 2) P = pressure within the reactor, Pa r = radial coordinate, m rp = mean pore radius, m RC = carbonaceous material consumption rate of oxygen, kg/ s RC,th = carbonaceous material consumption rate of oxygen in the thermobalance, kg/s Rp = particle radius, m Rg = perfect gas constant, J/mol·K Sh = Sherwood number (here taken equal to 2) t = time, s T = gas and sample temperature (TG), K Tg = gas temperature (entrained flow reactor), K Tm = mean temperature in the boundary layer around a particle, K T0 = reference gas temperature (573 K), K Tp = particle temperature (drop reactor), K TR = temperature of the reactor walls, K ug = gas velocity, m/s x = dimensionless radial coordinate within the particle xp = coordinate for particle position in the drop reactor, m XC = carbonaceous material conversion ratio XC,th = conversion ratio in the thermobalance XO2 = oxygen molar fraction (XO2)∞ = oxygen molar fraction far from the particle z = parameter for estimation of thermal capacity ΔRH0CO = reaction enthalpy for CO formation, J/mol ΔRH0CO2 = reaction enthalpy for CO2 formation, J/mol ηp = particle effectiveness factor εp = char particle emissivity (taken as 0.85) λg = gas thermal conductivity, W/m·K Φ = Thiele modulus of the particle ΦO2 = molar oxygen flux, mol/s Λ = molar stoichiometric factor, mol-C/mol-O2 ρC = density of carbonaceous particle material, kg/m3 ρcr = true density of carbon, kg/m3 ρp = particle density, kg/m3 ρ0p = initial apparent particle density, kg/m3 (ρbulk)rm = bulk density of the raw material, kg/m3 σ = Stefan−Boltzmann constant, W/m2·K4 τ = molar ratio CO/CO2 τp = tortuosity factor of the particle (here taken equal to 2) τ0ash = ash content in the initial char particle (τHC,W)rm = HC and water content within the raw material θp = particle porosity Ψ = structural parameter

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REFERENCES

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A Simplified Model Accounting for the Combustion of Pulverized Coal

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