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Direct Numerical Simulation of Pulverized Coal Combustion in a Hot Vitiated Co-ﬂow Kun Luo, Haiou Wang, Jianren Fan,* and Fuxing Yi State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou 310027, People’s Republic of China ABSTRACT: A compressible direct numerical simulation (DNS) solver for pulverized coal combustion has been developed and used to study a pulverized coal jet ﬂame with a Reynolds number of 28 284 based on the nozzle diameter. An eighth-order center diﬀerential scheme combined with an explicit tenth-order ﬁlter is used for spatial discretization. The classical fourth-order Runge−Kutta method is used for time integration. The characteristic non-reﬂecting boundary conditions are used to describe the boundary conditions. A comprehensive model for coal combustion is applied, and the reaction mechanism of CH4 with ﬁve species and two-step reactions is adopted for gas-phase combustion. The grid system has been carefully designed to make sure that turbulent scales and chemical reaction scales are reasonably resolved and the point-source assumptions of particles are valid. The simulation is partially validated against the experiment, and the particle behavior is investigated. It is found that, in the upstream region, the reaction rate is quite scattered and a single particle is found inside the burning ﬂame to form an individual particle combustion mode. While in the downstream region, the reaction zone is more continuous, with a large number of particles enclosed, which characterizes the group combustion mode. Conditional statistics with respect to the mixture fraction are also obtained to provide insights into coal combustion and the related models. For turbulent ﬂows and combustion, it is well-known that direct numerical simulation (DNS) has a unique advantage to resolve all length and time scales of turbulence and chemistry compared to other methods. With the fast development of super computers, it is now possible to study two-phase combustion using DNS. Nakamura et al.10 studied gas−liquid two-phase combustion ﬂow with a two-dimensional DNS method and found that the gas combustion region was approached to premixed combustion in the front of the hightemperature combustion zone. Reveillon and Demoulin11 used a three-dimensional DNS method to study the droplet gathering eﬀect on the inﬂuence of droplets evaporating in isotropic turbulence. It was found that, when the Stokes number was close to unity, the droplet gathering eﬀect was evident. With the increase of the droplet gathering eﬀect, the evaporation rates of the droplet group would decline. Baba and Kurose12 used DNS data to validate the ﬂamelet/progress variable model. The results showed that using the total enthalpy instead of the product mass fraction to present the progress variables could give better results. Luo et al.13 studied threedimensional spray swirl ﬂames with the DNS method. It was found that premixed combustion contributed more than 70% to the total heat release rate and the conditional mean evaporation rate increased almost linearly with the mixture fraction in the lean case and more complex in the rich case. Although there are more and more DNS studies of gas− liquid two-phase combustion, DNS of coal combustion is still very scarce. To the authors’ knowledge, the present work is the ﬁrst attempt to investigate a pulverized coal ﬂame using DNS. In the present study, a DNS solver is developed and applied to

1. INTRODUCTION Gas−solid/gas−liquid two-phase combustion exists in a large number of industrial applications, such as coal combustion in power plants and droplet combustion in internal engines. The underlying physics of two-phase combustion is very complicated. Usually, phase change, heat and mass transfer, chemical reaction, and interphase interactions coexist and are tightly coupled in the system. Understanding these unsteady multiphysics/multiscale interactions is obviously of paramount importance to two-phase combustion optimization and control. Some experimental work has been performed to study pulverized coal jet ﬂames. 1 However, because of the experimental diﬃculties in measuring complex two-phase ﬂow and combustion, the understanding of unsteady features of turbulent two-phase ﬂow and combustion relies heavily on the computational ﬂuid dynamics.2,3 In the early time, numerical methods based on the Reynolds-averaged Navier−Stokes (RANS) equation were used to study coal combustion.4−6 Murphy and Shaddix7 studied the relationship between the reaction rate of the coke and oxygen concentration and temperature of the gas phase in a jet ﬂame. The results showed that the single-ﬁlm model could describe the coke-burning feature in a proper way. Recently, large eddy simulation (LES) has also been applied in gas−solid two-phase combustion. Kurose and Makino8 showed that evaporation and released heat could increase gas-phase velocity in the center line as well as the jet half width but reduced the root-mean-square (rms) speed of the gas phase. The location of the largest concentration of particles would spread in the radical direction because of the increase of the axial velocity. Yamamoto et al.9 presented a LES of a pulverized coal jet ﬂame ignited by a preheated gas ﬂow. The agreement between the LES results and experimental data indicated that LES was able to correctly simulate pulverized coal combustion. © 2012 American Chemical Society

Received: May 28, 2012 Revised: September 26, 2012 Published: September 27, 2012 6128

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study a pulverized coal ﬂame in a hot vitiated co-ﬂow. The main objective is to analyze the ﬂame characteristics of pulverized coal combustion and provide statistical data that are meaningful to two-phase combustion models. The remainder of the paper is organized as follows. The numerical methods are introduced in section 2. The ﬂame characteristics and some conditional statistics are outlined in section 3. Finally, some conclusions are made in section 4.

Q = πd pNuλs(Tg − Tp)

Ω̇ = εσπd p2(Tg 4 − Tp 4)

∂t

∂ρg ug, i ∂xi

∂ρg ug, j

+

∂t

= Sṁ

∂ρg ug, iug, j ∂xi

(1)

∂ρg Yk ∂t

+

du p, i dt

∂τij ∂p =− + + ρg gj − β(ug, j − u p, j) ∂xj ∂xi ̇ j + S V,

∂ρg (ug, i + Vk , i)Yk ∂xi

ρg Cv

Dt

= ω̇ ″T − RTg − ρg

∂Tg ∂xi N

+ ρg

N

∂ (ρ ∂xi g

dm p

(3)

N

∑ (YkVk , iCp, k) + k=1

∂ug, i ∂ ⎛ ∂Tg ⎞ ⎜λ g ⎟ + σij ∂xi ⎝ ∂xi ⎠ ∂xj

∑ Ykfk , i Vk , i − Q − Ω̇ + SṪ k=1

(4)

p = ρg

Rr Tg W̅

ρg |ug⃗ − u p⃗ | 3 C D0 4 dp

C D0

= ṁ w + ṁ v + ṁ c

(12)

(13)

⎛ E ⎞ ⎛ E ⎞ ṁ v = − α1mdaf B1 exp⎜⎜ − 1 ⎟⎟ − α2mdaf B1 exp⎜⎜ − 2 ⎟⎟ ⎝ R rTp ⎠ ⎝ R rTp ⎠

(14)

where B is the pre-exponential factor and E is the activation energy. The values of these parameters are listed in Table 1. In conventional investigations, the volatile matter is usually treated as CxHyOz, where x, y, and z are determined from the element mass fractions. However, a large number of species must be solved in the equation system. To reduce the computational cost, the volatile matter is assumed to be CH4 in the present work. A previous study proved that reasonable results could be expected when this assumption was made.25

(6)

Re ≤ 1000 (7)

Table 1. Pre-factors and Activation Energies for Coal Devolatilization and Char Reactions

If there is mass exchange between the gas and solid phases, the variation of particle mass could decrease the drag force of the particle as

A C D = C D0 exp(A) − 1

(11)

where the coal is taken to be dry and ash free (daf). α1 and α2 are the mass stoichiometric coeﬃcients, which equal 0.234 and 0.8, respectively. The devolatilization reaction rate is expressed as

where dp is the diameter of a single coal particle and CD0 is the Stokes drag coeﬃcient, which can be expressed as ⎧ 24/Re , Re ≤ 1 ⎪ = ⎨ 24/Re(1.0 + 0.15Re 0.687), ⎪ ⎩ 0.44, Re > 1000

+ gi

⎧ ⎪(1 − α1)R1 + α1V1 coal⎨ ⎪ ⎩(1 − α2)R 2 + α2V2

(5)

The inﬂuence of particles is reﬂected in terms of source terms in the above equations, where Ṡm is the mass source term, Ṡk is the mass source term of the kth species, ṠV is the momentum source term, and ṠT is the energy source term. These terms will be discussed in more detail later. β is a parameter representing the applied force between the particles and the gas, which can be written as β=

τr

where ṁ w is the moisture vapor releasing rate, ṁ v is the volatile releasing rate, and ṁ c is the carbon releasing rate. The moisture evaporation rate is calculated by a diﬀusion model in a way similar to the droplet evaporation rate.24 The coal devolatilization reaction is simulated by a two competing reaction kinetic model suggested by Kobayashi et al.17 In this model, the devolatilization process consists of two competing ﬁrst-order reactions to account for the eﬀects of the heating rate on devolilization. Each describes the degradation of coal to residual chars (R1 and R2) and volatiles (V1 and V2) as

∑ YkVk , i/Wk) k=1

ug, i − u p, i

is the particle relaxation time. ρp is the particle where τr = density, and ηs is the viscosity at the coal surface. According to eq 11, the particle velocity can be obtained through integration 1 time, while the particle position can be obtained through integration 2 times. The total mass change of a coal particle can be calculated as

dt DTg

=

dp2ρp/18ηs

(2)

= ω̇ k + Sk̇

(10)

where ε is the blackness of coal particles and σ is the Stefan− Boltzmann constant. 2.2. Governing Equations for the Solid Phase. Because the density of the particle is almost 1000 times heavier than the density of the gas phase, only the Stokes resistance and gravity are considered in the particle dynamic equations. Moreover, the coal particles can be regarded as the dilute phase; therefore, the collisions between the particles are neglected. Under these assumptions, the motion equation of a single particle can be written as

2.1. Governing Equations for the Gas Phase. For the compressible gas−solid two-phase reacting ﬂow, the governing equations for mass, momentum, species, energy, and ideal gas state are written as +

(9)

where Nu is the particle Nusselt number and λs is the thermal conductivity at the coal surface. Ω̇ is the radiative heat between the particles and the gas and is calculated with a simple radiation model as

2. NUMERICAL APPROACH

∂ρg

A exp(A) − 1

B devolatilization reaction 1 devolatilization reaction 2 char reaction A char reaction B char reaction C

(8)

where A is a parameter related to the mass-transfer rate. Q is the conductive and convective heat exchanged between the particles and the gas phase 6129

3.7 1.46 1.255 1.813 7.351

× × × × ×

105 s−1 1013 s−1 103 m s−1 103 m s−1 103 m s−1

E 7.366 2.511 9.977 1.089 1.380

× × × × ×

104 105 104 105 105

J/mol J/mol J/mol J/mol J/mol

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On the surface of the coal particles, three kinds of heterogeneous char reactions would take place14,16

C + O2 → CO2

(15)

2C + O2 → 2CO

(16)

C + CO2 → 2CO

(17)

CO +

ṁ cB = −

ṁ cC = −

ṁ c =

(19)

⎛ E ⎞ 1 πd p2ρs YCO2,sBC exp⎜⎜− C ⎟⎟ γC ⎝ R rTp ⎠

(20)

ṁ cA

+

ṁ cB

6

Cc

dTp dt

(21)

(22)

where ρp is the coal density and kept as a constant of 2.2 × 10 kg/m3 and Cc is the heat capacity of the coal calculated as

Cc = (836.0 + 1.53(Tp − 273.0) − 5.4 × 10−4(Tp − 273.0)2 ) (23) Lw, Qv1, and Qv2 are the heat generation related to moisture evaporation, low-temperature devolatilization, and high-temperature devolatilization, respectively. ΔHc is the heat release related to char combustion on the coal particle surface. In the gas-phase equations, the source terms contributed by the particles for the exchange of mass, momentum, and energy are expressed as

̇ j=− S V,

SṪ = −

1 ΔV

1 ΔV

∑ ak k

∑ ak k

∑ ak k

(24)

H2O

N2

0.97

0.83

1.00

temperature (K)

axis velocity (m/s)

Y O2

YH2O

Y N2

320 1600

107 0.5

0.233 0.1200

0 0.1237

0.767 0.7563

moisture (%)

volatile matter (%)

ﬁxed carbon (%)

ash (%)

1.5

23.4

39.1

36.0

dmp, k u p, j dt

The time step is set as Δt = 6.5 × 10−8 s, and the computation advances over 160 000 time steps. The whole computational time is more than 11 ms, so that reliable statistics could be obtained. A total of 1024 central processing units (CPUs) at Shanghai Supercomputer Center are employed for 3 month to accomplish the simulation.

(25)

dmp, k CcTp, k dt

(26)

where ΔV is the volume of the mesh and ak is the distribution coeﬃcient of the kth particle source term in that mesh. 2.3. Numerical Algorithms. To solve the gas-phase governing equations, an eighth-order center diﬀerential scheme is used for spatial discretization. An explicit tenth-order ﬁlter is employed to remove any spurious high-frequency ﬂuctuations in the solutions. The classical fourth-order Runge−Kutta method is adopted for time integration.18 The characteristic non-reﬂecting boundary condition19,20 is used to describe the boundary conditions. The reaction mechanism of CH4 consists of ﬁve species (CH4, O2, CO, CO2, and H2O) and two steps15 CH4 +

CH4

1.39

Table 4. Proximate Analysis of Coal

dmp, k dt

CO2

mm. A random velocity disturbance is imposed.26 The Reynolds number based on the jet nozzle reaches Re = 28 284. The computational domain size is 35.6D in the streamwise direction and 20D in the other two directions. To satisfy the requirements for DNS and the point-source assumptions, a uniform grid spacing of 101.6 μm is used in the streamwise direction. In the other two directions, stretched grids are used with a minimum grid spacing of 76.2 μm. The resultant grid number is 1664 × 656 × 640 = 698 613 760. It should be noted that the Kolmogorov scale calculated in the stabilization region is about 91 μm based on the DNS database. Thus, it is believed that the turbulent scales are well-resolved in our simulation. Coal particles are assumed to distribute randomly at the nozzle exit with a diameter of 10 μm. Temperature and velocity of the pulverized coal are regarded as the same as those of the local ﬂuid. The mass rate of the pulverized coal is given as 0.211 12 × 10−3 kg/s, which implies that 14 coal particles are injected in the computational domain at each time step. The coal studied here is a typical coal produced in Huainan, China, and the components are listed in Table 4.

3

1 ΔV

CO 1.10

central jet co-ﬂow

= πd p2εσ(Tg 4 − Tp 4) + Q − ṁ w Lw − ṁ v1Q v1 − ṁ v2Q v2 + ΔHc

Sṁ = −

O2 1.11

2.4. Computational Details. Inﬂows of the gas phase can be divided into two streams, namely, the center jet and the hot co-ﬂow jet, with parameters speciﬁed in Table 3. The nozzle diameter is 4.57

where γ is the equivalent coeﬃcient,14,16 ρs is the gas density, YO2,s is the O2 mass fraction, and YCO2,s is the CO2 mass fraction at the coal surface. The coal energy equation is written as ρp πd p3

components Lewis number

Table 3. Inﬂow Conditions for the Gas Phase

ṁ cC

+

Table 2. Lewis Number of Each Species

(18)

⎛ E ⎞ 1 πd p2ρs YO2,sBB exp⎜⎜ − B ⎟⎟ γB ⎝ R rTp ⎠

(28)

Speciﬁc heat, enthalpy, and entropy are calculated according to the JANAF database.21 Viscosity and thermal conductivity are evaluated using the CHEMIKEN software. The diﬀusion coeﬃcient is solved through the Lewis number. The Lewis numbers of the species are listed in Table 2.

The heterogeneous char reaction rates are assumed to be ﬁrst-order in oxygen concentration and carbon dioxide concentration ⎛ E ⎞ 1 ṁ cA = − πd p2ρs YO2,sBA exp⎜⎜ − A ⎟⎟ γA ⎝ R rTp ⎠

1 O2 ↔ CO2 2

3 O2 → CO + H 2O 2

3. RESULTS AND DISCUSSION 3.1. General Characteristics. In our previous investigations,27,28 the DNS results and the experimental data agreed well for the gas−solid two-phase non-reacting ﬂow and the combustion of a single quiescent carbon in air was tested. In the present study, a comparison of axial distributions of the pulverized coal particle velocity between the simulation and the previous experiment1 is carried out to partially validate this simulation. However, because the experimental and numerical conditions are diﬀerent, only qualitative agreement could be expected.

(27) 6130

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Figure 1. Axial proﬁles of the (a) mean and (b) rms pulverized coal particle velocities.

Figure 2. Radial proﬁles of the mean particle (a) diameter and (b) velocity at three axial locations.

Figure 1a shows the axial proﬁles of the mean particle velocity. It is seen that the velocity decays faster in the experiment than that in the simulation in the upstream region. In the experiment, the inﬂow air for combustion is fully developed and the particles tend to follow the motion of the ﬂuid, while in the simulation, the particle velocity is kept unchanged until at x/D = 8, where turbulence begins to develop. However, in the downstream region, the predicted mean particle velocity is in accordance with the measured mean particle velocity. The axial proﬁles of the rms particle velocity are compared in Figure 1b. As mentioned above, turbulence in the simulation has not been developed in the upstream region, so that the rms particle velocity could be neglected. Again, in the downstream region, the predicted and measured rms particle velocities are in good agreement. To investigate the particle behavior in more detail, the mean particle diameter and velocity at three axial locations (x/D = 10, 20, and 30) are presented. The radial proﬁles of the mean particle diameter are shown in Figure 2a. Two points are evident. First, the mean particle diameter tends to decrease as the particles traverse downstream. It is explained by the fact that the coal particles are generally consumed through devolatilization and char combustion in the downstream

region. However, in some experiments,1 it was found that the mean particle diameter increased in the downstream region by the disappearance of small particles and the particle swelling. Thus, to further study pulverized coal combustion, the eﬀect of the particle swelling should be taken into account. Second, the mean diameter near the central line is larger than that with a large radial distance. It is speculated that smaller particles are apt to follow the large eddies in the ﬂame and move outward. The radial proﬁles of the mean particle velocity are displayed in Figure 2b. It is noted that, as the particles traverse downstream, the maximum particle velocity decreases and the maximum radial distance increases. The dispersion of the particles is strongly aﬀected by the vortex structure. Meanwhile, the ﬂow can be modulated by the particles. These interactions will be examined in the following. It is very interesting and important to study the particle− vortex interactions in the pulverized coal ﬂame. Figure 3 depicts the particle distributions and the q vortex isosurface. The q vortex deﬁned as q = (WijWij − SijSij)/2 is used to visualize the vortex structure, where Wij and Sij denote the asymmetric and symmetric parts of the velocity gradient tensor, respectively. Because the ﬂame is very slim, the sketch map is divided into 6131

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Figure 5. Distributions of the heat release rate and particles in a typical x−z plane. Figure 3. Particle distribution (red) and q vortex isosurface (yellow) in (a) x/D = 0−18 and (b) x/D = 18−35.

Figure 4. Distributions of the (a) particles superimposed with the particle temperature and (b) temperature in a typical x−z plane.

two parts (panels a and b of Figure 3). From Figure 3a, it is clear that, in the upstream region, the inﬂow disturbance and the Kelvin−Helmholtz instability result in a periodic street of vortex rings. Along the streamwise direction, a lot of counterrotating vortices or ribs are produced. After the formation of these ribs, the vortex structure becomes more and more threedimensional. Figure 3a shows that the particles are not dispersed in the radial direction in the upstream region. As the particles traverse downstream, they are observed to accumulate around the ribs, as shown in Figure 3b. It is worth noting that the particle Stokes number deﬁned as the ratio of the particle aerodynamic response time to the ﬂow characteristic time decreases as the particles traverse downstream because of the decrease of the particle diameter. Thus, it

Figure 6. Conditional (a) mean and (b) rms temperatures with respect to the mixture fraction.

is speculated that St is an important parameter in particle dispersion in pulverized coal combustion. The vortices are 6132

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tion, and sheath combustion. If the particle cloud is very dilute, then the oxygen mass fraction away from each particle will be almost the same as the ambient oxygen mass fraction, burning under which is called individual particle combustion. As the number of particles per unit volume is increased, group combustion could be expected. Figure 5 displays the particle distribution in a typical x−z plane, on which the heat release rate contour is also superimposed. At a glance, we could tell that the characteristic of combustion in the upstream region is quite diﬀerent from that in the downstream region. Two representative regions are outlined as regions A and B in Figure 5. A close examination of these regions is carried out. Region A is in the upstream region, where the reaction rate is quite scattered and a single particle is found inside the burning core. Thus, the combustion mode is individual particle combustion. As for region B, the reaction zone is more continue and envelopes a large number of particles, which characterizes the group combustion mode. 3.2. Conditional Statistics. If the ﬂuctuation of scalars, such as temperature and the species mass fractions, is mainly due to the ﬂuctuation of a key variable, such as mixture fraction, conditional averages may provide physical insight into combustion models. Here, the mixture fraction ξ is deﬁned as23 ξ = (2(YC − YC,2)/WC + (YH − YH,2)/2WH − (YO − YO,2)/WO)/(2(YC,1 − YC,2)/WC + (YH,1 − YH,2)/2WH − (YO,1 − YO,2)/WO)

(29)

where W is the molecular weight and Y is the element mass fraction. The subscript 1 denotes the coal particle, and the subscript 2 denotes the central jet. According to this deﬁnition, the maximum mixture fraction in the computational domain is about 0.4, which appears in the downstream region. Figure 6 shows the conditional mean and rms temperatures at three axial locations. It is seen from Figure 6a that the conditional mean of the temperature ﬁrst increases with ξ and peaks at the stoichiometric mixture fraction of about ξ = 0.12. At the axial location x/D = 10, the fuel mass fraction is low and the maximum mixture fraction is around 0.1. Figure 6b displays the conditional rms temperature. The variance of the temperature is low, and the maximum conditional rms temperature is below 200 K, which indicates that the conditional statistics are meaningful for pulverized coal combustion and subsequent submodels could be explored using the DNS database. It is seen that the peak of the conditional rms temperature shifts toward the richer mixtures as the jet advances. The conditional heat release rate is shown in Figure 7. At the axial location x/D = 10, the mean heat release rate is much higher than the rms heat release rate. However, at downstream locations, the rms heat release rate becomes comparable to the mean heat release rate. Thus, considerations should be given to model the heat release rate using conditional methods, such as conditional moment closure. From Figure 7a, it is clear that there are two peaks for the mean heat release rate at x/D = 20. One peak is close to the stoichiometric mixture fraction and the other peak is in the fuelrich side. This phenomenon is described in more detail in the following. Figure 8 displays the contours of the temperature, mixture fraction, and mass fractions of CH4 and CO2 at x/D = 20. In pulverized coal combustion, the heat release rate consists of two parts, which are the combustion of volatile and the

Figure 7. Conditional (a) mean and (b) rms heat release rates with respect to the mixture fraction.

found to be modiﬁed by the particles. From Figure 3a, it is seen that the periodic street is distorted by the particles and ﬁnally breaks down. However, in the downstream region, the vortices are seen to be less inﬂuenced by the particles. The distributions of the pulverized coal particles and the instantaneous temperature in a typical x−z plane are shown in Figure 4. The jet has been fully developed at this timing. The particles injected from a round nozzle have large axial velocities with neglecting radial and azimuthal velocities, so that there is little dispersion in the upstream region. As the jet advances, turbulence and vortices are developed. The particles follow the motion of the ﬂuid and show a large dispersion in the downstream region. The temperature of each particle is also presented in Figure 4a. In general, the temperature of the particle is close to that of the local ﬂuid. Hot particles are found in the shear layer, where the burning rate is the most intense. The particles outside the shear layer have a moderate temperature, while those along the centerline have the lowest temperature. To study the characteristics of particle combustion, the particle combustion regimes are examined. According to Annamalai and Ramalingam,22 three modes of combustion are identiﬁed: individual particle combustion, group combus6133

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Figure 8. Contours of the (a) temperature, (b) mixture fraction, (c) CH4 mass fraction, and (d) CO2 mass fraction at x/D = 20.

weight loss ratio also increases after x/D = 20, which is attributed to the diﬀerent devolatilization rates of the particles. In the present study, the combustion eﬃciency is relatively low because many coal particles leave the computational domain without signiﬁcant mass losses. A longer computational domain is needed to improve the combustion eﬃciency.

combustion of char. The volatile and oxygen react and generate heat near the stoichiometric mixture fraction, while the char combustion mainly takes place in fuel-rich regions, where the coal particles accumulate. Figure 8d shows the contour of the CH4 mass fraction. Most of the particles are distributed in regions with a high CH4 mass fraction, and these regions correspond to the second peak for the mean heat release rate at x/D = 20. The statistics conditioned on the centerline are studied. Figure 9 shows the particle number density and particle weight loss ratio along the centerline. The particle number density N is deﬁned as the number of particles in each grid, and the particle weight loss ratio is deﬁned as the ratio of the weights of particles to their initial values, which measures the degree of particle weight loss because of dehumidiﬁcation, devolatilization, and char combustion. There is an increase for the mean and rms particle number densities after x/D = 10. A peak is observed at x/D = 16 for the mean particle number density, followed by a quick decaying. While the rms particle number density peaks at about x/D = 20. The increase of the particle number density along the centerline is mainly caused by deceleration of the ﬂow, and the decrease of the particle number density is caused by the radial dispersion. As for the particle weight loss ratio, it is shown that there is a signiﬁcant weight loss after x/D = 20. The mass loss in this region is dominated by devolatilization. The rms value of the particle

4. CONCLUSION A compressible DNS solver for pulverized coal combustion has been developed and used to study a pulverized coal jet ﬂame with a Reynolds number of 28 284 based on the nozzle diameter. The co-ﬂow temperature is set as 1600 K, so that coal particles can be ignited by the vitiated co-ﬂow. The minimum grid size is designed as 76.2 μm, and the maximum coal particle diameter is 10 μm. This makes sure that the particle diameter is much less than the minimum grid size and the point-source assumptions of dispersed particles are reasonable. A comparison of axial distributions of the pulverized coal particle velocities between the simulation and the previous experiment1 is carried out to partially validate this simulation, and qualitative agreement is obtained. The particle diameter tends to decrease as the particle traverses downstream, and the mean diameter near the central line is larger than that with a large radial distance. The particle−vortex interactions in the pulverized coal ﬂame are examined. The particles are not 6134

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Figure 9. Axial proﬁles of the (a) particle number density and (b) particle weight loss ratio.

NOMENCLATURE ρ = density (kg/m3) u = velocity (m/s) Y = mass fraction V = diﬀusion velocity (m/s) ω̇ = reaction rate (kg m−3 s−1) τij = viscous tensor (kg m−1 s−2) σij = viscous and pressure tensor (kg m−1 s−2) Cv = heat capacity at constant volume (J kg−1 K−1) Cp = heat capacity at constant pressure (J kg−1 K−1) Cc = heat capacity of the coal (J kg−1 K−1) f = volume force (m/s2) p = pressure (kg m−1 s−2) T = temperature (K) g = gravity force (m/s2) ω̇ ″T = heat release rate (J m−3 s−1) W = molecular weight (kg/mol) Rr = ideal gas constant (J mol−1 K−1) Q = conductional and convective heat exchanged between particles and the gas phase (J m−3 s−1) Ω̇ = radiative heat exchanged between particles and the gas phase (J m−3 s−1) Ṡm = mass source term (kg m−3 s−1) Ṡk = mass source term of the kth species (kg m−3 s−1) ṠV = momentum source term (kg m−2 s−2) ṠT = energy source term (J m−3 s−1) ΔV = control volume for each node (m3) B = pre-exponential factor E = activation energy (J/mol) α = mass stoichiometric coeﬃcient γ = equivalent coeﬃcient a = distribution coeﬃcient Nu = Nusselt number Re = Reynolds number St = Stokes number ξ = mixture fraction

Subscript

dispersed in the radial direction in the upstream region, while they accumulate around the ribs in the downstream region. The vortex structure is found to be distorted by the particles. The DNS results show that the ﬂame characteristics in the upstream region are quite diﬀerent from those in the downstream region. In the upstream region, the reaction rate is quite scattered and single particle is found inside the burning ﬂame. While in the downstream region, the reaction zone is more continuous, with a large number of particles enclosed, which characterizes the group combustion mode. Some conditional statistics are also obtained to provide insights into two-phase coal combustion models.

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i and j = three components related to the Cartesian coordinates g and p = gas phase and solid phase s = coal surface k = parameters related to species k daf = dry and ash free

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AUTHOR INFORMATION

Corresponding Author

*Telephone/Fax: 86-571-87951764. E-mail: [email protected]. Notes

The authors declare no competing ﬁnancial interest.

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ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (50976098 and 51176170) and the Fundamental Research Funds for the Central Universities. 6135

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