Slip Velocity in Downer Reactors: Drag Coefficient ... - ACS Publications

Ind. Eng. Chem. Res. 2010, 49, 6735–6744

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Slip Velocity in Downer Reactors: Drag Coefficient and the Influence of Operational Variables Mohammad Ashraful Islam,* Stefan Krol, and Hugo I. de Lasa Chemical Reactor Engineering Centre, Chemical and Biochemical Engineering Department, Faculty of Engineering, UniVersity of Western Ontario, N6A 5B9 London, Ontario, Canada

Cluster formation in downer reactors is considered a most important factor influencing slip velocity between the flowing phases. This study investigates the formation of clusters using CREC-GS-Optiprobes. This probe is a nonintrusive device that records information from a well-defined optical measuring volume. This probe can provide accurate measurements tracking closely gas-solid flows in downer units. This study shows that cluster drag coefficients can be correlated as function of both Reynolds number and number of particles contained in agglomerates. This yields a close to linear relationship between cluster size and average slip velocity. 1. Introduction The great prospects of gas-solid down-flow reactors have stimulated research about the hydrodynamics, the mixing, the mass and heat transfer, the modeling, and the simulation of these units.1-4 It is in this respect important to establish the behavior of these gas-solid flows using as a basis the particle clusters or particle aggregates. According to the law of particle mechanics, the terminal settling velocity of a single particle can be calculated as a balance of forces exerted on a particle. In downflow reactors, however, where there is a population of particles moving simultaneously under gravitational force, particles experience an increase in slip velocity as observed by several authors.5-13 There are several possible causes that can be traced as contributing factors allowing particles to evolve with a velocity faster than the expected terminal settling velocity, such as (a) drag force reduction with mass increment due to particle agglomeration, (b) delayed wake separation due to increased particle roughness and leading to smaller drag, and (c) exchange of momentum and viscous forces between the gas and the particle flowing phases. There is, thus, a need for clarifying these matters using experimental measurements. Sobocinski et al.7 and Krol et al.10 advanced the view that the agglomerates are in fact strings of particles. In particular, Krol and de Lasa10,11 developed a theoretical analysis to explain the mechanisms of cluster formation in down-flow reactors and the unusual rise in slip velocity observed in these units. A number of techniques are available to detect and measure cluster velocity and cluster size in downer systems such as fiber optical probe,7,13 special micrograph probe, CCD miniature video camera,6,14 and needle capacitance probe.9,15 Among these techniques, the fiber optical sensors are relatively simple, inexpensive devices with significant potential.16 However, conventional fiber optic backscattering sensors suffer three major problems: (a) the short focal distance of the probe, (b) the low light intensity of the captured reflected light by the receiver fibers, and (c) the infinite measurement volumes from which the probes collect reflections. As a result, these probes often provide inaccurate measurements. However, with the incorporation of a GRIN lens, the advanced design of the Chemical Reactor Engineering Centre-Gas-Solid Optiprobe (CREC-GSOptiprobe) overcomes most of these limitations.17 * To whom correspondence should be addressed. E-mail: ashraf. [email protected].

The goal of the present study is to establish that cluster drag coefficients can be correlated as function of both Reynolds number and number of particles contained in agglomerates in a downer unit. This approach is considered for a wide range of solid circulation fluxes (3-108 kg/m2 s) and moderate gas velocities (0.5-3.1 m/s). 2. Experiments 2.1. Experimental Conditions. The experimental unit employed to simulate the hydrodynamics of a down-flow reactor is shown in Figure 1. Both the riser and downer unit are made from transparent acrylic pipes of internal diameter 0.0263 m. The height of the downer reactor is 3.0 m. A detailed description about the experimental unit is provided in Nova et al.13 The fiber optic probes were inserted into the downer column at 1.85 m below the air injection port considering that most of the particles will reach terminal settling velocity at this location. Figure 2 reports the dP/dz values for various solid fluxes showing that, below 1.6 m from the injector, dP/dz displays small and positive values. This is a confirmation of the applicability of the fully developed flow pattern model (clusters evolving at terminal velocity), an assumption required by eq 7. The operating conditions of the present study were set according to a combination of independent variables such as gas volumetric flow rate and solid mass flux as shown in Figure 3. The gas velocities and solid circulation rates were varied from 0.54 to 3.1 m/s and from 3 to 108 kg/m2 s, respectively. The temperature of the compressed air changed from 19 to 27 °C, while the absolute operating pressure of the downer section varied from 103 to 113 kPa. Fluid catalytic cracking (FCC) particles with the volume weighted mean particle diameter, dp,vol-avg of 84.42 µm (standard deviation (σdP) 33.62 µm) were employed as the solid phase while air was used as the fluidizing medium. The volume fractions in particle size distribution (PSD) of FCC particles were measured using MASTERSIZER 2000 Ver. 5.22 (MALVERN Instruments). Considering the appropriate size of particles, the number fractions were then calculated from the volume fractions. The CREC-GS-Optiprobe measurements were carried out at the centerline of the downer unit. For a single operating condition five signals were collected consecutively. Each signal waveform collected in 1 s contains 100 000 data points.

10.1021/ie901466p  2010 American Chemical Society Published on Web 12/29/2009

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Ind. Eng. Chem. Res., Vol. 49, No. 15, 2010

Figure 1. Adaptation of the schematic diagram of a riser/downer experimental unit (from Nova et al.13).

Figure 2. Changes of pressure drop gradient (dP/dz) along the downer axial position.


Figure 3. Operating conditions selected in the present study.

2.2. Techniques for the Measurement of “Cluster Velocity” and “Number of Particles in a Cluster”. The crosscorrelation function between the data of two time series can be used for estimation of the cluster velocity (Vcl). These time series can be obtained with sets of two CREC-GS-Optiprobes axially spaced along the flow stream in the downer section. The time delay, corresponding to the maximum value of the crosscorrelation function, provides the particle transit time (τ*) between the two sensors. The distance between the measuring regions of the upper and lower probes (L ) 0.6 cm) and the particle transit time between both sensors can be used to estimate the velocity of the cluster with the following expression: Vcl ) L/τ*

(1)

A more detailed calculation procedure about Vcl can be found in Nova et al.13 First, there is the superficial gas velocity (Vg) in the downer section that can be calculated using the gas flow rate (Qg), measured at the outlet of the downer section and the cross-sectional area of the column (Ac) in the following way: Vg ) Qg /Ac

(2)

The next step in the analysis is the evaluation of the solid holdup (εs) and the gas void fraction (ε). The solid holdup is the fraction of the down-flow unit volume occupied by the solid phase, while the rest of the volume is the gas void fraction. The following simplified model can be used for the calculation of solid holdup: εs ) 1 - ε )

Gs js FpV

(3)

where Gs and Fp represent average solid mass flux (kg/m2 s) j s is and average particle density (kg/m3), respectively, and V calculated as Nobs

∑ NV

cl

js ) V

1 Nobs

(3a)

∑N 1

where a cluster with N, number of equivalent mean particles, travels at a velocity of Vcl. Nobs is the number of observations in a given experimental condition.

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Using the approach of the present study, the evaluation of the cluster size can be based on measurements of the widths of the signal peaks. For each signal an average (X) and a standard deviation (σx) are calculated. A baseline at average plus 3 times the standard deviation (X + 3σx) is set to eliminate the likely uncorrelated optic signals from the main signal. Figure 4 can be used to illustrate the evaluation procedure. For example, in the case of the signal reported in Figure 4, signal average and standard deviation voltages are -0.042 and 0.559 V, respectively. Therefore, the baseline is set in this case at X + 3σx or 1.635 V. Peaks which appear above this baseline are considered appropriate for cluster size calculations, while the other signals are discarded for further analysis. Regarding the peaks meeting the condition of surpassing the baseline, the peak widths of these selected peaks are measured horizontally at the baseline. Using these procedures, the width of an individual peak, ts, is obtained as a time interval (e.g., ts ) x2 - x1 in Figure 4) in seconds corresponding to the time that the cluster took to cross the length of the diameter of the focal point of the probe (probe’s measuring region). Moreover, it is recommended to acquire from the upper and lower optic sensors at least five pairs of highly correlated signals with this condition considered as the minimum data for the calculation of number of particles (N) in a cluster. Since in many instances several peaks appeared above the baseline (X + 3σx) in the same data report, an average of peak widths, ts-avg, was considered for each experimental condition. With the average width, ts-avg, and an average particle diameter, dp,vol-avg, the cluster size expressed as N was evaluated using the following formula: ts-avgVcl - h + dp,vol-avg (4) N) dp,vol-avg where Vcl represents the cluster velocity at corresponding pair of time series and h is the characteristic transversal dimension of the sensing region (focal point).18 Thus a minimum set of 10 values of N were obtained for every single operating condition. The average values of Vcl and N are reported later on in the present study. The CREC-GS-Optiprobes require an optoelectronic system to perform the data acquisition. The design principles and calibration procedures of this system are described in Nova et al.17 In the context of the present study three programs were developed using LabVIEW (version 6.1) and were implemented for three purposes: the program “Probe Signal Sampling.vi” for data acquisition from the experimental unit, another program called “Cross Correlation Function Hi-Speed.vi” for estimation of the cluster velocity through calculation of the average transit time from the cross-correlation function, and the last one called “DataCov2txt.vi” for writing the waveform into a text file. A MATLAB (version 7.2.0.232) code was also developed to estimate the average (X), standard deviation (σx) and the widths of the selected peaks of the signals (ts and ts-avg). 3. Slip Velocity and Number of Particles A total of 54 experimental runs were performed using different combinations of gas flows and solid mass fluxes. The CREC-GS-Optiprobes were placed in the lower section of the downer unit (1.85 m below gas injection point) to secure stabilized particle velocity values. The slip velocities (Vslip), which is defined as the difference between the cluster velocity and the interstitial gas velocity, were calculated with the following relationship: V js - g (5) Vslip ) V ε

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Figure 4. Example of signal peak width determination from the CREC-GS-Optiprobe signals using the X + 3σx baseline definition. Peaks selected under this criterion differentiate clearly from the signal noise.

Figure 5. Change of slip velocity with the operating conditions: superficial gas velocity and solid mass flux.

In Figure 5, slip velocities for 54 experimental runs are reported as function of operating variables in a three-dimensional plot. On the basis of the reported results, it appears that slip velocity steadily increases with the increase of gas velocity. Figure 6 reports slip velocities as a function of solid concentrations as well as superficial gas velocities. For these experiments the solid holdup (εs) was in the 0.0006-0.0326 range. On the basis of these data one can conclude that the relationship of slip velocity with solid mass flux does not appear to be a straightforward one. The slip velocity changes significantly in the lower particle concentrations and higher gas superficial velocity ranges, becoming insensitive to these parameters when εs surpasses 0.010. To provide a phenomenological based explanation of gas-solid flows in down-flow reactors, one can consider particle agglomeration or cluster formation as the dominant flow feature influencing gas-solid slip velocity. In this respect, the CRECGS-Optiprobes is an advanced optical device allowing nonintrusive detection of clusters at the point of measurement (measuring volume) with this cluster characterization including both cluster size and cluster velocity. With the collected data from the CREC-GS-Optical probes, one can consider the changes of the slip velocity with the N average parameter (number of particles with average diameter

Figure 6. Slip velocity as a function of solid concentration and superficial gas velocity.

in the observed cluster) as will be shown and reviewed in the upcoming sections of this contribution. 4. Particle Cluster Drag Coefficient in Downer Units Regarding a possible fundamental approach to describe the fluid dynamics of clusters in down-flow reactors, one can consider an estimation of the slip velocities of particle agglomerates, using as a basis the balance of forces exerted in the cluster. It is in this respect generally recognized that the agglomerate drag coefficient, CD, can be expressed in terms of particle Reynolds number, Re, as well as one or more shape factors. In recent years several CD-vs-Re expressions have been proposed and summarized for different particle shapes in the technical literature.5,19-29 Haider and Levenspiel21 pioneered the development of a universally applicable and simple to use drag expression for spherical and nonspherical particles based on a single shape factor, the sphericity. Chhabra et al.23 compared five most promising drag expressions considering 1900 experimental data points from a wide range of particle shapes and hydrodynamic conditions. Based on the overall mean and maximum percent errors, Chhabra et al.23 recommended that the drag expression proposed by Ganser30 is the most appropriate method for CD


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where Vslip is the slip velocity; Fgand µ are the density and dynamic viscosity of gas, respectively. As a result, in the constant acceleration section of the downer where the drag force is counterbalanced by the gravitational and buoyancy forces, the experimental drag coefficients for a cluster consisting of N particles can be calculated by the following equation: CD-exp )

Figure 7. Comparison of the drag coefficients, CD, measured by different drag expressions. Table 1. Parameters Required for Ganser30 and Haider and Levenspiel21 Drag Expressions in Terms of N and Volume-Weighted Average Particle Diameter shape factors

expressions in terms of N

N1/2dp,vol-avg

ds

Re )

K1

2 1 + φ-0.5 3 3

(

)

-1

dv - 2.25 D

101.8148(-log φ)

K2

0.5743

calculation which uses the volume equivalent sphere diameter (dv), the sphericity of agglomerate (φ), and the pipe diameter (D). Apart from using volume equivalent diameter and sphericity for the determination of steady-state drag force on agglomerates, a second possible approach used by several researchers is to calculate the drag force by considering the actual projected surface area of the irregular particles.31,32 According to the cluster configuration proposed by Krol et al.,10 a cluster is likely formed with a single leading particle and several trailing particles. This particle arrangement is the likely one to be found for a fluid dynamically stable cluster. Thus, in the context of the present study, the cluster shape was assumed as a vertical chain of N number of spherical particles touching each other at their contact points. As a result, the following is adopted: the projected area equivalent diameter of a cluster is assumed to be the same as the area weighted average particle diameter, dp,area-avg. Instead of defining the Reynolds’ number based on the diameter of a sphere which has the same volume as the agglomerate, the area weighted average particle diameter, dp,area-avg was used as follows: Re )

dp,area-avgVslipFg µ

24 (1 + 0.1806Re0.6459) + Re

0.4251 6880.95 1+ Re

(8)

0.4305 (9) 3305 1+ (Re)K1K2

dp,area-avgVslipFg µ

Re

(7)

CD 24 ) (1 + 0.1118((Re)K1K2)0.6567) + K2 (Re)K1K2

(ds /dv)-2 ) N-1/3

φ

3Fgdp,area-avg2Vslip2

where dp,vol-avg is the volume weighted mean particle diameter, dp,area-avg is the area weighted average particle diameter, Vslip is the slip velocity, g is the gravitational acceleration, and Fg and Fp are the densities of gas and particle, respectively. On the basis of eq 7 the CD values were calculated for different cluster sizes as a function of the Reynolds number and as reported in Figure 7. Furthermore, also in Figure 7, the drag coefficients calculated using Stokes’ drag expression (CD ) 24/Re) and two widely used empirical equations proposed by Haider and Levenspiel21 (eq 8) and Ganser30 (eq 9) are presented for comparison purposes. CD )

N1/3dp,vol-avg

dv

4Ngdp,vol-avg3(Fp - Fg)

(6)

One can notice that in Stokes’ equation as well as in eqs 8 and 9 the Re number is based on an area weighted mean particle diameter, dp,area-avg, while Ganser’s30 drag coefficient expression involves a K1 Stokes’ shape factor and a K2 Newton’s shape factor. A list of different equivalent diameters and shape factors used in this paper is provided in Table 1 in terms of the N parameter. Reviewing the results reported in Figure 7, one can observe that the Stokes’ law and Ganser’s expression provide relatively lower values than the experimental results. On the other hand, the expression of Haider and Levenspiel provides a close trend with the results obtained experimentally. Few data points show relatively high drag coefficients due to the fact that at low gas velocity clusters with higher numbers of particles may likely not have a chainlike configuration. One can also notice from Figure 7 that the Haider and Levenspiel expression as a function of Re expressed in terms of area weighted mean particle diameter, dp,area-avg, can also be effective to calculate drag coefficients due to the fact that the drag force exerted on a cluster is mainly coming from its frontal projected area. 5. Particle Cluster Drag Coefficients for Cluster Sizes with Different Particle Configurations Regarding the calculation of CD estimations, one has to consider that the CREC Optiprobe detect an axial dimension for the agglomerates. This axial dimension, as “seen” by the CREC Optiprobe, is the result of different possible configurations of a “leading” particle followed by “trailing” particles. For instance as reported in Figure 8, the FCC particles used in this research displayed a PSD varying from 25 to 225 µm. Thus, one can expect that, in the vertical chainlike arrangement,

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gravitational and buoyancy forces for 225 µm particles could differ from the configuration of the 75 µm leading particle followed by the two other 75 µm trailing particles as well as a 112.5 µm leading particle chased by another 112.5 µm trailing particle. One can certainly argue that larger leading particles will cause more drag, so the slip velocity may decrease. However, one can also expect that larger particles also gain higher gravitational forces than the smaller particles. Table 2 provides a sample calculation for these three configuration of particles having the same N ) 3. It appears from Table 2 that a particle of 225 µm diameter experiences a higher terminal velocity with a corresponding increase in the Re and a reduction in the CD value following the expected trend of 24/Re. 6. Particle Cluster Drag Coefficients from Experimental Data Figure 8. Volume weighted particle size distribution of FCC particles used in this work. Table 2. Comparison of Three Configurations with N ) 3

an FCC particle with a 225 µm dimension will be detected by the CREC-GS-Optiprobes with an equivalent number of N ) 3 (3 × 75 µm ) 225 µm). Similarly, two 112.5 µm (2 × 112.5 µm ) 225 µm) particles will appear as an equivalent number of N ) 3 in front of a CREC-GS-Optiprobe. This “ambiguity” in the definition of the particle number is accompanied by other important differences: (a) the projected area as well as the

Figure 9. Drag coefficient measured using eq 7 for various cluster sizes of N from 2 to 8. Standard error deviations for the reported points are (1%. Note: reported data points represent average values resulting of a minimum of at least five repeats.

As a result of the data analysis reported and the CD values calculated using eq 7, it was felt that further consideration of the predictions of CD was required using the equivalent number of particles present in the agglomerates, as a parameter. On this basis, the data were segregated into families of subdata sets sharing the same N number of particle property, as shown in Figure 9. Given that there was only one data point for N ) 1 and 9, these data were excluded from Figure 9. Reviewing these families of subdata sets in a log CD-log Re plot, one can notice that the drag coefficients decrease consistently with the increase in slip velocities for clusters having the same size (same axial length represented by an equivalent number of average particle size, N), showing a very small spread in the data correlated. Coefficients of correlation in the log CD-log Re plot for all families of subdata sets were larger than 99%. As a result of this analysis one can see that consideration of CD as a combined function of Re, the particle Reynolds number, and N, the number of particles in a cluster, is a valuable approach for the fluid dynamic description of clusters evolving in gas-solid down-flow units. Regarding Figure 9, one can also notice the following important observations: (a) at the same particle Re number, increasing N leads to larger drag coefficients; (b) at a set N value, augmenting the particle Re number yields higher slip velocities given a reduced CD. Moreover, it is interesting to note from Figure 9 that the two clusters having the same axial length or N value can experience two different slip velocities. This uncertainty of slip velocity values for a given N is an inherent characteristic of gas-solid flows in downers and explains the apparent data spread shown in Figure 7 when one tries to correlate the slip velocity using N as a sole parameter functionality. It can perhaps be argued that there is still the possibility that this disparity may be explained given that clusters with the same N may not necessarily have the same shapes while descending in the column. In our view, this is unlikely to contribute given the need for the clusters to evolve down flow as stable particle structures offering a configuration where drag forces are minimized.10 Furthermore, CD and Re values for the three cluster configurations are reported in Figure 10 along with the experimental data points for all clusters with axial lengths equivalent to N ) 3. It can be noticed that the configuration of three particles (75 µm leading particle with two other 75 µm trailing particles) and two particles (112.5 µm leading particle with another 112.5 µm trailing particle) are well in the range of experimental points


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equivalent of N ) 8 (e.g., 8 × 84 µm ) 672 µm) gives a predicted drag coefficient of 0.03. However, by analyzing the PSD, the likelihood of having a particle size over 225 µm is essentially zero. Therefore, it can be expected that most of the experimental data points should remain above the drag coefficient value of 1.9. Given the above-described facts, one can see that two major parameters affect the drag coefficients for cluster structures: the Reynolds number (Re) and N (the number of particles in the cluster). Thus it can be written as CD ) f(Re, N)

(10)

On this basis a nonlinear regression was performed on A, R, and β as in eq 11. NR (11) Reβ After nonlinear regression the following equation is found with a regression coefficient (R2) higher than 97%. CD ) A

Figure 10. log CD versus log Re plot for three vertical chain configurations, which falls in the same line with experimental data points.

N1.3488 (12) Re1.9128 The experimental and predicted drag coefficients are plotted in Figure 12 with 95% upper and lower confidence levels. Table 3 lists cross-correlation coefficients of the parameters proposed in eq 12. The drag coefficient in eq 12 could be approximated with the Haider and Levenspiel21 drag expression as shown by CD ) 24

CD ) 24

Figure 11. Limiting drag coefficients for cluster sizes determined experimentally. Line 1 represents drag coefficients of a single particle having the size of multiple of average particle size; line 2 is the drag coefficient calculated using Stokes’ law for the particles in line 1.

N1.3488 24 0.4251 (1 + 0.1806Re0.6459) + ≈ 6880.95 Re1.9128 Re 1+ Re (13)

In the present study the maximum Reynolds’ number observed is 11. Thus, the second term in Haider and Levenspiel’s expression becomes for all practical purposes insignificant. In addition, and given the magnitude of dp,area-avg, Fg, and µ values, eq 13 is simplified as 4.09Vslip0.913 + 2.00Vslip1.56 ) N1.35

Table 3. Cross-Correlation Coefficients for Drag Expression Proposed in This Study parameter

A

R

β

A R β

1.00 –0.6154 –0.0982

–0.6154 1.00 0.6993

–0.0982 0.6993 1.00

measured with the CREC-GS-Optiprobes. In fact, from the particle size distribution plot (Figure 8) one can see that a single 225 µm FCC particle is unlikely to be present given the PSD of Figure 8. Therefore, it can be concluded that clusters having the same axial dimension but with different particle configurations will share the common feature of belonging to the same line, as shown in Figure 10. To enhance the analysis, further calculations were done for other N values. The drag coefficients in line 2 in Figure 11 were calculated considering Stokes’ law. The deviation from Stokes’ law is most likely due to the change of cluster shape from spherical to cylindrical form; thus, the projected area reduces as well as the drag and eventually the slip velocity increases. The terminal settling velocities for single particles having the diameter of equivalent N (e.g., 4 × 84 µm ) 336 µm) were also calculated to find out the limiting cases as shown by line 1 in Figure 11. For instance for a particle having the diameter

(14)

Numerical regression on eq 14 provides Vslip and N pairs that as reported in Figure 13 yield an essential linear relationship. 7. Particle Drag Coefficient for Solids with Particle Size Distributions In the context of the present study, it is observed that the slip velocity can provide an initial proximate estimation of the number of particles forming a cluster. However, there is an expected variation in the slip velocities for a given number of particles and this is a result of the intrinsic character of a cluster being formed by more than one possible configuration of particles in the chain. As demonstrated above, a given axial cluster can be constituted by larger particles having a higher slip velocity than a cluster of the same length consisting of smaller particles. In order to simulate various cluster configurations, clusters were randomly generated by assuming a vertical chain of spherical particles touching each other at their contact points. Using the MATLAB random number generator (RandomMat), the combination of particles matching the cluster’s axial length within 1% value as observed by the CREC-GS-Optiprobe was done (e.g., (4 × 84) ( 1% ) 332.64-339.36 µm). The pool of 1000

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Figure 12. Comparison of experimental CD drag coefficients and CD predicted drag coefficients with 95% confidence level.

Figure 13. Slip velocities of clusters in downer reactor as a function of number of particles present in descending clusters. Repeats for standard deviation calculations were seven as a minimum.

particles used in this analysis displayed the same frequency distribution as in the particle size distribution reported in Figure 8. For example, among 1000 particles 7% (frequency value in Figure 8 for particle size 50 µm) or 70 particles are of size 50 µm. The implemented program randomly picks particles one by one and checks the combined length. If the length is within 1% of the observed value, the program routine calculates the velocity of the chainlike cluster. The first particle considered in this random selection was always labeled as the “leading particle” and was used for the Reynolds number as well as for the projected area for drag calculations. This random selection

of particles was tried at least 100 000 times and this for every possible cluster length combination (e.g., N ) 1, 2, 3, 4, 5, 6, 7, 8, and 9). As a result, it was judged that this provided a good database to test the impact of the various cluster configurations on the correlation of slip velocities on number of particles. Furthermore, the drag coefficients for all created clusters were calculated according to eq 7. The observed average, maximum, and minimum values are reported in Figure 13. The maximum value was calculated by adding the standard deviation with the average value. On the other hand, the minimum value assessed


was calculated by subtracting the standard deviation with the average value. As a consequence, it can be concluded that the various possible cluster configuration necessary bring a spread of the data points in the slip velocity-number of particle graph representation with, however, all experimental data points being included inside the ranges as defined by the broken lines of Figure 13. In summary and on the basis of the data of this study, it can be concluded that if one tries to find the phenomenological relationship between slip velocity and axial cluster length, one can only establish a band (range) of correlated values (refer to Figure 13) rather than a tight correlation, with this being an intrinsic characteristic of the clusters in down-flow reactors offering different particle configurations.

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K1 ) Stokes’ shape factor K2 ) Newton’s shape factor L ) distance between the measuring regions of the upper and lower probes (0.6 cm) N ) number of particles in a cluster with diameter dp,vol-avg Nobs ) number of observations in an experimental condition Re ) Reynolds number based on area weighted mean particle diameter ) (dp,area-avgVslipFg)/µ Vg ) superficial gas velocity (m/s) Vcl ) cluster velocity (m/s) j s ) average cluster velocity as given by eq 3a V Vslip ) slip velocity (m/s) VT ) terminal settling velocity of a single particle (m/s) Qg ) gas flow rate (m3/s) ts-avg ) average of peak width (second) X ) signal average (volt)

8. Conclusion The following are the significant conclusions regarding the present study. (a) The CREC-GS-Optiprobe, of the reflective type with an incorporated GRIN lens, can be successfully applied to study the gas-solid flow behavior in a downer reactor under a wide range of hydrodynamic conditions. The gas velocities and solid mass fluxes were varied from 0.54 to 3.1 m/s and from 3 to 108 kg/m2 s, respectively. (b) The cluster slip velocity primarily depends on the gas velocity. The effect of solid concentration becomes insensitive as the solid concentrations exceed 0.01. Under these operating conditions, particles evolved as strings, clusters, or agglomerates with a relatively large distribution of cluster length, ranging from 96 to 713 µm (N from 1.3 to 8.61). (c) The CD values, calculated using the Stokes drag coefficient expression and two other promising empirical drag coefficient expressions, compare favorably with the drag coefficients calculated from the force balance at constant particle velocity condition. For these operating conditions the Haider and Levenspiel21 CD-vs-Re expression provides an average approximation of the drag coefficients. (d) A drag expression is proposed where the drag coefficient is a function of both the Reynolds number and the number of particles contained in agglomerates. This yields a close to linear relationship between cluster size and average slip velocity. (e) The intrinsic characters of the clusters offer different potential particle configurations. This yields a relationship between slip velocity and axial cluster length that must be viewed as a range of correlated values rather than a tight single correlation. Nomenclature Ac ) cross-sectional area of the column (m2) CD ) drag coefficient CD-exp ) experimentally determined drag coefficient D ) tube diameter (0.0263 m) dp,vol-avg ) volume weighted mean particle diameter (m) dp,area-avg ) area weighted mean particle diameter (m) ds ) diameter of a sphere with the same surface area as the object (m) dv ) diameter of a sphere which has the same volume as the agglomerate (m) Gs ) solid mass flux (kg/m2 s) g ) gravity acceleration (m/s2) h ) characteristic transversal dimension of the sensing region (focal point) (m)

Greek symbols ε ) gas void fraction εs ) solid holdup µ ) gas kinematic viscosity (kg/m.s) Fg ) density of the gas (kg/m3) Fp ) density of the particle (kg/m3) φ ) sphericity factor σdp ) standard deviation of particle size distribution (m) σx ) standard deviation of signal (volt) τ* ) Transit time

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ReceiVed for reView September 17, 2009 ReVised manuscript receiVed November 20, 2009 Accepted November 24, 2009 IE901466P

Slip Velocity in Downer Reactors: Drag Coefficient ... - ACS Publications

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