Ind. Eng. Chem. Res. 2002, 41, 4663-4673
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Estimation of Solids Mixing in a Fluidized-Bed Combustor Fredrik Niklasson,* Henrik Thunman, Filip Johnsson, and Bo Leckner Department of Energy Conversion, Chalmers University of Technology, S-412 96 Go¨ teborg, Sweden
A simple method is proposed for the estimation of the effective lateral dispersion of fuel in a fluidized-bed combustor. The method is based on a combination of a model for drying and devolatilization of the fuel and cross-sectional measurements of the water concentration above the bottom bed during steady-state operation. By correlation of the drying of the fuel particles and the measured moisture concentrations above the bed, the effective lateral dispersion coefficient of the fuel particles is determined. This coefficient was estimated to be on the order of 0.1 m2/s, which is considerably higher than that predicted by most of the expressions given in the literature. Introduction Bubbling (stationary) fluidized-bed boilers are widely used for combustion of wood waste, usually delivered as wood chips. Such fuels, originating from the forest, have moisture contents between 30 and 50%. When injected into the combustion chamber, the fuel dries, devolatilizes, and burns. The drying and devolatilization processes overlap in time for thermally thick fuel particles, such as those used in a fluidized-bed combustor. A substantial fraction of the combustion takes place above the fluidized bed because of the large volatile content (80-90% on dry basis). This behavior makes it possible to operate the boiler without internal heattransfer surfaces in the bed, and the problem connected to erosion on bottom-bed tubes is avoided. However, the release of a large amount of volatiles in the bed may result in lateral maldistribution of the combustible gases above the bed, which could be unfavorable for the performance of the combustor. To counteract this problem in commercial units, the fuel might be fed into the bed in more than one feed point, perhaps assisted by a spreader when the cross-sectional area of the bed is large. Each added feed point increases the cost of the installation, and it is necessary to reduce the number of feed points while maintaining fuel burnout and low emissions. When the number of feed points is being optimized, knowledge of the lateral mixing rate of the fuel in the bed is important. The mixing rate is difficult to determine experimentally during operation of a full-scale boiler, but on the other hand, results from small-scale experiments are probably not applicable to a boiler. Here, the lateral mixing rate is expressed as an effective (cross-sectional average) dispersion coefficient, which is estimated from a relatively simple experiment. The aim of the work is to develop a method to estimate the lateral dispersion of fuel, applicable to boilers. The Chalmers 12 MWth circulating fluidized-bed boiler1 was used. The boiler was operated under noncirculating conditions. The arrangement of the fuel feed is simple; the fuel is fed through a chute in a certain location onto the top of the bed, as shown in Figure 1. A semiempirical approach was used to determine the * To whom correspondence should be addressed. Phone: +46 31 7721438. Fax: +46 31 7723592. E-mail: frni@ entek.chalmers.se.
Figure 1. Bottom region of the Chalmers fluidized-bed boiler. The fuel enters from the fuel chute (F) and meets the splash zone before falling down in the dense bed. A-A is the height at which the water concentration in the gas was measured. The dashed arrows indicate circulation of solids.
dispersion coefficient. The method will be explained below, but first a thorough literature survey is presented with the purpose of answering the question, can we use the available information for the present problem? Dispersion of Solids The fluid dynamics of a fluidized bed are complex because of the interactions of particles and gas. The common view is that the mixing in the bed is caused by bubbles stirring the bed material as they rise through the bed and erupt at the surface,2 even if a certain mixing also is observed before the onset of bubbling at minimum fluidization velocity.3 To simplify the treatment of the complex system, the lateral solids mixing in a fluidized bed is commonly described as a random walk process, averaged by the diffusion equation2-13
∂C ) Dsr∇2C ∂t
(1)
where C is the concentration of tracer particles and Dsr is the lateral dispersion coefficient. Equation 1 is easy
10.1021/ie020173s CCC: $22.00 © 2002 American Chemical Society Published on Web 08/09/2002
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Ind. Eng. Chem. Res., Vol. 41, No. 18, 2002
to solve numerically, once the geometry with the boundary conditions and Dsr are defined. The dispersion coefficient depends on several parameters and varies with the location in the bed, in both horizontal and vertical directions. Small bubbles are formed at the air distributor in the bottom of the bed. Because of coalescence of bubbles on their way up through the bed, a zone with higher bubble concentration may develop in the vessel’s center with increasing height above the air distributor.8 When the bubbles burst at the surface of the bed, clusters of bed material, along with fuel particles, are projected up into the freeboard. This process has a major influence on the lateral dispersion of particles: it has been shown that the effective lateral dispersion is reduced by up to 65% when vertical baffles above the surface of the bed prevent lateral motion of projected particles.2,13 The laterally nonuniform bubble distribution creates a “Gulf Stream” of bed material, which moves upward in the regions with high bubble activity and downward in regions with low bubble activity (usually at the walls).12,14 A hypothetical solid circulation pattern is illustrated by the circular arrows in Figure 1. The motion of tracer particles is affected by the solid circulation pattern, which in turn depends on wall effects and the geometrical configuration of the reactor.8,12 All of the transport processes mentioned are usually embedded in the lateral dispersion coefficient in eq 1. However, there are exceptions; for example, Schlichthaerle and Werther15 introduced a convection term to account for the flow of externally recirculated solids in a circulating fluidized bed. Because the circulation pattern and bubble characteristics may vary from one fluidized-bed application to another, also the dispersion coefficient is affected. The dispersion coefficient depends not only on the particle size distribution and fluidization properties but also on the geometry and size of the vessel containing the bed. This observation should invoke caution when applying expressions derived from laboratory-scale experiments to full-scale fluidized-bed boiler applications. Nevertheless, literature on the lateral dispersion coefficient is reviewed below, showing the diversity of the forms of the available expressions. This diversity indicates that the physical processes behind the lateral dispersion are complex, different from case to case, or even unknown. On the basis of a bubbling bed model, Kunii and Levenspiel4 theoretically derived eq 2 to describe the lateral dispersion coefficient of solids
Dsr )
3 δ Umfdb 16 1 - δ mf
(
)
(2)
where Umf is the minimum fluidization velocity, mf is the voidage at Umf, δ is the bubble fraction, and db is the effective bubble size. The expression was verified by two test series where the bubble size was used as a fitting parameter. The authors recommended a restricted use of eq 2, to illustrate trends, until it has been further validated. Borodulya et al.6 formulated an expression based on the theory of turbulent mixing and verified it with extensive experiments employing heated tracer particles over a wide range of particle sizes and gas velocities in four different reactors. Although the results were scattered, an equation was given to fit the experimental results in dimensionless form:
Dsr (U - Umf)H0
) k2
( ) Dc H0
n1
Frn2
(3)
In eq 3, Dc denotes the equivalent diameter of the vessel, H0 the bed height at rest, U the superficial gas velocity, and Fr the Froude number. The values of the three fitting parameters were given as k2 ) 0.013, n1 ) 0.5, and n2 ) -0.15. Borodulya et al. actually measured the thermal diffusivity, and it should be noted that Salam et al.13 found the thermal diffusivity to be only 55% of the solid dispersion in a fluidized-bed reactor. Borodulya et al. introduced heated tracer particles in an isolated compartment of the bed and started the experiment by withdrawing baffles containing the heated particles. The dispersion coefficient was estimated from the time elapsed until thermocouples in different locations of the bed showed an increased temperature. In contrast to these transient experiments, Salam et al. carried out steady-state experiments, which were compared to the lateral dispersion coefficient of solids, estimated from transient experiments. The thermal diffusivity was determined from temperature profiles in the bed when heat was added with the fluidizing air to a part of the bed, whereas the dispersion of solids was estimated from concentrations of coal tracer particles in extracted samples of bed material. Bellgardt and Werther16 estimated the lateral dispersion coefficient of solids from gas concentration measurements over the surface of a fluidized bed, which was operated under ambient conditions and had a cross section of 2.0 × 0.3 m. They injected frozen carbon dioxide pellets near the air distributor while measuring the temperature gradient in the bed and the concentration of evaporated CO2 over the surface of the bed. From a fit to the experimental data in one direction, the authors expressed the horizontal dispersion coefficient as
∫0H1 -δ δxgdb3 dh
1 Dsr ) D0 + 0.023 H
(4)
where the dispersion coefficient at minimum fluidization velocity, D0, was determined from experiments as D0 ) 0.67 × 10-3 m2/s; g is the acceleration of gravity, and H is the expanded bed height. Berruti et al.8 measured the lateral dispersion coefficient at different heights above the air distributor in a fluidized bed with a diameter of 0.27 m and a height of 0.19 m. They found that the dispersion coefficient depends not only on the vertical position in the bed but also on the lateral position. They expanded an original expression of Yan and Fan2 accordingly:
Dsr ) 0.185 ×
[ ( [
)( ) ]
h r 5 (U - Umf)dp × 10-4 1 - 0.44 + 2.87 Hmf R (U - Umf)dpFf -0.25 h 1.45 Fp - Ff -0.43 (5) µf dp Ff
] [] [ ]
Equation 5 predicts a sharp decline of the dispersion coefficient when r/R g 0.75, and it even gives a negative dispersion coefficient at the surface of the bed (h ≈ Hmf) when r/R g 0.8. It seems unlikely that the wall effects would reach that far into the core of a bed of a size corresponding to a commercial fluidized-bed boiler. Tables 1 and 2 summarize the test conditions and the expressions mentioned, with addition of some experi-
no
no
yes, in Table 2
yes, in Table 2 yes, in Table 2 yes, in Table 2 no
7
11
6
10
a
12
8
15
16
0.8
1.0 × 0.3
tapered 2.5 × 0.15
0.19
0.4-0.9
2.0 × 0.3
L 0.27
0.16-0.30
0.05
0.5 × 0.1
0.90 × 0.15
0.07-0.40
0.20 × 0.010
0.125-0.40 0.055-0.35 0.237-0.375 0.14-0.20 0.25-0.35
0.02-0.05
0.60 × 0.05
L 0.70 0.60 × 0.20 0.40 × 0.25 0.50 × 0.05 2.6 × 1.6
0.6
bed height [m]
L 1.52
bed size [m]
Density is not given.
yes, in Table 2 yes, in Table 2
yes, in Table 2
2
13
yes, in Table 2
expression for Dsr
5
ref
material, size tracer particles
0.26 mm
resin, 0.90 mm, 1306 kg/m3
sand, 0.4 mm, 2470 kg/m3
quartz sand, 0.45 and 0.85 mm, 2600 kg/m3 quartz sand, 0.15 mm
sand,a
quartz sand, 0.24 mm quartz sand, 0.26 mm glass beads, 0.24 mm quartz sand, 0.26 mm sand, 1.42 mm, 2500 kg/m3
sand, 0.15-0.3 mm, 2700 kg/m3 silica gel, 0.22 mm, 1610 kg/m3 0.035-0.425 0.32-1.2 0.06-0.80 0.03-0.10 0.1-0.45
3-6
0.15-0.30
0.37-0.54 0.67-0.84
0.45-0.95
U - Umf [m/s]
partition in the center of the bed, half volume dyed particles; the dye was washed with water, and the concentration of the dye was measured a fraction of the fluidizing air was heated; the temperature profile in the bed was measured tracer particles were injected at one side of the bed; the time until they reached the other side was measured heated particles in a partition of the bed; measured transient temperature profile after removal of the partition wall
tracer particles released at the center of the bed; bed samples withdrawn at different locations
technique
particles,a
manual sampling and sieving to find the tracer concentration coal 0.8-1.6 mm U/Umf ) 2-5 heated material or tracer particles in the bed partition; compared solid and thermal diffusivity carbon dioxide ice particles, 0.1-0.7 measured CO2 concentration over the bed and 10 mm, cylinders temperature in the bed carbon dioxide ice particles, 3.0 measured CO2 concentration over the bed and 8 mm temperature in the bed potassium cyanide, 0.4 mm, 0.09-0.20 manual sampling at several positions in the bed 2210 kg/m3 Mung bean, 4.1 mm, 3-5 manual sampling at four positions at 0.5 m above 1263 kg/m3 the air distributor
coal particles,a 8-18 mm
heated bed material
coal particles, 8-13 mm, 1300 kg/m3
heated bed material
coal ash, 0.3 mm, 1530 kg/m3 coal, radioactive, (bulk density) 0.06-1.0 mm, 680-800 kg/m3 (bulk density) silica gel, 1.125 mm dyed bed material sand, 0.491 mm, 2600 kg/m3
bed particles
Table 1. Experimental Results on Lateral Dispersion in Fluidized Beds
0.04-0.40
0.0002-0.002
0.12
0.0007-0.0026
0.0005-0.002
0.001-0.01
not evident
0.0004-0.0016
0.0001-0.08
0.0001-0.0008
0.007-0.015
range of Dsr [m2/s]
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Ind. Eng. Chem. Res., Vol. 41, No. 18, 2002
Table 2. Expressions Predicting Lateral Dispersion in Fluidized Beds expression
5 2 6 10
Dsr ) (3/16)[δ/(1 - δ)]Umfdb/mfa Dsr/[(U - Umf)Hmf] ) 0.46 × 10-4[dpFg(U - Umf)/µf]-0.21(Hmf/dp)0.24[(Fp - Ff)/Ff]-0.43 Dsr/[H0(U - Umf)] ) k2(Dc/H0)n1Frn2 Dsr ) 0.54 × 10-3(dp/dt)0.224(Fp/Ft)0.35 + 0.926B0.767(dp/dt)0.66(Fp/Ft)1.17δ/(1 - δ) where B ) 90571300(Hmf/H0)0.9333(U/Umf)1.6324(Fp/Ff)-3.0877H0 Dsr ) 0.91 × 10-4[(U - Umf)/Umf]1.16H0.54(dt/dp)0.25; when dt > dp
13 16 8 12 a
prediction of Dsr for present boiler conditions (Table 3)
ref
1 Dsr ) D0 + 0.023 H
∫
10-4[1
δ gd 3 dh 1 - δx b
H
0
Dsr ) 0.185 × - (0.44 + 2.87h/Hmf)(r/R)5](U - Umf)dp[(U Umf)dpFf/µf]-0.25(h/dp)1.45[(Fp - Ff)/Ff]-0.43 DsrFf/µ ) 1369Rep0.7782Art-0.005236-0.2706
0.02 m2/s 0.03 m2/s 0.03 m2/s 0.007 m2/s 0.003 m2/s 0.01 m2/s 0.05 m2/s 1.3 m2/s
Expression from ref 4. Table 3. Boiler Operation Parameters
Figure 2. Test results on Dsr (filled symbols for circulating fluidized beds): (*) present experiment; (0) Bellgardt and Werther;16 (rotated 4) Berutti et al.;8 (+) Bi et al.;11 ()) Borodulya et al.;6 (X) Highley and Merrick;5 (4) Salam et al.;13 (9) Schlichthaerle and Werther;15 (×) Subbarao et al.;7 (O) Yan et al.;2 (3) Xiang et al.;10 (rotated 2) Xiao et al.12
ments7,11,15 and some derived empirical correlations.10,12,13 A great deal of work has been carried out, but in most cases the conditions were different from those in a boiler, and the expressions have to be extrapolated to estimate the lateral dispersion coefficient for the present conditions. Most of the experiments were performed at excess gas velocities that were less than 50% of that in the present fluidized-bed boiler. The exceptions are the test series of Xiao et al.,12 Schlichthaerle and Werther,15 and Bi et al.11 The former two works were carried out in circulating fluidized beds and the latter in a shallow fuel dryer. In addition to gas velocities, parameters that differ between the experiments are, for example, particle size distributions, particle and gas densities, bed height and geometry, and fluid viscosity. Altogether, it can be concluded that none of the experimental conditions were close to those of a stationary fluidized-bed boiler. A summary of the experimental results is shown in Figure 2, where the dispersion coefficient is plotted against the excess gas velocity multiplied by the height of the bed, giving a quantity of the same dimension as the dispersion coefficient. The set of data represented in Figure 2 is approximate because the definition of the reported bed heights varies, and values occasionally had to be read from diagrams. However, the figure illustrates that the dispersion coefficient increases with
parameter
value
fuel flow (kg/s) total air flow (kg/s) primary air flow (kg/s) recycled gas flow (kg/s) bed temperature (K) bed material (silica sand) mass mean diameter (mm) density (kg/m3) bed pressure drop (kPa) superficial gas velocity (m/s) min fluidization velocity (m/s)
0.5 2.1 1.36 0.22 1173 0.7 2600 5.8 2.3 0.17
bed height times gas velocity and that the experimental results are scattered over a wide range. Table 2 includes values of the lateral dispersion coefficient extrapolated to the present boiler conditions (Table 3). The results differ by more than 2 orders of magnitude. The lack of experiments under boiler conditions and the widespread results from extrapolation of available expressions are the reasons for the present investigation. The focus of the present work is on the lateral dispersion. However, the fuel particles might segregate in the vertical direction in the bed because of size and density differences between fuel particles and the bed material. The vertical dispersion and segregation are of secondary interest, but vertical segregation most likely influences the lateral fuel mixing because the lateral dispersion coefficient has been found to increase with the height in the bed.2,8,13 At low gas velocities tracer particles, considerably larger and with less density than the bed material, have been found to circulate close to the surface of the bed.14,17 Nienow et al.17 found that the penetration depth of tracer particles increases with the square root of the excess gas velocity (U - Umf):
X ) 1.2(U - Umf)0.5 H
(6)
This expression was verified for excess gas velocities up to 0.25 m/s, which should be compared with the corresponding excess gas velocity in the present boiler, 2.1 m/s. Equation 6 gives a larger penetration depth than the bed height, if extrapolated to excess gas velocities higher than 0.7 m/s, indicating that fuel particles will be present along the entire height of the bed in the present application. In addition, the lateral dispersion is at least 1 order of magnitude lower than the vertical dispersion.16,18,19 Thus, the lateral dispersion controls the fuel mixing in the bed of the present application.
Ind. Eng. Chem. Res., Vol. 41, No. 18, 2002 4667
Figure 3. Measured average pressure along the height in the fluidized-bed combustion chamber, measured at the wall (top diagram). The bottom graph shows estimated solids concentration, with the zones: (I) bottom bed, (II) splash zone, and (III) freeboard. A-A is the plane of measurement, and F is the height of the fuel inlet.
Experiments This section describes the experiments performed to validate models and to form the basis for the estimation of the lateral dispersion. The Boiler. Gas concentration profiles were measured in the Chalmers 12 MWth boiler, further described elsewhere.1,20-22 The size of the combustion chamber is 1.7 × 1.7 × 13.5 m. The vertical side walls are covered with a refractory lining (0.11 m thick) up to 2 m above the air distributor, where the fluidization air is introduced through 144 air-cap nozzles. The boiler was temporarily modified from a circulating fluidized-bed boiler to be operated under noncirculating conditions. The modifications included an increased pressure drop across the air distributor, an increased size of the bed material (quartz sand) to 0.7 mm, and the installation of a device to improve the gas mixing at the inlet of the secondary air injected through 12 tubes at two opposing side walls at 3.7 m above the air distributor. The fuel dispersion in the bed is assumed to be unaffected by the secondary air arrangement. The bed height was 0.5 m, calculated as the region with a constant pressure gradient below the splash zone and shown as region I in Figure 3. The fluidized bed was operated in the “bubbling fluidization regime” according to the regime diagrams of Bi and Grace.23 The bed temperature is controlled by recycling of the flue gas, which is mixed with the primary air below the air distributor. The recycled flue gas enters the bed as a cold gas and leaves the bed at bed temperature and, hence, lowers the bed temperature. In the present case, the amount of recycled flue gas was about 16% of the primary air. The fuel enters the combustion chamber by gravity from the fuel chute, shown to the left in Figure 1, at a height of 1.1 m above the air distributor. With a bed height of about 0.5 m during operation, the fuel had to fall 0.6 m through the splash zone before reaching the dense bed. Because of the strong fluctuations of both solid concentration and gas velocity within the splash zone, the fuel was dispersed around the fuel chute prior to entering the bed. It is likely that the fuel
Figure 4. Fluidized bed seen from above at cut A-A in Figure 1. Ly and Lx are the widths of the side walls. F is the width of the fuel chute. The positions where the gas was sampled are marked X.
particles were interacting with clusters of bed material because pressure drop measurements,20 shown in Figure 3, indicate the presence of a considerable amount of solid material between the fuel chute (F) and the plane of measurement (A-A). The fuel consisted of wood chips with a high moisture content (40-45%). The geometrical shapes of the particles of commercial wood chips are hard to define. The sizes are different from those of the bed particles and vary in a range of a few millimeters to some centimeters. The large variation in size and shape prevents the use of a simple expression to characterize drying and devolatilization. However, the particles can be represented by a batch consisting of more than 200 wood chips, randomly sampled from a large pile of fuel, with side lengths measured piece by piece and described as rectangular parallelepipeds. The resulting accumulative mass distribution vs specific surface (the surface area divided by volume of the parallelepipeds) followed a lognormal distribution with a mean of 710 m-1 and a standard deviation of 290 m-1. The operation parameters of the boiler during the tests are summarized in Table 3. Gas Analysis. Cross-sectional measurements of gas concentrations were performed in nine positions as shown in Figure 4. The gas was sampled just above the surface of the bed at 0.65 m above the air distributor (height at A-A in Figures 1 and 3), using a water-cooled probe with a ceramic filter at the tip. The concentrations in the cross-sectional positions will be referred to by index i, ranging from 1 to 9, as denoted in Figure 4. The temperature in the suction line was controlled to 170 °C to prevent condensation inside the probe. From the probe, the gas was lead through a heated tube to an FTIR instrument (Bomem MB 100), where the gas was analyzed with respect to H2O and CO2. Each position was sampled during 10 min, and the FTIR instrument recorded the average gas concentration once every minute. Because of the stochastic nature of a fluidized bed, the local gas concentrations fluctuated substantially in the region close to the bed despite the overall steady-state boiler operation. The time-average water vapor concentration at 0.65 m above the air
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Figure 5. Normalized mass distributions of water in the gas above the bottom bed: measured (solid bar); modeled (Dsr ) 0.12 m2/s; dashed bar). The water contribution from the recycled flue gas is subtracted. The fuel chute (F) is located on the left rear side in the diagram.
distributor is shown as solid bars in Figure 5, with the wall of the fuel chute marked by the arrow labeled “F”. In Figure 5, the background water vapor concentration from recycling of the flue gas has been subtracted and the magnitudes of the gas concentrations were scaled to give a cross-sectional average of unity. When a base water concentration, equal to the lowest measured concentration, is further subtracted, about 22% of the total water remains to form the nonuniform distribution over the cross section. Method The lateral dispersion of fuel in the bed is estimated from measured gas concentrations above the bed, while the concentration of fuel in the bed is modeled together with drying and pyrolysis. The two models, dispersion and drying pyrolysis, are solved independently and combined afterward. The fuel particles are tracked by the water concentration, which provides a more explicit gas concentration profile over the bed than, for example, CO or hydrocarbons because the water in the fuel is released at an initially high rate, rapidly decreasing with time. This method gives the total lateral dispersion rate of fuel in the bed and splash zone in conjunction (the lower dense region of the combustion chamber). Drying and Devolatilization Model. The drying of a biomass particle injected into a furnace is modeled by a wet shrinking core moving toward the center of the fuel particle. The drying consumes a considerable amount of the energy transferred to the particle, giving a steep increase of the temperature just outside the drying front. In the work of Palchonok et al.,24 the volatile gases were considered to be released at the pyrolysis front following the drying front. The fraction of volatiles released after drying is finished depends on the particle’s shape; for a spherical fuel particle, not much of the volatiles remain to be released after the drying is finished, but for a slab, there is a considerable fraction remaining. Accordingly, the last part of the devolatilization has to be modeled without a wet core. The model used in this work25 is based on the concept of the model by Palchonok et al.24 and validated by single-particle experiments.26 These experiments, with wood particles, showed that roughly 20% of the mass of the volatile gases consists of water. Hence, the
Figure 6. Calculated rate of water emitted per kilogram of fuel (solid line), after injection into the fluidized bed. The dashed line shows the fraction of total water released up to a certain time (logarithmic time scale).
contribution from the pyrolysis is included when modeling the total water emitted; it is not possible to distinguish water from drying and from pyrolysis in the measurement of the water concentration in the gas above the bed. The time-resolved water yield, Wj(t), emerging from the fuel particles during drying and pyrolysis was obtained from the particle conversion model25 for each of the 208 particles in the fuel sample. The heat transfer to the fuel particles was estimated for a particle surrounded by bed material having bed temperature because the region close to the fuel chute showed no significant drop of temperature. The time-resolved water yield per mass unit of fuel, W(t), is obtained by adding the contribution from each particle (j), scaled by the mass of the total fuel sample (mb):
W(t) )
1
n
∑Wj(t)
mbj)1
(7)
The modeled transient water yield is shown in Figure 6. Lateral Dispersion Model. The transient concentration of a batch of tracer (fuel) particles injected into the bed is obtained by a numerical solution of eq 1 in the horizontal plane. This concentration corresponds to the mass of fuel tracer particles per unit surface area, integrated over the height of the bed. The model is twodimensional, and all lateral transport processes are included in one effective cross-sectional average dispersion coefficient, Dsr. If the model (eq 1) would be extended into three dimensions, the gross convective flow of particles has to be included, and inevitably, this would involve more unknown parameters. The dispersion coefficient, Dsr, in eq 1 is removed as a parameter, and the solution is generalized when the equation is rewritten into dimensionless form. The numerical calculation of the dispersion equation is described in the appendix. It is assumed that the entire tracer batch enters the bed evenly distributed in an area at the fuel chute, marked as a rectangle in the horizontal plane, shown in Figure 7a. The amount of injected tracer particles is adjusted to give a cross-sectional average concentration of unity. The modeled progress of the concentration of the tracer particles is illustrated
Ind. Eng. Chem. Res., Vol. 41, No. 18, 2002 4669
the combustion air in the air plenum below the air distributor and is hence evenly distributed over the cross section of the bed:
Cim )
aE h i + CwGr Gt
(10)
Here, Gt is the total mass flow of gas through the fluidized bed and a is an unknown response factor, depending on the behavior of the flow of gas through the bed and on the time that the fuel spends above the plane of measurement. To avoid the estimation of a by means of the throughflow through bubbles (flow rate, frequency, and velocity), a is eliminated using a concentration of water (X), which is normalized with the quantity of water originating from drying and devolatilization of the fuel in the bed: Figure 7. Calculated tracer concentration (concentration values on the curves), seen from above, at different dimensionless times after reaching the bed: (a) initial distribution of tracer particles; (b-d) tracer concentration gradients when τ equals 0.02, 0.05, and 0.10, respectively.
by Figure 7a-d, where the tracer concentrations (given as numbers on the curves) are plotted for various dimensionless times, τ ()Dsrt/LxLy), after the injection of tracer particles into the furnace. Lateral Distribution of Moisture. The lateral distribution of water is calculated by multiplying the tracer concentration in the bed, obtained from eq 1, with the water yield of the fuel. Thus,
E(x,y,Dsr,t) ) W(t)
C(x,y,Dsr,t) LxLy
(8)
where E is the water yield per kilogram of fuel, resolved in the two-dimensional plane and in time. The water distribution from a continuous feed of fuel, E h , is obtained from integration of eq 8 over time, scaled with the actual fuel feed rate (Gf):
E h (x,y,Dsr) ) Gf
∫0 E(x,y,Dsr,t) dt ∞
(9)
The dispersion of the fuel, Dsr, is then determined by fitting eq 9 to the measured distribution of the water concentration. When the measured gas concentrations above the surface of the bed are compared with the results from the modeling, the following should be noted: (a) A large fraction of the primary air penetrates the bed as intermittent high-velocity throughflow during the eruptions of bubbles. (b) The recycled flue gas contains a known amount of water, which is added to the water leaving the fuel. (c) Some of the hydrocarbons in the volatile gases might have burned in the bubbles of the bed, thereby adding water to the flue gas, in addition to the water in the volatile gases leaving the fuel particles. (d) The fuel particles are occasionally projected from the bed up into the splash zone, above the plane of measurement, where emitted water escapes detection. The measured water concentration in position i (Cim) is the sum of two contributions: from the fuel in the bed (E h i) and from the recycled flue gas (CwGr), where Cw is the water concentration and Gr is the mass flow of recycled flue gas. The recycled flue gas is mixed with
Xi )
Cim - Cb Ca - Cb
(11)
where it has been assumed that a is constant over the cross section of the bed. Ca denotes the cross-sectional average of the measured H2O concentrations, and Cb is the contribution of water from the recycled flue gas:
Cb ) CwGr/Gt
(12)
The modeled water yield, eq 9, is normalized into a dimensionless concentration, Yi(Dsr), in the same way as the measured one, X in eq 11, but there is no background gas concentration to consider because no recycled gas has been introduced in the model. The total discrepancy (V) of the two normalized distributions is minimized with respect to Dsr to find the best-fit dispersion in the n ) 9 measurement points n
V(Dsr) )
|Xi - Yi(Dsr)| ∑ i)1
(13)
Before the results are presented, a brief discussion about the unknown influencing parameters is given. Factors Influencing the Result. The dispersion coefficient obtained from eq 13 is sensitive to the assumed initial lateral distribution of the fuel fed on top of the bed. Before the bed is reached, the fuel passes the splash zone (Figure 1). The residence time and the dispersion of the fuel in this zone define the initial conditions for modeling of the fuel dispersion in the bed. With a terminal velocity of at least 4 m/s and a superficial velocity of 2 m/s, the residence time of the fuel particles in the splash zone is less than 0.2 s if the instantaneous interactions of solids and the intermittent gas jets from erupting bubbles are disregarded. However, these interactions may increase the time for particles to reach the plane A-A. Another process that could influence the results is the unknown fraction of hydrogen combustion. Hydrogen leaves the dry fuel particles in the volatile gases either as H2, H2O, or hydrocarbons. About 6 wt % of the dry mass of wood fuel is hydrogen, of which about one-third is released as water in the volatile gases.26 This contribution is accounted for by the model.25 The remaining part of the hydrogen is burned somewhere in the combustion chamber, producing water that has not yet been accounted for. The fraction of the hydrogen com-
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bustion that occurs in the bottom bed influences the measured water vapor concentration at plane A-A, but it is likely that the fraction is small because the major part of the volatiles is known to burn above the bed. If the rate of hydrogen combustion in the bed is assumed to be proportional to that of water release, then it will not influence the results because it could be treated as an increase of the parameter a in eq 10, and a is eliminated by eq 11. On the other hand, because the devolatilization rate is lower than the drying rate, the distribution of emitted hydrogen is probably more homogeneous than that of water. If the hydrogen combustion in the bed is assumed to be homogeneously distributed over the horizontal plane, the water product could be accounted for by adding it to the variable Cb in eq 11. This is done below to analyze the sensitivity of the solution. Results While transported from the fuel chute to the dense bed, the fuel is assumed to spread and dry with the same rates as those in the dense bed. The time delay in the splash zone determines the lateral distribution and the degree of drying of the fuel as it reaches the bed. The influence of this time delay on the initial distribution of the fuel at the plane of measurement is illustrated by the normalized fuel distribution at four dimensionless times (τ) in Figure 7a-d. In this context, these distributions could be interpreted as the results of time delays in the splash zone. The moisture emitted from the fuel particles above the plane of measurement (during the time delay) is not detected in the measurements. Therefore, the modeled water yield from the fuel during the time delay is subtracted when the modeled moisture distribution is fitted to the measured distribution. Optimization of eq 13 yields the effective lateral dispersion coefficient for different assumed time delays in the zone between the fuel chute and the plane of measurement. A typical modeled distribution of emitted water is shown as dashed bars in Figure 5 (for a time delay of 0.4 s), compared to the distribution of the measured concentrations drawn as solid bars. The modeled water concentrations along the centerline of the furnace from the fuel chute to the rear wall are compared with measurements in Figure 8 for three time delays. A variation of the time delay between 0.1 and 0.6 s gives dispersion coefficients (Table 4) in the range of Dsr ) 0.09-0.20 m2/s. Obviously, the result is sensitive to the time delay. The sensitivity of the dispersion coefficient was also investigated by varying the rate of drying and devolatilization with (10% from the modeled rate (Figure 6), whereby the dispersion coefficient changed less than 10%. The influence of laterally homogeneous hydrogen combustion is illustrated by a case where 25% of the hydrogen in the dry fuel was assumed to burn evenly distributed in the bed. This reduced the dispersion coefficient by about 10% (Table 4, second column), which is a small quantity compared to the uncertainty caused by the time delay of the fuel in the splash zone. Thus, the accuracy of the estimation of the effective fuel dispersion coefficient is determined by the behavior of the fuel in the splash zone. Discussion The dispersion coefficient obtained by the method proposed is substantially higher than those predicted
Figure 8. Calculated relative concentration of water along the centerline of the furnace, from the fuel chute to the rear wall, for three assumptions on fuel time delay in the splash zone (td) using the corresponding optimized dispersion coefficient (Dsr). The boxes mark the measured concentrations. Table 4. Resulting Effective Lateral Dispersion Coefficient from Minimization of Equation 13 for Assumed Residence Times in the Splash Zone, with or without Accounting for Hydrogen Combustion in the Bed Dsr (m2/s) time delay (s)
without H2 combustion in the bed
with H2 combustion in the bed
0.1 0.2 0.4 0.6
0.20 0.16 0.12 0.10
0.17 0.14 0.11 0.09
from most published expressions (Table 2) for the conditions in the present bed (Table 3). The large scatter of predictions is probably due to the differences in the test conditions behind the various expressions, most of them far from the conditions in a fluidized-bed combustor. Several studies on the dispersion of particles in fluidized beds in Table 1 were conducted with tracer particles having properties (density and size) similar to those of the bed material,2,6-8 and many other features in those tests also differ from those of the present case. There are various reasons for the relatively high dispersion coefficient in the present work. As seen in Figure 6, the rate of drying of the fuel is very high during the first few seconds; about 25% of the total moisture is emitted within 3 s. Furthermore, the variation of moisture (Figure 5) over the cross section of the bed corresponds to about 22% of the total moisture emitted from the fuel. Clearly, the movement of the fuel particles during the first few seconds after injection is critical for the lateral distribution of the emitted water, and this explains why the solution is sensitive to assumptions regarding the behavior of the fuel before reaching the bed. During the first seconds after injection of the fuel, it will most likely be present in the splash zone or in the upper region of the dense bed, where the lateral dispersion coefficient is higher than that inside the dense bed.2,13 An optimization of eq 13, accounting for only three measurement points along the wall closest to the fuel chute, yields a local Dsr, which is 20-30 times higher than that from the three points closest to the rear wall. The fuel particles are probably dispersed
Ind. Eng. Chem. Res., Vol. 41, No. 18, 2002 4671
along the height of the bed while transported toward the rear wall, implying that the effective lateral dispersion of the fuel at the rear wall is inhibited by the lower dispersion rate inside the dense bed. This shows the importance of the splash zone for the overall dispersion process in a fluidized bed, at least when the fluidizing gas velocity is high. Consequently, it is not surprising that the effective lateral dispersion coefficient estimated by the present method is higher than when tracer particles are introduced below the bed surface. For fluidized-bed boilers firing wood fuels, it is technically preferable to feed the fuel on top of the bed, and this results in a higher initial lateral dispersion of the fuel than if the fuel were introduced in the dense bed. The measured concentration profile of water vapor showed a larger drop at the corners (Figure 5) than is predicted by the two-dimensional dispersion model with a constant dispersion coefficient. This discrepancy is probably caused by wall effects and unevenly distributed bubble activity, which were neglected in the dispersion model for simplicity.
Figure 9. Calculated tracer concentration gradients, seen from above, at different times after reaching the bed: (a) initial distribution of tracer particles; (b-d) tracer concentration gradients when τ equals 0.02, 0.05, and 0.10.
Conclusions From a comparison between measured concentrations of water and the modeled behavior of the fuel in the bed of a stationary bubbling fluidized-bed boiler, the effective lateral dispersion coefficient was found to be on the order of 0.1 m2/s. This is substantially higher than that reported from most previous experiments of particle dispersion in the bottom bed of laboratory-scale units. The most likely explanation for the discrepancy is that the method described, with fuel fed on top of the bed, includes a significant contribution from the dispersion of the fuel in the splash zone compared to most reported experiments, where the tracer particles are introduced into the dense bed. Another explanation is that the dispersion coefficient here is estimated over a larger bed surface at higher gas velocity than most previous experiments. Previous estimates of dispersion, such as those found in Table 2, cannot be expected to reliably represent the lateral dispersion coefficient outside of their experimental ranges. Furthermore, the use of a simple diffusivity equation to describe the motion of fuel particles within a fluidized bed is a coarse simplification, but it can still be used as a significant support for the design of fuel feed systems for fluidizedbed boilers. An advantage of the method presented is that it can be applied to full-scale boilers during normal steadystate operation. A drawback is the method’s sensitivity to the fuel behavior in the splash zone, before it reaches the bed below the plane of measurements. The motion of fuel particles in the splash zone is left as a subject for further studies.
which is solved over the horizontal cross section of the fluidized bed (Figure 4). The concentration (C) is that of fuel particles in the bed, mass per area unit, as a function of coordinates and time after injection of the particles into the bed. The gradual mass loss of fuel particles, due to conversion, is not considered at this stage. Hence, it is more correct to consider C as a concentration of tracer particles. As a first approach, the effective lateral dispersion coefficient (Dsr) is considered to be constant over the bed’s surface. Prior to numerical solution, eq A1 is rewritten into a dimensionless form and a source term (S) is added to represent the injection of tracer particles:
Ly ∂2C Lx ∂2C ∂C + +S) Lx ∂x 2 Ly ∂y 2 ∂τ
(A2)
xn ) x/Lx
(A3)
yn ) y/Ly
(A4)
τ ) Dsrt/LxLy
(A5)
n
n
where
The source term is applied as
lsx and Lx lsy lsy 1 1 + < yn < (A6) 2 2Ly 2 2Ly
S ) S0
for τ ) 0, xn
0.02
Nomenclature
Figure 10. Calculated tracer concentrations at the positions marked in Figure 9, as functions of dimensionless time (τ).
∂C )0 ∂xn
for xn ) 0 or xn ) 1
(A8)
∂C )0 ∂yn
for yn ) 0 or yn ) 1
(A9)
The domain of calculation is reduced by utilizing symmetry, parallel to the x axis through the center. The half domain is divided into 50 × 25 cells, and eq A2 is solved over each of these cells by applying the Crank-Nicolson scheme.27 A time step of ∆τ ) 4 × 10-6 is selected during the first part of the calculation, and then the time step is increased to ∆τ ) 8 × 10-4 during the remaining time span (0 e τ e 3). The problem has a smooth solution that is relatively insensitive to the increased time step. Parameters are given in Table 5. The gradient lines of the calculated tracer concentration at τ ) 0, 0.02, 0.05, and 0.1 are shown in Figure 9a-d, where the crosses (×) mark the positions of gas concentration measurements. The calculated timeresolved concentrations of tracer particles in positions 1-6 are shown in Figure 10 (the concentrations in positions 7-9 are the same as those in positions 1-3 because of symmetry). The output from the numerical calculation delivers one array for the dimensionless time (τ), one array for the coordinates (xn, yn), and one matrix containing the values of the concentration of tracer particles. That is, for every calculated concentration in the solution matrix, there are three corresponding parameters (τ, xn, yn). To transform the dimensionless solution to the real case, the values in the τ array are transformed by inserting numbers for Lx, Ly, and Dsr into eqs A3-A5, and the dimensionless coordinates are multiplied with Lx and Ly to yield the real dimensions. The matrix containing the tracer concentrations remains unaffected by the transformation, which is a benefit of using the dimen-
a ) constant Art ) Archimedes number for a tracer particle C ) concentration of tracer particles, kg/m2 Ca ) average of measured concentrations for all positions, kg/m3 Cb ) water mass fraction from recycled flue gas in bed, kg/m3 i Cm ) measured water mass concentration in position i, kg/m3 Cw ) water mass concentration in recycled flue gas, kg/m3 db ) equivalent spherical volume bubble diameter, m dp ) bed material diameter, m dt ) tracer particle diameter, m D0 ) lateral solid dispersion coefficient at Umf, m2/s Dc ) equivalent diameter of vessel containing the bed, m Dsr ) lateral solid dispersion coefficient, m2/s E ) transient water release rate distribution per kilogram of fuel, kg/(m2 s) E h ) continuous water release rate distribution, kg/(m2 s) Fr ) Froude number, Fr ) (U - Umf)2/gH0 g ) gravity constant, m/s2 h ) coordinate of the vertical position in the bed, m H ) expanded bed height, m Hmf ) bed height at minimum fluidization velocity, m H0 ) static bed height, m i ) index for the measurement locations (1-9) lx, ly ) widths of area of fuel feed on the bed, m Lx, Ly ) widths of the rectangular vessel containing the bed, m mb ) mass of the fuel sample, kg Gf ) fuel feed rate, kg/s Gr ) mass flow of recycled flue gas, kg/s Gt ) mass flow of the total flue gas, kg/s r ) coordinate of the radial position in the bed, m R ) radius of the vessel containing the bed, m Rep ) Reynolds number for bed material S ) sourceterm of tracer particles t ) time, s td ) time delay in the splash zone, s U ) superficial gas velocity, m/s Umf ) minimum superficial fluidizing velocity, m/s V ) discrepancy (Σ|Xi - Yi|) W ) rate of H2O release from fuel, kg/s x, y ) lateral coordinates in the rectangular bed, m xn, yn ) dimensionless coordinates in the rectangular bed X ) penetration depth of fuel particles, m Xi ) relative H2O concentration, measured Yi ) relative H2O concentration, modeled δ ) bubble fraction in the bed mf ) void fraction of the bed at minimum fluidizing velocity µf ) fluid viscosity, kg/(m s) Ff ) fluid density, kg/m3 Fp ) solid density, bed material, kg/m3 Ft ) solid density, tracer material, kg/m3 τ ) dimensionless time (Dsrt/LxLy)
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Received for review March 6, 2002 Revised manuscript received May 31, 2002 Accepted June 13, 2002 IE020173S
