A Chemical Reaction Engineering and Transport Model of Kraft Char

A numerical model for carbon burning in the char bed of a kraft recovery boiler ... from the gases immediately above the bed and for radiant heat tran...
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Ind. Eng. Chem. Res. 2002, 41, 1477-1483

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A Chemical Reaction Engineering and Transport Model of Kraft Char Bed Burning Jari Sutinen Fortum Power and Heat Oy, Technology Centre, Rajatorpantie 8, P. O. Box 20, Vantaa, FIN-00048 Fortum, Finland

Reijo Karvinen Department of Mechanical Engineering, Tampere University of Technology, P. O. Box 527, FIN-33101 Tampere, Finland

Wm. James Frederick, Jr.* School of Chemical Engineering, Chalmers University of Technology, Kemiva¨ gen 10, SE 412 96 Gothenburg, Sweden

A numerical model for carbon burning in the char bed of a kraft recovery boiler was developed that predicts the rate of char burning and the char bed temperature locally within the furnace. The model describes the char bed chemistry, accounting rigorously for chemical element and energy balances. Char burning rates are based on established kinetic rate data, and diffusion of reactant gases within the bed is taken into account. The chemical model is coupled to the gas field and furnace thermal environment above the bed to account for convective mass and heat transfer from the gases immediately above the bed and for radiant heat transfer from the furnace cavity. This model predicts, within 7% on average, the experimentally measured char burning rates reported by others in laboratory char bed studies, as well as predicting their reported char bed surface temperatures with an average error of 18 °C. It also predicts quantitatively the effects of gas composition and velocity on those rates with an average error of 7%. Introduction Black liquor is a biomass material resulting from the chemical conversion of wood to papermaking fiber. It is an important fuel in paper-manufacturing countries, providing roughly one-half of the energy consumed by pulp and paper millssabout 2.5 × 109 GJ/year worldwide. These fuels contain residual inorganic pulping chemicals along with the dissolved biomass, about onehalf of the wood processed in the production of fibers during pulping. They are burned in large, specially designed recovery boilers to recover the pulping chemicals and energy from the dissolved wood. One of the main objectives in black liquor combustion is to recover the spent pulping chemicals as Na2S and Na2CO3 for further conversion to the active pulping chemicals, Na2S and NaOH. For that reason, the concentrated liquors, containing 20-35% water, are fired as coarse sprays, with a mean particle diameter of 2-3 mm and a distribution ranging from 0.5 to 5 mm.1 These relatively large droplets are partially burned in flight and partially burned in a char bed at the bottom of the furnace. The char bed itself consists of a hot, porous, actively burning layer at the surface of the bed and a colder, denser carbon/inorganic mass below it. The molten slag (smelt) from the burning residue flows out of the boiler through ports at the base of the boiler walls. The char bed is essential for obtaining high conversion of the oxidized inorganic sulfur compounds in the liquor, mainly Na2SO4 and Na2S2O3, to Na2S. It also protects the Na2S in the smelt * Author to whom correspondence should be addressed.

in the bottom of the boiler from reoxidation to Na2SO4 by the combustion air. Numerical models are now used extensively in designing recovery boilers and in evaluating their performance.2-4 These models have handled the char bed in a very simplistic way, treating it as a source of combustible gases of specified composition and a sink for inorganic material. Important factors such as local char burning rates, accumulation or depletion of char and inorganic matter in the bed, variations in bed temperature and their influence on heat transfer, and interactions of the gas field and bed chemistry in controlling char bed shape have not been considered. Here, we present a numerical model for a char bed that predicts the rate of char burning and the char bed temperature locally within the furnace. The model developed here describes the char bed chemistry, accounting rigorously for chemical element and energy balances. Char burning rates are based on established kinetic rate data, and the diffusion of reactant gases within the bed is taken into account. The chemical model is coupled to the gas field and furnace thermal environment above the bed to account for convective mass and heat transfer from the gases immediately above the bed and for radiant heat transfer from the furnace cavity. This model predicts accurately the values of experimentally measured char burning rates, as well as the effects of gas composition and velocity on those rates. Char Bed Physical and Chemical Processes The processes that occur within a char bed and at its surface are illustrated in Figure 1. The bed consists of

10.1021/ie000393a CCC: $22.00 © 2002 American Chemical Society Published on Web 02/21/2002

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a low-density, active char layer several centimeters thick supported by a colder, denser char bed in which no significant amount of burning takes place.1,5-7 Char enters the bed from above as the residue from partially or completely dried, pyrolyzed, or burned black liquor droplets that contain carbon, organic matter, inorganic salts (primarily sodium-sulfur compounds), and water. O2, CO2, and water vapor are transported to the bed from the gas above it through the boundary layer surrounding the char bed. The main reactions in the char bed are the oxidation of carbon to CO by water vapor, CO2, and oxygen and the reduction of Na2SO4 to Na2S.1 Pyrolysis of organic matter to CO and light hydrocarbons, volatilization of sodium, and H2S and COS formation can also occur.8,9 The gases produced by the reactions in the char bed are transported away from it through the gas boundary layer, and they can react with gases within the boundary layer. The char bed reactions are net endothermic. Energy transferred to the bed from above provides the heat required to maintain a temperature high enough for carbon gasification and reduction of Na2SO4 compounds to proceed. The transport of mass and energy to the char bed strongly affects the rate at which the char bed reactions proceed. Energy is exchanged between the bed, the furnace gases, and the furnace cavity by radiation and convection and can be transported by conduction from the active char bed to the inactive bed below. The net endothermic reactions and evaporation of water within the char bed tend to cool the bed. The bed temperature is determined by an energy balance for these processes. The char bed processes are thus coupled with the gas-field phenomena, and the equations governing momentum, heat, and mass transport between the two must be solved simultaneously to determine the bed temperature and the rate at which char is burned. Char Bed Model Char Burning Rates under Chemical-KineticControlled Conditions. The conversion of carbon in black liquor char to CO and CO2 can occur by several paths. These include direct combustion of carbon, oxidation of carbon by sodium sulfate, and reactions with CO2 and water vapor, i.e.

2C + O2 w 2CO

(1)

4C + Na2SO4 w Na2S + 4CO

(2)

C + CO2 w 2CO

(3)

C + H2O S CO + H2

(4)

The reduction of Na2CO3 by carbon can also contribute to char carbon conversion, although this reaction is slow compared with other carbon conversion processes.9 It is not considered in this model. COS and H2S formation, both of which have a minor influence on the carbon conversion rates,8 are also not considered. The net rate of conversion of char carbon to gases, when controlled by chemical kinetics, is the sum of the rates of carbon conversion via reactions 1-4. In the model developed here, the rate for direct carbon oxidation (eq 1) was based on Wendt’s modification of Smith’s correlation of carbon combustion data.10 The specific surface area of the carbon in the char was taken as 122 m2/g on the basis of measurements of van Heinengen

Figure 1. Functions of the char bed near a smelt spout wall in a sloped-floor recovery boiler. Table 1. Chemical Kinetic Rate Equations for the Char Carbon Reactionsa i

ki [mol/(m3 s)]

O2 CO2 H2O sulfate reduction

1.9 × 104Acpox 32 × 106[Cc]pCO2/(pCO2 + 3.4pCO) 2.56 × 109[Cc]pH2O/(pH2O + 1.42pH2) 5240[Na2][SO4][Cc]/ (0.0011 + [SO4])

a

Ea,i/R (K) ref -17 086 -22 500 -25 300 -14 696

10 17 18 13

Rate equations are of the form RKi ) ki exp(-Ea/RTb).

et al.11 The direct carbon oxidation rate was reduced by a factor of 0.6 to account for the fact that molten sodium salts block access to some of the carbon surface.12 It will be shown later in this paper that the exact values of these parameters are not important with respect to the predicted burning rate, as direct carbon oxidation is limited by both mass transfer and consumption of O2 by gas-phase reactions in the boundary layer. Carbon conversion by sulfate reduction proceeds according to eq 2. In this model, the rate of sulfate reduction reaction under chemical-kinetic-limited conditions was based on the rate equation of Cameron and Grace13 and the stoichiometry shown in eq 2. An analysis of their rate equation indicated that it predicts a rate of sulfate reduction that is too slow to account for the high levels of sulfate reduction, typically 0.850.95 as Na2S/total sulfur mole ratio, that are observed in recovery boilers. However, it provides reasonable predictions when multiplied by a factor of about 50.14 Cameron and Grace’s data was obtained by mixing kraft char with molten Na2SO4-Na2CO3 mixtures. It might be that the low-density char might have floated to the surface of the melt, buoyed by the gases released during the sulfate reduction reaction; this would have resulted in a low rate of sulfate reduction. More recently, other researchers have reported much higher rates of reduction of sulfate to sulfide.15,16 [Note that these results became available after the work reported here had been completed.] In the present study, a multiplier of 50 was used with the Cameron and Grace rate equation for the reduction of Na2SO4. The kinetic rate equations for carbon conversion via reactions 1-4 are included in Table 1. Transport Effects to and Within the Char Bed. The effect of gas diffusion within the char bed on the overall rates of oxidation and gasification was taken into account by an effectiveness factor based on the Thiele modulus.19 The apparent first-order rate constant ki for each reaction was calculated as the rate of the reaction

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from the equations in Table 1 divided by the concentration of the gas species in the gas cell adjacent to the char bed surface. The thickness of the active char bed layer was taken as 0.05 m on the basis of experimental measurements by Brown et al.20 Because of the very low density of kraft char particles (30-50 kg/m3), the effective diffusivity was assumed to be the same as the diffusivity of the respective gases in nitrogen.21 Overall Rate of Char Burning. When simultaneous oxidation and char gasification occur in the char bed, some or all of the CO and H2 produced is oxidized to CO2 and water vapor in the gas film surrounding the char bed.6 In our model, this oxidation is assumed to occur within the boundary layer above the char bed. Thus, the rate of carbon removal is equal to the sum of the rate of oxygen consumption, adjusted for the ratio of CO/CO2 produced, plus the rate of consumption of CO2 and water vapor, plus the rate of carbon removal due to the net rate of sulfate reduction in the char bed. Sulfate reduction was assumed to occur at steady state, and the rate of oxygen consumption was calculated as the sum of the rates of carbon oxidation by sulfate reduction and by direct oxidation of carbon. Depending on the temperature of the active char bed zone, the overall rates of O2, CO2, and water vapor consumption in the char bed can be controlled by a combination of chemical kinetics, intrabed diffusion, and film mass transfer. The overall rates of O2, CO2, and water vapor consumption were calculated from the rates of film mass transfer and intrabed diffusion-limited chemical kinetics for the cases in which each is independently the rate-limiting process. The gas-film masstransfer coefficient was estimated from calculated gasfield velocity profiles and the Reynolds analogy between momentum and mass transport. Coupling of the Char Bed and the Gas Field. The rates of transport of heat and mass to the bed from the gas above it are obtained by treating the bed surface as a wall and calculating the velocity profile adjacent to it. Because the governing equations of velocity, temperature, and concentrations have the same form, it was possible to utilize the analogy in coupling the momentum, temperature, and concentration profiles to the wall. The problem is represented as a two-dimensional and stationary elliptic flow, for which the governing equations of velocity, temperature, and gas concentrations can be expressed in general form as

∂ ∂φ ∂φ ∂ ∂ ∂ (FVφ) + (FWφ) ) Γ + Γφ + Sφ ∂y ∂z ∂y φ ∂y ∂z ∂z

(

)

(

)

(5)

Turbulence is taken into account with the standard k- model. The rate of gas burning is assumed to be governed by mixing, using the eddy dissipation model of Magnussen and Hjaertager.22 Because the char bed surface grows near to the primary jet level, jets at the bed surface behave as turbulent wall jets. In the near-wall region, single jets act separately, but far enough from the nozzle wall, they converge and begin to behave as a plane wall jet. The behavior of the wall jet deviates from the behavior of the turbulent boundary layer as a result of the strong interaction between the near-wall region and the freejet boundary layer. Numerous studies of wall jets have been reviewed by Rajaratnam.23 Near the wall, wall functions were used to connect the flow-field calculations to the wall (i.e., the bed surface). This avoided the need to use frequent mesh

points in the viscous sublayer region in which steep gradients exist. Also, the turbulence model did not require the low-Reynolds-number correction that accounts for the effect of viscous forces on the turbulence structure. With the wall-function scheme, the calculation point nearest to the wall was placed just on the outside of the viscous sublayer in the logarithmic-law region. The connection of the wall conditions (shear stress and heat and mass flux) to the fully turbulent fluid was made by using the logarithmic law of the wall for momentum transfer and the logarithmic temperature law for heat transfer. Values of 0.411 and 4.90 were used for the von Karman constant and the empirical surface-roughness parameter, respectively. The wall functions were implemented into the momentum, energy, and scalar equations via the source term in eq 5. The friction factor for the turbulent flow was calculated iteratively. Convective heat transfer coefficients between the char bed surface and the gas field above it were calculated from Stanton number correlations for laminar and turbulent flow. The radiation heat flux from the furnace walls was taken into account in the energy balance for the bed, but the gas radiation was not included. Details of the model coupling the char bed surface and the gas flow, temperature, and concentration fields adjacent to it are available by request from the authors. Gas Reactions in the Boundary Layer. When simultaneous oxidation and char gasification occur in the char bed, the CO and H2 produced can be oxidized to CO2 and water vapor in the gas film surrounding the char bed. Grace et al.6 showed experimentally that, in the presence of water vapor, CO is completely oxidized in the boundary layer above a char bed, limited only by the availability of O2. In the absence of water vapor, however, more than half of the carbon leaving the char bed exited the reaction zone as CO. On the basis of the analysis of Grace et al.,6 the overall rate of conversion of char carbon to CO in the boundary layer was assumed to be complete when water vapor and oxygen were present. When no water vapor was present, the overall rate of char carbon conversion to CO and CO2 was calculated as the sum of the oxygen flux as O2 and the CO2 flux to the bed. In this case, the ratio of CO/CO2 in the product gases must be specified. Oxidation of char carbon can also occur through oxidation of the Na2S in the char bed and subsequent reduction by carbon of the Na2SO4 produced or through reduction of Na2CO3 in the char bed. In the char carbon oxidation model developed here, these effects were not included. Material and Energy Balances. Material balances were performed for a unit section of the char bed for each of the chemical elements C, H, O, Na, and S. The mass inputs were the black liquor char solids; any water remaining in the char if drying of the black liquor droplets was incomplete; and O2, CO2, and water vapor from the gas phase. The input rates of each element were calculated from the input rates and elemental composition of each component. The Na2SO4 content of the char reaching the bed was also specified. The outputs from the char are gases (CO, CO2, H2, H2O) and sodium salts (Na2CO3, Na2S, Na2SO4). All sulfur entering the char bed was assumed to leave as either Na2S or Na2SO4, and the fraction of sulfur leaving as Na2S was specified. The remaining sodium was assumed to leave as Na2CO3. The distribution of carbon

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exiting as gaseous species was determined by a modification of the criterion of Grace et al.,6 as discussed earlier. The rate of carbon accumulation was calculated as the difference between the carbon input and output rates and used to indicate whether the char bed was growing or shrinking. Hydrogen entering as char was assumed to leave as H2, while any water entering with the char was assumed to leave as water vapor. The local char bed temperature was obtained from an energy balance performed on a unit section of the char bed. The inputs were energy from the furnace walls and gases above the bed by convective transport, the energy content of the char and gases reaching into the bed, and the net energy generated by reactions 1-4. (Note that radiation from the gas combustion zone above the char bed is important in the energy balance for industrial recovery boilers. However, it was not important in the experiments used to validate the model developed here, and so, it was not included.) The reaction enthalpy changes were calculated as a function of temperature from enthalpy of formation data from the CHEMSAGE database.24 A value of zero was used for the heat of devolatilization. The radiant and convective energy inputs from the gases and wall were calculated iteratively with the unknown bed temperature as the input. The outputs were the gases leaving the bed, the inorganic salts leaving the bed, and the energy required to evaporate the water entering with the char. The unknown bed temperature was used as the reference temperature for the energy balance. Conduction of energy from the active char layer to the inactive bed below was neglected. Implementation of the Model. The PHOENICS code, a commercial computational fluid dynamics code, was used for the flow-field calculations. The char bed model was included as a subroutine within the PHOENICS code. Calculations were performed on a DEC Alpha-Station 220 computer. Results The model presented above was used to calculate the char bed burning rate and surface temperature under the same conditions as used by Brown et al.20 and Grace et al.6 in their experimental measurements. In their experiments, prepyrolyzed black liquor char was burned in a laboratory reactor designed to simulate a recovery boiler. The walls surrounding the char bed were heated electrically to 750-800 °C. A preheated gas mixture (250 °C), introduced through a slot jet (20 cm × 0.76 cm), swept across the char surface (20 cm × 10 cm). Gas velocities at the nozzle were varied systematically between 3.0 and 16.8 m/s in the experiments. The grid used included 875 cells in the domain within an area of 440 cm2. A body-fitted coordinate grid25 was used to ensure that the near-wall (bed) cells were in a suitable area for the wall function used to describe the gas velocity profile near the wall. The gas flow was assumed to be two-dimensional, with the plane jet sweeping across the 10-cm-long char bed. This is a reasonable approximation because the jet is relatively wide compared to its height. In one set of experiments, Brown et al. measured the influence of the O2 concentration on carbon consumption. Three different O2/N2 mixtures were used, with oxygen concentrations of 7, 14, and 21%. In Figure 2, our calculated mean values of carbon consumption rates are compared to those obtained in their experiments. The mean square differ-

Figure 2. Comparison of calculated average values of carbon consumption, indicated by the solid lines, and those measured by Brown et al.,20 indicated by the symbols. Table 2. Calculated versus Experimental Burning Rates and Char Bed Surface Temperatures for Four Different Inlet Gas Compositions with 14% O2 in the Entering Gas CO2 H2O(v) (mol %) (mol %) 0 0 10 10 a

0 10 0 10

rate of carbon consumption [mol/(m3 s)] measureda

calculated

0.244 0.325 0.249 0.281

0.236 0.291 0.241 0.301

char bed surface temperature (K) measureda calculated 1234 1205 1263 1240

1211 1171 1262 1224

Data are from Brown et al.20 and Grace et al.6

ence between the experimental and predicted molar fluxes was 0.04 mol/m2s; this is less than a 9% difference in terms of the average measured flux. The calculated slopes of the molar flux versus gas velocity were within 4% of the experimental values at all three oxygen concentrations. In another set of experiments, Grace et al. evaluated the effects of the CO2 and water vapor concentrations on the char burning rate in the presence of O2. In these experiments, four different gas mixtures were used. The oxygen mole fraction was kept constant (14%), and the CO2 and water vapor mole fractions were either 0 or 10%. The calculated mean values of carbon consumption and temperature are compared with the results from the Brown et al. experiments in Table 2. The model predicts correctly the effects of the presence or absence of water vapor, and the predicted overall burning rates agree with the experimental values, differing, on average, by less than 7%. The model also predicts accurately the experimentally measured char bed temperature in the active zone, with an average difference of 18 °C. To obtain the results shown in Figure 2 and Table 2, the Stanton number was corrected by multiplying it by a factor of 1.08. The correction is needed because the use of wall functions with wall jets is not as well understood as their use with turbulent boundary layers, as described below. Also, the wall (char bed surface) roughness is not known, and the bed was assumed to be smooth. A third possible cause is that the effective surface area of the bed was bigger than that of the rectangular plane described by the model. The results obtained after the correction are in good agreement with the experimental results, and the magnitude of the correction factor is reasonable. It should be noted that the burning rates are quite sensitive to the Stanton number. The 8% correction used increases the burning rates by about 60%. This is because the model predicts a temperature that is too low for the bed surface if the Stanton number is not corrected. Too low a temperature

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Figure 3. Calculated burning rate versus distance along the bed surface for three different values of the assumed active bed burning layer thickness.

Figure 5. Dependence of carbon consumption on inlet mass flow rate, inlet gas velocity, and temperature with constant mass flow rate as a function of the distance from the bed edge.

Figure 4. Calculated temperature of the active bed burning layer versus distance along the bed surface for three different values of the assumed active bed burning layer thickness.

Figure 6. Dependence of bed temperature on inlet mass flow rate, inlet gas velocity, and temperature with constant mass flow rate as a function of the distance from the bed edge.

slows the chemical kinetics of the reactions and causes a large decrease in carbon burning rates. If the roughness of the wall had been known, a correction factor might not have been needed. We conclude that the model gives qualitatively good results and predicts correct trends but that it includes some weaknesses, as described below, which could result in significant errors in the calculated results. The thickness of the active bed burning layer (L) is an input, rather than a calculated, parameter. Calculations were made with values of L from 0.01 to 0.09 m to determine its impact on the burning rate and active bed burning layer temperature. The results, presented in Figures 3 and 4, show that there is no significant effect of the assumed active bed burning layer on either the burning rate or the active bed burning layer temperature. The dependences of the carbon consumption rate on the inlet gas velocity (5.0-12.2 m/s), temperature (425525 K), and mass flow rate (4.4-5.4 kg/m2/s) are shown in Figures 5 and 6 as a function of the distance from the edge of the bed. These results were calculated using the same conditions as in some of the experiments of Brown et al. The gas composition at the jet was 14% O2, 10% CO2, 10% H2O, and 66% N2. For the calculated values, neither the inlet velocity nor the inlet gas temperature had a large influence on the carbon consumption rate or bed temperature when the mass flow rate was kept constant. A higher velocity at constant inlet gas temperature increases the mass flow rate at the inlet, and this leads to a higher carbon consumption rate and bed temperature. On the other hand, if the inlet velocity is kept constant and the temperature is decreased, the inlet mass flow rate increases because

Figure 7. Effect of CO2 and water vapor in the inlet gases on carbon consumption. The balance of the gas in each case is N2.

of the higher gas density. This increases the carbon consumption rate and temperature in the bed. The influences of the presence of CO2 and water vapor in the gas field above the bed on the carbon consumption rate and bed temperature are shown in Figures 7 and 8. The conditions are again the same as in the experiments by Brown et al. discussed earlier. If the case with the 14% O2/86% N2 inlet mixture is compared with the case in which a 14% O2/10% CO2/76% N2 inlet mixture is used, it can be seen that the presence of CO2 increases the char carbon consumption. This is because, in the absence of water vapor, CO is not oxidized to CO2 in the boundary layer and the rates of conversion of carbon at the char surface by oxygen and CO2 are additive. The bed temperature decreases because gasification is an endothermic reaction. In the case with a 14% O2/10% H2O/76% N2 mixture, the rate of carbon consumption at the leading edge of the char bed is lower than the rate with the O2/N2

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model developed in this study was tested only with data from experiments in which temperature levels and temperature differences were low compared to those in recovery boilers. No data are available for conditions more representative of the temperature levels and temperature differences in recovery boilers. This presents a limitation in our ability to assess the validity of the prediction of mass and heat transfer using wall functions. List of Symbols

Figure 8. Effect of CO2 and water vapor in the inlet gases on bed temperature. The balance of the gas in each case is N2.

mixture, but at the trailing edge, it is slightly higher. These two cases illustrate the important influence of reactions in the gas boundary layer on the overall rate of char carbon conversion. In the absence of water vapor in the gas phase, O2 reacts with the char to produce CO, not CO2, so that 2 mol of char carbon are converted per mole of O2 consumed. However, when water vapor is present, the CO is oxidized to CO2 in the boundary layer to the extent that O2 is available. The net effect is that no more than 1.7 mol of char carbon can be gasified per mole of O2 consumed when water vapor is present: 1.0 mol by O2 and at most 0.7 mol by water vapor. This is why the carbon consumption profile for the O2/H2O/N2 case in Figure 7 is below or never greatly above the profile for the O2/N2 mixture. In contrast, the carbon consumption at the downstream end of the char bed is almost the same level for both cases. In that region, O2 is becoming depleted, and less of it is available to convert CO to CO2. In addition, the CO2 produced upstream reacts with the char carbon to produce CO. The product gas from this region of the char bed contains both CO and CO2. The bed temperature in the O2/H2O/N2 case is at the highest level because oxidation to CO2 produces more heat than oxidation to CO, even though gasification with both H2O and CO2 is endothermic. In the fourth case, the inlet gas flow consists of a 14% O2/10% CO2/10% H2O/66% N2 mixture. The carbon consumption is the highest in this case because of the additional flux of oxidant (CO2) to the char surface. The bed temperature is lower than with the O2/H2O/N2 mixture because the additional endothermic effect of CO2 gasification decreases it. Conclusions The char bed model presented here, when coupled with the gas flow field via wall functions, predicts very well both the char bed burning rates and the active char layer temperature. When compared with the limited available experimental data, these parameters are predicted with average differences of 7% and 18 °C, respectively. The model predicts that increasing the mass flow rate of gas past the char bed surface increases the rate of carbon burning and decreases the char bed surface temperature. Changes in gas velocity and gas temperature per se, when occurring at constant gas mass flow rate, have negligible effects on the overall burning rate and bed temperature. The focus of the work reported here was the coupling between the bed and flow-field phenomena above it. The

Ac ) specific surface area of char, m2/g [Cc] ) carbon/Na2 mole ratio in char k ) parameter in k- model L ) thickness of active char burning layer, m [Na2] ) sodium content of char, kmol of Na2/kg p ) pressure, bar Sφ ) source term for mass, momentum, or energy [SO4] ) sulfate/Na2 mole ratio in char V, W ) velocity components in the y and z directions, respectively x, y, z ) spatial dimensions F ) density, kg/m3 φ ) state variable (velocity, temperature, or concentration)  ) parameter in k- model Γφ ) effective diffusivity coefficient Subscripts CO ) carbon monoxide CO2 ) carbon dioxide ox ) oxygen

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Received for review April 7, 2000 Revised manuscript received January 2, 2002 Accepted January 9, 2002 IE000393A