Energy & Fuels 2007, 21, 1493-1503
1493
Sensitivity Analysis of a Fixed Bed Combustion Model Robert Johansson,* Henrik Thunman, and Bo Leckner Department of Energy and EnVironment, Chalmers UniVersity of Technology, SE-412 96 Go¨teborg, Sweden ReceiVed October 6, 2006. ReVised Manuscript ReceiVed December 28, 2006
The ability to model the combustion of biofuels in a fixed bed is evaluated by a sensitivity analysis. The analysis treats the uncertainty of model parameters related to heat transport, reaction rates, and composition of volatiles. The scatter of the parameters is estimated from the differences between several published correlations. The results are compared with measurement data and possible model simplifications are discussed. It is shown that the bed model is able to reproduce the ignition rate and the maximum temperature. Prediction of these properties is relatively insensitive to the uncertainty of most of the parameters. Gas concentrations within the bed are more difficult to predict. They are greatly influenced by the composition of the volatile gas released during devolatilization. Also the composition of the volatile gas has a significant influence on the ignition of the gas in the model, affecting the ignition rate, particularly at low airflows. Moreover, the investigation shows that treatment of radiation can be simplified, and the number of gas species included in the model can be restricted without significant losses of model generality.
Introduction Biomass is burned in fixed beds in grate furnaces and domestic boilers, common techniques for small- to mediumscale heat production. Modeling of these beds improves the understanding of the combustion process and can support the design of furnaces and the investigation of problems related to the fuel bed. These models also provide boundary conditions at the surface of the bed for the modeling of combustion in the freeboard. Several bed models applied to conversion of biofuels can be found in the literature.1-7 These models have been used to investigate bed temperature, reaction rates, and ignition rate during the conversion process. The ignition rate, Ignbed, is defined as the velocity of the reaction front, ufront, multiplied by the initial density of the bed, Fbed, according to eq 1. The velocity of the front is calculated by dividing the vertical distance between two points in the bed (z2 - z1) by the difference in time ∆t when these points reach a temperature of 800 K.
Ignbed ) Fbed
z2 - z 1 ) Fbed ufront ∆t
(1)
This definition is consistent with that used in earlier works,7,8,9 although it was called reaction front rate in one case.7 Some * To whom correspondence should be addressed. Phone: + 46 31 772 5249. Fax: +46 31 722 3592. E-mail:
[email protected]. (1) Fatehi, M.; Kaviany, M. Int. J. Heat Mass Transfer 1997, 40, 26072620. (2) Shin, D.; Choi, S. Combust. Flame 2000, 121, 167-180. (3) Di Blasi, C. Chem. Eng. Sci. 2000, 55, 2931-2944. (4) Yang, Y. B.; Goh, Y. R.; Zakaria, R.; Nasserzadeh, V.; Swithenbank, J. Waste Manage. 2002, 22, 369-380. (5) Thunman, H.; Leckner, B. Fuel 2003, 82, 275-283. (6) Bruch, C.; Peters, B.; Nussbaumer, T. Fuel 2003, 82, 729-738. (7) Zhou, H.; Jensen, A. D.; Glarborg, P.; Jensen, P. A.; Kavaliauskas, A. Fuel 2005, 84, 389-403. (8) Gort, R. On the propagation of a reaction front in a packed bed. Ph.D. Thesis, Universiteit Twente, Enschede, The Netherlands, 1995. (9) Ro¨nnba¨ck, M.; Axell, M.; Gustavsson, L.; Thunman, H.; Leckner, B. Progress in Thermochemical Biomass Conversion, Bridgewater, A. V.; Ed; Blackwell Science: Oxford, 2001, 743-757.
other publications1,2,10 refer to the velocity of the front, but this property depends on fuel density. The reaction fronts in beds with different fuel densities have different velocities, while the ignition rates in general are comparable, as seen in Figure 3, which shows experiments on both wood pellets and wood chips. In a number of papers,11-13 the influence of the devolatilization rate, air flow, moisture content, particle size, fuel density, bed porosity, and heating value on temperature, conversion rate, and gas composition was examined. Other works2,14 looked at the influence of particle size, heating value, and density on the ignition rate. One model15 assumes local thermal equilibrium between gas and solid, and the combustion was approximated with a single-step reaction based on char kinetics. The authors primarily focused on differences between stoichiometric regimes, but they also examined the sensitivity of the ignition rate to variations of the heat of reaction, pre-exponential factor, and activation energy of the combustion reaction. In another work,7 the combustion of straw was modeled, and a parametric study was carried out on the solid conductivity, bed porosity, heat capacity of both the solid and gas, rate of devolatilization, straw diameter, heat- and mass-transfer coefficients, pyrolysis yields, tar combustion rate, and mixing of gaseous species. The results showed that, within the ranges investigated, these parameters do not affect the results more than 10%. However, the parametric ranges were not related to the span of the data found in the literature, and in general, the ranges were narrow. The present work is focused on the evaluation of uncertainties in the modeling of biofuel combustion in fixed beds with the (10) Saastamoinen, J. J.; Taipale, R.; Horttanainen, M.; Sarkomaa, P. Combust. Flame 2000, 123, 214-226. (11) Yang, Y. B.; Yamauchi, H.; Nasserzadeh, V.; Swithenbank, J. Fuel 2003, 82, 2205-2221. (12) Yang, Y. B.; Sharifi, V. N.; Swithenbank, J. Fuel 2004, 83, 15531562. (13) Yang, Y. B.; Ryu, C.; Khor, A.; Yates, N. E.; Sharifi, V. N.; Swithenbank, J. Fuel 2005, 84, 2116-2130. (14) Thunman, H.; Leckner, B. Proc. Comb. Inst. 2005, 30, 2939. (15) Fatehi, M.; Kaviany, M. Combust. Flame 1994, 99, 1-17.
10.1021/ef060500z CCC: $37.00 © 2007 American Chemical Society Published on Web 03/22/2007
1494 Energy & Fuels, Vol. 21, No. 3, 2007
goal of finding simplifications and assisting in further model development. Special emphasis is put on the evolution of gas concentration throughout the fuel bed and on the ability to reproduce measured concentrations inside the bed. The model deals with the same combustion system as the bed models just mentioned and many of the submodels and correlations applied are the same or similar to the ones included in these models. To simulate the conversion process, these models have to treat a complex set of equations with a large number of parameters. The parameters are based on data and correlations found in the literature, but there is some uncertainty in their values; therefore, there is a need to know how they influence the results. An improved understanding of the impact of these parameters is important for the quantitative and qualitative evaluation of model results. This work treats the sensitivity of the modeling results due to possible variation of the parameters related to effective reaction rates and heat transfer. The uncertainty of the parameters is estimated by comparison of several literature sources. The influence of the parameters on the ignition rate, temperature of the gas and solids, and gas concentration is evaluated, and the results are compared with measured data. The influence of the composition of the volatile gas is also addressed and possible model simplifications are discussed. Model The case selected for the sensitivity analysis consists of a bed of spherical biofuel particles with an initial size of 1 cm and a moisture content of 10% (on a wet basis). Combustion air is fed from below and ignition takes place at the surface of the bed, which has an initial height of 0.3 m. This creates a conversion situation that can be described in one dimension. After ignition, a reaction front propagates down into the bed; the fuel dries and devolatilizes, and the char, together with the released volatiles, burn as long as there is oxygen available. The speed and profile of the reaction front is stable during most of the propagation period, and the conversion mode is suitable for the comparison of models and measurements. This case has previously been thoroughly investigated both in experimental work8-10 and in modeling.1-7 The one-dimensional transient model employed in this work consists of the transport equations for energy and species, together with the continuity equation of the gaseous phase, and regards the bed as a porous medium. The energy equation is expressed for the solid and the gaseous phases including heat transfer between them. The energy equation of the solid phase accounts for radiation and conduction, while convection, dispersion, and diffusion of species with different specific heats are treated in the gas phase. Eight species are included in the gaseous phase: CO, CO2, H2O, H2, CH4, and CxHyOz in the volatile gas and O2 and N2 in the air. The gas species are transported by convection and dispersion. The density of the gas is given by the ideal gas law. Drying and devolatilization are modeled with Arrhenius expressions, while both diffusion and kinetics are considered in the conversion of the char. The homogeneous rates of conversion are the minimums of the kinetic and mixing rates. These conversion steps are expressed by source terms in the transport equations of energy and species. The equations are discretized in space according the finite volume method following the hybrid scheme. The time discretization is fully implicit to ensure numerical stability. Care has been taken to select step sizes, making the results independent of the time and space discretization. Shrinkage of the bed caused by conversion of the fuel is accounted for by recalculation of the sizes of the computational cells at each time step. The equations are summarized in Appendix 1, and a more detailed description of the bed model can be found elsewhere.16 (16) Johansson, R.; Thunman, H.; Leckner, B. Influence of intra-particle gradients in modelling of fixed bed combustion.Combust. Flame 2007, in press.
Johansson et al. The surface of the bed is a moving boundary that follows the shrinking bed and exchanges heat with the surroundings by radiation. The bed is ignited by a radiating flame, whose temperature is 1400 K with an effective area that is equal to the bed area. Once the reaction front has reached a few centimeters down into the bed, the flame is removed, and the surroundings are now assumed to radiate back with a temperature equal to that of the gases flowing out from the bed but with an upper limit of 1200 K. The gas temperature and the gas concentrations at the bed’s surface are assumed to have zero gradients, a Neumann condition, and the pressure is specified to be 1 bar. The grate on which the bed rests exchanges heat by conduction and radiation with the solids and transports heat by radiation and convection/conduction to the surroundings. The convective part of this heat is assumed to be absorbed by the entering air, and it is therefore returned to the bed by the air flow. The velocity and temperature of the entering air are specified. The initial temperature of the gas and solids in the bed and the temperature of the inlet air is 300 K. Procedure and Investigated Parameters. The sensitivity of the parameters included in the model has been investigated for two inlet air velocities: 0.05 m s-1 in Case I and 0.15 m s-1 in Case II. Both velocities correspond to substoichiometric conditions; all oxygen is consumed during the propagation of the front in the bed. Stoichiometric velocities, for which enough oxygen is supplied to convert all the ignited fuel, are in the range of 0.2-0.4 m s-1. The velocities investigated are in the lower range of primary air velocity in grate furnaces.17-18 The limits of the parameters cover the ranges of the correlations found in literature. Only one parameter at a time has been examined. For each parameter, a reference value has been chosen, which is applied when other parameters are investigated. The limits are summarized in Table 1 and graphically shown in Appendix 2, together with correlations from literature. In addition, the influence of the composition of the volatile gas has been investigated. The parameters seen in Table 1 were chosen as follows. Mixing Rate of Gaseous Species. Mixing controls the homogeneous reaction rates at high temperatures where the kinetics are fast. Three correlations for mixing have been compared. The first rate19 was derived using the Ergun equation as a starting point. The second14 correlation is a modification where the viscous term has been removed. This correlation is used as a reference. The third correlation7 is based on diffusion across a characteristic length, chosen as the diameter of a particle. The difference between the three mixing rates is very large. The rates have not been validated by experimental data, although the turbulent part in the first correlation was compared with CFD simulations. The third expression is highly dependent on the particle size: it was used to model straw combustion with the characteristic length taken as the straw diameter, resulting in a significant rate, despite being small compared with the other expressions. The first expression also depends highly on particle size and yields the highest rate. The rates of the third and first correlations have been chosen as the lower and upper limits, respectively. EffectiVe Surface Area. The effective surface area is that of the solids per unit of bed volume available for heat transfer and char conversion. It is important for heat transport between the phases and char combustion. The maximum effective area (the whole surface area of the particles per unit of bed volume) serves as reference value. This value is commonly used in the modeling of fixed beds. However, the particles in a packed bed are in close contact and, to some extent, cover each other, and the entire surface area may not contribute to the effective area. Therefore, the lower (17) van der Lans, R. P.; Pedersen, L. T.; Jensen, A.; Glarborg, P.; DamJohansen, K. Biomass Bioenergy 2000, 19, 199-208. (18) Kuo, J. T.; Hsu, W.-S.; Yo, T.-C. J. Energy Resour. Technol. 1997, 119, 120-128. (19) Yang, Y. B.; Goh, Y. R.; Nasserzadeh, V.; Swithenbank, J. Presented at the 3rd International Symposium on Incinaration and Flue Gas Treatment Technology, Brussels, Belgium, 2001.
Fixed Bed Combustion Model
Energy & Fuels, Vol. 21, No. 3, 2007 1495 Table 1. Parameter values for the simulations
parameter mixing rate of gaseous species
rmix ) 0.63 (ref 14)
effective surface area bed porosity axial dispersion
solid conductivity
extinction coefficient heat and mass transfer coefficients
shrinkage during devolatilization
(
ref value
) []
1.75ug(1 - ) Ci min dp Ωi
min value 2Dab
rmix ) ln(mO2,0/mO2,τm) (ref 7)
dp
2
0.4 0.6 × ref
(ref 19) Ap Asur ) Vp 0.6 1.2 × ref
0.6 × ref
1.5 × ref
0.7 × ref
1.3 × ref
Nu ) 2 + 1.1Pr1/3Re0.6
0.5 × ref
1.5 × ref
Sh ) 2 + 1.1Sc1/3Re0.6 (ref 23) 60 vol %
40 vol %
70 vol %
Asur )
Ap Vp
0.5 0.5dpu Deff,ax ) 0.8Dm + 0.5dpcp,gFgu kg,eff,ax ) 0.8kg,m + (ref 23) 0.2 W/mK for virgin wood λc ) (1 - c)(1.47 + Ts1.11 × 10-3) (ref 28) scaled projected surface area (ref 26)
Asur ) 0.6
limit of this parameter is set to 60% of the entire area, which accounts for significant contact areas of particles. Bed Porosity. The porosity of the bed varies with packing. A typical value for a packed bed is around 0.4,20-22 but some data obtained from a bed of biofuels indicate a slightly higher porosity.9 The reference value is chosen to 0.5 with lower and upper limits of 0.4 and 0.6. Axial Dispersion. Gas flow through a packed bed gives rise to dispersion, and the effective diffusion of heat and mass is much higher than the molecular counterpart. Increased heat transport enables the ignition of the gas earlier in the front, since the gas reaches the ignition temperature sooner. The reference values for effective heat and mass diffusion of the gas are taken from a standard correlation23 often applied in the modeling of fixed bed combustion.4,7,14 These values have been compared with empirical correlations for granular packed beds24 and packed bed reactors.25 The range of variation is found to be 60-120% of the reference value. Extinction Coefficient and Solid ConductiVity. Both radiative and conductive heat transport are important in the solid phase. They affect the temperature of the gas and solids and the propagation of the reaction front. The radiative heat transfer in the bed is modeled with a two-flux model, an approach used in other bed models.2,4,5 The extinction coefficient is equal to the projected surface area of the particles per unit bed volume, corrected by a scaling factor to account for dependent scattering.26 The influence of radiation has been investigated not only by changes of the extinction coefficient but also by comparison of the two-flux model with a radiative conductivity model,27 in which a radiative part is added to the conductivity of the solids. The lower and upper limits of the (20) Zou, R. P.; Yu, A. B. Powder Technol. 1996, 88, 71-79. (21) Sederman, A. J.; Alexander, P.; Gladden, L. F. Powder Technol. 2001, 117, 255-269. (22) Montillet, A.; Le, Coq, L. Powder Technol. 2001, 121, 138148. (23) Wakao, N.; Kaguei, S. Heat and Mass Transfer in Packed Beds; Gordon and Breach: New York, 1982. (24) Guedes, de Carvalho, J. R. F.; Delgado, J. M. P. Q. Chem. Eng. Sci. 2005, 60, 365-375. (25) Jacobsen, H. A.; Lindborg, H.; Handeland, V. Comput. Chem. Eng. 2002, 26, 333-357. (26) Singh, B. P.; Kaviany, M. Int. J. Heat Mass Transfer 1992, 35, 1397-1405. (27) Singh, B. P.; Kaviany, M. Int. J. Heat Mass Transfer 1994, 37, 2579-2583.
Ap Vp
(
rmix ) 0.65 150
max value Dab(1 - )2/3 dp2
+
)
1.75ug(1 - )1/3 min[CiΩi] dp
extinction coefficient are chosen to 70 and 130% of the reference value. The conductivity of the solids is calculated from a massweighted average of the conductivities of virgin wood and char. The conductivity of char depends on temperature.28 From an evaluation of literature data, the parameter range is estimated to be 60-150% of the reference conductivity. Heat- and Mass-Transfer Coefficients. The heat-transfer coefficient controls the temperature difference between the phases. Therefore it influences ignition of both the gas and solids. The masstransfer coefficient has a direct effect on the rate of char conversion in the diffusion-controlled regime. The reference values for the heatand mass-transfer coefficients are calculated according to correlations and fitted to an extensive amount of data from different experiments.23 They have been used in earlier work.1,2,5 The minimum is half of the reference, and the maximum is 1.5 times the reference value. This range covers almost all experimental data, forming the basis of the correlation.23 Shrinkage during DeVolatilization. Shrinkage during devolatilization affects char conversion, heat transfer between gas and solids, and mixing of gaseous species because these processes depend on the particle size. It also affects heat transfer in the bed: larger shrinkage reduces the distance between the hot combustion zone and the unreacted wood. During devolatilization, a biofuel particle shrinks, usually to a size corresponding to about 40% of the initial volume, but the shrinkage depends on the temperature and wood type.29 The shrinkage is varied between 40 and 70 vol %, and the reference value is 60 vol %. Composition of Volatile Gas. The influence of the composition of the volatile gas has been investigated by simulations with compositions taken from related works.3,30 The calculation method31 for the composition in the reference case involves two experimental ratios, which depend on the specific surface area of the fuel particles. Therefore, the influence of variations in these experimental ratios has also been investigated. No attempt has been made to estimate the upper or lower limits of these concentrations because (28) Thunman, H.; Leckner, B.; Niklasson, F.; Johnsson, F. Combust. Flame 2002, 129, 30-46. (29) Davidsson, K. O.; Pettersson, J. B. C. Fuel 2002, 81, 263-270. (30) Bryden, K. M.; Ragland, K. W. Energy Fuels 1996, 10, 269275. (31) Thunman, H.; Niklasson, F.; Johnsson, F.; Leckner, B. Energy Fuels 2001, 15, 1488-1497.
1496 Energy & Fuels, Vol. 21, No. 3, 2007
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Figure 1. Layout of the gas measurements in the reaction front.
Figure 3. Comparison of ignition rate in simulations and experiments: ×, Gort,8 wood blocks 10 mm, Xm,0 ) 0.1; O, Ro¨nnba¨ck et al.,9 wood cylinders 8 mm; 3, Saastamoinen et al.,10 wood 5-20 mm, Xm,0 ) 0.11. The lines are as follows: reference case (dashed-dotted), high mixing rate (solid), and low heat- and mass-transfer coefficients (dashed). The shaded area shows a change of less than 10% of the simulated ignition rate in the reference case. The solid straight line represents the stoichiometric air flow based on ignited fuel.
Figure 2. Simulated mole fractions of gas species at the position of the probe as a function of time. Parameters are those of reference case I. The shaded area is shown as a close up in Figure 4.
they depend on experimental conditions, and therefore, the results are hard to compare.
Results and Discussion The reaction front descends in the bed promoted by conductive and radiative heat transport in the solid phase, while the fuel is dried and devolatilized. The concentration of gases from the drying and devolatilization increases rapidly when the front passes at the same time that the concentration of oxygen decreases because of dilution. Figure 1. Figure 2 shows the gas composition at a position corresponding to the probe tip. As the reaction front progresses downward, the surface of the bed moves to a position below the gas probe; then, the simulated concentrations correspond to the gas leaving the surface of the bed. The char formed during devolatilization reacts with oxygen, which results in further heating of the solids and propagation of the front. In the downstream part of the front, the gas is colder than the solid phase, and heat transfer between the phases reduces the heating rate of the solids. Once the gas temperature is high enough for the released volatiles to ignite, the gas attains a higher temperature than the solids. Oxygen is consumed by the combustion of char and volatiles, and the concentration of the combustion products rises. If the air flow is low, there is not enough oxygen to complete the char conversion during propagation of the front, and a char layer builds up. This char is to some extent gasified by H2O and CO2. At 2200 s when the front reaches the grate, devolatilization is finished, and the concentration of volatile gases approaches zero, Figure 2. During the first period of the final char burnout, around 2600 s, there is a peak in the concentration of CO. In the second half of this conversion step, combustion is slower, and the concentration of oxygen increases again, while the fractions of CO and CO2 decrease.
Figure 4. Close up of measured32 and simulated gas concentrations at the position of the probe during passage of the reaction front. Simulated data are from reference case I.
Increased heat transport in the solid phase enhances the ignition rate and is followed by a small change of the temperature of the solids. When the rate increases, more fuel is heated per time unit, and since the heat release is more or less constant for a given air flow, the temperature tends to fall. A lower ignition rate, on the other hand, means that less fuel is heated per unit time, and it is, therefore, followed by a rise in temperature. This assumes that the heat release is independent of the ignition rate, which is only a reasonable approximation when all oxygen is consumed in the reaction front. For overstoichiometric conditions, there is more oxygen available. A higher ignition rate then means that more fuel is burned, and the heat release increases. Even for sub-stoichiometric conditions, a change of the ignition rate can affect the heat release if the fraction of oxygen needed for the gas-phase reactions changes compared to that consumed by the combustion of the solids. This is caused by the difference in heat release per mole of oxygen between heterogeneous and homogeneous combustion reactions. The effect is, however, of minor importance. The model predictions of ignition rate and maximum temperature agree well with experimental data,8-10 Figure 3. Figure 4 shows the corresponding agreement between simulated and measured32 gas concentrations in the reaction front. The comparison of the simulated and measured gas species is not straightforward because the local experimental air velocity at the centerline of the bed, where the probe is positioned, differs
Fixed Bed Combustion Model
Energy & Fuels, Vol. 21, No. 3, 2007 1497
from the cross-sectional average air velocity because of the higher velocities close to the reactor walls. The model assumes that the air flow is evenly distributed, that is, that the velocity is uniform over the cross-section. The simulated data are from reference case I, and the air velocity of this case corresponds to the experimental air velocity at the position of the probe. Since the average measured air flow is higher than the local velocity at the probe, the simulated ignition rate is 28% lower than the measured one. Therefore, the stoichiometries differ. The measured concentration profiles have been shifted along the time axis to make the initial reduction in the oxygen profile coincide with the simulated initial decrease to remove differences caused by the variation in ignition rates and initial bed heights. This is justified because the front propagates with a constant speed throughout most of the bed, and the temperature and concentration profiles in the front remain stable during this period. The purpose is to compare differences in the concentration profiles inside the front and not differences in the ignition rate. The increase in the concentration of water vapor in the initial stage, followed by a release of volatile gases, is well reproduced by the model. The sharp increase of CO2 compared to CO, seen in the measured profiles, is also evident in the simulated profiles. The trends of slightly increasing fractions of H2 and CO, together with a decreasing fraction of H2O during the propagation of the reaction front and a peak in the CO concentration during the final char burnout, are seen in both the experiments and simulations. Furthermore, the measured decrease of the water vapor concentration upstream of the reaction front, predominantly resulting from gasification reactions and the water-gas shift reaction, is also captured by the model. It should be pointed out that, when the entire conversion process is examined, the measured fraction of water vapor continues to decrease significantly during the period of front propagation while the simulated fraction remains more constant. The conversion of the lower hydrocarbons, mainly CH4, is not well captured by a model that predicts a much faster increase of the concentration than what is seen from the measurements. Figure 5 shows the relative difference in the ignition rates and peak temperatures of the gases and solids between the reference cases and the cases with the upper or lower limits of the parameters. The positions on the x axis where the lines reach a relative horizontal value of unity, corresponding to the reference case, are calculated from the relation between reference values and limits of the parameters (p)
xref ) xlow + (xhigh - xlow)
pref - plow phigh - plow
(2)
The rate of mixing of gaseous species is estimated because the rate changes during conversion. The correlation used for the lower limit of mixing does not depend on gas velocity, while the other two correlations do. Therefore the x value corresponding to the reference case differs between cases I and II. As shown in Figure 5, the influence of most parameters is less than 10% (shaded area), which is a small range compared to the scatter in measurement data in Figure 3. This is also the case for the influence on the gas concentration. Only the rate of mixing of the gaseous species and the heat- and mass-transfer coefficients affect the ignition rate and the temperatures more, while some of the gas species are also affected by the effective (32) Ro¨nnba¨ck, M.; Samuelsson, J.; Tullin, C.; Thunman, H.; Leckner, B. Presented at the Science in Thermal and Chemical Biomass Conversion Convention, Victoria, BC, Canada, 2004.
area. The effect of mixing and heat- and mass-transfer on ignition rate can also be seen in Figure 3. The composition of the volatile gas has an important effect on the ignition rate, as will be discussed below, and it significantly influences the concentration of gas species. Simulation with one of the alternative compositions30 results in small differences in the concentrations of H2O, CO, and CO2 and large differences, 20-70%, in the concentration of H2, CH4, and CxHyOz, whereas the second alternative composition3 gives large differences, >20%, in all gas species. Variation of the empirical ratios in the method31 for prediction of the composition in the reference cases changes the concentration of H2O, CO, and CO2 up to 15%, and the fractions of the other gas species are affected even more. The mixing rate influences the homogeneous reactions and the heat release in the gaseous phase: lower mixing rates give lower heat release, and the gas attains a lower temperature and less-steep temperature profile than it does at higher mixing rates. The temperature profiles for the two extreme values of mixing rates in Table 1 are illustrated in Figure 6. The slower heating at the lower mixing rate increases the cooling or, for some conditions, reduces the heating of the solids and thereby decreases the ignition rate. The effect is more pronounced in the low-velocity case, Case I, where the upper and lower limits of the mixing rate give significantly different ignition rates, as shown in Figure 5a-b. The low mixing rate reduces the rates of the gas reaction. Then, they consume less oxygen, which becomes available for char combustion, and the char is more completely converted. In Case II, this effect means that only a thin char layer builds up, resulting in a short residence time of the gas inside the bed. The gas, therefore, does not have enough time to reach its peak temperature before it leaves the surface of the bed. A higher mixing rate leads to a higher ignition rate and more fuel-rich gas. When the upper limit of the mixing rate is used, the stoichiometry approaches the experimental one. This results in the same trends in the simulated gas concentrations as in the reference case, but the fraction of water vapor is significantly higher and the fraction of CO is somewhat lower, as can be seen from a comparison of Figures 7 and 4. Heat and mass transfer between the gas and solid phases affects the ignition rate in two ways: lower heat transfer reduces the cooling of the gas, and lower mass transfer reduces the rate of char combustion. For the conditions investigated, the influence of the mass transfer dominates, and the lower heat- and mass-transfer coefficients markedly reduce the ignition rate; in case II, the ignition rate differs from that of the reference case by more than 10%, Figure 5b. The highest values tested do not change the ignition rate as much, and in all cases, the effect on the maximum temperatures is limited. The temperature difference between the phases is also influenced, as shown in Figure 8. The sensitivity of the heat transport mechanisms in the solid phase, conduction and radiation, is limited, but some comments should be made about an alternative radiation model. In the reference case, radiation was modeled by a two-flux model (Appendix I) but a radiative conductivity model was also tested for these conditions. This model requires less computational time and, for this reason, is a more efficient option. The radiative conductivity model yields a somewhat lower ignition rate ( 1 f R ) 1 if R1 < 1 f R ) R1
(A18)
(A19)
kc1 ) 1.715 Tgexp(-9000/Tg)
28
kc2 ) kc3 ) 3.42Tg exp(-15 000/Tg)
28
kc4 ) 3.42 × 10-3Tg exp(-15 600/Tg)
28
rg,i ) min[rkin,i rmix,i] for combustion reactions
14
rmix,i ) 0.63 (A20)
) []
(
1.75ug(1 - ) Ci min dp,h Ωi
CO + 1/2O2 f CO2
rkin,1 ) 1.3 × 108CO20.5CH2O0.5CCO exp(-15 100/Tg)
C1.16H4 + 1.58O2 f 1.16CO + 2H2O rkin,2 ) 1.585 × 1010CO20.8CCH40.7 exp(-24 392/Tg)
39
C6H6.2O0.2 + 2.8O2 f 3CO + 3.1H2 rkin,3 ) 20 700Tg CO2CCHO
40
0.3
H2 + /2O2 f H2O rkin,4 ) 10 CH2CO2 exp(-5050/Tg) 1
(A21)
38
11
0.5
exp(-9650/Tg)
3
CO + H2O f H2 + CO2 rkin,5 ) 2.78CH2OCCO exp(-1510/Tg)
41
H2 + CO2 f CO + H2O rkin,6 ) 93.69CCO2CH2 exp(-5604/Tg)
37
(1 - )2µgug (1 - )Fgug2 ∆P ) 180 +4 3 2 ∆z d 3d
42
p
p
1502 Energy & Fuels, Vol. 21, No. 3, 2007
Johansson et al.
Table A1 (Continued) equation Ap(1 - ) 6(1 - ) ) Vp dp
(A22)
Asur )
(A23)
Vp ) Vp,0 - Vp,0θdrying θcomb ) 1 -
(
Vp,0Fs,0Yash
[ (
Vp,0Fs,0Yc,θ/ Vp,0
(A25)
(A26)
( )
Ym 1 - θdrying - θdev Ym,θ
)]
(
)
Ym,0 Yash,0 ) Vp,0 1 - θdrying - θdev Ym,θ Yc,0
23
(Sdev + Sdrying)cp,g/Asur exp[(Sdev + Sdrying)cp,g/Asur/hT,0] - 1
hm,0 ) hm )
)
λg (2 + 1.1Pr1/3Re0.6) dp
hT,0 ) hT )
(
Ym,0 Ym Yv Yc 1- Vp,0θdev 1- Vp,0θcomb 1Ym,θ Ym,0 Yv,0 Yc,0
Ym,0 Vp,ash - θdrying - θdev Vp,0 Ym,θ
Vp,ash )
(A24)
)
ref
Dab (2 + 1.1Sc1/3Re0.6) dp
23
(Sdev + Sdrying)/Asur/Fg exp[(Sdev + Sdrying)/Asur/Fg/hm,0] - 1
λg,eff ) 0.8λg +0.5dpcp,gFgug/
23
Deff ) 0.8DAB + ugdp/
23
( )
(A27)
DAB ) DAB,ref
(A28)
cp,s )
1.75
37
′′′ + cp,cmc′′′) (cp,moistmdry mdry ′′′ + mc′′′
cp,moist )
A)
Tg 298
cp,sdry + 4190Ym/(1 - Ym) 1 + Ym/(1 - Ym)
+A
(23.55Ts - 1320Ym/(1 - Ym) - 6191)Ym
43
(1 - Ym)
where cp,sdry ) 4.206T - 37.7 (A29)
λs )
44
(λdrmdry ′′′ + λcmc′′′) mdry ′′′ + mc′′′
λc ) (1 - )(1.47 + Ts1.11 × 10-3)
28
Appendix 2 This section provides a comparison of correlations from different literature sources. Parameter ranges are indicated by the shaded areas in Figures A1-A5. Figure A1 shows the axial dispersion. The solid conductivities for char and virgin wood are shown by Figures A2 and A3, respectively. The extinction and heat-transfer coefficients are shown in Figures A4 and A5, respectively.
Figure A1. Axial dispersion: Wakao, ref 23, solid line; Guedes et al., ref 24, dashed line; Skaare et al., ref 25, dashed dotted line.
Figure A2. Solid conductivity for char: Thunman et al., ref 28, solid lines; Chan + radiative contribution, ref 36, thick lines; MacLean, ref 45, thin lines.
Fixed Bed Combustion Model
Figure A3. Solid conductivity for virgin wood: reference 0.2 W/mK, solid line; Thunman and Leckner, ref 46, dashed line; Maclean, ref 45, dotted line.
Figure A4. Extinction coefficent: Singh and Kaviany, ref 26, solid line; Chin and Choi, ref 2, dashed line.
Figure A5. Heat transfer coefficient: Wakao, ref 23, solid line; Bird, ref 47, dashed line.
Nomenclature Ap ) surface area of a particle, m2 Asur ) internal surface area, m2 m-3 C ) molar concentration of gas species, mol m-3 DAB ) binary diffusion coefficient, m2 s-1 Deff ) effective diffusion coefficient, m2 s-1 Ignbed ) ignition rate, kg m-2 s-1
Energy & Fuels, Vol. 21, No. 3, 2007 1503 J+, J- ) radiation intensity in positive and negative direction, W m-2 K ) extinction coefficient, [-] M ) molar mass, kg mol-1 S ) source term, unit dependent on equation R ) the gas constant, J mol-1 K-1 T ) temperature, K V ) volume, m3 X ) molar fraction, [-] Y ) mass fraction, [-] cp ) specific heat, J kg-1 K-1 dp ) diameter of particle, m hm ) mass transfer coefficient, m s-1 hT ) heat transfer coefficient, W m-2 K-1 kc ) reaction rate constant, char conversion, m s-1 m ) mass, kg p ) parameter value, unit depends on parameter r ) reaction rate, unit depends on equation t ) time, s u ) superficial velocity, m s-1 ufront ) reaction front velocity, m s-1 x ) position on vertical axis z ) bed height, m Ω ) stoichiometric coefficient, [-] R ) fraction oxygen consumed by hydrogen combustion, [-] ) porosity, [-] θ ) shrinkage factor, [-] κ ) optical thickness, [-] λ ) conductivity, W m-1 K-1 µg ) dynamic viscosity gas, Pa s F ) density, kg m-3 σ ) Stefan-Boltzmann constant, W m-2 K-4 φ ) dependent scattering scaling factor, [-] ω ) scattering albedo, [-] ∆h ) heat of reaction, J mol-1 ∆H ) heat of conversion, J kg-1 Subscripts 0 ) initial or inlet ash ) ash, dry wood basis bed ) property of the bed c ) char comb ) char combustion dev ) devolatilization dry ) dry wood eff ) effective g ) gas h ) heterogeneous reaction j ) homogeneous reaction j k ) gas species k in heterogeneous reaction l ) heterogeneous reaction l n ) gas species n in devolatilisation m ) moisture, wet wood basis p ) particle ref ) reference, 298 K s ) solid sur ) surface tot ) total v ) volatiles, dry wood basis vap ) evaporation EF060500Z
