Ind. Eng. Chem. Res. 2005, 44, 5397-5402
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Mathematical Model for the Prediction of Cycle-Time Distributions for the Wurster Column-Coating Process Travis Crites† and Richard Turton*,‡ Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, and Department of Chemical Engineering, West Virginia University, Morgantown, West Virginia 26506
The application of functional and nonfunctional coatings in the agricultural, food, and pharmaceutical industries is commonly carried out in the Wurster process, which is essentially a bottom-sprayed coating process occurring in a fluidized bed containing a central draft tube or partition. To predict the variability in the distribution of coating mass in a batch operation, information on the cycle-time distribution (CTD) of particles in the bed is required. The current work presents a compartment model to describe the CTD for the Wurster process. The basic model uses a plug flow compartment with dispersion, a mixed tank, and a second mixed tank with bypass, all connected in series. The resulting CTD is asymmetric with a sharp leading edge and a broader trailing edge with a significant tail. The addition of other compartments is used to model some of the anomalous results seen in published results from this type of equipment. Introduction The Wurster column-coating process has been used extensively for the film coating of particulates and for the application of controlled-release and enteric coatings on pharmaceutical and agricultural products. The precision and repeatability of active coatings for the pharmaceutical industry continues to be of interest for both research and manufacturing areas. The Wurster process, shown in Figure 1, utilizes a bottom-spray fluidized bed with a partition (the Wurster partition) or draft tube. This partition separates a zone where upward moving particles are sprayed with the coating solution from another zone where the particles recirculate through the bed and return to the spray region. The use of the partition or draft tube allows particles to be used that are not easily spouted (Geldart type A and B). When the number of passes the tablets make through the bed is controlled, the thickness of the coating imparted to the tablet may be controlled. Previous studies by Cheng and Turton1 and Shelukar et al.2 have shown that the major source of variation for coated products is due to the nonuniform spray coverage that individual particles receive each time they pass through the spray zone. Nevertheless, the variation in coating uniformity that is due to the spread of the number-ofpasses distribution may still be significant. Mann and co-workers3,4 have shown that the information contained in the cycle-time distribution (CTD) can be used to determine the number-of-passes distribution that is used directly to evaluate the uniformity of coating variation for a batch-coating operation. This work focuses on the quantification of the CTD in spouted fluidized beds with draft tubes using a compartment model approach with each compartment having a different residence time distribution (RTD). Several research groups have experimentally quantified the CTD * To whom correspondence should be addressed. E-mail:
[email protected]. † Carnegie Mellon University. ‡ West Virginia University.
Figure 1. Schematic diagram for the Wurster column-coating process.
of the particles through the use of magnetic particle detection. In such experiments, a magnetic particle is introduced into the bed, and a detector coil is placed around the partition. Each time the tracer particle makes a circuit around the bed, it is detected by the coil, and a signal is recorded. When the time between successive signals is measured, the distribution of circulation times may be obtained. Inferring the path of the tracer particle through the bed using this information is very difficult. However, compartment models are still an effective method for capturing data on CTDs and simulating behavior in fluidized beds. To verify independently the assumed compartmental structure of the equipment, techniques such as positron emission particle tomography (PEPT)5,6 that are capable of following the tracer particle everywhere in the equipment are required. Such verification is beyond the scope of the current work.
10.1021/ie049040s CCC: $30.25 © 2005 American Chemical Society Published on Web 02/17/2005
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Figure 2. Block flow diagram of the compartment model (dotted boxes represent compartments for anomalous flow behavior).
Model Development and Theory To develop a CTD model, the Wurster coating process was divided into several regions with distinct flow patterns: distributor, fountain regions, down-bed, dead zone, and what we refer to as anomalous flow regions. Figure 2 is a diagram showing the structure of the compartment model. For each compartment, an RTD was chosen that reflected the behavior of the region. A modeled CTD for the whole process was then synthesized by summing the residence-time contributions from each modeled compartment. The analysis of each compartment and its simulation methodology is given in the following sections. Fountain Region. As the circulating particles pass through the spray zone, they enter the up-bed or central region of the coater and then flow into the larger, disengaging section above the partition. The particles are forced upward through advective processes, but as the diameter of the bed increases, the particles start to disengage. This section of the equipment is referred to as the fountain region. The particles, upon leaving the fountain region, settle under the influence of gravity and move into the annular down-bed region on the other side of the partition. The RTD for this section is best described by a continuously stirred tank reactor (CSTR)where t is the amount of time the particles have spent
exp E(t) )
( ) -t τjFR-1
τjFR-1
∫0
t
E(ξ) dξ
t ) -τjFR-1 ln[1 - F(t)]
(2)
(3)
In simulating this and all other regions, the residence time is taken as a percentage of the modal residence time of the data set being analyzed, τjmode, which is a direct input to the model. A random number between 0 and 1 is then generated, corresponding to F(t), and the corresponding time spent in the fountain region during one cycle is calculated. The percentage of the modal residence time is the key simulation parameter in this region, which is adjusted to fit the data being analyzed. Down-bed, Up-bed, and Distributor. The downbed, up-bed, and distributor regions of the reactor are simulated as plug-flow regions (PFRs). The RTD function for a pulse input of particles is given by
(
E(t) ) δ t )
V )τ v˘
)
(4)
where δ(t ) τ) is the Dirac delta function. In our case, however, the distribution of residence times within the up-bed and down-bed will be smeared due to some amount of dispersion. In anticipation of these effects, the dimensionless dispersion number, Θ, is defined for the PFR regions as
(1)
in the region and τjFR-1 is the average residence time in the fountain region. A more useful function for the compartment model, however, is the cumulative probability function, F(t), which describes the fraction of particles that remain in the section for times shorter than t. It follows that
F(t) )
F(t) takes on values between 0 and 1 and is ideal for an algorithm involving random number generation. Integrating eq 2 and solving for t, the time spent by a single particle in the fountain region, we get
Θ)
D vL
(5)
where D is the dispersion coefficient, v is the velocity of the convective fluid (air), and L is the height of the bed. For small deviations from plug flow, Θ is typically less than 0.01. For this limiting case, Taylor dispersion is applicable; thus,
E(t) )
1
x4πΘ
[(
1-
exp -
t
jτPFR 4Θ
)
2
]
(6)
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This distribution is normal, having a variance equal to
σ2 ) 2Θ
(7)
For this analysis, Θ is used as an adjustable parameter, but typically assumes a value on the order of 10-3. Again, the value of t/τjPFR is taken to be a percentage of the modal experimental residence time and is varied to achieve a good fit. To predict the time that a particle spends in the up-bed and down-bed regions, random numbers are generated from the normal distribution having a mean of t/τjPFR and a variance given by eq 7. Since the majority of circulation time is spent in the up-bed and down-bed, this fraction is typically 0.75 or greater, unless other anomalous flow characteristics are present in the coater. The average residence time a particle spends in the distributor section is modeled simply as a fraction of the modal residence time. For all simulations, this parameter was held constant at 15% of the modal residence time. The average residence time for all three regions, then, is simply the sum of the time spent in the down-bed, up-bed, and distributor. Dead Zones. Dead zones, where particles undergo prolonged, localized recirculation, are apparent in all of the data sets where the CTDs have long tails. These long tails indicate that a small number of particles spend a much longer time in the bed per cycle than the average particle. Therefore, these regions are treated as CSTRs with residence times much longer than the residence time experienced by an average particle. The time that a particle spends in a dead zone, then, can be predicted using eq 3, where the residence time, τjDZ, is the parameter that is varied to fit the data. A typical value for this parameter is up to 1 order of magnitude greater than the modal circulation time. Regions of Anomalous Flow. Several of the CTDs studied here exhibit irregularities that are not easily rationalized. For example, some data sets indicate that regions in the coater exist where particles are trapped, experiencing long residence times. In distributions with long tails, this is manifested as small clusters or humps in the tail of the CTD. These regions have the characteristic shape of two CSTRs in series and may reflect end effects resulting from flow transitions from one region of the coating process to another. However, the goal of this work is not to rationalize these effects but, rather, to attempt to quantify the amount of particles affected by these flow patterns and their duration with respect to the mean circulation time. Applying eq 2 to two CSTRs in series with identical residence times, we see that
F(t) ) 1 -
( )
( )
-t -t t exp - exp τjAF,i τjAF,i τjAF,i
(8)
To simulate these secondary effects, a random number between 0 and 1 is generated for F(t), and the approximate modal value for the secondary effect is taken again to be a percentage of the modal experimental residence time. Values for the residence times in these regions are estimated, making them fully adjustable parameters. They are presented as τjAF,i. In some data sets, at the modal value of the CTD, multiple decaying peaks are seen. When this phenomenon is observed, it is simply modeled by a CSTR in series with the existing region that models the fountain
Figure 3. A typical CTD for the Wurster column-coating process.
region. In the model, a fraction of particles () is allowed to enter this secondary fountain region, with residence time τjFR-2, while the rest of the flow is directed around the region and enters the primary fountain region compartment. It is noted that the phenomenon associated with this flow anomaly may occur anywhere in the equipment and the description as a secondary fountain region is made only for convenience. The Full Model. For each simulation, a simulated particle is allowed to pass through the coater 1000 times. The cycle time per pass is computed by summing the residence time contributions between each successive pass through the PFR section of the model. The resulting cycle times are then binned to construct a simulated CTD and compared to the experimental results. A typical experimental CTD for the Wurster process is presented in Figure 3. The illustration shows the initial short tail that indicates deviations from plug flow due to dispersion (A). The steep rise that follows is attributed to fully developed plug flow in the up-bed, down-bed, and distributor (B). This reaches a maximum at the modal frequency of the distribution (C). From the mode, the distribution decays in an exponential fashion that is the result of the fountain regions in the coater (D). Finally, the distribution possesses a long tail that is the hallmark of the presence of a dead zone in the reactor (E). Humps in the long tail, not shown in Figure 3, are modeled using the anomalous flow regions. The compartment model, including key model outputs, was shown in Figure 2. Starting at the reactor labeled τjPFR, the simulated particles flow through the up-bed, down-bed, and distributor sections of the model. A fraction, , enters the secondary fountain region, while the remaining particles bypass this region and directly enter the main fountain region. After the particles leave the fountain regions, a fraction of particles, R, returns to the plug-flow regions while the remainder enters the dead zone, or the anomalous flow regions, with fractions β, γ, and δ, respectively. Thus, the average residence time predicted by the model is given by
τjmodel ) (τjPFR + τjFR-2 + τjFR-1) + (1 - R)(βτjDZ + γτjAF,1 + δτjAF,2) (9) Finally, to quantify the amount of time spent in each section by a particle relative to the predicted average residence time, a ratio of the time spent in a compart-
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Table 1. Best-Fit Model Parameters for Experimental Data from Cheng and Turton,1 Shelukar et al.,2 and Mann and Crosby4 parameter R β γ δ Θ τjmode (s) τjPFR (s) τjFR-1 (s) τjFR-2 (s) τjDZ (s) τjAR,1 (s) τjAR,2 (s)
Cheng and Turton 0.9 0.05 0.475 0.475
Shelukar et al. 250
Shelukar et al. 300
1.0
5 × 10-3 2.5 2.25 0.45
1.0 4 × 10-3 13.0 8.7 0.65 3.9
20 5.1 4.1 3.24 0.68 0.13
SSE/SSTO
0.9243
13.25 0.65 0.05 0.30
0.06 0.07 0.06
Mann and Crosby 238-0
Mann and Crosby 194-0
0.9 0.1
1.0
1.0
1.0
1 × 10-3 8.0 5.5 2.1
1 × 10-3 4.7 3.0 1.0
7.9 × 10-3 4.3 2.5 0.35
5 × 10-3 5.3 3.2 1.6
0.15 5 × 10-3 6.5 4.3 2.7 2.0
7.8 0.70 0.27 0.03
0.8553
Mann and Crosby 238-24.6
0.9 0.1
20
jτmodel (s) θPFR θFR θFR-2 θDZ θAR,1 θAR,2
Shelukar et al. 350
0.9755
12.69
Simulation Predictions 4.13 2.85 0.73 0.88 0.24 0.12
4.80 0.66 0.34
7.30 0.59 0.37 0.04
0.9757
0.9090
0.03
Error Calculations 0.9478
0.9316
ment to the predicted mean residence time is defined as
θregion )
fτjregion τjmodel
(10)
where θregion is the fractional residence time for a particular compartment and f is the fraction of particles in the simulation that pass through the region. The value of the fractional residence time ranges from 0 to 1. Simulation Error Analysis. A conventional sum of squares error analysis was used to quantify the accuracy of model predictions. The sum of squares error (SSE) is given as
SSE ) SSTO - SSR
(11)
h )2 and SSR ) ∑i(Y ˆ i - Yi)2; Yi where SSTO ) ∑i(Yi - Y denotes an experimental value, Y h denotes the average of all data points, and Y ˆ i denotes a model prediction. The ratio of the SSE to SSTO is reported in Table 1 for all the data sets considered in the current work. Results and Discussion Results of Cheng and Turton.1 In this work, Cheng and Turton1 measured the CTD of a magnetic tracer Nonpareil particle through a Wurster coating column, using a detector coil wound around the partition. Several runs were conducted at identical conditions to confirm the reproducibility of the experiments, and the CTDs obtained at fixed operating conditions were consistent from run to run. For the model, the average of the four experimental runs performed was analyzed. The experimental data and model predictions are presented in Figure 4. The experimental form shows evidence of dispersion and anomalous flow. Thus, the adopted model for this CTD consisted of a distributor, an up-bed and down-bed with dispersion, a fountain region, a dead zone, and two regions of anomalous flow. Two regions of anomalous flow were used in the model
Figure 4. Experimental data and modeled prediction for the CTD data of Cheng and Turton.1
due to the two prominent clusters present in the long tail. Their modeled residence times were chosen as the modes in the circulation time distribution at which the humps occurred. Simulated parameters and regression error analysis are given in Table 1. Results of Shelukar et al.2 In this work, the volumetric flow rate of the fluidizing gas was varied from 250 to 350 m3/h. The minimum flow needed for circulation in this column was 200 m3/h. Figure 5 shows the experimental and modeled data for each run. At a flow rate of 250 m3/h, there is substantial spread in the region of the CTD associated with the fountain region. This anomalous flow was modeled by using the secondary fountain region since it appears that particles are passing through two different fountain regions with different average residence times. As the flow rate of air is increased well above the minimum spouting value, the distributions become sharper, and CTDs consistent with a single disengaging section emerge. Although the distributions possess long tails, there were no humps in the tail; therefore, a simple dead-zone model was used
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Figure 5. Experimental data and modeled predictions for the data of Shelukar et al.2: (A) 250 m3/h, (B) 300 m3/h, and (C) 350 m3/h.
here. The simulation parameters for each of the distributions are given in Table 1. Results of Mann and Crosby.4 Mann and Crosby4 also used the magnetic particle tracer method to determine CTDs in the Wurster process. The CTD was measured for two different flow rates of air through the spout 238 and 194 m3/h. For the 238 m3/h flow rate, the CTD was measured for flow rates of 0 and 24.6 m3/h through the annular section of the bed. It can be seen from Figure 6 that the CTDs for the higher flow rate are sharper than those for the lower one, indicating less dead zones and behavior closer to plug flow. When air is introduced into the annular region, particle flow between the down- and up-beds is aided, and accordingly the distribution is sharper than when no annular air is added. Since no long tail is present, no dead zone was needed for the model. The CTD at the lower flow rate with no annular air is much broader and possesses two modal frequencies, which is consistent with a more difficult transition from the down- and up-beds. To model this effect, a secondary fountain region was incorporated into the model. The simulation parameters for each of the distributions are given in Table 1.
Figure 6. Experimental data and modeled predictions for the data of Mann and Crosby:4 (A) spout air flow rate, 238 m3/h; annular air flow rate, 24.6 m3/h. (B) spout air flow rate, 238 m3/h; annular air flow rate, 0 m3/h. (C) spout air flow rate, 194 m3/h; annular air flow rate, 0 m3/h.
Conclusions The current work presents the general structure of a compartment model that may be used to describe the CTDs for coating processes taking place in spouted-fluid bed equipment containing draft tube inserts or Wurster partitions. The basic model uses the following compartments connected in series: plug flow with dispersion, a mixed tank, and a second mixed tank with bypass. The resulting CTD for this basic model is asymmetric with a sharp leading edge and a broader trailing edge with a significant tail. Several secondary effects in the CTDs obtained by the three research groups were noted. The presence of small but significant humps in the tail of the distribution was modeled as two stirred tanks placed in parallel with the basic compartment model. A broadening of the distribution after the mode and the appearance of secondary peaks just after the mode were both modeled using a stirred tank with bypass in series with the basic compartment model.
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Acknowledgment Financial support for this work by the National Science Foundation (NSF Grant # CTS- 0073404) is gratefully acknowledged. Currently, Travis Crites holds a National Science Foundation Graduate Research Fellowship. R.T. would also like to take this opportunity to thank Professor Dudukovic for his guidance and friendship over the last 20 years. Notation E(t) ) residence time distribution function F(t) ) cumulative residence time distribution function f ) fraction of particles having traveled through a reactor section SSE ) sum of squares error SSTO ) total sum of squares SSR ) regression sum of squares t ) time (s) v˘ ) volumetric flow rate (m3/h) V ) volume of reactor section (m3) Yi ) experimental value Y ˆ i ) modeled value Y h i ) average of all Yi Greek Letters R ) fraction of particles entering PFR section β ) fraction of particles entering dead zone γ ) fraction of particles entering anomalous flow region 1 δ ) fraction of particles entering anomalous flow region 2 ) fraction of particles entering secondary fountain region θregion ) fractional residence time spent in a reactor section Θ ) dispersion parameter σ2 ) variance of a distribution τjmode ) time corresponding to mode of experimental CTD (s)
jτmodel ) modeled prediction for average circulation time (s) τjDZ ) average residence time of a particle in the dead zone of the model (s) τjAF,1 or 2 ) average residence time of a particle in the anomalous flow regions 1 or 2 of the model (s) τjFR-1 ) average residence time of a particle in the primary fountain region of the model (s) τjFR-2 ) average residence time of a particle in the secondary fountain region of the model (s) τjPFR ) average residence time of a particle in the plug flow region of the model (s)
Literature Cited (1) Cheng, X. X.; Turton, R. The Prediction of Variability Occurring in Fluidized Bed Coating Equipment Part 1: The Measurement of Particle Circulation Rates in a Bottom Spray Fluidized Bed Coater. Pharm. Dev. Technol. 2000, 5 (3), 311. (2) Shelukar, S.; Ho, J.; Zega, J.; Roland, E.; Yeh, N.; Quiram, D.; Nole, A.; Katdare, A.; Reynolds, S. Identification and Characterization of Factors Controlling Tablet Coating Uniformity in a Wurster Coating Process. Powder Technol. 2000, 110, 29. (3) Mann, U. Analysis of Spouted-Bed Coating and Granulation. 1. Batch Operation. Ind. Eng. Chem. Process Des. Dev. 1983, 22, 288. (4) Mann, U.; Crosby, E. J. Cycle Time Distribution Measurements in Spouted Beds. Can. J. Chem. Eng. 1975, 53, 579. (5) Seville, J. P. K.; Simons, S. J. R.; Broadbent, C. J.; Parker, D. J.; Beynon, T. D. Particle velocities in gas-fluidised beds. Proc. 1st Int. Part. Technol. Forum [Denver (AIChE)] 1994, 1, 493. (6) Stein, M.; Ding, Y. L.; Seville, J. P. K.; Parker, D. J. Solids motion in bubbling gas fluidised beds. Chem. Eng. Sci. 2000, 55, 5291.
Received for review October 3, 2004 Revised manuscript received December 23, 2004 Accepted December 29, 2004 IE049040S
