Experimental and CFD Analyses of Bubble Parameters in a Variable

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Ind. Eng. Chem. Res. 2010, 49, 5166–5173

Experimental and CFD Analyses of Bubble Parameters in a Variable-Thickness Fluidized Bed Robert W. Lyczkowski,*,† Jacques X. Bouillard,‡ Isaac K. Gamwo,§ Mark R. Torpey,| and Eugene D. Montrone⊥ Energy Systems DiVision, Argonne National Laboratory, 9700 S. Cass AVenue, Argonne, Illinois 60439-4815, Direction des Risques Accidentels, INERIS, Parc Technologique ALATA s B.P. No. 2, 60550 Verneuil-en-Halatte, France, National Energy Technology Laboratory, United States Department of Energy, P.O. Box 10940 Pittsburgh, PennsylVania 15236-0940, New York State Energy Research and DeVelopment Authority (NYSERDA), 17 Columbia Circle, Albany, New York, 12203, and John Blizard Research Center, Foster Wheeler Corporation, 12 Peach Tree Hill Road, LiVingston, New Jersey 07039

Bubble characteristics in a variable-thickness fluidized bed containing nine tubes were experimentally investigated by analyzing absolute and differential pressure fluctuations. The latter were obtained from vertically aligned probes traversing the bed interior for three bed thicknesses: thin, square, and full. The important bubble parameters, namely, frequencies, effective diameters, and velocities, were determined by analyzing autocorrelations and cross-correlations obtained from these differential pressure signals for the thin and square beds. Wall effects were assessed by comparing the pressure fluctuations as the bed thickness was increased from thin to square. It was found that bubbles move faster within and above the tube bank than below it. This behavior was also found to be more pronounced in the wall regions of the full bed, which might explain why some commercial fluidized-bed combustors experience unusual metal wastage near their tube supports. Although bubble sizes consistently agreed between thin and square beds, bubble velocity reduction was observed for the thin bed. The experimental thin-bed differential pressure measurements were analyzed using a two-phase computational fluid dynamics (CFD) hydrodynamic model. Excellent agreement was obtained between the experimental results and predictions from our hydrodynamic model for autocorrelations, cross-correlations, power spectral densities, and bubble parameters. 1. Introduction In the midst of ever-increasing environmental regulations, fluidized-bed combustion (FBC) coal technology is striving toward being recognized as both environmentally and economically viable. Erosion, in general and particularly in in-bed heatexchanger tubes and waterwalls in bubbling and circulating FBC boilers, respectively, remain unresolved issues as reviewed by Finnie,1 Stringer,2 and Lyczkowski and Bouillard.3 As more large-scale FBC units accumulate substantial numbers of operating hours, more instances of general, rather than localized, erosion have been encountered. Although remedial actions, such as the use of finned or pinned tubes and plasma-deposited coatings, have considerably alleviated this problem, they have increased costs. Not fully understood are the metal wastage inconsistencies between apparently similar units operated under similar conditions. No rational explanation for these inconsistencies exists. These issues were reviewed by Lyczkowski and Bouillard.3 Regardless of the erosion mechanism taking place, it is believed that the main factor influencing in-bed erosion is solids motion. Hence, a detailed knowledge of solids circulation and associated bubble motion is essential to understanding metal * To whom correspondence should be addressed. Address: Energy Systems Division, Argonne National Laboratory, 9700 S. Cass Ave, Argonne, IL, 60439-4815. Tel.: 630-960-5711. Fax: 630-252-1342. E-mail: [email protected]. † Argonne National Laboratory. ‡ INERIS. § United States Department of Energy. | New York State Energy Research and Development Authority (NYSERDA). ⊥ Foster Wheeler Corporation.

wastage in FBC units. The interaction among geometrical factors, including tube diameter and spacing and tube bank height, and operating factors, such as bed height, fluidizing velocity, temperature, and bed material, determine the quality of fluidization and the propensity for wear.4 A practical industrial concern in the development of FBC technology is the impact of scaleup on tube wastage.3 To provide a rational mechanism for understanding erosion of tube banks in fluidized beds, Argonne National Laboratory has investigated FBC metal wastages from a fundamental standpoint by studying the hydrodynamics of gas-solids two-phase flow and erosion.3 Three-dimensional CFD models of fluidized beds have been developed to account for three-dimensional effects present in thin, so-called “two-dimensional” beds.5 They have also been developed to analyze three-dimensional rectangular beds.6 Ding and Lyczkowski7 were able to show the difference in modeling hydrodynamics and erosion in two and three dimensions for a rectangular fluidized-bed experiment. However, because computing time remains extensive on serial computers, hydrodynamic and erosion modeling of fluidized beds has been almost exclusively limited to two dimensions.8-10 Unfortunately, any significant three-dimensional effects cannot be taken into account. Parallelization of computer codes using multicore processors will serve to reduce the long running times experienced in the past. We cite a specific example: Gustavssen and Almstedt8 conducted erosion experiments for the two-tube configuration used by Olowson11 to study fluidized-bed hydrodynamics and simulated this cold, pressurized, fluidized bed, rectangular in cross section with a width of 0.3 m. The pressure was varied from 0.1 to 1.6 MPa. The experiment was modeled in two dimensions without assuming symmetry. Agreement with the

10.1021/ie901294e  2010 American Chemical Society Published on Web 12/23/2009

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data for the five bubble parameters frequency, pierced length, rise velocity, volume fraction, and flow rate was better at elevated pressures. Trends predicted from the computations were in disagreement with the data for bubble frequency and flow rate. At low pressures, agreement was poor, with severe overpredictions of bubble frequency, flow rate, pierced bubble length, and rise velocity. The authors argued that the reasons for disagreement might be due to the coarse grid used and to problems modeling the distributor plate. The disagreement is more likely due to the fact that the computations were done in two dimensions. Ding and Lyczkowski7 showed that computed erosion results compared better with the experiment modeled in three dimensions than in two dimensions. To save on construction costs, thin two-dimensional experimental fluidized beds, which are frequently utilized,12,13 might not exhibit proper three-dimensional affects. Geldart and Kelsey14 appear to have been the first to investigate the bubble motion in thin two-dimensional and three-dimensional beds using capacitance probes in an attempt to correlate three- and two-dimensional bubble sizes. Kathuria and Saxena15 found the fluidizing velocity to decrease with increasing bed thickness; however, the pressure drop displayed an inconsistent dependence. They concluded that a bed thickness of about 9 cm would be sufficient to eliminate wall effects on bed thickness. In contrast, Glicksman and McAndrews16 found the bubble rise velocity to increase and the bubble chord length and bubble frequency to decrease with bed thickness up to about 60 cm. These observations demonstrate the inconsistencies in the effects of change of fluidizing velocities as the bed thickness is increased. Few investigations of three-dimensional effects in fluidized beds with internals have been reported in the literature. Glicksman and McAndrews16 found that the inclusion of a tube bank in a full 1.2 m × 1.2 m fluidized bed caused the bubbling behavior to approach that of a narrower bed. To provide insight into these issues, a cold, shallow, variablethickness fluidized bed containing nine immersed tubes was constructed at Foster Wheeler Development Corporation. The design of the experimental fluidized bed and the locations of the absolute and differential pressure probes were aided by experience analyzing a similar thin fluidized-bed experiment.12 Details of the Foster Wheeler experimental facility are provided in the next section. The purposes of the Foster Wheeler experimental facility were (1) to provide data from a thin fluidized bed for comparison with CFD predictions that had been previously validated through analysis of thin two-dimensional experiments13 and (2) to provide data from three-dimensional experiments to assess changes in bubble parameters. Although both pressure and erosion data were obtained in this experimental facility, only the pressure data are analyzed in this work to obtain the bubble parameters. The purposes of this article are (1) to assess three-dimensional effects by analyzing variations of the bubble parameters, namely, bubble frequency, bubble velocity, and effective bubble diameter, from the experimental differential pressure measurements as the third dimension (thickness) is increased using the theory developed by Sitnai17 for single-bubble analysis and (2) to analyze the thin-bed experiment using the two-phase, twodimensional CFD model. 2. Experimental Setup The experimental facility constructed at Foster Wheeler Development Corporation consists of an air compressor, the cold-model variable-thickness fluidized bed, a recycle cyclone, a bag house, and an exhaust stack, as shown schematically in

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Figure 1. Schematic of the Foster Wheeler cold variable-thickness fluidizedbed experimental facility.

Figure 2. Schematic of the cold shallow Foster Wheeler variable-thickness fluidized bed containing nine tubes.

Figure 1. Fluidizing air is supplied by a two-stage air-cooled 100 kW two-cylinder compressor. The compressed air enters the air plenum through poly(vinyl chloride) (PVC) pipes at a nominal temperature of 339 K (66 °C). A movable vertical steel plate is used to vary the bed thickness and to control the length of tubes exposed to the bed material. Nine 51-mm outsidediameter tubes are arranged in a staggered pitch configuration and kept perpendicular to the vertical partition. The total rectangular cross section of the unit is 72.4 cm in width by 165.1 cm in depth, and the height is 67.3 cm. Molochite particles having a mean effective diameter of approximately 1500 µm and a density of 2480 kg/m3 (Geldart Group D) were used to represent typical industrial shallow bubbling fluidized-bed combustor bed material. Figure 2 presents a schematic showing the front elevation view of the fluidized bed. Above the air plenum is a flat perforated steel distributor plate that is 3.2 mm thick and has a total of 7403 holes, each with a diameter of 2.8 mm, drilled in a square pattern. The front side of the fluidized bed is covered by 1.9-cm-thick transparent Lexan plastic to facilitate the observation of bubble motion. The side

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Ind. Eng. Chem. Res., Vol. 49, No. 11, 2010 Table 1. Locations (cm) of Differential Probes in the Fluidized-Bed Interior variable-thickness bed

Figure 3. Schematic of the absolute and differential pressure probes, associated electronics, and data acquisition system.

frame of the fluidized bed is constructed of carbon steel. The expanded fluidized-bed height is maintained nominally at 40.6 cm, yielding a bed height approximately 8 cm above the top of the top row of tubes. The freeboard height of approximately 26.3 cm is sufficient to keep particles from elutriating from the top of the unit. The movable vertical steel plate can slide perpendicularly along the tube bank to produce two bed thicknesses, namely, 10.2 and 82.5 cm, and when it is removed, the resulting thickness is 165.1 cm. Configurations using these three thicknesses are termed “thin”, “square” (actually 72.4 cm × 82.5 cm), and “full”, respectively. To limit gas leaks, the sliding contacts between the movable vertical partition and the sides are temporarily glued shut for each experiment. The tube bank is supported by the front and back panels and by the moving partition, when in place. As shown in Figure 2 the center-to-center spacing of the tubes is 75 mm in the vertical direction and 152 mm in the horizontal direction. The two rows of the tube bank consist of three different materials: aluminum, carbon steel, and PVC plastic. The distance from the bottom of the lower row of tubes to the distributor is 294.5 mm. The composition of the tube materials is not of importance in this work because we are dealing only with analysis of the hydrodynamic behavior of the fluidized bed. A future article will address the erosion analysis of these tubes. Penetrations of 1.1 cm diameter were drilled through the front panel to accommodate the pressure probes. Two sets of pressure probe holes were drilled: One set of three horizontal holes (A, B, and C indicated in Figure 2) was used for the absolute pressure probes, and the other set of three vertically aligned pairs of holes (1, 2, and 3 in Figure 2) was used for the differential pressure probes. 2.1. Instrumentation. A schematic of the absolute and differential pressure probes and associated data acquisition system is shown in Figure 3. The fluidizing air velocity was measured by monitoring the pressure drop though a calibrated orifice installed on the inlet air pipe. A temperature monitor was located at the air plenum intake and read approximately 339 K (66 °C) for all the tests. Both absolute and differential pressures were calibrated using differential pressure transducers. Absolute pressures were measured relative to a reference pressure kept constant in a reference pressure tank. The differential pressure transducer models used were the DP45-24 and DP45-18 models made by Valendyne Engineering Corporation and were selected to measure the absolute and differential pressure measurements, respectively. Their accuracy is ( 0.25%

probe location

thin

square

full

wall region bed interior

5 5

5 37

5 72

of full scale. The DP45-24 and DP45-18 pressure transducers incorporate strain gages bonded to a stainless steel diaphragm and are designed to measure 2.2 and 0.55 kPa, respectively, at full range. All of the differential pressure transducers were connected to the pressure probes by plastic tubing. The 25 mV/V fullscale transducer outputs were fed into an amplifier that normalizes the voltage between +10 and -10 V. An analog/digital converter coupled to a wave carrier demodulator was interfaced between the amplifier and the data acquisition system, which itself was interfaced to a personal computer for data storage. The sampling frequency for all tests was set to 300 Hz. During a typical test, 60 s or 18000 data points were recorded and stored on disk. The three absolute pressure probes were flush with the front wall, and the differential pressure probes were slid into the front wall at the distances listed in Table 1 to determine the extent of three-dimensional hydrodynamic effects. All of the data were transferred from Foster Wheeler Development Corporation to Argonne National Laboratory for spectral analysis. Figure 4 is a typical plot of the raw first 5000 differential pressure data points (16.7 s) for the thin bed for the three probe locations 1, 2, and 3 shown in Figure 2. The data are in units of volts, with -10 V being equal to +551.6 Pa (+0.08 psi). 2.2. Operating Conditions. All test runs were conducted with the bed initially filled with Molochite particles to a static bed height of 36 cm. The composition of Molochite is as follows:18 52-53% SiO2; 44% Al2O3; 1.5-2.0% K2O; 1% Fe2O3; and small traces of TiO2, CaO, MgO, and Na2O. The particles are irregularly shaped and brittle and have a wide size

Figure 4. Raw experimental differential pressure measurements (Xn) in volts for the Foster Wheeler thin fluidized bed for probe locations 1, 2, and 3. Sampling frequency ) 300 Hz, 5000 of 18000 data points (n). -10 V is equivalent to 551.6 Pa (0.08 psi).

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Figure 5. Particle size distribution of Molochite used in the experiments. Table 2. Operating Conditions for the Variable-Thickness Fluidized-Bed Experiments particle material mean particle diameter particle density particle sphericity expanded bed porosity, ε temperature fluid carrier minimum fluidizing velocity, Umf fluidization velocity, U U/Umf exit pressure a

Molochitea 1500 µm 2480 kg/m3 0.86 0.46 339 K air 97.3 cm/s 121.9 cm/s 1.25 108.9 kPa

Molochite is a British registered trademark: No. 568142.18

distribution. The particle size weight fraction distribution was determined by sieving at Argonne National Laboratory and is shown in Figure 5. The weight-average mean diameter was about 1.5 mm. All variable-thickness beds were started and operated under the conditions listed in Table 2. When fluidized at 121.9 cm/s, the three beds expanded to a bed height of approximately 41 cm. Intense slugging and bubbling were observed for these Geldart Group D particles through the transparent front panel, although the beds were fluidized at only 1.25Umf. 2.3. Differential Pressure Signal Interpretation. A schematic of a bubble passing a lower differential probe pair is shown in Figure 6a. The upper portion of Figure 6b shows a schematic of a differential pressure probe record for this lower probe. Figure 6a shows four instances that can be used to interpret the differential pressure probe record as a bubble passes through it. Prior to time t2, the probe measures the differential pressure DP ) E in the emulsion phase of the fluidized bed. At time t2, the top of a bubble strikes and encloses the bottom probe in the differential pressure probe set. At time t3, the bubble has risen to entirely surround the probe pair. At this point, the pressure drop becomes essentially zero, the pressure drop inside the bubble. When the tail of the bubble passes the bottom probe at time t4, the differential pressure rises again, subsequently decreasing to equal the pressure of the emulsion phase higher in the fluidized bed. The lower portion of Figure 6b represents the same scenario as just described at a somewhat later time for the upper probe pair. Bubbles that are smaller than the probe spacing can cause overshoots, as can hysteresis of the differential pressure probe transducers. 3. Application of Spectral Analysis to Determine Hydrodynamic Bubble Parameters The basic technique for the analysis of hydrodynamic bubble parameters in this article is based on a modification and refinement of Sitnai’s methodology.17 Standard time-series analysis of signals using autocorrelations, cross-correlations, and

Figure 6. Schematic of a bubble passing differential probe pairs with associated differential pressure records.

power spectral densities facilitates the evaluation of these properties as described by Chang et al.19 First, the autocorrelation is determined for each signal. Sitnai17 determined that, for the theoretical transient pressure field produced by the theoretical Davidson model20 for a single noninteracting bubble in the absence of internals, the time lag, ta, determined from the autocorrelation yields the ratio of the bubble diameter, Db, to the vertical bubble velocity, Vb, given by ta )

Db Vb

(1)

Equation 1 was used to determine the ratio of the bubble diameter to bubble velocity from the time lag determined from the experimental and computed autocorrelations. To determine the vertical bubble velocity, the cross-correlation between two of the vertically aligned differential pressure probe signals was determined; the time delay, tc, was estimated either from the time between the first two peaks or from an average of the number of peaks over the signal length time, usually taken to be 5 s. Next, the bubble velocity was determined by dividing the vertical probe spacing, y, by the cross-correlation time lag, tc, as Vb )

y tc

(2)

Then, the bubble diameter was determined from Db ) Vbta

(3)

The cross-correlation time delays between probes 1 and 2 and between probes 2 and 3 were assigned to probes 1 and 3, respectively. The cross-correlation time lag for probe 2 was obtained by averaging the other two time delays. Finally, the

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Ind. Eng. Chem. Res., Vol. 49, No. 11, 2010 Table 3. Hydrodynamic Bubble Parameters for the Thin and Square Fluidized Beds probe

square bed 37 cm from front 5 cm from front thin bed

Vb (cm/s) Db (cm) Vb (cm/s) Db (cm) Vb (cm/s) Db (cm)

3

2

1

22 4.0 16 4.5 12.8 3.2

26 5.2 33 6.5 13.5 3.0

31 5.8 49 7.8 14.4 2.6

Table 4. Bubble Velocities (cm/s) Computed from Davidson and Harrison Eqs 5 and 6

Figure 7. Experimental differential pressure traces at probe locations 1, 2, and 3, 37 cm into the square fluidized bed.

power spectral densities were computed from the Fourier transforms of autocorrelations to determine the major frequencies.19 4. Experimental Results Typical differential pressure fluctuations for the three probe locations 1, 2, and 3 indicated in Figure 3 are shown in Figure 7. After correction for pressure transducer offset values, the pressure differentials did not always become zero during the bubble passage, indicating that the bed was in a pseudoslugging regime. This finding confirms what other investigators have already observed for the fluidization of Geldart Group D particles.21,22 The bed can slug as it bubbles for these coarse particles, unlike Geldart Group B particles. Rockey et al.22 stated that slugs are not solids-free. This can be explained by solids raining from the slug ceiling as it rises. Following the procedure based on the Sitnai theory17 described in the previous section, bubble diameters (technically heights) and velocities were determined for the three differential probe locations 1, 2, and 3 shown in Figure 2. The distance used for the vertical probe spacing, y, in eq 2 was taken to be the center-to-center distance between two vertical pairs of probes, 7.62 cm (3 in.). Table 3 summarizes the bubble parameters obtained for the thin and square beds. The major bed frequencies of 2.3 and 2.7 Hz determined for the thin and square beds, respectively, were obtained from the power spectral densities for the absolute pressure data. The theoretical bed frequency, f, can be estimated using the following theoretical expression, first derived by Verloop and Heertjes23 f)

1 2π

 Hg (2 -ε ε)

(4)

Db (cm)

eq 6

eq 5

3.5 4.0 5.0 6.0 7.0 8.0

20 22 24 27 29 31

40 44 48 54 58 72

where H and ε are the expanded bed height and porosity, respectively, and g is the acceleration due to gravity, 9.8 m/s2. Equation 4 predicts a frequency of about 1.4 Hz, which is in better agreement with the experimental value for the thin bed. Equation 4 was developed for a fluidized bed without internals and, hence, might not be valid for a bed containing tubes. Assuming that the bubble velocity varies roughly as Dbf, one can expect greater bubble velocities in the square bed because of its higher frequency. This is indeed the case, as shown in Table 3, which also shows that the bubble velocities are greater near the front wall than in the bed interior for the square bed. The bubble velocities above and within the tube bank are higher than those below it, and those for the square bed are greater than those for the thin bed. Wall effects in the thin bed reduce the bubble velocities by as much as a factor of 4-5. This finding has a significant implication. Because tube wastage is roughly proportional to the square of the solids velocity (which is nearly the same as the bubble velocity upon impact with a tube surface), tube wastage obtained in three-dimensional beds would be significantly higher than that measured in a corresponding thin twodimensional bed. This indeed turned out to be the case for 400-h runs that obtained tube wastage for the thin and full beds.24 A comparison was made between the experimentally determined bubble velocities given in Table 3 and those estimated from well-known theoretical equations given by Davidson and Harrison20 for a single bubble as Vb ) 0.71√gDb

(5)

and for a wall slug given as Vb ) 0.35√gDb

(6)

The sole entry in Table 4 is the bubble diameter, Db, which lies in the range of experimentally measured diameters. As can be seen by comparing Tables 3 and 4, the experimentally measured bubble velocities agree more closely with those evaluated by the wall-slug expression given by eq 6 for both the thin and square fluidized beds. This further reinforces our previous surmise that the bed operated in a slugging regime. Although the bubble velocities are quite a bit lower for the thin bed because of wall effects, the bubble sizes are surprisingly similar to those determined for the square bed. Bubble diameters

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Figure 8. Time-averaged (from 0.3 to 1.5 s) solids velocity vectors and porosity contours computed for the Foster Wheeler thin bed containing nine tubes.

for both the thin and square beds are close to the spacing between the tubes, as also observed by Fitzgerald.21 5. Comparison of Experimental Bubble Parameters with Computer Simulation for the Thin Fluidized Bed The Foster Wheeler thin-bed experiment was simulated using the experience gained by computer modeling and by analysis of the University Illinois at Urbana-Champaign (UIUC) computer-aided particle tracking facility (CAPTF) two-dimensional cold fluidized-bed experiment containing five tubes having the same diameters and spacings.12,25 In fact, the analysis of this experiment was extremely helpful in designing the Foster Wheeler fluidized variable-thickness fluidized-bed facility. Initially, the CAPTF experiment was analyzed using square tubes having 4 × 4 nodes.12 Subsequent reanalysis25 used more nearly round tubes with 6 × 6 nodes as a better approximation, and that is what was used in the present simulation. The CFD model equations and difference scheme are described in Gamwo et al.25 A uniform Cartesian mesh of 45 × 122 ) 5490 nodes was used in the present simulation, with cell sizes of 0.85 cm in each direction. These totals include the “dummy” cells around the periphery of the physical boundary. The number of computational cells was therefore 43 × 120 ) 5160 with the assumption of symmetry. As indicated in Gamwo et al.,25 the present results are felt to be essentially gridindependent. A solids viscosity of 0.1 Pa · s was used. A uniform time step of 2.5 × 10-5 s was used to compute the transient out to 1.58 s. A plot of the time-averaged solids velocity vectors and porosity contours is shown in Figure 8. The results compare

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qualitatively with the CAPTF simulation with solids recirculation below and above the tube bank and acceleration between the tubes. The only bubbles observed visually through the Lexan plastic front of the fluidized bed were single nearly spherical bubbles emanating from the tube surfaces at about 30° from their bottoms. This phenomenon was computed by the CFD model as shown in Figure 8 as indicated by the contours of porosity equal to 0.5. Because the differential pressure probes were located directly in the region of bubble formation, the probes were in direct contact with these single bubbles, and therefore, the application of the Sitnai theory17 is wholly justified without the need for any correction. The experimental data were checked for classification by calculating moving averages. The data for the differential pressure became stationary after about 2 s, reaching a very nearly steady-state periodic signal type.26 Therefore, standard spectral analysis techniques as used for the experimental data in section 3 are now justified and can confidently be applied to process the data. The Sitnai methodology17 described in section 3 was used to determine the bubble parameters. A more careful analysis was undertaken in this section because of the much more detailed computational results. The number of data points in the computed time series was increased from the experimental value of 300 per second to as much as 40000 per second [1/(2.5 × 10-5 s)]. Table 5 summarizes the comparison of experimental and predicted bubble parameters for the three probe locations, 1, 2, and 3. Because the lengths of the computed signals were less than 2 s, due to the long computer run time, the determination of predicted autocorrelations and cross-correlations was limited to this time interval. Comparison of the experimental and predicted autocorrelation time lag is shown in Figure 9 for probe location 1. Note that the autocorrelation became periodic within 1 s, justifying cutting off the computatins at about 2 s. As can be seen, the agreement is excellent. The agreement for probes 2 and 3 was also excellent, as shown in Table 5. The predicted cross-correlation time lag between probes 1 and 2, Ttransient, was determined from the time lag between the first two peaks as shown in the top plot in Figure 10, which compares the results with the experiment. The agreement with the experiment is not as good as for the autocorrelation time lag and might partly be a result of neglecting wall effects in the computations. It might also be due to the procedure for determining the experimental cross-correlation time lag, (Ttransient)avg, by counting peaks. This value depends on whether only major peaks should be considered or whether minor peaks should be included. The value of 0.53 s was determined by counting only major peaks. If the minor peaks are included, the cross-correlation time lag decreases to about 0.36 s and agrees much better with the predicted value. This would increase the bubble velocity to 21.1 cm/s and the bubble diameter to 4.2 cm. This remark also applies to the other two cross-correlations. As a cross check, the bubble velocity for probe 1 can be estimated from the product of the bubble velocity, Db, and the frequency, f (assumed to be the bubble frequency), to obtain 17.5 cm/s, which is in good agreement with both the predicted and experimental values. The length of the computed and experimental signals used in the analysis also affects the accurate determination of the autocorrelations and cross-correlations. Therefore, the error in the values of bubble diameters and velocities was estimated to be about 50%. As already mentioned above, the two-dimensional computations did not include the wall effects, and consequently, the

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Table 5. Summary of Experimental and Predicted Bubble Parameters for the Thin Fluidized Bed probe 1

bubble frequency from spectral analysis (Hz) bubble frequency from pressure oscillations from plots (Hz) transit time from cross-correlation plot (s) time lag from autocorrelation (s) bubble velocity (cm/s) bubble diameter (cm)

probe 2

probe 3

experiment

prediction

experiment

prediction

experiment

prediction

2.34 2.5 0.53 0.18 14.4 2.6

1.56 2.71 0.386 0.20 19.8 4.0

2.3 2.5 0.57 0.22 13.5 3.0

3.12 2.34 0.40 0.21 19.2 4.0

2.34 2.5 0.6 0.25 12.8 3.2

3.12 2.73 0.41 0.24 18.6 4.5

predicted bubble velocities are higher than the experimental values. The experimental bubble diameters shown in Table 5 decrease monotonically as the bubbles rise from probe 3 to probe 1. This is the same trend as shown by the calculations, which agree to within the estimated 50% error. The bubble diameter is roughly one-half of the minimum distance between the tubes (5.64 cm) and about one-quarter of the horizontal tube pitch, 15.2 cm. This is smaller than that determined visually by Anderson et al.27 in a thin two-dimensional 7 cm × 68 cm

fluidized bed containing tubes. However, their maximum particle diameter was about one-half (79 mm) of that used in the Foster Wheeler fluidized-bed experiment. They found that the maximum bubble diameter became smaller with larger particles. The power spectra were computed from the autocorrelations to determine the major frequencies, as shown in Table 5, which compares the computed and experimental results. Surprisingly, at each location, there is only one frequency and no higher

Figure 9. Comparison of (a) predicted and (b) experimental autocorrelations of differential pressures at probe location 1.

Figure 10. Comparison of (a) predicted and (b) experimental crosscorrelations of differential pressures between probes 1 and 2.

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harmonics. In addition, the frequencies for all three probe locations are basically the same. The major frequencies were also estimated by averaging the numbers of peaks per second from the signals themselves called “from plots” and the former frequencies “from spectral analysis”. The agreement of the predicted and experimental frequencies is much better. 6. Conclusions Sitnai’s theory17 was utilized to analyze bubble parameters in a fluidized bed containing obstacles. The theory was used to successfully compute bubble parameters from experimental differential pressure measurements in the Foster Wheeler variable-thickness fluidized bed and predictions from the twodimensional two-phase CFD analysis of the thin bed. It was found that, in the interior and wall regions, bubbles move generally faster within and above the tube bank than below it. It was also found that this behavior is more pronounced in the region near the vertical front wall of the square bed, which might explain why some industrial fluidized beds experience unusual metal wastage near the tube supports. The highest solids velocity was found in the region of the tubes, where the flow area is reduced, at about 30° from the bottom of tube centers, resulting in the generally highest erosion.24 This is the same place where bubbles were found to form, agreeing with visual observations through the Lexan plastic front wall. Although bubble diameters consistently agreed between thin and square beds, bubble velocities in the thin bed were found to be lower than those measured in the square bed, indicating the existence of significant three-dimensional hydrodynamic effects as well as wall friction. Although excellent agreement was found between the two-dimensional CFD predictions and the data for the thin fluidized bed for autocorrelation, cross-correlations, and power spectral densities, such models are generally not sufficiently accurate to describe the hydrodynamics and metal wastage in large-scale industrial units exhibiting three-dimensional effects. Acknowledgment This work was originally supported by the U.S. Department of Energy, Assistant Secretary for Fossil Energy, Morgantown Energy Technology Center [now National Energy Technology Laboratory (NETL)], under Contract W-31-109-ENG-38, and the Cooperative Research and Development Venture “Erosion of FBC Heat Transfer Tubes”. Members of the venture were the U.S. Department of Energy, National Energy Technology Laboratory, Electric Power Research Institute, State of Illinois Center for Research on Sulfur in Coal (now the Illinois Clean Coal Institute), Foster Wheeler Development Corp., ASEA Babcock PFBC, ABB Combustion Engineering, Inc., Tennessee Valley Authority, British Coal Corporation, CISE, and Argonne National Laboratory. Literature Cited (1) Finnie, W. Some Reflections on the Past and Future of Erosion. Wear 1995, 186/187, 1. (2) Stringer, J. Practical Experience with Wastage at Elevated Temperatures in Coal Combustion Systems. Wear 1995, 186/187, 11. (3) Lyczkowski, R. W.; Bouillard, J. X. State-of-the-Art Review of Erosion Modeling in Fluid/Solids Systems. Prog. Energy Combust. Sci. 2002, 28 (6), 543. (4) Lyczkowski, R. W.; Podolski, W. F.; Bouillard, J. X.; Folga, S. M. Metal Wastage Design Guidelines for Bubbling Fluidized Bed Combustors; Report no. DOE/MC/24193-3491/NTIS no. DE94000024; Argonne National Laboratory: Argonne, IL, 1992. (5) Gamwo, I. K.; Gidaspow, D.; Lyczkowski, R. W.; Soong, Y. ThreeDimensional Hydrodynamic Modeling of a Bubbling Fluidized Bed. In

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ReceiVed for reView August 17, 2009 ReVised manuscript receiVed December 7, 2009 Accepted December 8, 2009 IE901294E