Stabilization of Continuous Fluidized Bed Spray ...pubman.mpdl.mpg.de/pubman/item/escidoc:1755994:6/component/es… · uidized bed spray granulation - a Lyapunov approach Stefan Palis

Stabilization of continuous fluidized bedspray granulation - a Lyapunov approach

Stefan Palis1, Achim Kienle1,2

1 Otto-von-Guericke-Universitat,Universitatsplatz 2, D-39106 Magdeburg, Germany(corresponding author e-mail: [email protected])

2 Max-Planck-Institut fur Dynamik komplexer technischer SystemeSandtorstrasse 1, D-39106 Magdeburg, Germany

Abstract: This article deals with the stabilization of continuous fluidized bed spray granulationwith external product classification. Using Lyapunov stability theory a control law is derivedguaranteeing exponential convergence of the error particle size distribution in a norm associatedto the second moment. In contrast to other approaches popular for particle process control nomodel reduction is used.

Keywords: Chemical Process; Lyapunov Methods

1. INTRODUCTION

Granulation is one of the most important manufacturingprocesses in food and pharmaceutical industries. It is usedto produce granules from liquid products, e.g. solutions orsuspensions.In Fig. 1 a typical process flow sheet is shown. It consistsof a granulation chamber, two sieves to remove under andoversized particles from the product and a mill. It has beenshown [2] that this configuration of continuous fluidizedbed spray granulation with external product classificationand material recycles shows unstable behavior for certainranges of the operating parameters. The instability resultsin nonlinear oscillations of the particle size distribution,which gives normally undesired time variance of productproperties. Possible solutions are a redesign, i.e. avoidingparameter combinations connected to a region of instabil-ity, or a stabilization via feedback.The second approach should be preferred as it gives thepossibility to operate the process in the full range ofoperating parameters. In addition to that feedback controlallows a direct adjustment of the desired product proper-ties and rejects unforeseen disturbances.The main difficulties regarding the control design arisefrom the complicated plant model, as the dynamicsare described by a first order nonlinear partial integro-differential equation, the population balance equation,with sinks and sources in the domain. For continuouscrystallization comparable dynamical behavior has beenobserved. Here the main approaches for control are mostlybased on model reduction, i.e. through discretization [4],approximation of the particle size distribution with a seriesexpansion [5], or linearization [6].In this contribution a completely different approach basedon Lyapunov stability theory for distributed parametersystems as in [7], [8] is proposed not using any modelreduction.

Fig. 1. Scheme of the granulation process

2. MODEL OF A CONTINUOUS FLUIDIZED BEDSPRAY GRANULATION

The main assumptions for the model of a continuousfluidized bed spray granulation are:

• nonporous, spherical particles,• no agglomeration or breakage,• ideal mixing.

The granulator consists of a granulation chamber, wherethe particle population is fluidized through an air streamand coated by injecting a suspension me. The associatedparticle growth has been described in [1].

G =2me

%A=

2me

%πµ2(1)

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In the continuous configuration of the fluidized bed spraygranulation particles are continuously removed in orderto achieve a constant bed mass, which correlates to aconstant third moment of the particle number distributionµ3 =

∫∞0L3ndL. The particle flux being removed from the

granulator is:

nout = Kn. (2)

where K is the drain which follows from the assumptionof a constant bed mass and which is derived later. The re-moved particles are then sieved in two sieves and separatedinto three classes:

(1) Fine particles, which are directly fed back to thegranulator,

nfines = (1− T2) (1− T1) nout (3)

(2) Product particles, which are removed from the wholeprocess,

nprod = T2 (1− T1) nout (4)

(3) Oversized particles, which are grinded in a mill andfed back to the ganulator.

noversize = T1nout (5)

Here the separation functions T1 and T2 (Fig. 2) for thetwo screens are as follows:

T1/2 =

∫ L

0e−

(L−µ1/2)2

2σ21/2 dL

∫∞0e−

(L−µ1/2)2

2σ21/2 dL

. (6)

Fig. 2. Separation functions T1 and T2

To describe the process, a population balance model forthe particle size distribution has been derived in [2]. In thiscontribution a simplified modell is used neglecting externalseeding, attrition and overspray. Nevertheless, open-loopsimulation gives comparable results.

In the model fine particles are directly fed back to thegranulator, which results in a cancellation of the associatedsink and source. Hence the simplified population balanceequation consists of the following particle fluxes:

• nprod particle flux due to product removal,• noversize particle flux due to oversize removal,• nmill particle flux due to particles fed back from mill,

Fig. 3. Distribution of milled particles nmill for µM = 0.7

and particle growth associated with the size independentgrowth rate G.

∂n

∂t= −G∂n

∂L− nprod − noversize + nmill (7)

The particle distribution fed back from the mill is assumedto be a normal distribution, where the mean diameter µM

represents the mill grade. In the following the mill gradeµM will be used as the control input. For physical reasonsµM should not exceed a lower (0.5mm) and upper limit(0.9mm).

nmill = 6e−

(L−µM )2

2σ2M

√2ππ%σM

∫ ∞0

L3noversizedL (8)

Assuming ideal mass control the drain K is calculated suchthat the first time derivative of µ3 becomes zero implyinga constant bed mass.

µ3 =

∫ ∞0

L3 ∂n

∂tdL = 0 (9)

=

∫ ∞0

L3

[−G∂n

∂L− noversize − nprod + nmill

]dL(10)

Because the mill is assumed to be mass conserving thethird moments of the oversize flux and mill flux are equalresulting in:

0 =

∫ ∞0

L3

[−G∂n

∂L− nprod

]dL (11)

= 3G

∫ ∞0

L2ndL−K∫ ∞0

L3T2(1− T1)ndL. (12)

Here the fact that the particle size distribution vanishs atthe boundary (n(0, t) = lim

L→∞n(L, t) = 0) has been used

for integration by parts. Solving equation (12) for the drainK gives:

K =3G∫∞0L2ndL∫∞

0L3T2(1− T1)ndL

. (13)

For an extended model, i.e. with external seeding, attri-tion and overspray, an extensive bifurcation study has

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been undertaken in [3]. It has been shown, that for suf-ficiently small mill grades µM self-sustained oscillationsoccur. This circumstance can be explained by the fact,that for fine grinding the population surface excessivelyincreases resulting in a very small growth rate. Thus theparticle flux from oversize fraction vanishes, which givesa slowly growing population of small particles. When thispopulation reaches the oversize fraction the whole processrepeats. Analogous patterns of behavior are observed forthe simplified model considered in this contribution.Therefore for a specific mill grade, i.e. at the bifurcationpoint, the stationary particle size distribution loses itsstability and an additional limit cycle occurs [3]. In thefollowing a control law is designed in order to stabilize thestationary particle size distribution in the whole range ofµM .

3. CONTROLLER DESIGN

In order to design a control law the error e is defined as:

e =

∫ ∞0

L2 (nd − n) dL. (14)

It should be mentioned, that the error is an integralquantity of the difference between desired particle sizedistribution nd and the process particle size distribution.The choice is motivated by the fact, that the particlegrowth depends on the surface area of the particle sizedistribution, which is strongly connected to its secondmoment. To derive a controller the following candidateLyapunov function is introduced:

V =1

2e2. (15)

The time derivative of V along the system trajectories (7)is:

V = ee = −e∫ ∞0

L2 ∂n

∂tdL, (16)

=−e[∫ ∞

0

2LGn− L2 (T1Kn+ T2(1− T1)Kn) dL

+

∫ ∞0

L3T1KndL

∫ ∞0

L2nmilldL

]. (17)

Here nmill is the shape of the particle size distributiongenerated by the mill depending on the mill grade µM . Asthe second moment of nmill cannot be directly solved forthe mill grade µM the characteristic curve (Fig. 3) has beeninverted pointwise (Fig. 3). In the following the secondmoment of nmill will therefore be used as a virtual controluvirt.

uvirt =

∫ ∞0

L2nmill(µM )dL (18)

Using (17) the negative definiteness of the time derivativeof the candidate Lyapunov function V can be guaranteedchoosing the following virtual control law.

uvirt =1∫∞

0L3T1KndL

[ce−

∫ ∞0

2LGn

−L2 (T1Kn+ T2(1− T1)Kn) dL]

(19)

In addition to stability the control law (19) even guaran-tees exponential convergence of V .

V = −ce2 = −2cV (20)

The resulting control scheme (Fig. 3) therefore con-sists of the control law (19) using the virtual controluvirt and an inversion of the characteristic curve from∫∞0L2nmill(µM )dL to µM .

4. SIMULATION

For the numerical simulation the population balance hasbeen spatially discretized using a finite volume methodwith an upwind scheme. The spatial mesh contains 160equidistant points in the domain of 0 < L < 4mm.

Parametersρ 1.610−3 g

mm3

me1003.6

gs

minit 100kgµ1 1.4mmσ1 0.055mmµ2 1mmσ2 0.065mmµM 0.9mmσM 0.1mm

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To test the proposed control law the process is startedin the region of stability (for example µM = 0.9). Thenthe particle size distribution is shifted into the region ofinstability by continuously increasing the desired secondmoment µ2,d or decreasing the mill grade µM in open-loop operation. As can be seen in Fig. 4 the process be-comes unstable in open-loop operation showing increasingoscillations, which would finally reach the associated limitcycle [3]. Applying the proposed control law the process isstabilized and oscillations occurring during the shifting aredamped. The particle size distribution and all its momentsµ0, µ1, µ2 (Fig. 5) are stabilized with reasonable controleffort.For an implementation on the real plant either the appro-priate moments from equation (19) have to be measuredor calculated from the measured particle size distribution.A particle size distribution measurement can be achievedby for example by spatial filter velocimetry measurement,focused beam reflectance measurement or in-process videomicroscopy. Depending on the measurement principle antransformation from chord length distribution to particlesize distribution might by necessary, which is uncriticalas in fluidized bed spray granulation typically sphericalparticles are produced.

5. CONCLUSION AND FUTURE WORK

A controller for continuous fluidized bed spray granulationwith external product classification has been derived. Ithas been shown to give exponential convergence in anorm associated to the second moment. Simulation resultsindicate that although having convergence in a norm thewhole particle size distribution and its moments µ0 andµ1 are stabilized. So far the proposed control law assuresconvergence only in an idealized setting, i.e. no parameterand no model uncertainty. Thus further investigations willbe made on robustification and adaption.

REFERENCES

[1] L. Morl & M. Mittelstrass & J. Sachse, Zum Kugelwachstum beider Wirbelschichttrocknung von Losungen oder Suspensionen.Chem. Techn. 29, 1977, Heft 10, pp. 540-542.

[2] S. Heinrich & M. Peglow & M. Ihlow & M. Henneberg & L.Morl, Analysis of the start-up process in continuous fluidizedbed spray granulation by population balnce modelling, Chem.Eng. Sci. 57, 2002, pp. 4369-4390.

[3] R. Radichkov, T. Muller, A. Kienle, S. Heinrich, M. Peglow,L. Morl, A numerical bifurcation analysis of continuous flu-idized bed spray granulation with external product classification,Chem. Eng. and Proc., vol. 45, issue 10, Oct. 2006, pp. 826-837.

[4] R. Eek, Control and dynamic modelling of industrial suspensioncrystallizers, Ph.D. thesis, TU Delft, 1995.

[5] P. Christofides & N. El-Farra & M. Li & P. Mhaskar, Model-based control of particulate processes, Chem. Eng. Sci. 63, 2008,pp. 1156-1172.

[6] U. Vollmer & J. Raisch, Population balance modelling and H∞-controller design for a crystallization process, Chem. Eng. Sci.,vol. 57, issue 20, Oct. 2002, pp. 4401-4414.

[7] T. Sirasetdinov, Stability of systems with distributed parameters[in Russian], 1987.

[8] A. Martynjuk & R. Gutovski, Integral inequalities and stabilityof motion [in Russian], Naukowa Dumka, 1979.

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Fig. 4. Open loop system response

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Fig. 5. Open loop (blue) and closed loop (green) system response

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