hydrodynamics of liquid fluidized beds including the distributor region

Chemical Engineering Science, Vol. 47, NO. 15/16, pp. 4155-4166.1992. 000-2509/92 $5.00 + 0.00 Printed in Great Britain. Q 1992 Pergamon Press Ltd

HYDRODYNAMICS OF LIQUID FLUIDIZED BEDS INCLUDING THE DISTRIBUTOR REGION

MOHAMMAD ASIF, NICOLAS KALOGERAKIS and LEO A. BEHIE+ Pharmaceutical Production Research Facility, Faculty of Engineering, The University of Calgary,

Calgary, Alberta, Canada T2N lN4

(First received 18 February 1991; accepted for publication in revised form 17 January 1992)

Abstract-An important application of liquid-solid fluidized beds has been developed recently in bio- technology, namely, immobilized biocatalyst bioresctors. For this application, experiments have been carried out to investigate the effect of distributor-induced flow nonuniformities on the hydrodynamics of liquid fluidized beds. The influence of important variables associated with the design of distributors has been studied including the effect of the density of the solid particles on the distributor region flow behavior. In the presence of the low-density particles (pa = 1.61 g/cm3). the influence of the distributor region was quite significant whereas, for high-density particles @, = 2.46 g/cm3), it was negligible. A new model has been proposed which accounts for the stirring effects of the high-velocity orifice jets in the distributor _ region.

INTRODUCTION

Proper characterization of the hydrodynamics is an important aspect of chemical and biochemical reactor design and scaleup. Realistic models are required to correctly predict the level of conversion in fluidized- bed reaction systems as pointed out by Oertzen et al. (1989) and Ching and Ho (1984).

There are two factors which primarily influence the hydrodynamics of liquid-solid fluidized beds. One is the interaction between the solid and the liquid phase, and the other is the distributor effects. The effect of liquid-solid interactions is generally quantified in terms of the axial liquid dispersion coefficient which provides information about the degree of fluid mixing existing in the bed. Residence time distribution studies have proved to be an indispensable tool in this respect whereby a tracer is injected at some location in the bed and its concentration is monitored at a point downstream. The concentration distribution data is then processed to extract quantitative information about the dispersion characteristics of the bed. A significant amount of information is available in the literature on this subject (Chung and Wen, 1968; Krishnaswamy et al., 1978; Kikuchi et al., 1984, Tang and Fan, 1990).

The distributor effects can cause severe distortions in the fluidized-bed hydrodynamics which ultimately leads to incorrect estimation of the design parameters of the bed. Here, we are not interested in large-scale convective circulation patterns which are known to develop in the liquid fluidized beds when the pressure drop across the distributor drop is small compared with that across the bed (Agarwal et al., 1980). The issue of the present investigation is the effect of distributor-induced flow nonuniformities itself and its influence on the bed flow dynamics. Most studies on fluidized-bed hydrodynamics have recommended us-

‘Author to whom correspondence should be addressed.

ing a porous material as a distributor in order to eliminate distributor effects. Nevertheless, it is evident from these studies that these distributor designs are often difficult to realize in practice and could lead to serious operational problems. Moreover, these tech- niques do not necessarily guarantee elimination of distributor effects. For example, Carlos and Richardson (1968) found that the mean axial velocities of glass particles fluidized with dimethyl phthalate in the vicinity of a well-designed distributor composed of porous materials is much higher than the rest of the bed. In fact, an appropriate picture of distributor effects and the basic understanding of their nature and cause is still lacking in the literature.

In order to be useful, it is essential for any laborat- ory-scale study of distributor effects to examine a distributor and its pertinent design parameters which could easily be scaled for industrial application. A perforated-plate distributor is very suitable for such purposes, since it is inexpensive but strong enough to withstand weights usually encountered in the industry. Moreover, the pressure drop, an important variable of distributor design, can be easily controlled by varying the fractional open area or the hole density. Furthermore, distributors made of perforated plates are less susceptible to clogging.

The present study addresses the following key issues regarding distributor effects in a liquid fluidized bed. Firstly, how the various parameters associated with the design of the distributor influence the overall hydrodynamics of the fluidized bed. Secondly, the key effect of the density of the solid phase present in the bed on the hydrodynamics in the distributor region is examined. In addition, the cause of the nonidealities that makes the distributor region behave differently from the rest of the bed is characterized and a simple, but realistic model is proposed.

To this end, the following strategy was adopted. Two different sets of residence time distribution experiments were carried out. In one case, a tracer pulse

4155

4156 MOHAMMAD ASIF et al.

was injected away from the distributor so that the response was free from any kind of flow nonuniformities existing in the distributor region and reflected only the effect of the liquid-solid interaction in the bed itself (i.e. bed region). In another set of experiments, the tracer was injected immediately above the distributor. The response recorded for this latter experiment provided information about the combined effect of the distributor and the bed (i.e. distributor/bed). The response data obtained from these two experiments were processed in conjunction with appropriate models to extract the quantitative information about the effect of the distributor on the bed hydrodynamics. Experiments were designed in such a way that the distance between the tracer injection and the tracer measurement points were the same for both cases. This precaution was taken in order to avoid the influence of the time-height-dependent hydrodynamic parameters on the results, ample evidence of which is found in the work of Han et al. (1985) for packed beds and Asif (1991) for fluidized beds.

EXPERIMENTAL APPARATUS

The fluidized bed was a Plexiglas column of 7.68 cm diameter. A distributor was placed at the bottom of the bed which was preceded by a calming section. The physical properties of solid particles present in the fluid&d bed are given in Table 1. The main features of the experimental apparatus employed in the present study are described below.

Tracer injection system A plane source of tracer injection was achieved

using a 3 mm OD steel tube 10 cm long and sealed at one end, and connected to the tracer tank through the other end. On both sides of the tube, ten equidistant 1.0 mm holes were drilled across the axis in a length of 7.5 cm. The tube was placed inside the ‘fluidized bed along the diameter of the column in such a way that the tracer ejecting from the holes of the tube under pressure provided a uniform-plane source of tracer injection. This configuration was found satisfactory as reported by Tang and Fan (1990). The tracer, methylene blue, was kept air pressurized in a one-liter tank. Time and duration of tracer injection was controlled with an onoff valve operated with a Taurus computer data acquisition and contrcl system inter- faced with an IBM personal computer. For the present experiments, the pulse time was kept around 120’ms. The amount of tracer injected was controlled with the help of the air pressure in the

tracer tank (32&200 kPa) so that its concentration stayed within the measurement range of the measuring equipment.

It is worthwhile to point out here that the tracer injection system employed by the present study ensured a uniform injection across the cross-section of the column. Furthermore, due to the geometry of the experimental’system, any radial nonuniformity, even if present in the tracer injection will not influence the results obtained here. This is shown in Appendix C.

Tracer measurement system The concentration measurement system consisted

of a Brinkmann probe calorimeter (Model 700). It was based on fiber optic technology to measure the color of the solution and utilized a fiber-light guide that transmitted phase shifted, AC modulated light to the probe tip. One-half of the fiber-optic bundles transmitted light into the test solution. The other half returned the light to the instrument where it passed through an interference filter prior to impinging on the silicon detector cell. The light was returned by the probe tip, which had a mirror set at a fixed distance from the end of the fiber optics. The effective path length in the fluid was twice that distance, since the light travelled through the fluid two times-first to the mirror, and then back to the fiber-optic assembly. A glass window prevented fluid from coming into contact with the fiber optics, assuring that only the probe tip comes into contact with the test solution. The electronic circuitry of the instrument was designed in such a way that the extraneous or ambient light can not affect its readings.

The output of the instrument which was in terms of the absorbance, A, was directly proportional to the concentration of the solution. It exhibited very good linearity in the range of absorbance A = O-1.0. An- other useful feature of the probe calorimeter lies in its fast response to any change in the color of the solution. The total response time of the measuring and the recording instrument was found less than 120 ms. This was important since we were interested in recording time vs concentration data. Any delay caused by the measuring equipment in the concentration recording would lead to incorrect prediction of the parameters. These two features made this technique more attractive than the commonly used conductivity measurement technique where a salt solution is used as a tracer. In order to avoid the interference of solid particles with the probe measurement, the light path (10 mm long) of the probe was covered with a thin

Table 1. Physical properties of the fluidized-bed particles

Shape

Polypropylene (PP) Cylindrical 3.0 3.0 0.87 2.6 1.61 Glass (G2) Spherical 2.0 - 1.00 2.0 2.46 Glass (G3) Spherical 3.0 1.00 3.0 2.46

Hydrodynamics of liquid fluidized

sheet of Plexiglas on which five cuts of 1 mm width were made parallel to the direction of flow to only allow the free passage of the liquid through the light path. The analog response of the calorimeter was digitized with the Taurus computer interface and stored in the computer.

Inlet liquid-Jlow conjiguration The fluidizing medium was tap water pumped to

the column from a storage tank. The flow rate was controlled with the help of two calibrated rotameters. Before entering the column, the water was passed through a large capacity tank (0.3 m3) to dampen any flow fluctuations caused by the pump. The inlet water pipe was connected axially to a calming section and was capped with a small sealed Plexiglas cylinder with small holes around the periphery near the top in order to achieve uniform distribution of the water across the cross section of the calming section. The calming section, below the distributor, was a Plexiglas column of 12.7 cm diameter and 25 cm in length, packed with 3 mm glass beads.

Distributor details The distributors were perforated plates of Plexiglas.

As shown in Table 2, the distributor fractional open area, For, was 4% for most cases in order to have a good distribution without causing an excessively high pressure drops and to eliminate large-scale convective circulation patterns in the bed, a phenomenon associated with low pressure drop distributors. Another important distributor design parameter studied was the distributor hole density, N,,, or the number of holes. The value of N,, was varied over a wide range of 0.02 ‘to 1.33 holes/cm2 which caused the number of distributor holes to vary from 1 to 60. Selecting these two parameters, i.e. N,, and F,,, automatically fixed the diameter of the holes as evident from eq. (2). A configuration of square-edged holes, arranged equi-

Table 2. Specification of design parameters for five distributors

Holes (ho$&) F, 00,

(cm)

1 0.02 0.04 1.52 4 0.09 0.04 0.76

20 0.44 0.04 0.36 60 1.33 0.04 0.20

1 0.02 0.01 0.76

beds including the distributor region 4157

laterally, was chosen to keep the number of holes per unit area uniform throughout the distributor. This was to ensure uniform distribution of water across the cross section of the column. The obvious exception was the distributor with 0.22 holes/cm’ in which a single hole was drilled at the center of the plate. Since fractional open area was an important parameter affecting the orifice liquid velocity and the distributor pressure drop, one set of experiments was carried out with a single hole (0.02-N,,) distributor of 0.01 fractional open area. A very low N,, distributor was selected in this case study an extreme case of distributor effects. Other variables of design interest, e.g. orifice velocity and distributor pressure drop, could be varied by controlling the liquid flow in the column as shown by eqs (3) and (4). The range of these variables used here for different beds and distributors is shown in Tables 2 and 3.

Hole density, N,, = No. of holes, n

Area of column, A * (1)

Orifice diameter, D,, = .

Orifice velocity, U,, = F ( >

. 0,

(3)

Distributor pressue drop, AP,, = 0.5~~ ( >

% 2

(4) d

where, C, is the orifice coefficient. Its value was obtained from Kunii and Levenspiel(l969). It is worthwhile to point out here that the relevant parameter for the scaleup of distributors is the hole density not the number of holes.

Experimental methodology In the present work, experiments were carried out

to study the following three cases.

Case 1: combined distributor/bed response. These experiments were designed to study the effect of the distributor region on the bed hydrodynamics. As shown in Fig. 1 (a), this was achieved by keeping the tracer injection immediately above the distributor so that the response recorded in this case included the effects of the bed region and the distributor. The location of the tracer detection point was always kept at least 15 cm below the top of the fluidized bed to keep the measurement free from boundary effects. Other details of the experiments are given in Table 4.

Table 3. Range of important distributor variables for the fluidized beds

Bed

PP E

G3

% u.l, F0, (cm/s) @u/s) &$ (Z)

0.04 2.3-9.00 57.0-225.0 0.40-7.0 4.20 0.01 0.04 2.3-9.00 2.3-12.1 230.&900.0 57.0-303.0 0.4-l 7.&l 3.0 12.0 10.3 4.20

0.04 4.0-16.0 100.0-400.0 1.4-22.2 10.3


I 6

(a) (b) w-1

Fig. 1. Schematic diagram of the setup for three experimental studies: (1) column, (2) distributor, (3) calming section, (4) tracer measurement point, (5) tracer injection point,

(6) water inlet, (7) inlet section.

Case 2: bed-region response. These experiments

were carried out to study the hydrodynamics of the bed region of the fluidized bed in the absence of any kine of influence of the distributor. The setup employed for the purpose is shown in Fig. l(b). The tracer injection, in this case, was 25 cm away from a high hole-density distributor (1.33 - N,,) in order to keep it free from distributor effects. A high hole- density distributor was always found to have negligible distributor effects. The injection at z,, = 25 cm further ensured the elimination of distributor effects in the tracer input.

Case 3: effect of the location of tracer measurement point. In these experiments, the validity of the dispersion model to account for distributor effects, was examined. The setup is shown in Fig. l(c). The configuration was exactly the same as the case 1 except that the tracer measurement point was only 70 cm away from the tracer injection as opposed to 105 cm in case 1.

DISPERSION MODEL

The concentration distribution in the fluid phase of a liquid fluidized bed containing inert, nonadsorbing and nonporous solid particles is given by the following dispersion equation:

(5)

where, D, is the dispersion coefficient and Vi is the liquid interstitial velocity.

The location of the tracer injection and measurement points has important implications on the choice of the boundary conditions to solve eq. (5). This has been described in detail by Levenspiel (1984).

For the present experimental setup, the tracer measurement point was always at least 15 cm below the exit boundary of the bed, which justifies the assumption of infinitely long bed, i.e.

C and X/& + 0 as z--,oc). (6)

Similarly, for the experiments where the injection point was located away from the bed inlet, the following boundary conditions will be applicable:

C and dC/az + 0 as Z-+-co. (7)

In this case of infinite media, the solution of eq. (5) subject to the boundary conditions (6) and (7) and a pulse tracer injection is given by

vi c(t) = (&D,t)0.5 exp

(L - uity - 4D,t 1 (8)

where, L is the distance between tracer injection and measurement point.

Table 4. Summary of experiments

Particles Case (2) =I

(cm) (r& H.

(cm) Distributor (N‘W F0,)

PP PP PP PP PP

PP

PP

G2, G3 G2, G3

G2, G3

1 0 105 105 120 (0.02, 0.04) I 0 105 105 120 (0.09, 0.04) 1 0 105 105 120 (0.44, 0.04) 1 0 105 105 120 (1.33,0.04) 1 0 105 105 120 (0.02, 0.01)

2 25 130 105 145 (1.33, 0.04)

3 0 70 70 120 (0.02, 0.04)

1 0 105 105 120 (0.02, 0.04) 1 0 105 105 120 (1.33, 0.04)

2 25 130 105 145 (1.33, 0.04)

Hydrodynamics of liquid fluidized beds including the distributor region 4159

For the case where the tracer was injected immediately above the distributor, the physical situation represented a closed boundary condition which can be best described by the following equation (Nauman and BaulIham, 1983):

-D,, E ( >

+ UiC = U,s(t) at 2 = 0. (9)

The above equation assumes a finite degree of dispersion in the bed and zero dispersion in the entering liquid stream. This is, therefore, compatible with the physical situation considered here.

The solution of equation (5) subject to boundary conditions (6) and (9) is given by

PARAMETER ESTIMATION

The evaluation of the axial dispersion coefficient from the raw data involved two steps. First, the experimental measurements from seven replicate experiments were averaged, normalized and the standard deviation was computed for each measurement point. The latter is needed for the statistically correct determination of D, from the experimental data.

In the second step, D,, is determined by minimizing a suitable objective function. By considering not only the error in the measured concentration, but also in the measured interstitial velocity, the most appropriate objective function to be minimized from a statist- ical point of view (error-in-variable method, Reilly and Patino-Leal, 1981) is given by

(11)

where, cr, is the standard error in the measurement of U,, and a, is the standard error in the average concentration measurement at time t,. A value of 0.05Ui, was used for o, representing the uncertainty in the measurement of the interstitial velocity. a, is given by (cr,,.fi), where c$ is the variance computed by the seven replicate experiments. Ui, is the experimentally measured value of interstitial velocity and Vi is its fitted value. The objective function was minimized over D, and Vi simultaneously by solving the resulting normal equations

= 0 and = 0. (12)

The maximum deviation in the estimated true values of the liquid interstitial velocities never ex- ceeded 10% of the measured value. Moreover, Rangaiah and Krishnaswamy (1990) have recently

shown that better results are obtained by not treating U, as a constant in the minimization procedure. This is due to the fact that D, is very sensitive to U,. This approach was also followed by Tang and Fan (1990) and Goebel et al. (1986). In this paper, we also search for D, and U,. However, we minimize the statistically correct objective function based on the error-in-variable method.

RESULTS AND DISCUSSION

Low-density polypropylene particles Figure 2 shows the residence time distribution

curves obtained for cases 1 and 2 considered here (i.e. bed region and distributor/bed experiments). As explained previously, the response obtained for the combined distributor/bed region includes the effect of both the distributor and the bed. On the other hand, the response obtained for the bed region shows the effect of the dispersion in the bed only and is free from distributor effects. It is evident from the figure that the response of the 1.33 - N,, distributor closely follows the actual bed response. This is, however, not the case for the low hole-density distributor as is evident from diminished peaks and long tails of the response curves. As the distributor hole density, N,,, decreases, the distributor-induced disturbances leads to a larger deviation in the distributor/bed response compared to the bed-region response. In order to quantify the degree of mixing, these response curves were used to evaluate the axial liquid dispersion coefficient. For the distributor/bed experiments (case l), an apparent axial dispersion coefficient, D,, was computed. On the other hand, the dispersion coefficient obtained for the bed-region experiments (case 2) gave the degree of dispersion actually existing in the bed and was, therefore, called the actual dispersion coefficient, D,.

The effect of liquid superficial velocity, U,, on D, is shown in Fig. 3. The bed-region values of D, show close agreement with the following correlation of Tang and Fan (1990):

D = 0 575U~~2a. aE * (13)

7

0 Bed Rsglon

Distributor/Bsd

N,, (holes/cm2)

. 0.02 A 0.09 . 0.44 0 1.33

4 6 0 10 12 14 16 t6 20 Time (8)

Fig. 2. Effect of distributors on the dynamic response of the liquid fluidized bed (lJ, = 9.0 cm/s, U, = 225.Ocm/s, AP,

= 7.0 kPa, F,, = 4%).


It is obvious from Fig. 3 that the effect of distributors becomes more pronounced as the liquid superficial velocity increases. -This can be attributed to stirring effects of high-momentum orifice jets caused by high orifice velocities existing at high We as evident from eq. (3). These jets have orifice velocities U,, = 225 cm/s when U, = 9 cm/s. The deviation from

the actual bed-region values is almost absent for high hole-density distributors. Apparently, small diameter holes reduce distributor effects since for distributors of constant fractional open area, F,,, increasing the hole density results in smaller D,, as evident from Table 2.

Figure 4 shows the effect of the distributor pressure drop on the ratio of the apparent dispersion coefficient to the actual dispersion coefficient. This ratio can also be interpreted as the degree of distortion introduced by the distributor on the flow dynamics of the bed. A high value of Da/D,, as shown for the l- hole and 4-hole distributors, indicates a greater distortion due to distributor effects. Moreover, for the same pressure drop, fluidized beds with high N,, distributors show a small deviation from the actual bed behavior free from distributor effects.

On the other hand, Fig. 5 seems to convey a different message. Here, the effect of D,, is compared for distributors of different fractional open areas. Though distributor effects decrease as D,, decreases for the same fractional open area, D, suddenly increases even for the small D, distributor when F,, is low. This

35

t

q Bed Region

Diatributor/Bsd 0

,E N 25 - or

0 0.02 5 . P l 0.09 0.44 0 0 .

cp 15- Al.33 o

.

5- . . m

Tang and-fan Correlation

-5 0 2 4 6 8

Liquid Supcrfirbl Valecity. U, (em/s)

Fig. 3. Effect of liquid velocity and distributor hole density on the apparent dispersion coefficient (F, = 4%).

6

5

4

$3

. 0 . l . 2

1

01 I 0.4

Di&i;::oor pressure drop, APd

10.0

&Pa)

0.4 1.0 10.0 100.; Distributor Pressure Drop. 6Pd (kPa)

Fig. 4. Effect of distributor pressure drop and hole density Fig. 6. Effect of distributor pressure drop and fractional on the apparent dispersion coefficient (F., = 4%). open area on the apparent dispersion coefficient.

anomaly is explained by Fig. 6 which shows that the distributor with the lower fractional open area causes a higher distributor pressure drop. Reducing F,,, from 4 to 1% results in a four-fold increase in the orifice liquid velocity which, in turn, reflects in a higher pressure drop across the distributor. This causes a larger portion of the bed to be aflected by the disturbances caused by the high-velocity liquid jets. In this case, the improvement expected by reducing the D,,

from 1.52 to 0.76 cm is greatly overwhelmed by the increased U,,. It is interesting to note here that like Froude number, distributor pressure drop has a similar quadratic dependence on the orifice velocity. The use of Froude number is quite common in the study of the distributor region in gas fluidized beds where it is found to strongly influence its height as shown by Yates et al. (1986) and Yang and Keairns (1979).

Validity of the dispersion model

From the foregoing discussion, it is evident that distributor effects lead to a greater degree of mixing in the region of the bed close to the distributor. This is very clearly reflected in the overprediction of D, by the dispersion model. However, a key assumption of the dispersion model is the constant dispersion coefficient which implies uniform mixing characteristics throughout the bed. The assumption does not hold for a case where regions of different mixing characteristics exist. To investigate this issue further, we carried

0 Bed Region

Distributor/Bad

Polypropyl~na PortiolcoI

A-~4% 0.76 .--.1X 0.76

10

Fig. 5. Effect of distributor fractional open area and orifice diameter on the bed hydrodynamics.

8 Polypropylene Particles

0 D or

1 s2 0.76 0.78

.

N or 0.02 0.09 0.02

0 0

. .

Hydrodynamics of liquid fluidized beds including the distributor region 4161

out residence time distribution (RTD) experiments by moving the tracer measurement point, zi , closer to the injection point, ze (case 3 z0 = 0 cm, zi = 70 cm). The response obtained in this case was again analyzed with the dispersion model. The assumption of constant D, in the dispersion model will require case 1 (z. = 0 cm, zr = 105 cm) and case 3 to provide identical values of the dispersion coefficient. This is, however, not observed here as evident from Fig. 7. Disper- sion coefficients obtained for case 3 are generally higher than the ones of case 1. This shows that the dispersion model is not valid for modeling the flow dynamics of fluidized beds with distributor effects. Moreover, these results also imply greater mixing in the distributor region due to distributor effects since moving zi close to z0 or to the distributor region gives higher D,.

It is also evident from Fig. 7 that, unlike the distributor/bed case, similar experiments for the bed region (ze = 25 cm, zi = 105 cm and to = 25 cm, zi = 70cm) show good agreement in the predicted

values of the dispersion coefficient, justifying the use of the dispersion model for representing the flow behavior of the bed region.

Development of two-region CSTR-dispersion (CD)

model

From the results of Fig. 7, the following conclusions can be drawn. First, there .exists two distinct regions with widely different mixing characteristics in the liquid fluidized bed when distributor effects are significant. Second, the region of the bed close to the distributor, i.e. the distributor region has a much greater degree of mixing than the- rest of the bed. Third, although the dispersion model is valid for representing the flow dynamics of the bed free from distributor effects, it is not suitable for a case where distributor effects are significant.

A careful visual observation of the tracer spread close to the distributor also strongly supports the presence of a zone of intense mixing existing in this region. This mixing is caused by stirring effects of the high-velocity liquid jets (U,, = 60-900 cm/s) issuing forth from the orifice holes of the distributor. This

Fig. 7. Effect of the location of the sampling point on the prediction of the apparent dispersion coefficient under the influence of the distributor region (F,, = 4%, O,, = 1.52 cm). CES 47:15/16-v

behavior closely resembles that of a CSTR. Con- sequently, any realistic modeling of a fluidized bed with significant distributor effects will involve ac- counting for both the CSTR-like behavior in the distributor region and the dispersion characteristics of the rest of the bed. This will involve two parameters. One will be the height of the CSTR or the distributor region, y. Another parameter will be the dispersion coefficient quantifying the degree of dispersion in the bed region.

It is important to specify the proper boundary condition at the interface of the CSTR and the dispersion model. This boundary condition will be used as the inlet boundary condition in the solution of the dispersion model. It is given as

+U,C=UiCstr at z=y (14)

where C,,, is the response of the CSTR. It is worthwhile to note here that, although the above equation looks similar to the Danckwerts type of boundary condition, the .underlying assumption is the absence of a concentration gradient in the CSTR not zero dispersion.

The response of this model, subject to boundary conditions (6) and (14) is given as (see Appendix A)

C = OSU,exp [P,(L - y) - U,t/y]

x{(P, +a)exp[cc(L--y)]erfc(l,)

+ (PI - a) exp [ - a(L - r)] erfc (A,)}

- 2P, exp [2P,(L - y)] erfc (R3). (15)

For the CSTR-dispersion model, the corresponding objective function to be minimized is

S(D,, Y, vi) = 2 CC,(t,) - Cl(tk, vi, Da, ~11” 5 Is=1

+ (U, - ui,+. (16) 0”

In this case, the height of the distributor region y and Vi were varied to get the best fit. The value of D,

was not varied, but was kept fixed at the value obtained from the bed-region experiments.

In order to examine the validity of the proposed two-region CD model, we consider the same case which was considered in Fig. 7. As shown in Fig. 8, there is no difference in the height of the distributor region (y) even though the location of the tracer measurement point was changed drastically. This is physically realistic and proves the validity of the proposed two-region model to account for the flow dynamics of fluidized beds with distributor effects.

Figure 9 compares the prediction of the dispersion model and the CSTR-dispersion (CD) model. It is evident from the figure that the greater the height of the distributor region the larger the distortion in the bed hydrodynamics which is reflected by the over prediction of the value of D, by the dispersion model [Fig. 9(a)]. As the value of the apparent dispersion


50 .

E Polypropylens Particles

40- 0-o L= 105 cm

c‘ 0-m L- 70cm .e p 30 -

b $ 20-

$j j@- z lo-

r 0

0 Li,‘,id SuperAal Velocity. U, (crn~s)

10

Fig. 8. Effect of the location of the sampling point on the prediction of the distributor region height, y (F., = 4%, D,,

= 1.52 cm).

lo POIypropykm POrtklea

.

loo- \ .3

. n

* 10 - \/-

looo,, .

501 I

3 3O-

/

Poiypmpylana Par&lea o--o Dl~r/Elad

s zo-

l -• Bnd Region

B P

l”t~---- (a) I 01 _- I

2 4 6 8 10 uquid Supwficial Vslocity. U, (cm/s)

Fig. 9. Comparison of the the dispersion model and the CD model (F,,, = lo!, D,,, = 0.76 cm).

La .

3 3

Polypropylans Porticlsa Distributor/Bed

F Dor Nor o--o z O.ZP 1.33

0.20 - a-*4% 0.36 0.44 A-64% 0.76 0.00 r--4% 1.52 0.02 q --07% 0.76 0.02

coefficient approaches the actual dispersion coefficient at low U,, a rather small part of the bed is affected by the distributor region. On comparing the magnitude of error involved in the prediction of the CD model and the dispersion model [Fig. 9(c)], it becomes clear that the CD model is considerably better than the dispersion model.

10.0 100.0

Distributor Pressure Drop. APd (kPa)

It is important to establish at this stage that the Fig. 10. Effect of distributor pressure drop and hole density distortion in the bed hydrodynamics is not due to the on the delay time.

presence of dead zones in the distributor region. This conclusion might bc drawn from the long tails in the response in cases where distributor effects are significant. In order to investigate this issue further, we compute the delay time for distributor/bed cases as follows:

t delay = fdistributor - tbed (17)

where, &ributor is the mean residence time computed from the response curve that includes distributor effects, and Fbed is the mean residence time computed from the response curve which is free from distributor zone irregularities. The presence of dead zones will manifest itself in the form of large values of the delay time as pointed out by Levenspiel (1972). This is due to the fact that tracer particles get trapped in the dead spaces of the distributor region which cause a long tail in the response, thereby making Fdidistributor much larger than ted_ This is, however, not true if the distortion in the response curve is caused by a CSTR-like behavior in the distributor region. Despite the presence of long tails in the response as evident from Fig. 2, the delay time will be zero as shown in Appendix B.

The ratio oft dt,ay/Fis plotted against the distributor pressure drop in Fig. 10. It is clear that tdelay increases at low pressure drop for low hole-density distributors indicating the presence of dead zones in the distributor region. However, dead zones are absent at high AP,. In fact, high U,, liquid jets existing at high AP, cause greater stirring effects in the distributor region. Consequently, there are no dead zones and the value

Of tdelay is close to zero. On the other hand, at low AP, where U,, are not very high, dead zones starts to develop in the distributor region of low N,, distributors causing the value of tdclpy to rise significantly. Most important, the over prediction in the values of D, cannot be attributed to the development of dead zones. The reason is obvious here. Higher values of D,/D,, or larger deviations in D, are evident at high pressure drop (Figs 3-6) where tdclny is negligible or, more specifically, dead zones are absent. This further substantiates the validity of the CD model which assumes an intense mixing prevailing in the distributor region not the presence of dead zones. However,

0.30 )

Hydrodynamics of liquid fluidized beds including the distributor region

the possibility of dead zones influencing the flow dynamics of the fluid&d bed at very low AP, cannot be ruled out. In fact, it has been shown by Asif (1991) that for very low-density particles of polystyrene (pa = 1.05 g/cm3), dead zones significantly distort the bed flow dynamics which can be eliminated by using distributors with N,, > 1 hole/cm*.

Figure 11 shows the response of the bed in the presence of distributor effects. The predictions of the CD model are in close agreement with the experimental data. On the other hand, the response obtained by using D, evaluated from the Tang and Fan correlation [eq. (13)] show large deviations from the experimental data. This figure highlights the fact that fluidized beds with distributor effects cannot be modeled accurately with the help of existing cor- relations.

0 I 2 4 6 6 10 12 14 16

liquid Superficial Velocity, U, (cm/s)

Fig. 13. Effect of liquid velocity on the hydrodynamics of the bed of glass particles.

ing high-density solid particles. In this case, a change of U,, from 4.0 to 16.0 cm/s causes variation of U, from 100 to 400 cm/s and distributor pressure drop from 0.5 to 22 kPa. Comparing these results with those for low-density particles, it appears surprising at first glance that distributor effects are absent for fluidized beds of both 2 and 3 mm glass particles over such a wide range of U,, and AP,. This may be attributed to the random movement of heavy particles which tend to dissipate quickly the momentum of high-velocity jets in the distributor region.

High-density glass particles When the liquid fluid&d bed contains high-density

glass beads (p, = 2.46 g/cm3) of 2 mm diameter, no distributor effects whatsoever are visible in the response, as evident from Fig. 12. Similarly, for glass beads of 3 mm diameter, not much difference is observed in the prediction of D, for 0.02 - N,, and 1.33 - N,, distributors which are very close to D, values

of the bed region (Fig. 13). This means that distributor effects become insignificant in fluidized beds contain-

Polypropylane Particles

0 Experimenhal Dota

- CD Modal (O,,-6 cmZ/s. y-34 cm)

- -Tong and Fan Cormlation

0.00 5 10 15 20 25

Time (a)

Fig. 11. Comparison of the prediction of the dispersion model and CD model on the dynamic response of the liquid

fluid&d bed (F,, = l%, D, = 0.76 cm, U, = 6.25 cm).

0.4

s jj 0.3 E

d 0.2 P g

z 0.1 s

2 mm Glass Portklas 0 Bad Fhglon

2& Distributor/Bad

* “+. “or Do,

. 1.33 0.20

0 dh A 0.02 1.52

16

Fig. 12. Effect of distributor hole density on the response of the bed of glass particles (U, = 12.1 cm/s, U, = 302.5 cm/s,

AP, = 12.5 kPa. F,, = 4%).

4163

CONCLUSIONS

RTD experiments were used to quantify the effect of the distributor region on the hydrodynamics of a liquid fluidized bed. Two different sets of RTD experiments employed reveal how to identify this effect separately from the dispersion characteristics of the rest of the iluidized bed.

The density of the solid phase present in the bed plays a critically important role in influencing the distributor-induced ffow disturbances. In the fluid&d beds containing low-density particles of polypropylene (pa = 1.61 g/cm”), distributor effects were found to be significant for distributors of low hole density and high distributor pressure drop. These effects were confined to a region close to the distributor and, were attributed to stirring effects of high- velocity liquid jets issuing forth from distributor orifi- ces. The dispersion model was found to be unsuitable for modeling the behavior of the fluidized bed including distributor effects. A simple but more realistic two-region model was proposed. This model used a CSTR to account for the distributor region behavior while the rest of the best was modeled with the conventional dispersion model.

On the other hand, for fluidized bed containing high-density glass particles (PI = 2.46 g/cm3), distributor effects were found to be negligible over a wide range of distributor hole density and the distributor pressure drop. This is due to the fact that the movement of the high-density glass particles tend to over- whelm any distortion of the flow dynamics caused by the distributor.


Acknowledgement-This work was supported by the Natural c Sciences and Engineering Research Council of Canada

parameter defined by eq. (A21)

(NSERC). 8, roots of the eq. (C5) Y height of the distributor region.(i.e. CSTR)

A

C

cd

C,(t,)

cross-sectional area of column (= 46.3), cm2 concentration of the tracer orifice coefficient

C&At,)

c

C *u D,

D6U

Ir, D, D, FC., Jo

normalized average concentration (experimental value) model prediction of the concentration at time tk Laplace transform of concentration C response of the CSTR apparent axial liquid dispersion coefficient, Cll12/S

actual axial liquid dispersion coefficient, cm2/s particle diameter, mm equivalent particle diameter, mm radial dispersion coefficient, cm2/s distributor fractional open area Bessel’s function of the first kind of zero order

k 4 L

;

NO, PI PZ AP, AP, Pe

4 Q

lz S

t t

tdelny

vi

uie

first-order rate constant, l/s length of the particles, mm distance between tracer injection and tracer measurement point, cm total number of holes on the distributor total number of data point in an experiment distributor hole density, holes/cm2 parameter defined as (= 0.5 x UJD,), l/cm parameter ( = 0.25 x U f/D,), l/s bed pressure drop, kPa distributor pressure drop, kPa P&let number [ = ( U,L/D,)] variable defined by eq. (A15) volumetric flow rate, cm3/s dimensionless radial coordinates [eq. (Cl)] radius of the column (= 3X4), cm quadratic objective function defined by eqs (11) and (16) time, s mean residence time, s time delay, s liquid interstitial velocity, cm/s experimentally measured liquid interstitial velocity, cm/s

UO liquid superficial velocity, cm/s

VW liquid distributor orifice velocity, cm/s V volume of the CSTR in the bed, cm3 W variable defined by eq. (A7) w Laplace transform of W z axial coordinate, cm

20 location of the tracer injection

21 location of the tracer detection z axial coordinate in CD model [ = (z - r)],

NOTATION in the fluidized bed Dirac delta function parameter defined by eq. (B4) parameter defined by eq. (A18) parameter defined by eq. (A19) parameter defined by eq. (A20) first moment of the response curve fluid density, g/cm3 solid particle density, g/cm3 standard error in the measurement of concentration at time t, standard error in the measurement of U, parameter defined by eq. (All) sphericity

Abbreviations CD CSTR-dispersion model G2 glass particles of 2 mm diameter G3 glass particles of 3 mm diameter PP polypropylene RTD residence time distribution

REFERENCES

Agarwal, G. P., Hudson, J. L. and Jackson, R., 1980, Fluid mechanical description of fluid&d beds. Experimental investigation of convective instabilities in bounded beds. Ind. Engng Chem. Fundam. 19, 59-66.

Asif, M.. 1991, Hydrodynamics of liquid fluidized beds including distributor effects. Ph.D thesis. The Univeristv of Calgary, Calgary.

Carlos, C. R. and Richardson, J. F., 1968, Solid movement in liquid fluid&d beds-I._ Particle velocity distribution. Chem. Engng Sci. 23,8X3-824.

Carslaw. H. S. and Jaeger, J. C., 1959, Conduction of Heat in Solids, pp. 494-496. Clarendon Press, Oxford.

Ching, C. B. and Ho, Y. Y., 1984, Flow dynamics of immobilized enzyme reactors. Appl. Microbial. Biotechnol. 20, 303-309.

Chung, S. F. and Wen, C. Y., 1968, Longitudinal dispersion of liquid flowing through fixed and fluid&d beds. A.Z.Ch.E. J. 14, 857-866. -

Goebel, J. C., Booij, K. and Fortuin, J. M. H., 1986, Axial dispersion in single-phase flow in pulsed packed columns. Chem. Engng Sci. 41, 3197-3203.

Han. N. W., Bhakta. J. and Carbonell, R. G., 1985, Longitud- inal and lateral dispersion in packed bed effect of column length and particle size distribution. A.1.Ch.E. J. 31, 275287. -

Kikuchi, K. I., Konno, H., Kakutani, S., Sugawara, T. and Ohashi, H., 1984, Axial dispersion of liquid in liquid fluidized beds in the low Reynolds number region. J. them. Engng Japan 17, 362-367.

Krishnaswamy, P. R., Ganapathy, R. and Shemilt, L. W., 1978, Correlating parameters for axial dispersion in liquid fluidized systems. Can. J. them. Engng 56, 550-553.

Kunii, D. and Levenspiel, O., 1969, Fluidization Engineering, pp. 88-90. Wiley, New York.

L&&spiel, O., 1972, Chemical Reaction Engineering, pp. 296-301. Wiley. New York.

Levenspiel, O., 1984, 7’he Chemical Reactor Omnibook+, pp. 64.1-64.14. OSU Book Store, Corvallis, OR.

Nauman, E. B. and B-am, B. A., 1983, Mixing in Continu- ous Flow Systems, pp. 98-100. Wiley, New York.

Oertzen, G. A., Kalogerakis, N., Behie, L. A., Kiesser, T. and Bauer, W., 1989, Dynamic and steady state modelling of

Greek letters 0: parameter defined by eq. (Al 1)

Hydrodynamics of liquid fluidixed beds including the distributor region 4165

an immobilized glucose isomerase fluid&d bed bio-

reactor, in Fluidization (Edited by J. R. Grace, L. W. Shemilt and M. A. Bergougnou), Vol. 6, pp. 491498. Engineering Foundation, New York.

Rangaiah, G. P. and Krishnaswatny, P. R., 1990, Application of time domain curve-fitting to parameter estimation in RTD models. J. them. Eagng Japan 23, 124-130.

Reilly, P. M. and Patino-Leal, H., 1981, A Bayosian study of the error-in-variables model. Technontetrics 23, 221-231..

Tang, W.-T. and Fan, L.-S., 1990, Axial liquid mixing in liquid-solid and gas-liquid-solid fluid&d beds containing low density particles. Chem. Engng Sci. 45, 543-551.

Yang, W. C. and Keairns, D. L., 1979, Estimating the jet penetration depth of multiple vertical grid jets. Ind. Engng Chem. Fundam. 18, 317-320.

Yates, J. G., Bejcek. V. and Cheesman, D. J., 1986, Jet penetration into fluidized beds at elevated pressure, in Fluidization (Edited by IL. Ostegraad and A. Sorensen), Vol. 5, pp. 79-86. Engineering Foundation, New York.

APPENDIX A: RESPONSE OF CSTR-DISPERSION MODEL

SUBJECT TO A PULSE OF TRACER INJECTION

The response of a stirred tank to a pulse tracer input is given as

G=(4)exp( -ft)

=(:)exp( -$t)

(Al)

642)

where Q is the volumetric flow rate, v the volume of the CSTR and y is the height of the CSTR (i.e. the distributor region).

The mass balance at the interface of the CSTR and the rest of the bed yields the eq. (14), which in conjunction with eq. (AZ) can be written as

Equation (A3) is the inlet boundary condition at .r = y for the dispersion equation (5) to model the rest of the bed, which is given as

ac -4&g-U,$. at

Define

Z=(z-y).

We can now rewrite eqs (5) and (A3) as

2’ = 0,

De&e

where

C(.f, t) = W(F, t) exp (PI.2 - P2t) (A7)

(5)

(A4)

(A5)

u, -t . Y >

(fw

(AS)

Substituting eqs (A7) and (AS) in to eqs (A5) and (A6), we obtain

where

and ct=(Pz -$). (Al I)

Taking the Laplace transform of eqs (A9) (AlO), we obtain

(Al3

where, w represents the Laplace transform of W. The solution of eq. (A12) for semi-infinite media with

boundary condition (A13) is given as

W=t#l exp ( - 9Z) (4 + pl)(s - a)

(AI4)

where

(AI5)

The inverse Laplace transform of eq. (A14) could be found in Carslaw and Jaeger (1959). It is given as

W = 0.54 exp (at) C

exp ( - CSZ) (p, + a) erfc(l2)

exp (o?Z) + merferfc(&) 1 -(*) exp(P,? + P2t) erfc(1,). 6416)

Using eq. (A7), we can write

C=0.5U,exp(P1Z- U,t/y)[(P, +g)exp(@Z)erfc(&)

+ (PI - h) cxp ( - ZZ) erfc (A,)]

- 2P2 exp(2PiF) erfc (2,) (AI7)

where

4 = t4Da~)o.5 + (at)0.4 [ - 1

12 = & - (atY

[ - 1 A3 = (4&x" [ - + (P2 tp.5 1

(AI9)

G = (a/D,)O-“; a = (P2 - Vi/y) (A21)

P, = 0.5&/D,; P2 = 0.25U:/D,. (A22)

APPENDIX Bc CALCULATION OF DELAY TIME FOR

CSTR-DISPERSION MODEL

The Laplace transform of eqs (A5) and (A6) is given as - -

$-2P$- + c=o ( > 0

atZ=O

where c is the transformed variable. The solution of eqs (Bl) and (B2) for a semi-infinite media

is given as

+ PI)-’ exp [(PI - 4Fl

(B3)

(A9)

- +PIW=+exp(at) at?=0 (AIO)


where described as follows:

(B4)

The first moment pi of eq. (B3) is given as

Recall that z = (Z + y) and z in the present case is the distance L between the tracer injection and tracer measurement points.

In the absence of the distributor, a similar manipulation for dispersion equation (5) and boundary condition (9) will yield

e = 2P,(f1- PI)-’ exp [(Pi - q)Z].

In this case, p1 is given as

(B6)

~1 =(;+$)=($+$)=(t;..,+$). (B7)

Now, we compute the time delay defined by eq. (17) as follows:

*delay = *di,,ributc.r - tbed-

Since D, and Vi are the same for both of the models, we write

From eqs (BS) and (B7”), we can write now

t de1.y = .h.dis,ributc.or -.h,,d =O. WI

For computing tdcuy, eq. (B9) is preferred over eq. (BS), since it is much easier to calculate first moments in the present case than the mean residence time.

APPENDIX c: INFLUENCE OF RADIAL NONUNIFORMITIES IN THE TRACER INJECTION ON THE CONCENTRATION DJSTRLBUTION IN THE BED

We intend to prove here that radial nonuniformities in the tracer injection, even if present, will not affect the conccntra- tion profile obtained at the tracer measurement location in the present experimental setup. In order to show this, two extreme cases of tracer injection are considered. In one case, the tracer is injected uniformly along the cross section of the bed. As a result, there are no radial nonuniformities in the bed, and the one-dimensional dispersion model is valid for this case. In the other case, the tracer is injected at a point along the axis of the column instead of a uniform injection across the cross section of the bed. Obviously, the conccntra- tion distribution in the bed will now be governed by the following two-dimensional dispersion model:

Here the r-coordinates have been normalized with respect to the radius (R) of the column. And D, is the radial dispersion coefficient. The boundary conditions for the system can be

at r = 1.0 dC ( > - dr

= 0.0 for - al < Z < cc (C2)

at r=O.O C=finite for-cc <z< co. (C3)

In the axial direction, we consider the boundary conditions given by cqs (6) and (7).

The solution of eq. (Cl) with aforementioned boundary conditions and an impulse tracer input is given as

1 Text ( - WWR*) I> * WI

Here, Jo@,,,) is the Bessel’s function of the first kind of zero order and &s are the roots of the following equation:

JbUW = 0. (C5)

Here, the objective is to examine the radial nonuniformities in the concentration profile at the tracer detection point (2,) subject to a pulse tracer input. For this, we will compare concentration profiles at two different locations along the radius of the column. One is at the center of the bed (T = 0) and the other is at the wall (r = 1). Since D, is an important parameter governing the magnitude of the radial mixing, it is important to also examine its influence on radial concentration gradients. A higher value of D, will lead to a smaller concentration gradient in the radial direction. Here again, two widely different values of D, are considered. In one case D, is equal to D,, and O.lD, in the other case. The results are shown in Fig. 14. It is surprising that normalized concentration profiles for these cases mentioned above are identical, which means that there are no radial concentration gradients present in the column for L = 105 cm for a wide range of D, from 1 to 10.5 cm’/s. Even more interesting is the fact that the normalized concentration profile obtained for the two-dimensional case is identical to the one-dimensional case. These results present incontrovertible evidence that, in the present study, any nonuniformity in the tracer injection is homogenized by the radial dispersion of the concentration and there are no concentration gradients present at the tracer measurement location.

0.30 I -5 0.25

3 5 0.20

6 ” 3 0.15

.g 0 E

0.10

B 0.05

=I .

. .

. I

I

. I

0, Dr r 0 10.5 - - . 10.5 1.0 0.0 * 10.5 1.0 t.0 A 10.5 10.5 0.0 P 10.5 10.5 1.0

ui - 10.75 cm/s

. I

o.ocl -- ’ I.,

5 5 11 14 17

Time (m)

Fig. 14. Influence of radial nonuniformity in the tracer injection on the concentration distribution at the tracer detection point [zO = 0.0 cm, .zl = 105 cm, Vi = 10.76 cm/s, R

= 3.84 cm; all data points line up on (I) symbol].

hydrodynamics of liquid fluidized beds including the distributor region

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