Faculteit Bio-ingenieurswetenschappen

Academiejaar 2011-2012

Model reduction in preparation for a PBM for thepharmaceutical granule drying process

Timothy Van DaelePromotor: Prof. Dr. ir. Ingmar NopensTutor: ir. Severine Mortier

Masterproef voorgedragen tot het behalen van de graad vanMaster in de bio-ingenieurswetenschappen: Milieutechnologie

De auteur en promotor geven de toelating deze scriptie voor consultatie beschikbaar te stellenen delen ervan te kopieren voor persoonlijk gebruik. Elk ander gebruik valt onder de beperkin-gen van het aueursrecht, in het bijzonder met betrekking tot de verplichting uitdrukkelijk debron te vermelden bij het aanhalen van resultaten uit deze scriptie.

The author and promoter give the permission to use this thesis for consultation and to copyparts of it for personal use. Every other use is subject to the copyright laws, more specificallythe source must be extensively specified when using results from this thesis.

Ghent, June 2012

The promoter, The tutor, The author,

Prof. dr. ir. Ingmar Nopens ir. Severine Mortier Timothy Van Daele

Woord vooraf

A good simulation, be it a religious myth or scientifictheory, gives us a sense of mastery over experience. Torepresent something symbolically, as we do when we speakor write, is somehow to capture it, thus making it one’sown. But with this appropriation comes the realizationthat we have denied the immediacy of reality and that increating a substitute we have but spun another thread inthe web of our grand illusion.

Heinz Rudolf Pagels

Momenteel sluipt de deadline dichter bij, maar geen probleem want te dicht zal die nooitkomen. Omdat deze thesis niet door een enkele persoon gerealiseerd werd, wil ik nog enkelemensen bedanken. Allereerst wil ik mijn begeleidster Severine bedanken voor de vele feedbacken om mij in het begin van de thesis snel op weg te zetten in het toen nog onbekende terrein.Ook wil ik Stijn bedanken voor de nuttige tips en de andere inzichten die hij aanbracht.Ingmar wil ik bedanken voor de uitgebreide feedback en het snelle leeswerk dat hij realiseerde.Tenslotte wil ik mijn ouders en familie bedanken voor hun steun en interesse.

Bedankt!

Timothy

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Summary

Traditionally, the pharmaceutical industry mainly relies on batch processing. However, cur-rently the intention and opportunity exist to take the step towards continuous productionprocesses. Continuous production processes offer several advantages, i.e. easier to under-stand, improved quality, reduced generation of waste, increased safety, etc (Mortier et al.,2011, 2012).

In this study only a part of the entire continuous drying process of granules was investigated,i.e. the drying process. The drying is performed by means of a fluidized bed, which iswidely used in industrial drying processes because of its specific characteristics. The shifttowards continuous production processes largely depends on improved process knowledge. Forthis purpose, a coupled population balance modelling (PBM) - computational fluid dynamics(CFD) model can be developed. In this work a preparatory step to achieve this is studied,i.e. the reduction of a drying model in preparation of incorporation in a PBM.

The considered granules consist of a porous solid which is surrounded by a small layer ofwater. A suitable single particle drying model was proposed by Mezhericher et al. (2008), whodivides the drying process in two conceptual phases, i.e. the first phase which is consideredas droplet drying and the second phase in which the water is evaporating out of the solidthrough the pores. This single particle model was already fitted to experimental data byMortier et al. (2012). However, the considered model showed a significant computational loadto calculate the drying process of one single particle. Therefore, the direct use of the modelin the PBE would result in a high computational burden with far too long calculation times.To counter this computational limitations, a model reduction framework was proposed whichcould describe the most important dynamics of the original mechanistic model.

In order to gain more insight in the mechanistic drying model and to determine the mostimportant degrees of freedom, a local and global sensitivity analysis were performed. Bothsensitivity analyses showed the same result, i.e. the gas temperature was the most importantprocess variable for both drying drying phases, followed by the gas velocity. For the firstdrying phase both the gas temperature and the gas velocity were incorporated in the empiricalmodel. The constructed model had only a mean relative deviation of 1.10% in comparisonwith the mechanistic model. For the second drying phase only the gas temperature wasincorporated, because the influence of the gas velocity was too limited for an extension of theempirical model (a maximum deviation of ± 1.5% of the drying time was observed betweenthe output at standard conditions and at maximum gas velocity). The empirical model ofthe second drying phase showed a mean relative deviation of 1.97% with the mechanistic

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drying model. For both drying phase the same approach was followed and showed reliableresults and, hence, the developed model reduction technique can be considered more widelyapplicable. The obtained empirical model was used in different numerical methods for thePBE in order to determine its applicability for PBM. The four numerical implementations, i.e.finite difference method (FD), method of characteristics (MOC), method of moments (MOM)and high-resolution finite volume method (HRFV), were compared in terms of calculationtime and accuracy of the evolution of both the particle density distributions and the moments.MOC showed a reasonable accuracy and calculation time, making it a promising techniquefor further studies of the PBM for a drying population of particles.

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Samenvatting

Traditioneel maakt de farmaceutische industrie vooral gebruik van batchgewijze productie vangeneesmiddelen. Echter, op dit moment bestaat de intentie en mogelijkheid om de overstap temaken naar continue productieprocessen. Continue productieprocessen bieden verschillendevoordelen: ze zijn eenvoudiger te begrijpen, bieden een verbeterde kwaliteit van de producten,leveren een vermindering van de hoeveelheid geproduceerd afval, zijn veiliger,... (Mortier et al.,2011, 2012).

In deze thesis werd slechts een deel van het gehele productieproces bestudeerd, namelijk hetdroogproces van farmaceutische granules. Het drogen wordt uitgevoerd met behulp van eengefluıdiseerd bed, deze worden reeds op grote schaal gebruikt in industriele droogprocessenvanwege hun specifieke kenmerken. De verschuiving naar continue productieprocessen hangtgrotendeels af of de processen beter begrepen kunnen worden. Om dit te bereiken kan eengekoppeld population balance modelling (PBM) - computational fluid dynamics (CFD) modelworden ontwikkeld. In dit werk wordt een voorbereidende stap onderzocht, meer bepaald hetverlagen van de model complexiteit om dit gereduceerd model nadien te kunnen gebruiken ineen PBM.

De beschouwde granules bestaan uit een poreuze kern die omgeven is door een waterlaagje.Een geschikt droogmodel voor een enkel partikel is geopperd door Mezhericher et al. (2008),die het droogproces in twee conceptuele fasen splitst. In de eerste fase wordt de granulebeschouwd als druppel en in de tweede fase verdampt het water uit de porien van het partikel.Dit model werd al uitgebreid en gefit aan experimentele data door Mortier et al. (2012). Deberekening van het beschouwde model vereiste echter relatief veel rekenkracht voor slechtseen enkel deeltje. Het rechtstreekse gebruik van het droogmodel in de PBE zou dan ookresulteren in veel te lange rekentijden waardoor het praktisch niet mogelijk zou zijn om dePBE te berekenen. Om deze computationele beperkingen weg te werken, werd een algemeenkader voor model reductie gecreeerd die als doel heeft de belangrijkste dynamiek van deoorspronkelijke mechanistische model te beschrijven.

Om meer inzicht te krijgen in het mechanistisch droogmodel en de belangrijkste vrijheids-graden te bepalen, werden een lokale en globale gevoeligheidsanalyse uitgevoerd. Beide gevoe-ligheidsanalyses hadden hetzelfde resultaat, namelijk dat voor beide droogfasen de gastem-peratuur de belangrijkste procesvariabele was, dit gevolgd door de gassnelheid. Voor de eerstedroogfase werden zowel de gastemperatuur als de gassnelheid opgenomen in het empirischemodel. Het samengestelde model had slechts een gemiddelde relatieve afwijking van 1.10%in vergelijking met het originele mechanistisch model. Voor de tweede droogfase werd enkel

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de gastemperatuur opgenomen, omdat de invloed van de gassnelheid op de droogsnelheid tebeperkt was om het empirische model effectief uit te breiden (de maximale afwijking vande vereiste droogtijd bedroeg slechts ± 1.5% tussen de maximale gassnelheid en de stan-daardsituatie van 200 m3/h). Het empirische model van de tweede droogfase had een gemid-delde relatieve afwijking van 1,97% met het mechanistische droogmodel. Voor beide droog-fase werd dezelfde procedure gevolgd en toonde betrouwbare resultaten. Daarom werd eenruimer kader ontwikkeld voor de toepassing van de gebruikte modelreductie. Het verkregenempirische model werd gebruikt in verschillende numerieke methoden voor de PBE om detoepasbaarheid van PBM bepalen. De vier numerieke implementaties, meer bepaald de fi-nite difference methode (FD), de method of characteristics (MOC), de method of moments(MOM) en de high-resolution finite volume methode (HRFV), werden vergeleken op basis vanrekentijd en de nauwkeurigheid van de evolutie van de zowel de dichtheidsdistributie als demomenten. MOC liet een goede nauwkeurigheid en rekentijd optekenen, waardoor het eenveelbelovende techniek is om PBM mee uit te voeren.

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Contents

List of Symbols xiii

List of Abbreviations xv

List of Figures xvii

List of Tables xxi

Introduction 1

1 Literature Review 31.1 Production of pharmaceutical tablets . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Fluidized bed drying process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Single Particle drying model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 First drying phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.2 Second drying phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Model Reduction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Population Balance Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5.1 One-dimensional population balances . . . . . . . . . . . . . . . . . . . . 111.5.2 Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Materials and Methods 192.1 Stepwise Model Reduction Approach . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Local Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Global Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.2 Standardized Regression Coefficients . . . . . . . . . . . . . . . . . . . . 24

2.4 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 Population Balance Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Results and discussion 313.1 Local Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 First drying phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.2 Second drying phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Global Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.1 Sensitivity of moisture content towards the process variables . . . . . . 373.2.2 Sensitivity of moisture content towards the physical parameters . . . . . 40

3.3 Model Reduction: First Drying Phase . . . . . . . . . . . . . . . . . . . . . . . 41

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3.3.1 Fitting the growth term in function of the wet radius . . . . . . . . . . 413.3.2 Extension of the model with the gas temperature . . . . . . . . . . . . . 433.3.3 Extension of the model with the gas velocity . . . . . . . . . . . . . . . 48

3.4 Model Reduction: Second Drying Phase . . . . . . . . . . . . . . . . . . . . . . 503.4.1 Adaptive time step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.4.2 Fitting the growth term in function of the wet radius . . . . . . . . . . 513.4.3 Extension of the model with the gas temperature . . . . . . . . . . . . . 543.4.4 Extension of the model with the gas velocity . . . . . . . . . . . . . . . 57

3.5 Population Balance Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.5.1 Finite difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.5.2 Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 603.5.3 High Resolution Finite Volume . . . . . . . . . . . . . . . . . . . . . . . 613.5.4 Method of moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.5.5 Comparison between the different numerical implementations . . . . . . 62

4 Conclusions and perspectives 654.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Bibliography 67

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List of Symbols

β power coefficient of the porosity ε

cp,g specific heat of the drying agentcp,s specific heat of the solid

Dv,cr coefficient of vapour diffusion in the crust pores

ε emissivity of the solidη dynamic viscosity of the drying agent

Gr Growth

h net birth ratehD mass transfer coefficient

kd thermal conductivity of the particlekg thermal conductivity of the drying agentks thermal conductivity of solid in the wet core and crustkw thermal conductivity of liquid in the wet core

mk kth moment of a particle size distributionmv change of vapour mass in timeMw molecular weight of the liquid

Pg gas pressurepg pressure of the drying agentϕ porosity of the particle crust regionpv,i partial vapour pressure at the crust-wet core interfacepv,∞ partial vapour pressure in the ambient

< universal gas constantR external coordinatesRd radius of the dried droplet

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Rd,0 initial radius of dried dropletρd,w density of liquid fraction in the dropletρg density of the drying agentρs density of the solid in the wet core and the crustρv,∞ partial vapour density in the ambientρv,s partial vapour density over the droplet surfaceρwc,w density of the liquid component in the particle wet coreRi radius of the crust-wet core interfaceRp radius of the solid particleRw wet radius of the particle

Tcr,s temperature of the crust outer surfaceTg gas temperatureTp,0 initial temperature of the particleTwc critical temperature of waterTwc,s temperature of the crust-wet core interface

Vg gas velocity

wi weighted function

X internal coordinates

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List of Abbreviations

RH relative humidity

API active pharmaceutical ingredientAREM advanced rate elimination method

FD finite difference

GSA global sensitivity analysis

HRFV high resolution finite volume method

LHS latin hypercube samplingLSA local sensitivity analysis

MOC method of characteristicsMOM method of momentsMRE maximum relative errorMWR method of weighted residuals

ODE ordinary differential equation

PA pseudo-analyticalPBE population balance equationPBM population balance modellingPCA principal component analysisPD product-differencePDF probability distribution functionPSD particle size distribution

QMOM quadrature method of moments

REM rate elimination methodRMSE root mean square error

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SMM standard method of momentsSRC standardized regression coefficientSVD singular value decomposition

TRS total relative sensitivity

VSM variable simplification method

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List of Figures

1 Structure of the fluidized bed drying model (Mortier et al., 2011). . . . . . . . 1

1.1 Scheme of typical processing of pharmaceutical products during the formulationof tablets (Mortier et al., 2011). . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 ConsiGma fluidized bed dryer (Mortier et al., 2012). . . . . . . . . . . . . . . 4

1.3 Scanning Electron Microscopy photographs of granules after the drying process(Hegedus and Pintye-Hodi, 2007). . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Two phase drying of a droplet containing solids (Mezhericher et al., 2008) . . . 6

1.5 The evolution of the wet radius Rw in function of the time for both dryingphases at standard conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6 An illustration of the QMOM (Yeoh and Tu, 2009). . . . . . . . . . . . . . . . 13

2.1 Schematic illustration of the followed strategy for constructing a PBM for sim-ulating the drying behaviour of a population of particles (Mortier et al.). . . . . 20

2.2 The visualisation of the LHS for a normal probability distribution. In thisexample five sample have to be taken, so at the y axis the probability rangeis split in five equal parts. This ensures that the entire range is sampled withrespect to the probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Scatterplots of Y versus Z1 and Z2 after Saltelli et al. (2007) . . . . . . . . . . 25

2.4 The evolution of the wet radius Rw and the growth term Gr in function of thetime for both drying phases at nominal parameter values. . . . . . . . . . . . . 27

2.5 Scheme of the steps taken during the model complexity reduction (Mortier et al.). 28

3.1 Sensitivity functions of the first drying phase for all process variables. . . . . . 32

3.2 The mean quadratic TRS for the first drying phase for the different processvariables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 The quadratic TRS for all physical parameters. . . . . . . . . . . . . . . . . . . 34

3.4 Sensitivity functions of the second drying phase for all process variables. . . . . 35

3.5 The quadratic TRS of the second drying phase for all process variables. . . . . 36

3.6 The quadratic TRS of the physical parameters for the second drying phase. . . 36

3.7 Monte Carlo simulations for both drying phases. . . . . . . . . . . . . . . . . . 37

3.8 The linear SRC model and the rescaled generated growth rate of the first dryingphase after 3 seconds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.9 The linear SRC model and the rescaled generated growth rate of the seconddrying phase after 1 second. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.10 The growth term Gr and the first three derivatives in function of the wet radiusRw at nominal values (Table 2.1). . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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3.11 The growth rate, calculated with the mechanistic model, in function of Rw andTg for the first drying phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.12 Coefficients of the empirical model for the first drying phase in function of thegas temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.13 The effect of an increase of the coefficients by 10% on Gr. D shows the highestmaximum sensitivity of all coefficients. . . . . . . . . . . . . . . . . . . . . . . . 45

3.14 The values of the D and C coefficients after incorporating the second orderpolynomial for describing the evolution of D in function of the gas temperature. 46

3.15 The values of the C coefficient before and after the implementation of the Cfunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.16 The nonlinear regression of the B and A coefficients. . . . . . . . . . . . . . . . 47

3.17 The relative deviation between the mechanistic and empirical model in functionof Tg and Rw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.18 The relative drying rate at 55 for different gas velocities compared to thestandard case of 200 m3/h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.19 The relative deviation of the empirical model Gr,1(Rw,nor, Tg, Vg) in function ofTg and Vg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.20 The estimation of the drying time needed at a specific temperature by Formula(3.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.21 The growth rate of the second drying phase, for particles with R′w,nor smallerthen 0.5, can be linearised by logarithmic transformation. The suggested modelstructure for the left part of the second drying phase is A′ · (R′w,nor)B

′. . . . . . 52

3.22 The growth rate of the second drying phase, for particles with R′w,nor greaterthan 0.5, cannot be linearised by logarithmic transformation. The suggestedmodel structure (C ′ · (1−R′w,nor)E

′) needs to be extended. . . . . . . . . . . . 53

3.23 The linearised logarithmic transformation by using C ′ · (1+D′ ·R′w,nor)E′, with

D′ equal to −0.98 to illustrate the effect of the coefficient. . . . . . . . . . . . . 53

3.24 The growth rate, calculated with the mechanistic model, in function of Rw andTg for the second drying phases. . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.25 The growth rate of the second drying phase was approximated well for alltemperatures by the proposed model structure (Equation 3.9). . . . . . . . . . 54

3.26 The influence of the offset between the mechanistic and empirical model on therequired drying time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.27 Evolution of the relative offset in function of the gas temperature before arelative offset reduction was implemented. . . . . . . . . . . . . . . . . . . . . . 55

3.28 The relative deviation of the empirical model in function of Tg and Rw. . . . . 57

3.29 The evolution of Rw for the second drying phase in time for both the mecha-nistic model and the empirical model. . . . . . . . . . . . . . . . . . . . . . . . 58

3.30 The influence of Vg on the total drying time of the second drying phase isminimal. The relative offset is given for the two most extreme values of Vgwith respect to the reference value of 200 m3/h. . . . . . . . . . . . . . . . . . . 58

3.31 The particle density distribution of FD (100 classes and ∆t = 0.05 s.) after 6and 14 seconds and the pseudo-analytical solution with HRFV. . . . . . . . . . 60

3.32 The particle density distribution of MOC after 6 and 14 seconds and the pseudo-analytical solution with HRFV. The approximation of MOC with 200 classesis considered as good. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.33 The particle density distribution of HRFV for 200, 500 and 5000 classes. Alarge number of classes is needed to obtain an accurate density distribution,this leads to high computational times. . . . . . . . . . . . . . . . . . . . . . . . 61

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3.34 The evolution of the zeroth moment for the different numerical methods. Thealgorithms using a fixed grid (HRFV and QMOM) show a constant value overthe entire simulation, which should be the case because no nucleation, breakageor aggregation occur and no particles are removed or added to the control volume. 63

3.35 The evolution of the first moment for the different numerical implementations.The evolution shows that the population is actually drying, however the fixedgrid implementations (HRFV and QMOM) predict a faster drying than themoving grid implementations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.36 The evolution of the first moment for the FD implementation and the pseudo-analytical solution of the HRFV method. . . . . . . . . . . . . . . . . . . . . . 64

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List of Tables

2.1 Nominal values of the process variables used in the LSA. . . . . . . . . . . . . . 202.2 Nominal values of the physical parameters used in LSA. . . . . . . . . . . . . . 212.3 Range of the process variables for the Monte Carlo simulations. . . . . . . . . . 24

3.1 Values of the MRE criterion for all process variables for the first drying phase. 313.2 Values of the MRE criterion for all process variables for the second drying phase. 323.3 The SRC values of the process variables for both drying phases. . . . . . . . . . 393.4 Values of the eliminated process variable samples and the corresponding mass

transfer rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.5 The range of all physical parameters for generating 2000 samples by using LHS. 403.6 The ranked SRC for all physical parameters for the first drying phase after one

second. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.7 The ranked SRC for all physical parameters for the second drying phase after

one second. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.8 Functions for incorporating the influence of Tg for the first drying phase. . . . . 473.9 Empirical coefficients for describing the influence of Rw and Tg for the growth

rate of the first drying phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.10 The coefficients of the empirical model G∗r,1. Differences in the coefficients

between Gr,1 and G∗r,1 are indicated. . . . . . . . . . . . . . . . . . . . . . . . . 493.11 Theoretical drying times for different gas temperatures. . . . . . . . . . . . . . 513.12 Functions of the different coefficients for describing the second drying phase

with dependency of Rw and Tg. . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.13 Empirical coefficients of the second drying phase. . . . . . . . . . . . . . . . . . 563.14 The mean deviation between the drying time needed . . . . . . . . . . . . . . . 573.15 The computational time of the FD in function of the number of classes and the

time step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.16 The computational time and drying time for MOC in function of the number

of classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.17 The drying time of the HRFV converges to a value of 470.6 seconds, which

can be considered as a pseudo-analytical solution. The computational timedrastically increases by using a large number of classes. . . . . . . . . . . . . . 62

3.18 The increase of the number of classes for QMOM leads to a slight elevation ofthe calculation and drying time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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Introduction

Problem statement

Conventional pharmaceutical manufacturing is generally accomplished using batch process-ing with off-line time-consuming and less efficient laboratory testing conducted on randomlycollected samples to evaluate product quality after each batch step. Hence, limited rele-vant information is obtained after the process, making process control difficult and inducingunnecessary batch losses. Furthermore, the different batch processes themselves are poorlyunderstood inefficient black-boxes. In this respect, availability of a mechanistic model wouldincrease the understanding of the fundamental scientific phenoma and processes. The advan-tages of continuous production increase when a model is available to support the developmentof a real-time process control (Mortier et al., 2011). However, nowadays still some problemsare encountered, e.g. difficulties to meet the high product quality standards, the conser-vatism and negativism of the regulatory authorities towards continuous processes as well aslack of detailed process knowledge needed to define and implement such control. To curethese problems an improved process understanding is needed in order to launch continuousproduction with on-line control, reliable sensors at informative locations, adapted equipmentand of course a validated model (Mortier et al., 2011). This study deals with different aspectsof a mechanistic model for a fluidized bed dryer.

Figure 1: Structure of the fluidizedbed drying model (Mortier et al.,2011).

A fluidized bed dryer is not homogeneous, at the bot-tom the gas velocity will be higher, the air humiditywill be lower because the incoming air is still fresh,etc. All these local ambient conditions have an in-fluence on the drying rate and the movement of theparticles. In order to model the fluidized bed dryer ina realistic way, a coupled population balance method(PBM)-computational fluid dynamics (CFD) can beused (Figure 1). CFD can be used for modelling thespatial heterogeneity of such a reactor and the motionof the granules. On the other hand, PBM can be usedto calculate the evolution of the moisture content fora population of granules potentially with different sizeand degree of ’dryness’. Both techniques can also beconnected: CFD predictions influence the PBM be-cause the drying rate is dependent of the local ambient conditions. Vice versa, the PBMaffects the CFD, because the moisture content of the particle determines its weight, thus also

1

its motion as density effects are accounted for in CFD. The coupled PBM-CFD will, hence,account for both spatial and population heterogeneity (Mortier et al., 2011). By using PBM,a more realistic calculation of an entire population can be accomplished. A population bal-ance equation (PBE) can calculate how a population of particles will evolve. The dynamicsof the moisture content of a single particle, e.g. the drying rate, should be easily computablebecause otherwise the calculation PBE will be too slow and it cannot be used for embeddingin a CFD-model. In order to achieve such a quick model, the complexity of the considereddrying model should be reduced. Figure 1 indicates the different steps to be taken in order toobtain this combined PBM-CFD model. This work will focus on a preparatory step betweenthe drying model and the PBM. Indeed, the current drying model is too complex to incorpo-rate as such in the PBM and requires a reduction. The PBM is also briefly tested using theobtained reduced model.

Objectives

In this study, the main objective is to perform a model reduction of a validated drying modelwhich can be implemented in a PBE. To achieve this, several steps are conducted. First, a localand a global sensitivity analysis is performed in order to obtain more insight in the mechanisticdrying model and to determine the most important process variables to be included in thereduced model. Second, the most important process variables will be used to construct areduced empirical model. Finally, this reduced model is implemented in a PBM to test itsapplicability. Several different solution methods for solving the type of PBM at hand arebriefly tested as well.

Outline of the thesis

In the first chapter, some background information from literature is provided on fluidized beddrying, the considered drying model, different model reduction techniques, PBM and solu-tion techniques. In the materials and methods chapter, the sensitivity analysis proceduresare explained and the model reduction procedure as well as the generalized model reductionframework are introduced. Moreover, some practical measures for comparing numerical solu-tions of the population balance equations (PBE) are given. The third chapter contains theresults of the sensitivity analyses, a detailed description of the model reduction and prelimi-nary results of the PBE, including comparison of some applicable solution methods. Finally,in chapter four conclusions and perspectives are given.

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1Literature Review

1.1 Production of pharmaceutical tablets

A pharmaceutical product typically consists of two types of ingredients, active pharmaceuticalingredients (APIs) and excipients. An API is a material that has a therapeutical effect andwhich is produced through either chemical or biological processing or a combination of both.The production of this active ingredients in bulk is often considered as the primary phase.Excipients on the other hand have no therapeutical effect but are needed to ensure that thefinal dosage of the active ingredients is correct. These admixtures are often produced at adifferent production site or by another company. Typical examples of excipients are water,lactose, starch, sugar and colouring agents, but in practice many other excipients are alsobeing used (Plumb, 2005; Mortier et al., 2011).

The second phase begins with the formulation of the final product in bulk when one or moreAPIs are mixed with a number of excipients. This process is followed by several consecutivesteps (Figure 1.1), which will eventually lead to tabletting in over 80% of the cases (Plumb,2005; Mortier et al., 2011). These can either be executed in batch or continuous mode (e.g.ConsiGma;Figure 1.2) The step following the formulation is the granulation. Granulation isa size-enlargement process in which small particles are combined into larger, physically strongagglomerates in which the original particles can still be identified (Hegedus and Pintye-Hodi,2007; Mortier et al., 2011). Agglomerates are more suitable for further processing. Thegranulation namely leads to improved powder flow properties, enhanced compressibility andreduced demixing and dust formation (Muzzio, 2002; Mortier et al., 2011).

The next step in the pharmaceutical process is the drying of the granules. The drying processrequires thermal energy, because moisture must be removed from solid material through evap-oration. This energy can be applied by convection, conduction or vacuum drying. Convectionis achieved by means of a flowing gaseous medium, in which the gaseous molecules transmitheat. This principle is for instance used in fluidized bed drying. Conduction can be realised byheat exchange between adjacent particles of matter, e.g. heat transfer through a jacked bowlwall. With the use of vacuum drying the heat necessary to remove the moisture is directlyapplied to the aggregate. The latter process has a major advantage because it can be appliedat lower temperatures. This could be very useful when heat-sensitive APIs are to be dried.However a drawback is the longer drying time which is needed (Mortier et al., 2011).

The fluidized bed granulation and drying, the high-shear granulation with fluidized bed dry-ing and the single-pot equipment with a drying unit are the methods most commonly used toproduce granules in production-scale pharmaceutical manufacturing. Single-pot technology

3

Figure 1.1: Scheme of typical processing ofpharmaceutical products during the formu-lation of tablets (Mortier et al., 2011).

Figure 1.2: ConsiGma fluidized bed dryer(Mortier et al., 2012).

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has grown in popularity because of the transfer of moist granules from the high-shear gran-ulation to the fluidized bed dryer is critical. The granules which are dried with the fluidizedbed are smaller than those dried with microwaves. This consequence is due to the two majorcharacteristics of the fluidized bed drying. The aggregates are colliding with each other andwith the wall of the fluidized bed dryer. The second reason is that fluidized bed drying in-volves a more rapid change in temperature, which causes a faster evaporation of the moisture.Due to the much higher evaporation rate the granules experience more internal stress and willtherefore break in smaller parts. These two main properties of fluidized bed drying result insmaller aggregates that are more rugged (Figure 1.3). The advantage of fluidized bed dryingis the shorter drying time (Hegedus and Pintye-Hodi, 2007; Mortier et al., 2011).

(a) Microwave drying (b) Fluidized bed drying

Figure 1.3: Scanning Electron Microscopy photographs of granules after the drying process(Hegedus and Pintye-Hodi, 2007).

1.2 Fluidized bed drying process

Fluidized bed drying is a technology which has already been used for many years by theindustry for drying of wet solid particles. Many products, such as maize (Mourad et al.,1995), coconut (Niamnuy and Devahastin, 2005), baker’s yeast (Turker et al., 2006), beans(Nitz and Taranto, 2007), black tea (Temple, 1999), nylon (Ng and Tan, 2008) and coal (Potterand Keogh, 1981), can be successfully dried by using fluidized bed dryers. This technologyis widely used in industrial drying processes because of its specific characteristics. A highrate of heat and mass transfer and a high rate of solid transport to or from the dryer arerequired by the industry. Fluidized bed drying meets this requirements because it provides alarge contact surface area between solids and gas, high thermal inertia of solids, good degreeof solids mixing and rapid transfer of heat and moisture between solids and gas that shortensdrying time considerably without damaging heat sensitive materials. Furthermore, fluidizedsolid particles can be easily transported in or out of the dryer. However, the applicationof fluidized bed drying also has some serious drawbacks that lower the fluidization qualityand drying performance. There are problems such as attrition or pulverization of solids,agglomeration of fine particles, higher capital and operational costs of fluidized bed dryingin comparison to other drying techniques and the poor fluidization and non-uniform productcontrol of particles of the Geldart group C and D. The particles, which are part of theGeldart group C, are extremely fine (20-30 µm) and therefore the most cohesive ones becausethe surface forces become extremely important. The powder of Geldart group C will lift as aplug in small diameter tubes and lead to bad fluidization. The Geldart group D consists ofparticles which are quite big (>600 µm) and have high densities, therefore the fluidization of

5

group D requires very high fluid energies, causes a lot of abrasion and will lead to a spoutedfluidized bed (Geldart, 1973; Mortier et al., 2011). The higher operational costs, which werementioned earlier, are due to the higher pressure drop across the bed and the higher capitalcosts are caused by the wall friction of the fluidized particles. Another problem encounteredwith fluidized bed drying is the scaling-up. Nowadays for each specific material a pilot-planthas to be set up in order to obtain empirical data for the scaling-up, because no reliablemathematical models are available for fluidization (Daud, 2008; Mortier et al., 2011). For thedata collection, the ConsiGma continuous form-powder-to-tablet production line from GEAPharma Systems (Collette, Wommelgem, Belgium) was used. The continuous line consistsof three parts: a continuous twin-screw granulator (high shear), followed by a six-segmentedfluidized bed dryer system and a discharge system (Mortier et al., 2012).

1.3 Single Particle drying model

Though a single-droplet (containing a granule) drying model does not take into account manyof the complex interactions encountered during fluidized bed drying, it offers a practical anddirect way of observing the rates of evaporation and morphological changes at the drop-wise orparticulate level. According to the morphology of the individual droplets, the drying processcan be divided into two conceptual phases (Figure 1.4). In the first drying phase the droplet,that contains solids, enters the drying volume, gets or gives sensible heat and the evaporationoccurs resulting in the shrinking of the droplets’ diameter. The second drying phase starts atthe point where a dry solid crust, surrounding the wet core, is formed. In this phase waterevaporation takes place inside the wet particle at the receding interval between the crust andthe wet core. The vapour, which is generated over the interface, diffuses through the crustpores outside (Mezhericher et al., 2007, 2008; Mortier et al., 2011).

Figure 1.4: Two phase drying of a droplet containing solids (Mezhericher et al., 2008)

In previous models the distinction was often made between a single droplet with dissolvedsolids and a single droplet with insoluble solids. This distinction is no longer needed, becausethe proposed model can cope with both. The optimised model proposed by Mezhericheret al. (2007) solves the main problems which were encountered in previous models. Theseproblems were: bad formulation of the steady-state conditions, ignoring of the initial heating-up period, unjustified neglecting the droplet/wet particle temperature profile, disregarding

6

crust porosity and inaccurate calculation of mass transfer rate. Also the heat absorption bythe crust was neglected in some models. A recently further optimised model also accounts forthe temperature dependence of the physical properties of the droplet/particle (Mortier et al.,2011).

1.3.1 First drying phase

At the beginning of the drying process, a typical droplet consists of large amounts of liquidand insoluble or dissolved solids. The excessive liquid forms a layer over the particle surface,so the first drying phase is similar to the evaporation of a pure liquid droplet (Mezhericheret al., 2008). The rate of change of the radius of the drying droplet Rd is given by:

d(Rd)

dt= − 1

ρd,w4π(Rd)2mv,1 (1.1)

with ρd,w the density of the fluid of the droplet and mv,1 the mass transfer rate of the firstdrying phase.

Because the first drying phase is similar to the evaporation of a pure liquid droplet, the massconvection law can be used to describe the mass transfer rate from the droplet surface to theambient environment, i.e. the surrounding gas phase (Mezhericher et al., 2008):

mv,1 = hD(ρv,s − ρv,∞)4π(Rd)2 (1.2)

with hD the mass transfer coefficient, ρv,s the partial vapour density over the droplet surfaceand ρv,∞ the partial vapour density in the ambient. The evolution of the droplet radius Rdat standard conditions is given in Figure 1.5(a).

1.3.2 Second drying phase

During the second drying phase, the wet particle is conceptually considered as a sphere withisotropic physical properties and temperature-independent crust thermal conductivity. Thecrust region is treated as one pierced by a large number of identical straight cylindrical capil-laries, and the wet core is assumed to be a sphere with liquid and solids. Following expressionhas been proposed for the crust-wet core interface receding rate (see Figure 1.4):

d(Ri)

dt= − 1

ϕρwc,w4πR2i

mv,2 (1.3)

with Ri the radius of the crust-wet core interface, ϕ the porosity of the particle crust regionand ρwc,w the density of the liquid component in the particle wet core. The mass transfer ratemv,2 between the crust-wet core interface and the ambient is given by:

mv,2 = −8πϕβDv,crMwpg<(Tcr,s + Twc,s)

RpRiRp −Ri

ln

pg − pv,ipg −

(<

4πMwhDR2pmv +

pv,∞Tg

)Tp,s

(1.4)

with Rp the radius of the solid particle, Dv,cr the coefficient of vapour diffusion in the crustpores, Mw the molecular weight of the liquid, pg the pressure of the drying agent, pv,i andpv,∞ the partial vapour pressures at the crust-wet core interface and in the ambient, Twc,sand Tcr,s the temperatures of the crust-wet core interface and the crust outer surface, < the

7

universal gas constant and β an empirical power coefficient. It is noteworthy that Equation 1.4is an implicit relationship, a solution of the mass transfer rate mv can only be found in aniterative way. A model extension was performed by Mortier et al. (2012) who found a relationbetween β and Tg from experimental data. The relation was fitted by an exponential function(Equation 1.5).

β = β1 · expβ2·Tg (1.5)

with β1 and β2 representing empirical parameters. The evolution of Ri for the extended modelat standard conditions is given in Figure 1.5(b).

0 2 4 6 8 105.95

6

6.05

6.1

6.15·10−4

t (s)

Rw

(m)

(a) First drying phase

0 100 200 300 400 5000

2

4

6·10−4

t (s)

Rw

(m)

(b) Second drying phase

Figure 1.5: The evolution of the wet radius Rw in function of the time for both drying phasesat standard conditions.

1.4 Model Reduction Methods

In order to solve the drying process of a population of particles, the complexity of the mecha-nistic drying model has to be reduced for an efficient evaluation because such detailed physi-cally based mathematical models are time consuming to solve and require the use of sophisti-cated hardware and software resources (Banerjee et al., 1998). Multiple techniques have beendeveloped to reduce complex models, e.g. semiconductor devices, forecast models, molecularsystems (Antoulas, 2005; Mortier et al.). In what follows, a short review of reduction methodsproposed in the literature is given.

The first group of model reduction techniques are the heuristic model reduction methods.However the main disadvantage is that they require a lot of user input. A detailed analysis ofthe model behaviour with regard to the selected set of parameters is needed. The interactionand feedback between model components to identify key processes of the system should beassessed. The changes in model structure must be done by domain experts (Van Nes andScheffer, 2005; Mortier et al.).

The second group is the most simple reduction technique, i.e. methods that are based on thelinearisation or the reduced-order series expansion of the system’s nonlinearities using Tayloror Volterra series. However the application of this techniques is only possible for weaklynonlinear systems (Phillips, 2000). This group of techniques can be considered as not usefulfor the considered model, because very strong nonlinearities are present in the drying rate.

The third group of model reduction techniques are those based on mathematical concepts.Several of those techniques are projection based, i.e. where the systems are projected onto a

8

lower-dimensional subspace and the model equations are solved for the substituted projectedstates (Bernhardt, 2008). The methods used most in this area will be briefly described. Thefirst method is the singular value decomposition (SVD) which cannot be applied to highlycomplex systems. The second method, i.e. principal component analysis (PCA) (also knownas proper orthogonal decomposition (POD)), can be used for nonlinear systems (Antoulas,2005) and aims to obtain low-dimensional descriptions that capture much of the phenomena ofinterest, thereby using the covariance matrix of the data to determine the leading eigenvectors.In fact the purpose of the two methods is the same but there are two main differences. TheSVD is the discrete form of the PCA and it can be computed for non-square matrices while thePCA can only be calculated for square matrices (Chatterjee, 2000; Mortier et al.). Moreoverthe PCA can be obtained from the SVD of mean centered data. The mean centering isthe important difference between PCA and SVD and can yield qualitatively different resultsfor datasets where the mean is not equal to zero. An important advantage by using theSVD instead of constructing the covariance matrix directly from the data to perform PCA,because this can lead to loss of precision when encountering small numbers in the datamatrix(Dhillon, 2001). This technique can be applied to the considered drying model, however acomplex model structure will be produced.

The next group is a data adaptive model reduction scheme described by Bernhardt (2008);Mortier et al., which can be applied to the transformation and reduction of systems of ordinarydifferential equations (ODEs). It is a multistep approach using the a low dimensional projec-tion of the model data followed by a Genetic Program/Genetic Algorithm hybrid method toevolve the new model systems. PCA and parameter tuning importance are two techniquesthat were compared by (Mortier et al.). The PCA approach offers the opportunity to be usedas a self-controlled routine, meaning that this procedure can be repeated automatically untila predefined upper-limit of the error functional is achieved. By using the parameter tuningimportance, no unique criterion could be identified because even parameters with small pa-rameter tuning importance caused significant model prediction discrepancies. A step-by-stepprocedure must be followed when using this procedure and the model reduction result mustbe studied critically (Mortier et al.). This technique can be applied to the considered dryingmodel, however a complex model structure will be produced.

The last group of model reduction techniques are those based on evaluating the sensitivityof the performance indicators to a parameter vector. The advanced rate elimination method(AREM) belongs to this category and focuses on the importance of individual rates, small ratesare subsequently eliminated from the model. A drawback of the AREM is that the method hasto be carried out for all rates, which is particular the case for models with large computationaltimes. The not-’advanced’ rate elimination method (REM) proposed by Lawrie (2008) couldbe of use, by first screening out some of the uninfluential rates, and afterwhile using the morecomputationally intense AREM to the remaining rates. The variable simplification method(VSM) concerns identifying state variables that can be set to constants by assessing theirimportance of variation. The reduction techniques of this group have been applied for thereduction of ecosystem models and showed significant reductions in model complexity, e.g.Lawrie and Hearne (2007) applied AREM for a environmental model and eliminated 85% ofthe rates which were originally incorporated. The advantage of the techniques of the lattergroup is the automatical reduction of the model complexity, however with the use of thismethods there is no guarantee that the obtained model for a given diagnostic is the simplestone (Lawrie and Hearne, 2007; Mortier et al.).

Not one non-heuristic technique is suited to reduce the model complexity of the originalimplicit model and replacing it by a model structure which is easy accessible. Therefore itwas decided to perform a heuristic model reduction in which the new created model should

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incorporate the most important variables. The methodology which was developed during themodel reduction was used to create a framework.

1.5 Population Balance Models

In systems where particles are present, the system dynamics are often significantly affectedby the behaviour of these particles (Ramkrishna, 2000). This is also the case for fluidized beddrying, many heterogeneities are encountered, e.g. particles have different degrees of wetness,they encounter other ambient conditions, etc. The incorporation of these heterogeneities canbe realised by using population balance modelling (PBM).

Firstly the behaviour of the single particles in their local environment is needed to analysethe behaviour of the entire population of particles. This population can be described bya density distribution of a suitable variable such as the number of particles, the mass, thevolume etc. To characterize the dynamics of this distribution, variables affecting them needto be identified. Those variables can be divided in two major classes, more specific: theinternal and the external coordinates. The external coordinates represent the spatial locationof the considered population of particles. On the other hand the internal coordinates containa quantitative characterization of certain properties of the considered particle, except itsphysical location. The combination of the internal and the external coordinates makes upthe particle state space. The properties of the particle population state space can be bothdiscrete (e.g. discrete sites in a certain space) or continuous (e.g. wetness). Depending onthe number of internal coordinates, a PBM can be one- or multidimensional.

The local environment around the particle also influences the behaviour of the particle. Thiscontinuous phase consist of several variables which quantitatively describe the local environ-ment (e.g. the gas velocity, gas temperature, humidity, etc.) and is indicated by vector Y.The continuous phase is only dependent of the external coordinates and time.

Mathematical formulation

A general population balance equation (PBE) is given by Equation 1.6 which describes thechange of the density distribution f(X,R, t) in time and consists of several terms:

∂

∂tf(x, r, t) +∇x · X f(x, r, t) +∇r · R f(x, r, t) = h(x, r,Y, t) (1.6)

Firstly, the two divergence terms represent the convective transport, one along the internalcoordinates X and one along the external coordinates R. These terms are continuous intime, and describe the evolution of X and R. These processes are convective because theyresult from convective motion in the particle state space. Furthermore no change in the totalnumber of particles in the system is caused, except when particles depart from the boundariesof the system. The net birth rate h is responsible for the change in number of particlesdue to birth and death processes, like aggregation, nucleation or breakage. This last term isdiscontinuous, as particles appear and disappear at discrete time steps. This equation has tobe supplemented by initial and boundary conditions (Ramkrishna, 2000). An example of a netbirth term is showed in Equation 1.7, the physical meaning of each term has been indicated(Kumar et al., 2008).

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h(x, r,Y, t) =

birth due to aggregation︷ ︸︸ ︷1

2

∫ x

0β(x− x′,x′, t)f(x− x′, t)f(x′, t)dx′

death due to aggregation︷ ︸︸ ︷−f(x, t)

∫ ∞0

β(x,x′)f(x′, t)dx′ (1.7)

birth due to breakage︷ ︸︸ ︷+

∫ ∞x

γ(x′)b(x′,x)p(x,x′)dx′

death due to breakage︷ ︸︸ ︷−b(x,x′)f(x, t)

1.5.1 One-dimensional population balances

When the general PBE mentioned in Equation 1.6 is rewritten in a one-dimensional form, anonhomogeneous PBE for size-dependent growth is obtained (Gunawan et al., 2004):

∂

∂tf(x, t) +

∂[G(x, t)f(x, t)]

∂x= h(x, t) (1.8)

where G(x, t) is the growth or decay rate of the considered internal coordinate. In this studyh(x, t) is equal to zero, because it is presumed that nucleation, breakage or aggregation donot take place in the fluidized bed. Also no particles are entering or leaving the system. Theone-dimensional homogeneous population balance will be treated more in detail, because onlythis type of population balances will be used in this thesis. The solution of Equation 1.8is not straightforward, therefore multiple numerical algorithms have been developed. Thosealgorithms are treated in more detail in the next section.

1.5.2 Numerical Solutions

The accurate simulation of the dynamics of the distribution can be challenging, becausethe distribution can extend many orders of magnitude in size and time, and the gradientsof the distribution can be very sharp. Therefore several specialised algorithms were devel-oped to solve this type of PBE numerically, those algorithms can be roughly divided in fivedifferent classes (Gunawan et al., 2004): Method of Moments, Method of Characteristics,Method of Weight Residuals/Orthogonal collocation, Monte Carlo Method and Finite differ-ence schemes/discrete population balances.

1) Method of Moments

Standard method of moments (SMM) Instead of tracking a whole distribution of par-ticles, Hulburt and Katz (1964) developed a technique to transform the problem into thetracking of some lower order moments of the size distribution. The moments of the particlesize distribution (PSD) are given by Equation 1.9:

mk(x, r, t) =

∫ ∞0

f(x, r, t)xkdx (1.9)

with mk the kth moment of the PSD.

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The first few integer moments are important, because they describe important physical quan-tities like the total number of particles, the size, the surface, the volume (Yeoh and Tu, 2009).The primary advantage of the SMM is its numerical economy, because it only tracks theevolution of a small number of moments (Frenklach, 2002). The second advantage is thatit does not suffer from truncation errors in the PSD approximation. Mathematically, thetransformation from a PSD space to a space of moments is rigorous. However only undercertain conditions, the moment equations are closed. Under certain conditions this momentscan be closed and as a consequence the differential equations of the lower order moments donot depend on the values of the higher order moments. When this condition is fulfilled thesmall number of ODEs can be solved very efficiently and with very high accuracy. However,this moments closure conditions are often violated for more complex models (Gunawan et al.,2004; Kariwala et al., 2012). Another weakness of the method is that by only calculating themoments the PSD is not accessible at every time step. Reconstruction techniques can be used,but those are far from ideal. Nowadays, mostly a particle density distribution is assumed apriori so only a limited number of moments are needed. Other reconstruction techniques areavailable but also show drawbacks which limit their applicability, e.g. simplification of theproblem, extensive computational calculation time (John et al., 2007).

quadrature method of moments (QMOM) To overcome the closure problems whichwere encountered with SMM McGraw (1997) introduced a method which was based on theGaussian quadrature. This technique is based on the numerical approximation of the PSDby a finite set of Dirac’s delta functions as shown in Equation 1.10. The transformationfrom a PSD space to a space of moments is therefore no longer needed, the approximation isalso illustrated in Figure 1.6. It should be noted that the Gaussian quadrature suffers fromtruncation errors, however it successfully eliminates the problem of fractional moments forwhich the special closure consideration is usually required (Yeoh and Tu, 2009).

f(x, r, t) ≈N∑i=1

wi(r, t)δ(x− xi(r, t)) (1.10)

with wi the weights.

The N abscissas xi and the N weights wi are determined from 2N moments of f(r) by inversionof:

µk ≡∫ ∞

0xkf(x)dx =

N∑i=1

xkiwi (1.11)

with k = 0, 1, · · · , 2N − 1.

It can be easily seen that for example the first six moments are needed to determine allxi and wi for a QMOM approximation of an order of three. For polynomial PSD of order2N-1 the quadrature is exact, for other functions it can be approximated very well. Forthe determination of the different weights and abscissas several algorithms have been devel-oped. The algorithm originally used is the product-difference (PD) algorithm which is basedon the theory of orthogonal polynomials. The coefficients of the ’three-term-recurrence re-lationship’, that is satisfied by orthogonal polynomials, are calculated using a sequence ofmoments. The coefficients are thereafter used to calculate the quadrature points or abscissasand the weights. This algorithm however is numerically ill-conditioned for computing theGauss quadrature rule (Lambin and Gaspard, 1982). Furthermore the computation of thequadrature rule is unstable and sensitive to small errors, especially when a large number ofmoments is used. The exponential growth of higher order moments causes instability because

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the algorithm multiplies large moments a large number of times (Upadhyay, 2011). Otheralgorithms have been proposed, like the Jacobian Matrix Transformation, Direct QuadratureMethod of Moments, Principal-Component-Analysis-QMOM, etc. which avoid the instabilityof the PD algorithm, but for the determination of the initial conditions the PD algorithm isstill needed. The method of moments (MOM) can also be used for multidimensional PBEs,but then it loses its simplicity.

Recently, Upadhyay (2011) proposed the Chebyshev algorithm which determines the abcissasand weights directly from recursive formulae without the use of the PD algorithm for thedeterimation of the initial conditions. Therefore the algorithm would be more appropriatedfor higher-order quadrature.

Figure 1.6: An illustration of the QMOM (Yeoh and Tu, 2009).

2) Method of Characteristics

The method of characteristics (MOC) offers a technique which is in general a powerful toolfor solving linear growth processes. It has the capability to overcome numerical diffusion anddispersion, it is computationally efficient and it gives highly resolved solutions. In contrastto other techniques MOC avoids the numerical dissipation error caused by the growth termdiscretization (Kumar and Ramkrishna, 1997). The technique is also very suitable to dealwith the hyperbolic nature of the growth term.

∂

∂tf +∇z · Z f = h(z, t) (1.12)

with z = [x, r] the combination of the internal and external coordinates, this combination hasno mathematical consequences. When the previous Equation 1.12 is rewritten with the useof the product rule in Equation 1.13, Equation 1.14 is found.

∇z · (Zf) = (∇zf) · Z + f(∇z · Z) (1.13)

∂

∂tf + Z · ∇z f = h(z, t)− (∇z · Z) f (1.14)

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When the differential equation described in Equation 1.14 is solved in a particular way, namelyalong characteristic curves, then ODEs are obtained which can be easily solved. The solutionhas to obey the following properties to obtain characteristic curves:

dz

ds= Z(z, t),

dt

ds= 1, z(0) = z0, t(0) = t0

with s the parameter along characteristic curve, the parameter is assumed to vanish at thebeginning of the characteristic curve. Equation 1.14 can be written as

df

ds= h(z, t)− (∇z · Z) f (1.15)

While the method is highly efficient when the physics are simple, the approach does notgeneralize to complex physics. For solving the PBE with only growth involved the MOC is agood choice, when aggregation or nucleation has to be incorporated then another technique(e.g. finite difference scheme) has to be supplemented (Qamar and Warnecke, 2007).

3) Method of Weighted Residuals/Orthogonal collocation

For the method of weighted residuals (MWR) the distribution is approximated by a finite setof basis functions. The set of functions is generally obtained by truncating an orthonormalfamily. The basic idea of the method is to approximate a function u by a function u, whichis a linear combination of basis functions chosen from a linearly independent set (HartleyGrandin, 1991):

u ∼= u =n∑i=1

aiφi (1.16)

When function u is approximated by u it will be very probably that an error E or residual Rwill exist

E(x) = R(x) = D(u(x))−D(u(x)) 6= 0 (1.17)

with D the derivative of the specified function. However the error is not zero the MWR hasto be forced in such way that the average sense over the domain is zero.∫

XR(x)widx = 0 i = 1, 2, · · · , n (1.18)

where n is equal to the number of unknown constants ai in u. Several algorithms havebeen developed to determine the different coefficients, dependent of the choice of wi anotheralgorithm is being used. The most used algorithms are: the collocation method, the sub-domain method, the least squares method, the Galerkin method and the method of moments.The three last methods will be discussed in more detail, because the method of moments isused in this thesis and the Galerkin method however is based on the least squares method(Hartley Grandin, 1991).

Least squares method For this algorithm a continuous summation of all the squaredresiduals is minimized. The mathematical form is given by:

S =

∫XR(x)R(x)dx =

∫XR2(X)dx (1.19)

14

To achieve a minimum of all the squared residuals, the previous function has to be differenti-ated to all the different parameters. Therefore:

∂S

∂ai= 0 (1.20)

= 2

∫XR(x)

∂R

∂aidx (1.21)

It can be easily seen that Equation 1.21 is equal to Equation 1.18 when the weight functionwi is equal tot the partial derivative ∂R

∂ai.

Galerkin method As mentioned before the Galerkin method is a modification of the leastsquares method. Instead of using the derivative of the residual with respect to an unknownparameter ai, the derivative of the approximated function u is used. The weight functionsbecome:

wi =∂u

∂ai(1.22)

By defining the weight functions by Equation 1.22, it can be shown that the weight functionsbecome equal to the set of linearly independent basis functions defined in Equation 1.16.

wi =∂u

∂ai= φi (1.23)

Method of Moments The MOM has already been discussed in subsubsection 1.5.2, butas previously stated the MOM is a special case of the more general Galerkin method. Theweight functions are now defined as:

wi = xi−1 i = 1, 2, · · · , N (1.24)

When the basis functions for the approximation are chosen as polynomials, then the MOMmay be equal to the Galerkin method (Hartley Grandin, 1991).

The main weakness of this method is that the basis functions have to be carefully tuned toeach particular system if only a few ODEs are desired (Gunawan et al., 2004).

4) Monte Carlo Method

Monte Carlo simulations track the evolution of a set of random sampled particles, so onlya part of the particles is being followed. This will result in a set of stochastic populationbalance equations instead of deterministic equations. Monte Carlo has showed to be veryuseful, because it can handle any number of state variable and can also incorporate a com-plex interactions and systems (Ramkrishna, 1985; Ethayaraja and Bandyopadhyaya, 2006).However the method is computationally expensive (Gunawan et al., 2004).

5) Finite difference schemes/discrete population balances

Finite Difference Finite difference schemes were the first used by Euler (1707-1783) tofind approximate solutions of differential equations. Therefore the technique is also known as

15

the Euler method (D’Acunto, 2004). Three possible approximations by finite difference exist,namely the forward, backward or central.

dF (t)

dt= lim

∆t→0

F (t+ ∆t)− F (t)

∆t(1.25)

dF (t)

dt= lim

∆t→0

F (t)− F (t−∆t)

∆t(1.26)

dF (t)

dt= lim

∆t→0

F (t+ ∆t)− F (t−∆t)

2∆t(1.27)

All those approximations converge to the same derivative as ∆t → 0. When ∆t is small butfinite, the previous equations become:

dF (t)

dt≈ F (t+ ∆t)− F (t)

∆t(1.28)

dF (t)

dt≈ F (t)− F (t−∆t)

∆t(1.29)

dF (t)

dt≈ F (t+ ∆t)− F (t−∆t)

2∆t(1.30)

The approximations stated above are not converging to the same derivative, the solution andthe error will differ. The local truncation error can be found by approximating the derivativelocally by a Taylor expansion. Finite difference methods are easy to implement, but alsohave some disadvantages. They are problematic in handling complicated geometries, whencompared to other discretization methods like the finite element method for example. Howeverthe topology of the physical state space can be thought as that of a hypercube, which doesnot require special treatment from a numerical approximation point of view. The secondproblem is the sensitivity to nonlinearities. Finally, in order to improve the accuracy of thenumerical approximations, they frequently require numerical boundary conditions (Mantzariset al., 2001). This numerical boundary conditions have no physical meaning, but are neededfor solving the PBE properly. There exists many different finite difference implementations,which all have to be compared. The best implementation should be chosen (Wojcik and Jones,1998). When the conservation of the total number of particles and mass is wanted, this canonly be guaranteed in the limit of infinite resolution (Patankar, 1980).

High resolution finite volume method (HRFV) The high resolution schemes originatefrom the compressible gas dynamics, which are the state-of-the-art methods for aerodynam-ics, astrophysics, detonation waves, and related fields where shock waves occur (LeVequeet al., 1998). They were developed with the purpose to provide high accuracy while avoidingnumerical diffusion (which is smearing) and numerical dispersion (which are nonphysical os-cillations). The problems stated before are encountered with other finite difference and finiteelement methods. Instead of developing new algorithms, the successful method was adaptedfor solving PBEs. This is possible because the nature of the equations which have to be solvedare both hyperbolic. As an advantage the experience and efforts which were already under-taken could be transferred. Furthermore the high resolutions algorithms are very general, sothey can be easily been modified to any particle problem of interest. (Gunawan et al., 2004)

In LeVeque (2002) the extension of the high-resolution algorithms was made to simulatevariable-coefficient linear systems. These systems have the same mathematical structure as

16

the PBEs with size-dependent growth and, therefore the high-resolution methods are directlyapplicable to these equations. The algorithms can provide at least an second-order accuracyfor all smooth regions (That is in actually the reason why they are called high-resolution meth-ods), and avoid numerical diffusion associated with first-order methods, and the numericaldispersion near sharp gradients or discontinuities associated with other second-order methods(Gunawan et al., 2004). The modern high-resolution methods are derived from an integralrepresentation for the underlying conservation equations, as a consequence this ensures thatthe main property of the distribution (e.g. total volume, total energy, total momentum) isexactly conserved and an accurate simulation of the growth rate kinetics (LeVeque et al.,1998), without the need for specialised procedures as typically required for finite differencemethods.

For the one-dimensional case, let kt denote the time, ks the size interval and fmn the approxi-mation of the average population density (Gunawan et al., 2004):

fmn ≈∫ nks

(n−1)ks

f(x,mkt)dx (1.31)

with m,n integers such that m ≥ 0 and 0 ≤ n ≤ N . Where N is the total number of classes.For the specific case that is treated in this thesis, the PBE is a one-dimensional case whichwas already mentioned in Equation 1.8. This time however a homogeneous PBE is considered(no particles are entering or leaving the system, no particles are produced by breakage andno particles are removed by coagulation, i.e. the net birth rate h(x, t) is equal to zero). Apossible high-resolution scheme is (1.32):

fm+1n =fmn −

ktks

(Gnfmn −Gn−1f

mn−1)−

[ktGn2ks

(1− ktGn

ks

)(fmn+1 − fmn )ψn

− ktGn−1

2ks

(1− ktGn−1

ks

)(fmn − fmn−1)ψn−1

](1.32)

where ψn represents the van Leer flux limiter. The purpose of the flux limiter is to avoidspurious oscillations that would otherwise occur with high-order spatial discretization schemesbecause of shocks, discontinuities or sharp changes in the solution domain (Gunawan et al.,2004). It is given by

ψ(θn) =|θn|+ θn1 + |θn|

(1.33)

with

θn =fmn − fmn−1

fmn+1 − fmn(1.34)

For solving PBEs the high resolution finite volume method has gained popularity over thelast years, because of its advantages mentioned before (Qamar et al., 2006).

17

18

2Materials and Methods

By three methods we may learn wisdom: first, byreflection, which is noblest; second, by imitation, which iseasiest; and third, by experience, which is the most bitter.

Confucius

The techniques which are available to reduce model complexity were described in section 1.4.However, all the non-heuristic techniques were not appropriate for the drying model studiedin this work in order to reduce model complexity in such a way that an easy accessible modelcould be generated. Therefore, an alternative method was developed. In the next section thestepwise approach of the developed model reduction technique will be introduced.

2.1 Stepwise Model Reduction Approach

The used drying model requires too much computational time, therefore the direct applicationof the model for PBM is not possible. To counter this problem a stepwise model reductionwas performed, which will be explained briefly. First, a local sensitivity analysis (LSA) anda global sensitivity analysis (GSA) is performed. In this way the importance of the differentdegrees of freedom on the output value (i.e. moisture content) can be determined. In thisway, more insight can be gained and a certain number of most important variables can beselected to be incorporated in the reduced model. The next part consists of the actual modelreduction, this will be explained more detailed in section 2.4. Finally, the reduced model canbe implemented in the PBE and several numerical techniques can be used to solve the PBE,in order to verify its whether it correctly simulated the drying behaviour of a populationof particles and how accurate the solutions are. The schematical overview of the followedprocedure is given in Figure 2.1.

The MATLAB code of both the drying model and the GSA were already implemented priorto this work.

2.2 Local Sensitivity Analysis

The purpose of LSA is to find out what kind of effect a change in the value of a degree offreedom has on the output locally around a certain point in parameter space. In contrast aGSA investigates the impact globally over the entire parameter space. The LSA and GSA

19

Figure 2.1: Schematic illustration of the followed strategy for constructing a PBM for simu-lating the drying behaviour of a population of particles (Mortier et al.).

performed in this study are only used in a qualitative way in order to determine the mostimportant degrees of freedom. Those degrees of freedom can be divided in two subgroups,namely the process variables and the physical parameters. The latter contains propertiesof the particles or gas which are (rather) static, e.g. the critical temperature of water, theporosity of the particles, etc. The former contains variables which are dynamic and cantherefore be used to control the drying process, e.g. gas temperature, relative humidity.

LSA is performed for five process variables: the gas temperature (Tg), the initial temperatureof the particle (Tp,0), the velocity of the incoming gas (Vg), the relative humidity (RH) andthe pressure of the gas (Pg). The process variables can be influenced to a certain extent by theoperator or could be adaptable by extending the current installation. In the current systemit is possible to adapt the gas temperature and the gas velocity. Relative humidity would bechangeable when a dehumidifier is installed where the air would be dehumified before enteringthe fluidized bed dryer. The initial temperature of the particle is influenced by the granulatorwhich precedes the fluidized bed dryer. The gas velocity could possibly accelerate the dryingprocess. The assigned values of the process variables are shown in Table 2.1, these values willbe called the nominal conditions in this thesis.

Table 2.1: Nominal values of the process variables used in the LSA.

Process variable Nominal value

Tg () 55Tp,0 () 25RH (%) 7Pg (Pa) 101000Vg (m3/h) 200

The derivative of a function y(t) with respect to a variable θj can be approximated by a finitedifference form (as shown in Equation 2.1), i.e. linear approximations, which can be used fornumerical differentiation methods. The approximation becomes more accurate for small ∆θ.

∂y(t)

∂θj= lim

∆θj→0

y(t, θj + ∆θj)− y(t, θj)

∆θj(2.1)

∆θj can be rewritten as a function of θj , introducing the perturbation factor p.

∆θj = p θj (2.2)

20

Table 2.2: Nominal values of the physical parameters used in LSA.

Parameter Value Unit Parameter Value Unit

Rp 0.6 mm Twc 647.13 Kkw 0.63 W/(mK) ks 0.75 W/(m· K)ϕ 0.05 - η 2e-5 kg/(m· s)ρg 1.2 kg/m3 kg 0.0285 W/(m· K)cp,g 1009 J/(kg· K) Mw 18.015 g/molρwc,w 1000 kg/m3 ρs 1525 kg/m3

kd 0.07 W/(m· K) cp,s 1252 J/(kg· K)Rd,0 0.615 mm ε 0.8 -

The determination of the most optimal perturbation factor is important, because it needs tobe verified that the exactness of the approximation of the finite difference is good. From atheoretical point of view this perturbation factor would have to be as small as possible becausethe approximation of the differential equation by the finite difference will be better for non-linear models. However the numerical approximation is limited to the accuracy of the programand the computer on which the calculation is performed. The most optimal perturbationfactor is therefore obtained when the difference between the positive (Equation 2.3) and thenegative perturbation (Equation 2.4) is minimized.

∂Gr(t)

∂θj

∣∣∣∣+

=Gr(t, θj + p · θj)−Gr(t, θj)

p · θj(2.3)

∂Gr(t)

∂θj

∣∣∣∣−

=Gr(t, θj)−Gr(t, θj − p · θj)

p · θj(2.4)

Different criteria can be used to determine the most optimal perturbation factor:

Sum of squared errors (SSE)

SSE =

t∑t=0

(∂Gr(t)

∂θj

∣∣∣∣+

− ∂Gr(t)

∂θj

∣∣∣∣−

)2

(2.5)

Sum of absolute errors (SAE)

SAE =t∑t=0

∣∣∣∣∂Gr(t)∂θj

∣∣∣∣+

− ∂Gr(t)

∂θj

∣∣∣∣−

∣∣∣∣ (2.6)

Maximum relative error (MRE)

MRE = max

∣∣∣∣∣∣∣∂Gr(t)∂θj

∣∣∣+− ∂Gr(t)

∂θj

∣∣∣−

Gr(t)

∣∣∣∣∣∣∣ (2.7)

Ratio of the positive and negative perturbated sensitivity functions (RPN)

RPN(t) =

∂Gr(t)∂θj

∣∣∣+

∂Gr(t)∂θj

∣∣∣−

(2.8)

21

In this thesis the criterium of the maximum relative error (MRE) (Equation 2.7) is used toperform the determination of the optimal perturbation factor. In this way an easier compar-ison between the first and second drying phase is possible, because it is a relative measure ofthe error.

After determining the most optimal perturbation factor, the LSA can be performed. The LSAis performed using the nominal values given in Table 2.1 and the determined perturbationfactor. The central difference scheme can be used to determine the local sensitivity function:

∂Gr(t)

∂θj=Gr(t, θj + ∆θj)−Gr(t, θj −∆θj)

2∆θj(2.9)

It must be noticed that the values of local sensitivities are still absolute at this moment, thisis important when we want to compare the sensitivity of the different degrees of freedomon the model output. For example when two degrees of freedom θ1 and θ2 have the samesensitivity by using the same perturbation factor, but θ2 is ten times larger than θ1, theabsolute sensitivity measure will be affected by that. By using a relative sensitivity measurelike the relative sensitivity against a degree of freedom or total relative sensitivity (TRS)the sensitivity of θ1 will become ten times higher and be conform with the expected output.Different relative sensitivity functions are defined:

Relative sensitivity against a degree of freedom, allowing to compare sensitivity of oneoutput for different degrees of freedom:

∂yi(t)

∂θj· θj (2.10)

Relative sensitivity against an output, allowing to compare sensitivity of different out-puts for 1 degree of freedom:

∂yi(t)

∂θj· 1

yi(t)(2.11)

TRS, allowing to compare all sensitivities of different outputs to different degrees offreedom:

∂yi(t)

∂θj· θjyi(t)

(2.12)

For the evaluation of the local sensitivity both the relative sensitivity against a degree offreedom (Equation 2.10) is used. The method offers the advantage that the value of thedifferent degrees of freedom are normalized, what makes it possible to compare the degrees offreedom. Moreover the only output that is taken into account is the growth rate Gr, so theuse of the TRS would give no additional information.

2.3 Global Sensitivity Analysis

The LSA, performed in section 2.2, can be considered as a useful technique, but the maindrawback of this technique is its limited validity as the LSA might yield different results atdifferent locations in parameter space and will hence be dependent on the choice of the nominalvalues. This is certainly the case for non-linear models. To cure this limitation a GSA canbe executed to determine how the uncertainty in the output of the model can be apportionedto the different sources of uncertainty in the model input averaged over the entire parameterspace (Saltelli et al., 2004). A GSA typically consist of two steps, namely the sampling

22

and the analysis. Different analysis techniques are available, however in this study only thestandardized regression coefficient (SRC) is used because the technique is straightforward andthe sensitivity analysis is only used in a qualitative way, i.e. to determine the most importantvariables. Therefore the need for a more rigorous analysis is not deemed necessary, for moreanalysis techniques Saltelli et al. (2007) provides a nice and detailed overview. For samplinga Monte Carlo method is used, this technique relies on repeated sampling to compute themodel results (Saltelli et al., 2007).

2.3.1 Monte Carlo

The Monte Carlo simulation technique typically consist of several steps:

1. As a deterministic model constituent now becomes stochastic, different values for theconstituent need to be generated. If several stochastic constituents zi are considered,then for each constituent N samples (Monte Carlo sample size) have to be taken and tobe combined in the following matrix:

M =

z

(1)1 z

(1)2 · · · z

(1)r

z(2)1 z

(2)2 · · · z

(2)r

· · · · · · · · · · · ·z

(N−1)1 z

(N−1)2 · · · z

(N−1)r

z(N)1 z

(N)2 · · · z

(N)r

(2.13)

in which each column represents N samples for constituent i (ranging from 1 to r).

For the generation of the matrix in expression 2.13 many different algorithms can beused. Two general algorithms will be discussed shortly. First of all the most obvioussampling method is random sampling. However there is no certainty that the entiresample space will be sampled properly. A more advanced algorithm is used, namelythe latin hypercube sampling (LHS). This method relies on the probability distributionfunctions (PDFs) of the different degrees of freedom. This distributions have to bedetermined or to be assumed. Afterwards the cumulative distribution function of eachparameter is subdivided in N disjoint strata, so for each sample a disjoint intervalSi is created. Also two additional conditions are set, firstly every disjoint strata Sirepresents the same marginal probability 1/N and secondly a sample is taken from eachstratum (McKay et al., 1979). For example, when the range of a certain input parameterX is subdivided in five compartments and a uniform PDF is presumed, then the fivesubdivisions Si will all cover one fifth of the probability (1/N) and also one fifth ofthe original range S. However if a normal probability distribution is presumed insteadof the uniform distribution, then the previous statement is no longer true. When Sis subdivided in five parts using LHS, then the five subparts will all cover 20% of theprobability, but not of the range. The outer parts will cover more than 20% and theinner parts less. This is shown in Figure 2.2.

23

0 1 2 3 4 5 6

0

0.2

0.4

0.6

0.8

1

X

f(x

)

Figure 2.2: The visualisation of the LHS for a normal probability distribution. In this examplefive sample have to be taken, so at the y axis the probability range is split in five equal parts.This ensures that the entire range is sampled with respect to the probability.

As a consequence sampling the entire range the estimate of the distribution functionwill be more stable for LHS than it is by performing random sampling.

2. The model is run for all the generated samples (each row of matrix M) and N outputvalues are produced, which forms the output of the Monte Carlo analysis:

Y =

y(1)

y(2)

· · ·y(N−1)

y(N)

(2.14)

Since no prior knowledge on values for the process variables is available, the process variablesare sampled from a uniform distribution between the chosen minimum and maximum values.The sampling space of the process variables used in this work is showed in Table 2.3.

Table 2.3: Range of the process variables for the Monte Carlo simulations.

Process variable Range

Tg () 20 - 80Tp,0 () 25 - 50RH (%) 1 - 15Pg (Pa) 100500 - 101700Vg (m3/h) 150 - 500

2.3.2 Standardized Regression Coefficients

When the Monte Carlo simulation is performed for the model under study, then for all thedegrees of freedom N samples are taken from their distribution. As such, an input vectoris constructed, the output can be computed for each row in matrix M to obtain the desiredoutput vector (Equation 2.14).

24

Now a ranking of the different degrees of freedom has to be performed. In Figure 2.3 anillustration of the technique is presented. When looking at Figure 2.3, it can be easily noticedthat the influence of Z1 on the output Y is larger than the influence of Z2 on the modeloutput, because the variance in the output of Y is lower for Z1 than for Z2. However, a moreobjective and mathematical measure has to be formulated.

−4 −2 0 2 4

−4

−2

0

2

4

Z1

Y

−4 −2 0 2 4

−4

−2

0

2

4

Z2

Y

Figure 2.3: Scatterplots of Y versus Z1 and Z2 after Saltelli et al. (2007)

If a linear model is assumed, then intuitively the partial differential of Y to Zi would be usedto describe the relative importance of the degrees of freedom. Because a higher value for thepartial differential leads to a higher importance of that specific input parameter on the modeloutput Y (Equation 2.15).

SpZi=∂Y

∂Zi(2.15)

with SpZithe sensitivity based on the partial differential (superscript p) from input parameter

Zi. However, this ’solution’ would possibly give wrong results. Looking back at the illustrationin Figure 2.3, both partial differentials would be equal because the slope is the same (SpZi

= Ωi

and Ω1 = Ω2). Using Formula 2.15 would give a different sensitivity as there should beexpected. This problem can be countered by incorporating the standard deviation of thedegree of freedom Zi and normalizing this deviation by dividing it with the standard deviationof the output Y .

Sσzi =σzi∂Y

σY ∂zi(2.16)

with SσZithe variance based sensitivity (superscript σ).

Now the calculation of the relative importance of the different parameters shows more rea-sonable results. The variance of Y is better described by the variance of Z1 than it is bythe variance of Z2. Thus the importance of Z1 on the model output Y will be higher. Thisoutcome matches the expectations. The calculation of the importance of the different degreesof freedom can be carried out by performing a linear regression. By taking the estimatedvariances of the different degrees of freedom into account, it can be shown that:

βzi =bzi σziσY

∼= σzi∂Y

σY ∂zi(2.17)

with βZi the estimated SRC and bZi the regression coefficient of degree of freedom Zi.

25

The SRCs are quite easy to interpret, however the method has some drawbacks. First of all, itcan only cope with linear models, so it is recommended to use the method when the coefficientof determination R2 is larger than or equal to 0.70. The coefficient of determination is givenby:

R2 =

∑mi=1(yi − y)2∑mi=1(yi − y)2

(2.18)

with yi the estimated output of the linear model, yi the output of the original model and ythe mean of the original outputs.

However for nonlinear monotonic models the ranking of the coefficients can be quite effective,irrespective of the degree of nonlinearity (Saltelli et al., 2007). The second remark is thatthe model does not cope with interactions between the different input parameters. Otherranking methods like elementary effects, variance based, etc. can cope with nonlinearity andinteractions. As mentioned before, in this thesis the GSA is used to get rather a qualitativethan a quantitative view on the ranking of the coefficients.

The proposed technique is surely applicable for the mechanistic drying model. When lookingto Figure 1.5, it can be easily seen that the first drying phase is a nonlinear monotonic function,so the method is surely applicable for this part. For the second drying phase, however thereis a nonmonotonic nonlinearity, so it could be expected that the method of the SRC is notuseful anymore. The growth curve however shows a symmetry around half the drying timethat is needed to perform a total drying of the particle. If all the samples of the Monte Carlosimulations are taken when all the drying functions are still in the first part of the seconddrying phase. Then the function is monotonic and the suggested method is still applicable.

2.4 Model Reduction

After performing both the LSA and GSA, the importance of all the degrees of freedom isknown and a greater insight has been gained. The mechanistic drying model however istoo slow to use in the PBE and therefore it has to be simplified or replaced by a morestraigthforward model structure. From the sensitivity analyses we now know which are theimportant parameters that need to be retained during the model reduction. As mentionedbefore, a heuristic framework is proposed which relies on the outcome of the GSA. The reducedempirical model should be able to predict the growth term Gr which is required in the PBE.The evolution of the growth rate Gr, which is the derivative of Rw is visualised in Figure 2.4.

26

0 2 4 6 8 105.95

6

6.05

6.1

6.15·10−4

t (s)

Rw

(m)

0 2 4 6 8 10−2

−1.5

−1

−0.5·10−6

t (s)

Gr

(m/s

)

(a) First drying phase

0 100 200 300 400 5000

2

4

6·10−4

t (s)

Rw

(m)

0 100 200 300 400 500−4

−3

−2

−1

0·10−5

t (s)

Gr

(m/s

)

(b) Second drying phase

Figure 2.4: The evolution of the wet radius Rw and the growth term Gr in function of thetime for both drying phases at nominal parameter values.

In order to determine the model structure the following approach was set up and used (Figure2.5). First, the original mechanistic model is used as data generator, it is presumed that themodel can reasonably predict the reality. This gives the ability to generate all the informationwhich could be needed for our model reduction. Second, the growth rate Gr has to bedescribed in function of Rw. For both the first and the second drying phase an empiricalmodel structure g(x, P,D) is developed, with x the internal coordinate of interest (Rw), Pa vector of empirical parameters and D the range of the most sensitive degrees of freedomselected for the model reduction. The initial development of the model is performed at thenominal parameter values (Table 2.1) in order to reveal a function g which is only dependentof the internal coordinate. Afterwards the model is compared with the original model andthe empirical parameters are determined. At this stage the model cannot take changes ofenvironmental conditions into account. However, the empirical parameters can be used tointroduce those effects. One or more parameters pi will be selected and the most sensitivedegree of freedom D is varied. The evolution of the values of pi with i ∈ 1, 2, ..., n can befitted by a new empirical function hi(ci, D) and be incorporated in the empirical model. Afterthis incorporation, another empirical parameter pj with j ∈ 1, 2, ..., n \ i can be fitted infunction of D. This procedure can be repeated until the effects of the degree of freedom arerepresented in an acceptable way. Afterwards the new empirical parameters ci can be used tointroduce the influence of a new degree of freedom.

27

Figure 2.5: Scheme of the steps taken during the model complexity reduction (Mortier et al.).

2.5 Population Balance Modelling

Implementation

In order to verify if the proposed reduced model structures are useful and feasible for thenumerical calculation of the PBE, the empirical model of the first and the second drying phaseare implemented together. different numerical methods. In this way a basic comparison canbe made between the implemented solutions. The implemented methods are: QMOM, MOC,HRFV and the finite difference technique. The first three methods were already implemented,but for QMOM the Chebyshev algorithm was implemented for determining the abscissas andweights instead of the PD algorithm (Upadhyay, 2011). The finite difference technique wasfully implemented, by using the algorithm proposed by Hu et al. (2005).

28

Comparison between the different numerical solutions

The four implementations will be compared for the most important and straigthforward char-acteristics.

Calculation Time: The proposed algorithms have to be as accurate as possible, butalso the computational time is important. These are usually two competing criteriaand a trade-off is usually required. For online control the computational time has to besmall, moreover for the use of PBM this condition also has to be fulfilled.

Zeroth Moment: The zeroth moment expresses the total number of particles whichare present at a certain time in a control volume. When no nucleation, breakage oraggregation appeares and no particles are entering or leaving the control volume, thenthe zeroth moment should remain constant. In this way it can be verified if the numberand mass balances are closed. In this study the normalized zeroth will be used, so forclosed mass and number balances a constant value of one should be obtained.

M0,t =

∑Ni=1 ni,t∑Ni=1 ni,0

(2.19)

First Moment: The first moment is equal to the weighted mean of Rw with the numberof particles as weighting factors. The evolution of the first moment visualises howthe drying is progressing, furthermore it gives the opportunity for an easy comparisonbetween the different numerical methods.

M1,t =

∑Ni=1 ni,t · (Rw)i,t∑N

i=1 ni,t(2.20)

29

30

3Results and discussion

And now for something completely different.

Monty Python

3.1 Local Sensitivity Analysis

The optimal perturbation factor to be used in the local sensitivity analysis for the moisturecontent was determined for the different process variables at nominal values (Table 2.1), morespecific: Tg, Vg, RH, Pg and Tp,0. The perturbation factor was varied between 10-8 and 10-2

with a logarithmic step of one. The results are presented in Table 3.1 (first drying phase)and Table 3.2 (second drying phase). To determine the most optimal perturbation factor theMRE criterion was used (Equation 2.7). The use of a logarithmic step size of one allows tofind a good estimation of the perturbation factor quickly. Further optimization of the differentperturbation factors was not deemed necessary because the sensitivity analysis is only usedin a qualitative way. For each process variable the perturbation factor with the lowest MREis indicated in bold.

Table 3.1: Values of the MRE criterion for all process variables for the first drying phase.

Input Perturbation factor10-2 10-3 10-4 10-5 10-6 10-7 10-8

Tg 4,61E-04 4,61E-05 4,61E-06 4,68E-07 1,12E-07 1,12E-06 1,12E-05Vg 1,14E-05 1,14E-06 1,14E-07 1,34E-08 3,08E-08 3,08E-07 3,08E-06RH 9,00E-06 8,99E-07 9,42E-08 6,33E-08 1,27E-06 8,81E-06 9,08E-05Pg 1,63E-08 1,63E-09 1,62E-10 2,19E-11 8,77E-11 6,29E-10 5,93E-09Tp,0 0,013473 1,35E-03 1,35E-04 1,35E-05 1,35E-06 2,01E-07 2,07E-06

For all process variables the most optimal perturbation factor is situated near 10-6 and 10-5,this is true for both drying phases. These are quite usual perturbation factors as they avoid theimpact of non-linear effects in the model (larger perturbation factors) and numerical effects(too small perturbation factors). The most optimal perturbation of a specific process variableis not necessarily the same for both drying phases. A final global perturbation factor of 10-6

was chosen for performing the LSA. This perturbation factor will also be used for performingthe LSA for the physical parameters.

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Table 3.2: Values of the MRE criterion for all process variables for the second drying phase.

Input Perturbation factor10-2 10-3 10-4 10-5 10-6 10-7 10-8

Tg 0,006742 0,066887 2,47E-04 2,47E-05 2,57E-06 1,91E-05 1,98E-04Vg 7,33E-05 7,69E-05 7,69E-06 7,70E-07 5,37E-07 6,73E-06 5,47E-05RH 0,05279 1,80E-04 1,80E-05 1,80E-06 1,50E-05 1,69E-04 0,001628Pg 3,67E-06 1,15E-08 1,15E-09 1,29E-10 1,06E-09 1,26E-08 1,02E-07Tp,0 2,34E-04 1,06E-04 9,60E-05 9,60E-06 9,64E-07 3,51E-06 4,38E-05

The LSA was performed for all degrees of freedom in the model. The performance of theLSA for the physical parameters was done to check whether the model was sensitive to theseparameters, sensitive parameters should be determined more precisely in order to have a wellperforming model.

3.1.1 First drying phase

Sensitivity of the moisture content towards the process variables

The sensitivity functions for the different process variables are shown in Figure 3.1 and givealready more insight in the first drying phase. The TRS measure was used to determine thesensitivity (Equation 2.12).

0 2 4 6 8 10

0.4

0.45

0.5

t (s)

TRS

Vg

0 2 4 6 8 10

−1.5

−1

−0.5

0

0.5

1

1.5

t (s)

TRS

Tg

0 2 4 6 8 100.1

0.15

0.2

0.25

t (s)

TRS

Pg

0 2 4 6 8 100

5

10

15

20

25

30

t (s)

TRS

Tp,0

0 2 4 6 8 10−0.5

−0.4

−0.3

−0.2

−0.1

t (s)

TRS

RH

Figure 3.1: Sensitivity functions of the first drying phase for all process variables.

Gas temperature (Tg): The sensitivity function of Tg is increasing in the beginningand stagnates after a few seconds. Tg has a strong positive influence on the dryingprocess, but it takes a few seconds before the liquid layer has reached the temperatureof the gas. Therefore, in the beginning the sensitivity to Tg is lower.

Gas velocity (Vg): An elevation of Vg has a positive influence on the drying process,because the air surrounding the particle is renewed more frequently. The fresh air is less

32

saturated than the present air thus enhancing the drying process. A possible explanationof the observed peak is that by increasing Vg the heat transfer from the air to the particleis raised, leading to a higher drying rate.

Relative humidity (RH): The drying rate is reduced when RH of the fresh entering airis increased. This effect is higher in the beginning of the first drying phase, because theparticle has to acclimatize to the ambient gas temperature. The negative effect of RHis due to the higher moisture content of the air which prevents an efficient vaporizationof the water layer around the particle.

Gas pressure (Pg): Pg has a higher positive impact in the beginning of the first dryingphase, but the absolute sensitivity function is decreasing to 0.1 where it levels off. Apossible explanation is that an elevated pressure leads to a higher retaining capacity forwater per air volume unit, thus increasing the drying rate.

Initial temperature particle (Tp,0): When the initial temperature of the particle isincreased, the drying rate is strongly raised in the early drying stage. However, theeffect is fading out after a few seconds. After a few seconds, Tg is the most importantfactor because the gas is heating the particle no matter what the initial temperaturewas. From that moment Tp,0 has no added value anymore.

In Figure 3.2 the mean of the quadratic TRS instead of the non-quadratic TRS is plotted, assuch it is easier to interpret the importance of each process variable. Figure 3.2 summarizesFigure 3.1 and clearly illustrates that the descending order of importance is: Tp,0, Tg, Vg,RH and Pg. However, it is noteworthy that not all process variables can be controlled in thesystem. The initial temperature of the particle can be influenced by adapting the temperatureat which the granulation is performed. The gas temperature and gas velocity can easily beset and controlled in the fluidized bed dryer itself. As mentioned in section 2.2, the humiditycan be controlled, but the current installation should be adapted for this purpose. On theother hand the gas pressure cannot be easily controlled.

10−2 10−1 100 101 102

Tg

Vg

RH

Pg

Tp,0

QTRS

Process

Variable

Figure 3.2: The mean quadratic TRS for the first drying phase for the different processvariables.

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Sensitivity of moisture content towards the physical parameters

The LSA is also performed for the physical parameters in the drying model. The quadraticTRS of the different parameters is shown in Figure 3.3. The values are quite high for kd,ρwc,w, Mw and Rd,0, but it is still lower than the sensitivity of process variables Tp,0 and Tg.The other parameters show a rather low impact on the model output. The sensitivity to themodel in respect with cp,s and ks is even equal to zero. This is conform with the expectations,because in the first drying phase the drying is considered as droplet drying so the propertiesof the solid will not influence the drying rate.

10−6 10−5 10−4 10−3 10−2 10−1 100 101

φ

Rp

Rd,0

η

ρg

kg

cp,g

Mw

ρw

ρs

kd

kw

ks

cp,s

Twc

ε

QTRS

Param

eter

Figure 3.3: The quadratic TRS for all physical parameters.

3.1.2 Second drying phase

Sensitivity of moisture content towards the process variables

The sensitivity functions of the second drying phase (Figure 3.4) are different when comparedto the first phase, this could be expected because the model structure is also different. Duringthe first seconds, for all process variables the sensitivity drastically changes. This is due tothe transition from the first to the second drying phase.

Gas temperature (Tg): The model is very sensitive with respect to Tg. The sensitivityis high in the beginning, then suddenly drops and after about five seconds the sensitivityfunction again increases in a linear way until the end of the drying phase. By increasingTg, the air can hold more water and the diffusion rate of the air molecules is enhancedleading to a higher drying rate in the pores (which leads to a linear increase of thesensitivity).

Gas velocity (Vg): A high sensitivity prevails at the start of the second drying phase,but this is rapidly decreasing. The influence of Vg is fading out, because the air flow

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0 20 40 600

0.2

0.4

0.6

t (s)

TRS

Vg

0 20 40 605

5.5

6

6.5

t (s)

TRS

Tg

0 20 40 60

−0.05

0

0.05

0.1

t (s)

TRS

Pg

0 20 40 600

1

2

3

t (s)

TRS

Tp,0

0 20 40 60

−0.2

−0.15

−0.1

−0.05

t (s)

TRS

RH

Figure 3.4: Sensitivity functions of the second drying phase for all process variables.

cannot penetrate in the pores of the particles to enhance the drying process. Therefore,the influence of Vg will be limited to the outer region of the particle.

Relative humidity (RH): The relative humidity has a negative effect on the dryingprocess, the sensitivity curve is increasing in the beginning until a value of -0.05 andfrom then it is decreasing slightly. An elevated RH reduces the drying rate, becausethe water content of the surrounding air is higher, thereby preventing a more efficientevaporation.

Gas pressure (Pg): The sensitivity function of Pg is positive in the beginning but issharply declining to -0.05 and from there it is smoothly decreasing in time. The elevatedpressure seems to block the air present in the pores, thereby reducing the drying rate.

Initial temperature of the particle (Tp,0): The initial temperature of the particlehas a positive influence on the drying process, but the effect is fading out soon. Theparticle has already acclimatized to the prevalent conditions during the first dryingphase, therefore the influence of Tp,0 is lower than it is during the first drying phase.

In Figure 3.5, the quadratic TRS of the second drying phase is shown for the different processvariables. To summarize the LSA: the Tg has become more important than in the first dryingphase, whereas Vg has lost much of its impact on the particle which can be easily explained.The drying process takes place inside the particle, i.e. in the pores of the particle, the effectof the Vg almost only applies on the outer surface of the particle. Inside the pores the effectwill be minimal, in contrast to Tg which has a major impact on the drying rate. The RHretains its negative impact, but the Pg now has a negative impact in contrast to its positiveimpact on the first drying phase. Tp,0 has a positive sensitivity for both drying phases, thisimpact is drastically reduced for the second drying phase. By choosing MRE as the measurefor sensitivity for LSA, the evolution of the sensitivity is not considered. However, the rankingof the process by LSA was not changed by using another sensitivity measure. In conclusion,Tg is the most influential process variable on the drying of particles.

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10−4 10−3 10−2 10−1 100 101 102

Tg

Vg

RH

Pg

Tp,0

QTRS

Process

Variable

Figure 3.5: The quadratic TRS of the second drying phase for all process variables.

Sensitivity of moisture content towards the physical parameters

The sensitivity functions of the parameters for the second drying phase are given in Figure 3.6.Both Rd,0 and Rp show a high influence on the drying rate, which is not surprising becausethe first derivative of the evolution of the wet radius Ri is the model output. There is also acoupling between Rd,0 and Rp which explains why they both show such a high impact on themodel. In contrast to the first drying phase, cp,s and ks now show an impact on the growthrate.

10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101

φ

Rp

Rd,0

η

ρg

kg

cp,g

Mw

ρw

ρs

kd

kw

ks

cp,s

Twc

ε

QTRS

Parameter

Figure 3.6: The quadratic TRS of the physical parameters for the second drying phase.

For the second drying phase Rp and Rd,0 have to be determined more precisely, because theirsensitivity is about ten times higher than the other physical parameters and their order ofmagnitude of the sensitivity value is the same as for the most important process variable, i.e.Tg.

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3.2 Global Sensitivity Analysis

Since the LSA only provides a snapshot of sensitivity at the nominal values of the degreesof freedom, a GSA can help in determining an ’average’ sensitivity over the entire space ofthe degrees of freedom. A Monte Carlo simulation was performed and 2000 samples (N)were generated for both the process variables and the parameters. The output of the MonteCarlo simulations for the process variables is presented in Figure 3.7. The drying time of thedifferent simulations in Figure 3.7 is varying greatly, because the sampled ambient conditionswill determine the drying rate. Drying performed under more favourable conditions will resultin shorter drying times.

0 5 10 15 20 25 30−8

−6

−4

−2

0·10−6

t (s)

Gr(m

/s)

(a) Evolution of the different samples for the firstdrying phase, the black line visualizes at which time(after three seconds) the GSA is performed.

0 50 100 150 200 250 300 350−2

−1.5

−1

−0.5

0·10−4

t (s)

Gr(m

/s)

(b) Evolution of the different samples for the firstdrying phase, the GSA was performed after one sec-ond.

Figure 3.7: Monte Carlo simulations for both drying phases.

3.2.1 Sensitivity of moisture content towards the process variables

First Drying Phase

The first drying phase is a nonlinear monotonic curve, as mentioned before the method of SRCshould show good results. Based on Figure 3.8, in which the mechanistic model outputs arecompared with the SRC model outputs, it can be decided that the linear model, constructedby using the SRC method, performs well. The correlation coefficient R2 is about 0.97 whichcan be considered as good. The ranking of the different process variables is given in Table 3.3.As expected the most important process variable is Tg followed by Vg and RH. The outputof the GSA is somewhat different from the LSA which was performed earlier. The Tp,0 isno longer the most important process variable (Figure 3.2), but is now almost irrelevant fordescribing the observed variance of the model output. Tg,Vg,Pg and Tp,0 show a negativesensitivity, thus an increase of one of these process variables will lower the growth rate or inother words increase the drying rate. On the other hand, an increase in RH will lead to lowerdrying rates.

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R2 = 0.96911

0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000−4

−3

−2

−1

0

1

2

3

Monte Carlo Simulation index

Res

cale

dGr,

1

Mechanistic model outputsSRC model outputs

Figure 3.8: The linear SRC model and the rescaled generated growth rate of the first dryingphase after 3 seconds.

Second Drying Phase

The second drying phase is a nonlinear non-monotonic curve, but as stated before this wouldnot be a problem if the building of the SRC model is performed when all the curves are stillmonotonic. This is the case during the first seconds of the second drying phase, because athigher gas temperatures the drying rate is much higher and after a few seconds the particleswill already enter the second part of the second drying phase. To be more precise the seconddrying phase of a particle which is performed at standard conditions, with as only change anelevation of the gas temperature to 80, only takes about 12.6 seconds. So when a GSA isperformed after 6.3 seconds not all the curves will be monotonic anymore. By performing theGSA after 11 seconds a correlation coefficient R2 with a value of only 0.54 is obtained, whichis too low for considering the model as linear.

By constructing the linear model when all the growth curves are still monotic, Figure 3.9 isobtained (i.e. after 1 s into the second drying phase). The retrieved R2 is about 0.75 whichis above the minimum requirement of 0.7. The linear model approximation with SRC can beconsidered as acceptable. The same remark as in the previous section can be made, i.e. Tp,0 isno longer the most significant process variable. The only process variable which has a strongimpact on the drying rate is the gas temperature which has a SRC of 0.87.

The output of the GSA for all process variables is shown in Table 3.3, the influence of thechosen time on the performance of the GSA can clearly be noticed.

In some particular cases the LHS generated samples which led to negative mass transfers.

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R2 = 0.75406

0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000−4

−3

−2

−1

0

1

2

Monte Carlo Simulation index

GP

art

2Simulation model outputsLinear model outputs

Figure 3.9: The linear SRC model and the rescaled generated growth rate of the second dryingphase after 1 second.

As a consequence, the moisture content of the particle was increased, these cases were elim-inated as they are not realistic. Table 3.4 clearly shows that this particular problem wasonly encountered at roughly the same conditions. The explanation of this particular issue isstraightforward (Equation 1.2). Initially the liquid layer is at a low temperature (i.e. Tp,0)and Tg and RH are quite high. As a consequence the partial vapor density ρv,s over thedroplet surface is low and the partial vapor density ρv,∞ is rather high, therefore a net masstransfer of liquid from the ambient to the droplet surface is obtained. Probably this situationwill continue until the temperature of the particle Tp has risen, from that moment the netmass transfer of the liquid will be performed in the desired direction and the drying processwill really start. However, in practice this situation will not be encountered, because whendealing with high humidity a dehumidifier will be installed. Therefore, those samples were

Table 3.3: The SRC values of the process variables for both drying phases.

Process VariableDrying Phase

First Second3s 1s 11s

Tg -0.93 -0.87 -0.73Vg -0.29 -0.02 -0.03RH 0.13 0.02 0.05Pg -0.00 -0.00 -0.01Tp,0 -0.02 -0.03 -0.03

R2 0.97 0.75 0.54

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ignored for performing the GSA. A reduction of the sampling space could prevent the sam-pling of process variables which lead to negative mass transfer rates. A reduction of the upperboundary of Tg from 80to 70could be a possibility. However, by choosing a larger samplespace these problems will be encountered and give more insight how the ambient conditionsare interacting.

Table 3.4: Values of the eliminated process variable samples and the corresponding masstransfer rate.

Vg () Vg (m3/h) RH (%) Pg (Pa) Tp,0 (K) mv (kg/s)

72.00 238.99 13.34 100575.12 301.01 -1.32E-0979.68 439.99 13.95 100686.96 304.54 -8.67E-0979.71 458.15 10.04 100519.75 298.21 -6.75E-0978.48 204.35 14.63 100604.93 306.66 -3.15E-0971.11 254.54 14.00 100841.47 298.56 -4.86E-0977.52 253.47 12.78 101465.32 302.96 -3.56E-0975.47 232.84 13.84 101094.92 301.81 -5.11E-0978.31 353.56 14.25 100878.15 301.95 -1.12E-0877.59 256.19 14.54 101300.80 304.84 -5.09E-0975.70 472.98 11.66 101497.60 298.33 -6.51E-0973.45 242.71 13.40 100817.51 299.83 -4.56E-0977.43 467.11 14.90 101151.05 299.08 -1.78E-0874.97 207.20 11.05 101031.76 299.98 -8.44E-1076.53 429.23 14.18 101564.21 300.73 -1.10E-0873.39 296.82 14.91 101281.27 302.50 -4.29E-0979.62 272.24 11.82 101320.07 300.36 -7.54E-0977.67 219.45 11.30 101486.87 298.71 -5.56E-0973.59 495.30 12.15 101369.12 299.50 -3.34E-0971.06 458.32 13.73 100972.39 298.62 -5.30E-09

3.2.2 Sensitivity of moisture content towards the physical parameters

The global sensitivity of all physical parameters was investigated. For each of the parametersa range was set such that by sampling the maximum difference between the nominal valueand the sample value was 10% (Table 3.5).

Table 3.5: The range of all physical parameters for generating 2000 samples by using LHS.

Parameter Range Unit Parameter Range Unit

Rp 0.54-0.66 mm Twc 582.4-711.8 Kkw 0.57-0.69 W/(mK) ks 0.675-0.825 W/(m· K)ϕ 0.045-0.055 - η 1.8e-5-2.2e-5 kg/(m· s)ρg 1.08-1.32 kg/m3 kg 0.0257-0.0314 W/(m· K)cp,g 908-1109.9 J/(kg· K) Mw 16.21-19.82 g/molρwc,w 900-1100 kg/m3 ρs 1373-1678 kg/m3

kd 0.063-0.077 W/(m· K) cp,s 1127-1377 J/(kg· K)Rd,0 0.554-0.677 mm ε 0.72-0.88 -

For the first drying phase (Table 3.6) no major differences were found between the LSA and

40

GSA. For the second drying phase (Table 3.7) the most sensitive parameters of the LSA, i.e.Rd,0 and Rp, had become less important indicating that the LSA emphasized observations ina particular point in the space of the degrees of freedom. However, many similarities couldbe noticed between the outputs of the two methods.

Table 3.6: The ranked SRC for all physicalparameters for the first drying phase afterone second.

Parameter SRC

ρwc,w 0.59Rd,0 0.46Mw 0.39kd 0.30ρg 0.19ρs 0.12Rp 0.10kg 0.09η 0.09cp,g 0.09Twc 0.05kw 0.04cp,s 0.00ks 0.00ε 0.00ϕ 0.00

R2 0.99

Table 3.7: The ranked SRC for all physicalparameters for the second drying phase afterone second.

Parameter SRC

ρwc,w 0.58Mw 0.52ϕ 0.50Rp 0.19ρs 0.16ks 0.12cp,s 0.11kd 0.09Rd,0 0.05ρg 0.03Twc 0.03kg 0.03cp,g 0.02η 0.01ε 0.00kw 0.00

R2 0.97

3.3 Model Reduction: First Drying Phase

The results, obtained by performing the LSA and GSA, are used to develop a model structurewhich incorporates the most important process variables (Tg and Vg). The values of thedrying rate were generated for the entire range of the most important process variables byuse of the mechanistic drying model. In order to determine an appropriate model structure, alinear relationship between the growth rate and the input will be constructed by performinga transformation which yet has to be determined. The advantage of such a linearized systemis that it is straightforward and a first estimation of the coefficients is possible. Afterwards,the proposed model structure can be verified and the empirical coefficients can be optimizedby using an optimization algorithm.

3.3.1 Fitting the growth term in function of the wet radius

First hypothesis

The shape of the drying curve showed an exponential behaviour, the first hypothesis was toconstruct the curve on a log-scale in order to transform it to a linear curve. However, a linearbehaviour was not the observed, thus a pure exponential relationship was not sufficient todescribe the data.

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Second hypothesis

The exponential shape of the growth term had to be explained with an exponential term, butthe addition of other terms could help to solve the nonlinear relation on the log-scale. Thenew hypothesis was that a numerical derivation of the growth term would eliminate all termsexcept the exponential one if the extra terms have a polynomial form. Plotting the derivativeson a log-scale would finally yields a linear curve. The visualization of this reflection is givenin Figure 3.10.

0 0.2 0.4 0.6 0.8 1

−1.8

−1.6

−1.4

−1.2

−1

·10−6

Rw,nor(−)

Gr(m/s

)

(a) Evolution of the growth term.

0 0.2 0.4 0.6 0.8 110−8

10−7

10−6

10−5

10−4

Rw,nor(−)

dG

r,1

dR

w,n

or

(b) First derivative on a log-scale.

0 0.2 0.4 0.6 0.8 110−9

10−8

10−7

10−6

10−5

10−4

10−3

Rw,nor(−)

d2G

r,1

dR

2 w,n

or

(c) Second derivative on a log-scale.

0 0.2 0.4 0.6 0.8 110−8

10−7

10−6

10−5

10−4

10−3

10−2

Rw,nor(−)

d3G

r,1

dR

3 w,n

or

(d) Third derivative on a log-scale.

Figure 3.10: The growth term Gr and the first three derivatives in function of the wet radiusRw at nominal values (Table 2.1).

The first derivative of the growth term did not show a linear behaviour on a log-scale, but thesecond derivative did. The proposed hypothesis was confirmed and verified at other conditionstoo. More specific, the proposed model structure was compared with the growth term at othergas temperatures because Tg is the most important process variable based on the sensitivity.The formula for describing Gr in function of the wet radius Rw is given by Equation 3.1.

Gr,1(Rw,nor) = A+B ·Rw,nor + C · eD·Rw,nor (3.1)

42

Rw,nor =Rw −RpRw,0 −Rp

(3.2)

with Gr,1 the growth rate of the first drying phase. Rw,nor is the normalized wet radius ofthe first drying phase and was introduced for two reasons: By normalizing the wet radius theformula could easily be used at other conditions and a transformation of the wet radius wasneeded, because of the numerical limitations that were encountered in Matlab.

A first estimation of the different parameters (A, B, C, D) of Equation 3.1 was made basedon the different derivatives:

dGr,1(Rw,nor)

dRw,nor= B + C ·D · eD·Rw,nor (3.3a)

d2Gr,1(Rw,nor)

dR2w,nor

= C ·D2 · eD·Rw,nor (3.3b)

d3Gr,1(Rw,nor)

dR3w,nor

= C ·D3 · eD·Rw,nor (3.3c)

D can be obtained by dividing Formula (3.3c) by Formula (3.3b). C can be obtained out ofFormula (3.3b) or (3.3c) if D is known. B can be obtained out of Formula (3.3a). A canbe obtained out of Formula (3.1). The obtained coefficients are then used as initial guessfor a nonlinear optimisation with the use of ’fminsearchbound’, this function is available onthe MATLAB Central and combines the fminsearch function with the possibility to setboundaries for the different coefficients, to reduce the root mean square error (RMSE).

The proposed empirical model structure is less complicated than the original mechanisticdrying model. In contrast to the original model, the empirical model does not take anyambient conditions into account (yet). To increase the utility of the empirical model, theinfluence of the most important process variables has to be Incorporated in the empiricalmodel.

3.3.2 Extension of the model with the gas temperature

In Figure 3.11 the growth term is given in function of Rw and Tg for both drying phases.In order to incorporate Tg and to preserve the proposed model structure, the 4 coefficientsthat were introduced in Equation 3.1 are described in function of Tg. These coefficients needto be fitted, but by fitting the coefficients by a certain relationship additional errors willbe introduced. To minimize those deviations the coefficients were not fitted simultaneously,but one by one. First the D-coefficient was fitted in function of Tg and with the introducedrelation that describes D the other coefficients were optimised again. Errors incorporated byintroducing the D-relation are minimized in that way. The schematic overview of the followedprocedure was already showed in Figure 2.5.

Determining temperature dependent functions of the coefficients

For every temperature between 20 and 80 with a step of 0.5, the proposed empiricalmodel structure was fitted to the evolution of the drying rate at that specific temperature.

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6

6.05

6.1

6.15

·10−4

20

40

60

80

−4

−3

−2

−1

0

·10−6

Rw (m) Tg (C)

Gr(m

/s)

Figure 3.11: The growth rate, calculated with the mechanistic model, in function of Rw andTg for the first drying phases.

20 30 40 50 60 70 80

−3

−2

−1

·10−6

Tg ()

(a) A

20 30 40 50 60 70 800

0.5

1

1.5

2

2.5·10−7

Tg ()

(b) B

20 30 40 50 60 70 80

0

2

4

·10−12

Tg ()

(c) C

20 30 40 50 60 70 8012

13

14

15

16

17

Tg ()

(d) D

Figure 3.12: Coefficients of the empirical model for the first drying phase in function of thegas temperature.

44

Figure 3.11 illustrates that the drying rate at a specific Rw depends on Tg. Therefore anevolution of the empirical coefficients could be expected (Figure 3.12).

D-coefficient Coefficient D was fitted first, because the sensitivity of the growth term tothe power D will be higher compared to the other coefficients (Figure 3.13).

0.7 0.75 0.8 0.85 0.9 0.95 1−2

−1.5

−1

−0.5

0

0.5

1

1.5·10−6

Rw,nor (-)

Gr(m

/s)

Original1.1 ·A1.1 ·B1.1 · C1.1 ·D

Figure 3.13: The effect of an increase of the coefficients by 10% on Gr. D shows the highestmaximum sensitivity of all coefficients.

D showed a parabolic behaviour, with a peak around 38. However, the appearance of thepeak is due to the value of coefficient C at 38, being equal to zero so the system of linearequations is undetermined. This anomaly will therefore be neglected and filtered out whenperforming a nonlinear regression on the shape of D later on in the procedure. To describethe coefficient in function of Tg two polynomial functions were considered, namely one ofsecond order and the other of fourth order. The deviation between the proposed functionand the determined D values was lower with the last polynomial, though the second orderpolynomial was preserved. The error that is introduced by using a function with a lowerperformance can and will be eliminated by the regression of the other coefficients. This isshown in Figure 3.14(b). The values of C were updated to reduce the error between themechanistic and the empirical model. The other coefficients showed no change, which couldhave been expected because C and D are highly correlated.

C-coefficient After describing D, the positive part of C showed many similarities with ageneralized normal distribution. C determines if the derivative of the exponential part of thecurve is positive or negative. It is very important that the turning point of this exponentialpart is at the right point. Therefore, different hypotheses were developed:

In the first hypothesis it was tried to approach the positive part of C by a normaldistribution. This approach worked out well for the positive part, but no turning pointwas incorporated. So for lower temperatures, this function was useless.

The second hypothesis was an expansion of the first hypothesis, so for the positive partthe normal distribution was kept but this was added with a linear function. The purposeof this linear function was to describe the negative part of C. The solution was also notconvincing, because the turning point was not at the right point and due to the linear

45

20 30 40 50 60 70 8012

13

14

15

16

17

Tg ()

D originalD function

(a) D

20 30 40 50 60 70 80−1

0

1

2

3

4

5·10−12

Tg ()

Original C valuesUpdated C values

(b) C

Figure 3.14: The values of the D and C coefficients after incorporating the second orderpolynomial for describing the evolution of D in function of the gas temperature.

function the normal function was biased. This bias led to a bad approximation of thepositive part and the the negative part was also not approximated in a proper wayeither. Therefore this solution was dismissed.

The last hypothesis was that an expansion of the first hypothesis was still needed, butthis additive had to fade out at higher temperatures. Therefore it was considered thatthe first hypothesis could be expanded with another generalized normal distribution,but this time for the negative part. This solution seemed to work out very well, theturning point was approximated almost perfectly without forcing the algorithm towardsthis solution. The positive and negative part were both predicted well (Figure 3.15).

20 30 40 50 60 70 80−1

0

1

2

3

4

5·10−12

Tg ()

Updated C valuesFit

Figure 3.15: The values of the C coefficient before and after the implementation of the Cfunction.

46

B-coefficient The values of the B did not change significantly by fitting D and C, so Figure3.12(b) could still be used to determine the functionality between B and Tg. The evolution of Bwas approximated by a polynomial of the third order. No other possibilities were investigated,because the polynomial regression performed well. A small offset at a temperature of 20was introduced, this could be minimized by elevating the order of the polynomial to four. Aglobal optimisation would be performed later, so this was considered as a temporarily problemand secondly the small error that is introduced by using a function with a slightly lowerperformance will show negligible when compared with the deviation between the mechanisticmodel and the experimental results (Mortier et al., 2012).

A-coefficient Coefficient A was approximated in the same way as B, but now a fourthorder polynomial was opted for. It was possible to approximate the values by a lower orderpolynomial (Figure 3.16(b)), however an ’overfitting’ was considered as useful. Indeed, thevalues of A are fitted in the last step of the procedure, so a more exact approach is neededfor keeping the model performance at a high level.

20 30 40 50 60 70 800

0.5

1

1.5

2

2.5·10−7

Tg ()

DataFit

(a) B

20 30 40 50 60 70 80−3.5

−3

−2.5

−2

−1.5

−1

−0.5·10−6

Tg ()

DataFit

(b) A

Figure 3.16: The nonlinear regression of the B and A coefficients.

Table 3.8: Functions for incorporating the influence of Tg for the first drying phase.

Coefficient Function

A = a1 · T 4g + a2 · T 3

g + a3 · T 2g + a4 · Tg + a5

B = b1 · T 3g + b2 · T 2

g + b3 · Tg + b4C = c1 · exp(−(Tg + c2)2/c3) + c4 · exp(−(Tg + c5)2/c6)D = d1 · T 2

g + d2 · Tg + d3

Goodness of fit of the reduced model

The mean relative weighted error between the results produced by the mechanistic model andthe reduced empirical model fitted to it (including dependence of Tg) is 0.53%. The relative

47

6 6.02 6.04 6.06 6.08 6.1 6.12 6.14

·10−4

20

30

40

50

60

70

80

Rw (m)

Tg

()

0

1

2

3

4

5·10−3

Figure 3.17: The relative deviation between the mechanistic and empirical model in functionof Tg and Rw.

deviation of the empirical model in function of Tg and Rw is visualised in Figure 3.17. Thecoefficients of the optimised model are given in Table 3.9.

Table 3.9: Empirical coefficients for describing the influence of Rw and Tg for the growth rateof the first drying phase.

Subscript Coefficienta b c d

1 -6.43E-15 1.45E-12 4.97E-12 0.00372 -2.74E-12 -1.30E-10 62.43 -0.413 -5.28E-10 -3.99E-09 202.82 23.494 -7.24E-09 -2.57E-08 -3.05E-13 -5 -2.35E-07 - 36.43 -6 - - 96.14 -

3.3.3 Extension of the model with the gas velocity

In order to further improve the empirical model and to incorporate more flexibility towardsthe ambient conditions, the growth rate can also be made function of Vg (note that theempirical model developed in the previous section will not be able to predict impacts of Vg asthis was not considered in the reduction). The gas velocity is chosen as it is the second mostimportant input of the mechanistic model (Table 3.3) and can be easily adapted in operationalconditions. Therefore, the effect of Vg on the growth rate was plotted relative to the standardconditions at which the empirical model was fitted, i.e. 200 m3/h (Figure 3.18).

The relative effect of the growth rate differs only little when considering Rw and Tg. Therefore,a very straightforward strategy is developed, i.e. find a supplementary function which is onlydependent of Vg, but is able to predict the growth rate with high accuracy. A polynomial

48

6 6.02 6.04 6.06 6.08 6.1 6.12 6.14

·10−4

−0.2

0

0.2

0.4

0.6

Rw (m)

Gr,1(R

w,T

g,V

g)

Gr,1(R

w,T

g,2

00)(-)

Vg = 150Vg = 200Vg = 250Vg = 300Vg = 350Vg = 400Vg = 450Vg = 500

Figure 3.18: The relative drying rate at 55 for different gas velocities compared to thestandard case of 200 m3/h.

function of second order was implemented (Equation 3.4) and showed good results. Thecoefficients of the polynomial function were optimised together with the coefficients of theoriginal empirical function Gw(Rw, Tg) in order to reduce the error.

G∗r,1(Rw, Tg, Vg) = (v1 · V 2g + v2 · Vg + v3) ·Gr,1(Rw, Tg) (3.4)

Goodness of fit of the reduced model

The mean relative weighted error between the mechanistic model and the empirical fit is1.10%, which is very reasonable. The relative deviation, in function of Tg and Vg, is shown inFigure 3.19. The coefficients of the optimised empirical model G∗r,1 are given in Table 3.10.

Table 3.10: The coefficients of the empirical model G∗r,1. Differences in the coefficients betweenGr,1 and G∗r,1 are indicated.

Subscript Coefficienta b c d v

1 -6.43E-15 1.45E-12 4.97E-12 0.0037 -1.28E-062 2.74E-12 -1.35E-10 62.67 -0.41 0.0023843 -5.28E-10 3.99E-09 202.82 23.49 -0.427114 -7.24E-09 -2.57E-08 -3.05E-13 - -5 -2.35E-07 - 36.43 - -6 - - 96.29 - -

49

20 30 40 50 60 70 80150

200

250

300

350

400

450

500

Tg ()

Vg

(m3/h

)

0.5

1

1.5

2

2.5

3

3.5

4

4.5·10−2

Figure 3.19: The relative deviation of the empirical model Gr,1(Rw,nor, Tg, Vg) in function ofTg and Vg.

3.4 Model Reduction: Second Drying Phase

By performing the calculations of the drying process of the second phase, a drawback of thenumerical simulation was discovered. The time step had to be manually adapted in order toobtain accurate results within an acceptable time. To circumvent this drawback an adaptivetime step was proposed which is function of the most important process variable, i.e. the gastemperature. This procedure will first be outlined before the model reduction is treated.

3.4.1 Adaptive time step

The theoretical required drying time is highly dependent on Tg, e.g. for a gas temperature of20 it takes about 38 days to obtain dried particles. Whereas the drying at 80 only takesabout 12.6 seconds (Table 3.11). By using a fixed value for the time step, the drying processwas not finished at low Tg when using a relatively small value for the number of time steps.By increasing the value of the time step the accuracy of the algorithm drastically lowered athigher temperatures. To have an efficient algorithm, an adaptive time step was introduced.The aim of the implementation of an adaptive time step is to adjust the time step in functionof Tg. In this way for each temperature an optimal time step will be used. This approachhas several advantages, first of all the need for an accurate and efficient algorithm is fulfilled.Secondly, a more modest advantage is that the number of time steps which are needed willall be in the same range. So when performing a global optimisation after the incorporationof the influence of Tg, the weight of each temperature will already be more or less equal.

In order to develop a function for describing the time step, the drying time at each temperaturehad to be calculated or estimated. This calculation was performed for a limited number oftemperatures. The following approach was followed: first, for a specific temperature a certaintime step was set manually. This time step was chosen by trial and error and had to besufficiently low in order to have at least several thousand time steps. Then the algorithmwas executed and after finishing the calculation, the number of iterations could be easilydetermined. The total theoretical drying time which was needed is equal to the multiplication

50

of the number of iterations with the time step which had been set. The results of this methodare given in Table 3.11. The data are rather rough, but for this purpose the determination ofthe order of magnitude is sufficient.

Table 3.11: Theoretical drying times for different gas temperatures.

Tg () Iterations ∆t (s) Drying Time (s)

20 5572 600 3343200

25 6721 100 672100

30 6882 20 137640

35 6992 5 34960

40 5057 2 10114

50 2363 0.5 1181.5

60 9965 0.02 199.5

70 8925 0.005 44.63

80 12586 0.001 12.59

Now, an easy function was looked for to describe the theoretical drying time in function ofTg. The data became linear when the logarithm of both the theoretical drying time as thegas temperature was taken. Therefore, the function used to describe the data was:

tdry = a · T bg (3.5)

mln(tdry) = ln(a) + b · ln(Tg) (3.6)

with tdry the drying time, a and b empirical parameters. Equation 3.5 can be easily trans-formed to a linear form (Equation 3.6) and is very useful to describe the data. The optimisedrelationship between the total drying time and the gas temperature is given by:

tdry = 4.1817E18 · T−9.1721g (3.7)

A nonlinear optmisation was not performed because the accuracy of the time step is notdeterminative for the accuracy of the drying rate when the number of time steps is not toolow. The approximation of the data given in Table 3.11 by the function of Formula (3.7) isvisualised in Figure 3.20.

3.4.2 Fitting the growth term in function of the wet radius

A similar strategy for model reduction is followed for the second drying phase. However,the shape of the growth term for the second drying phase seems to be more complicatedcompared to the first drying phase, but a certain symmetry can be noticed around half thedrying time (Figure 2.4(b)). In order to develop an empirical model for the second dryingphase, the growth term was split into two conceptual parts, one part left and one part right ofthe symmetry border. In order to construct an empirical model, Rw was normalised relativelyto Rp (Equation 3.8).

R′w,nor =RwRp

(3.8)

51

20 30 40 50 60 70 80101

102

103

104

105

106

107

Tg ()

Tot

ald

ryin

gti

me

(s)

datafit

Figure 3.20: The estimation of the drying time needed at a specific temperature by Formula(3.7).

First, the regression was performed for the growth term at low values of Rw. This is illustratedin subfigure 3.21(a), only the normalised radius R′w,nor between 0 en 0.2 is used in order toreduce the influence of the right part. By using a logarithmic transformation, Figure 3.21(b)is obtained and shows a linear behaviour. Therefore, the left part of the growth term can bedescribed by a power law: A′ · (R′w,nor)B

′.

0 0.5 1 1.5 2

·10−1

−7

−6

−5

−4

−3

−2

−1

0·10−5

R′w,nor (-)

Gr,

2(m

/s)

(a) A piece of the left part of the evolution of thegrowth term at standard conditions.

−5 −4 −3 −2−14

−13

−12

−11

−10

log(R′w,nor)

log(−Gr,

2)

(b) The logarithmic transformation of subfigure3.21(a).

Figure 3.21: The growth rate of the second drying phase, for particles with R′w,nor smallerthen 0.5, can be linearised by logarithmic transformation. The suggested model structure forthe left part of the second drying phase is A′ · (R′w,nor)B

′.

Because the right part showed some similarities with the left part of Gr,2, the same approachwas investigated. R′w,nor had to be reflected around 0.5 in order to obtain the same curve.To obtain this mathematically, R′w,nor was deducted from a value of one. This is illustratedin subfigure 3.22(a). The logarithmic transformation, visualised in subfigure 3.22(b), shows a

52

nonlinearity. The first thought would be to dismiss the model structure of the right part, i.e.C ′ · (1−R′w,nor)E

′. However, the proposed model structure can be improved by incorporating

a new coefficient, the suggestion is to adapt the model in such a way that the right part isdescribed by C ′ · (1 + D′ · R′w,nor)E

′. This additional coefficient introduces the possibility to

linearise the logarithmic transformation (Figure 3.23).

0 0.5 1 1.5 2

·10−1

−1.5

−1

−0.5

0·10−5

1−R′w,nor (-)

Gr,

2(m

/s)

(a) A piece of the right part of the evolution of thegrowth term at standard conditions.

−6 −5 −4 −3 −2−14

−13.5

−13

−12.5

−12

−11.5

−11

−10.5

log(1−R′w,nor)

log(−Gr,

2)

(b) The logarithmic transformation of Figure3.22(a).

Figure 3.22: The growth rate of the second drying phase, for particles with R′w,nor greaterthan 0.5, cannot be linearised by logarithmic transformation. The suggested model structure(C ′ · (1−R′w,nor)E

′) needs to be extended.

−4 −3.5 −3 −2.5 −2 −1.5−14

−13.5

−13

−12.5

−12

−11.5

−11

−10.5

log(1−R′w,nor)

log(−Gr,

2)

Figure 3.23: The linearised logarithmic transformation by using C ′ · (1 +D′ ·R′w,nor)E′, with

D′ equal to −0.98 to illustrate the effect of the coefficient.

The most optimal fitting function for the entire second drying phase can be found by combiningthe two conceptual functions outlined above:

Gr,2(R′w,nor) = A′ · (R′w,nor)B′+ C ′ ·

(1 +D′ ·R′w,nor

)E′(3.9)

53

3.4.3 Extension of the model with the gas temperature

To incorporate the influence of the Tg on the drying rate of the particle, the same strategyas adopted for the first drying phase was followed (Figure 3.24). So for the entire rangeof Tg (20-80) the function Gr,2 was fitted to the mechanistic drying data (obtained bysimulating the mechanistic drying model) and the different coefficients were saved. A verygood approximation of the growth term was achieved for all temperatures (Figure 3.25). Incontrast to the previous drying phase, the evolution of the coefficients in function of Tg didnot smooth out. Therefore, the approximation of the parameters by a certain function wasnot feasible. Another approach was used by setting coefficients B′ and E′ to −1, in this waya smooth evolution was achieved.

0

2

4

6

·10−4

20

40

60

80

2

4

6

8

10

Rw (m) Tg (C)

−log(−G

r)

Figure 3.24: The growth rate, calculated with the mechanistic model, in function of Rw andTg for the second drying phases.

0 0.2 0.4 0.6 0.8 1−7

−6

−5

−4

−3

−2

−1

0·10−9

R′w,nor (-)

Gr

(m/s

)

datafit

(a) 20

0 0.2 0.4 0.6 0.8 1−7

−6

−5

−4

−3

−2

−1

0·10−5

R′w,nor (-)

Gr

(m/s

)

datafit

(b) 55

Figure 3.25: The growth rate of the second drying phase was approximated well for all tem-peratures by the proposed model structure (Equation 3.9).

54

However, by fitting the growth term by only three adaptable coefficients an offset was intro-duced. This is illustrated in Figure 3.26, the offset has a huge impact on the required dryingtime. By reintroducing coefficients B′ and E′ after the optimisation of the other coefficients,the relationship between the latter and Tg was undone. A solution was sought by introducinga coefficient F ′ for reducing the offset in order to reintroduce the remaining coefficients B′ andE′. Afterwards F ′ could be possibly ignored. By reducing this offset a new problem occurred,i.e. the offset at higher temperatures is higher in absolute values but lower in relative values(Figure 3.27). By optimising the coefficients using RMSE the fitting will be favoured at highertemperatures.

0 0.2 0.4 0.6 0.8 1

−6

−4

−2

0

·10−9

R′w,nor (-)

Gr

(m/s

)

datafit

0.45 0.5 0.55

−1.2

−1

−0.8

−0.6

−0.4

·10−10

(a) The offset between the data and the empiricalmodel proposed in Equation 3.9 at 20.

0 0.2 0.4 0.6 0.8 1 1.2

·107

0

1

2

3

4

5

6·10−4

t (s)

Rw

(m)

datafit

(b) The evolution of the moisture content for the em-pirical model at 20differs seriously from the mech-anistic model.

Figure 3.26: The influence of the offset between the mechanistic and empirical model on therequired drying time.

20 30 40 50 60 70 80−0.5

0

0.5

1

1.5

2

2.5

Tg ()

Gr,m

ech−G

r,em

p

Gr,em

p

datafit

Figure 3.27: Evolution of the relative offset in function of the gas temperature before a relativeoffset reduction was implemented.

The minimal drying rate in function of the Tg is between −1.2 ∗ 10−11 (20) and −3.4 ∗ 10−5

(80). Therefore, the introduction of a new parameter F ′ worked well for high temperatures,

55

but no improvement could be seen at low temperatures. To solve this problem a new approachwas developed. A relative offset was introduced that uses the value of the empirical model ata normalised wet radius of 0.5, which is multiplied with a function. This relative offset is thenadded to the original empirical model (Equation 3.10). After the successful implementationand optimisation of Rf , coefficients B′ and E′ could be optimised too and a good empiricalfit was obtained. The eventual acquired model is given by:

Gr,2,fix(Tg) = Rf ∗ (A′ · 0.5)B′+ C ′ ·

(1 +D′ · 0.5

)E′) (3.10)

Rf = Rf,1 · TRf,2g +Rf,3 (3.11)

with Rf,1, Rf,2 and Rf,3 empirical coefficients. So the correct empirical model of the seconddrying phase is given by Formula 3.12. After introducing this relative offset, all the coefficientscould be optimised simultaneously and the relationships given in Table 3.12 were obtained.

Gr,2,glob(Rw, Tg) = Gr,2(Rw, Tg) +Gr,2,fix(Tg) (3.12)

Table 3.12: Functions of the different coefficients for describing the second drying phase withdependency of Rw and Tg.

Coefficient Function

A′ = − exp(a′1) · T a′2

g

B′ = b′1 · exp(b′2 · (Tg − 20))− 1

C ′ = − exp(c′1) · T c′2g

D′ = exp(d′1) · exp(d′2 · Tg)− 1E′ = e′1 · exp(e′2 · (Tg − 20))− 1

Goodness of fit of the reduced mode

The mean relative weighted error between the model en the empirical fit is 1.97%. Thedistribution of this error in function of the wet radius and the gas temperature is visualisedin Figure 3.28. The values of the empirical coefficients are showed in Table 3.13.

Table 3.13: Empirical coefficients of the second drying phase.

Subscript Coefficienta′ b′ c′ d′ e′ Rf

1 9.93 -0.13 10.33 0.10 -0.10 7.24E042 -55.12 -0.38 -56.66 -10.08 -0.49 -3.513 - - - - - -0.11

56

1 2 3 4 5

·10−4

20

30

40

50

60

70

80

Rw (m)

Tg

()

0

0.5

1

1.5

2

2.5

3·10−2

Figure 3.28: The relative deviation of the empirical model in function of Tg and Rw.

In order to verify if the proposed model structures performed well, the total drying time iscalculated. This is the time needed to reach a moisture content of zero. A comparison wasmade between the original mechanistic drying model and the empirical drying model overthe entire range of the gas temperature. Based on Table 3.14, it can be concluded that thesuggested model is sufficient. As visualisation of the evolution of Rw at a gas temperature of20 and 55 is showed in Figure 3.29.

Table 3.14: The mean deviation between the drying time needed

Gas Temp () Drying Time (s) Rel. Deviation (%)Mechanistic Empirical

20 3.34E+06 3.37E+06 0.6330 1.38E+05 1.38E+05 0.0040 1.01E+04 1.02E+04 0.3150 1.18E+03 1.19E+03 0.3555 4.67E+02 4.67E+02 0.1060 1.99E+02 1.99E+02 -0.3670 4.47E+01 4.41E+01 -1.2380 1.26E+01 1.26E+01 0.55

Global -0.16

3.4.4 Extension of the model with the gas velocity

An extension of the model with the gas velocity was not considered, because the deviationbetween the experimental results and the mechanistic drying model is larger than the relativeinfluence of Vg (Figure 3.30).

57

0 0.5 1 1.5 2 2.5 3 3.5

·106

0

1

2

3

4

5

6·10−4

t (s)

Rw

(m)

EmpiricalMechanistic

(a) Evolution at 20.

0 100 200 300 400 5000

1

2

3

4

5

6·10−4

t (s)

Rw

(m)

EmpiricalMechanistic

(b) Evolution at 55.

Figure 3.29: The evolution of Rw for the second drying phase in time for both the mechanisticmodel and the empirical model.

20 30 40 50 60 70 80−2

−1.5

−1

−0.5

0

0.5

1

Tg (C)

Rel

ativ

eoff

set

(%)

Vg = 150 m3/h

Vg = 500 m3/h

Figure 3.30: The influence of Vg on the total drying time of the second drying phase is minimal.The relative offset is given for the two most extreme values of Vg with respect to the referencevalue of 200 m3/h.

58

3.5 Population Balance Modelling

In the previous sections a reduced model for the drying rate was derived. This now allowsimplementing this in a PBM. Four different numerical solution methods of the PBE arecompared in their performance to model the drying process of a population of particles.. Theimplementations will be compared in terms of calculation time, the evolution of the zeroth andfirst moment and their number density distribution (as function of the internal coordinate, i.e.wet diameter). For the numerical calculations the drying time is the time needed to achieve awet radius of 3E-5 m. This value has been chosen because at this value the drying time wasstill available for all implementations. The ODE-solver of the MOC always quit earlier, evenif the absolute and relative error tolerance were drastically reduced. In order to perform anumerical calculation, the range of the internal coordinate has to be discretized. Two typesof discretizations were used, i.e. fixed grid and moving grid discretization. The fixed grid,which is used in the HRFV and the QMOM, divides the entire range of the internal coordinateinto N classes. The particles are moving along those classes in order to adapt their internalcoordinate. The moving grid, which is used in finite difference (FD) and MOC, divides onlya certain part of the entire range into N classes. The particles can still move to adjoiningclasses, but the classes themselves are also moving. The advantage of this latter is that onlythe part of interest has to be calculated. The reduction of the calculation time and requirednumber of classes is possible, while preserving at least the same accuracy as the fixed gridmethod. Every numerical solution can be used with both discretization techniques.

3.5.1 Finite difference

The FD method is the only method that does not need an ODE-solver to solve the PBE.The time step had to be set manually and a time steps of 0.5 and 0.05 seconds were chosen.Different numbers of classes N were chosen (20, 100, 200). The drying time was respectively470.5 and 470.65 seconds. The evolution of the zeroth and first moment did not show anydependency on the number of classes. However, the density distribution is worthless for thelargest time step, because a lot of (negative) numerical noise is introduced by converting theclasses to a density distribution. This can be seen in Figure 3.31 where a comparison witha pseudo-analytical method is provided too. This effect is reduced when using smaller timesteps, but not entirely solved. By increasing the number of classes, the number of peaksremains the same, but they become more steep. The use of a more advanced conversion couldcounter this problem, but the calculation time would significantly increase. The computationaltime in function of the number of classes and the chosen time step is showed in Table 3.15.

Table 3.15: The computational time of the FD in function of the number of classes and thetime step.

N Computational Time (s)

∆t = 0.5 s ∆t = 0.05 s

20 0.41 21.75

100 1.66 119.16

200 8.14 235.28

59

5.6 5.7 5.8 5.9 6 6.1

·10−4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4·105

Rw (m)

Number

ofparticles

(-)

PA HRFVFD

(a) After 6 seconds.

5.35 5.4 5.45 5.5 5.55 5.6 5.65 5.7 5.75

·10−4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4·105

Rw (m)

Number

ofparticles

(-)

PA HRFVFD

(b) After 14 seconds

Figure 3.31: The particle density distribution of FD (100 classes and ∆t = 0.05 s.) after 6and 14 seconds and the pseudo-analytical solution with HRFV.

3.5.2 Method of Characteristics

The MOC shows a stable drying time, independent of the number of classes (Table 3.16).By converting the classes to a density distribution a smooth distribution is obtained despitethe fact that the same conversion algorithm as with FD is used. The differences between thedensity distributions of 100 and 200 classes were small, therefore only the density distributionof 200 classes is visualised (Figure 3.32). Another difference between the MOC and FD isthat after 14 seconds the density distribution of the MOC is slightly drier than that of theFD.

5.7 5.8 5.9 6 6.1

·10−4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4·105

Rw (m)

Number

ofparticles(-)

PA HRFVMOC: 20 classesMOC: 200 classes

(a) After 6 seconds.

5.3 5.35 5.4 5.45 5.5 5.55 5.6 5.65 5.7

·10−4

−2

0

2

4

6

8

·104

Rw (m)

Number

ofparticles(-)

PA HRFVMOC: 20 classesMOC: 200 classes

(b) After 14 seconds

Figure 3.32: The particle density distribution of MOC after 6 and 14 seconds and the pseudo-analytical solution with HRFV. The approximation of MOC with 200 classes is considered asgood.

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Table 3.16: The computational time and drying time for MOC in function of the number ofclasses.

N Computational Time (s) Drying Time (s)

20 2.064 472.22

100 21.64 470.86

200 96.94 470.93

3.5.3 High Resolution Finite Volume

In Figure 3.33 the evolution of the number density distribution in time for the HRFV methodis showed for different number of classes, i.e. 200, 500 and 5000 (pseudo-analytical (PA)).By drastically increasing the number of classes, the approximation of the analytical solu-tion by the discretized form will be better and the obtained solution can be considered aspseudo-analytical (PA). It can be concluded that the HRFV converges rather slowly to thePA solution. This will lead to a drastical increase of the computational time in order toobtain an accurate solution of the PBE (Table 3.17). Figure 3.33 also nicely illustrates theoccurrence of numerical diffusion.

5.6 5.7 5.8 5.9 6 6.1

·10−4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4·105

Rw (m)

Number

ofparticles

(-)

PA HRFVHRFV 200HRFV 500

(a) After 6 seconds.

5.35 5.4 5.45 5.5 5.55 5.6 5.65 5.7 5.75

·10−4

0

2

4

6

8·104

Rw (m)

Number

ofparticles(-)

PA HRFVHRFV 200HRFV 500

(b) After 14 seconds

Figure 3.33: The particle density distribution of HRFV for 200, 500 and 5000 classes. Alarge number of classes is needed to obtain an accurate density distribution, this leads to highcomputational times.

3.5.4 Method of moments

For the implementation of QMOM at least 100 classes were needed in order to have a smoothevolution of the first moment. QMOM is a very efficient calculation method (Table 3.18) andthe number of classes can be strongly increased without significantly increasing the calculationtime. Main drawback of the algorithm remains the construction of the density distributionout of a limited number of moments.

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Table 3.17: The drying time of the HRFV converges to a value of 470.6 seconds, which canbe considered as a pseudo-analytical solution. The computational time drastically increasesby using a large number of classes.

N Computational Time (s) Drying Time (s)

100 27.22 467.16

200 174.82 460.48

500 1 868.61 470.20

5000 2 332 107.13* 470.64

* This is equal to almost 27 days of calculation!

Table 3.18: The increase of the number of classes for QMOM leads to a slight elevation of thecalculation and drying time.

N Computational Time (s) Drying Time (s)

1E2 0.78 442.82

1E3 0.98 459.00

1E4 0.99 459.31

1E7 2.05 459.47

3.5.5 Comparison between the different numerical implementations

A comparison is made between the FD (100 classes and ∆t = 0.5 s), MOC (100 classes),HRFV (200 classes) and QMOM (1E7 classes), based on the zeroth and first moments andthe calculation time.

Zeroth Moment

The evolution of the zeroth moment is visualised in Figure 3.34 and clearly shows that thealgorithms using the moving grid are not constant in time. The normalized zeroth momentshould be constant at a value of 1 at all times, because no nucleation, breakage or aggregationoccurs and no particles are entering or leaving the control volume. The number of particlesof the fixed grid methods stay constant however, the observed deviations are rather small(maximum about 1%).

First Moment

In Figure 3.35 the progress of the first moment is given, this value reflects the weighted meanof the wet radius of the entire population. The algorithms using a moving grid calculate adrying time of 490 seconds, while the methods using a fixed grid show a lower drying time.This is due to numerical diffusion which artificially (but erroneously) speeds up the dryingprocess. The HRFV algorithm shows the same behaviour as QMOM, but at the end thedrying rate of the HRFV scheme is reduced and the drying time approximates the dryingtime of the moving grid methods. Different solutions are obtained and there is interest todetermine which is the right solution. Therefore, the implementation of HRFV is expanded;

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0 50 100 150 200 250 300 350 400 450 5000.988

0.99

0.992

0.994

0.996

0.998

1

1.002

1.004

1.006

t (s)

M0

FDHRFVMOCQMOM

Figure 3.34: The evolution of the zeroth moment for the different numerical methods. Thealgorithms using a fixed grid (HRFV and QMOM) show a constant value over the entiresimulation, which should be the case because no nucleation, breakage or aggregation occurand no particles are removed or added to the control volume.

more classes are used for the calculation. In literature this method is considered as accurate(Gunawan et al., 2004; Qamar et al., 2006), so by enlarging the number of classes a pseudo-analytical solution could be obtained because the approximation of the differential equationconverges to the real solution.

The HRFV method was executed again for 5000 classes. The calculation took about 2.33million seconds or in a more understandable measure: almost 27 days. So this extensivealgorithm is not useful for online control, but is now used for determining the correct solution.Figure 3.36 clearly shows that the evolution of the moving grid algorithms are more accuratein terms of drying time. Because enlarging the number of classes worked out well for theHRFV method, this was also tried with the QMOM. The increased number of classes had apositive impact on the drying time, but the offset with the other methods could not be totallyeliminated.

Calculation Time

A fundamental difference in calculation time was found between the fixed and moving gridmethods. The fixed grid methods performed badly, because many calculations are performedfor classes without particles. In order to obtain the same accuracy as the moving grid tech-nique, the number of classes had to be increased, leading to higher computational time. Aswitch to more advanced grid techniques could reduce this computational burden. In liter-ature other techniques are available, e.g. the adaptive moving mesh method (van Dam andZegeling, 2006).

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0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

6

·10−4

t (s)

M1

FDHRFVMOCQMOM

440 450 460 470 480

0

0.5

1

·10−4

Figure 3.35: The evolution of the first moment for the different numerical implementations.The evolution shows that the population is actually drying, however the fixed grid implemen-tations (HRFV and QMOM) predict a faster drying than the moving grid implementations.

0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

6

·10−4

t (s)

M1

Pseudo-analytical HRFVFD

440 450 460 470 480

0

0.5

1

·10−4

Figure 3.36: The evolution of the first moment for the FD implementation and the pseudo-analytical solution of the HRFV method.

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4Conclusions and perspectives

4.1 Conclusions

This study aimed to perform a model reduction of an existing validated drying model forpharmaceutical applications. This was needed in view of its implementation in a PBM,requiring an easy structure and fast solution.

The drying model was first analyzed in order to gain more insight in the model dynamics andto determine the most important degrees of freedom. Both LSA and GSA were performed.Based on the results, it could be concluded that the gas temperature was the most importantprocess variable for both drying phases followed by the gas velocity Vg. Therefore, it wasdecided to maintain these 2 process variables in the model reduction.

A model reduction was performed for both conceptual drying phases. The followed procedureconsists of several steps: First, the model structure was determined. Second, for the mostimportant process variable the evolution of the empirical coefficients is observed when varyingthe considered process variable in order to realize a good estimation of the original model.Finally, for each coefficient a function is proposed and implemented in order to incorporatethe influence of the process variable on the model output. The followed procedure was used tocreate a generalized framework for model reduction. For the first drying phase, the influenceof both Tg and Vg were incorporated in the empirical model. The mean relative devationbetween the empirical and the original model was 1.10%, which can be considered as good.For the empirical model of the second drying phase only the effect of Tg was incorporated,because the influence of Vg on the second drying phase was too small (maximum change of± 1.5% in the required drying time in comparison with the standard condition of 200 m3/h),thus this process variable was not appended to the model. For the second drying phase, thedeviation between the empirical and the original model was 1.97%.

The empirical model was implemented in the PBE and performed well. However, the compu-tational time had to be reduced. Therefore, 4 different numerical solution methods of the PBEwere compared with respect to computational time and accuracy. The MOC was consideredas useful method, because of the limited calculation time and acceptable accuracy.

In order to conclude, the results of this study can be used for incorporating the drying processin a PBM. In this way, more information and insight can be gained about the drying char-acteristics of a fluidized bed. Also, the developed framework for model reduction is rather

65

generic and can be used in a broad variety of cases were a high computational burden isobserved.

4.2 Perspectives

The current empirical model could be extended with more process variables, i.e. Vg (onlyfor the second drying phase), RH, Pg and Tp,0. This would increase the applicability of theempirical model, but comes at a cost of increased computational load. For the first dryingphase, currently the additive effect of both Tg and Vg are taken into account. However, it isvery likely that there is a kind of combined effect. To incorporate this combined effect, aninteraction term could be supplemented to the current model.

The PBM was only investigated for some straightforward characteristics, but further inves-tigation is needed. A fast and accurate PBM method should be developed for pure growthprocesses, probably the MOC with an adaptive grid mesh can be used. In order to verify theresults of the numerical solutions of the PBE experimental results should be gathered, andconfronted with the model predictions.

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