Radial gas mixing in the upper dilute core of a circulating fluidized bed

293

Powder Technology, 70 (1992) 293-301

Radial gas mixing in the upper dilute core of a circulating fluidized bed J. Werther,

E.-U. Hartge

and M. Kruse

Technical. University Hamburg-Harbulg, D 2100 Hamburg 90 (Germany)

Abstract Based on the assumption of a core-annulus structure, radial gas mixing characteristics were experimentally studied in the core of the upper dilute zone of a pilot scale circulating fluidized bed (CFB), 0.4 m in diameter and 9 m in height. CO2 as a tracer was injected continuously into the center of the bed by a point source. Radial tracer gas concentration profiles were measured in several planes downstream of the injection point. An analytical solution derived by Klinkenberg et al. (Ind. Eng. Chem., 6 (1953) 1202) [l] for the description of gas mixing in turbulent single phase flow was successfully applied to determine the Peclet number, Pe,., for radial gas mixing in the core zone. Pe, c was found to be independent of the superficial gas velocity u, which is in agreement with the gas mixing beha;iour in turbulent single phase with other authors’ measurements in single phase rates G,, up to 70 kg m-’ s-l. The fact that Pe,. mean solids concentration in the core zone and significant influence on the turbulent intensity of

flow. A mean value of Per,. =465, which is also in agreement flow, was calculated from measurements at different solids was found to be independent of G, is attributed to the low to the size of the particles, which is too small to cause a the gas phase.

Introduction Many researchers (e.g. [2-91) have shown that the flow structure in the upper dilute zone of a circulating fluidized bed is characterized by the presence of two phases. A dilute suspension flows upwards preferentially in the center of the bed, whereas in the wall region a dense phase descends in form of particle clusters or strands. Several mechanisms have been suggested to account for the formation of this characteristic core-annulus structure [5, 8, 10, 111. Compared to the number of investigations dealing with fluid mechanical properties, the number of papers dealing with gas mixing in the CFB is fairly small. From their backmixing measurements Cankurt and Yerushalmi [12] have concluded that backmixing of gas in the CFB is negligible compared to convective gas flow. The same conclusion was later confirmed by other authors [3, 131. On the other hand, measurements of gas residence time distributions by Brereton ef al. [14] have shown that the CFB as a whole exhibits a considerable amount of backmixing in the gas phase. Li and Weinstein [15] have already pointed out that this apparent contradiction may be explained by the existence of different flow regions inside the CFB. Recently Li and Wu [16] showed that the extent of axial gas dispersion depends on bed voidage, which is mainly influenced by the gas velocity u and the solids rate G,.

0032-5910/92/$5.00

Concerning lateral mixing of gas in the riser of the CFB, Yang et al. [17] found that the intensity of radial mixing, which is characterized by the radial dispersion coefficient, decreases with increasing gas velocity u and decreasing solids rate G,. Contrary to this investigation are the results of Adams [18] who indicated an increasing horizontal dispersion coefficient with increasing gas velocity and decreasing solids rate. From the work of Bader et al. [3] no relationship between radial gas mixing and gas velocity or solids rate was obtained because gas velocity and solids rate have been changed simultaneously. Martin et al. [ 191 introduced a turbulent flow profile of gas into their gas mixing model to describe their measurements. The calculated radial dispersion coefficients showed a slight increase with increasing solids rate, whereas the influence of gas velocity was not studied at any constant solids rate. In the present investigation a concept is suggested and experimentally tested which divides the CFB into a core and an annulus region. Both experiments and theoretical interpretation are focused here on the study of the radial mixing of gas in the core zone. This latter property is of significance for large-scale CFBs where gaseous reactants are introduced into the bed. An example is the addition of secondary air into the upper part of the combustion chamber in circulating fluidized bed combustors (CFBC).

0 1992 - Elsevier

Sequoia.

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294

D ax. c was varied over a wide range of physically rea-

Theory

Based on the idea of the core-annulus structure, the CFB is divided into a core and an annulus zone with different gas mixing characteristics (Fig. 1). The dispersion coefficients are D,, c and D, c in the core zone and L,. and Q,. in the annular zone, respectively. The boundary between the two zones is located at the radius R* (a cylindrical vessel is assumed). In the core zone the dilute suspension with a solids volume concentration c,, c flows upward at a superficial gas velocity U, which is based on the empty cross-section of the core zone. The wall zone is characterized by a solids volume concentration c,, a and a gas velocity U, based on the empty cross-section of the annulus. The mass transfer coefficient p describes the mass transfer between the two zones. In the present work the investigation is restricted to gas mixing in the core zone. The interaction between the core and annulus is the subject of a separate investigation [20]. A mass balance on a differential volume element, (2~7 dr dz) of the core zone leads under conditions of steady state to eqn. (1) if the voidage term (1 -c,, J is set to unity, which may be justified by the low mean solids concentration c,., in the core zone [2]: 0=

-uc$ +Dax,c$ s2c

+Dr,+;;

(1)

Thevariablez denotes the distance between the injection plane and the measuring plane which is positive in the upward direction; I refers to the radial distance from the vessel axis. Equation (1) was suggested by several authors [l, 21, 221 as a description of gas mixing in turbulent single phase flow in a tube. In the comprehensive work by Klinkenberg et al. [l] several analytical solutions are given for different special cases of eqn. (1). First calculations in the course of the present work with the solution of eqn. (1) showed that the influence of the axial dispersion term on the calculated gas concentration profiles in the core zone of the CFB is negligible in comparison to the term which describes convective transport in the axial direction. Although

sonable values, its influence on the shape of the calculated radial concentration profiles was found to be insignificant. Without loss in accuracy eqn. (1) may therefore be reduced in the present application to:

o=_u6c,+D “sz with the boundary conditions: z-

--03

c,=o

!!?!

r=R*

Sr

=O

and together with the condition that there is a source of constant strength in the origin. The analytical solution derived by Klinkenberg et al. [l] is:

(5) with l42R* *

Per,.=

(6) (7)

z*=

4

(8)

R*

J1(u,)=O

(9)

The average concentration calculated from:

C,, in the present case is

R* G=

2mC, dr

s2 s 0

(10)

In the present application it is important to fulfill the boundary condition, eqn. (4), because this means that no transfer of tracer gas into the wall zone is allowed. This is, strictly considered, not practicable because the cylindrical interface at r =R* does not consist of an impenetrable barrier. The solution (5) can therefore only be applied to the measurements if the front of the tracer gas, which has been introduced on the axis of the riser, has not yet reached the imaginary wall at r=R*. This condition must be accounted for during the measurements. The mixing parameter Pe,. is determined by fitting the solution (5) to the measured gas concentration profiles by the method of least squares. Experimental

Fig. 1. The core-annulus

model for gas mixing in the CFB.

The experiments were carried out in a pilot scale CFB facility with a riser of 0.4 m diameter and a total

295

height of 9 m. A detailed description of the unit and of the measuring techniques for determination of gas velocity, solids mass flow, local solids concentration and local solids velocity is given in [2]. Sand with a surface mean diameter d,, of 0.13 mm and a density of 2 600 kg rnp3 was used as the bed material. The gas velocity at the point of minimum fluidization was measured to U ,r=O.O2 m s-l. The fluidizing velocity was varied between 3 and 6.2 m s-’ whereas the solids rate was varied between 0 and 70 kg me2 s-l. COZ, which was used as the tracer gas, was taken from a tank. The volumetric flow of CO2 was between 0.00133 and 0.00274 m3 s-l (1 bar, 20 “C). The experimental study of radial gas mixing was performed by injection of the tracer on the axis of the riser. Radial profiles of the CO, concentration were measured downstream of the injection point which was located at a height of 5.53 m above the gas distributor. Gas sampling was achieved using sampling probes which could be moved across the diameter of the bed. At the tips of the probes ceramic filter elements were fixed to prevent solids from entering the sampling line. The concentration of tracer gas was measured by an infrared analyzer (Rosemount Binos). The experimental set-up is shown in Fig. 2.

6

7

6

point of injection+

tracer

5 u = 3.1 m/s, Gs = 31 kg/m2s h,m

____t__~

4

u = 3.1 m/s, G, = 70 kg/m25

0

0.05

0.1

0.15

0.2

0.25

%-

Fig. 3. Axial profiles of mean solids fraction at different operating conditions of the CFB.

“‘a”

25 20 15

Results and discussion Before starting gas mixing investigations, axial profiles of mean solids fraction were calculated from pressure drop measurements. Figure 3 shows that, for the range of operating conditions used in the present investigation, the point of tracer injection always remains in the upper dilute zone of the CFB. Next, gas mixing in the empty column was investigated. In order to check the influence of the velocity of tracer gas u0 at the outlet of the injection tube, concentration profiles were measured at a constant gas velocity in the column of 3 m s-’ for different values of u,,. Figure

-1

-0.5

0

1

0.5

r/R Fig. 4. Tracer gas concentration profile in the empty riser (u = 3 m s-l, u,=3.6 m s-’ t = 0.58 m), points: measurement, solid line calculated from e&r. (5) with R*=R.

0

1

2

3

4

5

6

7

8

uo,m/s

Fig. 5. Influence of the tracer gas velocity u0 at the outlet of the injection tube on the Peclet number Pe,, in single-phase flow (empty riser, R* =R).

Fig. 2. Experimental mixing.

set-up for the investigation

of radial gas

4 shows as an example the measured concentration profile and the analytical solution, eqn. (5). Figure 5 shows the relationship between ZQ,and the Peclet number which has been calculated from the measured profiles. It can be seen that there is no significant influence on

296

the Peclet number for outlet velocities u0 up to 3.6 m which represents the maximum velocity u,,, in the centerline of the fully developed turbulent single phase flow for a superficial gas velocity u of 3 m s-l and a Reynolds number of 80 000 [23]. For higher values of u0 the Peclet number decreases sharply. The resulting increase of the radial dispersion coefficient D, may be explained by the increasing momentum of the jet which forms at the outlet of the injection tube for velocities uo > %I,,. In order to achieve a nearly isokinetic introduction of the tracer gas, an inlet gas velocity u. = u,,, was used in the empty tube experiments whereas the velocity was adjusted to uo=u, in the case of the measurements in the CFB. Values for U, may be found in Table 1. In order to check the measuring techniques, gas mixing in the empty tube was studied first. Measurements were taken at different levels between 0.58 and 1.46 m downstream of the injection point. The resulting Pe,. numbers are listed in Table 1 and are plotted in Fig. 6. The Peclet numbers range from 430 to 540. The resulting mean value of 482 is in good agreement with other authors’ results [24]. After testing the installation in the empty tube, the gas mixing in the core zone of the circulating fluidized bed was investigated. An evaluation of the tracer gas concentration profiles requires knowledge of the radius R* of the core zone. In a previous investigation [2] local net mass fluxes G,* were calculated from local

measurements of solids concentrations and velocities. In Fig. 7 plots of G,* W. the radial distance r from the vessel centerline are shown which have been calculated from measurements with the quartz sand used here. It can be seen that the boundary between the net upward flowing suspension in the core zone and the net downward flow of solids in the annular zone is located at roughly R* =0.85 R. Although there is some scatter in the numerical values of R *, no significant tendency of variation with gas velocity u and solids rate G, was found. The value of R*IR = 0.85 is in agreement with experimental findings of other authors

S -l,

TABLE

1. Peclet numbers

of radial dispersion

Experiment no. empty tube (R*=R)

1 2 3 4

[41Figure 8 shows as an example the tracer gas profile measured in the core zone of the CFB at a gas velocity u of 3 m s-l and at a solids rate G, of 70 kg me2 s-l. The measurements are compared with the model calculations according to eqns. (1) and (2), respectively. In the former case the Peclet number Pe,. and the ratio p =D,, JD, c were optimized simultaneously to fit the measurements. The best fit is in this case obtained with Pe,. =486 and p = 100. In the latter case, the solution, eqn. (5), of eqn. (2) leads to a best fit for Pe,.=473. Compared to other inaccuracies, the difference between these two numerical values of the Peclet number was not felt to be significant. In all the measurements in the course of the present work, the value of the parameter p varied between 0 and 300 with a mean value around 100. Since the relative

in single phase flow and in the CFB U

L

(m s-r)

(m)

3 3 3 3

0.58 0.73 1.05 1.46

0 0 0 0

3 3 3 3 3 3 3 3 3.1 3 3 3 3 3.5 4 5 6.2

0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.73 0.9 1.05 1.46 0.58 0.58 0.58 0.58

4 11 15 28 38 42 56 70 51 31 30 32 33 29 33 32 28

UC

:g

me2 s-‘)

512.2 427 447.6 541.8 482

mean Pe,, circulating fluidized bed (R * = 0.85 R)

mean Pe,,

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 (without no. 17)

Per, c

(m s-l)

3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.6 3.5 3.5 3.5 3.5 4.1 4.6 5.8 7.2

448.8 393.4 446.4 488.7 532.2 490.7 502.8 472.5 465.6 451 397.6 427.3 282.4 478 464.4 554 421.8 465

297

19, 0

0.2

0.4

0.6

0.8

1

1.2

1.4

0

1.6

z ,m

Fig. 6. Influence of the distancez between injection and measuring height on the Peclet number Pe,, (0) in the core of the CFB (28-32 kg m-’ s-‘) (R* =0.85 R) and (0) in single-phase flow (R* =R), respectively.

Fig. 7. Radial profiles of local specific net mass flow of solids calculated from local measurements of solids concentrations and velocities.

"'a, 201

5

-1

-0.5

0 r/R+

0.5

1

Fig. 8. Tracer gas concentration profile in the core zone of the CFB (U =3 m s-l, G,=70 kg m-’ s-i, z=O.58 m), points: measurement, solid line calculated from eqn. (5), dashed line is solution of eqn. (1).

deviation in the Peclet numbers obtained from eqns. (1) and (2), respectively, was always less than lo%, it was decided to neglect the axial dispersion effect in comparison to convective transport in the axial direction, and the analytical solution, eqn. (5), of the simplified model, eqn. (2), was taken as the basis of all further calculations. The model of Klinkenberg et al. [l] for the single phase flow seems also to be applicable to the core zone of the CFB. This is confirmed by the plot of Fig. 9 where all the tracer gas profiles, which were measured under conditions of the CFB, are plotted in a dimen-

0

1

C(r,) =

2

3

C0/2i

’ 4

r/r m

Fig. 9. Dimensionless tracer gas profile in the core zone of the CFB. Comparison of measuring points with the theoretical solution for single-phase flow, eqn. (11).

sionless form. and r,,, is that concentration represents the

Co denotes the concentration at r=O, distance from the centerline where the has fallen to half of C,,. The solid line equation:

CJC, = 0.5(“‘+

(11)

which is also given by Klinkenberg et al. [l] to describe the shape of tracer gas concentration profiles. Equation (11) is seen to give a good description of the measurements. The operating conditions of all measurements in the CFB and the Pe,. values resulting from the application of eqn. (5) are also listed in Table 1. In a next step, tracer gas profiles were measured in the CFB at different distances z downstream of the injection point. In Fig. 6 the Peclet numbers under CFB conditions are compared with those calculated from the measurements in the empty tube. There is no significant difference between single-phase flow and two-phase conditions up to values z of 1.05 m. For z= 1.46 m the measurement in the empty tube yields roughly the same value as before, whereas in the CFB the Peclet number is much lower, indicating a higher mixing coefficient. An explanation of this result is given in Fig. 10 where the measured profiles at different values of z are compared with the calculations (solid lines) using the mean value of Pe, c = 465 (cfi Table 1). The profiles measured at distances of 0.58, 0.73 and 1.05 m, respectively, are well described by the calculated profiles, whereas a significant difference is observed in the case of the largest distance, 1.46 m. In this latter case the measured concentration profile is much flatter than the calculated one, the reason being that in this case the tracer front has already reached the boundary of the core zone at r= R * and as a consequence tracer material has been transferred into the wall region. The transfer of material across this boundary violates the boundary condition, eqn. (4), which means that the model underlying both eqns. (2) and (1) is no longer applicable. As a consequence, the Peclet number resulting from the ap-

298

c/ca”

20

20 z = 0,73

“cZi” ..____~

mp--I

m

15

z = I,46

m

15

o-=

-

-1

-0,5

0 . r/R

Fig. 10. Comparison

:..~-dcEx.

0,5

of measured

1

concentration

-1

Pe, C ’

I

500

i -7 LA...

._._ a

400 300 ;:l;

-

,

;

1:-_.~-/-! A

0

1

2

3 4 u ,m/s

5

I 6

0 r/R’

profiles with calculations

plication of eqn. (5) to this measurement (run no. 17) has not been considered in calculating the mean value of Pe, C in Table 1. Figure 11 shows the influence of the superficial gas velocity u on the Peclet number Pe, c for gas velocities between 3 and 6.2 m s-l. The solids rate G, was kept at a constant value of 30 kg m-2 SC’. From this plot no significant variation of Pe,. with u can be seen. The result that the Peclet number is independent of the gas velocity is in agreement with the findings for horizontal gas mixing in turbulent single phase flow of other authors (c$ [24]). The relationship between the Peclet number and the solids rate G, is depicted in Fig. 12. In this test series the gas velocity u was set equal to 3 m s-l. Although there is some scatter, no significant influence of the

6OOi

.1 -0,5

7

Fig. 11. Influence of gas velocity II on the Peclet number Pe,, in the core zone of the CFB (G,=28...32 kg m-* s-*,2=0.58... 1.05 m).

0

O,5

with Per,,=465

IO

20

30

1

(u=3 m s-l,

40

G, ,kg/m

250

60

G,=28...32

70

kg m-’ s-l).

80

s

Fig. 12. Influence of solids circulation rate G, on the Peclet number Pe, E in the core zone of the CFB (u=3 m s-‘, .7=0.58...1.05 m).

solids rate G, on the Peclet number up to a G, value of 70 kg mm2 s-l can be seen. This result is quite unexpected because, according to previous investigators [17, 181, a change in the solids rate and therefore in the solids concentration in the two-phase flow in the core zone should have an influence on the gas mixing characteristics. In order to find a physical explanation for this behaviour, local probability density distributions of the solids concentration were measured. Figure 13 shows an increase of the local mean value of the solids concentrations in the core zone with increasing G,. An extrapolation of the measurements shown in Fig. 13 to higher G, values of e.g. 70 kg me2 s-l would lead to relatively high values of c,, =. This strong dependence of c,, C on G, is in sharp contrast to the complete lack of influence of G, on the Peclet number.

299

d,Jl, = 0.00013 m/0.034 m = 0.0038

0

5

10

15

20

25

30

35

40

Gs, kg/m%

Fig. 13. Influence of the solids rate G, on the mean solids concentration in the core zone of the CFB (fiber optical measurements at h=6.11 m, r=O, u=3 m s-r).

Fig. 14. Percentage change in turbulent intensity at the transition from single- to two-phase flow as a function of length scale ratio d,/l, (after Gore and Crowe [25]). The arrow indicates the d& value of the present CFB system.

In the second step the role of particle size was investigated. In the fluid mechanics literature, measurements of different authors concerning the influence of the size of the particles on the turbulence structure are available. Gore and Crowe [25] have recently given a comprehensive representation of numerous authors’ data. In their representation of the change in turbulent intensity as a function of the ratio of particle size d, to the size 1, of the turbulent eddies, they used results obtained by Hutchinson et al. [26] on the relationship between Z, and the tube radius R. These latter authors found that for single-phase flow in pipes the ratio 1,/R was approximately constant for varying values of r/R and a numerical mean value of ZJR=O.22 for Reynolds numbers between 5X lo4 and 5X 105 was obtained. A numerical value of ZJR = 0.2 was accordingly used by Gore and Crowe [26] in their re-evaluation of other authors’ data. The result of their investigation is represented in Fig. 14, where the relationship between the ratio d,/l, and the percentage change in turbulent intensity at the transition from single- to two-phase flow is plotted. In the present application I, is obtained from: Z,=O.2R*=O.2~0.17

m=0.034 m

It follows with dp,s= 0.13 mm:

For dp/Ze= 0.004 it is seen from Fig. 14 that the turbulent intensity in the case of the circulating fluidized bed should only be slightly reduced when compared to the single-phase flow case. This is in complete agreement with the findings of the present tracer gas experiments. It should be noted that on the basis of Gore and Crowe’s results the gas mixing characteristics in circulating fluidized beds should be scale-dependent. Large scale installations with big bed diameters will always be characterized by small values of d,/l,, which will lead to a minor reduction in turbulent intensity, i.e. the radial mixing characteristics will be more or less the same as in single-phase gas flow. Caution should be exercised, however, with the use of small-scale CFBs. In the case of laboratory columns of some centimeters in diameter, the ratio dp/Ze might exceed the border line indicated in Fig. 14 and radial gas mixing in the CFB could then be much better than in single-phase flow. The mean value of Fe, c = 465 which has been derived from the measurements has been used to predict radial gas mixing in a large scale CFB with d,= 8 m. Taking simply the same value of 0.85, which has been obtained in our measurements for the ratio R*/R, the diameter of the core zone turns out to be 6.8 m in this case. For the case of the injection of the reactant at the center of the bed, concentration profiles have been calculated at 8, 15 and 20 m, respectively, downstream of the injection point. Figure 15 shows that it takes long distances to flatten the profiles. Even at 20 m downstream of injection level the local concentration at the center is still ten times higher than the mean concentration. This simulation shows that the degree of gas mixing in the upper core region of a CFB is not high. It is therefore important in practice to obtain efficient distribution of a reactant at the injection level by proper design of the feeding system.

c/c,,

I

25 20

,z -

= 8 m

+pz;,5m

15

( z=20m+

I-

IO , ,, !

5 30> 0-1

-0.5

~

‘,) ‘_\ __

0 r/R'

0.5

1

Fig. 15. Calculation of radial gas mizing on an industrial scale CFB (d,= 8 m, u =3 m s-r, calculation with Pe,,=465, injection of the reactant at the center of the bed).

300

Conclusions

cc

Gas mixing experiments were carried out in the upper dilute zone of a pilot scale CFB facility at the Technical University Hamburg-Harburg. CO, as a tracer was injected at the center of the bed and tracer gas concentration profiles were measured at several distances downstream of the injection point. Based on the assumption of the core-annulus structure, a model for the gas mixing in turbulent single-phase flow was applied to the core zone of the CFB. It was shown that the measurements were well described by this model. The present investigation showed further that it is important to include the core-annulus structure into the interpretation of the data, because the mixing behaviour in the wall zone is quite different from that of the core zone. Evaluation of the mixing parameter Pe,. from measurements at gas velocities u between 3 and 6.2 m s-l yielded no significant variation of Per,. with u. The result that the Peclet number Pe,. in the core zone of the CFB is independent of the gas velocity is in agreement with the gas mixing behaviour in turbulent single-phase flow. No effect of the solids rate G, on the Peclet number was found for solids rates in the range between 0 and 70 kg me2 s-l. In order to find a physical explanation, local solids concentrations in the CFB and fluid/particle interactions were considered. The solids concentration in the core zone turns out to be too low and the size of particles to be too small to cause a significant change in the turbulent intensity of the gas flow which is responsible for the dispersion of gas. The mean value of Pe, .=465 was used to calculate concentration profiles of a gaseous reactant which is injected at the centerline of an industrial scale CFB. The calculations show that it takes long distances to flatten the profiles.

CZ.”

co

S--l

h 4 P Pe, C r r* ml R R* u

U max

uo

z

Z*

List of symbols ith positive root for which the Bessel function J,(u,) is zero, local solids volume concentration, cross-section average solids volume concentration, -

height above gas distributor, m characteristic size of turbulent eddies, m ratio of D,,. to D, C, Peclet number defined by eqn. (6), distance from the vessel centerline, m dimensionless distance from the centerline defined by eqn. (7), distance from the centerline where the concentration of tracer gas is equal to C,/2, m radius of fluidization column, m radius of core zone, m superficial gas velocity based on the empty column, m s-l gas velocity based on the empty cross-section of the annular zone, m s-l gas velocity based on the empty cross-section of the core zone, m s-l maximum gas velocity on the centerline of the fully developed, turbulent single-phase flow, m S-l

U mf

Acknowledgement This work was supported by the Deutsche Forschungsgemeinschaft within the Sonderforschungsbereich 238. The authors would like to thank Britta Wrage for her valuable help in the course of experimental work.

volume concentration of tracer gas in the core zone, km01 mm3 average volume concentration of tracer gas defined by eqn. (12), km01 me3 volume concentration of tracer gas at r=O, kmol mm3 diameter of particle, m surface mean diameter of particles, m diameter of fluidization column, m coefficient of axial gas dispersion, m2 s-l coefficient of radial gas dispersion, m2 s-l specific solids circulation rate, kg m-’ s-l local specific net mass flow of solids, kg m-*

P

superficial gas velocity at minimum fluidization, m s-’ tracer gas velocity at the outlet of the injection tube, m s-l axial distance between point of tracer injection and measuring plane, m dimensionless distance defined by eqn. (8), mass transfer coefficient, m s-’

indices

a C

annulus core

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