Article pubs.acs.org/EF
Numerical Studies on the Inclined Flame Front Break of Filtration Combustion in Porous Media Junchun Zhang, Leming Cheng,* Chenghang Zheng, Zhongyang Luo, and Mingjiang Ni State Key Laboratory Clean Energy Utilization, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China ABSTRACT: The inclined flame front break in the filtration gas combustion of a lean methane (CH4)/air mixture during propagation downstream in an inert porous medium was investigated using a two-temperature and two-dimensional numerical model. The development of flame inclination was studied by focusing on the flame front break, which splits into two or three waves. The essence of the flame front instability and inclined flame front break mechanisms was discussed. The physical parameters, such as flame structure, gas temperature distributions, and effects of system properties on flame characteristics, were studied in detail. The results show that the flame front break easily takes place in higher Lewis number mixtures, higher inlet velocity, lower equivalence ratio, and larger heat loss regimes. Higher inlet velocity and lower equivalence ratios result in higher flame instability in the combustion process. wavefront inclination grew approximately linearly until flame front break occurs and the flame front splits into two hightemperature zones. However, the flame front break phenomena in FGC were mainly studied by experiments, while no satisfactory theoretical or numerical description of this process has yet been reported. There are several attempts to analyze the nature of flame front inclination instability theoretically.13−15 Kakutkina14 used a thermal model for the analysis of some aspects of the flame front inclination instability. Stabilization of a flame front inclination at a definite slope angle can be adequately explained by the proposed model of instability development; however, the flame front break conditions were not considered. Dobrego and Zhdanok15 solved flame front inclination instability of the small perturbations using the equations of heat transfer in the heating region and concluded that, in case the thickness of the hightemperature zone is smaller than the 2/π(qu/qλ)D value, the flame front is broken, where qu and qλ are the heat fluxes of the incoming gas stream and high-temperature regions, respectively, and D is the transverse size of the porous body. Characteristics regimes of premixed gas combustion in highporosity microfibrous porous media were numerically studied by a two-dimensional (2D) model,16 and only some wrinkles on the flame front surface were observed. Dobrego et al.17 investigated the dynamics of flame front inclination instability caused by perturbation using 2D numerical simulation and showed that the startup transient, linear growth, and perturbation compensation phases of the perturbation evolution process may be distinguished from each other. Zheng et al.18 used a 2D numerical model to study the FGC wavefront inclinational instability and found that the reaction zone in the middle of the combustor becomes weaker as the flame front inclination instability develops. However, although these models studied
1. INTRODUCTION The filtration gas combustion (FGC) in an inert porous medium and low-velocity regime has numerous technological applications, such as volatile organic compound (VOC) oxidization, burning of lean combustible mixtures in the superadiabatic combustion regime, heat exchangers, hydrogen production, power engineering, etc.1−3 Experimental and numerical studies have been conducted on the combustion characteristics of gas mixtures, including thermal performances, flame stabilization, gaseous emissions, and heat-transfer mechanism.4,5 In the present study, the FGC regimes are mostly considered with the formation of a quasi-steady combustion front in the medium because of a narrow heat release zone and intense interphase heat transfer. However, the FGC wave can be hydrodynamiclly unstable when propagating in the direction of filtration. Many studies have reported on the instability of the flame front observed in their experiments,6−12 and this poses a severe problem in the industrial application of the FGC process. Studies on FGC flame front instability have been conducted over the last 2 decades.13−18 Two types of FGC flame front instability (e.g., hot-spot and inclination) were observed and studied widely. Hot-spot instability manifests itself as an initially continuous flame front broken into two or more separate fragments, and inclination instability is expressed by the plane combustion flame front inclined at an angle relative to the normal in the filtration direction. The inclination angle increases as the flame front propagates. In many cases, the inclined flame front stabilizes at a definite inclination angle. However, in case the stabilization does not occur before an inclination angle of 54− 60° is reached, flame front break (or quenching of the lower flame wave) takes place.12 A double wave structure was first observed by Saveliev et al.6 for lean hydrogen/air mixtures in a packed bed. Yang et al.9 observed that the combustion wave splits into two and more parts during downstream wave propagation under higher velocity conditions in small diameter quartz tubes filled with high-porosity microfibrous media. Dobrego et al.7 and Shi et al.10,12 experimentally found that, in some tests, FGC flame front inclination instability reached its saturation and showed no tendency to grow any further, while in some other tests, the © 2013 American Chemical Society
Received: April 23, 2013 Revised: June 13, 2013 Published: July 25, 2013 4969
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the evolution of flame front inclination, they failed to predict the flame front break in the porous media combustor. From the previous studies, we know that the experimental and numerical studies mentioned above mostly focused on inclined flame evolution in the packing bed; however, the inclined flame front break in porous media has not been explored. These instability evolution mechanisms are still not clear, and it is necessary to take more investigation on this topic. Knowledge of the dynamics of inclined flame front break is of fundamental importance to overcome the flame instability issues, and these results will be very instructive for designing and operating FGC systems in the practice. In this study, the combustion of lean methane (CH4)/air mixtures under FGC conditions in an inert porous medium was investigated by a two-temperature and 2D model. The development of flame inclination was studied by focusing on the flame front break, which splits into two or three waves as the combustion wave propagates downstream. The mechanisms of flame instability and inclined flame front break were discussed. Furthermore, the physical parameters, such as flame structure, gas temperature distributions, and effects of system properties on flame characteristics, were studied in detail.
where K1 is the permeability coefficient and K2 is the inertial loss coefficient. They are constants for different foam types under constant hydrodynamic conditions.19 λeff‑g is the effective thermal conductivity of the gas mixture and expressed as follows: λeff‑g = λg + cgρgD∥d, where gas radiation is neglected and D∥d is the thermal diffusivity in porous media.20 λeff‑s is the effective thermal conductivity of solid and expressed as follows: λeff‑s = (1 − m)λs + λrad = (1 − m)λs + (16/3α)σTs2, where Rosseland approximation is used by assuming an optically thick region around the high-temperature zone and α is the extinction coefficient.22 Dm,i is the mass diffusion coefficients, assuming that the Lewis number Le for the CH4/air mixture equals 1 unless any specific statement is mentioned.6,21 The thermal diffusivity and mass diffusion coefficient are determined as follows:14,17 D d = 0.5ug d ,
h v = 6(1 − m)
cg = 947.0e0.000183Tg
P = ρg RTg
(11)
Q in = −εinσ(Ts 4 − T0 4) outlet
Q out = −εoutσ(Ts 4 − T0 4)
(2)
+ mcgρg u∇Tg
k=1
Q wall = − β(Ts − T0) − εwall′σ(Ts 4 − T0 4)
(3)
∂Ts = (1 − m)∇(λeff‐s∇Ts) − hv (Ts − Tg) ∂t (4)
species transport equation
+ ρg u∇Yi = ρg, i ∇(Dm, i∇Yi ) + ωiWi
(5)
Nsp
i = 1, ..., Nsp − 1
(13)
∑ Yi = 1 i=1
(14)
At the solid walls, the no-slip condition for the velocity components is implemented. Further, at the outlet boundary, the gradients of all variables are set to zero. Other parameters used in the standard case are as follows: porosity, m = 0.43; outlet pressure, p0 = 1.013 × 105 Pa; tube length, L = 0.8 m; burner diameter, D = 0.06 m; bedding particle diameter, d = 0.003 m; solid density, ρs = 3707 kg/m3; specific heat, cs = 774 J kg−1 K−1; methane heat content, H = 5.3 × 107 J/kg; Le = 1; β = 5 W m−2 K−1; and εwall = 0.1. The governing equations are solved using the CFD software Fluent 6.3, by changing the one-temperature model to a two-temperature model with the help of the user-define function and user-define scalar. The problem does not allow for a steady-state solution because of the combustion wave propagation. The time integration was performed using an implicit method. Initially, the perturbation was introduced as an inclined gas and solid temperature at T = 1400 K between L = 30 and 70 mm.
solid-phase energy equation
(1 − m)csρs
(12)
The combustion zone is assumed to be not insulated. Further, the heat loss at burner sides can be expressed as follows:
= m∇(λeff‐g ∇Tg) − m ∑ ωihiWi + hv (Ts − Tg)
∂t
(10)
inlet
+ mρg u∇u
Nsp
∂(ρg Yi )
(9)
Hsu and Matthews found that a single-step chemical reaction mechanism is adequate for predicting the flame characteristics, except the emissions for very lean conditions. Therefore, the global single-step kinetics has been used in this study. The radiation emitted from the inlet and outlet cross-section of the solid phase is expressed by the following computations:
gas-phase energy equation
∂t
(8)
The equation of state for an ideal gas is as follows:
(1)
ρg ⎞ ⎛μ m | u | ⎟u = −m∇P + mμ∇2 u − m⎜ + K2 ⎠ ⎝ K1
mcgρg
Nu v = 2 + 1.1Re 0.6Pr1/3
The change in specific heat capacity for the gas phase because of a change in temperature changing is expressed as follows:23
linear momentum equation
∂Tg
,
24
+ ∇(ρg u) = 0
∂t
d2
μg = 3.37 × 10−7T 0.7
continuity equation
∂(mρg u)
Nu v λg
The viscosity changes for the gas phase because of a change in the temperature are expressed as follows:23
A two-temperature and 2D model was established to investigate the evolution of FGC flame front perturbation. The thermal model and kinetic flame structure are based on local volume-averaged treatment and well-mixed pore approximations. The inert porous media are assumed to be non-catalytic and optically thick with an impermeable fluid−solid interface. The porosity variation near the tube wall was neglected. The governing equations for incompressible reactive flow while considering combustion and heat-transfer effects can be expressed as follows:
∂t
(7)
Local thermal nonequilibrium is considered here and the energy equations of the gas and solid phases are coupled by the volumetric heat exchange coefficient defined as follows:17
2. NUMERICAL MODEL
∂(ρg )
Le = Dm /D d
(6) 4970
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Figure 1. Contours of inclined flame front break characteristics under conditions of ug = 0.4 m/s and φ = 0.3.
3. RESULTS AND DISCUSSION Figure 1 shows the contours of flame shape, gas temperature, velocity in the x direction, and diffusion coefficient of inclined flame front break in a packed bed under the following conditions: ug = 0.4 m/s and φ = 0.3. The heat of reaction was used to show the flame shape. Figure 1a shows the variation of the flame shape as the flame front inclinational instability develops when the combustion wave propagates downstream. The flame is S-shaped in appearance. The flame shape near the wall is flat under the noslip and thermal boundary conditions assumed in the numerical model. The inclination angle increases as the combustion wave propagates downstream. Two waves are observed after t = 1500 s. The flame front splits into two parts between t = 1500 and 2000 s. Then, the two waves propagate downstream with constant velocities. The propagating velocity of the lower flame is smaller than the upper flame, and the crack area gradually increases with time. Further, another “S-shape” upper flame is formed, and the inclination angle of the upper flame decreases and eventually stabilizes with a definite inclination angle. The critical breakup inclinational angle is 64°. Shi et al.10,12 experimentally found that inclined flame front breakups always happened when the inclinational angle reaches 54−60°. The numerical predictions are relatively in good agreement with the experimental results. Figure 1b shows the gas temperature variation in the tube at t = 2500 s. There are two split high-temperature zones, and the upper high-temperature area is larger than the lower hightemperature area. Kakutkina14 concluded that, during motion of an inclined wave, the presence of a virtual source and a virtual sink gives rise to a heat flux along the front. The temperature at the lower part is lower than that of the upper part. The flame front breakup probably occurred at the place with lower temperature and higher lateral surface through which there is heat exchange with the ambient medium. Figure 1c shows the velocity in the x direction in the tube at t = 2500 s. The fresh gas flow has a left to right change in velocity when it passes through the lower flame front, and that may lead to a higher filtration velocity in the upper flame zone. It is because of the pressureinclined distribution caused by thermal non-uniformity. For the boundary effects, the product gas flow of the upper flame zone has a right to left change in velocity. Figure 1d shows the variation in diffusion coefficients in the tube at t = 2500 s. The mixture diffusion coefficient is obviously higher for the lateral fresh
mixture zone, whereas it is lower in the lateral product zone of the flame front. There is a transverse supply of gas-mixture components, whose concentration is lower in the products compared to the fresh mixture. Figure 2 shows the schematic diagram of the inclined flame front break. The upper part shows the gas temperature contours
Figure 2. Schematic diagram of the inclined flame front break.
of the split wave, and the lower part shows the flame shape. The flame front is continuous at t1, while it breaks at t2. However, there is a curved front at t1 toward the products in zone B, where it seems to be the source of the crack formed. Next, there are lower and upper flames with wave-propagating velocities of uw1 and uw2, respectively. There is a temperature variation along the flame front because of thermal non-uniformity. The effect of variation in the length of the flame front on the development of instability of an inclined flame during the wave propagation was studied by Kakutkina.14 It was proven that the inclination angle would become larger as the wave propagates. When a cold fresh gas mixture flows through the lateral surfaces of the inclined flame front, the fuel and oxygen concentrations are lower in the products in zone A than that in the fresh mixture. Thus, right to left supplies of the fuel and 4971
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not occur when the value of Le = 0.5. Gas mixtures characterized by a lower Lewis number are always stable; however, gas mixtures characterized by Le = 1 (air mixtures with methane or acetylene) or a higher Lewis number (air mixtures with hydrogen) are properly unstable. A higher Lewis number results in a higher convective mass transfer and higher decrease of the fuel and oxygen concentrations at zone B. Thus, the flame front break properly takes place. However, more investigation on the flame front breakup instability criterion is needed. According to Zheng et al.,18 the inclination growth increment is proportional to the diameter of the combustor D. We denote the H/D ratio as the correlation for the inclination perturbation growth, where H is the distance between the lower and upper flame fronts, as shown in Figure 2. Figure 4 shows the inclination
oxygen concentrations occur by mass transfer. Consequently, the fuel and oxygen concentrations at zones B and C are lower than those at zone A. Furthermore, a larger inclination angle would lead to lower fuel and oxygen concentrations at zones B and C. The flame front area increases as the inclination instability develops, and that leads to a slower chemical reaction rate at the flame front. However, heat is removed in the flame zone because of the temperature difference between the fresh gas mixture and products. Further, more and more heat is removed as the lateral surface increases, especially at zone B, where the inclinational angle is the largest. Therefore, there is a competition between the heat release by chemical reaction and heat removal processes at the flame front. In cases where the heat release by chemical reaction is insufficient, a curved flame front is generated toward the products. When the inclination angle approaches 90°, flame extinction occurs at zone B. Subsequently, the flame zone splits into two hot zones near the walls. The supply of the mixture components to the lateral surface of the flame front is determined by not only the component concentration in the initial mixture but also its diffusion coefficient. Convective heat transfer occurs along the flame front, in both the combustion-product zone and the heating zone of the fresh mixture, while convective mass transfer occurs only in the heating zone of the fresh mixture. The effect of external heat and mass transfer can be represented by the Lewis number Le = Dm/D∥d (Dm is the mass diffusion coefficient, and D∥d is thermal diffusivity because of dispersion). The Lewis number Le for the CH4/air mixture equals 1. However, to study the effects of the Lewis number on the flame front evolution, we increased or decreased the mass diffusion coefficient Dm to let the Lewis number Le equal 0.5 and 1.5. Figure 3 shows the effects of the Lewis number on the flame front evolution under the following conditions: ug = 0.4 m/s and φ = 0.3. The flame front break does
Figure 4. Inclination perturbation growth under conditions of ug = 0.4 m/s and φ = 0.3.
perturbation growth under the following conditions: ug = 0.4 m/s and φ = 0.3. The H/D ratio increases as the flame propagates downstream; however, the growth rate decreases until it reaches a definite value because the correlation curve becomes linear after breakup. Because of the difference in wave-propagating velocities between zones A and B, the crack area of the flame front gradually increases. The relationship between the propagating velocity and the combustion temperature of a flat combustion flame can be expressed as follows:6 u w = ut{1 − [(ΔTa /ΔTc)2 − 4βv λeff‐s /(ugρg cg)2 ]1/2 }
(15)
where uw is the combustion wave-propagating velocity, ut is the thermal wave velocity, ΔTa and ΔTc are the adiabatic and combustion temperature rise of the mixture, respectively, and βv is the effective volumetric heat-transfer coefficient of the combustor with surroundings. The average combustion temperature rise ΔTc decreases as the flame inclinational angle increases, owing to the increase in the flame front area. The wave-propagating velocity difference between zones A and B decreases as the flame propagates downstream. Thus, the growth rate of the H/D ratio decreases. However, when the flame is split into two parts, the flame front area does not increase anymore, two combustion waves propagate downstream with a constant velocity, and the correlation curve of H/D becomes linear.
Figure 3. Effects of Lewis number on the flame front evolution under conditions of ug = 0.4 m/s and φ = 0.3. 4972
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Figure 5. Three-wave formation and inclination perturbation growth at φ = 0.2.
Table 1. Numerical Estimation of Characteristics for the Inclined Flame number
φ
ug (m/s)
uw1 (×104, m/s)
uw2 (×104, m/s)
H/Dmax
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.4 0.4 0.3 0.3 0.3 0.3
0.2 0.3 0.4 0.6 0.2 0.3 0.4 0.6 0.3 0.4 0.4 0.4 0.4 0.4
0.93 1.38 1.70 2.38 0.63 0.91 1.18 1.81 0.60 0.88 1.18 1.17 1.29 1.18
0.93 1.48 2.03 3.33 0.63 1.03 1.44 2.47 0.60 0.88 1.44 1.43 1.29 1.59
1.20 break, three waves break, three waves break, three waves 1.15 break break break 1.47 1.87 break break 2.30 break
H/Dbreak
specificity
1.55 1.85 2.33 1.83 2.08 2.67
2.67 2.17 2.28
standard case
without heat loss εout = 0, β = 5 W m−2 K−1 Le = 0.5 Le = 1.5
Figure 6. Effects of various physical parameters on the flame front evolution.
Another type of flame containing three waves is observed at φ = 0.2 as the equivalence ratio of the mixture is reduced. Figure 5 shows the three-wave formation and inclination perturbation growth of the flame front. In Figure 5a, the flame front splits into lower and upper flames at t = 2000 s, followed by the upper flame front splitting into two parts at t = 3500 s, under the following
conditions: ug = 0.3 m/s and φ = 0.2. Similarly, in Figure 5b, the flame front break occurs at t = 1500 s, followed by the upper flame front break occurring at t = 2750 s, under the following conditions: ug = 0.4 m/s and φ = 0.2. In comparison to the flame front evolution at φ = 0.3, as shown in Figure 1, the lower flame zone eventually becomes smaller and smaller. It can be explained 4973
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ratio is proportional to the inlet velocity. A larger inlet velocity leads to higher instability. The flame front slope does not increase until the growth rate of the front slope becomes zero. Otherwise, the flame front break takes place, and the growth rate of the H/D ratio becomes eventually constant. The effects of the equivalence ratio on the inclined flame front evolution can be studied by comparison to entries 2, 6, and 9 or 3, 7, and 10 in Table 1. As the equivalence ratio increases, the respective wave-propagating velocities decrease. This phenomenon of flame front break easily takes place at lower equivalence ratio regimes. The critical break value of the H/D ratio increases with an increasing equivalence ratio. Figure 8 shows the
by the fact that the heat release in the combustion zone becomes smaller as the heat content of the mixture decreases and, therefore, becomes insufficient for the heat loss through the wall at zone A. Similar observations in the quenching of the inclined wave that begins with the lower flame zone have been reported in previous experimental results.10,12,13 The upper flame is also Sshaped in appearance. In case the heat release by chemical reaction in the middle zone of the upper flame becomes insufficient for the heat loss, the flame front break occurs in the same manner as discussed above and three waves propagate downstream with constant velocities. However, as shown in Figure 5a, the lower and middle flames meet to form a wider flame at t = 4500 s. The gas filtration combustion wave characteristics for the inclined flame were obtained numerically for all of the simulation cases (Table 1) and used further for comparative analysis. In case of eventually stabilized inclined flame, a maximum ratio of the H/ D value is obtained. However, in case of eventually breakup flame, the H/D ratio showed a gradual increase without attaining a maximum value but there is a critical value of the H/D ratio at which the flame front breaks out. The wave-propagating velocities of lower and upper flames are represented by uw1 and uw2, respectively, and the velocities were calculated after the flame front break takes place. The physical parameters, such as inlet velocity, equivalence ratio, heat loss from the burner side wall, and Lewis numbers for gas mixtures, were investigated. Figure 6 shows the effects of various physical parameters on the flame front evolution. As shown in Table 1, as the inlet velocity increases, the respective wave-propagating velocities of the lower and upper flames also increase dramatically. Moreover, the velocity differences in the flames also increase. At lower velocity regimes, eventually, the inclined flame is easily stabilized without any flame front break. The critical flame front break value represented by the H/D ratio increases as the inlet velocity increases. Panels a−c of Figure 6 show the effect of the inlet velocity on the flame front evolution. As shown in Figure 6a, the inclination angle increases at first and eventually stabilizes at a constant value after t = 1500 s under the following conditions: ug = 0.2 m/s. Panels b and c of Figure 6 show that, as the inlet velocity increases, the flame front break appears at ug = 0.3 and 0.4 m/s, respectively. Figure 7 shows the inclination perturbation growth for different inlet velocities. The growth rate of the H/D
Figure 8. Inclination perturbation growth for different equivalence ratios.
inclination perturbation growth at different equivalence ratios. The growth rate of the H/D ratio is inversely proportional to the equivalence ratio. A lower equivalence ratio leads to higher instability. Panels b, d, and e of Figure 6 show the effect of the equivalence ratio on the flame front evolution. The flames eventually become stabilized at a higher equivalence ratio of φ = 0.4 under both of these conditions: ug = 0.3 and 0.4 m/s. Another type of flame containing three waves was observed at lower equivalence ratio regimes, as discussed above. Inclination instability develops fast at lower equivalence ratio regimes, while it decreases with the increase of the equivalence ratio. As shown in Table 1, the effect of heat loss from the burner wall on the respective wave-propagating velocities of lower and upper flames is negligible when compared to the entries 7, 11, and 12; however, the critical break value of the H/D ratio decreases with an increase of the heat loss, and that means flame front break occurs earlier in larger heat loss regimes. Figure 9 shows the inclination perturbation growth for different side thermal losses. The growth rate of the H/D ratio is a little higher with less side thermal losses. The curve seems to be parallel for the same wave-propagating velocities in lower and upper flames. In comparison to panels c and f of Figure 6, the area of the lower flame zone decreases, while that of the upper flame zone increases, with an increase of the heat loss through the side wall. The flame front seems to break out at the middle of the combustor without any heat loss. However, the results show that the effect of heat loss is small because the effect of heat loss on inclined flame front evolution is small in low-velocity regimes.10
Figure 7. Inclination perturbation growth for different inlet velocities. 4974
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R = universal gas constant (J mol−1 K−1) t = time (s) T = temperature (K) u = velocity vector (m s−1) ug = inlet velocity of gas (m s−1) ut = thermal wave velocity (m s−1) uw = combustion wave-propagating velocity (m s−1) uw1 = lower flame front propagating velocity (m s−1) uw2 = upper flame front propagating velocity (m s−1) Wi = molecular mass of species i Yi = local mass fraction of species i ΔTa = adiabatic temperature rise of the mixture (K) ΔTc = combustion temperature rise of the mixture (K) Greek Symbols
β = convective heat-transfer coefficient through the wall (W m−2 K−1) βv = effective volumetric heat-transfer coefficient of the combustor with surroundings (W m−3 K−1) λ = thermal conductivity (W m−1 K−1) μ = dynamic viscosity (Pa s) ρ = density (kg m−3) σ = Stefan−Boltzmann constant φ = equivalence ratio ωi = molar rate of production of species i (mol s−1)
Figure 9. Inclination perturbation growth for different side thermal losses.
4. CONCLUSION Inclined flame evolution for lean CH4/air mixtures in the gas filtration combustion process during a propagation downstream in porous medium was investigated by a two-temperature and 2D numerical model. Flame front break that splits into two or three waves was captured successfully. The H/D ratio value increases as the flame propagates downstream; however, the growth rate decreases to zero or a definite value at which the flame front breaks up. Gas mixtures characterized by Le = 1 (air mixtures with methane or acetylene) or a higher Lewis number (air mixtures with hydrogen) are properly unstable. The results predict that the phenomenon of flame front break easily takes place at higher inlet velocity, lower equivalence ratio, and larger heat loss regimes. Higher inlet velocity and lower equivalence ratios result in higher flame instability.
■
Subscripts
■
g = gas s = solid in = inlet out = outlet wall = burner wall
REFERENCES
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AUTHOR INFORMATION
Corresponding Author
*Telephone: +86-571-87952802. Fax: +86-571-87951616. Email:
[email protected]. Notes
The authors declare no competing financial interest.
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NOMENCLATURE c = specific heat (J kg−1 K−1) d = pellet diameter (m) D = burner diameter (m) D∥d = thermal diffusivity because of dispersion (m2 s−1) Dm,i = diffusion coefficient of species i (m2 s−1) H = distance between the lower and upper flame fronts (m) hi = molar enthalpy of formation of species i (J mol−1) hv = volume convective heat-transfer coefficient (W m−3 K−1) L = tube length (m) Le = Lewis number m = porosity Nu = Nusselt number P = pressure (Pa) Pr = Prandtl number qu = heat fluxes taken by incoming gas stream (W) qλ = heat fluxes taken from the high-temperature region (W) Q = heat fluxes between the combustor and surroundings (W/ m2) 4975
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dx.doi.org/10.1021/ef400745s | Energy Fuels 2013, 27, 4969−4976
