Investigation of inhomogeneity in prepa

Friedrich-Alexander-Universität Erlangen-Nürnberg

Lehrstuhl für Thermische Verfahrenstechnik

Untersuchung von Inhomogenitäten in präparativen flüssigchromatographischen

Säulen

-

Investigation of inhomogeneity in prepara-tive liquid chromatographic columns

Der Technischen Fakultät der Universität Erlangen-Nürnberg vorgelegt zur Erlangung des

Grades

DOKTOR INGENIEUR

vorgelegt von Diplom-Ingenieur Dirk-Uwe Astrath

Erlangen - 2007

D.-U. Astrath Investigation of inhomogeneity in preparative liquid chromatographic columns

B

Als Dissertation genehmigt von der

Technischen Fakultät der Universität Erlangen-Nürnberg

Tag der Einreichung: 16. Oktober 2006

Tag der Promotion: 05. Juni 2007

Dekan: Prof. Dr.-Ing. Alfred Leipertz

Berichterstatter: Prof. Dr.-Ing. Wolfgang Arlt, Prof. Erling H. Stenby


i

Danksagung

Zunächst und an erster Stelle möchte ich meinen Eltern Rita und Dietrich Astrath danken, die

mir mein Studium ermöglichten und in ihrem elterlichen Grundvertrauen („Dirk-Uwe wird es

schon richtig machen“) stets zu mir standen. Meinem Bruder Detlef Astrath danke ich für die

steten, stillen Hinweise, mich selbst, das Leben und die Wissenschaft nicht ganz so ernst zu

nehmen.

Kerstin Albrecht, die ein weites Stück Weges an meiner Seite ging, danke ich für den Trost

und die Aufmunterung, die sie mir in mancher Stunde des Zweifelns spendete.

Wolfgang Arlt rechne ich neben dem Vertrauen, welches er mit der Annahme zum Doktoran-

ten in mich setzte, und den großen Freiräumen, die er mir bei der Bewältigung meiner Aufga-

ben ließ, vor allem seine menschliche Art, mit Kritik und einer konträren Meinung umzuge-

hen, hoch an.

Erling Stenby schulde ich viel Dank für die zahlreichen wertvollen Laborstunden, in denen

ich in seinem Institut in Lyngby forschen durfte. Es ehrt mich sehr, dass er bereitwillig die

Mühe und den weiten Weg auf sich nahm, um meiner Promotion als Gutachter beiwohnen zu

können.

Ulrich Rüde danke ich neben seiner spontanen Bereitschaft, als fachfremder Prüfer zu fungie-

ren, für die Flexibilität bei der Terminfindung für die Verteidigung. Axel König bin ich für die

Art verbunden, mit der er mich als Prüfungsvorsitzender sicher und souverän durch unruhiges

Wasser geleitete und zur gegebenen Zeit erlöste.

Sollte die vorliegende Arbeit ein wenig Ruhm und Ehre verdienen, so gebührt ein großer Teil

davon meinen Studien-, Diplom- und Masterarbeitern Andreas Schwarz, Aina Carreras Moli-

na, Su Wong Sik, Alexander Buchele und Florian Lottes. Ohne ihr großes persönliches Enga-

gement wären die folgenden Seiten leer geblieben.

Matthias Buggert, Martin Drescher, Hans Geus, Carsten Jork, Jörn Rolker, Matthias Seidel,

Irina Smirnova, Oliver Spuhl, Supakij Suttiruengwong, Dirk Weckesser und vielen weiteren

Berliner beziehungsweise Erlanger Lehrstuhlangehörigen danke ich für die schönen fünf Jah-

re. Es hat Spaß gemacht mit Euch!

Stefanie Herzog danke ich für viel geteiltes Leid über Fernbeziehungen zwischen Berlin und

Erlangen. Bei Steffi Hiller stehe ich für Ihre selbstlosen Mühen, aus einem Assistentenfrisch-


ii

ling einen wissenschaftlichen Mitarbeiter zu machen, in tiefer Schuld. Thomas Schneider bin

ich für die entstandene wissenschaftliche Fernfreundschaft sehr verbunden. Jing Lan Wu dan-

ke ich für die Einladung nach China. Ich werde kommen.

Ich habe mich sehr über die Teilnahme meiner Freunde Eugen Duvnjak, Sascha Ersel und

Sebastian Nowak an meiner Verteidigung gefreut. Die Erinnerung an den gewaltigen Schall-

druck von Eugens Singstimme wird mir erhalten bleiben.

Zum Abschluss, in die Zukunft blickend, möchte ich Hannah Böing danken. Es ist schön, dass

es Dich gibt.

Es ist vollbracht. Auf zu neuen Ufern ...


iii

Zusammenfassung (deutsch)

Die Flüssigchromatographie ist eine thermische Grundoperation deren hohe Trenneffizienz

stark von der homogenen Struktur des gepackten Bettes abhängt. Aus diesem Grunde wurden

im Rahmen der vorliegenden Arbeit zwei unterschiedliche Ansätze zur einflussfreien Beurtei-

lung der Packungseigenschaften chromatographischer Säulen untersucht.

Die erste Ansatz umfasste die Untersuchung der Eignung tomographischer Meßmethoden zur

Bereitstellung relevanter Informationen über die Packungsstruktur der Säulen. Anhand von

experimentellen Untersuchungen konnte hierbei nachgewiesen werden, dass sowohl compu-

tertomographische Untersuchungen als auch Magnetresonanztomographiemessungen zur Be-

reitstellung von Informationen über die Packungseigenschaften geeignet sind.

Der andere Ansatz beinhaltete die Studie des Einflusses von unregelmäßigen Packungen auf

die Messsignale, die mittels peripherer Sensoren aufgezeichnet werden. Zur experimentellen

Untersuchung dieser Einflüsse wurden erfolgreich Vorgehensweisen entwickelt, Packungsfeh-

ler definiert im Experiment nachzustellen. Anhand der anschließenden batchchromatographi-

schen Untersuchungen ließ sich ableiten, dass unterschiedliche Inhomogenitätsmerkmale zu

unterschiedlichen Mustern in der Signalverschiebung der Sensoren führen und somit eine

Identifikation erlauben.

Zusätzlich zu den experimentellen Untersuchungen wurden zur Erweiterung der verfügbaren

Datenbasis Computational Fluid Dynamics Simulationen durchgeführt. Hierzu wurden zu-

nächst die Programmmerkmale des kommerziellen CFD-Codes StarCD mittels Usercoding so

erweitert, dass eine Beschreibung chromatographischer Vorgänge möglich war. Anschließend

wurde die Modellierung inhomogener Säulen anhand der vorhandenen Messergebnisse erfolg-

reich validiert. Weiterführende Simulationen erlaubten die gezielte parametrische Untersu-

chungen des Einflusses von Eigenschaften der Fehlstelle auf das Trennergebnis.


iv

Summary

Liquid chromatography is a unit operation with a high separation efficiency that is based on a

homogenous structure of the packed bed. Because of this, the work is devoted to study the

suitability of two different, non-invasive approaches for the evaluation of the packing proper-

ties.

The first approach comprised the investigation of the suitability of tomographic measurement

techniques to yield relevant information about the packing structure. On the basis of experi-

mental investigations it could be shown that computed tomography measurements as well as

nuclear magnetic resonance studies are suited to get information about the properties of the

packed bed.

The second approach was about the influence of irregular column packings on the signals

monitored by peripheral sensors. In order to study these effects experimentally, procedures to

mimic inhomogeneities in the experiment were successfully developed. From the following

batch chromatographic investigations it could be learned that different kinds of column bed

inhomogeneity result in distinct patterns of the signal shift monitored by the peripheral sen-

sors. This renders the identification of the type of irregularity possible.

In addition to the experimental studies, computational fluid dynamics simulations were car-

ried out in order to expand the available data base. Firstly, the feasibilities of the commercial

CFD code StarCD were expended by means of usercoding to allow the modeling of chroma-

tographic processes. Subsequently the modeling of inhomogeneous columns was successfully

validated by comparison of the simulation and the experimental results. Additional simula-

tions enabled parameter studies of the effect of the inhomogeneity’s properties on the separa-

tion result.


v

Inhaltsverzeichnis

DANKSAGUNG (DEUTSCH) .............................................................................................................................. I

ZUSAMMENFASSUNG (DEUTSCH)..............................................................................................................III

ZUSAMMENFASSUNG (ENGLISCH) ............................................................................................................ IV

INHALTSVERZEICHNIS (DEUTSCH) ............................................................................................................V

INHALTSVERZEICHNIS (ENGLISCH) .....................................................................................................VIII

0 EINLEITUNG (DEUTSCH)....................................................................................................................... 1

1 EINLEITUNG (ENGLISCH) ..................................................................................................................... 4

2 GRUNDLAGEN DER CHROMATOGRAPHIE ..................................................................................... 7

2.1 ADSORPTION AUS DER FLÜSSIGPHASE ................................................................................................... 7

2.2 POROSITÄT ........................................................................................................................................... 9

2.3 DAS CHROMATOGRAMM UND ABGELEITETE PARAMETER ....................................................................11

2.3.1 Retentionszeit und verwandte Größen............................................................................................11

2.3.2 Peakbreite und verwandte Größen................................................................................................ 12

2.3.3 Auflösung ...................................................................................................................................... 14

2.4 DAS STUFENKONZEPT ......................................................................................................................... 14

2.5 MODELLIERUNG DER CHROMATOGRAPHIE .......................................................................................... 15

2.5.1 Das ideale Modell der Chromatographie...................................................................................... 15

2.5.2 Axiale Dispersion .......................................................................................................................... 18

2.5.3 Stoffübergangswiderstand ............................................................................................................. 20

2.5.4 Van Deemter Gleichung ................................................................................................................ 21

2.5.5 Das Gleichgewichts-Dispersionsmodell........................................................................................ 22

2.6 DRUCKVERLUST ................................................................................................................................. 23

2.7 SLURRY PACKVERFAHREN ................................................................................................................... 24

2.8 INHOMOGENE PACKUNGSSTRUKTUREN IN DER SÄULE......................................................................... 25

2.8.1 Inhomogenitäten aufgrund des Slurry Packverfahrens ................................................................. 26

2.8.2 Inhomogenitäten aufgrund instabiler Regionen innerhalb der Säule............................................ 27

2.9 CFD-MODELLIERUNG ........................................................................................................................ 27

2.9.1 Die finite Volumen Methode .......................................................................................................... 28

2.9.2 Diskretisierung des dispersiven Terms .......................................................................................... 29

2.9.3 Diskretisierung des konvektiven Terms.......................................................................................... 30

2.9.4 Zeitliche Diskretisierung ............................................................................................................... 30

2.9.5 Der SIMPLE Alghoritmus ............................................................................................................. 31


vi

2.9.6 Dimensionslose Kennzahlen.......................................................................................................... 31

3 GRUNDLAGEN TOMOGRAPHISCHER MESSVERFAHREN......................................................... 33

3.1 MOTIVATION FÜR NICHTINVASIVE MESSUNGEN ................................................................................... 33

3.2 COMPUTERTOMOGRAPHIE FÜR PACKUNGEN UND PORÖSE MEDIEN ...................................................... 34

3.2.1 Grundlagen der Computertomographie........................................................................................ 34

3.2.2 Computertomographie im Zusammenhang mit porösen Medien................................................... 35

3.3 GESCHWINDIGKEITSMESSUNGEN MIT MAGNETRESONANZTOMOGRAPHIE ........................................... 36

3.3.1 Grundlagen der Magnetresonanztomographie.............................................................................. 36

3.3.2 Phasencodierte Geschwindigkeitsmessungen................................................................................ 38

4 EXPERIMENTELLES ............................................................................................................................. 42

4.1 EXPERIMENTELLER AUFBAU FÜR EXPERIMENTE MIT KÜNSTLICHEN INHOMOGENITÄTEN .................... 42

4.1.1 Preparative Chromatographiesystem............................................................................................ 42

4.1.2 Säulen und stationäre Phasen ....................................................................................................... 43

4.1.3 Mobile Phasen und Tracerstoffe.................................................................................................... 44

4.1.4 Lokale Inhomogenitäten................................................................................................................ 45

4.2 EXPERIMENTELLE ERGEBNISSE .......................................................................................................... 47

4.2.1 Referenzmessungen ....................................................................................................................... 47

4.2.2 Lokale Inhomogenitäten................................................................................................................ 47

4.2.3 Einlasskavität ................................................................................................................................ 49

4.2.4 Entstehung von Feinpartikeln durch Abrieb.................................................................................. 50

4.3 CHARACTERISIERUNG VON PACKUNGSINHOMOGENITÄTEN DURCH COMPUTERTOMOGRAPHIE............ 51

4.3.1 Experimenteller Aufbau................................................................................................................. 51

4.3.2 Kalibrierung des CT-Scanners ...................................................................................................... 54

4.3.3 Bandenprofile und interne Durchbruchskurven ............................................................................ 55

4.3.4 Radiale Homogenität der Säulen .................................................................................................. 58

4.3.5 Effizienz der Säulen....................................................................................................................... 60

4.4 GESCHWINDIGKEITSMESSUNGEN MITTELS MAGNETRESONANZTOMOGRAPHIE ................................... 62

4.4.1 Experimenteller Aufbau................................................................................................................. 62

4.4.2 Vergleichsmessungen..................................................................................................................... 63

4.4.3 Geschwindigkeitsprofile ................................................................................................................ 64

5 CFD SIMULATIONEN ............................................................................................................................ 66

5.1 NUMERISCHE BETRACHTUNGEN ......................................................................................................... 66

5.1.1 Skalierung der partiellen Differentialgleichungen ........................................................................ 66

5.1.2 Wahl von Orts- und Zeitschrittweite.............................................................................................. 67

5.1.3 Permeabilität und Druckverlust .................................................................................................... 70

5.2 IMPLEMENTIERUNG DER ADSORPTION................................................................................................. 71

5.3 GLEICHGEWICHTS-DISPERSIONSMODELL............................................................................................. 72

5.4 STOFFÜBERGANGSMODELL................................................................................................................. 74


vii

5.5 VALIDIERUNG DER ISOTHERMEN ......................................................................................................... 76

5.6 CFD-MODELLIERUNG LOKALER INHOMOGENITÄTEN ......................................................................... 77

5.6.1 Vergleich zwischen Experiment und Simulation für lokale Leerstellen ......................................... 77

5.6.2 Einfluss lokaler Inhomogenitäten auf die Permeabilität der Säule............................................... 82

5.6.3 Einfluss lokaler Inhomogenitäten auf das erste Moment .............................................................. 84

5.6.4 Einfluss lokaler Inhomogenitäten auf die Effizienz der Säule ....................................................... 85

5.7 CFD-MODELLIERUNG EINER EINLASSKAVITÄT................................................................................... 86

5.7.1 Einfluss einer Einlasskavität auf die Effizienz der Säule............................................................... 87

5.7.2 Einfluss einer Einlasskavität auf das erste Moment..................................................................... 89

5.7.3 Einfluss einer Einlasskavität auf die Permabilität der Säule ........................................................ 90

5.8 CFD-MODELLIERUNG EINER WANDREGION BASIEREND AUF COMPUTERTOMOGRAPHIE ..................... 91

5.8.1 Modellentwicklung basierend auf computertomographischen Experimenten .............................. 91

5.8.2 Vorgehensweise bei der Simulation ............................................................................................... 94

5.8.3 Simulationsergebnisse ................................................................................................................... 94

5.8.4 Auswirkungen radialer Inhomogenitäten...................................................................................... 95

5.9 AUSWIRKUNGEN EINER WANDREGION BEZÜGLICH CHARAKTERISTISCHER PARAMETER...................... 97

5.9.1 Einfluss einer Wandregion auf die Permabilität der Säule............................................................ 97

5.9.2 Einfluss einer Wandregion auf auf das erste Moment ................................................................... 98

5.9.3 Einfluss einer Wandregion auf die Effizienz der Säule .................................................................. 98

6 ZUSAMMENFASSUNG ......................................................................................................................... 101

6.1 EINFLUSS AUF DIE SIGNALE PERIPHERER SENSOREN.......................................................................... 101

6.2 TOMOGRAPHISCHE MESSTECHNIKEN ................................................................................................ 104

SYMBOLVERZEICHNIS................................................................................................................................ 106

LITERATURSTELLEN....................................................................................................................................111

APPENDIX I ..................................................................................................................................................... 123

APPENDIX II.................................................................................................................................................... 123

APPENDIX III .................................................................................................................................................. 124

APPENDIX IV .................................................................................................................................................. 125

APPENDIX V .................................................................................................................................................... 126

APPENDIX VI................................................................................................................................................... 128


viii

Table of contents

DANKSAGUNG (DEUTSCH) .............................................................................................................................. I

ZUSAMMENFASSUNG (DEUTSCH)..............................................................................................................III

SUMMARY ......................................................................................................................................................... IV

INHALTSVERZEICHNIS ...................................................................................................................................V

TABLE OF CONTENTS .................................................................................................................................VIII

0 EINLEITUNG (DEUTSCH)....................................................................................................................... 1

1 MOTIVATION AND GOAL....................................................................................................................... 4

2 BASICS OF CHROMATOGRAPHY ........................................................................................................ 7

2.1 ADSORPTION FROM THE LIQUID PHASE ................................................................................................. 7

2.2 FRACTIONAL MOBILE PHASE VOLUME................................................................................................... 9

2.3 THE CHROMATOGRAM AND DERIVED PARAMETERS ..............................................................................11

2.3.1 Retention time and related quantities.............................................................................................11

2.3.2 Peak width and related quantities ................................................................................................. 12

2.3.3 Resolution...................................................................................................................................... 14

2.4 THE STAGE CONCEPT........................................................................................................................... 14

2.5 MODELING OF CHROMATOGRAPHY ..................................................................................................... 15

2.5.1 Ideal Model of Chromatography ................................................................................................... 15

2.5.2 Axial dispersion............................................................................................................................. 18

2.5.3 Mass transfer resistance................................................................................................................ 20

2.5.4 Van Deemter equation ................................................................................................................... 21

2.5.5 The Equilibrium-Dispersive-Model............................................................................................... 22

2.6 PRESSURE DROP .................................................................................................................................. 23

2.7 SLURRY PACKING ................................................................................................................................ 24

2.8 INHOMOGENEOUS COLUMN BED STRUCTURE ...................................................................................... 25

2.8.1 Inhomogeneities associated to slurry packing techniques............................................................. 26

2.8.2 Inhomogeneities associated to unstable column regions............................................................... 27

2.9 CFD-MODELING................................................................................................................................. 27

2.9.1 The finite volume method .............................................................................................................. 28

2.9.2 Discretisation of the dispersive term ............................................................................................. 29

2.9.3 Discretisation of the convective term ............................................................................................ 30

2.9.4 Temporal discretisation ................................................................................................................. 30

2.9.5 SIMPLE algorithm ........................................................................................................................ 31

2.9.6 Dimensionless numbers................................................................................................................. 31


ix

3 BASICS OF TOMOGRAPHIC MEASUREMENT TECHNIQUES .................................................... 33

3.1 MOTIVATION FOR NON-INVASIVE MEASUREMENTS .............................................................................. 33

3.2 COMPUTED TOMOGRAPHY FOR PACKED BEDS AND POROUS MEDIA ..................................................... 34

3.2.1 Basics of Computed Tomography .................................................................................................. 34

3.2.2 Computed Tomography in conjunction with porous media ........................................................... 35

3.3 NUCLEAR MAGNETIC RESONANCE VELOCIMETRY ............................................................................... 36

3.3.1 Basics of nuclear magnetic resonance .......................................................................................... 36

3.3.2 Phase encoded velocity measurements.......................................................................................... 38

4 EXPERIMENTAL..................................................................................................................................... 42

4.1 EXPERIMENTAL SETUP FOR THE EXPERIMENTS WITH ARTIFICIALLY CREATED INHOMOGENEITIES ........ 42

4.1.1 Preparative chromatography system............................................................................................. 42

4.1.2 Columns and stationary phase materials ...................................................................................... 43

4.1.3 Mobile phases and tracer component ........................................................................................... 44

4.1.4 Local inhomogeneities................................................................................................................... 45

4.2 EXPERIMENTAL RESULTS..................................................................................................................... 47

4.2.1 Reference measurements ............................................................................................................... 47

4.2.2 Local inhomogeneities................................................................................................................... 47

4.2.3 Inlet void ....................................................................................................................................... 49

4.2.4 Creation of fines ............................................................................................................................ 50

4.3 CHARACTERIZATION OF PACKING HOMOGENEITIES BY MEANS OF COMPUTED TOMOGRAPHY.............. 51

4.3.1 Experimental set-up....................................................................................................................... 51

4.3.2 Calibration of the CT-scanner....................................................................................................... 54

4.3.3 Band profiles and intra column breakthrough curves ................................................................... 55

4.3.4 Radial homogeneity of the columns............................................................................................... 58

4.3.5 Column efficiency.......................................................................................................................... 60

4.4 VELOCITY MEASUREMENTS BY NUCLEAR MAGNETIC RESONANCE IMAGING ....................................... 62

4.4.1 Experimental set-up....................................................................................................................... 62

4.4.2 Comparative measurements .......................................................................................................... 63

4.4.3 Velocity profiles ............................................................................................................................. 64

5 CFD SIMULATIONS................................................................................................................................ 66

5.1 NUMERICAL CONSIDERATIONS ............................................................................................................ 66

5.1.1 Scaling of the partial differential equations .................................................................................. 66

5.1.2 Choice of mesh density and time step............................................................................................ 67

5.1.3 Permeability and pressure drop .................................................................................................... 70

5.2 IMPLEMENTATION OF ADSORPTION...................................................................................................... 71

5.3 EQUILIBRIUM DISPERSIVE MODEL ....................................................................................................... 72

5.4 MASS TRANSFER MODEL ..................................................................................................................... 74

5.5 VALIDATION OF ISOTHERMS ................................................................................................................ 76


x

5.6 CFD-MODELING OF LOCAL COLUMN INHOMOGENEITIES .................................................................... 77

5.6.1 Comparison between experiment and simulation result for hollow regions.................................. 77

5.6.2 Influence of local inhomogeneities on the column permeability ................................................... 82

5.6.3 Influence of local inhomogeneities on the first moment ................................................................ 84

5.6.4 Influence of local inhomogeneities on the column efficiency ........................................................ 85

5.7 CFD-MODELING OF AN INLET VOID .................................................................................................... 86

5.7.1 Influence of an inlet void on the column efficiency ...................................................................... 87

5.7.2 Influence of an inlet void on the first moment .............................................................................. 89

5.7.3 Influence of an inlet void on the column permeability ................................................................. 90

5.8 CFD-MODELING OF A WALL REGION BASED ON COMPUTED TOMOGRAPHY EXPERIMENTS .................. 91

5.8.1 Model set-up based on computed tomography experiments.......................................................... 91

5.8.2 Simulation processing ................................................................................................................... 94

5.8.3 Simulation results .......................................................................................................................... 94

5.8.4 Consequences of the radial column inhomogeneity ...................................................................... 95

5.9 CONSEQUENCES OF A WALL REGION REGARDING CHARACTERISTIC PARAMETERS ............................... 97

5.9.1 Influence of a column wall region on the column permeability .................................................... 97

5.9.2 Influence of a column wall region on the first moment ................................................................. 98

5.9.3 Influence of a column wall region on the column efficiency ......................................................... 98

6 RÉSUMÉ.................................................................................................................................................. 101

6.1 INFLUENCE ON PERIPHERAL SENSOR SIGNALS ................................................................................... 101

6.2 TOMOGRAPHIC MEASUREMENT TECHNIQUES .................................................................................... 104

LIST OF SYMBOLS......................................................................................................................................... 106

REFERENCES...................................................................................................................................................111

APPENDIX I ..................................................................................................................................................... 123

APPENDIX II.................................................................................................................................................... 123

APPENDIX III .................................................................................................................................................. 124

APPENDIX IV .................................................................................................................................................. 125

APPENDIX V .................................................................................................................................................... 126

APPENDIX VI................................................................................................................................................... 128


1

0 Einleitung (deutsch)

Die Flüssigchromatographie, wohlbekannt als eine Standardanalyseanwendung, ist ebenfalls

eine thermische Grundoperation bei der zwei Hilfsphasen verwendet werden, um die Auftren-

nung von zwei oder mehr Komponenten zu erreichen. Üblicherweise ist die erste Hilfsphase

eine Flüssigkeit während die zweite Hilfsphase fest ist. Die zu trennenden Substanzen werden

in der flüssigen (bzw. mobilen) Phase gelöst und von dieser durch ein dicht gepacktes Bett

bestehend aus porösen Partikeln transportiert. Dieses Bett wird als stationäre Phase bezeich-

net. Aufgrund unterschiedlicher Affinitäten der gelösten Substanzen zur stationären Phase

werden einige Komponenten stärker retardiert und eluieren später. Dieser Unterschied in der

Retentionszeit ermöglicht die Trennung. Die Methode wurde von dem russischen Wissen-

schaftler Mikhael Tswett Anfang des zwanzigsten Jahrhunderts entwickelt, vorangetrieben

und zunächst für präparative Anwendungen eingesetzt [Tswett1906].

Die dynamische Entwicklung der Biotechnologie, der pharmazeutischen Industrie und der

Produktion von Feinchemikalien haben innerhalb der letzten zwei Dekaden einen merklich

Anstieg der Bedeutung der präparativen und industriellen Chromatographie verursacht [Guio-

chon1994]. In diesen Industriezweigen, die ein großes Spektrum an Substanzen vermarkten

und hohe, teils von den Regulierungsbehören (z.B. die FDA [FDA1992]) auferlegte Rein-

heitsanforderungen erfüllen müssen, ist die Flüssigchromatographie als vielseitige und ther-

misch schonende Trennoperation im Vergleich zu den klassischen Grundoperationen prädesti-

niert.

Um industrielle chromatographische Prozesse ökonomisch zu betreiben, muss das chromato-

graphische System eine hohe Trenneffizienz aufweisen, die nur durch eine homogene Struktur

des gepackten Bettes erreicht werden kann [Mann1998]. Bereits während der Anfänge der

Chromatographie erkannte M. Tswett die Bedeutung einer regelmäßigen Packungsstruktur für

die hohe Effizienz, die für flüssigchromatographische Anwendungen typisch ist [Tswett1906],

[Tswett1967]: „Die homogene Textur der adsorbierenden Säule ist sehr wichtig sonst gestal-

ten sich die verschiedenen Adsorptionszonen zu sehr unregelmäßigen Gebilden, was deren

mechanische Trennung höchst erschwert”.

Das Ziel dieser Arbeit ist die Entwicklung von Methoden für die einflussfreie Bestimmung

der Eigenschaften des gepackten Bettes, dass die Effizienz des chromatographischen Prozes-

ses bestimmt. Die Arbeit beschränkt sich hierbei auf Slurry-Packmethoden, die die am weite-


2

sten verbreitete Methode der Säulenherstellung darstellen [Dingenen1994]. In diesem Zu-

sammenhang ist unter einflussfrei eine Beurteilungsmethode gemeint, die weder auf der Zer-

störung des gepackten Bettes beruhen noch die Hydrodynamik innerhalb der Säule in gleich

welcher Form beeinflussen darf (z.B. durch Platzierung von Messfühlern innerhalb der Säu-

le). Im Folgenden werden zwei Ansätze verfolgt. Der erste Ansatz beschäftigt sich mit der

Eignung nichtinvasiver, tomographischer Messmethoden (Röntgencomputertomographie,

Magnetresonanztomographie) Informationen über die Packungsstruktur bereitzustellen.

Der andere Ansatz umfasst die Untersuchung des Einflusses von unregelmäßigen Packungen

auf die Messsignale, die mittels externer, peripherer Sensoren wie dem Detektor bzw. von

Manometern aufgezeichnet werden. Diese Art von Sensoren wird üblicherweise zur Überwa-

chung und Kontrolle des chromatographischen Prozesses eingesetzt beeinflusst die Hydrody-

namik innerhalb der Säule nicht. Zur Untersuchung der Signalveränderungen die durch Inho-

mogenitäten verursacht werden, wurden hierfür sowohl im Experiment als auch in der

Simulation künstliche unregelmäßige Strukturen innerhalb der Säule geschaffen.

Zur besseren Verständlichkeit der Arbeit wird in Kapitel 2 eine kurze Einführung in die

Chromatographie und deren Modellierung mittels Computational Fluid Dynamics (CFD) ge-

geben. Besondere Beachtung erfahren Erkenntnisse über die Packungsstruktur von slurry-

gepackten Säulen. Kapitel 3 deckt die theoretischen Grundlagen, die zum Verständnis von

Experimenten im Zusammenhang mit Röntgencomputertomographie und porösen Medien

sowie Flussuntersuchungen mittels Magnetresonanztomographie notwendig sind, ab.

Kapitel 4 beleuchtet die experimentellen Untersuchungen, die im Rahmen dieser Arbeit

durchgeführt wurden. Dies umfasst sowohl die computertomographischen Untersuchungen

zur Packungsstruktur und die Magnetresonanzstudien zu Flussprofilen als auch batchchroma-

tographische Untersuchungen zum Einfluss der künstlich geschaffenen Inhomogenitäten auf

die aufgezeichneten Signale.

Kapitel 5 geht auf die Modellierung unregelmäßiger Bettstrukturen in chromatographischen

Säulen mit Hilfe des Computational Fluid Dynamics ein. Zunächst werden die Modellkapazi-

täten des verwendeten CFD-codes mittels Usercoding so erweitert, dass die Simulation chro-

matographischer Prozesse möglich ist. Anschließend – im Zusammenhang mit inhomogenen

Packungen in der Säule – erlauben die CFD-Rechnungen einerseits die im Vergleich zum Ex-

periment raschere Untersuchung des Einflusses von Parametern wie der Größe und der An-


3

ordnung der Inhomogenität und helfen somit, die experimentelle Datenbasis zu erweitern.

Andererseits ermöglichen die CFD-Untersuchungen auch die Studie von Problemen, die expe-

rimentell nur schwer oder gar nicht nachstellbar gewesen wären.

Das abschließende Kapitel 6 dient der Zusammenfassung der Erkenntnisse der vorangegange-

nen Kapitel in einer übersichtlichen und anschaulichen Form. Die Ergebnisse werden zusam-

mengestellt und kritisch beleuchtet Weiterhin wird eine kurze Übersicht über weiterführende

Fragestellungen, die aus der vorgelegten Arbeit hervorgehen und offene Probleme, die weiter-

hin auf eine Lösung warten, gegeben.


4

1 Motivation and goal

Liquid chromatography, well known as a standard analytical method, is also a unit operation

that uses two auxiliary phases to achieve separation of two or more components. Normally,

the first auxiliary phase is a liquid with the second auxiliary phase being a solid. The sub-

stances to be separated are dissolved in the liquid (or mobile) phase which then conveys them

through a bed of tightly packed, porous particles comprising the so-called stationary phase.

Due to different adsorption affinities of the components to the stationary phase, some com-

pounds are retained more strongly and – consequently - elute later. This difference in retention

time allows for the separation. The method was developed and pioneered by the Russian sci-

entist Mikhael Tswett at the end of the twentieth century and initially used for the preparative

separation of plant pigments [Tswett1906].

The dynamic developments in biotechnology, the pharmaceutical industry, as well as in the

production of fine chemicals are responsible for the significant increase in the importance of

preparative and industrial chromatography within the last two decades [Guiochon1994].

These branches of industry market a huge spectrum of substances and have to fulfill high pu-

rity demands, which are partly imposed by the regulation authorities (e.g. the FDA

[FDA1992]). In this setting liquid chromatography as a versatile and thermally benign separa-

tion process is destined to prevail over the classical unit operations.

In order to run industrial scale chromatographic processes economically, the chromatographic

system must have a high separation efficiency that can only be achieved in the presence of a

homogenous structure of the packed bed [Mann1998]. Already during the early stages of

chromatography M. Tswett became aware of the importance of a regular packing structure for

attaining higher efficiency for liquid chromatographic applications [Tswett1906],

[Tswett1967]: „The homogenous texture of the adsorbent is very important, otherwise the

various adsorption zones are formed so very irregularly that their mechanical separation is

very difficult.”

The goal of this work is to develop methods for the non-invasive evaluation of the packed

bed’s characteristics which determine the efficiency of the chromatographic process. The

work is restricted to columns packed by slurry methods, being the most widespread mode for

packing manufacture [Dingenen1994]. In this context non-invasive means that the evaluation

method must not rely on the destruction of the packed bed nor must it influence the hydrody-


5

namics inside the column in any way (e.g. by placing sensors inside the packed bed). In the

following two different approaches are discussed. Within the first approach the feasibility of

non-invasive tomographic measurement techniques (such as X-ray Computed Tomography

and Nuclear Magnetic Resonance Imaging) to yield information about the packing structure

of chromatographic columns is studied.

The second approach involves the investigation of the influence of uneven column packings

on measurement signals acquired by external, peripheral sensors like the detector or pressure

gauges. These kinds of sensors are commonly used to monitor and control chromatographic

processes and do not influence the hydrodynamics inside the column. In order to investigate

the signal shift caused by the inhomogeneities, artificial irregularities were generated in the

experimental as well as in the simulation studies.

As a brief introduction the basics of chromatography and its modeling by means of computa-

tional fluid dynamic (CFD) methods are reviewed in chapter 2. Special attention is paid to

considerations of the packing structure of chromatographic columns packed by means of

slurry techniques. Chapter 3 covers the theoretical fundamentals needed for the understanding

of x-ray computed tomography experiments in conjunction with porous media and the inves-

tigation of flow structures by means of nuclear magnetic resonance imaging.

Chapter 4 illuminates the experimental investigations carried out within the scope of this

work. This comprises the (x-ray computed) tomographic experiments on packing structures

and the nuclear magnetic resonance studies of the flow profile as well as the batch chroma-

tographic investigations of the influence of artificially created irregularities on monitored sig-

nals.

Chapter 5 discusses the modeling of unevenly distributed bed structures in chromatographic

columns by means of computational fluid dynamics. For a start, the modeling capabilities of

the CFD code are extended by user coding to enable the simulation of chromatographic proc-

esses. Thereafter – in the context of inhomogeneous column packings – the CFD calculations

allow on the one hand to study the influence of certain parameters like the size or the location

of the irregularity more swiftly thereby expanding the experimental data base. On the other

hand they render possible the investigation of problems that are hardly accessible by meas-

urements.


6

The final chapter 6 serves to summarize the findings of the previous chapters in a concise and

illustrative form. The results are compiled and thoughtfully discussed. Furthermore, a brief

survey of continuative problems arising from the present work as well as open questions is

presented.


7

2 Basics of chromatography

The following sections serve to acquaint the reader with the basics of chromatography and its

modeling. The adsorption process is briefly introduced as the underlying thermodynamic

principle of chromatographic separations and then the chromatogram as the most common

depiction of the separation result is discussed. Attention is paid to the modeling of chroma-

tographic processes and the description of effects contributing to the broadening of solute

peaks inside the chromatographic column.

Slurry techniques as the most common methods to pack chromatographic columns are ex-

plained and a brief literature review about packing inhomogeneities associated with these

packing techniques given.

Finally, computational fluid dynamics is introduced as an alternative to make feasible a more

rigorous modeling of chromatographic processes.

2.1 Adsorption from the liquid phase

In liquid chromatography, the components to be separated are dissolved in a liquid (the mo-

bile phase) which percolates through a column packed with solid, porous particles (the sta-

tionary phase or adsorbent). The separation principle of the process bases on the difference in

the liquid-solid adsorption equilibria of the components to be separated. The differences in

adsorption affinity result in distinct migration speeds and render purification possible. In this

context adsorption means the binding of the dissolved molecules from the mobile phase to the

surface of the porous adsorbent.

Adsorption affinity is commonly described by means of loading isotherms (or adsorption iso-

therms) which correlate the rate of the mobile phase concentration ic and the stationary phase

concentration iq in the state of equilibrium. Hirsch [Hirsch2000] demonstrated that adsorp-

tion isotherms describe thermodynamic state variables.


8

Figure 2.1: Nomenclature of the adsorption process [Johannsen2004]

At low solute concentrations, the dependence of the equilibrium concentration in the adsorb-

ent is linear and the simplest form of loading isotherm describes the adsorption behaviour

well. *iq denotes the equilibrium concentration of the stationary phase, iH is the so-called

Henry-constant of the isotherm.

iii cHq ⋅=* (2.1)

At higher concentrations the concentration overload leads to non-linear adsorption behaviour

as the number of adsorption sites becomes restricted. The most prevalent non-linear adsorp-

tion equilibrium relation is the Langmuir isotherm, which accounts for the effects of solute

interactions and sorbent saturation [Dünnebier2000].

ii

iiii

cb

cbqq

⋅+

⋅⋅=1

*max,

* (2.2)

According to the competitive Langmuir approach the number of sites at which the solute

molecules can adsorb is limited and the total concentration in the stationary phase can not

exceed a limiting concentration *max,iq (loadability). When this concentration is reached, all

available adsorption sites are covered and additional molecules can not adsorb, even if the

concentration in the bulk phase is further increased.


9

Figure 2.2: Langmuir isotherm

2.2 Fractional mobile phase volume

The total volume CV of a chromatographic column can be divided into three parts: i) the vol-

ume extV between the porous stationary phase particles, ii) the total pore volume intV of the

stationary phase, and iii) the particle volume solV without pores. Using these volumes differ-

ent porosities can then be defined [SeidelMorgenstern1995].

Figure 2.3: Fractional volumes inside a chromatographic column


10

The total porosity, defined as the ratio of the entire volume occupied by the mobile phase and

the total column volume, is given by

C

exttot

V

VV += intε (2.3)

It is can be determined from tracer experiments with a component small enough to enter the

pore system but not such that it adsorbs on the surface of the stationary phase.

The external or interstitial bed porosity extε or bed voidage is defined as the ratio of the inter-

particle volume extV and the column volume.

C

extext

V

V=ε (2.4)

It can be determined from tracer experiments with a non-adsorbed component whose molecu-

lar structure is sufficiently large to prevent it from entering the pore system of the stationary

phase. Depending on the choice of whether the intraparticle pore volume intV is considered as

part of the sorbent phase or as part of the mobile phase volume, the fractional mobile phase

volume ε equals either the external porosity extε or the total porosity totε . The ratio of the

stationary or sorbent phase volume and the mobile phase volume is commonly referred to as

the phase ratio F . It can be expressed in terms of the fractional volume ε .

ε

ε−=

1F (2.5)

In the following, the intraparticle pore volume is assumed to be part of the stationary phase

volume. Consequently all molecules within the pore system are considered adsorbed even if

they are dissolved in the liquid inside the pores. Since the structure of the stationary phase is

not homogenous, the sorbent concentrations are averaged over the whole particle volume.

It is worth noting that the external porosity extε and the total porosity totε are not independent

of each other but coupled by the following equation.

( ) int1 εεεε ⋅−+= extexttot (2.6)

intε is the internal porosity which is defined as the ratio of the intraparticle pore volume intV

and the particle volume PV .


11

PV

Vintint =ε (2.7)

2.3 The chromatogram and derived parameters

2.3.1 Retention time and related quantities

In order to evaluate the quality of a chromatographic separation, the mobile phase is trans-

ported through a detector that records the dissolved components based on a certain physical

principle. The recording of the detector signal over time is called a chromatogram. The deflec-

tions corresponding to the detected components are referred to as peaks [Lenz2003].

Figure 2.4: Chromatogram for the pulse injection of a three component mixture containing two retained and one unretained compounds

The chromatogram allows to obtain basic information for the development of a chroma-

tographic separation process [Schulte2005]. In cases where the peaks are symmetrical, the

time from the start of the measurement (time of injection) to the maximum of a peak is called

the retention time of a component.

For asymmetrical peaks, the concept to measure the retention time at the apex of a peak no

longer holds true. For these peaks where the retention time of the peak apex does not coincide


12

with the retention time of the peak’s center of gravity, the retention time is more generally

defined as the first moment of the peak.

( )

( )∫

∫∞

∞

⋅

⋅⋅

=

0

0,1

dttc

dtttc

i

i

iµ (2.8)

It is proportional to the components affinity to the stationary phase.

( )ioi HFt ⋅+⋅= 1,1µ (2.9)

The use of retention time or the first moment to describe a certain chromatographic system

lacks from the disadvantage that it depends on the flow velocity of the mobile phase

[Schulte2005]. Consequently the capacity factor 'ik which can be evaluated from the retention

time of the component and the retention time 0t of a non-adsorbed substance is defined as a

purely thermodynamic parameter. It depends only on the distribution of the component be-

tween the two auxiliary phases.

iiR

i HFt

ttk ⋅=

−=

0

0,' (2.10)

In analogy to other separation techniques, a separation factor is introduced. It is given as the

ratio of the capacity factors of either component and can be determined from the retention

times.

1

2

01,

02,1,2

H

H

tt

tt

R

R=

−

−=α (2.11)

Unfortunately a high separation factor is no guarantee for satisfactory separation results. The

separation factor simply gives information on whether a separation is possible from a purely

thermodynamic point of view. The broadness of the peaks is not taken into consideration.

2.3.2 Peak width and related quantities

Another important quantity to describe a peak is the peak width iω which is a measure for

peak broadening inside the column and closely related to the efficiency of the process. Se-

peration being the stated goal of chromatography, it is beneficial if the bands are narrow. The


13

peak width can be measured at different positions relative to the peak height. The peak width

1.0,iω at 10% peak height and 5.0,iω at 50% peak height are most frequently used.

As previously mentioned the method of moments provides a more general way to describe the

spreading of a peak. The second central moment, which is identical to the variance 2iσ of the

peak, takes into account the total degree of dispersion independent of a certain, arbitrarily

chosen peak height.

∫

∫∞

∞

−

=

0

0

2,1

2

)(

dtc

dttc

i

ii

i

µ

σ (2.12)

Neither the peakwidth iω nor the variance of the peak describe the skewness, or degree of

asymmetry, of a peak. This is usually accounted for by the so called tailing factor iT compar-

ing the width of the two peak halves at 10% peak height [Meyer1992].

Figure 2.5: Evaluation of the tailing factor (based on [Schulte2005])


14

1.0,

1.0,

i

i

ia

bT = (2.13)

2.3.3 Resolution

The resolution is a measure well suited to assess the effectiveness of the entire chroma-

tographic separation system. It combines thermodynamics (difference in retention time) and

column efficiency (peak width) and defines the degree of separations of two analytes or

peaks, respectively.

( ) 2/21

2,1,

ωω +

−= RR tt

R (2.14)

2.4 The stage concept

Martin and Synge [Martin1941] were the first to use the terms number of theoretical plates

iN and height equivalent to a theoretical plate iHETP for chromatographic applications.

They modeled a chromatographic column with a cascade of iN ideally stirred tanks or plates

in order to describe their separation results. Nowadays, iN is a synonym for the efficiency of

a chromatographic column. The iHETP , which is the height of a layer equivalent to a theo-

retical plate, is equal to the column length divided by the efficiency iN .

i

Ci

N

LHETP = (2.15)

The most accurate way to calculate the number of theoretical plates is the method of mo-

ments. iN can be calculated from the first absolute moment and second central moment of a

peak.

2

2,1

i

i

iNσ

µ= (2.16)

Several short-cut methods can also be used, two of which are mentioned here. If the peak is

symmetrical, iN can be determined as follows [SeidelMorgenstern1995].


15

2

,5.0

,54.5

⋅=

i

iR

i

tN

ω (2.17)

If the peak is asymmetrical, the following method which was proposed by Foley et al.

[Foley1983] can be applied. The asymmetry of the peak is taken into account by use of the

tailing factor iT .

( )i

i

iR

i Tt

N +

⋅= 25.17.41

2

,1.0

,

ω (2.18)

2.5 Modeling of chromatography

2.5.1 Ideal Model of Chromatography

If the assumptions of ideal chromatography - ideal plug flow, axially uniform volumetric flow

rate, mass transfer in axial direction by convection only, local equilibrium between stationary

and mobile phase throughout the column, incompressible mobile phase - are taken for

granted, the species mass balance for a differential slice of the column can be sketched as

shown in figure 2.6.

Figure 2.6: Mass balance for a differential slice of a chromatographic column. The assump-tions of ideal chromatography are taken for granted.


16

As this work centers around liquid chromatography, the pressure drop along the column does

nearly not effect the density and the assumption of an incompressible fluid is justified.

According to the assumptions, the accumulation of mass inside the differential slice is due to

convection only and can be expresses as:

( ) ( )dzzmzmt

mconviconvi

acci+−=

∂

∂,,

,&& (2.19)

Due to the assumption that the fluid is incompressible, the volumetric flow rate V& is constant

throughout the column. The accumulated mass accim , can be divided into the mass contained

in the mobile phase volume MPaccim , and the mass contained in the stationary phase volume

SPaccim , .

( ) ( )[ ]dzzczcVt

m

t

mii

SPacci

MPacci +−⋅−=

∂

∂+

∂

∂&,, (2.20)

Dividing both sides of equation (2.20) by the slice volume ( )dzAC ⋅ yields,

( ) ( ) ( )[ ]dzzczcdzA

V

t

q

t

cii

C

ii +−⋅⋅

−=∂

∂⋅−+

∂

∂⋅

&εε 1 (2.21)

where ic and iq denote the volume averaged mean concentrations inside the slice. The ratio

of the volumetric mobile phase flow and the column cross section is generally referred to as

the superficial velocity SFu , a hypothetical velocity based on the assumption of an empty col-

umn.

Since a local equilibrium between the two phases is taken for granted, the stationary phase

concentration iq equals the equilibrium concentration ( )ii cfq =* given by the adsorption

isotherm.

The difference in the convective terms at positions z and dzz + can be approximated by

means of a truncated, first order Taylor series expansion. For 0→dz , the error becomes neg-

ligible. Furthermore, the mean concentration of the slice converges towards the concentration

( )zc at position z .


17

( )( )

( )dz

dzz

cu

t

zq

t

zciSF

ii

∂

∂⋅−

=∂

∂⋅−+

∂

∂⋅ εε 1

(2.22)

Dividing equation (2.22) by the fractional mobile phase volume ε yields the common form of

the Ideal Model of Chromatography, a partial differential equation which contains two inde-

pendent variables - time t and spatial co-ordinate z – only.

0*

=∂

∂⋅+

∂

∂⋅+

∂

∂

z

cu

t

qF

t

c iSF

ii

ε (2.23)

Equation (2.23) allows to derive an expression for the velocity ( )icu with which a certain

concentration ic migrates through the chromatographic column. This concentration velocity

is given by [Helfferich1993]:

( )

⋅+

=

i

i

SF

i

dc

dqF

ucu

*

1

ε

(2.24)

For linear adsorption isotherms or for highly diluted samples whose concentrations are still

within the linear range of the isotherm, the slope ii dcdq* of the isotherm is constant and

equals the Henry-coefficient iH . Consequently the band profile does not alter during the mi-

gration process and the elution profile is identical to the injection profile.


18

Figure 2.7: Influence of the isotherm type on peak shape [Johannsen2005]

For non-linear isotherms the shape of the adsorption isotherm influences the profile of the

eluting band. For Langmuir-like isotherms, the slope and thereby the concentration velocity

are continuously decreasing functions of concentration. This forces the band tail to spread out

even in the absence of dispersive influences. For the peak front this would lead to the physi-

cally unrealistic overhanging of the concentration profile [Ruthven1984]. Instead the peak

front travels along the column as a sharp concentration shock whose velocity ( )icu ∆ is given

by [Helfferich1993]

( )

∆

∆⋅+

=∆

i

i

SF

i

c

qF

ucu

*

1

ε

(2.25)

For the sake of clarity the influence of the isotherm type on the elution profile is sketched in

figure 2.7.

2.5.2 Axial dispersion

As already mentioned, the premises of ideal chromatography give rise to the preservation of

the injection profile in linear chromatography and the of sharp fronts in non-linear chroma-


19

tography. In non-ideal chromatography several effects contribute to the broadening of a peak

and the relaxation of shock fronts. Among these effects are molecular and eddy diffusion.

Eddy diffusion comprises the effects of band broadening that are due to the stochastic nature

of a packed bed. Due to the stochastic nature, there are local permeability differences which

have an impact on the fluid flow distribution [Guiochon2003]. Two further contributions are

illustrated in figure 2.8. As the packing is random, molecules can move on distinct paths. The

differences in the length of the passageways shown in figure 2.8a are reflected in the band

broadening. Figure 2.8b shows the velocity distribution between two particles of the station-

ary phase. Due to the tight packing and the relatively small size of the particles, the interstitial

velocity is bound to the laminar regime. The parabolic velocity profile typical for laminar

flow is adversarial as it causes the peaks to broaden.

Figure 2.8: Contributions to eddy diffusion (based on [Schulte2005])

The combined effects of molecular and eddy diffusion are commonly addressed as axial dis-

persion and can be taken into account by an additional term in the differential mass balance

that is defined in analogy to Fick’s law of diffusion.

z

cDj i

axdisp∂

∂⋅−= (2.26)

axD is the axial dispersion coefficient. According to [Ruthven1984] it can be assumed that the

contributions of molecular and eddy diffusion are approximately additive.


20

εγγ

SF

Pmax

udDD ⋅⋅+⋅= 21 (2.27)

The first term on the right hand side of equation (2.27) represents the effects of molecular

diffusion. mD is the molecular diffusivity. 1γ is related to the bed tortuosity and accounts for

the hindrance of diffusion due to the presence of the particles obstructing the diffusion paths

of the molecules. The second term represents eddy diffusion whose effects are proportional to

the particle size and flow velocity. 2γ is a geometrical constant.

2.5.3 Mass transfer resistance

In non-ideal chromatography, the mass transfer between the mobile and the stationary phase is

not instantaneous and the two phases are not in local equilibrium throughout the column. The

adsorption process can be subdivided into different substeps [Schulte2005]: convective and

diffusive transport towards the particle, film diffusion, pore and surface diffusion, adsorption

and desorption step. The kinetics of each of these steps contribute to the total band broadening

caused by mass transfer resistances.

Figure 2.9: Mass transfer phenomena during the adsorption of a molecule [Schulte2005]

Most models accounting for mass transfer resistance in chromatography take only one of the

substeps into account, the contributions of the others are assumed to be negligible. An over-

view over the variety of available mass transfer models is given by Guiochon et al. [Guio-

chon1994]. The commonly used, so-called solid film linear driving force model shall be cited

as an example. According to this model, the mass transfer resistance is located in a thin film in

which the stationary phase concentration varies from the equilibrium concentration *iq at the


21

phase boundary to the stationary phase concentration iq . All effects of mass transfer resis-

tance are lumped into the apparent mass transfer coefficient imk , which is assumed to be inde-

pendent of concentration.

( )iiimi qqk

t

q−⋅=

∂

∂ *, (2.28)

2.5.4 Van Deemter equation

Van Deemter et al. [VanDeemter1956] derived a relationship between the axial dispersion

coefficient, the mass transfer coefficient and the height equivalent to a theoretical plate. For

that reason they compared the elution profile predicted by the Martin and Synge plate model

[Martin1941] with an analytical solution of the dispersed plug flow model which was derived

by Lapidus and Amundson [Lapidus1952]. Van Deemter et al. demonstrated that both solu-

tions reduce to a Gaussian profile if the number of theoretical plates is not too low. Since both

analytical solutions, from Martin and Synge as well as from Lapidus and Amundson, were

derived for linear adsorption isotherms, the relationship cannot be applied in the non-linear

region of the isotherm. According to Guiochon et al. [Guiochon1994], Van Deemter’s equa-

tion can be expressed as follows

imi

SF

i

i

SF

mP

SF

SFikk

u

k

k

u

Dd

uC

u

BAHETP

,'

2

'

'1

21

22

2⋅

⋅

+⋅+

⋅⋅+⋅⋅=⋅++=

ε

ε

γγ

εε

(2.29)

In the context of this work it is worth mentioning that the A -term in the Van Deemter equa-

tion which represents eddy diffusion is an indicator of the column homogeneity and density

[Stanley1997] where high values indicate a poor packing. It does not depend on the interstitial

velocity.

The B -term representing the effects of molecular diffusion is dominant at low interstitial ve-

locities where the residence times of the substances to be separated are long. The long resi-

dence times allow diffusion to become the dominating factor for band broadening. Increasing

the mobile phase flow results in a hyperbolic decrease of the consequences of diffusion in

terms of the HETP . At high flow velocities the contribution of diffusion to the total HETP

becomes negligible.


22

Figure 2.10: Van Deemter curve

The C -Term in the Van Deemter equation which accounts for the effects of mass transfer re-

sistance depends linearly on the mobile phase velocity. Finite mass transfer rates in between

the two phases allow molecules contained in the bulk mobile phase to proceed whereas mole-

cules contained in the stationary phase stay back. The finite mass transfer rates prevent them

from being conveyed back into the bulk mobile phase instantaneously. These local differences

in the molecules’ migration velocities cause the band to spread. This effect becomes more

influential at higher velocities, where the velocity differences in between molecules contained

in the stationary and the mobile phase are more significant.

2.5.5 The Equilibrium-Dispersive-Model

It can be seen from the Van Deemter equation that the effects of axial dispersion and mass

transfer resistance are linearly additive [Ruthven1984]. This is the basis for the equilibrium

dispersive model which bases on the assumption that the stationary and the mobile phase are

in a permanent local equilibrium throughout the column. All non-idealities, including band

broadening due to mass transfer resistance are lumped into an apparent axial dispersion coef-

ficient appaxD .


23

2

2*

z

cD

z

cu

t

qF

t

c iappax

iSF

ii

∂

∂⋅=

∂

∂⋅+

∂

∂⋅+

∂

∂

ε (2.30)

The circumstance that the two phases are assumed to be in equilibrium makes a mass balance

for the stationary phase superfluous. The stationary phase concentration can be calculated

from the mobile phase concentration via the adsorption isotherm. This is the reason for the

simplicity of the equilibrium dispersive model. It is worth noting that the apparent axial dis-

persion coefficient is not constant but a function of the interstitial velocity as given by the Van

Deemter equation.

2

HETPuD

SFappax ⋅=

ε (2.31)

2.6 Pressure drop

In order to model the pressure drop LP∆ inside chromatographic columns, Darcy’s law for

the description of fluid flow through porous media is used [Lage1998]. Based on empirical

findings, it correlates the dependency of the pressure drop on the superficial velocity u and

the viscosity η of the percolating fluid.

κ

η⋅−=

∆ SFu

L

P (2.32)

The application of Darcy’s law is restricted to the laminar flow regime. The laminar flow

condition is given as [Nicoud1993]:

( )1

1

6Re

2,1

<<⋅−

⋅⋅

=ηε

ρ

ext

SFd

u

(2.33)

The Reynolds numbers for chromatographic applications are always lower than 0.2 [Baumeis-

ter1995]. This is the range of creeping flow, in which Darcy law is clearly valid.

The permeability κ is a property of the fixed bed and depends on the Sauter mean diameter

2,1d , the external or interstitial porosity extε and the shape of the particles. It can be estimated

from the Kozeny-Kármán equation that founds on the representation of the porous, tortuous

bed as a bundle of non-connected capillaries.


24

( ) ( )

( )2

32

2,1

136

1

ext

ext

K

d

h ε

εκ

−

⋅⋅

⋅= (2.34)

For most bed consisting of particulate materials the Kozeny-coefficient Kh depending on the

particle shape is in the following range [Nicoud1993]:

15.4 ±=Kh (2.35)

2.7 Slurry packing

Generally there exist two methods to pack chromatographic columns [Dingenen1998], dry

packing and slurry packing methods. The latter ones are also referred to as filtration tech-

niques. For the reason that the small particle diameters used in liquid chromatography abet

adhesion and the formation of agglomerates which are adversarial for the column perform-

ance, slurry packing methods prevailed and are nowadays the most commonly used methods

to fill liquid chromatographic columns [Dingenen1994]. While using these methods, the parti-

cles are firstly uniformly dispersed in a solvent thereby preventing the agglomeration of the

particles. For analytical and smaller scale columns the dispersion is subsequently pumped into

the column through the open inlet at pressures up to 1000bar. Analogous to a filtration proc-

ess, the particles are hold back by the outlet frit and gradually form the packing. The high

pressure drop applied and the resulting shear stresses give rise to a dense column packing.

Due to technical difficulties associated with the realization of high pressure for larger scale

columns, e.g. thickness of the column wall or the dimensions and costs of a pump conveying

the necessary flow rates, modified packing technologies were developed for these kind of

columns. The most common one is the so-called axial compression process [Unger2005].

Hereby the slurry is filled into the column through the open end. Afterwards the column is

closed. The needed compression of the packing is attained through a movable piston. The

principle is sketched in figure 2.11.


25

Figure 2.11: Principle of the axial compression technology [Dingenen1998]

Several parameters like the slurry concentration, the choice of the solvents used for dispersing

(slurry solvent) and compressing (pushing solvent) the particles, the packing pressure, etc.

have an influence on the result of either slurry packing method.

2.8 Inhomogeneous column bed structure

Macroscopic maldistributions of the fluid phase caused by non-uniformities in the structure of

the packed bed have an impairing effect on the efficiency of chromatographic separations and

result in a drift in key performance parameters, such as productivity [ONeil2004]. Theoretical

investigations carried out by Yun et al. [Yun1994], [Yun1996] and Lenz [Lenz2003] have

shown that velocity variations in a chromatographic column as caused by inhomogeneous

packing have a similar effect as an increased extent of effective axial dispersion and can lead

to a severe distortion of the detected bands. The profiles of the eluting peaks are significantly

affected and the productivity in preparative applications impaired.

Experimental evidence that the packing structure in chromatographic columns is not necessar-

ily homogeneous was summarized in a review article by Guiochon et al. [Guiochon1997].

According to Heuer et al. [Heuer1996], the neglect of non-uniform packing structures in the

models commonly used for the simulation of chromatographic separations lead to discrepan-


26

cies between the model forecast and the experimental findings while scaling up chroma-

tographic columns.

On the one hand the above mentioned, macroscopic packing inhomogeneities can be intrinsi-

cally tied to the packing technique applied. It is well known that slurry packed columns offer

a rather homogeneous core region surrounded by a denser and less permeable region in the

vicinity of the column wall whereas dry packing techniques lead to an increased permeability

of wall region due to size segregation of the particles during the packing [Guiochon1997]. On

the other hand the inhomogeneities may also be the result of a reorganization of the bed

caused by the formation of unstable regions during the column packing.

2.8.1 Inhomogeneities associated to slurry packing techniques

The presence of a rigid wall in contact with the particulate stationary phase material is the

main source for the of systematic radial variations of the column properties inside slurry

packed chromatographic columns [Guiochon1999]. The column wall supports the bed of

packing material and, in the packing process, induces radial heterogeneity of the bed. Without

the friction between the stationary phase material and the column wall during the packing

process, the bed would be homogeneous. The friction causes the distribution of the stress and

strain to be uneven throughout the bed under formation. As a consequence, the porosity and

the permeability distributions in the readily packed column are not homogeneous and the col-

umn efficiency of the wall region is remarkably lower than that of the core region [Cher-

rak2001], [Farkas1994]. Experimental observations demonstrate the intensity of the friction

between the bed of a packed column and its wall [Guiochon1999].

Experimental evidence that the bed structure of slurry packed chromatographic is inhomoge-

neous is numerous. Baur et al. [Baur1988] and Farkas et al. [Farkas1994,1996,1997a&b]

measured local eluent histories at different radial positions of the column outlet. They found

that the migration velocity is higher close to the column center compared to the column wall

and concluded that the wall region offers a reduced permeability. At the same time the ob-

served bands corresponding to radial outlet positions close to the column wall were more

strongly dispersed and the efficiency close to the column wall consequently lower. Yun et al.

[Yun1997] as well as Brandt et al. [Brandt1996] investigated the band profiles of dye-tracers

as well as the shape of dyed layers of packing material after unpacking the column. The re-

sults were in good agreement with the findings of the local eluent history measurements.


27

2.8.2 Inhomogeneities associated to unstable column regions

A problem associated with the use of preparative scale columns is the stability of the packed

bed [Colin1993]. During the packing process, unstable regions (e.g. particle bridges) develop.

These regions remain unaffected during the packing procedure. Due to the high shear forces

resulting from the mobile phase flow and the lack of wall support in the center of larger di-

ameter columns, the bed eventually reorganizes during the operation thereby giving rise to the

of voids, channels, and zones with deviant packing density [Dingenen1998], [ONeil2004].

The reorganization can occur in form of catastrophic events that take place for no apparent

reason [Guiochon1995] or as a long term degradation of the packing quality. The high me-

chanical stress is also responsible for the destruction of particles during the operation, a prob-

lem that often applies to angular materials which tend to produce fines by the breaking of

sharp corners [Colin1993].

A commonly observed phenomena is the formation of a void at the inlet of the column

[Sarker1995a,b], [Stanley1997], [Kaminski1992] that is accompanied by a drastic loss in col-

umn efficiency. A void volume of 1% of the column length only can cause a loss in column

efficiency as huge as 66% [Guiochon1999].

Stanley et al. [Stanley1997] report the loss of performance for several semi-preparative col-

umns during operation due to a shift in packing structure. Marme et al. [Marme1992] ob-

served packing inhomogeneities with lower packing density containing up to 20% more mo-

bile phase in preparative columns packed with silica gel.

Evidence of particle fragmentation and alteration in the particle size distribution after column

operation was observed by Marme et al. [Marme1992] and by Sarker et al. [Sarker1996].

2.9 CFD-Modeling

The distribution of the fluid flow inside a chromatographic column is strongly determined by

the structure of the stationary phase packing. To take the uneven flow distribution into ac-

count, it must be modeled using equations describing the hydrodynamic behaviour of the col-

umn in two (axial, radial) or even three (axial, radial, angular) spatial dimensions. Due to the

small scale geometrical structure of a chromatographic bed consisting of tiny particles, the

hydrodynamic effects cannot be numerically resolved but are modeled by the use of distrib-

uted momentum resistances like Darcy’s equation (2.32). The velocity is no longer assumed to


28

have a plug like, evenly distributed profile but is a mutable quantity depending on the local

column characteristics in terms of porosity and permeability.

Pµ

κu ∇−=r

(2.36)

Besides the equation of motion, the continuity equation for an incompressible fluid

0=⋅∇ ur

(2.37)

as well as the species equations must be solved numerically.

iaxiii cDc

u

t

qF

t

c 2∇=∇⋅+∂

∂⋅+

∂

∂

ε

r

(2.38)

For this reason the computational effort to integrate the system of partial differential equations

increases significantly compared to the merely one-dimensional models discussed earlier. For

the numerical integration of the system of partial differential equations (PDE) given above,

the use of a commercial computational fluid dynamics (CFD) code comprising different

schemes for the numerical integration of the PDE-system and features for mesh generation as

well as graphical pre- and post-processing is favorable [Boysen2004]. In this context compu-

tational fluid dynamics comprises the analysis of a system involving fluid flow by means of

computer-based simulations [Versteeg1995]. The CFD code StarCD (StarCD V3.20) devel-

oped by CD-Adapco was chosen as a well established commercial code. It provides interfaces

for the extension of the original modeling capabilities by means of user subroutines thereby

allowing for the implementation of ad- and desorption which are not readily available.

A detailed description of all numerical integration schemes implemented in the code would go

far beyond the scope of this work. Because of this, the following section is restricted to a ba-

sic illustration of the most significant schemes.

2.9.1 The finite volume method

Most of the commercial CFD codes, such as StarCD or Fluent, are based on the finite volume

method (FVM). One of the reasons why it has succeeded over other methods is that the

scheme guarantees local and global conservation of mass, species, and momentum [Ver-

steeg1995]. The method shall be illustrated by applying it to the species equation (2.38). For

the sake of clarity all the following derivations are restricted to a spatially one dimensional

case (axial) for a non-adsorbing species ( 0=iq ) inside a cylindrical column with cross sec-


29

tion A . All parameters (ε , axD ) and the superficial velocity u are assumed to be constant

throughout the column.

Figure 2.12: Sketch of the finite volume nomenclature for a 1D- problem

In order to discretize the partial differential equation, the domain is subdivided into a number

of finite control volumes CV over which the partial differential equations is integrated. By

applying the Gauss-Theorem, the volume integrals of the transport terms can be converted

into surface integrals over the closed surface CS of the control volumes [Meyberg1991].

( ) SdcDSdcu

dVt

c

CS

iax

CS

i

CV

irr

r

⋅∇⋅=⋅

⋅+

∂

∂∫∫∫ ε

(2.39)

Within one control volume ( zAV CV ∆⋅= ), all properties including the concentrations are

assumed to be constant and equal to the concentration at the cell centered node P . The con-

centrations on the cell faces are assumed to be constant and equal to the concentration at the

face centered nodes w (west) and e (east). Sr

is the normal vector perpendicular to the cell

surface. Due to the assumptions the scalar product of the normal vector and the velocity vec-

tor/ concentration gradient reduce to zero on all cell faces but the column cross section. Ac-

cordingly it can be written

( ) Az

c

z

cDAcc

uzA

t

cw

i

e

iax

wi

ei

Pi ⋅

∂

∂−

∂

∂⋅=⋅−⋅+∆⋅⋅

∂

∂

ε (2.40)

2.9.2 Discretisation of the dispersive term

Central differencing approximation: A central differencing approximation can be applied to

discretize the dispersion term in equation (2.40). According to the central differencing ap-

proximation, the partial derivative is approximated with a central difference quotient. This

yields


30

( )( )WPE

axWPPEax

we

ax cccx

D

x

cc

x

cc

x

D

x

c

x

c

x

D+−⋅=

−−

−⋅=

∂

∂−

∂

∂⋅ 2

2δδδδδ

(2.41)

2.9.3 Discretisation of the convective term

Central differencing scheme: The integration of the convective term yielded a result which

still contains concentrations at the face centered nodes e and w . In order to derive a set of

equations which can be solved numerically, these quantities must be replaced by concentra-

tions at the cell centered nodes W , P , and E . In analogy to the central differencing ap-

proximation used for the dispersive term, linear interpolation can be used to estimate the con-

centrations on the cell surface. As the central differencing approximation this so-called central

differencing scheme is of second order accuracy. For the reason that a linear interpolation

cannot account for the direction of convective transport (upstream and downstream node are

equally weighted), the scheme is only partially suited for the discretisation of the convective

term.

Upwind differencing scheme: According to the upwind differencing scheme the concentra-

tions ec as well as wc are assumed to be equal to concentration of the upstream (or upwind)

node in reversed flow direction thereby taking the flow direction into account. The numerical

accuracy of the upwind scheme is of the first order only.

( ) ( )εε

u

z

ccu

z

cc PEwe ⋅∆

−=⋅

∆

− (2.42)

MARS scheme: Frequently high order schemes offer a good accuracy but lack numerical sta-

bility. MARS is a robust second order scheme recommended for StarCD calculations

[CDAdapco2005] operating in two separate steps [CDAdapco2004]. Due to company pub-

lishing politics the details of the MARS differencing scheme are not open to the public do-

main. The MARS scheme was used for all simulations unless marked otherwise.

2.9.4 Temporal discretisation

In order to solve transient problems by means of CFD calculations, the relevant ordinary dif-

ferential equations resulting from the spatial discretisation described above must be integrated

over time. In a simplified form it can be written:


31

( ) ( )[ ] ( )∫∫++

=∆⋅−∆+=∂

∂tt

t

Pi

Pi

tt

t

Pi dttfttcttcdtt

cδδ

(2.43)

( )tf comprises the discretized transport terms. The time integration of ( )tf is not straight

forward as it is an unknown function of time (e.g. the dependency of the concentrations on

time is not known). StarCD offers two different time integration schemes.

Implicit Scheme: ( )tf is considered to be fixed at an unknown value at the final time tt δ+

of the time step. The implicit scheme is of first order accuracy but less restrictive in terms of

the time step size than the explicit scheme

Crank-Nicholson scheme: ( )tf is considered to vary linearly with time in between t and

tt δ+ . The Crank-Nicholson scheme is of second order accuracy.

The Implicit and the Crank-Nicholson scheme may also be combined (blended schemes).

2.9.5 SIMPLE algorithm

In order to determine the velocity as well as the pressure field inside a chromatographic col-

umn by means of CFD, the equation of continuity as well as the momentum equation em-

ployed must be solved simultaneously. This can be achieved by use of the SIMPLE algorithm.

SIMPLE stands for Semi-Implicit Method for Pressure-Linked Equations. The algorithm was

originally put forward by Patankar and Spalding [Versteeg95] and is essentially a guess pro-

cedure for the calculation of the pressure on the staggered grid. After an initial guess, the al-

gorithm performs three steps which are i) solving discretized momentum equations, ii) solve

pressure correction equation, iii) correct pressure and velocities.

The last step provides for an improved guess. It has to be mentioned that the SIMPLE algo-

rithm is suited for stationary problems only. In the following, the velocity field as well as the

pressure field were assumed to be independent of time and calculated in a stationary run. The

transient calculations for the concentration field were carried out based on the results for pres-

sure and velocity fields from the stationary run as suggested by Boysen [Boysen2004]. Be-

cause of this, the SIMPLE algorithm was well suited.

2.9.6 Dimensionless numbers

From a numerical point of view, the problem can be described by the use of two dimen-

sionless numbers. The first of which is the so called Courant number Co [Guiochon2003]


32

that provides information how many cells of mesh width z∆ are passed per time step t∆ . For

retained components, the concentration velocity given by equation (2.24) and thereby the

Courant number depends on the slope of the isotherm [Boysen2004].

( ) z

t

dcdqF

uCo

∆

∆⋅

⋅+=

1

ε (2.44)

The second number is equivalent to the commonly used Diffusion number and will be called

Dispersion number Di in the following.

( ) ( )21 z

t

dcdqF

DDi ax

∆

∆⋅

⋅+= (2.45)

It is worth noticing that the ratio of the Courant number and the Dispersion number yields the

cell Peclet number CellPe .

ax

CellD

zuPe

∆⋅=

ε (2.46)


33

3 Basics of tomographic measurement techniques

In this chapter the reader shall be familiarized with the basics of x-ray computed tomography

(CT) experiments for porous media and the determination of velocity profiles by means of

nuclear magnetic resonance techniques.

Firstly, a brief motivation is given why the application of non-invasive measurement tech-

niques in conjunction with chromatographic columns seems promising.

Subsequently, the general principles of computed tomography are explained. The section

about computed tomography is concluded by showing how CT-experiments can yield relevant

information about porous media.

Finally it is illustrated how nuclei with a magnetic moment behave inside an external mag-

netic field in the presence and the absence of electromagnetic irradiation. The chapter is con-

cluded by demonstrating how this behaviour may be used to measure velocity profiles in a

non-invasive way by the use of Nuclear Magnetic Resonance techniques.

3.1 Motivation for non-invasive measurements

Experimental investigations in the field of chemical engineering frequently deal with the de-

termination of quantities such as velocity or concentrations depending on the local position

inside the measurement domain as well as on time. This gives rise to the demand for measur-

ing these quantities simultaneously at all spatial positions inside the measurement domain to

obtain a representative profile of the dependent variables at each point of time. Even with

microscopic sensors potentially not unfavorably influencing the investigated profile, it is pos-

sible to monitor at single locations only thereby not meeting the above mentioned requirement

of a spatially resolved result [Mewes1991]. In the case of chromatography, the vast majority

of studies on the column heterogeneity carried out so far has been made by measuring the

radial distribution of the column properties at the column exit [Koh1998].

Within the last years this demand was enforced by the advancement of CFD methods allowing

for a spatially resolved numerical analysis of the problem at hand and the lack of experimental

methods providing data well suited for the validation and further development of these mod-

els [Nietzsche2002].

X-ray computed tomography (CT) techniques and magnetic resonance tomography (MRT)

also referred to as nuclear magnetic resonance imaging (NMRI) have been used in medicine


34

as diagnostic tools for several decades since the first developments by Hounsfield (CT) as

well as Lauterbur and Mansfield (NMRI) in the 1970s. The techniques allow to acquire two or

even three-dimensional images of the internals of an object from a large series of x-ray pro-

jections. In recent years the methods have been increasingly used for engineering applica-

tions.

The following discussion of the principles is restricted to the fundamentals needed for the

understanding of the investigation of chromatographic columns by means of computed tomo-

graphy and nuclear magnetic resonance imaging, respectively. A more comprehensive discus-

sion of computed tomography [Kalender2005], [McCullough1977] or nuclear magnetic reso-

nance techniques [Callaghan1991] can be found elsewhere.

3.2 Computed tomography for packed beds and porous media

3.2.1 Basics of Computed Tomography

Computed tomography is based on the ability of x-rays to pass through almost all matter al-

beit with some attenuation. The degree of attenuation depends on the local linear attenuation

coefficients as well as on the length of the object and is given by Lambert-Beer’s law. The

local linear attenuation coefficient µ is a property of the particular material. It should be

mentioned that equation (3.1) is a simplification based on the supposition that the x-ray beams

are monochromatic whereas most beams are polychromatic [Mogensen2001].

∫ ⋅−=

ObjS

dxI

I

00

ln µ (3.1)

The intensity of the attenuated beams passing through the object is monitored to obtain a pro-

jection of the object (s. figure 3.1).


35

Figure 3.1: Principle of CT-Scanning

From the projection data an image file that consists of an array of CT numbers covering a

specified range and representing the relative radiodensity of the object can be reconstructed.

For this purpose mathematical algorithms which have implemented the discovery by Radon

that the distribution of a material property in an object layer can be determined from the inte-

gral values (projections) along an infinite number of lines passing through this layer are used

[Kalender2005]. Commonly the CT-number is defined as a dimensionless (in terms of SI-

units) attenuation coefficient relative to pure water.

10002

2 ⋅−

=OH

OHCT

µ

µµ (3.2)

3.2.2 Computed Tomography in conjunction with porous media

For a porous object like the packing in a chromatographic column, the CT-number is given as

the weighted mean of the CT-Number of the two phases. In particular the weighted mean of

the CT-numbers of the mobile Phase MP and the stationary phase SP [Kantzas1994].

( )[ ]SPMP CTCTCT ⋅−+⋅= εε 1 (3.3)

During a breakthrough experiment the column is initially saturated with a mobile phase of

known composition ( 1MP ). Subsequently the initial mobile phase is displaced by a second

mobile phase ( 2MP ) having a distinct composition. During the duration of the breakthrough

the CT-Number depends on the mobile phase saturation S and is given by [Peters1990]:


36

( ) [ ] ( )[ ]SPMPMPMPMP CTCTSCTStCT ⋅−+⋅+⋅⋅= εε 12211 (3.4)

The mobile phase saturation S is given by the volume fraction of the corresponding mobile

phase. Consequently the local saturations S during a breakthrough can be determined from

three CT images representing either saturated state and the breakthrough. Furthermore, it must

be provided that the CT-numbers of the pure mobile phases 1MP and 2MP are known and

that their contrast in the attenuation coefficient is sufficiently large.

3.3 Nuclear magnetic resonance velocimetry

3.3.1 Basics of nuclear magnetic resonance

The physical basis of magnetic resonance phenomena is the spinning movement of the nuclear

particles that gives rise to a magnetic moment µr

in case that the number of protons and/or

neutrons in the nucleus is odd. In the following the nucleus of light hydrogen H1 shown in

figure 3.2 that served as the resonance nucleus in our experiments is quoted for the sake of

illustration.

If the proton H1 is exposed to an external magnetic field 0Br

, its magnetic moment µr

will

align with the external magnetic field by precessing around the field lines (s. figure 3.2). The

precession frequency that is called Larmour frequency 0ω is a function solely of the field

strength 0Br

and the magnetogyric ratio γ being a proportionality constant typical for the nu-

cleus of interest.

00 B⋅= γω (3.5)


37

Figure 3.2: Precession of the magnetic moment in an external magnetic field

The net vector of the precessing moment can align either parallel or anti-parallel to the field

lines of the external magnetic field. These magnetic moments no longer have the same energy.

The vector parallel to the field is lower in terms of energy while the vector opposing the ex-

ternal field has a higher energy. The energy difference in between the two states depends on

the Larmour frequency 0ω and is given by

π

ω

2

0⋅=∆

hE (3.6)

In the absence of irradiation, the ratio of nuclei in the low energy state and the high energy

state is given by a Boltzmann distribution,

∆−=

kT

E

N

N

low

highexp (3.7)

all magnetic moments precess with the Larmour frequency, and the orientation of the mag-

netic moments is degenerate or non-coherent, respectively.


38

In order to disturb the equilibrium characterized by the Boltzmann distribution and to bring

additional nuclei from the low energy state to the high energy state, they must be excited by

electromagnetic irradiation of energy E∆ . But exposing the nuclei to irradiation of the right

frequency will not only increase the total energy by “flipping” nuclei into the high energy

state. It will also generate a circling magnetic moment in the plane perpendicular to the exter-

nal field by forcing the magnetic moments to become phase coherent.

Figure 3.3: Precession of a collection of nuclei around an external magnetic field 0B .

M represents the vector sum of all nuclear magnetic moments: a) Before irradiation; b) Orientation of the rotating magnetic field in the xy-plane (irradiation with Larmour fre-quency); c) During irradiation - coherent nuclear magnetic moments give rise to a circling

magnetic moment in the xy-plane (based on [Macomber1997]).

As soon as the irradiation ends, the distribution of the nuclei among the energy states will

exponentially decay towards the equilibrium state, a nuclei relaxing into the low energy state

emits a photon of energy E∆ . The characteristic time of the decay is commonly called 1T .

Furthermore, the magnetic moments will eventually dephase. The decay of phase coherence is

also given by an exponential function. The characteristic time is called 2T .

The opportunity to excite nuclei into the high energy state and to monitor the relaxation proc-

ess or the circling magnetic moment in the plane perpendicular the external field, respectively,

is the basis for nuclear magnetic resonance techniques.

3.3.2 Phase encoded velocity measurements

In order to determine displacement velocities of the spins by means of phase sensitive nuclear

magnetic resonance measurements, additional transient gradients gr

are superimposed on the


39

external magnetic field 0Br

. Consequently the field strength is a function of the spatial coordi-

nate z in the direction of the field lines during the time that the gradient is impressed.

( ) zgBzB ⋅+=rvr

0 (3.8)

Furthermore, the Larmour frequency depends on the z -coordinate:

( ) zgz ⋅⋅+= γωω 0 (3.9)

If a system exposed to a homogenous magnetic field is firstly excited by means of electro-

magnetic irradiation, all nuclear magnetic moments will precess in phase coherence at the

same Larmour frequency 0ω independent from their location.

Figure 3.4: Spin labeling [Jakob2001]

Secondly, the system is exposed to a magnetic field gradient gv

for a very short period of time

δ (it should be small enough to allow the neglect of nuclei displacement during δ ). During

this time, the Larmour frequency is a function of the coordinate z (s. equation 3.9). Nuclei

exposed to a higher field strength will precess faster. After time δ , the gradient is switched

off for a time δ>>∆ . In this time, the nuclei will displace, e.g. in the presence of flow. For

the reason that no gradient is enforced, all nuclei precess with the same frequency 0ω albeit

they have a phase shift depending on their original location z .


40

After time ∆ , the system is exposed to a reversed gradient gr

− for time δ . For stationary

spins, the reversed gradient will result in the nullification of the phase shift as shown in fig-

ure 3.4 . By contrast, spins that were displaced during time ∆ are not exposed to the same

magnitude of the magnetic field so that a phase shift ∆Φ is conserved. The monitored signal

yields this phase shift which is proportional to the length of displacement in the direction of

the gradient and allows the calculation of the displacement velocity.

zg ∆⋅⋅=∆Φ γ (3.9)

An illustration of such an NMR-sequence, commonly referred to as Pulsed-Gradient-Spin-

Echo-sequences (PGSE) [Stejskal1965], is given in figure 3.5.

Figure 3.5: Pulsed Gradient Spin Echo sequence. The combination of a °y180 -excitation and

a positive gradient gr

+ is equivalent to a negative gradient gr

− . TE denotes the echo-time

(time from the center of the 90°-pulse to the center of the echo); HF denotes high frequency irradiation [Jakob2001]

It should be mentioned that phase encoded velocity measurements are sensitive within a cer-

tain range of velocities only. The maximum phase shift of 180° is associated to a maximum

velocity VENC (velocity encoding) while the minimal phase shift of -180° corresponds to the

smallest velocity, -VENC, that can be measured. Higher (or smaller) velocities associated


41

with phase shifts exceeding ±180° are reprojected to smaller angles and identified as veloci-

ties with a smaller magnitude (aliasing effect).


42

4 Experimental

This chapter shall acquaint the reader with the experimental approaches used to investigate

the consequences of packing inhomogeneity.

In the first sections of this chapter, the packing and measurement techniques that were used to

investigate the consequences of artificially created packing inhomogeneities in terms of peak

shape, column efficiency, and pressure drop are explained. Furthermore, the apparatuses for

this kind of batch chromatographic experiments are introduced.

The last sections are about the series of measurements that were carried out with the non-

invasive measurement techniques introduced in the previous chapter. Computed tomography

was used to monitor the breakthrough of tracer fronts in situ whereas nuclear magnetic reso-

nance imaging rendered the determination of intra-column velocity profiles possible.

4.1 Experimental setup for the experiments with artificially created inhomogeneities

4.1.1 Preparative chromatography system

A Merck-Prepbar2 preparative liquid chromatography system was used for the experiments

with artificially created inhomogeneities in the column packing (s. figure 4.1).

Figure 4.1: Set-up of the preparative chromatography system

The system comprises two 30l solvent reservoirs for mobile phase storage. The reservoirs can

be connected individually or jointly to the pump by two valves. A membrane pump (pm


43

400ex, Merck, Germany) with pulsation suppression conveys the mobile phases towards the

column. A filter is implemented into the piping in order to hold back solid mobile phase impu-

rities. The mobile phase can be thermostated via a heat exchanger. A six-port valve (Valco

Instruments) equipped with a 5ml sample loop allows for sample injection. A comparatively

small sample loop volume was chosen for the reason that the purpose of the experiments was

column testing rather than the production of pure compounds [Laiblin2002]. Behind the six-

port valve the mobile phase stream flows into the column. Two pressure gauges (4AD-30,

Jumo, Germany) located in front and behind the chromatographic column afford to monitor

the pressure drop over the column length. A UV-detector (UV-Detektor 64, Knauer, Germany)

connected in series measures the tracer concentration at the outlet of the column. The detector

is equipped with a preparative flow cell and was calibrated regularly [Schwarz2004], [Car-

rerasMolina2005]. Finally an assembly of fraction valves allow to recycle a pure mobile

phase stream to the solvent reservoirs or to cut out the tracer peaks as waste fractions in order

to prevent mobile phase contamination. The fraction valves are also diverted from its intended

use to determine the mobile phase flow rate by volume-time-measurements [Schwarz2004],

[CarrerasMolina2005]. Resistance thermometers (Pt100, Conatex, Germany) implemented

into the tubing enable the measurement of the mobile phase temperature at the column inlet

and the column outlet as well as the temperature of the heat transfer medium at either end of

the heat exchanger. All sensor signals (UV-detector, pressure gauges, thermometers) are

transmitted to a computer by a data acquisition system with an acquisition frequency of 1Hz.

The software FlexPro-Control is employed for data processing. A detailed description of the

data acquisition system is given by Schaarschuh [Schaarschuh2000] and Laiblin [Laib-

lin2002].

The system was optimized in terms of extra column dead volume [Schwarz2005], [Buch-

ele2005] in order minimize extra column contributions to the overall band broadening. The

dead volume of the system as well as its contribution to the overall band broadening were

determined with the column removed from the system [Buchele2005].

4.1.2 Columns and stationary phase materials

Two different axial compression columns were used for the experiments with artificially cre-

ated inhomogeneities.

The first column which was integrated into the preparative chromatographic system for the

experiments with local inhomogeneities and inlet voids [CarrerasMolina2005] was a self


44

packing device with an inner diameter of ID = 100mm (self packing device id. 100, Merck,

Germany) made of stainless steel. The column was packed with non-porous glass beads (Mi-

cro-Glaskugeln Typ S, SiLi, Germany) with a particle size distribution in between 40 and

70µm.

The second type of column employed for the investigation of the influence of fines

[Schwarz2005] was an axial compression column made of glass with an inner diameter of

ID = 50mm (SuperCompact 50, Götec Labortechnik, Germany). The column was packed with

two irregular, porous silica gel phases (C-Gel 560 40-63µm & 15-35µm, Uetikon, Switzer-

land) with a particle size distribution from 40 to 63µm and 15 to 35µm, respectively. The

phases were also mixed prior to packing in order to mimic the results of particle breakage.

Irregular, angular materials produced by a milling process were chosen because of the ten-

dency of these materials to produce fines [Colin1993].

Figure 4.2: Scanning Electron Microscope (SEM) images of the irregular silica-gel phase [Laiblin2002]

Each column was packed according to the instructions of the column manufacturer

[Merck1996], [Götec1998].

4.1.3 Mobile phases and tracer component

Methanol (MeOH) and isopropanol (IPA) were used as mobile phases for the experiments on

the preparative chromatography system. Each solvent was filtered before filling it into the

solvent reservoirs of the apparatus. Furthermore IPA was used as the slurry and pushing sol-

vent for packing the columns [CarrerasMolina2005], [Schwarz2005].


45

Diethyl phthalate is a standard compound for the testing of chromatographic columns

[Unger2005]. It was chosen as the tracer component for the experiments due to its liquid state

at ambient temperature, the good miscibility with alcohols and the aromatic ring which en-

ables the UV-detection of the substance.

Figure 4.3: Molecular structure of diethyl phthalate

4.1.4 Local inhomogeneities

In order to imitate local column inhomogeneities with increased permeability, hollow spheres

consisting of ceramic foams [Meyer2003], [Garrn2004] were integrated into the column bed

during the packing process. These spheres were developed in conjunction with the Institute of

Materialscience and Technologies - Department Ceramic of the Technische Universität Berlin.

Figure 4.4: Hollow spheres made of ceramic foam

The foams are produced from a ceramic suspension of water and a dispersing agent. A protein

(Bovine Serum Albumin) is used as a binder in the suspension which is mixed in a planetary

ball mill. The resulting milky-white, viscous mash is transferred into a form made of Teflon

and warmed to 200°C inside a microwave. The warming leads to an evaporation of water

which expands the pore volume as well as to a stabilization of the porous structure due to the

solidification of the proteins. Afterwards the green compact is made by exposing the structure


46

to 600°C thereby decomposing the proteins. The final sintering process is carried out at a

temperature of 1200°C. The resulting foams are highly porous and have an interconnected

pore structure enabling percolation.

Figure 4.5: SEM- images of the ceramic foams pore structure

The foams fulfill the essential conditions such that on the one hand side the diameter of the

interconnections in between the pores (not the pore diameter) are predominantly smaller than

the glass beads used for the chromatographic experiments (s. figure 4.5) which have a size

distribution in between 40-70µm. This ensures that the stationary phase particles are effec-

tively hindered from penetrating the interior of the hollow region. On the other hand side the

permeability of the foams is significantly higher than the permeability of a packed bed con-

sisting of the glass beads as can be seen by comparing the respective data in table 4.1 and 4.2,

respectively. This ensures that the spheres are well suited to imitate packing regions with in-

creased permeability. The permeability of the foams was determined by measuring the mass

flow of water induced by water columns of defined height [Schwarz2005].


47

Table 4.1: Properties of the hollow, porous spheres

Mean value Standard deviation

Diameter [m] 0,0262 0,00048

Shell thickness [m] 0,00383 0,00007

Permeability κ [10-12 m²] 38 3,7

The size of the sphere was chosen to be comparatively small compared to the column volume

(~0.5%) in order to ensure the integrity of the shell during the operation of the column as well

as to keep the side effects on the packing process small.

4.2 Experimental results

4.2.1 Reference measurements

In order to have reference values for the measurements with artificial local inhomogeneities

and inlet voids, several unaltered reference columns with glass beads were packed with the

packing device. The columns were operated with different flow rates in order to measure

Van Deemter curves and permeabilities [CarrerasMolina2005]. As to be expected from equa-

tion (2.29), the Van Deemter curves for the non-porous, inert (no C-Term in equation (2.29))

glass beads offered a rather constant HETP as a function of interstitial velocities high enough

to neglect the effects of molecular diffusion (no B-term in equation (2.29)) [Carreras-

Molina2005]. An example of the chromatograms for the reference columns is given together

with an elution history of an inhomogeneous column in figure 4.6.

4.2.2 Local inhomogeneities

Figure 4.6 shows a typical chromatogram for the columns packed with artificial inhomogenei-

ties in contrast to the elution history of one of the reference columns. The run of the two

chromatograms is decidedly different. Whereas the regularly packed columns exhibit almost

symmetrical peaks, the artificial inhomogeneities give rise to the occurrence of shoulders on

the front of the eluting peaks. This kind of pattern was found to be typical for these kinds of

columns [CarrerasMolina2005].

The peak shape is due to the high permeability of the hollow spheres. Within the spherical

region of increased permeability, the fluid velocity is higher and the tracer band precedes

thereby giving rise to the of the shoulder on the front. Nevertheless, the shoulders do not


48

have a significantly impairing effect on the efficiency of the packing if calculated with the

method of moments as can be seen in table 4.2. The use of any of the short-cut methods for

the determination of the efficiency would be meaningless for the reason that the peak shape is

distorted other than a mere tailing.

The reason that the efficiency is unaffected within the limits of the experimental accuracy is

explained by the comparatively small part of the column volume (~ 0.5%) covered by the

hollow spheres.

Figure 4.6: Comparison of peak-shapes for homogenous and inhomogeneous columns. Differ-ent retention times due to distinct flow rates.

The small part of the column volume occupied by the hollow spheres made from ceramic

foam is also the explanation for the non-significant influence on the permeability of the

sphere. The pressure drop over the column length is mainly determined by the column volume

exhibiting the same hydrodynamic properties like the reference columns. Consequently either

type of column offers permeabilities with the same order of magnitude and the discrepancies

were found to be below the column to column reproducibility in terms of permeability.


49

Table 4.2: Comparison of chromatographic parameters for regularly packed columns and columns with spherical inhomogeneities

Columns with

spherical inhomoge-

neities

Reference columns Column to column

reproducibility

HETP [m] 0,0008 0,0009 ±0,00018

Permeability κ

[10-12 m²] 2,5 2,5 ±0,21

4.2.3 Inlet void

Columns with an inlet void in between the inlet frit and the chromatographic bed were packed

by slightly lifting the piston of the self packing device after completing the compression proc-

ess. During this operation the column was connected to the mobile phase reservoirs to assure

that the void was filled with mobile phase while lifting the piston.

Figure 4.7: Influence of an inlet void on the peak shape of the eluting band. Column length ≈ 0,335m, Flowrate ≈ 5,5ml/s


50

The experiments are well suited to mimic the influence of an inlet void formed by the settling

of the packing during operation on the HETP and the general shape of the detected peak. The

experiments cannot provide information about the effect of void formation on the pressure

drop or the permeability for the reason that the packing does not settle and the interstitial void

fraction of the bed remains unaltered.

Figure 4.7 compares the chromatograms of the columns packed to have an inlet void with a

chromatogram monitored on a homogenous reference column. It is evident that the formation

of an inlet void is reflected by a characteristic, severe broadening of the peaks accompanied

by a strong tailing effect. A similar effect of void formation on the appearance of the elution

history was observed by Lenz who ascribed the significant band broadening caused by the

inlet void to a strong back mixing inside the void region. The void region behaves similar to a

stirred vessel [Lenz2003].

In table 4.3 it can be seen that the formation of an inlet void due to onward settling of the

chromatographic bed during the operation results in a dramatic deterioration of the separation

performance. The efficiency of the columns dropped about 85% because of an inlet void

comprising approximately 3% of the column length. The results are supported by findings

made by Colin [Colin1993] who reports a loss in efficiency about 80% due to the formation

of a void region next to the column inlet during the continuous operation of an axial compres-

sion column filled with 10µm angular particles.

Table 4.3: Influence of an inlet void on the efficiency of a preparative chromatographic col-umn. Column length ≈ 0,335m, Flowrate ≈ 5,5ml/s

Length of inlet void [m] 0,00

(Reference) 0,01 0,02 0,04

HETP [m] 0,0009 0,0058 0,0050 0,0088

Column to column reproducibility [m] ±0,00018 ±0,0014

4.2.4 Creation of fines

Table 4.4 summarizes the characteristics of the columns packed with different stationary

phase compositions in terms of particle size distributions for mimicking abrasion. A maxi-

mum amount of 20% of the finer stationary phase material was not exceeded during the ex-

periments [Schwarz2005] for the reason that the occurrence of higher amounts of fines during


51

the normal operation of a column is unlikely due to the strong decrease in permeability asso-

ciated with the abrasion of the stationary phase particles.

As to be expected (s. section 2.6), the pressure drop and the permeability of the columns, re-

spectively, turned out to be good discriminators for the amount of fines. A fraction of 20%

finer material already causes a permeability decrease of almost 50%. This is in good agree-

ment with earlier experimental findings made by Dewaele et al. [Dewaele1983], who ob-

served a strong increase in the column back pressure with an increasing amount of smaller

particles.

On the contrary, the column efficiency in terms of the HETP is ill-suited to serve as a good

discriminator in order to recognize the formation of a wider particle size distribution due to

particle breakage inside the column. The HETP is rather constant within the limits of the

experimental accuracy for different shares of the finer stationary phase material. Only col-

umns packed with pure C-Gel 560 15-35µm material offered a significantly lower HETP .

This is in good agreement with conventional chromatographic wisdom that the efficiency of a

column is mainly determined by the size of the larger particles [Halasz1971], [Done1972].

Table 4.4: Comparison of chromatographic parameters for columns with different particle size distributions

Weight percentage of the packing

C-Gel 560 40-63µm C-Gel 560 15-35µm HETP [m]

Permeability κ

[10-12 m²]

100 0 0,0022 3,4

90 10 0,0020 2,8

80 20 0,0025 1,8

0 100 0,0010 0,5

Column to column reproducibility 0,00034 0,22

4.3 Characterization of packing homogeneities by means of computed tomography

4.3.1 Experimental set-up

Methanol (MP1) and solutions of potassium iodide (KI) in methanol (MP2) were used as mo-

bile phases for the computed tomography experiments that were carried out in conjunction


52

with the Center of Phase Equilibria and Separation Processes – Department of Chemical En-

gineering of the Technical University of Denmark [Astrath2004], [Astrath2006]. The concen-

trations of the solutions were 20g/l (abbreviation: 2%) and 40g/l (4%), respectively. Potas-

sium iodide was chosen as the tracer compound for the reason that the high atomic mass of

iodine allows for a good contrast in the CT images.

Figure 4.8: Experimental setup

The mobile phases were conveyed by a high pressure gradient chromatography system with

two pumps. The setup of the high pressure gradient chromatography system is sketched in

figure 4.8.

A double piston HPLC pump (K-120, Knauer, Germany) delivered methanol/ potassium io-

dide solutions (MP2) at a flow rate of 10ml/min. Another double headed HPLC pump with

pulsation suppression (K-1800, Knauer, Germany) conveyed pure methanol (MP1) for re-

flushing the columns at higher flow rates. A six port valve (Valco) allowed for the mobile

phase transition. The valve was placed shortly before the column inlet in order to reduce extra

column band broadening to a minimum. The column, exposed to ambient temperature, was

mounted to the patient table of the CT scanner which allowed for moving the column along its

axis during the experiments (s. figure 4.9).


53

Figure 4.9: Assembly of the chromatographic columns inside the gantry of the CT-Scanner

During each of the frontal analysis experiments, the breakthrough behaviour of the progress-

ing front was monitored in four column cross sections at reduced axial positions Lzz =* of

05.0*1 =z , 35.0*

2 =z , 65.0*3 =z , and 95.0*

4 =z , respectively. Scans of each cross section were

taken until the front had passed before moving the column to the next monitoring position.

The maximum acquisition frequency was approximately Hz2.0 . After each breakthrough

experiment, the column was equilibrated for at least two column volumes before the images

of the saturated state were taken at all monitoring positions. A flow meter was located behind

the column to observe the flow stability of the pumps. The volumetric flow rates where found

to be accurate within 2% of the indicated value for all experiments.

Static axial compression columns made of glass (Goetec Labortechnik, Germany) with inner

diameters of ID = 26mm and ID = 50mm and column lengths of 240mm and 350mm were

investigated experimentally. Steel columns could not be used for the reason that steel causes

artifacts in the CT images. The distribution system consisted of a combination of a sintered

glass plate and a tissue filter for holding back the stationary phase material. The columns were

packed with a polydisperse, hydrophobic octadecyl stationary phase (ODS) based on a silica

gel matrix (C-Gel 560 C18 40-63µm, Zeochem AG, Switzerland). The particle size distribu-

tion and the Sauter mean diameter (SMD) µmd 3.532,1 = of the stationary phase material

were determined by photoelectric sedimentation analysis, the effective internal porosity for


54

methanol 265.0int =ε was measured using a titration technique [Mottlau1962], [Lottes2005]

and the density was determined gravimetrically 341.1 cmg=ρ [Lottes2005].

A non-polar, reversed phase system was chosen to reduce interactions between the ionic tracer

and the stationary phase to a minimum. Both columns were slurry packed with isopropanol

for the slurry as well as the pushing solvent [Lottes2005]. The isopropanol was filtrated and

degassed before use. The compression pressure during the packing operation was 10bar. This

was close to the maximum operation pressure of the large diameter column. The packing was

allowed to settle under the maximum packing pressure for at least one hour.

A Siemens Somatom plus 4th Generation CT-Scanner was used for all experiments. The set-

tings for data acquisition of the CT scanner were as follows: The slice thickness was 2mm, the

scan time was 2s, the energy parameters were 137kW and 255mA.

4.3.2 Calibration of the CT-scanner

The dependencies of the CT numbers on the mobile phases concentration of potassium iodide

were determined for the packed chromatographic columns. In either column they exhibited a

linear relationship against the tracer salt concentration allowing for the use of equation (3.4)

for the interpretation of the CT images.


55

Figure 4.10: Dependency of the CT-number on the mobile phase composition inside the packed chromatographic columns

4.3.3 Band profiles and intra column breakthrough curves

Representative images of the intra column “breakthrough” behaviour are shown in figure 4.11

where white regions correspond to the breakthrough front. It can be observed that the intra

column concentration profile is not evenly distributed over the column cross. The concentra-

tion step is firstly visible in the middle of the cross section. At the margins the concentration

lags behind. Similar band shapes were found e.g. by Brandt et al. [Brandt1996]. The warped,

parabolic shape of the band is partly due to a non-optimal design of the inlet distribution sys-

tem of the glass columns. Transporting the tracer towards the wall region of the glass columns

takes additional time. Thus the concentration front close to the wall falls behind during the

distribution process. This contributes to the concave shape of the migrating band. It shall be

emphasized that the molecules in the center of the column elute in the front part of the band.

Those lagging behind, eluting from the larger region close to the column wall, form the tail

and present a major amount of the substance. Consequently, the band exhibits tailing if re-

corded with a bulk detector at the column outlet.


56

Figure 4.11: Successive breakthrough of a KI/MeOH-Solution replacing MeOH in the ID 50

column at 35.0* =z . White regions refer to KI/MeOH-Solution; dark refers to pure MeOH.

Equation 3.4 allows to determine intra column breakthrough curves of the salt solutions which

provide information about the efficiency of the chromatographic process in different zones of

the chromatographic column. For this purpose, the CT images were firstly processed using the

public image processing program ImageJ. The image sections showing the packed cross sec-

tion of the column were extracted as an array consisting of x- and y-coordinates as well as

CT-numbers. MATLAB was used for further data processing and the calculation of the satura-

tions [Lottes2005].

Figure 4.12 illustrates the band behaviour inside the columns. As expected, the tracer bands

were continuously spreading while progressing through the columns giving rise to a mono-

tonic increase of the variance 2σ of the bands. The parameters of the equilibrium dispersive

model (EDM) given by equation (2.30) were fitted to the experimental data in order to quan-

tify the results. For non-adsorbing components, an analytical solution of the EDM exists. The

analytical solution is a function of two parameters, the Peclet number zPe and the dimen-

sionless time τ , only [Guiochon2003].

appax

zD

zuPe

⋅= (4.1)


57

Rzt

t=τ (4.2)

The analytical solution is given by:

( ) ( ) ( )

+⋅⋅+

−⋅+= ττττ 1

2exp5.01

25.05.0 Z

zZ Pe

erfcPePe

erfS (4.3)

Representatives of the matched curves are shown together with the experimental data in figure

4.12. The parameters of the EDM are given in Appendix I.

Figure 4.12: Experimental and simulated (EDM) intra column saturation fronts for the ID 50 column at different axial positions. KI/MeOH-Solution replaces pure MeOH.

The fitted curves represent the experimental curves essentially well with the exception of the

rear of the bands or the upper part of the lines in Figure 4.12, respectively. In these regions

deviations from the experimental curves were found at some axial positions. This indicates

radial non-uniformities in the column packing [Guiochon1994].


58

4.3.4 Radial homogeneity of the columns

In order to investigate the homogeneity and the radial dependency of the column properties in

a more detailed manner, each of the three column segments enframed by the monitoring posi-

tions was equidistantly (with respect to the columns radius) subdivided into ten annuli. The

EDM was then fitted to the breakthrough curves corresponding to each of the annulus seg-

ments. An illustration of the results for core and wall region is given in figure 4.13. It is evi-

dent that the bands travel faster in the central region than in the vicinity of the column wall.

Accordingly the mobile phase preferentially percolates through the core region of the column

where the permeability is higher. The increased permeability which gives rise to higher linear

velocities is reflected in shorter residence times in the interior of the chromatographic col-

umns. The observation that the packing in the core and the vicinity column of the column ex-

hibit different properties is commonly referred to as the column wall effect which has its cau-

sation in the friction of the packing against the column wall (s. section 2.8).

Figure 4.13: Comparison of the saturation fronts in the core and the wall region of the col-umn at different axial positions of the ID 50 column. KI/MeOH-Solution replaces pure MeOH


59

Moreover, the concentration profiles close to the column wall have much stronger spread than

the corresponding profiles in the inner annulus. The observation that the wall region possesses

a heightened influence of dispersion phenomena and a stronger spreading of the tracer bands

is in good agreement with earlier findings, e.g. by Östergren et al. [Östergren2000].

The dependency of the local column properties was investigated more closely by making use

of the additivity of the retention time and the variance of the tracer bands. The additivity of

the first moment was used to determine linear tracer velocities KIu as a function of the col-

umn radius for the single column sections.

( )( ) ( )

3..1,,,

,1

1 =−

−=∆

+

+ izrtzrt

zzzru

iFRi

FR

iiKI (4.4)

Figure 4.14: Linear velocity profiles of the potassium iodide bands in different axial zones of the ID 26 column. Deviations are given with respect to the core velocity of the column.

Results of the analysis are shown in figure 4.14 and figure 4.15, respectively. The core veloci-

ties used to normalize the data are given in Appendix II. With the exception of the down-

stream zone ( 95.0..65.0=Lz ) of the ID=50mm column (s. figure 4.15), all zones exhibit the


60

same, typical parabolic dependency of the axial linear velocity on the radial position inside

the column. The velocity profile is rather homogeneous in the central region of the column.

With increasing radius the linear velocities lessen monotonically and reach their minimum

next to the wall. Similar results concerning the shape of the velocity and corresponding con-

centration columns were found in several previous investigations and are discussed in sec-

tion 2.8.

In the downstream zone of the ID=50mm column, the velocity profile is atypically rather uni-

form throughout the column cross section. The deviations from the core velocity are less than

3%.

Figure 4.15: Linear velocity profiles of the potassium iodide bands in different axial zones of the ID 50 column. Deviations are given with respect to the core velocity of the column.

4.3.5 Column efficiency

The additivity of the variance [Lode1998] was used to determine the column efficiency in

terms of the HETP for the annulus segments in between the first and the last monitoring posi-

tion.


61

( ) ( ) ( )4...12

12

42 ,,, zrzrzr ∆+= σσσ (4.5)

The variances as well as the retention times of the intra column breakthrough curves were

obtained from the fitted EDM curves. Subsequently the HETP values of the sections of inter-

est were calculated along the lines of equations (2.15) and (2.16), respectively.

( )

( ) ( )( )( )142

14

4...12

,,

,zz

zrtzrt

zrHETP

FR

FR

−⋅−

∆=

σ (4.6)

As can be seen in figure 4.16, the dispersive effects are more strongly pronounced in the re-

gion next to the column wall in either column where the HETP is higher. Whereas the trend

of the HETP is rather flat and even in the central region of the column, it rises significantly

approaching the column wall. The homogeneous, central region in the columns expands to

approximately 65% of the column diameter which is in excellent accordance with findings

from Farkas et al. [18] who report that in larger scale columns the central core region has a

diameter on the order of two-thirds of the column diameter.

Figure 4.16: Radial variations of the HETP for columns with different diameter. The disper-sion measurements were carried out between the first and the last monitoring position.


62

It is worth noting that the HETP values recorded for the single annulus sections are most

generally lower than the overall HETP given in Appendix III. This is due to the wide velocity

distributions causing the molecules in the central region to proceed. Because of this warpage

the apparent total band width in axial direction is larger than the local bandwidth. This under-

lines the impact of velocity and concentration distribution on the column efficiency. The per-

formance of a column which offers a locally efficient behaviour can deteriorate severely when

conjoined by a maldistributed, parabolic distribution. The enlarged differences in the retention

times of the core region (peak front) and wall region (peak tail) are reflected by broader peaks

and reduced efficiencies.

4.4 Velocity measurements by nuclear magnetic resonance imaging

4.4.1 Experimental set-up

Pure, deionised water was used as the mobile phase for the nuclear magnetic resonance imag-

ing (NMRI) experiments for the reason that common NMRI-systems rely on the relaxation

properties of excited hydrogen nuclei in the water molecule. A contrast agent was neither em-

ployed as a tracer compound nor as a mobile phase additive.

A preparative HPLC pump (K-1800, Knauer, Germany) described in section 4.3.1 delivered

the water at high volumetric flowrates. The pump was connected to the columns in closed

circuit by exceptionally long tubes that allowed to maintain the pump setup well outside the

magnetic field of the NMRI-system.

A static axial compression column made of glass (Goetec Labortechnik, Germany) with an

inner diameters of ID = 26mm (s. section 4.3.1 for details) was examined regarding the veloc-

ity profile inside the column. For the reason that no integral part of the column must be made

out of metal, the pistons of the column were homemade of polyvinyl chloride (PVC). The

columns were packed with a polydisperse, hydrophilic silica gel phase (C-Gel 560 40-63µm,

Zeochem AG, Switzerland). The column was slurry packed with isopropanol for the slurry as

well as the pushing solvent. The isopropanol was filtrated and degassed before use. The com-

pression pressure of 40bar was chosen close to the maximum operation pressure of the glass

column.

During the experiments, the column which was exposed to ambient temperature, was encap-

sulated by a jacket made of PVC that should have protected the NMRI-system in case of col-


63

umn leakage or breakage. The column together with the jacket were mounted inside the ex-

tremity coil of the NMRI-system which fitted the geometrical measures of the jacket nicely. A

reference tube with recumbent water was placed inside the jacket together with the column to

have a reference that allows phase and velocity correction, respectively.

The non-invasive velocity measurements were carried out on a 1.5 Tesla Siemens Magnetom

Sonata NMRI-System hosted by Schering AG Berlin. The thickness of the investigated slice

located in the central region of the column was 4mm. A fast low angle shot (FLASH) se-

quence with an echotime TE =19ms and a repetition time of TR =62ms was selected to ac-

quire two dimensional, phase encoded images of the velocity distribution over the column

cross section.

Figure 4.17: Phase encoded image of the velocity profile. Recumbent reference in the left up-per corner; column in the right bottom corner

4.4.2 Comparative measurements

In order to test whether a medical NMRI-system is capable to measure flow-velocities which

have a small order of magnitude compared to in vivo blood-flow [Laiblin2002], the velocity

distribution inside an empty, cylindrical column with a conically shaped inlet geometry was

monitored in the laminar regime [Lightfood1995, Sedermann1997]. For this regime, the ve-

locity profile ( )rv is parabolic and obeys the law of Hagen-Poiseuille.


64

( )

−⋅⋅=

2

2

12R

rurv (4.7)

As can be seen in figure 4.18, the experimental results match the parabolic flow profile pre-

dicted by equation 4.7. The order of magnitude as well as the general shape of the profile are

well represented. Low velocities close to the column wall are reproduced extremely well

whereas higher velocities close to the maximum in the column center are slightly underesti-

mated by the measurements. This might be due to the aliasing effect that velocities exceeding

the velocity encoding (VENC) representing the maximal velocity associated to a phase shift

of 180° are reprojected to smaller angles and identified as lower velocities.

Figure 4.18: Nuclear magnetic resonance measurements of the velocity profile in an empty

column with conical inlet geometry. Laminar regime: OHCdu2

Re ν⋅= ≈ 82.

4.4.3 Velocity profiles

An example of the velocity profiles measured by means of NMRI is given together with the

superficial velocity and the interstitial velocity in figure 4.19. The flow profiles were found to

be well in the reasonable range in between the superficial velocity (lower bound) and intersti-

tial velocity (upper bound). The measured velocities must be lower than the interstitial veloc-

ity due to the stagnant water inside the pore volume that contributes to the detected phase

shift.


65

The velocity profiles measured with the packed chromatographic columns exhibited a rather

plug like velocity profile in the core of the column. The fluctuations of the velocity in the cen-

tral region reflect the stochastic nature of the packed bed. Similar fluctuations were observed

by Harding et al. [Harding2001] who investigated velocity profiles in analytical scale chroma-

tographic columns by means of NMRI.

Towards the column wall the velocity decreased significantly as ascertained for the computed

tomography experiments. Contrary to these experiments, the size of the wall region with re-

duced permeability is significantly lower, occupying approximately eight percent of the col-

umn radius. As several influencing parameters were varied in between the CT and the NMRI

experiments, the cause for the difference in the area near the column wall cannot be clarified

unambiguously. E.g. the packing pressure and the hydrophilic properties of the stationary

phase material were different in either series of measurements.

Figure 4.19: Nuclear magnetic resonance measurements of the velocity profile in a chroma-tographic column packed with Ueticon C-Gel 560 40-63µm. Superficial and interstitial veloc-

ity for a hypothetical external porosity of ε =0.4 are given for comparison.


66

5 CFD Simulations

This chapter presents the results of modeling inhomogeneous columns by means of the com-

putational fluid dynamics code StarCD. Firstly, it explains how the modeling capabilities of

the code were extended by user coding to make the simulation of chromatographic processes

feasible. Moreover, attention was paid to the correlation of numerical accuracy and mesh den-

sity.

Thereafter it is demonstrated how the CFD calculations rendered the investigation of the in-

fluence of the properties of the separation performance possible. Special attention was paid to

the shift in typical chromatographic parameters, namely the efficiency, the retention time, and

the permeability of the column. It must be noted that the method of moments was conse-

quently used to calculate for the first two parameters because the band profiles were predomi-

nantly asymmetric.

The simulation results were critically compared to the experimental findings presented in the

previous chapter in order to evaluate the validity of the computational outcomes. Furthermore,

the local packing parameters derived from the computed tomography experiments were used

to set up a two dimensional model that is more akin to a real column than the typical one di-

mensional models.

5.1 Numerical considerations

5.1.1 Scaling of the partial differential equations

The partial differential equations that shall be solved by the use of the commercial CFD code

StarCD depend on two types of independent variables: i) spatial coordinates e.g. axial and

radial coordinate and ii) time. Characteristic quantities corresponding to the independent vari-

ables which e.g. could be used to make the variables dimensionless are i) the column length

CL or column diameter Cd and ii) the column dead time 0t .

In StarCD, the standard system of units for calculations is the International System of Units

(SI-System). In the SI-system, the characteristic spatial quantities have typical length scales of

between 10-2m to 100m, whereas the order of magnitude of the retention time is 102s to 103s.

Accordingly, the proportion of the characteristic properties in terms of numbers is (by ap-

proximation) in between 102 to 105. From a numerical point of view the high order of magni-

tude of the ratios is adversarial and amplifies numerical dispersion [Boysen2004]. For this


67

reason, all variables and parameters in the partial differential equations were rescaled into the

so-called CGS-System of units (centimeter, gram, second).

5.1.2 Choice of mesh density and time step

Due to the steep concentration gradients associated with the high efficiency of chroma-

tographic columns which frequently exceeds several hundred theoretical plates, computer

simulations of chromatographic separations are prone to numerical dispersion. As a conse-

quence, the total variance of a simulated peak ( )Simulation2σ is a result of two contributions.

Firstly, band broadening due to physical effects which are accounted for by the model equa-

tions ( )Model2σ (e.g. axial dispersion or mass transfer resistance). Secondly, band broaden-

ing due to numerical effects ( )ErrorNumerical2σ that are not described by the model equa-

tions but are a consequence of the discretisation of the original partial differential equations.

( ) ( ) ( )ErrorNumericalModelSimulation 222 σσσ += (5.1)

The accuracy of the simulation results depends on mesh density z∆ , the width of the time

step t∆ as well as on the choice of the discretisation schemes (s. section 2.9). In general,

higher mesh densities, smaller time steps, and high order discretisation schemes yield a reduc-

tion in the band broadening to numerical effects.

( ) ( )tzfErrorNumerical ∆∆= ,2σ (5.2)

Conversely, these have to be paid off by a high computational effort in terms of computation

time and memory capacity. Because of this it is necessary to find a compromise between

computational accuracy and computation costs.

One approach is to use the effects of numerical dispersion to represent band broadening due to

physical effects. The phenomena contributing to the spreading of the peaks are no longer

modeled. In other words, the partial differential equations lack terms that account for band

broadening. Instead the mesh density and the time step are adapted to yield the correct vari-

ance of the bands. Such an approach was e.g. followed by Lenz [Lenz2003] and Lisso

[Lisso2002]. It has the advantage that the computational effort is restricted to a minimum.

Another approach is to use a sufficiently high mesh density and a sufficiently small time step

to make the band variance due to numerical errors negligible.


68

( ) ( )ErrorNumericalModel 22 σσ >> (5.3)

The approach offers the advantage that it is physically more consistent. For instance, for an

n -component separation it renders the modeling of different extents of peak spreading for

each component possible. Due to its physical consistency, the latter approach was applied

within the frame of this work. Consequently, a systematic approach to select mesh density z∆

and time step t∆ is required. The approach should assure that neither the influence of numeri-

cal dispersion is dominant nor that the computational effort is needlessly large.

For transient calculations, the company who developed StarCD, CD-Adapco, recommends to

choose the time step and the spatial mesh density to yield Courant numbers defined by equa-

tion 2.44 smaller than one hundred [CDAdapco2004].

Contrariwise, Boysen [Boysen2004] found that for computational fluid dynamics simulations

of chromatographic separations, the Courant number has to be close to unity to keep the influ-

ence of numerical dispersion sufficiently low. The condition of Boysen is apparently more

restrictive and demanding in terms of computing power.

( )1

1≅

∆

∆⋅

⋅+=

z

t

dcdqF

uCo

ε (5.4)

Furthermore it can be seen from equation 5.4, that either restriction for the Courant number

merely allows to chose a well suited ratio of the time step and mesh width while a second

restriction would be needed to actually determine either numerical parameter.

( ) ( )εε SFSF u

dcdqF

u

dcdqFCo

z

t ⋅+≅

⋅+⋅=

∆

∆ 11 (5.5)

It is also evident that the lightest slope of the adsorption isotherm dcdq determines the ratio

zt ∆∆ whereas the demands on the time step would be less restrictive for concentrations cor-

responding to steeper slopes. This kind of numerical behaviour is commonly referred to as

stiffness [Werner1986].

In order to investigate the influence of mesh width and time step on numerical dispersion and

to derive a relationship that enables the determination of a well suited mesh density, one di-

mensional models of chromatography similar to the model discussed in section 2.9.1 were


69

investigated [Su2005]. In this context, multi-dimensional models lack analytical solutions that

allow to quantify the accuracy of the numerical simulation result.

Figure 5.1 shows the ratio of the simulated peak variance ( )Simulation2σ and the variance

due to physical sources of band broadening accounted for by the model ( )Model2σ as a func-

tion of the Courant Number Co and the cell Peclet Number CellPe as defined by equation

(2.46).

Figure 5.1: Dependency of numerical dispersion on the Courant- and the Dispersion-Number. Dark blue regions refer to a variance ratio of ≈ 1.

Beginning with the Courant number, the results given in figure 5.1 confirm the findings of

Boysen [Boysen2005]. The variance ratio reduces to unity only if Co approaches unity. Re-

garding the cell Peclet Number it can be seen that the ratio of the variances approaches unity

only if the Peclet number itself is also close to unity. This can be used to deduce a relationship

for the determination of the mesh density:

1≅∆

⋅=effax

CellD

zuPe

ε (5.6)

εε u

D

u

DPez

effax

effax

Cell ≅⋅=∆ (5.7)


70

It is worth noting that the investigated model considers dispersion to be the only source of

band broadening. Nevertheless, the applicability of the findings is more general. For models

where mass transfer resistance is modeled explicitly, the additivity of the effects can be taken

advantage of (s. sections 2.5.4 and 2.5.5) in order to determine well suited numerical settings.

5.1.3 Permeability and pressure drop

In StarCD, the pressure drop in porous media is modeled by the following equation:

SF

jj

SF

j

SF

jj

j

uuudx

dP⋅+⋅⋅=− βα (5.8)

jα as well as jβ are coefficients that must be defined by the user. j denotes the spatial direc-

tion. As explained in section 2.6, the flow regime in chromatography is laminar and the pres-

sure drop is a linear function of the superficial velocity as given by the Darcy’s law repro-

duced in equation (2.32). Consequently, jα reduces to zero.

0=jα (5.9)

According to Darcy’s law, the jβ -coefficient is given by

κ

ηββ ==j

(5.10)

Due to the stochastic structure of the packed bed, the permeability and thereby the

β -coefficient(s) are assumed to behave isotropically. The β -values were either determined

from the experimental pressure readings (s. section 4.2) or calculated with the Kozeny-

Kármán equation (2.34). The coefficient Kh was set to 61.4=Kh in order to yield

15036 =⋅ Kh as the most common value (e.g. known from the Blake-Kozeny equation or the

laminar term in the Ergun equation [Bird2002]) for the proportionality constant.

The pressure correlation given by equation (5.8) and the continuity equation (2.37) form a

system of coupled partial differential equations. In order to assure that the pressure – and

thereby the velocity – calculations were carried out correctly, it was double-checked that fore-

casted and simulated pressure drops over the column length are in good agreement. It was

found that the column length must be discretised into a minimum number of approximately

five hundred cells to fulfill the requirement [Su2005].


71

Due to the restriction for the mesh density that is necessitated by equation (5.7) in combina-

tion with the low order of magnitude of the dispersion coefficients in chromatography, equa-

tion (5.7) also ensures the correctness of the pressure and velocity profile calculations.

5.2 Implementation of adsorption

StarCD as a commercial Computational Fluid Dynamics code was developed to solve the

(species) mass and moment transport equations to provide information on the velocity, con-

centration, and pressure profiles inside the computational domain. In this context, the code

enables the modeling of moment and species transport in a fluid percolating through a porous

medium but lacks the ability to account for accumulation in the porous medium itself. In other

words, the code does not enable the user to implement adsorption in a straightforward manner.

Boysen et al. [Boysen2002], [Boysen2003], [Boysen2004] describe two distinct strategies to

consider the influence of adsorption for chromatography simulations using Fluent, another

commercial CFD package.

Strategy A: Adsorption is not modeled directly but represented by reducing other parameters,

namely the interstitial porosity and the dispersion coefficient, of the simulations. This strategy

suffers from the disadvantages that i) it is limited to the equilibrium dispersive model of

chromatography (mass transfer cannot be accounted for) with linear adsorption isotherms and

ii) the simulation of an n -component separation requires n simulation runs e.g. for the reason

that a specific velocity that depends on the Henry coefficient is needed for each component.

Strategy B: Adsorption is implemented via a user subroutine that is designated to implement

homogenous reactions in the fluid phase.

In this work, the more universally applicable approach of strategy B was chosen to model

adsorption in the CFD simulation. From this it follows that the differential species mass bal-

ance given by equation (2.38) must be rewritten in order to implement adsorption in StarCD:

ScDcut

ciaxi

i &r=∇⋅−∇⋅+

∂

∂⋅ 2εε (5.11)

S& is the reaction source term which shall allow for the implementation of homogenous

chemical reactions. For chromatographic applications, S& must equal


72

( )t

qS i

∂

∂⋅−−= ε1& (5.12)

In StarCD the source term S& for homogenous mobile phase reactions is implemented via the

subroutine code SORSCA.f. This subroutine enables the user to specify source terms per unit

volume for species in linearised form:

MPiPP wSSS ⋅−= 21

& (5.13)

PS1 and PS2 can either be constant or an arbitrary function of parameters (e.g. density, tem-

perature, etc.). iw is the mass fraction of component i in the mobile phase.

StarCD does not enable the definition of derivatives with respect to time in SORSCA.f. Be-

cause of this, the accumulation term has to be represented by a difference quotient [Su2005].

( ) ( )t

ttwtwS

SPi

SPiMP

P∆

∆−−⋅⋅−−= ρε )1(1 (5.14)

02 =PS (5.15)

MPρ is the mobile phase density. SPiw is a superficial quantity that equals a stationary phase

mass fraction in the mobile phase. The mass fractions iw can be passed to and from other

subroutines. An example for the subroutine SORSCA.f for single component adsorption is

provided in Appendix IV.

5.3 Equilibrium dispersive model

In the EDM, all effects of band broadening due to mass transfer resistance are lumped into an

apparent axial dispersion coefficient (s. section 2.5.5). Consequently, the stationary phase

concentration/ mass fraction is given by the adsorption isotherm ( )cf or ( )MPwf , respec-

tively. The stationary phase mass fractions ( )twSPi are assigned to hypothetical components

within the StarCD calculations.

( ) ( )niSPi wwwftw ,...,...,1= (5.16)


73

Before calculating a new value for ( )twSPi , the old value is preserved using a second hypo-

thetical species for each component. Consequently, n2 hypothetical components must be de-

fined for an n -component separation [Su2005].

The assignment of the stationary phase mass fractions is implemented using the user subrou-

tine POSDAT.f which is intended to enable the output of results at the end of each time step.

For the reason that POSDAT.f is called only at the end of each time step, the values ascribed

to the hypothetical velocities are preserved during the iteration process. A similar strategy was

followed by Schneider [Schneider2006] in order to implement a source term (describing Joule

Heat generated in electrochromatography) into the energy equation.

An example of the subroutine POSDAT.f modeling linear single component adsorption by

means of the equilibrium dispersive model is provided in Appendix V.

Figure 5.2: Chromatogram of a two component mixture in case of Langmuir isotherms. 1D-StarCD-Model with user coding for EDM. Parameters of the simulation as follows:

mLC 3.0= , 4.0=ε ,s

mu SF 4105 −⋅= ,s

mD effBA

27, 10−= ,

mlgc inj

BA 1.0, = , st 1.0=∆ ,

mz 5103 −⋅=∆


74

An example of a chromatogram simulated with EDM-user-coding is given in Figure 5.2. The

example shows that the user coding can also account for the tailing peak behaviour in case of

degressive or Langmuir-type isotherms.

It is worth noting that the use of the equilibrium dispersive model in combination with a linear

isotherms leads to diverging solutions unless the Henry coefficient is smaller than the recipro-

cal of the phase ratio ( FH i 1< ). Otherwise the amount of component ( ) iiSPi cHm ⋅⋅−= ε1

adsorbed on the stationary phase material during the initialising time step is larger than the

amount of component iMPi cm ⋅= ε that is actually contained in the mobile phase volume. This

leads to negative mobile phase concentration and to diverging solutions.

5.4 Mass transfer model

Due to the mass transfer model introduced in section 2.5.3, the mass transfer resistance in

between the two phases is accounted for by a lumped mass transfer coefficient. Within the

StarCD terminology, the equation can be written as follows:

)( *,

SPi

SPiim

SPi wwkt

w−=

∂

∂ (5.17)

By using the explicit scheme [Knabner2000] for temporal discretisation, the equation can be

integrated to solve for the stationary phase concentration at time t . Similar to the approach

used for the EDM, the stationary phase mass fractions ( )twSPi are assigned to hypothetical

components within the StarCD calculations.

( ) ( ) ( )[ ] ( )ttwttwttwtktw SPi

SPi

SPiim

SPi ∆−+∆−−∆−⋅∆⋅= *

, (5.18)

Before computing new values for ( )twSPi , the old values of the mobile MP

iw and stationary

phase mass fraction SPiw are preserved using two more hypothetical species for each compo-

nent. Consequently, n3 hypothetical components must be defined for an n -component sepa-

ration [Su2005].


75

Figure 5.3: Chromatograms for different orders of magnitude of the mass transfer coefficient

imk , . 1D-StarCD-Model with user coding for LDFM and linear isotherm. Parameters of the

simulation as follows: mLC 3.0= , 4.0=ε ,s

mu SF 4105 −⋅= ,s

mD effBA

27, 105 −⋅= , 2.0=iH ,

mlgc inj

BA 5.0, = , st 2.0=∆ , mz 5103 −⋅=∆ . For tsk im ∆== 10.5, the LDFM reduces to the

EDM.

The solid film linear driving force model (LDFM) was also implemented through the user

subroutine POSDAT.f. An example of the subroutine POSDAT.f modeling linear single com-

ponent adsorption by means of the solid film linear driving force model is provided in Appen-

dix VI.

It must be mentioned that the use of the explicit scheme enforces another restriction on the

choice of the time step t∆ . For imkt ,1<∆ the scheme is stable. For imkt ,1=∆ equation

(5.18) reduces to the implementation of the EDM described in the previous section. For

imkt ,1>∆ the explicit scheme becomes unstable.

An example for the correct representation of the influence of the mass transfer coefficient on

the overall broadness and shape of the peaks by the user coding is given in figure 5.3. A de-

crease of the mass transfer coefficient results in an efficiency loss reflected by broader peaks.


76

In agreement with chromatographic theory [Guiochon1994], very low mass transfer coeffi-

cients result in asymmetric, tailing peaks. The retention time of the peaks’ center of gravity

determined by the adsorption isotherm is not affected. In accordance with equation (2.9) all

peaks shown in figure 5.3 have the same first moment of si 313,1 =µ .

5.5 Validation of isotherms

In order to validate the implementation of different isotherms models, frontal analysis ex-

periments [Guiochon1994, SeidelMorgenstern2004] as a dynamic method to determine ad-

sorption isotherms by step experiments were simulated. The simulations were carried out with

different step concentrations to investigate whether the StarCD models extended by user cod-

ing for implementation of adsorption are capable of restoring the original isotherms. Initially

the simulated columns were free of solute.

The equilibrium concentration of the stationary phase that corresponds to the respective step

concentrations of the mobile phase was determined from the corresponding breakthrough

curves ( )tc .

( )( )( )

−

−

⋅=∫∞

= stept

step

step ct

dttcc

Fcq

0

0* 1 (5.19)

Examples for linear as well as Langmuir isotherms are provided in figure 5.4 where lines rep-

resent the original isotherm equations, whereas the points denote the equilibrium concentra-

tion restored from the frontal analysis simulations, respectively.

As can be seen, the simulated points coincide well with the curves of the original isotherms

thereby proving that the incorporation of adsorption by means of user coding indeed describes

the thermodynamics of the column correctly.


77

Figure 5.4: Comparison between original isotherm equations and equilibrium data derived from Frontal Analysis simulations in StarCD for linear and degressive isotherms. Data to the left of the abscissa break refer to the left ordinate, whereas data to the right of the abscissa

break refer to the right ordinate.

5.6 CFD-Modeling of local column inhomogeneities

5.6.1 Comparison between experiment and simulation result for hollow regions

In order to assess and validate the predictions of CFD-simulations for columns with local in-

homogeneities, a two dimensional model with a hollow, spherical region in the column center

was set up and compared with the experimental data given in section 4.2.2. The dimensions of

the hollow region matched the dimensions of the ceramic spheres used in the experiments

(table 4.1). The rest of the packing was assumed to be homogenous [CarrerasMolina2005].

The packing parameters were chosen to yield the permeability and efficiency values of the

homogenous reference columns as determined in the experiments (table 4.2). A comparison

between simulated and experimental chromatogram is given in figure 5.5.


78

Figure 5.5: Comparison of eluting bands’ shape for experimental and simulated columns with

a spherical, hollow inhomogeneity of cmd sphere 6.2= . Experimental conditions as described

in section 4.2.2. Parameters of the simulation as follows:

mLC 325.0= , 47.0=ε ,s

mu SF 4104.3 −⋅= ,s

mDax

28105.8 −⋅= ,

st 1=∆ , mrz 4103 −⋅=∆=∆ .

It is clearly apparent that the CFD-model – although it matches the experimental peak nicely

in terms of peak widthness (s. figure 5.5) and efficiency (s. table 5.1) – is unable to account

for the exact shape of the experimental chromatogram. The nearly symmetrical simulation

result shows no shoulder on the front of the peak.

Table 5.1: Comparison of experimental and simulated HETP for columns with a spherical,

hollow inhomogeneity of cmd sphere 6.2= . Same parameters as for figure 5.6.

experimental StarCD

HETP [m] 0,0008 (±0,00018) 0,00076


79

To evaluate whether the relatively small inhomogeneity (0.5% of the column volume as in the

experiments) has any impact on the simulation result at all, the chromatogram was compared

to the effluent history of a homogenous column. The comparison is provided in figure 5.6.

Figure 5.6: Comparison of eluting bands’ shape for simulated columns. i) homogenous case;

ii) with a spherical, hollow inhomogeneity of cmd sphere 6.2= . Parameters of the simulation

as follows: mLC 325.0= , 47.0=ε ,s

mu SF 4104.3 −⋅= ,s

mDax

28105.8 −⋅= ,

st 1=∆ , mrz 4103 −⋅=∆=∆ .

It can be seen that the presence of a hollow region is not insignificant for the simulation result

bearing analogy with the experimental observations. In either case (computational and labora-

tory findings), the inhomogeneity predominantly affects the front of the band. As already

mentioned in section 4.2.2, the alteration of the peak front is due to the increased permeability

of the hollow region where the fluid velocity is higher and the tracer band precedes thereby

giving rise to i) the of a shoulder on the front in the experiments or ii) an increased amount of

fronting in the simulations.


80

Although the inhomogeneous model is unfit to account for the formation of a shoulder, it pre-

dicts an increased fronting of the bands owing to the disturbed hydrodynamics close to and

inside the hollow region. The absence of the shoulder is believed to be due to restrictions in

the modeling capabilities of the StarCD code. These restrictions are associated to effects that

were intensively studied in connection with so-called “Taylor Dispersion” [Bird2002]. Taylor

[Taylor1953] and Aris [Aris1956] studied the spreading of bands in steady laminar flow

through straight tubes. They showed that any kind of band profile (in this case a peak with a

shoulder) will eventually relax to a Gaussian, cross sectionally averaged concentration profile

sufficiently far downstream of the injection point (or in this case of the point of distortion).

The relaxation process is enhanced by diffusion/dispersion in the radial direction. According

to Shankar and Lenhoff [Shankar1989], the length required to relax the original profile is

given by

m

relaxationD

RuL

2max ⋅

> (5.20)

where mD describes transport in the radial direction by means of Fick’s law. It is noted that

diffusion into the axial direction is not accounted for in the studies of Taylor [Bird2002]. It is

obvious that the relaxation towards a Gaussian band shape is attenuated by large orders of

magnitude of mD .

In chromatography, dispersive transport in the radial direction, that is otherwise seldomly ac-

counted for, is described by means of a radial (or transverse) dispersion coefficient rD . It is

well known, that this radial dispersion coefficient is much lower than the axial dispersion co-

efficient. Experimental results by Eon [Eon1978], Baumeister et al. [Baumeister1995], and

Tallarek [Tallarek1996] showed that at mobile phase velocities 204 * << u used in prepara-

tive chromatography the ratio rax DD is larger than 3 and often in between 5 and 10. At

higher velocities the ratio can even exceed these values. *u is the reduced mobile phase ve-

locity defined as:

m

PSF

D

duu ⋅=

ε* (5.21)

For the test system of diethyl phthalate as the tracer, isopropanol as the mobile phase, and

glass beads of µmd P 55≈ described in section 4.1.2 and section 4.1.3, the reduced mobile


81

phase velocity is 100* ≈u . Porosity and superficial velocity are given in figure 5.6, the mo-

lecular diffusion coefficient smDm210104.3 −⋅= of diethyl phthalate in isopropanol was

estimated with a correlation given by Danner and Daubert [Danner1983]. The thermophysical

properties needed for the correlation method were determined using a data compilation from

the same authors [Daubert1989].

Up to the last version of StarCD (StarCD V4) an anisotropy in terms of the dispersion coeffi-

cient could not be defined [CDAdapco2006] and the value of the axial dispersion coefficient

also characterises radial dispersion in the simulations. Hence, the influence of radial disper-

sion is overrated in the chromatograms computed with StarCD and the anomaly found on the

front of the experimental peak relaxes too swiftly in the simulation studies that yield smooth

fronts.

Figure 5.7: Comparison of eluting bands’ shape for simulated columns with spherical, hollow inhomogeneity of different dimensions. Parameters of the simulation as follows:

mLC 325.0= , 47.0=ε ,s

mu SF 4104.3 −⋅= ,s

mDax

28105.8 −⋅= ,

st 1=∆ , mrz 4103 −⋅=∆=∆ .


82

In order to point out that the developed StarCD models are in principal capable to predict the

formation of shoulders due to local inhomogeneities, additional simulations with an increased

extent of the inhomogeneity were carried out. The enlargement of the hollow region aimed to

add weight to the significance of the hydrodynamic disturbances associated with the inho-

mogeneity while keeping the influence of radial dispersion constant. The results are given in

figure 5.7.

The results in figure 5.7 show that the models indeed predict “shouldering” if the hydrody-

namic distortions of the flow profile are sufficiently large not to be smoothed out by the ef-

fects of radial dispersion.

It may be concluded that the models developed to reproduce inhomogeneous column behav-

iour represent physical columns well in terms of common chromatographic parameters like

efficiency, band width, or retention times. Despite the good corroboration in terms of these

overall column properties, the bands predicted by the StarCD simulations are generally

smoother than the corresponding peaks monitored on a physical column. The relaxation of

characteristic patterns on the peak due to overrating of radial dispersion is less accurate the

smaller the hydrodynamic distortion due to the inhomogeneity.

5.6.2 Influence of local inhomogeneities on the column permeability

In order to investigate the influence of local inhomogeneities on common chromatographic

parameters (efficiency, permeability) derived from the detected signals (chromatogram, pres-

sure drop), spherical regions with different properties were implemented into the column

models [Su2005]. The parameters altered were the porosity difference εε −SP in between the

spherical region and the rest of the packing, the size SPd of the inhomogeneity, the axial posi-

tion Po of the sphere, the length L of the column, as well as the overall dispersion coeffi-

cient. The column diameter md 1.0= and the mean external porosity 4.0=ε of all columns

were kept constant. A constant mean porosity ε of the columns was chosen in order to mimic

inhomogeneity due to rearrangement processes of the stationary phase inside the column dur-

ing the operation. Here the amount of stationary phase and thereby the mean porosity of the

column remain unaltered.

A sketch of this type of model is provided in figure 5.8:


83

Figure 5.8: Sketch of the inhomogeneous column models used to investigate the consequences of different degrees of local inhomogeneities. LPo ⋅= 25.01 , LPo ⋅= 50.02 , LPo ⋅= 75.03

In a parameter study, packing parameters potentially affecting the properties of the column

were varied.

i) The influence of the size spd of the inhomogeneity was investigated while the column

length L as well as the position Po of the sphere remained unaltered.

ii) The influence of the length L of the inhomogeneous column was studied while the size

spd and the position Po of the inhomogeneity were kept constant.

iii) The influence of the position Po of the inhomogeneity was investigated while the column

length L and the size of the inhomogeneity remained constant.

Additionally the effect of the porosity difference εε −SP between the inhomogeneity and the

column mean was studied in all cases. The results of the simulation runs are given in ta-

ble 5.2.

Firstly, it was found that the pressure drop is not a well suited discriminator for local column

inhomogeneity. The shift in overall column permeability is generally low as can be seen in

table 5.2. The maximum deviations in pressure add up to approximately -9% and +4% com-

pared to the homogenous reference column only (marked bold in table 5.2) even though the

corresponding inhomogeneities had remarkable dimensions. Their diameter equaled 88% of

the column diameter and their porosities differed eight digits from the mean porosity.

Secondly, the influence of certain column and inhomogeneity parameters on the pressure drop

could be educed. It was found that the overall column permeability is a monotonic functions

of the porosity difference in between the inhomogeneity and the rest of the packing as can be

seen from the “table”-column data of all simulated columns in table 5.2. Interestingly, inho-


84

mogeneities with increased porosities εε >SP engender better permeable packings. Obvi-

ously, the region of inhomogeneity whose porosity is altered significantly has a stronger im-

pact on the overall column permeability than the moderate, contrarious adjustment of the po-

rosity of the larger rest of the column needed to keep the mean porosity constant. It should be

reemphasised that the dimension of the permeability shift is generally small.

Table 5.2: Dependency of the column permeability on different packing parameters. The num-bers i)...iii) refer to the cases of the parameter study explained above. All columns have the

same mean porosity 4.0=ε . The table cells contain data on the permeability ratio 0κκ be-

tween the inhomogeneous and the reference column.

i) dd SP / # ii) Ld SP / ‡ iii) Po †

εε −SP 0.44 0.66 0.88 0.075 0.15 0.3 1 2 3

-0.08 0.994 0.977 0.923 0.997 0.994 0.989 0.905 0.923 0.920

-0.04 0.997 0.987 0.963 0.999 0.997 0.993 0.953 0.963 0.959

0.00

(Reference) 1 1 1 1 1 1 1 1 1

0.04 1.004 1.013 1.028 1.002 1.004 1.007 1.024 1.028 1.026

0.08 1.007 1.023 1.043 1.004 1.007 1.015 1.036 1.043 1.042

# 3.0/ =Ld , 2=Po ; ‡ 44.0=dd SP , 2=Po ; † 88.0=dd SP , 15.0/ =Ld SP

The relative size of the inhomogeneity has the expected impact. Larger relative diameters

dd SP / and Ld SP / cause larger permeability changes as can be seen from the row data in

columns 1-3 and 4-6, respectively.

The influence of the position Po on the pressure drop of the column is not distinct. As can be

concluded from the row data of columns 7-9 in table 5.2, the influence of the position is nei-

ther pronounced nor is it clearly correlated with the axial coordinate. It may be concluded that

the permeability is free of influence from the position.

5.6.3 Influence of local inhomogeneities on the first moment

The effect of local column inhomogeneities on the first moment of the peaks was investigated

in a similar fashion. It was also found that the first moment i,1µ of a peak is not a well suited


85

discriminator for local column inhomogeneity. As to be expected, the maximum shift of the

first moment was found for the combination of the largest investigated porosity difference

08.0=− εε SP and the largest size of the inhomogeneity 88.0=dd SP . The deviation from

the first moment of the homogenous column was found to be less than 2%. The position of the

inhomogeneity showed no influence on the first moment.

5.6.4 Influence of local inhomogeneities on the column efficiency

Figure 5.9: Dependency of the column efficiency on the difference between the porosity of the

inhomogeneity SPε and the mean porosity 0εε = for columns with different dispersion coeffi-

cients axD .

A difference in behaviour may be observed for the impact of local column inhomogeneities on

the efficiency of chromatographic columns. The influence is clear but depends on the overall

efficiency of the column as can be found in figure 5.9. The efficiency loss caused by the in-

homogeneity is moderate for a low efficiency column characterised by a dispersion coefficient

of smDax26105.2 −⋅= whereas the efficiency loss associated with a similar inhomogeneity


86

is dramatic for a column whose efficiency is one order of magnitude higher

( smDax27105.2 −⋅= ). This observation corroborates well with findings from Lenz

[Lenz2003].

The relative size of the inhomogeneity again has the expected impact. Larger relative diame-

ters dd SP / and Ld SP / cause larger efficiency drops as can be seen from the row data in col-

umns 1-3 and 4-6, respectively.

As for the permeability, the influence of the position Po on the efficiency of the column is

not explicit. The influence of the position on the overall column efficiency is neither distinct

nor is it clearly correlated with the axial coordinate. It may again be concluded that also the

efficiency is free of influence from the position.

Table 5.3: Dependency of the column efficiency on different packing parameters. The numbers i)...iii) refer to the cases of the parameter study explained in section 5.6.2. All columns have

the same mean porosity 4.0=ε and the same dispersion coefficient smDax207105.2 −⋅= .

The table cells contain data on the efficiency ratio 0NN between the inhomogeneous and the

reference column.

i) dd SP / # ii) Ld SP / ‡ iii) Po †

εε −SP 0.44 0.66 0.88 0.075 0.15 0.3 1 2 3

-0,08 0,699 0,346 0,212 0,821 0,699 0,535 0,244 0,212 0,219

-0,04 0,904 0,688 0,539 0,948 0,904 0,823 0,585 0,539 0,556

0,00

(Reference) 1 1 1 1 1 1 1 1 1

0,04 0,913 0,722 0,592 0,953 0,913 0,840 0,634 0,592 0,606

0,08 0,744 0,422 0,290 0,850 0,744 0,594 0,326 0,290 0,292

# 3.0/ =Ld , 2=Po ; ‡ 44.0=dd SP , 2=Po ; † 88.0=dd SP , 15.0/ =Ld SP

5.7 CFD-Modeling of an inlet void

In order to learn more about the behaviour of columns with an inlet void due to a sacked

packing, congeneric two dimensional CFD-models were set up and compared with the ex-

perimental data given in section 4.2.2 [CarrerasMolina2005]. The dimensions of the cavities


87

next to the column inlet matched the artificially created voids, as examined in section 4.2.3

for comparison. The inlet void was modeled as an empty tube with laminar flow. The model

region representing the (sacked) packing was assumed to be homogenous. The efficiency of

the (sacked) packing was chosen to yield the efficiency of the homogenous reference column

as given in table 4.2. Conversely the porosity and the permeability of the simulated, sacked

packings were deliberately chosen not to match the permeability of the reference column.

As explained in section 4.2.2, the inlet void in the experiments was formed by slightly lifting

the piston of the self packing device. Because of this, the experiments were well suited to

mimic the influence of an inlet void on the efficiency. However, the experiments were unable

to yield information about the effect of void formation on the permeability as the packing did

not settle and the interstitial void fraction of the bed remained unaltered.

In the simulations, the influence of a sacked bed on the interstitial void volume and the per-

meability was accounted for. Only the Kozeny-coefficient Kh as defined in equation (2.34)

was derived from the experimental results. The external porosity and the permeability were

then calculated from equation (2.34) taking into account that the external porosity of the pack-

ing reduces due to the sacking process:

( )( )

1

10

−

−⋅=

=

void

C

void

Cvoidext

voidext

L

L

L

LL

L

ε

ε (5.22)

CL and voidL are the length of the column and the length of the void, respectively.

5.7.1 Influence of an inlet void on the column efficiency

The simulation results are in good agreement with the experimental findings in terms of i) the

general peak shape and b) the severe efficiency loss due to the formation of an inhomogeneity.

As in the experiments, the simulations show a cavity next to the inlet causing peak tailing.

The tailing gets stronger with increasing void size.

Figure 5.10 shows a comparison between the shapes of a simulated and an experimental peak

for a void length of 4cm. Due to the adaption of the external porosity to the simulation studies

(s. Equation (2.34)), experimental and simulated peaks must have distinct retention times.


88

Consequently, also the peak height must be different for the reason that a longer retained peak

disperses more strongly provided that the effective efficiencies are equal (s. equation (2.16)).

Figure 5.10: Comparison of experimental and simulated peak shapes for columns having an inlet void with a length of 4cm. Experimental conditions as described in section 4.2.3. Pa-

rameters of the simulation as follows: mLC 33.0= , 46.0=ε , s

mu SF 4106.6 −⋅= ,

smDax

28109.7 −⋅= , st 1.0=∆ , mrz 4103 −⋅=∆=∆ .

In order to compare simulated and experimental bands in a well suited fashion, values are

displayed in dimensionless form in figure 5.10. The dimensionless time τ defined by equa-

tion 4.2 is given on the abscissa whereas the dimensionless concentration 0)( Cc τ is given on

the ordinate. The concentration 0C is related to the area of the peak and given by

( )∫∞

⋅=0

0 ττ dcC (5.23)

This form of the plot was chosen because equations (2.34), (2.12), (2.8) that jointly define the

efficiency of a column can be rearranged to yield


89

( ) ( ) τττ

dC

cN ⋅−⋅= ∫

∞2

0 0

1 (5.24)

It can be seen that the efficiency is a function solely of two dimensionless variables. Accord-

ingly, two matching bands in the dimensionless plot make for similar column properties in

terms of separation efficiency.

As can be seen, the simulation result is in good qualitative agreement with the experimental

findings even though some quantitative differences may be observed. The tailing behaviour

characteristic for the existence of an inlet void is very well matched by the computed band.

Nevertheless it should be mentioned that the model slightly overestimates the peak height in

terms of dimensionless concentration resulting in a higher efficient column.

Table 5.4 summarises the efficiency data for all degrees of inlet void formation investigated. It

can be observed that the models and the experimental findings corroborate well in terms of

the magnitude of the HETP even though the exact bandwidth of the experimental data is not

perfectly adhered to in all cases.

Table 5.4: Comparison of experimentally and computationally determined column efficiencies in terms of the HETP for columns with inlet voids of different sizes.

Length of inlet void [m] HETP [m]

(simulation)

HETP [m]

(experimental incl. column to column

reproducibility)

0,01 0,00497 0,0044 – 0,0072

0,02 0,00712 0,0036 – 0,0064

0,04 0,01034 0,0074 – 0,0102

5.7.2 Influence of an inlet void on the first moment

Also the influence of an inlet void on the first moment of the peaks was borne in mind during

the simulation studies. As for local inhomogeneities, the first moment i,1µ of a peak was

found to be not a well suited discriminator for this kind of column inhomogeneity. The maxi-

mum shift of the first moment was found for the largest extent of the inlet void. The deviation

from the first moment of the homogenous column was found to be only -3%.


90

5.7.3 Influence of an inlet void on the column permeability

As mentioned above, the laboratory investigations concerning the appearance of an inlet void

did not allow to gain information about the influence on the permeability. By contrast, the

StarCD models described in the previous sections were setup with consideration of the varia-

tions in interstitial porosity and column permeability due to void formation.

According to the simulation results, the formation of an inlet cavity significantly affects the

permeability unfavourably. The simulation results for the investigated columns are given in

figure 5.11. The permeability of the column packing is plotted for comparison.

Figure 5.11: Dependency of the column and the packing permeability on the length of the inlet void. The permeabilities κ of the inhomogeneous columns are displayed relative to the per-

meability of the homogenous column 0κ . All columns have the same mean porosity 0εε = .

Parameters of the simulation as follows: mLC 33.0= , 46.0=ε , s

mu SF 4106.6 −⋅= ,

mrz 4103 −⋅=∆=∆ .


91

It can be seen that the high permeability of the packing-free inlet void damps the degradation

of the packing permeability due to the reduction of the external porosity in terms of the over-

all column permeability. The overall column permeability is not as strongly affected even

though the hydrodynamic influence of the cavity zone is far from compensating the perme-

ability loss. Recapitulating it may be noted that a loss in column permeability can have a share

in discriminating a void formation next to the column inlet.

5.8 CFD-Modeling of a wall region based on computed tomography experiments

A computational fluid dynamic model of a chromatographic column employing two spatial

dimensions allowed to study the consequences of column heterogeneity and the existence of a

wall region for a real column. The model was based on the results of the computed tomogra-

phy experiments by deducing local packing properties like e.g. the permeability from the

monitored data described in section 4.3. The model was then used to investigate the spreading

of tracer fronts as well as the effect the column heterogeneity on the resolution.

5.8.1 Model set-up based on computed tomography experiments

The permeability data characterising the regional hydrodynamic behaviour of the segments as

well as the dispersion coefficients that represent the efficiency dependence on the radial coor-

dinate given in figure 5.12 were implemented into the model.

The determination of the dispersion coefficients for the single annuli based on the HETP data

for the ID=50mm column given in figure 4.16 as well as the linear velocity data provided in

figure 4.15 and Appendix II, respectively. The HETP values were converted into dispersion

coefficients via equation (2.31).


92

Figure 5.12: Permeability and Dispersion data for different regions of the ID=50mm column. The data were determined based on the results of CT-experiments in section 4.3.

The permeability data for the single annuli was determined with a MATLAB algorithm

[Lottes2005]:

1. The mean porosity KIε for each of the three axial segments was determined from the re-

tention times of the cross sectionally averaged “intra-column” breakthrough curves given

in Appendix I. The mean porosities KIε were converted into interstitial (or external) po-

rosities extε by means of equation 2.6 for the reason that the interstitial porosities deter-

mine the hydrodynamics of the column.

2. An initial guess for the pressure drop in each of the three axial segments was made based

on equation (2.34). The coefficient Kh was again set to 61.4=Kh as explained in sec-

tion 5.1.3.

3. The pressure drop estimation in combination with the velocity data for the single annulus

segments given in figure 4.15 and Appendix II allowed to determine initial guesses for the

external porosity of the annulus segments by the use of equation (2.34).


93

4. It was checked whether the sum of the interstitial volumes of the annuli yields the mean

interstitial porosity of the axial segment. The MATLAB algorithm fulfilled the criterion in

equation (5.25) after the first loop in all cases. Otherwise the iteration must return to

step 2 to achieve an improved guess for the pressure drop over the axial segment.

4

10

1,

10−= =<

⋅

−∑

criterionstopA

A

C

iiexti

ext

ε

ε (5.25)

A symmetry boundary condition that utilises the rotational symmetry of the cylindrical col-

umn geometry was exploited in order to keep the computational effort reasonable. The model

consisted of thirty regions with different properties. The number corresponds to the three axial

column segments enclosed by the four monitoring positions which were further subdivided

into ten annuli.

All of the thirty column regions were discretised by a structured, equidistant, and – due to 2D-

approach – Cartesian mesh. To ensure high numerical accuracy, the spatial mesh width of the

cells as well as the time step were chosen according to equation (5.5) and equation (5.7), re-

spectively. The parameters used were the effective mean porosity for potassium iodide of the

column 53.0=ε as determined from the retention data and the arithmetic mean of the disper-

sion coefficients smDax28101.3 −⋅= given in figure 5.12, respectively. The spatial mesh

width was found to be mz 4108.1 −⋅=∆ . Accordingly, every region was discretised by 590

(axial) x 14 (radial) cells. Altogether the model consisted of 247800 cells. The time step used

was st 1=∆ .

It should be emphasised that two different porosities must be distinguished for the sake of the

CFD-simulations (s. section 2.2). On the one hand, the interstitial porosity that defines the

hydrodynamic behaviour of the column is given by equation 2.36. In order to compute the

pressure and velocity profiles correctly, the interstitial porosity was used for setting up the

StarCD-model.

On the other hand, the external porosity is not the characteristic porosity of potassium iodide

for the reason that the sizes of the tracer ions are too small to be effectively excluded from the

internal pore system of the stationary phase particles. In order to account for the penetration

of the intra-particle pore space by the tracer, the ions within the particles’ pore system were


94

considered to be adsorbed. The resulting adsorption isotherm is linear, the Henry-Coefficient

equals the intra-particle porosity 265.0=SPε .

ccq SP ⋅=⋅= 265.0* ε (5.26)

Within the StarCD environment, the adsorption was implemented as described in section 5.2

and section 5.3 (equilibrium dispersive model), respectively.

5.8.2 Simulation processing

As set forth in section 2.9.5, the calculations were carried out in two successive steps. Firstly,

the velocity and pressure profiles inside the column were computed in a stationary run by

solving the momentum equation for distributed resistance in porous media (equation 2.36) as

well as the continuity equation. For this purpose, the material properties (density, viscosity) of

pure methanol were used.

In a second, transient calculation, the material balance for potassium iodide was solved based

on the velocity data acquired during the stationary run. It should be mentioned that this simu-

lation approach takes concentration independent material properties (density, viscosity) of the

mobile phase for granted. Due to the relatively low concentration of potassium iodide in MP2,

this assumption holds true as an approximation.

5.8.3 Simulation results

The result of the StarCD-simulation in terms of the saturation history of potassium iodide at

the two downstream positions is provided in Figure 5.13. The CFD-simulation results are

compared with the experimental data as well as the curve of the EDM that matched the ex-

perimental results best. For the sake of clarity, only the rears of the bands, where the EDM

failed to account for the tailing observed in the experiments (s. section 4.3.3), are shown.


95

Figure 5.13: Comparison of the simulation results for a 1D-EDM and a 2D-StarCD-Model

with the experimental data at the two downstream positions inside the ID 50mm column. The

superficial velocity is smu SF 5105.8 −⋅= .

It is obvious that the more sophisticated StarCD-model which accounts for the local hydrody-

namic properties of the column provides a superior match of the experimental data than the

more common 1D-EDM. The StarCD-simulation results reflect the band tailing which was

found in the experiments well. Only marginal deviations from the experimental data may be

found.

It should be mentioned that also in these simulations the influence of radial dispersion was

overestimated for the very same reasons illustrated in section 5.6.1. But the significance of the

overestimation is small for the reason that the hydrodynamic distortions due to the existence

of a wall region are strong.

5.8.4 Consequences of the radial column inhomogeneity

In order to illustrate the significance of the effects of column radial heterogeneity on the sepa-

ration performance, the resolution R in between two peaks was determined as a function of


96

the separation factor 12α in case of linear isotherms. The resolution R and the separation fac-

tor 12α are defined by equation (2.14) and equation (2.11), respectively.

The CFD-model described above remained unchanged in terms of permeability and dispersion

coefficients. Only the Henry-coefficients were altered for the single runs to yield different

separation factors. The resolution of a homogenous column that has i) the same mean porosity

as the inhomogeneous column and ii) a constant dispersion coefficient equaling the high effi-

ciency of the column core (s. figure 5.12) is given for comparison. Compared to the inhomo-

geneous column, the efficiency of the reference column is 100% higher

( smD effax

28104.1 −⋅= compared to smD effax

28109.2 −⋅= ).

Figure 5.14: Comparison of the resolution as a function of the separation factor for a) the

radially heterogeneous column as characterised by means of CT (StarCD-model) and b) a

homogenous column having the same hydrodynamic properties as the column core of the in-

homogeneous column.

The choice of a reference column having similar properties as the column core of the inhomo-

geneous column goes back to the concept of the ‘infinite diameter column’ [Knox1969,


97

Knox1976]. Knox et al. suggested to use a column with a column bore wide enough to pre-

vent that a centrally injected sample ever penetrate the less efficient wall region thereby en-

hancing the effective efficiency.

It is evident that the deviations of the packing properties in between the core and the wall of

the investigated column cause a tremendous loss in separation performance. Within the inves-

tigated range of separation factors, the resolution of the inhomogeneous column is reduced by

almost 30% compared to the separation performance that may be achieved by the column core

only (infinite diameter column).

5.9 Consequences of a wall region regarding characteristic parameters

5.9.1 Influence of a column wall region on the column permeability

In order to investigate the influence of the existence of a wall region with properties different

from the column core on standard chromatographic parameters systematically, models con-

sisting of two regions – i) a core region and ii) a wall region – with different properties were

developed [Su2005]. The parameters altered were the porosity difference between the wall

and the core region as well as the geometrical dimensions of the wall region. The column di-

ameter md 1.0= , the column length mL 3.0= and the mean external porosity 4.0=ε of all

columns were kept constant. A constant mean porosity ε of the columns was chosen for the

same reason as for the local inhomogeneities. The formation of inhomogeneities due to rear-

rangement processes of the stationary phase inside the column during the operation would

have no effect on the mean porosity of the column for the reason that the total amount of ad-

sorbent inside the column remains constant.

A sketch of this type of model is provided in figure 5.15.

Figure 5.15: Sketch of the inhomogeneous column models used to investigate the conse-quences of a column wall effect.


98

The existence of a column wall region has a marked effect on the permeability of the columns

investigated by means of simulation studies. For a wall region comprising 30% of the column

radius ( 3.02 =⋅ da ) similar to the results gained from the computed tomography investiga-

tions in section 4.3, the permeability gain was up to 25% (s. table 5.4) in our simulations. It is

important to note that – different from local inhomogeneities – the overall permeability of the

column enhances irrespective of the region that has the higher permeability (usually the core

region). The permeability gain is proportional to the absolute value of the porosity difference

as well as on the dimensions of the wall region.

Table 5.5: Dependency of the column permeability on the porosity difference εε −SP be-

tween the wall region and the column mean; the size of the inhomogeneity spd ; the column

length L ; the position of the inhomogeneity Po . The table cells contain data on the perme-

ability ratio 0κκ between the inhomogeneous and the reference column.

da⋅2

εε −Wall 0.1 0.2 0.3 0.4

-0.08 1,036 1,111 1,243 1,732

-0.04 1,001 1,023 1,055 1,117

0.00

(Reference) 1 1 1 1

0.04 1,010 1,023 1,035 1,041

0.08 1,034 1,129 1,164 1,420

5.9.2 Influence of a column wall region on the first moment

In analogy to the approach used for local inhomogeneities, the effect of a column wall region

on the first moment of the peaks was investigated. Again it resulted in a bands first moment

i,1µ not being effected by the existence of an inhomogeneity.

5.9.3 Influence of a column wall region on the column efficiency

The efficiency of the chromatographic is influenced negatively through the existence of a wall

region with hydrodynamic properties different from the column core. Similar to the findings

for regional inhomogeneities, the degree of the efficiency loss depends on the initial effi-


99

ciency of the column that is e.g. characterised by the dispersion coefficient. This behaviour is

illustrated in figure 5.16. Even though the efficiency loss caused by the wall region is already

clearly pronounced for a column with moderate efficiency ( smDax26105.2 −⋅= ), the effi-

ciency loss associated to a wall region with similar dimensions is still more dramatic for a

column with an increased efficiency ( smDax27105.2 −⋅= ).

Figure 5.16: Dependency of the column efficiency on the difference between the porosity of

the wall region wallε and the mean porosity 0εε = for columns with different dispersion coef-

ficients axD . Extend of wall region: 2.02 =da .

As can be seen in figure 5.16 and from the column data in table 5.6, the drop in terms of col-

umn performance does not depend of the algebraic sign of the porosity difference even though

the ordinate in figure 5.17 is an axis of symmetry by approximation only. In this context is

worth noting that for 3.02112 ≈−=⋅ ad where the cross sections of wall and core are

equal, the efficiency loss is indeed independent of the algebraic sign.


100

The extent ad⋅2 of the wall region has a manifest impact on the performance, larger wall

regions make for less efficient columns.

Table 5.6: Dependency of the column permeability on the porosity difference εε −wall be-

tween the inhomogeneity and the reference column; the size of the inhomogeneity spd ; the

column length L ; the position of the inhomogeneity Po . The table cells contain data on the

permeability ratio 0κκ between the inhomogeneous and the reference column.

da⋅2

0εε −SP 0.1 0.2 0.3 0.4

-0.08 0,059 0,015 0,006 0,003

-0.04 0,154 0,047 0,022 0,011

-0.02 0,394 0,157 0,082 0,046

0.00

(Reference) 1 1 1 1

0.02 0,341 0,147 0,083 0,052

0.04 0,103 0,040 0,022 0,015

0.08 0,024 0,010 0,006 0,005


101

6 Résumé

The goal of this work was to develop methods for the non-invasive evaluation of the packed

bed’s characteristics of preparative chromatographic columns. Two different strategies were

mapped out: i) the investigation of the influence of uneven column packings on the signals of

peripheral sensors and ii) the use of non-invasive tomographic measurement techniques to

gain information about the packing structure. The achievements of this work concerning both

strategies are briefly summarised and discussed in the following sections.

For the sake of computational modeling of chromatographic processes in inhomogeneous

packed beds, the CFD code StarCD was adapted to render the simulation of chromatography

possible. Of particular importance was the development of strategies to implement adsorption

into the coding. Despite some limitations of the StarCD code concerning the correct represen-

tation of anisotropic dispersion in chromatographic beds, the results of the CFD simulation

represented the experimental principally results well and allowed the investigation of effects

that could hardly have been studied in the laboratory.

6.1 Influence on peripheral sensor signals

Four different kinds of column inhomogeneity – local inhomogeneities, a column wall region,

the formation of an inlet void, and the abrasion of particles – were investigated experimentally

as well as by means of computational fluid dynamic simulations. In order to carry out the ex-

perimental investigations, methods to artificially mimic column bed heterogeneity were de-

veloped.

The studies aimed to find patterns in the shift of the response signals of the sensors that may

serve to identify the sort of inhomogeneity in order to learn more about the malfunction of the

chromatographic process. For the sake of simplicity, the sensor signals were not evaluated

directly but converted into common chromatographic parameters. Only three of the most basic

parameters used in chromatography – permeability, retention time, and efficiency – were in-

vestigated within this work. Other, less basic parameters could be consulted in order to

achieve an even more detailed resolution of the fault patterns that was not required to achieve

the ends of this work. The effect of each of the inhomogeneities on these three parameters is

illustrated in table 6.1.


102

Table 6.1: Influence of different kinds of column inhomogeneities on chromatographic pa-rameters. ↑ = increase; → = constant; ↓ = decrease

permeability

κ

retention time

1µ

efficiency

N

local inhomogeneities → → ↓

wall effect ↑ → ↓

inlet void ↓ → ↓

particle abrasion ↓ → →

It is obvious that on the one hand the occurrence of any kind of packed bed inhomogeneity

results in a typical mutation in the combination of the i) column efficiency and ii) column

permeability. A qualitative illustration of the mutations that form fault indicating patterns

[Freyermuth1994] is given in figure 6.1. On the other hand, the first moment of the bands is

not so much affected by the kinds of faults associated to inhomogeneities of the packing that

were studied in this work. As a parameter that is independent of the packing structure and

closely related to the thermodynamics of the process it is revealed to be useful as a discrimi-

nator for the chemical degradation of the stationary phase surface to aging, fouling, or non-

inert mobile phase additives.

Beside the patterns sketched in figure 6.1, two of the findings regarding the effects of inho-

mogeneities are worth special attention. Firstly, the results indicate that the position of locally

confined inhomogeneities are almost free of influence on the efficiency. This is in good

agreement with statements from Lenz [Lenz2003] and alleviates the difficulties in identifying

the sort of irregularity. Secondly, it was found that the extent of the efficiency loss depends on

the original efficiency of the column. Highly efficient columns are especially prone to effi-

ciency losses due to the formation of uneven packing structures.

In preparative chromatographic practice, two different ways how to identify the sort of inho-

mogeneity are conceivable. The first, less sophisticated way relies test runs under defined

conditions in terms of i) a mobile phase with a well known viscosity (preferably a pure sol-

vent) in order to determine the permeability of the process and ii) the injection of a tracer un-

der linear isotherm conditions for estimating the efficiency. Substances already present in the

process should be favoured not to violate the requirements of good manufacturing practice


103

[EU1999]. Test runs could either be carried out in regular intervals in order to recognise the

development of inhomogeneities in good time or post to the occurrence of a process malfunc-

tion to speed up the cause study.

Figure 6.1: Qualitative sketch of the shift in permeability and efficiency due to the existence of a column inhomogeneity

A second way rests upon model based parameter estimation [Deibert1994]. For the purposes

of this work, the parameters to be estimated are the coefficients – e.g. in the equilibrium dis-

persive model of chromatography – and again the permeability. The effective dispersion coef-

ficient is directly coupled to the efficiency. The isotherm parameters (e.g. Henry or Langmuir-

coefficients) have to be estimated to achieve a complete set even though the thermodynamic

quantities do not yield a benefit concerning the identification of packing irregularities as men-

tioned above. Far from it, non-linear isotherms complicate the identification for the reason

that methods for the identification of linear systems can no longer be applied. An example for

the parameter estimation for chromatographic processes is e.g. given by Klatt et al.

[Klatt2000].


104

6.2 Tomographic measurement techniques

Within the second approach the feasibility of non-invasive tomographic measurement tech-

niques to yield information about the packing structure of preparative chromatographic col-

umns was studied.

X-ray computed tomography and nuclear magnetic resonance imaging were used to observe

the breakthrough behaviour and the velocity distribution inside preparative columns made of

glass in situ. Both measurement techniques proved themselves to be well suited tools for the

non-invasive monitoring of columns which are opaque for the respective technique. For x-ray

computed tomography, local column properties like the permeability and the efficiency could

be derived from the intra-column breakthrough curves.

Although caution should be exercised regarding the interpretation of results based on a limited

number of columns, either non-invasive measurement technique independently furnished

proof that the properties of the chromatographic packing are not evenly distributed within the

column. The packed beds under investigation offered a denser and less efficient wall region

compared to the column core. This is in good agreement with earlier findings [Guiochon1997]

providing additional evidence that the porous bed in slurry packed columns is heterogeneous

in the radial direction.

The local column parameters derived from the computed tomography experiments were suc-

cessfully implemented in a two dimensional computational fluid dynamics model of a chro-

matographic column. The band shapes predicted with the rigorous two dimensional model

matched the laboratory results significantly better than the plots calculated from a common

one dimensional model. The results suggest that the radial dimension should be accounted for

while modeling preparative chromatographic processes e.g. for the sake of scale-up.

Despite the benefit for this work the results of the non-invasive measurement techniques offer,

an insight into the packing structures of the columns on the scale of the particle diameter

could not be achieved.

Consequently, the analysis of the chromatographic bed’s configuration by means of high reso-

lution imaging techniques remains a worthy subject for future research. In combination with

Lattice-Boltzmann methods as a modeling tool for the direct simulation of Newtonian flow in


105

complex structures, it should be beneficial to understand the fundamental processes contribut-

ing to band broadening in chromatography in sharper detail.


106

List of symbols

symbol SI-Unit meaning

A [m] constant in the Van Deemter curve

CA [m²] column cross section

1,2α [-] separation factor

ib [m³/kg] Langmuir parameter

B [m²/s] constant in the Van Deemter curve

0B [T] magnetic field strength

ic [kg/m³] mobile phase concentration of species i

C [s/m] constant in the Van Deemter curve

0C [kg/m³] peak area for a dimensionless time plot

Co [-] Courant number

CT [-] CT-number, dimensionless attenuation coefficient (x-ray)

Cd [m] column diameter

Pd [m] particle diameter

SPd [m] diameter of spherical column inhomogeneities

2,1d [m] Sauter mean diameter

axD [m²/s] axial dispersion coefficient

appaxD [m²/s] apparent axial dispersion coefficient

mD [m²/s] molecular diffusion coefficient

Di [-] Dispersion number


107


E [J] energy

ε [-] porosity

extε [-] external or interstitial porosity

intε [-] intra particle or internal porosity

totε [-] total porosity

F [-] phase ratio

Φ [rad] phase difference

g [T/m] magnetic field gradient

γ [-] magnetogyric ratio

1γ [-] packing property (related to tortuosity)

2γ [-] geometrical constant characterizing the packing

h [J*s] Planck’s constant; Jsh 34106261.6 −⋅=

Kh [-] Kozeny coefficient

iH [-] Henry coefficient of species i

iHETP [m] height equivalent to a theoretical plate (for species i)

η [Pa*s] dynamic viscosity

I [cd] intensity of x-ray light

j [kg/m²s] mass flux

dispj [kg/m²s] mass flux due to dispersion


108


k [J/K] Boltzmann constant; KJk 23103807.1 −⋅=

'ik [-] capacity factor of component i

imk . [-] mass transfer coefficient

κ [m²] permeability

CL [m] column length

L [m] length of column inlet void

m [kg] mass

accim , [kg] accumulated mass of species i

MPaccim , [kg] mass of species i accumulated in the mobile phase

SPaccim , [kg] mass of species i accumulated in the stationary phase

convm& [kg/s] mass flow due to convection

µ [1/m] local linear attenuation coefficient (x-ray)

OH 2µ [1/m] local linear attenuation coefficient of water (x-ray)

i,1µ [s] first moment of the peak of component i

iN [-] number of plates or efficiency of a column (for species i)

P [Pa] pressure

Pe [-] Peclet number

cellPe [-] cell Peclet number

ZPe [-] local Peclet number


109


iq [kg/m³] stationary phase concentration of species i

*iq [kg/m³] equilibrium stationary phase concentration of species i

*max,iq [kg/m³] stationary phase loadability of species i

R [m] column radius

R [-] peak resolution

Re [-] Reynolds number

S [-] mobile phase saturation/ volume fraction

PS1 [kg/m³*s] constant in the reaction source term in StarCD

PS2 [kg/m³*s] constant in the reaction source term in StarCD

S& [kg/m³*s] reaction source term in StarCD

2iσ [s²] variance of the peak of species i

t [s] time

0t [s] retention time of a non retained tracer/ column dead time

iRt , [s] retention time of species i

T [K] temperature

1T [s] characteristic time of energy relaxation (NMR)

2T [s] characteristic time of the decay of phase coherence (NMR)

iT [-] tailing factor of the peak of species i


110


TE [s] echo time (NMR)

τ [-] dimensionless time

u [m/s] velocity

KIu [m/s] tracer velocity of potassium iodide

SFu [m/s] superficial velocity

( )icu [m/s] concentration velocity

( )icu ∆ [m/s] velocity of a concentration shock

CV [m³] column volume

extV [m³] external or interstitial volume

intV [m³] intra particle pore volume

PV [m³] particle volume

V& [m³/s] volumetric flow

iw [-] mass fraction

0ω [1/s] Larmour frequency

iω [s] peak width of species i

1.0,iω [s] peak width of species i at 10% peak height

5.0,iω [s] peak width of species i at 50% peak height

z [m] axial coordinate of a column


111

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123

Appendix I

Parameters of the EDM fitted to the intra column breakthrough curves measured by x-ray

computed tomography. Results e.g. shown in figure 4.12.

ID 26 ID 50

position Rzt [s] zPe R

zt [s] zPe

05.0*1 =z 95 131 255 252

35.0*1 =z 390 365 900 518

65.0*1 =z 687 337 1544 614

95.0*1 =z 979 345 2180 1082

Appendix II

Core velocities for different axial zones used to normalize data in figure 4.14 and 4.15, re-

spectively. Core velocities are calculated based on the intra column retention times of the re-

spective annulus segments as determined from the x-ray computed tomography experiments.

( )coreuKI [m/s]

zone ID 26 ID 50

35.0...05.0*1 =z 0.000256 0.000173

65.0...35.0*1 =z 0.000264 0.000167

95.0...65.0*1 =z 0.000263 0.000173


124

Appendix III

Column overall HETP values as derived from the x-ray computed tomography experiments.

ID 26 ID 50

overall HETP [µm] 740 352


125

Appendix IV

C*************************************************************************

SUBROUTINE SORSCA(S1P,S2P)

C Source-term for scalar species

C*************************************************************************

C STAR VERSION 3.20.000

C*************************************************************************

INCLUDE 'comdb.inc'

COMMON/USR001/INTFLG(100)

INCLUDE 'usrdat.inc'

DIMENSION SCALAR(50)

EQUIVALENCE( UDAT12(001), ICTID )

EQUIVALENCE( UDAT03(001), CON )

EQUIVALENCE( UDAT03(002), TAU )

EQUIVALENCE( UDAT03(009), DUDX )

EQUIVALENCE( UDAT03(010), DVDX )

EQUIVALENCE( UDAT03(011), DWDX )

EQUIVALENCE( UDAT03(012), DUDY )

EQUIVALENCE( UDAT03(013), DVDY )

EQUIVALENCE( UDAT03(014), DWDY )

EQUIVALENCE( UDAT03(015), DUDZ )

EQUIVALENCE( UDAT03(016), DVDZ )

EQUIVALENCE( UDAT03(017), DWDZ )

EQUIVALENCE( UDAT03(019), VOLP )

EQUIVALENCE( UDAT04(001), CP )

EQUIVALENCE( UDAT04(002), DEN )

EQUIVALENCE( UDAT04(003), ED )

EQUIVALENCE( UDAT04(004), HP )

EQUIVALENCE( UDAT04(006), P )

EQUIVALENCE( UDAT04(008), TE )

EQUIVALENCE( UDAT04(009), SCALAR(01) )

EQUIVALENCE( UDAT04(059), U )

EQUIVALENCE( UDAT04(060), V )

EQUIVALENCE( UDAT04(061), W )

EQUIVALENCE( UDAT04(062), VISM )

EQUIVALENCE( UDAT04(063), VIST )

EQUIVALENCE( UDAT04(007), T )

EQUIVALENCE( UDAT04(067), X )

EQUIVALENCE( UDAT04(068), Y )

EQUIVALENCE( UDAT04(069), Z )

EQUIVALENCE( UDAT09(001), IS )

C-------------------------------------------------------------------------

CC USER CODING FROM HERE ON!

CC Userfile defines a source term in the transport equation

CC The source term represents accumulation in the stationary phase

C-------------------------------------------------------------------------

CC (Definition of Porosity)

Porosity=0.4

CC (If IS equals the Scalar02 representing MOBILE PHASE Concentration)

CC (SCALAR03: Old SP-Concentration)

CC (SCALAR04: New SP-Concentration)

IF(IS.EQ.2.) THEN

S1P=-(SCALAR(04)-SCALAR(03))*DEN*(1-Porosity)/DT

S2P=0

ENDIF

C-------------------------------------------------------------------------

RETURN

END


126

Appendix V

C*************************************************************************

SUBROUTINE POSDAT(KEY,VOL,U,TE,ED,T,P,VIST,DEN,CP,VISM,CON,

* F,ICLMAP,ICTID,RESOR,VF,FORCB,IRN,PREFM,LEVEL)

C Post-process data

C*************************************************************************

C--------------------------------------------------------------------------

*

C STAR RELEASE 3.150

*

C--------------------------------------------------------------------------

*

INCLUDE 'comdb.inc'


DIMENSION KEY(-NBMAXU:NCTMXU),VOL(NCTMXU),U(3,-NBMAXU:NCMAXU),

* TE(-NBMAXU:NCMAXU),ED(-NBMAXU:NCMAXU),T(-NBMAXU:NCTMXU,1+NSCU),

* P(-NBMAXU:NCMAXU),VIST(-NBMAXU:NCMAXU),DEN(-NBMAXU:NCTMXU),

* CP(-NBMAXU:NCTMXU),VISM(-NBMXVU:NCMXVU),CON(-NBMXCU:NCMXCU),

* F(3,-NBMAXU:NCMAXU),ICLMAP(NCTMXU),ICTID(NCTMXU),

* RESOR(63,-100:100),VF(NCDMXU),

* FORCB(3,NWLMX),IRN(NWLMX)

DOUBLE PRECISION P

DIMENSION PREFM(4)


C-------------------------------------------------------------------------

CC USERCODING FROM HERE ON!

CC USERFILE will

CC A) create output files containing the chromatogram & Pressure drop

CC B) define Scalars according to the equilibrium model of chromatography

C-------------------------------------------------------------------------

CC*************************************************************************

CC******************* PART A: Output files ********************************

CC*************************************************************************

CC Definition of 2 vectors with cell-numbers of all inlet/outlet cells

CC MONI1: Inlet; MONI2 Outlet

DIMENSION MONI1(1)

DIMENSION MONI2(1)

DATA MONI1/1/

DATA MONI2/1000/

CC LEVEL=2: At the end of the iteration/time step

IF(LEVEL.EQ.2) THEN

CC INTFLG(1)=0: FIRST ITERATION

IF(INTFLG(1).EQ.0) THEN

CC Create & Open Outputfiles

OPEN(86,FILE='Chromatogram',FORM='FORMATTED', STATUS='UNKNOWN')

OPEN(87,FILE='PressDrop',FORM='FORMATTED', STATUS='UNKNOWN')

CC INTFLG(1)=1: Files exist and are open

INTFLG(1)=1

ENDIF

CC (Definition of "Help-Variables ZOUT")

CC (ZOUT1&2: Chromatogram --> determination of mean concentration)


127

CC (ZOUT3: Pressure Drop)

ZOUT1=0

ZOUT2=0

ZOUT3=0

CC (Summation over all inlet/outlet cells)

DO 200 I=1,1

ZOUT1=ZOUT1+(T(MONI2(I),3)*U(3,MONI2(I)))

ZOUT2=ZOUT2+U(3,MONI2(I))

ZOUT3=ZOUT3+((P(MONI1(I)))-(P(MONI2(I))))

200 CONTINUE

CC Determination of mean concentration/pressure drop

ZOUT1=ZOUT1/ZOUT2

ZOUT3=ZOUT3/1

CC Write mean concentration & pressure drop to file

WRITE(86,801) ITER,ZOUT1


CC**********************************************************************

CC************** PART B: Equilibrium Dispersive Model ******************

CC**********************************************************************

CC Definition of Henry constant

Henry=0.2

CC Do for all cells

DO 300 I=1,1000

CC T(I,2)=Scalar01 not used in EDM

T(I,2)=0.0

CC T(I,4)=Scalar03=Old SP concentration

T(I,4)=T(I,5)

CC T(I,5)=Scalar04=New SP concentration

T(I,5)=T(I,3)*Henry

300 CONTINUE

C***********************************************************************

C ******************Write formats below*********************************

C-**********************************************************************

800 FORMAT(I5,1x,1(E10.4,1x))

801 FORMAT(I5,1x,1E10.4)

ENDIF

RETURN

END


128

Appendix VI

C*************************************************************************

SUBROUTINE POSDAT(KEY,VOL,U,TE,ED,T,P,VIST,DEN,CP,VISM,CON,

* F,ICLMAP,ICTID,RESOR,VF,FORCB,IRN,PREFM,LEVEL)

C Post-process data

C*************************************************************************

C--------------------------------------------------------------------------

*

C STAR RELEASE 3.150

*

C--------------------------------------------------------------------------

*

INCLUDE 'comdb.inc'


DIMENSION KEY(-NBMAXU:NCTMXU),VOL(NCTMXU),U(3,-NBMAXU:NCMAXU),

* TE(-NBMAXU:NCMAXU),ED(-NBMAXU:NCMAXU),T(-NBMAXU:NCTMXU,1+NSCU),

* P(-NBMAXU:NCMAXU),VIST(-NBMAXU:NCMAXU),DEN(-NBMAXU:NCTMXU),

* CP(-NBMAXU:NCTMXU),VISM(-NBMXVU:NCMXVU),CON(-NBMXCU:NCMXCU),

* F(3,-NBMAXU:NCMAXU),ICLMAP(NCTMXU),ICTID(NCTMXU),

* RESOR(63,-100:100),VF(NCDMXU),

* FORCB(3,NWLMX),IRN(NWLMX)

DOUBLE PRECISION P

DIMENSION PREFM(4)


C-------------------------------------------------------------------------

CC USERCODING FROM HERE ON!

CC USERFILE will

CC A) create output files containing the chromatogram & Pressure drop

CC B) define Scalars according to the Linear Driving Force Model (LDFM)

C-------------------------------------------------------------------------

CC*************************************************************************

CC******************* PART A: Output files ********************************

CC*************************************************************************

CC Definition of 2 vectors with cell-numbers of all inlet/outlet cells

CC MONI1: Inlet; MONI2 Outlet

DIMENSION MONI1(1)

DIMENSION MONI2(1)

DATA MONI1/1/

DATA MONI2/1000/

CC LEVEL=2: At the end of the iteration/time step

IF(LEVEL.EQ.2) THEN

CC INTFLG(1)=0: FIRST ITERATION

IF(INTFLG(1).EQ.0) THEN

CC Create & Open Outputfiles

OPEN(86,FILE='Chromatogram',FORM='FORMATTED', STATUS='UNKNOWN')

OPEN(87,FILE='PressDrop',FORM='FORMATTED', STATUS='UNKNOWN')

CC INTFLG(1)=1: Files exist and are open

INTFLG(1)=1

ENDIF

CC (Definition of "Help-Variables ZOUT")

CC (ZOUT1&2: Chromatogram --> determination of mean concentration)


129

CC (ZOUT3: Pressure Drop)

ZOUT1=0

ZOUT2=0

ZOUT3=0

CC (Summation over all inlet/outlet cells)

DO 200 I=1,1

ZOUT1=ZOUT1+(T(MONI2(I),3)*U(3,MONI2(I)))

ZOUT2=ZOUT2+U(3,MONI2(I))

ZOUT3=ZOUT3+((P(MONI1(I)))-(P(MONI2(I))))

200 CONTINUE

CC Determination of mean concentration/pressure drop

ZOUT1=ZOUT1/ZOUT2

ZOUT3=ZOUT3/1

CC Write mean concentration & pressure drop to file



CC**********************************************************************

CC************** PART B: Equilibrium Dispersive Model ******************

CC**********************************************************************

CC Definition of Henry constant

Henry=0.2

CC Definition of Mass transfer coefficient

Coeff=0.5

CC Do for all cells

DO 300 I=1,1000

T(I,2)=Scalar01=Old MP concentration

T(I,2)=T(I,3)

CC T(I,4)=Scalar03=Old SP concentration

T(I,4)=T(I,5)

CC T(I,5)=Scalar04=New SP concentration

T(I,5)=Coeff*DT*(Henry*T(I,2)-T(I,4))+T(I,4)

300 CONTINUE

C***********************************************************************

C ******************Write formats below*********************************

C-**********************************************************************

800 FORMAT(I5,1x,1(E10.4,1x))

801 FORMAT(I5,1x,1E10.4)

ENDIF

RETURN

END

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Investigation of inhomogeneity in prepa

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