Modeling the Axial and Lateral Mixing of Solids in Fluidized Beds

This description is revisited in this work to develop a mathematical model for both the axial and the lateral mixing of solids in fluidized beds. The ...
0 downloads 0 Views 541KB Size
Download PDF
5656

Ind. Eng. Chem. Res. 2001, 40, 5656-5665

Modeling the Axial and Lateral Mixing of Solids in Fluidized Beds Juan C. Abanades* and Gemma S. Grasa Department of Energy and Environment, Instituto de Carboquı´mica (CSIC), Miguel Luesma Castan 4, 50015 Zaragoza, Spain

The solid flow patterns in fluidized beds are often depicted as a number of convective currents induced by rising bubbles. This description is revisited in this work to develop a mathematical model for both the axial and the lateral mixing of solids in fluidized beds. The model uses concepts similar to those used in the countercurrent backmixing model, which is widely used for axial mixing only. Mixing experiments using coal and PVC (as a white tracer) were carried out to obtain experimental concentration maps for model validation. The model reproduces well the general features observed during the mixing experiments, as well as the effects of gas velocity, particle size, and the presence of internals. The selection of most model parameters can be justified with observations of bed and bubble properties. The choice of the exchange rate of solids between countercurrent phases is discussed in light of the new data derived in this work, previously published data, and a sensitivity analysis of the model predictions for this parameter. 1. Introduction Some of the main advantages of fluidized beds as chemical reactors arise from the rapid mixing of solids that is typical of these systems. In most cases, the assumption of instantaneous and perfect mixing of solids is correct, and the system can be operated and controlled in almost isothermal conditions. However, in fluidized beds where heat-intensive reactions are taking place, these assumptions might no longer be practical at relevant time and length scales. Mixing-related problems can lead to undesired temperature profiles, feed and product maldistributions, and poor unit performance, particularly in large industrial units such as fluidized-bed combustors and gasifiers. Solid mixing in fluidized beds has been extensively investigated in the past,1-5 and a number of reviews exist on both experimental techniques6,7 and modeling approaches.3,5,6 In general, the semiempirical models used to interpret tracer RTD curves with a combination of ideal reactors8 are good only for specific units and conditions, and more mechanistic approaches are required when a certain generality is needed. The solid diffusion model first applied by May9 is still the preferred option to account for solid mixing in large fluidized beds. Recent examples of the application of this model to large units are the experiments by Ito et al.10 in fluidized beds with dense tube packings and the investigations of Knoebig et al.11 on the bottom region of a large circulating fluidized bed combustor. Despite the success of the diffusion model in interpreting solid mixing data, the use of a purely diffusional mechanism to describe solid mixing is counter to observations9 of macroscopic convective patterns of solids in fluidized beds. These strong solid currents induced by bubbles cannot be adequately represented by a diffusion law and a countercurrent backmixing model12 (CCBM) is more appropriated.3,6 A clear advantage of this model is its mechanistic character, because it accounts for the upflow of solids transported by the bubbles and the slow downflow in the emulsion phase that are widely observed in experience. The model is very versatile and has been successfully used by several authors3,6,7,12-21 * Author to whom correspondence should be addressed. Tel.: +34 976 733977. Fax: +34 976 733318. E-mail: [email protected].

to describe solid mixing and segregation of solids in the axial direction of fluidized beds. In addition, it can be well integrated with equivalent submodels for the gas phase in more comprehensive reactor models.1-3,6 The practical limitations of the CCBM model arise from an inability to model lateral mixing (except when it is coupled with a lateral diffusion model13) and the lack of a reliable correlation for the parameters, especially for the exchange of solids between the countercurrent phases.3,6 The purpose of this work is to generalize the countercurrent backmixing model to make it suitable for both the axial and the lateral mixing of solids in fluidized beds, as well as to derive procedures for parameter estimation to make the model more practical over a wider range of conditions and scales. This is done maintaining the simple description of the solid flow patterns in the bubbling fluidized bed as a convective flow of solid streams induced by bubbles. 2. Experimental Section The experimental investigation of solid mixing in this work is based on the application of an image-analysis technique to mixing experiments in a 2D fluidized bed. Coal was used as the main solid in the bed, and PVC powder (white) was used as the optical tracer. The technique is based on the video recording and analysis of the mixing process of the tracer in the bed. This technique was pioneered with other colored solids by Lim et al.,7 who reviewed a number of alternatives and highlighted the benefits of this technique for mixing experiments. The cold facility used in this work is shown in Figure 1. The space occupied by the bed can be up to 2 m in height, 0.9 m in length, and 0.08 m in width. The front Plexiglas window is 20 mm thick. It is suspended in two rails that facilitate easy opening and allow for changes of geometry, distributors, bed dimensions, and location of internals. Most of the experiments were conducted with dimensions of 2 × 0.6 × 0.025 because of limitations in the air flow supply. The air enters the plenum chamber (0.4 m in height) on opposite sides and finds a distributor with 3% free area, made up of holes of 1-mm diameter. The fluidizing solids were selected to differ in density (1140 kg/m3 for PVC, 1430 kg/m3 for coal) to allow for separation by flotation in a saturated solution

10.1021/ie0009278 CCC: $20.00 © 2001 American Chemical Society Published on Web 10/13/2001

Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5657

Figure 1. Scheme of the fluidized bed rig and the arrangement of internals used in the mixing experiments.

Figure 3. Experimental maps of tracer volume fractions derived from images in Figure 2 using eq 1.

tionship between the physical input signal and this image numeric property. A calibration exercise is therefore required. After other forms of calibration curves for solid concentration were reviewed , the following equation was chosen22 because it avoided the need for tedious recalibrations when the solid optical properties, light conditions, camera settings, etc., were changed

I Imin C) Imax log Imin log

Figure 2. Example of experimental images from an experiment with internals. u - umf ) 0.94 m/s, dpc ) 0.95 mm, dpPVC ) 1.09 mm, hmf ) 0.5 m.

of NaCl in water. The materials could be then used in new experiments after being washed and dried. In a typical experiment, a loosely packed bed is carefully arranged with a square of tracer in a bottom corner of the bed. The bed is allowed to fluidize at the preset gas velocity, and the mixing process between tracer and main solid is recorded in a SVHS videotape. Examples of images in a typical experiment with internals are shown in Figure 2. The 25 images per second are then examined, and selected images are digitized to the computer. The image-analysis software (Optimas 6.1) allows for the identification and spatial location of the bubbles, as well as the bulk portions of the bed from which concentrations are derived. After the bubbles are deleted, the remaining portions are relocated, and the dimensions are normalized to the initial height at minimum fluidization (hmf). The tracer concentration in a given region of the bed is derived from the brightness in the image. This is given in the gray scale of 0-255 of luminance (I) by the image-analysis software. Cameras and other devices for capturing images do not usually provide a well-established rela-

(1)

The limitations of our experimental setup and optical equipment allowed for a spatial resolution in the experimental mesh of 0.05 × 0.05 m. An example of the concentration maps obtained from Figure 2 by the application of eq 1, is plotted in Figure 3. A number of experiments were carried out for superficial gas velocities varying from 0.7 to 1.5 m/s, using two particle size distributions both for coal (dpc ) 0.50 and 0.95 mm) and for PVC (dpPVC ) 0.85 and 1.09 mm). The individual umf values of coal were 0.12 and 0.32 m/s, respectively, and those of PVC were 0.23 and 0.35 m/s, respectively. The final mixtures (15 vol % in PVC) had umf values of 0.15 and 0.33 m/s, respectively. Finally, to investigate the presence of internals, two configurations of internals occupying a similar percentage of the area (7%) as steam tubes in PFBC units23 were used. The internals had two different diameters: 55 mm (16 internals) and 30 mm (49 internals). A perfect mixing of tracer was observed at stationary state in all experiments. Therefore, the segregation mechanisms were assumed to be negligible with respect to the mixing mechanisms. 3. Results and Discussion The solid flow patterns observed in our experiments agreed with well-known descriptions of bubbling fluid-

5658

Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001

Table 1. Bubble and Bed Properties Observed during the Mixing Experiments type of experiment

Db (m)24

ub (m/s)

without internals

Db ) (u - umf)0.5h0.75g-0.25

ub ) u - umf + 0.4xgDb

with internals

Db ≈ 0.2

ized beds:1,2 the solids move upward in the wake and drift regions behind the bubbles. The solids flow slowly down near the wall and, less frequently, in the center of the bed and other regions without bubbles. There is an intense splashing zone that produces very rapid mixing of the tracer at the top of the bed. The imageanalysis technique allowed for the estimation of bubble properties in our system and Table 1 summarizes the correlation and range of values that best agree with our bubble observations and measurements. The equation of Rowe24 for Db was the most appropriate for the experiments without internals, with average diameters between 0.2 and 0.35 m. The average bubble size diminishes slightly (around 0.2 m) when internals are present and does not change much with velocity in the interval studied. The equation for the bubble rise velocity includes a constant K ) 0.4 for the bed without internals that is similar to the value obtained by Hull et al.25 for a 2D fluidized bed of a different size and K ) 0.3 for the bed containing internals. A detailed investigation of these aspects was not the subject of this work. It is recognized that they are, to some extent, dependent on the installation used in this work and that they should be reconsidered when the model is applied to other systems. However, as discussed below, the information in Table 1 will be useful in establishing a better link between bubble properties and more general parameters in the mixing model. 3.1. Convective Model of Solids Mixing. In the bubbling-bed model1,2 and in previous versions of the CCBM for axial solids mixing,3,6,7,12-21 the entire bed is represented as a single column with one or more solid phases moving upward and downward. However, in shallow and/or large fluidized beds, several of these mixing columns can exist simultaneously in different parts of the bed. These distinctive flow structures are also called gulf streams or vortex rings,2 individual circulation cells,26 or bubble regions.4 The model for solids mixing proposed in this work makes use of this widely observed feature, adapting the CCBM to model lateral mixing in beds with more than one of these macroscopic flow structures. Therefore, it is assumed that the bed can be divided into several of these mixing columns, each of which has an identical solid flow structure, with solids moving upward driven by the bubble activity and downward in the emulsion regions. The number and characteristics of these mixing columns are strongly affected by the distributor design, the bed aspect ratio, and the operating conditions.2,4,26 Here, an estimation of the number of mixing columns, Nc, is possible through direct visual observation during the experiments. A number of Nc ) 2 was selected as representative of the gross solid flow structure observed in all experiments in the 0.6-m-wide fluidized bed. The selection of this integer number is a simplification that makes the solid flow structure rigid and unaffected by time, in contrast with the experimental observation of a more flexible and random behavior of bubble movement and solid flow structure. However, it agrees reasonably well with the average behavior of the bed and fits with the observations of bubble properties reported in Table 1. When direct observation is not

δ2

δ)

u - umf ub + umf

ub ) u - umf + 0.3xgDb

possible, it is recommended that the ratio between the bed width and the average bubble size at the top of the bed be used to estimate Nc

Nc ≈

Lb Dbmax

(2)

The formulation of the model for any integer value of Nc is represented in Figure 4 in terms of volume flows of bulk solids. It is assumed that each mixing column has three phases, the central one moving upward at velocity u1 and the two adjacent ones move downward at velocity udown ) u1f1/(1 - f1), where f1 is the fraction of ascending phase in the bed. There is an exchange coefficient, Kw, that governs the transverse volume flow of solids between the central phase and the lateral ones. This exchange mechanism is defined as

(

)

solids flow between (Adz) ) fK two adjacent cells in 2Nc 1 w countercurrent phases

(3)

This volume flow of bulk solids between the two cells is responsible for the lateral mixing of tracer within the mixing column. In the previous version of the CCBM model for axial mixing only, the difference in concentration between adjacent phases is ignored when the axial concentration profiles are calculated. In this work, this lateral solid flow between cells is used to model and calculate the lateral tracer concentration profiles within a mixing column, thus allowing for different concentrations in adjacent cells at the same height. The averaging of concentrations is only carried out with the cells placed at the top of the bed, where intense splashing is observed during the experiments and perfect mixing between adjacent cells is assumed (see eq A6 in the Appendix). To connect the different mixing columns to each other, a second exchange coefficient, Kd, is defined between each pair of cells in cocurrent downward phases (Figure 4)

(

)

solids flow between (Adz) two adjacent cells in ) (1 - f1)Kd 2Nc cocurrent phases

(4)

Two extreme assumptions can be adopted to define Kd and, hence, the solids exchange between cocurrent phases. These depend on the stability with time of the assumption that the rigid flow structure depicted in Figure 4 can be used to describe the solid flow patterns in the bed. In fluidized beds with well-defined and stable solid flow patterns (for example, in a shallow fluidized bed with a gas distributor with a few tuyere), the streams of bubbles, and hence the spatial locations of the upward and downward solid flowing regions, are very stable and do not change with time.2,4,26 Therefore, the exchange of solids between the cocurrent phases moving downward should be very poor, and Kd ) 0 should be a good choice for this parameter. Under these conditions, the lateral movement of solids in the bed would present discontinuities in the lateral direction,

Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5659

Figure 4. Solids flow patterns and tracer volume flows considered in the model (the term (1 - mf)A/Nc has been excluded in all the solid flows).

as the solids could only be exchanged from the upflowing phases toward adjacent downflowing phases. The mixing columns would be practically isolated, and the macroscopic lateral mixing between adjacent mixing columns would rely on the mixing of solids in the splashing of solids above the bed surface.4 In contrast, in freely bubbling fluidized beds such as the one used in this work, the flow structure depicted in Figure 4 needs to be understood as an average representation of a much more transient flow structure. Regions that contain a portion of the bed moving downward can contain, instants later, the wake or drift of a rising bubble. Also, the rapid growth of bubbles displaces solids laterally that can rapidly be replaced by other solids after the bubbles’ passage. Although the average behavior of the bed over time is still reasonably represented by Figure 4, the real bed does not show such a rigid flow structure, and a certain exchange of solids between cocurrent phases from adjacent mixing column is allowed. This transport mechanism is driven by the same bubble activity responsible for Kw. Therefore, it cannot be higher in absolute terms than the flow of solids exchanged between countercurrent phases

(

)(

)

solids flow between solids flow between g two adjacent cells in two adjacent cells in countercurrent phases cocurrent phases (5)

The maximum value for Kd is obtained when total continuity in the lateral flow of solid in the bed is assumed

Kd ) Kwf1/(1 - f1)

(6)

This relation has proven useful in interpreting the experimental data available in this work and was

adopted for the definition of Kd in fitting our experimental data. In other applications of the model, where a choice between the two extremes described above (Kd ) 0 or Kd from eq 6) is not possible, this parameter will have to be used as an adjustable parameter. The system of partial differential equations obtained by writing tracer mass balances in differential volume elements of the phases is presented in the Appendix. A rapid solution methodology is also described that makes use of the limited range of values that the model parameters can take when their physical meaning are considered. The full version of the model requires five parameters that need numerical definitions to produce a solution: Nc, Kd, f1, u1, and Kw. With the previous definitions of Nc and Kd, the proposed model faces the same degree of uncertainty as previous versions of the countercurrent backmixing model for axial mixing only. There are three remaining coefficients, namely, f1, u1, and Kw, to be defined. The first two are related to the bubble wake fraction and the bubble rise velocities in the expanded bubbling bed as follows

δ f1 ) fw (1 - δ)

(7)

u1 ) ub(1 - δ)

(8)

In the absence of direct empirical information for these parameters, a number of estimations and correlations can be used in eqs 7 and 8 for the bubble properties.1,2 However, during the experiments carried out in this work, direct measurements of u1 were possible. They were obtained by measuring the time taken by the tracer to move a given distance at the beginning of the experiment. The times used to determine the upward velocity of the solids were the break-

5660

Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001

Table 2. Compilation of Literature Correlation and Data Range for the Exchange Parameter of Solids between Countercurrent Phases Kw (s-1) author

correlation

constant Db

Lim et al.7 Shen and Zhang20

1 × 107

Hoffmann et al.14 Chiba and Kobayashi15 Sitnai16 Basesme and Levi 17 This work

(

0.081 (2mfDb) 4 umf π mfDb

Fgµgmf

(Fp - Fg)Fpdpumf

range of fitted values

)x

g Db

-

Figure 5. Solids velocities measured during the mixing experiments vs those calculated from bubble velocities and eq 8.

through times of tracer to the top surface of the bed. Similarly, it was also possible to measure the times taken by a thin layer of tracer placed at the top lefthand side of the bed to settle to the bottom of the bed. A similar technique was also used by Sitnai et al.16 and Lim et al.7 Figure 5 shows a series of measurements of upward and downward solid velocities against the expected upward velocity from eq 8, applied with the data and correlation of Table 1. Although the data dispersion is very important in Figure 5, it is clear that a gross discrepancy exists between the experimental values of the solid velocities and those expected from the bubble velocities. This is in qualitative agreement with recent results obtained by Seville et al.,27 who used a precise technique to track the movement of individual particles in the bed. They found that the average solid velocities were approximately 0.5 times the bubble rise velocities. Figure 5 shows that the experimental solid velocities are better correlated with corrected forms of eq 8

u1 ) 0.44ub(1 - δ)

(9)

udown ) 0.21ub(1 - δ)

(10)

The fact that these phase velocities are lower than the measured bubble rise velocities (Table 1) can be explained by attending to the contribution of the slowly moving drift phase to the upward solid transport. This

ub, u1 (m/s)

Db (m)

1.01-6.45

experimental

experimental

0.23-1.38

eq Johnsson et al.28

eq Darton et al.29

0.63-3.17

eq Nicklin 30

eq Geldart 31

0.03-0.12

experimental

experimental

0.25 0.5-3.5 1-3

experimental experimental experimental

experimental Table 1

Figure 6. Compilation of values for the exchange coefficient between phases obtained by different authors (see Table 2).

phase becomes more important in relatively large particle systems, where the bubble wakes become smaller2 and their participation in the upward solid transport diminishes. In the absence of information for quantifying the relative importance of wakes and drifts in the upward solid transport, eq 9 lumps the contribution of both transport mechanisms into an average solid velocity. Finally, the fraction of solids in the upflowing phase can be estimated directly from the measurements of the moving phases (eqs 9 and 10)

f1 )

udown (u1 + udown)

≈ 0.3

(11)

The previous discussion leaves the exchange coefficient between the countercurrent solid flows, Kw, as the only adjustable parameter in the model. Table 2 and Figure 6 summarize the available information from a literature review on this parameter. Also included are the ranges of values that the different authors found when fitting their experimental data. In this work, the best-fit values were consistently between 1 and 3 s-1 with eqs 2, 6, 9, and 11 used to define the rest of model parameters. Table 2 highlights the important discrepancies found in the values and in the correlation proposed for Kw in the revised bibliography. Chiba and Kobayashi15 pro-

Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5661

Figure 7. Sensitivity analysis of model predictions to different Kw values at two representative times of the mixing process: (a) t ) 1.3 s; (b) t ) 2.9 s. f1 ) 0.3, u1 ) 0.6 m/s, Nc ) 2.

posed a linear dependency of Kw on umf, whereas Shen and Zhang20 found it to be inversely proportional. Basesme and Levy17 refuted some of the results of Chiba and Kobayashi15 while using a similar technique to estimate Kw in incipient bubbling beds. Hoffmann et al.14 also argue that Chiba and Kobayashi15 worked with solids for which umf was too close to derive clear dependencies. Hoffman et al.14 and Lim et al.7 suggested a simpler and more general relation for Kw of the form

Kw )

constant Db

(12)

Shen and Zhang20 proposed a similar dependency but using Db0.5. However, this dependency with bubble diameter is also weak in an analysis of the individual data sets from authors as Lim et al.7 and Basesme and Levy.17 (Shen and Zhang,20 and Hoffmann et al.14 do not report individual values of Kw, and only show the final adjusted equation.) In addition, eq 12 was unable to fit our experimental data with reasonable quality. The best-fit values of Kw obtained in this work are consistently in the range of 1-3 s-1 despite the different conditions used in the experiments. Figure 6 shows that the data obtained in this work compare well with some other data previously reported in a plot of Kw against the ascending phase velocity u1. This equals ub when no other information is available in the references of Table 2. Most of the values extracted from the references analyzed in Table 2 fall within the same relatively narrow interval. Parts of the data from Basesme and Levy17 and Chiba and Kobayashi15 are overpredicted, but this can be attributed to the conditions of incipient fluidization under which these experiments were carried out. Also, the very low values of exchange coefficients derived by Sitnai et al.16 do not fit any equation found in the literature. This is probably a result of the particular flow distribution adopted to interpret their experiments, with two downward dense phases moving at very different velocities. Some series of data in Figure 6 show a certain autocorrelation between u1 and Kw. This can occur because, when experimental data are being fit to the CCBM model, the assumption of high values of ascending solid flow tend to generate low

Figure 8. Profiles simulated with the model, working at the conditions of Figure 2 and with optimum model parameters: Kw ) 2.1 s-1, u1 ) 0.61 m/s, f1 ) 0.3, Nc ) 2.

values of the exchange coefficient between phases and vice versa.18 With the previous considerations in mind, it can be argued that Figure 6 shows a surprisingly good consistency of values of Kw between 1 and 3 s-1, considering the wide variety of solid characteristics and bed dimensions and the different techniques and approaches used to derive Kw in the different works reported in Table 2. From a practical point of view, a reasonable choice is a constant value of Kw ) 2 s-1 for all of the experiments presented in this work. This apparently gross simplification in the choice of this parameter is supported by the sensitivity analysis presented in Figure 7. As can be seen in this figure, the concentration maps predicted by the model at two representative times in the mixing process are sensitive to the values of Kw, but only when

5662

Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001

Figure 9. Comparison of the experimental and calculated tracer volume fractions for a spatial resolution of 0.3 × 0.25 m (bed divided into four sections).

this parameter varies in a range of 1 order of magnitude. The predictions for Kw ) 1 and Kw ) 3 with the rest of the parameters kept constant are rather close to the central value. This is stressed below in more detail, when model predictions are compared with experience and when the experimental uncertainties determining concentration maps are taken into account in the discussion. 3.2. Comparison of Model Predictions with Experience. The solution of the final version of the model, with parameters as defined in the previous paragraphs, is compared here with the experimental concentration maps obtained with coal and PVC. Figure 8 represents a simulation of the experiments included in Figures 1 and 2. The comparison of experimental and predicted concentrations in this example yields a regression coefficient of 0.82. This relatively low value of regression is typical for these types of experiments and can be

explained by considering the random bed behavior at length and time scales much lower than those of the macroscopic flow patterns, as discussed above. The modest regression coefficient is in contrast with the satisfactory comparison of more general features in the mixing experiment, including the general flow patterns of the tracer, the almost perfect mixing achieved in less than 4 s, and the apparent rate of mixing of, for instance, the fraction of the bed initially occupied by the tracer. In Figure 9, a comparison between all of the experimental and calculated values of the tracer concentration is shown at a larger spatial resolution of 0.3 × 0.25 m to reduce the local discrepancies and facilitate an overall comparison with experience. As expected, the comparison under these conditions is better (regression coefficient close to 0.95), although the resolution of the model is still at the same time and length scale as in Figure 8 (see Appendix). The model predicts well the general trends of the mixing experiments when the gas velocity, particle sizes, and two types of internals used in this work are varied. The mixing rates are very sensitive to the gas velocity, as can be seen in the example in Figure 10, where a series of experimental and simulated concentrations maps at different gas velocities are presented. Similarly, a strong effect of particle size on mixing rates was observed in both experimental and predicted results, especially at the low gas velocities. The only changes needed in the model to incorporate the effect of these operating condition are the effect of superficial gas velocities and minimum fluidization velocities on the bubble size and rise velocities (as in Table 1) and, hence, on the velocities of the phases (eqs 9 and 10). The effect of the internals on the observed mixing rates can also be explained by attending to their effect on bubble properties. The inclusion of internals reduced mixing rates (and also produced smaller and slower bubbles). However, between the two configurations investigated, no relevant differences were observed in the mixing process. This is also in agreement with the observation (Table 1) that bubble properties were not clearly affected

Figure 10. Effect of gas velocity on solids mixing. Tracer concentration maps 2.3 s from the begining of the experiment: (a) experimental, (b) simulated. Bed without internals, dpc ) 0.95 mm, dpPVC ) 1.09 mm, hmf ) 0.5 m.

Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5663

by the changes in internals configuration over the interval of conditions studied. As a final remark, it is necessary to highlight that the examples of the model calculations represented in Figures 8-10 were obtained using values for the model parameters that can be justified from independent measurements of bubble size and velocity (f1, u1) and macroscopic bed properties (Nc). Also, the chosen value for Kw, in agreement with independent measurements reported by some other authors, seems to be less important than expected when a reasonable degree of uncertainty in the calculations is accepted. This ability to incorporate in the model a number of parameters that can be independently estimated for specific systems and conditions is an important advantage of the countercurrent backmixing model with respect to other alternative approaches to the modeling of mixing in fluidized beds.

tracer mass balances between any adjacent cells at height z produce a system of 3 × Nc linear partial differential equations

4. Conclusion The axial and lateral mixing of solids in fluidized beds can be modeled by attending to the gross convective currents of solids or “mixing columns” that can appear simultaneously in different parts of the bed. The individual solids flows in each mixing column can be quantified using the same concepts as in previous versions of the CCBM model for axial mixing only: the solids flow upward driven by the bubble activity and downward in the emulsion region, with a certain exchange of solids between the countercurrent phases. The observation with an image-analysis technique of the bubble properties in the bed has provided simple definitions for some model parameters such as the number of mixing columns, the rise velocity of the solids, and the phase volume fractions. It was found that the average solid rise velocity is around 0.44 times the value expected from the measured bubble rise velocities. The exchange coefficient between countercurrent phases is the main adjustable parameter when the model is fit to the database generated from experiments at different velocities, two particle sizes, and with and without two types of internals. In this work, this parameter was found to fall between 1 and 3 s-1. Within the same interval are several other published values for this parameter, obtained from very different installations and procedures. A sensitivity analysis shows that the model is relatively insensitive within this parametric interval when predictions are compared with the intrinsic scatter in the experimental data. The model comparison with experience suffers when results are compared at small time and spatial scales but it improves remarkably when the scales become wider and the uncertainties in the experimental data are taken into account. In its final version, the model provides reasonable predictions using sets of parameters that are easily justified from independent measurements of bed and bubble properties and a single adjustable parameter (Kw ) 2 s-1) that is in agreement with other values published in the literature.

f1 f1 ∂Cdr,i ∂Cdr,i (C - Cdr,i) ) u1 + Kw ∂t (1 - f1) ∂z (1 - f1) u,i Kd(Cdr,i - Cdl,i+1)

Acknowledgment This work is part of a project partially funded by the European Coal and Steel Community (7220-ED-081) and the Spanish CICYT (QUI98-0592). Help from E. Soley in the experimental work is also acknowledged. Appendix The model divides the lateral dimension of the bed into Nc columns with three phases each. Therefore, the

∂Cdl,1 f1 f1 ∂Cdl,1 ) u1 - Kw (C - Cu,1) ∂t (1 - f1) ∂z (1 - f1) dl,1 ... f1 ∂Cdl,i ∂Cdl,i ) u1 + Kd(Cdr,i-1 - Cdl,i) ∂t (1 - f1) ∂z f1 Kw (C - Cu,i) (1 - f1) dl,i Kw ∂Cu,i Kw ∂Cu,i ) -u1 + (C - Cu,i) (C - Cdr,i) ∂t ∂z 2 dl,i 2 u,i

... ∂Cdr,Nc ∂t

∂Cdr,Nc f1 f1 ) u1 + Kw - Cdr,Nc) (C (1 - f1) ∂z (1 - f1) u,Nc for i ) 1, 2, ..., Nc

(A1)

To facilitate the notation and the comparison with experimental data, all solid flows and dimensions are referred to an equivalent bed without bubbles (z ) hmf). The tracer concentration map at t ) 0 in the mixing experiments (C ) 0 for the regions with coal and C ) 1 for the regions with PVC) provides the initial conditions. The solution of the system of partial differential equations is carried out by decoupling the movement of the phases and the exchange of solids between phases. This is a similar approach to the “cinematic method”32 used to solve the axial version of the CCBM model. Using this method, the bed height is divided into a number of cells in the axial direction, Nz. These cells exchange solids with the adjacent cells at the same vertical position during a certain time (∆t), and then they jump and are relocated. The distance they jump is the product of the phase velocity and the time spent exchanging solids. For convenience, the slowest phase is the one used to define the unit of these increments

hmf Nz

(A2)

∆zd(1 - f1) u1f1

(A3)

∆zd ) ∆t )

During this time increment, the fastest phase jumps a higher distance (∆zu) or higher number of cells than the slower phase

m)

∆zu (1 - f1) ) ∆zd f1

(A4)

In the program code used to solve the model, this ratio is an integer, m, that represents the number of cells by which the ascending phase moves up when the descend-

5664

Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001

ing phase moves down one cell. This integer can be changed (1 unit in different relocations to make sure that the average over time is close to the real number (1 - f1)/f1. After every relocation of cells, the system of 3Nc linear differential equations for every height generated when the convective terms in eq A1 are deleted needs to be solved. The integer m is also used in the program code to apply the boundary conditions of the model during the relocation of cells. At z ) 0, the leftand right-hand-side cells of each mixing column move during the relocation of cells to the central upflowing phase to occupy m cells

Cu,i,j )

Cdl,i,1 + Cdr,i,1 for i ) 1, ..., Nc; j ) 1, ..., m 2 (A5)

At the top of the bed, the volume occuppied by the cells at z ) hmf in the descending phase is replaced by the volume of the m cells in the ascending phase that move up during relocation. Because of the perfect mixing assumed in the splashing zone, the concentration of tracer in the m cells of the ascending phase at the top of the bed in each mixing column is averaged to calculate the tracer concentrations in the descending cells at j ) Nz in all of the mixing columns (two in this work) Nc

∑ ∑ i)1j)N -m

Cdl,i,Nz ) Cdr,i,Nz )

Cu,i,j)

z

mNc

for i ) 1, ..., Nc (A6)

With these equations and conditions, the model can be solved numerically to provide the tracer concentrations in the different cells at different times. To obtain stable solutions (unaffected by the choice of ∆t), very small time increments are required, especially in the first stages of the mixing process where the gradients of concentration after the relocation of cells are high. Application of the theory of eigenvalues and eigenvectors can provide solutions, allowing larger increments of time, of the form33

C(to+∆t) ) e∆tAC(to) for any cell at height z (A7) where C(to+∆t) is the vector of dimension 3Nc of tracer concentrations calculated after solids exchange between cells at every height z. A is the matrix (3Nc × 3Nc) of coefficients of the differential equation resulting from eq A1 after the terms with u1 have been deleted

(

e∆tA ≡ I + ∆tA +

∆t2A2

+

2!

∆t3A3 3!

∞

+ ... )

∑

m)0

∆tmAm m! (A9)

The numerical solution of the model now relies on the development of eq A9. The use of the first two terms requires very small time increments to obtain acceptable solutions. To test more precise alternatives, eq A9 was expanded with the help of algebraic software (Maple V) to terms three and four. After these studies, an approximation to a second-order polynomial for the exponential of the matrix (three terms in eq A9), with typical time steps of 0.1 s and 100 horizontal cell divisions (Nz) for the slow phase, proved satisfactory. These were tested for a range of reasonable values for model parameters (f1 between 0.05 and 0.4, u1 between 0 and 1.5 m/s, and Kw between 0 and 5 s-1). Within these ranges of parameter values, the simulation results are unaffected by further reductions in the time and height increments (or increases in Nz). Notation

Nz

(

Appropriate solutions of eqs A7 and A8 of practical interest can be investigated with the following expression33

A ) matrix of coefficients of the system of equations after decoupling (eq A1) A ) bed cross-sectional area, m2 C ) solid volume fraction of tracer Cdl,i,j ) tracer volume fraction in a cell in the left-hand side of the downward phase of mixing column i at height jhmf/ Nz Cdr,i,j ) tracer volume fraction in a cell in the right-hand side of the downward phase of mixing column i at height jhmf/Nz Cu,i,j ) tracer volume fraction in a cell of the upward phase in the mixing column i at height jhmf/Nz Db ) bubble diameter, m Dbmax ) maximum bubble diameter, m f1 ) volume fraction of ascending phase in a bed without voids fw ) wake volume fraction based on the bubble volume hmf ) bed height at minimum fluidization, m I ) identity matrix I ) gray-scale value in a bed image Imax ) gray-scale value in a bed made up of tracer only Imin ) gray-scale value in a bed without tracer Kd ) exchange coefficient of solids between cocurrent phases, 1/s

ANc×Nc ) f1 f1 -Kw K 0 ... ... (1 - f1) w(1 - f1) ... ... ... ... ... ...

...

...

...

...

...

...

...

...

...

... ... f1

f1

K ... Kd -Kd - Kw (1 - f1) w(1 - f1) Kw -Kw ... 0 2 f1 Kw ... ... 0 (1 - f1) ... ... ... ... ... ... ...

...

...

... ... ... ...

...

...

... ... ... ...

...

0

... ... ... ...

...

Kw 2

... ... ... ...

...

f1 Kd ... ... ... -Kd - Kw (1 - f1) ... ... ... ... ... ...

... ... f1

f1

-Kw ... ... 0 Kw (1 - f1) (1 - f1)

)

(A8)

Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5665 Kw ) exchange coefficient of solids between countercurrent phases, 1/s Lb ) lateral bed dimension or bed width, m m ) number of cells that a cell in the ascending phase moves up during a relocation Nc ) number of mixing columns into which the bed is divided in the lateral direction Nz ) number of cells into which the bed is divided in the axial direction ∆t ) increment of time during which exchange of solids is allowed between adjacent cells in the lateral direction, s u ) superficial gas velocity, m/s umf ) minimum fluidization velocity, m/s u1 ) velocity of the ascending phase in a bed without voids, m/s udown ) velocity of the descending phase in a bed without voids, m/s ub ) bubble rise velocity, m/s z ) height in a bed without voids, m ∆zd ) increment of height for a cell moving down during relocations, m ∆zu ) increment of height for a cell moving up during relocations, m Greek Symbols mf ) voidage at minimum fluidization δ ) fraction of the bed taken up by the bubble void Fp ) particle density, kg/m3 Fg ) gas density, kg/m3 µg ) gas viscosity, kg/(m s)

Literature Cited (1) Kunii, D.; Levenspiel O. Fluidization Engineering; John Wiley & Sons: New York, 1969. (2) Kunii, D.; Levenspiel, O. Fluidization Engineering, 2nd ed.; Butterworth-Heinemann: Woburn, MA, 1991. (3) van Deemter, J. J. Mixing. In Fluidization, 2nd ed.; Davidson, J. F., Clift, R., Harrison, D., Eds.; Academic Press: New York, 1985. (4) Baeyens, J.; Geldart, D. Solids Mixing. In Gas Fluidization Technology; Geldar, D., Ed.; John Wiley: New York, 1986. (5) Potter, O. E. Mixing. In Fluidization; Davidson, J. F., Harrison, D., Eds.; Academic Press: New York, 1971. (6) Lim, K. S.; Zhu, J. X.; Grace, J. R. Hydrodynamics of gassolid fluidization. Int. J. Multiphase Flow 1995, 21, 141. (7) Lim, K.; Gururajan, V. S.; Agarwal, P. K. Mixing of homogeneous solids in bubbling fluidized beds: Theoretical modeling and experimental investigation using digital image analysis. Chem. Eng. Sci. 1993, 12, 2251. (8) Levenspiel, O. The Chemical Reactor Omnibook; Octave Levenspiel: Corvallis, OR, 1979. (9) May, W. G. Fluidized bed reactors studies. Chem. Eng. Prog. 1959, 55, 49. (10) Ito, O.; Kawabe, R.; Miyamoto, T.; Orita H.; Mizumoto M.; Miyadera H. Direct Measurement of Particle Motion in a LargeScale FBC Boiler Model. In Proceedings of the 15th International Conference on Fluidized Bed Combustion, Savannah, GA, May 1619, 1999; American Society of Mechanical Engineers: New York, 1999; Paper 0023. (11) Knoebig, T.; Luecke, K.; Werther, J. Mixing and reaction in the circulating fluidized bed. A three-dimensional combustor model. Chem. Eng. Sci. 1999, 54, 2151.

(12) van Deemter J. J. Mixing and contacting in gas-solid fluidized beds. Chem. Eng. Sci. 1961, 13, 143. (13) Shen, L.; Zhang, M.; Xu, Y. Solids mixing in fluidized beds. Powder Technol. 1995, 84, 207. (14) Hoffmann, A. C.; Janssen, L. P. B. M., Prins, J. Particle segregation in fluidized binary mixtures. Chem. Eng. Sci. 1993, 48, 1583. (15) Chiba, T.; Kobayashi, H. Solid exchange between the bubble wake and the emulsion phase in a gas-fluidized bed. J. Chem. Eng. Jpn. 1977, 10, 206. (16) Sitnai, O. Solids mixing in a fluidized bed with horizontal tubes. Ind. Eng. Chem. Process Des. Dev. 1981, 20, 533. (17) Basesme, E. A.; Levy, E. K. Solids exchange between the bubble wake and the emulsion phase in a two-dimensional gasfluidized bed. Powder Technol. 1992, 72, 45. (18) Abanades, J. C.; Kelly, S.; Reed, G. P. A mathematical model for segregation of limestone-coal mixtures in slugging fluidized beds. Chem. Eng. Sci. 1994, 49, 3943. (19) Bilbao, R.; Lezaun, J.; Mene´ndez, M.; Izquierdo, M. T. Segregation of straw/sand mixtures in fluidized beds in non-steady state. Powder Technol. 1991, 69, 31. (20) Shen, L.; Zhang, M. Effect of particle size on solids mixing in bubbling fluidized beds. Powder Technol. 1998, 97, 170. (21) Hull, A. S.; Chen, Z.; Agarwal, P. K. Influence of horizontal tube banks on the behavior of bubbling fluidized beds. 2. Mixing of solids. Powder Technol. 2000, 111, 192. (22) Grasa, G.; Abanades, J. C. A calibration procedure to obtain solid concentration from digital images of bulk powders. Powder Technol. 2001, 114, 125. (23) Goidich S. J.; Robertson A. The pressurized combustor. In Pressurized Fluidized Bed Combustion; Alvarez Cuenca, M., Anthony, E. J., Eds; Chapman and Hall: London, 1995. (24) Rowe, P. N. Prediction of bubble size in a gas fluidized bed. Chem. Eng. Sci. 1976, 31, 285. (25) Hull, A. S.; Chen, Z.; Fritz, J. W.; Agarwal, P. K. Influence of horizontal tube banks on the behavior of bubbling fluidized beds. 1. Bubble hydrodynamics. Powder Technol. 1999, 103, 230. (26) Whitehead, A. B. Distributor Characteristics and Bed Properties. In Fluidization, 2nd ed.; Davidson, J. F., Clift, R., Harrison, D., Eds.; Academic Press: New York, 1985. (27) Seville, J. P. K.; Ding, Y. L.; Stein, M. Particle motion in bubbling fluidized beds. In Proceedings of the 3rd European Conference on Fluidization, Toulouse, France, May 29-31, 2000; SFGP: Nancy, France, 2000; p 21. (28) Johnsson, P.; Andersson, S.; Leckner, B. Expansion of a bubbling fluidized bed. Powder Technol. 1991, 68, 117. (29) Darton, R. C.; La Nauze, J. F.; Davidson J. F.; Harrison, D. Bubble growth due to coalescence in fluidized beds. Trans. Inst. Chem. Eng. 1977, 55, 274. (30) Nicklin, D. J. Two-phase bubble flow. Chem. Eng. Sci. 1962, 17, 693. (31) Geldart, D. The effect of particle size and size distribution on the behaviour of gas-fluidised beds. Powder Technol. 1972, 6, 201. (32) Lakshmanan, C. C.; Potter, O. E. Numerical simulation of the dynamics of solids mixing in fluidized beds. Chem. Eng. Sci. 1990, 45, 519. (33) Hubbard, J. H.; West, B. H. Differential Equations: A Dynamical Systems Approach; Springer-Verlag: New York, 1995.

Received for review October 30, 2000 Accepted August 23, 2001 IE0009278