Microstructured Devices for Chemical Processing [1 ed.] 352733128X, 9783527331284

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Madhvanand N. Kashid, Albert Renken, and Lioubov Kiwi-Minsker Microstructured Devices for Chemical Processing

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Madhvanand N. Kashid, Albert Renken, and Lioubov Kiwi-Minsker

Microstructured Devices for Chemical Processing

The Authors Dr. Madhvanand N. Kashid

Ecole Polytechnique Fédérale de Lausanne EPFL-SB-ISIC-GGRC 1015 Lausanne Switzerland

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

and Library of Congress Card No.: applied for

Syngenta Crop Protection Monthey SA Route de l’Ile au Bois 1870 Monthey Switzerland

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

Prof. Dr. Albert Renken

Ecole Polytechnique Fédérale de Lausanne EPFL-SB ISIC-LGRC, Station 6 1015 Lausanne Switzerland Prof. Dr. Lioubov Kiwi-Minsker

Ecole Polytechnique Fédérale EPFL-SB ISIC-LGRC, Sation 6 1015 Lausanne Switzerland

Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at . © 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Print ISBN: 978-3-527-33128-4 ePDF ISBN: 978-3-527-68519-6 ePub ISBN: 978-3-527-68518-9 Mobi ISBN: 978-3-527-68523-3 oBook ISBN: 978-3-527-68522-6 Cover Design Formgeber, Mannheim,

Germany Typesetting Laserwords Private Limited,

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Printed on acid-free paper

V

Contents Preface XI List of Symbols 1

1.1 1.2 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.4 1.5 1.5.1 1.5.2 1.6 1.7

2

2.1 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.4

XIII

Overview of Micro Reaction Engineering 1 Introduction 1 What are Microstructured Devices? 2 Advantages of Microstructured Devices 2 Enhancement of Transfer Rates 2 Enhanced Process Safety 5 Novel Operating Window 7 Numbering-Up Instead of Scale-Up 7 Materials and Methods for Fabrication of Microstructured Devices 9 Applications of Microstructured Devices 10 Microstructured Reactors as Research Tool 11 Industrial/Commercial Applications 11 Structure of the Book 13 Summary 13 References 14 Basis of Chemical Reactor Design and Engineering 19 Mass and Energy Balance 19 Formal Kinetics of Homogenous Reactions 21 Formal Kinetics of Single Homogenous Reactions 22 Formal Kinetics of Multiple Homogenous Reactions 24 Reaction Mechanism 25 Homogenous Catalytic Reactions 26 Ideal Reactors and Their Design Equations 29 Performance Parameters 29 Batch Wise-Operated Stirred Tank Reactor (BSTR) 30 Continuous Stirred Tank Reactor (CSTR) 35 Plug Flow or Ideal Tubular Reactor (PFR) 39 Homogenous Catalytic Reactions in Biphasic Systems 45

VI

Contents

2.5 2.5.1 2.5.1.1 2.5.1.2 2.5.2 2.6 2.6.1 2.6.1.1 2.6.2 2.6.2.1 2.6.2.2 2.6.2.3 2.6.2.4 2.6.2.5 2.6.2.6 2.6.3 2.7 2.8

3

3.1 3.2 3.2.1 3.2.2 3.3 3.3.1 3.3.2 3.3.3 3.4 3.4.1 3.4.2 3.4.2.1 3.4.2.2 3.4.3 3.5 3.5.1 3.5.2 3.5.2.1 3.5.2.2 3.6 3.6.1 3.6.2 3.6.3

Heterogenous Catalytic Reactions 49 Rate Equations for Intrinsic Surface Reactions 50 The Langmuir Adsorption Isotherms 51 Basic Kinetic Models of Catalytic Heterogenous Reactions 53 Deactivation of Heterogenous Catalysts 57 Mass and Heat Transfer Effects on Heterogenous Catalytic Reactions 59 External Mass and Heat Transfer 60 Isothermal Pellet 60 Internal Mass and Heat Transfer 69 Isothermal Pellet 69 Nonisothermal Pellet 77 Combination of External and Internal Transfer Resistances 79 Internal and External Mass Transport in Isothermal Pellets 79 The Temperature Dependence of the Effective Reaction Rate 81 External and Internal Temperature Gradient 82 Criteria for the Estimation of Transport Effects 83 Summary 84 List of Symbols 86 References 87 89 Nonideal Flow Pattern and Definition of RTD 89 Experimental Determination of RTD in Flow Reactors 91 Step Function Stimulus-Response Method 92 Pulse Function Stimulus-Response Method 93 RTD in Ideal Homogenous Reactors 95 Ideal Plug Flow Reactor 95 Ideal Continuously Operated Stirred Tank Reactor (CSTR) 95 Cascade of Ideal CSTR 96 RTD in Nonideal Homogeneous Reactors 98 Laminar Flow Tubular Reactors 98 RTD Models for Real Reactors 100 Tanks in Series Model 100 Dispersion Model 101 Estimation of RTD in Tubular Reactors 105 Influence of RTD on the Reactor Performance 107 Performance Estimation Based on Measured RTD 108 Performance Estimation Based on RTD Models 110 Dispersion Model 111 Tanks in Series Model 112 RTD in Microchannel Reactors 115 RTD of Gas Flow in Microchannels 117 RTD of Liquid Flow in Microchannels 118 RTD of Multiphase Flow in Microchannels 122

Real Reactors and Residence Time Distribution (RTD)

Contents

3.7

List of Symbols 126 References 127

4

4.1 4.2 4.3 4.4 4.4.1 4.4.1.1 4.4.1.2 4.4.1.3 4.4.1.4 4.4.1.5 4.4.1.6 4.4.1.7 4.4.1.8 4.4.2 4.4.2.1 4.4.2.2 4.4.2.3 4.4.2.4 4.4.2.5 4.4.2.6 4.4.2.7 4.5 4.5.1 4.5.2 4.5.2.1 4.6 4.7 4.8

Micromixing Devices 129 Role of Mixing for the Performance of Chemical Reactors 129 Flow Pattern and Mixing in Microchannel Reactors 136 Theory of Mixing in Microchannels with Laminar Flow 137 Types of Micromixers and Mixing Principles 143 Passive Micromixer 144 Single-Channel Micromixers 144 Multilamination Mixers 146 Split-and-Recombine (SAR) Flow Configurations 148 Mixers with Structured Internals 149 Chaotic Mixing 149 Colliding Jet Configurations 150 Moving Droplet Mixers 151 Miscellaneous Flow Configurations 153 Active Micromixers 154 Pressure Induced Disturbances 154 Elektrokinetic Instability 155 Electrowetting-Induced Droplet Shaking 156 Ultrasound/Piezoelectric Membrane Action 156 Acoustic Fluid Shaking 157 Microstirrers 157 Miscellaneous Active Micromixers 158 Experimental Characterization of Mixing Efficiency 158 Physical Methods 158 Chemical Methods 159 Competitive Chemical Reactions 159 Mixer Efficiency and Energy Consumption 171 Summary 172 List of Symbols 173 References 173

5

Heat Management by Microdevices

5.1 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.3 5.3.1 5.3.2

179 Introduction 179 Heat Transfer in Microstructured Devices 181 Straight Microchannels 181 Curved Channel Geometry 189 Complex Channel Geometries 191 Multichannel Micro Heat Exchanger 191 Microchannels with Two Phase Flow 193 Temperature Control in Chemical Microstructured Reactors Axial Temperature Profiles in Microchannel Reactors 197 Parametric Sensitivity 201

195

VII

VIII

Contents

5.3.3 5.3.3.1 5.3.3.2 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.5 5.6

Multi-injection Microstructured Reactors 212 Mass and Energy Balance in Multi-injection Microstructured Reactors 213 Reduction of Hot Spot in Multi-injection Reactors 218 Case Studies 221 Synthesis of 1,3-Dimethylimidazolium-Triflate 221 Nitration of Dialkyl-Substituted Thioureas 222 Reduction of Methyl Butyrate 223 Reactions with Grignard Reagent in Multi-injection Reactor 224 Summary 226 List of Symbols 226 References 228

6.1 6.2 6.3 6.3.1 6.3.2 6.3.3 6.4 6.5 6.5.1 6.5.2 6.5.3 6.5.4 6.6 6.6.1 6.6.2 6.6.3 6.6.4 6.6.5 6.6.6 6.7 6.8

Microstructured Reactors for Fluid–Solid Systems 231 Introduction 231 Microstructured Reactors for Fluid–Solid Reactions 232 Microstructured Reactors for Catalytic Gas-Phase Reactions 233 Randomly Micro Packed Beds 233 Structured Catalytic Micro-Beds 235 Catalytic Wall Microstructured Reactors 238 Hydrodynamics in Fluid–Solid Microstructured Reactors 239 Mass Transfer in Catalytic Microstructured Reactors 243 Randomly Packed Bed Catalytic Microstructured Reactors 244 Catalytic Foam Microstructured Reactors 245 Catalytic Wall Microstructured Reactors 246 Choice of Catalytic Microstructured Reactors 253 Case Studies 255 Catalytic Partial Oxidations 255 Selective (De)Hydrogenations 257 Catalytic Dehydration 259 Ethylene Oxide Synthesis 259 Steam Reforming 260 Fischer–Tropsch Synthesis 261 Summary 261 List of Symbols 262 References 262

7

Microstructured Reactors for Fluid–Fluid Reactions

6

7.1 7.2 7.2.1 7.2.2 7.2.2.1 7.2.2.2

267 Conventional Equipment for Fluid–Fluid Systems 267 Microstructured Devices for Fluid–Fluid Systems 268 Micromixers 269 Microchannels 271 Microchannels with Inlet T, Y, and Concentric Contactor 271 Microchannels with Partial Two-Fluid Contact 271

Contents

7.2.2.3 7.2.2.4 7.2.2.5 7.2.3 7.3 7.3.1 7.3.1.1 7.3.1.2 7.3.1.3 7.3.1.4 7.3.1.5 7.3.2 7.3.2.1 7.3.2.2 7.3.2.3 7.3.2.4 7.3.2.5 7.3.2.6 7.3.2.7 7.4 7.4.1 7.4.2 7.4.3 7.4.3.1 7.4.3.2 7.4.3.3 7.4.4 7.4.4.1 7.4.4.2 7.4.4.3 7.4.4.4 7.4.5 7.5 7.5.1 7.5.2 7.5.2.1 7.5.2.2 7.5.2.3 7.6 7.6.1 7.6.2 7.6.2.1 7.6.2.2

Microchannels with Mesh or Sieve-Like Interfacial Support Contactors 271 Microchannels with Static Mixers 272 Parallel Microchannels with Internal Redispersion Units 272 Microstructured Falling Film Reactor for Gas–Liquid Reactions 272 Flow Patterns in Fluid–Fluid Systems 273 Gas–Liquid Flow Patterns 273 Bubbly Flow 273 Taylor Flow 274 Slug Bubbly Flow 279 Churn Flow 279 Annular and Parallel Flow 280 Liquid–Liquid Flow Patterns 280 Drop Flow 281 Slug Flow 281 Slug-Drop Flow 282 Deformed Interface Flow 282 Annular and Parallel Flow 283 Slug-Dispersed Flow 283 Dispersed Flow 283 Mass Transfer 284 Mass Transfer Models 285 Characterization of Mass Transfer in Fluid–Fluid Systems 286 Mass Transfer in Gas–Liquid Microstructured Devices 287 Mass Transfer in Taylor Flow 287 Mass Transfer in Slug Annular and Churn Flow Regime 292 Mass Transfer in Microstructured Falling Film Reactors 293 Mass Transfer in Liquid–Liquid Microstructured Devices 296 Slug Flow (Taylor Flow) 296 Slug-Drop and Deformed Interface Flow 297 Annular and Parallel Flow 297 Slug-Dispersed and Dispersed Flow 298 Comparison with Conventional Contactors 299 Pressure Drop in Fluid–Fluid Microstructured Channels 300 Pressure Drop in Gas–Liquid Flow 301 Pressure Drop in Liquid–Liquid Flow 304 Pressure Drop – Without Film 304 Pressure Drop – With Film 305 Power Dissipation in Liquid/Liquid Reactors 307 Flow Separation in Liquid–Liquid Microstructured Reactors 307 Conventional Separators 308 Types of Microstructured Separators 308 Geometrical Modifications 309 Wettability Based Flow Splitters 310

IX

X

Contents

7.6.3 7.7 7.7.1 7.7.1.1 7.7.1.2 7.7.2 7.7.2.1 7.7.2.2 7.7.2.3 7.7.2.4 7.7.2.5 7.8 7.9

Conventional Separator Adapted for Microstructured Devices 315 Fluid–Fluid Reactions in Microstructured Devices 315 Examples of Gas–Liquid Reactions 317 Halogenation 317 Nitration, Oxidations, Sulfonation, and Hydrogenation 318 Examples of Liquid–Liquid Reactions 319 Nitration Reaction 319 Transesterification: Biodiesel Production 320 Vitamin Precursor Synthesis 320 Phase Transfer Catalysis (PTC) 321 Enzymatic Reactions 322 Summary 323 List of Symbols 324 References 325

8

Three-Phase Systems 331 Introduction 331 Gas–Liquid–Solid Systems 331 Conventional Gas–Liquid–Solid Reactors 331 Microstructured Gas–Liquid–Solid Reactors 333 Continuous Phase Microstructured Reactors 333 Dispersed Phase Microstructured Reactors 334 Mass Transfer and Chemical Reaction 336 Reaction Examples 341 Gas–Liquid–Liquid Systems 346 Summary 347 List of Symbols 347 References 348

8.1 8.2 8.2.1 8.2.2 8.2.2.1 8.2.2.2 8.2.2.3 8.2.2.4 8.3 8.4 8.5

Index

351

XI

Preface This book is written based on the potential use of microstructured devices in chemical equipment and the intensification of chemical processes. The term “microstructured devices” is coined based on their characteristic dimensions that are in the submillimeter range and on their different types such as mixers, reactors, heat exchangers, and separators. Owing to the small characteristic dimensions, diffusion times are short and the influence of transport phenomena on the rate of chemical reactions is efficiently reduced. Heat transfer is greatly enhanced compared to conventional systems, allowing a strict control of temperature and concentration gradients leading to an improved product yield and selectivity. In addition, safe reactor operation is possible under unconventional conditions such as high reaction temperatures and reactant concentrations. As a consequence, novel process windows can be opened, but not accessible with traditional systems. Therefore, microstructured devices are versatile tools for the development of sustainable chemical processes. This book focuses on reaction engineering aspects, such as design and characterization, for homogeneous and multiphase reactions. On the basis of chemical reaction engineering fundamentals, it addresses the conditions under which these devices are beneficial, how they should be designed, and how such devices can be integrated or applied in a chemical process. Designed as a pedagogical tool with target audience of university students and industrial professionals, it seeks to bring readers with no prior experience of these subjects to the point where they can comfortably enter into the current scientific and technical developments in the area. However, this book does not include the cross-disciplinary subjects such as fabrication techniques of these devices, integration of sensors and actuators, and their use for biological applications. To facilitate comprehension, the topics are developed beginning with fundamentals in chemical reaction engineering with ample cross-referencing. The understanding of concepts is facilitated by clear descriptions of examples, supplied by exercises including solutions, and provided by figures and illustrations.

XII

Preface

Finally, the authors want to highlight the complexity of microreaction engineering in particular. Therefore, this book must be viewed as a tool for stimulation of novel and meaningful solutions for the complex chemical reaction realities. It is also important to note that the growing interests and complementary developments of this subject require periodic updates. Lausanne, Switzerland May 2014

Madhvanand Kashid, Albert Renken, Lioubov Kiwi-Minsker

XIII

List of Symbols

Commonly Used Symbols

This is a list of commonly used symbols. Besides, there are some special symbols used for each chapter which are listed chapterwise. Symbols

Significance

Unit

A a

Exchange or surface area Specific interfacial area or catalytic surface area per reactor volume Cross-section area Bond number Bodenstein number Biot number (mass), Biot number (thermal) Dimensionless concentration Capillary (=) or Carberry (=) number Concentration of molecule Ai Heat capacity of fluid or mixture First Damköhler number Second Damköhler number Second Damköhler number for mixing Axial dispersion coefficient Dean number Effective molecular diffusion coefficient, molecular diffusion coefficient Hydraulic diameter Diameter of channel (or tube) Intrinsic activation energy, apparent activation energy of reaction j Ratio of residual concentration to initial Fourier number Gravitational acceleration Height

m2 m2 m−3

Acs Bo Bo Bim , Bith C Ca ci cp DaI DaII DaII mx Dax De Deff , Dm dh dt E, Ea f Fo g H

m2 — — — — — mol m−3 J kg−1 K−1 — — — m2 s−1 — m2 s−1 m m J mol−1 — — m2 s−1 m (continued overleaf)

XIV

List of Symbols

Symbols

Significance

Unit

h Ha Ji k, kr , kj

Heat transfer coefficient Hatta number Molar flux of species i Reaction rate constant for homogeneous and quasi-homogenous, constant of heterogenous reaction, constant of reaction j Pre-exponential or frequency factor

W m−2 K−1 — mol m−2 s−1 variable (s−1 (mol m−3 )−(n−1) )

k0 KC K kG k GL kL kL a km k ov L, Lc , Le , Lt ṁ Nu ni n ni ṅ i p Pi Pr Pe Q Q̇ ̇ q̇ r , q̇ ex q, R R Re Ri rj , reff rads , rdes Sk, i sk, i Se

Reaction equilibrium constant thermodynamic equilibrium constant Mass transfer coefficient in gas phase Mass transfer coefficient in gas–liquid system Mass transfer coefficient in liquid phase Volumetric mass transfer coefficient Mass transfer coefficient of heterogeneous reactions Overall mass transfer coefficient Length, characteristic length, length of entrance zone, length of tube or channel Mass flow rate Nusselt number Reaction order with respect to species Ai Overall reaction order No of moles of molecule Ai Molar flow rate of molecule Ai Pressure Rate of production Prandtl number Péclet number Energy Rate of heat flow Specific heat rate, of reaction, of heat exchange/transfer Ideal gas law constant Radius Reynolds number Overall reaction/transformation rate of molecule Ai Rate of reaction/transformation of reaction j, effective reaction rate Rates of adsorption, of desorption Selectivity of product k with respect to reactant i Instantaneous selectivity of product k with respect to reactant i Semenov number

variable (s−1 (mol m−3 )−(n−1) ) variable — m s−1 m s−1 m s−1 s−1 m s−1 m s−1 m kg s−1 — — — mol mol s−1 Pa mol s−1 — — J W J m−3 s−1 J mol−1 K−1 m — mol m−3 s−1 mol m−3 s−1 — — — —

List of Symbols

Symbols

Significance

Unit

Sc Sh T, Tb , Ts

Schmidt number Sherwood number Temperature, bulk temperature, surface temperature Time, characteristic cooling time, diffusion time, reaction time, mass transfer time, mixing time, axial dispersion time, axial molecular diffusion time, radial diffusion time Mean residence time Overall heat transfer coefficient Internal energy Overall volumetric heat transfer coefficient Superficial velocity, velocity of gas bubble (slug), velocity at radial position r, superficial flow velocity of gas phase, superficial velocity of liquid phase Volume, internal (reaction) volume Volumetric flow rate Width Rate of work done, by flow, by shaft Conversion Yield of product k with respect to reactant i Dimensionless length Length

— — K

t, tc , tD , tr , tm , tmx , tax , t D, ax , tD, rad

t U Ui Uv u, ub , u(r), uG , uL

V , VR V̇ W Ẇ , Ẇ f , Ẇ s X Y k, i Z z Greek symbols � � �(z) � � �̇ Δ ΔG ΔHr , ΔHa Δp ΔS ΔTad � �p , �bed � � �, �eff , �f , �wall

Thermal diffusivity Prater number Dirac pulse Film thickness, catalytic layer or boundary layer Arrhenius number Shear rate Symbol of difference Gibbs free energy Heat of reaction, heat of adsorption Pressure drop Entropy Adiabatic temperature rise Specific power dissipation Porosity of catalyst pallet, of randomly packed bed Efficiency factor Dimensionless time Thermal conductivity, effective, of fluid, of wall

s

s W m−2 K−1 J W m−3 K−1 m s−1

m3 m3 s−1 m J s−1 — — — m m2 s−1 — — m — s−1 — J mol−1 J mol−1 Pa J mol−1 K−1 K W kg−1 — — — W m−1 K−1 (continued overleaf)

XV

XVI

List of Symbols

Symbols

Significance

Unit

� � �i,j

Dynamic viscosity Kinematic viscosity Stoichiometric coefficient of species i in reaction j Geometric factor Density Interfacial tension Residence time, of plug flow reactor, of reactor, residence time referred to reaction volume

Pa s m2 s−1 —

� � � �, � PFR , �R

— kg m−3 N m−1 s

Common Indices Subscript

0 ∞ app av Ax b c cap cat eff eq ex film gen I II in max min out op ov P s v

Initial value Asymptotic or infinite value Apparent or observed Average Axial Bulk Cooling Hemispherical cap Catalyst Effective Equilibrium External Wall film General Phase I Phase II Inlet Maximum Minimum Outlet Optimum Overall Pallet Surface Volumetric

Superscript

0

Values at standard conditions

List of Symbols

Dimensionless Numbers Dimensionless number

Significance

Definition

Adiabatic temperature rise Arrhenius number

Property of reaction mixture, represent temperature rise in worst case and is independent of reactor type/reaction rate Relative importance of activation temperature (E/R) to system bulk temperature (Tb ) Relates external mass or heat transfer rates at catalyst pallet surface to diffusion or conduction inside the pallet

ΔTad =

Biot number (mass) Biot number (thermal) Bodenstein number Carberry number

Capillary number First Damköhler number Second Damköhler number Second mixing Damköhler number Dean number

Efficiency (reactor) factor (fluid–fluid system) Effectiveness factor (porous catalyst)

�=

(−ΔHr )cb �cp

E RTb

tD tm

Bim =

=

L2c k a De m p Bith = h⋅L �e u⋅L Dax

Ratio of convective transport rate to (axial) diffusion transport rate It gives effective reaction rate over mass transfer rate in catalytic reactions where no internal (pellet) mass and heat transfer resistances are considered Used in fluid–fluid systems. It is ratio of viscous forces to surface tension acting across an interface, that is, interfacial tension Used to set design criteria – ratio of residence time in the reactor to the characteristic reaction time Used to set design criteria – ratio of reaction rate to mass transfer rate

Bo =

Used to set design criteria – ratio of reaction rate to mixing rate

DaIImx =

Used to characterize the flow in curved channels – it is product of Re and square root of channel diameter to curvature radius Ratio of effective reaction rate and the maximal rate referred to the reactor volume corresponding to the maximum concentration in the reacting phase

De = Re

Ratio of effective reaction rate and the rate of reaction at bulk concentration and temperature

�p =

Ca = �ex DaII

Cai =

ub ⋅�i �

DaI =

� tr

DaII =

tm tr

(

�=

tmx tr

dh R′′

)0.5

reff rmax

Jeff = Js De cs ∕L⋅� tanh(�) kr c s L � = tanh �

(continued overleaf)

XVII

XVIII

List of Symbols

Dimensionless number

Significance

Definition

Effectiveness factor (mass transfer) or trade-off index Euler number

Used to access mass transfer performance with energy input

�m =

It is ratio of pressure drop in a given reactor length to kinetic energy. It is ratio of residence time to diffusion time

Eu =

Δp �⋅u2

Fo =

� tD

Fourier number Hatta number

Nusseltnumber Peclet number

Prandtl number Prater number Reynolds number Reynolds number (particle) Reynolds number (foam) Schmidt number Sherwood number (particle) Sherwood number

Used for fluid–fluid systems and signifies whether the reaction takes place in the bulk or near the interface (of reaction phase). It is ratio of reaction rate to interfacial mass transfer rate Use to characterize relative importance of convective heat transfer over conductive heat transfer Ratio of rate of convection to rate of diffusion/dispersion

Used to characterize momentum and heat diffusion – ratio of momentum (viscous) diffusion to molecular diffusion Ratio of maximum temperature difference catalyst center and surface temperature to the surface temperature Most commonly used to characterize the fluid flow – gives relative importance of inertial forces over viscous forces

DaIm = Eu km aR ⋅L �⋅u2s ⋅ Δp us

√ tm Ha = = tr √ kr′ �II D = i,II √ kr′ Di,II

kL,II

h⋅dh �

Nu =

Peax = Peax =

u⋅dt (tube) Dax u⋅dp εbed Dax

(packed bed) � � � = �� = �∕(�c ) p

ΔTmax Ts (−ΔHr )cs De Ts �e

�=

Re =

�udt �

Rep =

(u dp ) �

Refoam =

u⋅ds ⋅� �

� Dm

Used to characterize momentum and mass diffusion – ratio of momentum (viscous) diffusion to molecular diffusion

Sc =

Use to characterize relative importance of convective mass transfer over diffusional mass transfer

Shp =

Sh =

=

d p km Dm

km ⋅dh Dm

List of Symbols

Dimensionless number

Significance

Definition

Thiele modulus

Ratio of characteristic diffusion time in the catalyst and the characteristic reaction time

�2 =

tD tr

Vp

kr c s De

√ k � = L Dr ; first order reaction; �gen = √ (n−1)

Ap

√ Weisz modulus

Bond number First Damköhler number (mass transfer)

Used to measure influence of transport process on reaction kinetics experimentally – ratio of effective reaction rate to (effective) diffusion rate

Relates body forces to surface tension forces Ratio of residence time in the reactor to the characteristic mass transfer time

L2 k De

=

�s2

n+1 2

=

R2sphere

tD tr,eff

⋅

=

cs = De rp,eff 2 �p �s 2 = tD = �gen t ( )2 r,eff Vp n+1 rp,eff Ap 2 De cs �p �2gen �gdh2

BO = � � DaIm = t R = km aR ⋅L u

m

Abbreviations BSTR CSTR CVD LIGA MASI MSR PFR PRL PVD RTD SMF SLPC SCR, SAR, SHR

Batchwise-operated stirred tank reactor Continuously-operated stirred tank reactor Chemical vapor deposition Lithography, galvanization, and molding most abundant surface intermediate Microstructured reactors Plug flow reactor Power rate law Physical vapor deposition Residence time distribution Sintered metal fiber Supported liquid phase catalyst Serpentine channel reactor, split and recombine reactor, staggered herringbone reactor

=

XIX

1

1 Overview of Micro Reaction Engineering This chapter is a comprehensive introduction to the field of micro reaction engineering – an increasingly relevant and rapidly expanding segment of Chemical Reaction Engineering and Process Intensification. Here emphasis is placed on the definition of the term “micro-reactor,” which is often used in various contexts to describe different equipments such as micro-mixers and micro-heat-exchangers. The more well-recognized term is microstructured devices. The advantages and limitations of these microstructured devices are compared to conventional chemical production equipments.

1.1 Introduction

Every industrial process is designed to produce a desired product in the most economical way. The large-scale production of chemicals is mostly carried out using different equipments, such as mixers, reactors, and separators with typical dimensions up to a few meters. The process classification is often referred to as “scale” and depends on the volume and quality of the product. The classifications are bulk chemicals, intermediates, and fine chemicals processes. The bulk chemicals are produced in large quantities in dedicated production units. The intermediate scale products and fine chemicals are produced in the plants mostly dominated by batch processing. Batch reactors are flexible and can be easily shared between multiple products. Therefore, they are considered to be suitable over centuries and there has been no radical change in the batch processing technology. However, in many cases conventional equipment is not sufficiently efficient. In this context, there is a need to develop chemical industries implementing sustainable technology. There are two main approaches to reach this target: chemical and engineering. In the first one, the improvements are achieved by alternative synthesis and processing routes, for example, developing highly selective catalysts and using special reaction media – a typical chemical approach. In the second one, the mass- and heat-transport rates are improved, for example, by increasing the specific interfacial area and thus reducing the diffusion path lengths. This in Microstructured Devices for Chemical Processing, First Edition. Madhvanand N. Kashid, Albert Renken and Lioubov Kiwi-Minsker. © 2015 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2015 by Wiley-VCH Verlag GmbH & Co. KGaA.

2

1 Overview of Micro Reaction Engineering

turn helps to enhance the safety by virtue of the lower hold-up and superior temperature control, even for strongly exothermic reactions. In addition to this, the reactor performance is enhanced operating reactors dynamically [1, 2], and using non-conventional energy sources. This overall development is often referred to as “Process Intensification”, which can be defined in various ways depending on the application involved. However, a generic definition summarising the above discussion is given in the following [3]: “Any chemical engineering development that leads to a substantially smaller, cleaner and more energy efficient technology is Process Intensification!” Micro-technology is one of the powerful tools to attain the goals of process intensification. 1.2 What are Microstructured Devices?

The concept of process intensification using miniaturized equipments was pioneered by Professor Ramshaw and his group at Imperial Chemical Industries (ICI), UK, in the late 1970s, who considered how one might reduce equipment size by several orders of magnitude while keeping the same production rate [3]. The objectives were to reduce cost (smaller equipment, reduced piping, low energy, increased reactivity – higher yields/selectivity, reduced waste, etc.), to enhance safety (low hold-up and controlled reaction conditions), to make a compact size of the plant (much higher production capacity and/or number of products per unit of manufacturing area), and to reduce plant erection time and commissioning time (time to market). These miniaturized systems are the chemical processing systems in three-dimensional structures with internal dimension in submillimeter range. They are referred to as microstructured devices, microstructured reactors, or microreactors, and the research field is referred to as “microreactor” or “microreaction” technology. The advantages and limitations of these devices come from the dimensions increasing greatly the transport processes and the high specific surface area (surface to volume ration). This is described in the following subsection.

1.3 Advantages of Microstructured Devices 1.3.1 Enhancement of Transfer Rates

Let us consider Fourier’s law to describe the influence of transfer scales on heat transfer rates. For simplicity, Fourier’s law for the flux in one-dimensional space can be written as dQ dT = Q̇ = � ⋅ A ⋅ dt dz

(1.1)

1.3

Advantages of Microstructured Devices

where Q is the heat energy (J), � is thermal conductivity (W mK−1 ), and A is the heat transfer surface area (m2 ). The temperature gradient dT∕dz is the driving force for heat transfer. From Equation 1.1, for a given temperature difference, a decrease in the characteristic dimension results in an increase in these gradients and thus in higher heat transfer rates. The same analogy of concentration and momentum gradient could be applied to mass and momentum transfer resulting in higher mass transfer rates. Besides the effect of decreasing linear dimensions on the corresponding gradients, the effective surface area for exchange processes has to be considered. Let us integrate Equation 1.1 for a unit volume of reactor: Q̇ A = q̇ = U ⋅ ⋅ ΔT = U ⋅ a ⋅ ΔT V V

(1.2)

where U (= k∕Δz) is the overall heat transfer coefficient (W⋅m−2 K−1 ) and a is the specific surface area (surface area per unit volume, m2 ⋅m−3 ). For a circular tube, a = 4∕dt , where dt is the tube diameter. Thus, with decreasing characteristic dimensions, the specific surface area of the system increases leading to higher overall performances. The surface to volume ratio for microdevices can be as high as 50 000 m2 m−3 [4]. For comparison, the specific surface area of typical laboratory and production vessels seldom exceed 100 m2 m−3 . Moreover, because of the laminar flow regime within microcapillaries, the internal heat transfer coefficient is inversely proportional to the channel diameter. Therefore, overall heat transfer coefficients up to 25 000 W m−2 K−1 can be obtained, exceeding those of conventional heat exchangers by at least 1 order of magnitude [5]. Indeed, conventional heat exchangers have overall heat transfer coefficients of less than 2000 W m−2 K−1 [6]. Similar performance enhancement could be realized by the miniaturization for mass transfer leading to efficient mixing. For multiphase systems within microdevices, the interfacial surface to volume ratio between the two fluids is notably increased. Indeed, the miniaturized systems possess high interfacial area up to 30 000 m2 m−3 . The traditional bubble columns do not exceed a few 100 m2 m−3 [7]. The characteristic time of chemical reactions, tr , which is defined by intrinsic reaction kinetics, can vary from hours (for slow organic or biological reactions) to milliseconds (for high temperature oxidation reactions) (Figure 1.1). When the reaction is carried out in an eventual reactor, heat and mass transfer interfere with the reaction kinetics. The transfer rates presented above results in the characteristic time of physical processes (heat/mass transfer) in conventional reactors ranging from about 1 to 102 s. This means that relatively slow reactions (tr ≫ 10 s) are carried out in the kinetic regime, and the global performance of the reactor is controlled by the intrinsic reaction kinetics. The chemical reactor is designed and dimensioned to get the required product yield and conversion of the raw material. The attainable reactant conversion in the kinetic regime depends on the ratio of the residence time in the reactor to the characteristic reaction time (tr ).

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1 Overview of Micro Reaction Engineering

10–3

10–2

10–1

100

101

102

Conventional organic synthesis

Many organic reactions

SN2 reactions

Alkylation, nitration

Low-T Grignard addition

Metal/halogen exchange Grignard ketone addition

Characteristic intrinsic reaction time (examples)

Hydrolysis

4

103

104

Char. time, transport phenomena Mixing, heat exchange μ-structured reactors

Mixing, heat transfer conventional reactors

Time scale (s)

Mass and heat transport influenced

Figure 1.1 Time scale of chemical and physical processes [8]. (Adapted with permission from Elsevier.)

Depending on the kinetics and the type of the reactor, the residence time should be several times higher than the characteristic reaction time to get conversions >90% [9, 10]. For fast chemical reactions, the characteristic reaction time is in the same order of magnitude as the characteristic time for the physical processes (Figure 1.1). The performance of a conventional reactor is influenced in this case by mass and/or heat transfer. For very fast reactions, the global transformation rate may be completely controlled by transfer phenomena. As a result, the reactor performance is diminished as compared to the maximal performance attainable in the kinetic regime, and the product yield and selectivity is very often reduced. To avoid mass and heat transfer resistances in practice, the characteristic transfer time should be roughly 1 order of magnitude smaller compared to the characteristic reaction time. As the mass and heat transfer performance in microstructured reactors (MSR) is up to 2 orders of magnitude higher compared to conventional tubular reactors, the reactor performance can be considerably increased leading to the desired intensification of the process. In addition, consecutive reactions can be efficiently suppressed because of a strict control of residence time and narrow residence time distribution (discussed in Chapter 3). Elimination of transport resistances allows the reaction to achieve its chemical potential in the optimal temperature and concentration window. Therefore, fast reactions carried out in MSR show higher product selectivity and yield. The relative heat and mass transfer performance of microstructured reactors with respect to conventional reactors is depicted in Figure 1.2. As can be seen, both in terms of heat and mass transfer, as explained above, microstructured devices offer superior performance. A simplified algorithm for a single step homogenous reaction that could help in choosing conventional and microstructured devices based on kinetics, thermodynamics, and transport rates is presented in Figure 1.3. Here tr , ΔH r, and t mx

1.3

Advantages of Microstructured Devices

Rotating packed bed

Micro-structured reactor

c1 = L

c2 c4

Mass transfer

Static mixer Pulsed column

c3

h

Static mixer/ plate exchanger

b

s2 s1

Spinning disc reactor

Injector Loop reactor

Plate heat exchanger Stirred tank

Heat transfer Figure 1.2 Benchmarking of microstructured reactors. (Adapted from Ref. [11]. Copyright © 2009, John Wiley and Sons.)

are characteristic reaction time, heat of reaction, and characteristic mixing time, respectively. In the case of heterogenous reactions, mixing time would be replaced by characteristic mass transfer time. For a thermodynamically favored reaction, the chemical kinetics could be obtained for different operating conditions such as temperature, pressure, concentrations rendering a reaction rate equation allowing process optimization. As described before, the limiting factor can be the intrinsic kinetics, the thermodynamics, or the heat and mass transfer of the reacting system. The characteristic reaction time of the reaction is then obtained for the operating conditions where the reaction can be operated under temperature control and the product is not decomposed. If the characteristic reaction time is less than 1 s and the heat of reaction is more than −50 kJ mol−1 , the use of microstructured devices is proposed. However, even if the reaction time is high and heat of reaction is relatively low, the microstructured devices could be used to enhance the mixing leading to higher productivity. 1.3.2 Enhanced Process Safety

Process safety is an important issue for chemical industry in general and for exothermic reactions and reactions involving hazardous chemicals in particular. High hold-up of reactants in conventional batch reactors leads to very high impact in the case of accidents. A common approach to handle fast exothermic

5

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1 Overview of Micro Reaction Engineering

A1 + A2 → A3 Chemical kinetics thermodynamics

If (tr < 10s) and/or (–ΔHr > 50 kJ mol–1)

Yes

No

If (tm < 1s)

Yes

No

Microstructured devices

Conventional devices Figure 1.3 An algorithm showing choice of reactor based on reaction kinetics, thermodynamics, and mixing rates for a homogeneous reaction.

reactions is through dilution of the reactants by solvents or using semibatch mode, which is the slow addition of one of the reactants. Microstructured devices are safer than conventional devices because of the small amount of reactants and products inside the reactor. Indeed, in case of failure, the small amount of eventual toxic chemicals released can easily be neutralized [6]. The high heat transfer performance of microdevices allows rapid heating and cooling of the reaction mixture, avoiding hot or cold spots and providing nearly isothermal conditions [5]. Under the predominant laminar regime, the volumetric heat transfer resistance at the reactor microchannel side is proportional to the square of the reactor diameter. In principle, by using the strong dependence of the heat transfer rates on the reactor diameter, any exothermic reaction can be controlled by adjusting the reactor diameter [12].

1.3

Advantages of Microstructured Devices

1.3.3 Novel Operating Window

In the case of slow reactions, the transformation rate is limited by intrinsic kinetics. A drastic increase of the temperature allows exponential acceleration of the reaction rate in agreement with the Arrhenius Law. Moreover, the pressure can be advantageous to accelerate reactions, to shift equilibrium, to increase gas solubility, to enhance conversion and selectivity, to avoid solvent evaporation, and to obtain single-phase processes [8, 13]. The overall transformation rate of such reactions could be significantly increased in these novel operating windows. Using microstructured devices, these reactions could be performed in novel operating windows under more aggressive conditions than in conventional devices. The pressure can easily be increased to several hundred bars because of the small reaction volumes and low mechanical stress. The microdevices allow an easy control of process parameters such as pressure, temperature, and residence time. Thus, an unconventional operating window, that is, high temperature, pressure, and concentrations could be used within microstructured devices even in explosive and thermal runaway regimes [8]. Operating MSR under novel process windows, the key performance parameters can be increased by a few orders of magnitude. A few examples are presented here. In the case of esterification of phthalic anhydride with methanol 53-fold higher reaction rate between 1 and 110 bar for a fixed temperature of 333 K was observed [14]. A multiphase (gas/liquid) explosive reaction of oxidation of cyclohexane under pure oxygen at elevated pressure and temperature (>200 ∘ C and 25 bar) in a transparent silicon/glass MSR increased the productivity fourfold. This reaction under conventional conditions is carried out with air [15]. Another example is for the synthesis of 3-chloro-2-hydroxypropyl pivaloate: a capillary tube of 1/8 in. operated at 533 K and 35 bar, superheated pressurized processing much above the boiling point, allowed to decrease reaction time 5760-fold as compared to standard batch operation [16]. The condensation of o-phenylenediamine with acetic acid to 2-methylbenzimidazole in an MSR is an impressive example of the reduced reaction time from 9 weeks at room temperature to 30 s at 543 K and 130 bar [17]. 1.3.4 Numbering-Up Instead of Scale-Up

Microstructured devices bring in fundamental changes in the approach toward the step from laboratory to industrial scale. Conventionally, the size of the laboratory reactor or flask is upgraded to a few cubic meters to meet the target productivity through different steps including pilot scale studies. This involves cost and time expense scaling up. The numbering-up (also referred to as scale-out) concept consists of an increase in the number of parallel operating units preserving the advantages of MSR, particularly their high surface to volume ratio. This approach

7

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1 Overview of Micro Reaction Engineering

is simpler and faster than the conventional processes (no redesign and pilot plant experiments), thus, decreasing considerably the time between discovery and production and hence shortens the time to market. The break-even point of the cash flow curve could be reached at an earlier point of time, which renders the whole concept more appealing. Moreover, the numbering-up strategy allows to adapt the production to the market demand by increasing or decreasing the number of units as well as an earlier start of production resulting in a lower cost. There are two ways of numbering-up of microstructured devices: internal and external (Figure 1.4). For external numbering-up, multiple identical units are operated in parallel. The advantage is that each single unit is independent of the others and performs as the developed lab-scale unit. However, as each unit will need individual equipment (such as pumps, tubing, flow meters), the costs of external numbering-up are considerable. When numbering-up is carried out internally, the amount of equipment is reduced and thus the cost is lower. The fluids in this case are contacted in a mixing zone and subsequently are distributed into the reaction channels, where conditions are similar to the lab-scale single channel device. The plates or chips fabricated or the standard microtubes that are used as MSR are assembled in two types of geometries: monolith geometry and multiplate geometry. In the former case the inlet stream is distributed simply between all the channels through a large distributor, while in the second case, the inlet stream is first divided into different plates/layers and then distributed into channel plates. The main problem for internal numbering-up to overcome is the equal distribution of fluids to the multiple channels. Equal distribution is indispensable to Mixing zone

Distributors One oulet from each distributor goes to one of the following channel

Microchannels Microchannels

Product collection and analysis

(a)

Product collection and analysis

(b)

Figure 1.4 Schematics of Numbering-up of microstructured reactors: (a) external numbering-up, (b) internal numbering-up [18]. (Adapted with permission from Elsevier.)

1.4

Materials and Methods for Fabrication of Microstructured Devices

9

obtain identical reaction conditions in each channel to ensure high reactor performance and safety. The eventual maldistribution in the microchannels leads to a broad residence time distribution and can even result in the clogging of some of the channels, thus affecting product yield and selectivity [18]. The flow nonuniformities generally occur because of two reasons: a poor reactor design and manufacturing tolerances. The manufacturing tolerances usually cause variations of local temperature in the range of about 5%; the former reason can cause flow ratios in different channels more than a factor of 4 [19]. A further disadvantage of internal numbering-up is the absence of reaction control in the mixing zone. As soon as both reactants are contacted, reaction is initiated and heat is generated. However, heat can be efficiently removed from the channels placed after the distribution section. Possible solutions to overcome the mentioned problems are discussed in chapter 5.

1.4 Materials and Methods for Fabrication of Microstructured Devices

The steps in the selection and use of suitable microstructured devices for a particular chemical production are depicted in Figure 1.5. Majority of steps corresponds to the procedure that is followed for conventional equipments. However, more emphasis is placed on fabrication techniques as it involves structures in micro-, nanometer scale requiring very precise fabrication techniques. In addition, they should be able to accommodate, either individually or combined, these structures and sensors that are required to control the process. Different materials such as metals, glass, polymers, and ceramics are used to fabricate the microstructured devices. Various techniques such as etching, lithography, electroplating, molding, polymer microinjection molding and embossing are applied to make the microscale channels of different cross-sectional geometries (e.g., circular, rectangular, square). Some of the commonly used techniques to fabricate complex structures of microdevices are summarized briefly in the following. 1) The LIGA (German acronym Lithographie, Galvanoformung und Abformung of Lithography, Electroplating, and Molding) process: The LIGA process can be expected to be superior to other methods for fabrication microstructures with high aspect ratios (ratio of height to width) and to produce microstructures of complex shapes [21]. This technique is suitable for a broad range of Chemistry/process, thermodynanics

Material of construction and operating window

Set of geometries through computer simulation/graphics

Choice of geometry based on fabrication and control method

Assembling/ numbering-up and final tests

Figure 1.5 Steps in the selection and use of microstructured devices for chemical production.

Commercial use

10

1 Overview of Micro Reaction Engineering

2)

3)

4)

5)

materials, including metals, alloys, polymers, as well as ceramics and composites to fabricate net-shaped high aspect ratio components. In this technique a layer of photo resisting resin is deposited on a base plate and exposed to high-energy X-rays to the area that is to be etched [22]. The irradiated portions of the photoresist are dissolved and eliminated with a solvent obtaining a three-dimensional structure. The metal is deposited on this structure by electroforming. This metal structure can either be used as it is or it can serve as a mold for making a large number of parts through injection molding or stamping. This technique also uses an inexpensive Ultraviolet (UV) light source, but precision is an issue in this case. Micro-EDM ( Electrical Discharge Machining): This technique uses electric discharge or sparks created between a workpiece and an electrode in dielectric fluid. When the workpiece and the electrode are separated by a small gap, called as spark gap, a pulsed discharge occurs, which removes material from the workpiece through melting and evaporation [23]. It is possible to etch as fine as 50 μm on the surface of the contacting plate using the programmable movement of the electrode [24]. A metal form obtained with another process can also be used as an electrode in order to obtain the desired surface on the substrate. Wet etching: This is a highly selective technique and involves chemical reactions. The metal plate to be etched is first covered with a resist using spincoating or lamination technique. The resist is further structured via different techniques, for example, it is irradiated with UV light through a mask giving a structured resist layer partly covering the metal. Etching of the metal in the uncovered area with subsequent removal of resist gives microstructures [25]. Dry etching: In this technique, an ion beam is directed on the surface to be etched. This process is easy to use, but it is rather expensive [26] and limited to a certain materials (e.g. silicon), which is not always suitable for a given application. Recently a novel manufacturing technique for microstructured reactors was proposed [27]. This technique is based on a cheap and resource-efficient production of structured plates by using roll embossing. The stacked plates are joined by laser welding or vacuum brazing. The method allows manufactoring microstructured reactors in a wide range of throughput, pressure and temperature for homogeneous and multi-phase reactions.

1.5 Applications of Microstructured Devices

Microstructured devices have been successfully used for the continuous processing of fast and highly exothermic reactions [28] and chemical transformations involving toxic, sensitive, and explosive chemicals such as nitration [29], hydrogenations [30, 31], polymerization [32, 33], oxidation [34], halogenations [4], alkylation [35], tetrazole synthesis [36], and reaction of diazomethane [37]. With

1.5 Applications of Microstructured Devices

newly developed online monitoring techniques, MSR can be used as a powerful laboratory tool to investigate the reaction mechanism and kinetics. 1.5.1 Microstructured Reactors as Research Tool

Over the past two decades, MSR have been used in the laboratory as well as in industries. Such activities are regularly reported on various scientific meetings and conferences. One of the most important conferences, IMRET (International Conference on Microreaction Technology), was started in 1997 (Frankfurt, Germany) and is being followed up successful till date. Information on this research area is available in the form of books [7, 38–40]. In addition to the usual update through scientific journal papers, quite a few reviews are published on MSR [4, 38, 41–46]). Microstructured devices are replacing conventional flasks or laboratory reactors as a laboratory tool. Let us take the example of an investigation of reaction mechanism and kinetics of fast and highly exothermic reactions. It is conventionally carried out using the measurements from either continuous online monitoring (e.g., stopped-flow technique [47]) or offline analysis [48]. Two difficulties arise in the former case: integration of efficient cooling to suppress temperature rise because of spectroscopic devices and separation of intermediates for identification. Therefore, only offline method combined with sample quenching (freeze/chemical) can allow catching intermediates providing information about reaction mechanism. The quenching for offline analysis is done either by adding an agent to the continuous stream [36] (referred to as quenched-flow technique) or to a container that collects the sample. MSR have been successfully used to overcome such problem. A slugflow MSR binds both reactants in the dispersed microliter droplets, which are carried by an inert fluid within a microcapillary. Each isolated slug acts as a micro-batch reactor. Because of the small size of the reactor assembly, it can be incorporated into commercially available calorimeters with precise temperature control. This concept has been applied for the investigation of enzymatic reaction kinetics [49], bromination of styrene [50], hydrolysis of pnitrophenyl acetate [51], high-throughput catalyst screening [52], and cyclization of Pseudoionone [53]. Besides, there are several reports on mixing and kinetics studies involving handling of hazardous reactants using both invasive and noninvasive techniques. 1.5.2 Industrial/Commercial Applications

The prominent industries that benefited from microstructured devices are pharmaceutical, specialty, and fine chemical industry. A few micro-plants have been developed and successfully tested in the laboratory as well as on commercial sites

11

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1 Overview of Micro Reaction Engineering

Table 1.1 Examples of industries/institutes active in microprocess engineering. MSR activities

Company/institute

MSR/mixer design and fabrication, process development

Fraunhofer ICT-IMM Mainz [57, 58], Forschungszentrum Karlsruhe GmbH [59], Ehrfeld Mikrotechnik BTS [60], Microinnova Engineering GmbH [61]

MSR design and fabrication, development of laboratory systems

Mikroglas GmbH [62], Mikronit microfluidics [63], Little Things Factory [64], Syrris [65]

Engineering services of MSR

Bayer Technology Services [66], Alfa Laval [67]

Development of MSR materials

Corning [68, 69]

MSR process development and demonstration of industrial production

Merck [70], SK Chemicals [71], Ampac Fine Chemicals [72], Phoenix Chemicals [73], Clariant GmbH [74], DSM [69, 75, 76], Lonza [43, 44], Sigma-Aldrich [77]

Table 1.1. The industries that are working in this area are Cellular Process Chemistry Systems GmbH, Fraunhofer ICT-IMM, Micronit Microfluidics, Ehrfeld BTS GmbH, Systanix, Inc., Styrris, to name but a few. In 2005 Fraunhofer ICT-IMM, Germany, has made successful runs of a continuous microstructured nitroglycerine plant (capacity 15 kg h−1 ) at the site of Xi’an Chemical Industrial Group HAC in China [54]. This plant consists of three main parts: the mixing of sulfuric acid with nitric acid (both highly concentrated fuming liquids), the reactor, and the phase separation (washing and purification devices). On demand, glycerine and the acid mixture are fed separately into an MSR where mixing occurs within milliseconds. Such micro-plant has also been developed for the polymerization of Methyl methacrylate [32] and showed significant improvement in the control of molecular weight because of superior heat transfer efficiency. The plant constructed by numbering-up eight tubes (microreactors) was continuously operated for 6 days without any problems. Microreactor technology is an important part of the decentralized mobile plant concept. Such plants have already been developed for biological applications (e.g., miniature analytical thermal cycling instrument, MATCI, to amplify and detect DNA via the polymerase chain reaction in real-time [6]). The objective behind chemical mobile plants is to reduce the risk associated with transporting hazardous chemicals. Rather than transporting hazardous chemicals, a distributed production strategy may be used with economic manufacturing on consumers’ site, as is currently performed for oxygen and nitrogen [7]. Recently, a chemical plant has been developed by Ehrfeld Mikrotechnik BTS GmbH, Germany, in a briefcase, which can be used for a variety of applications. Microstructured devices have several applications in pharmaceutical and fine chemicals where production amounts are often less than a few metric tons per

1.7 Summary

year [55]. The processes that rely on batch or semibatch mode could be operated continuously with multiple advantages over the batch processes. A review on the benefits of such devices for pharmaceutical industrial processes cites that 50% of the reactions in fine chemical/pharmaceutical industry could benefit from a continuous process [43]. In drug industry, there is always a strong time pressure to bring new molecules on the market to maximize the profit because of the manufacturing patent life of 20 years [56]. The process development time could be reduced significantly (Table 1.1).

1.6 Structure of the Book

The purpose of this book is to present the engineering aspects of microstructured devices. It addresses these questions: under which conditions microstructured devices are beneficial; how the devices should be designed; and, finally, in which way microstructured devices can be integrated in a chemical process. It also includes several theoretical and practical design examples on which the industry personnel have been working for several years. Some of these examples are included as exercise for the master and doctoral students in the curriculum at Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland. This book is divided into two parts: homogenous and multiphase (heterogenous) systems. Prior to explaining homogenous systems, a chapter on the fundamentals of chemical reaction engineering is presented, which is the basis for the discussions throughout the book. Homogeneous reactions part consists of three chapters that concentrate on mixing, residence time distribution, and heat management. The chapters in the second part elaborate on fluid–solid reactions, fluid–fluid reactions, and three-phase reactions. Each chapter contains the introduction, the types of devices used in a particular application, the basic design equations, the examples depicting the design methodology for a particular application, and chemical examples.

1.7 Summary

In this chapter, the microstructured devices are introduced underlying their potential benefits for the process industries. The reduced scale facilitates the temperature control giving an opportunity to maintain the temperature within any window required. Enhanced (heat/mass) transfer rates allow control of highly exothermic and hazardous reactions. It also increases production rates and thus reduces the total processing volume. In addition, microreactors can be simply numbered up for large-scale production, avoiding the problem of scale-up of conventional reactors.

13

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Thus, these devices are suitable when the following case-specific drawbacks are faced in the conventional processing options:

• • • • •

severe transport limitations (heat or mass transfer) low yields and high wastes because of multistep reactions safety issues for hazardous materials poor control of reaction parameters failure to meet market quality demand.

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and Yoshida, J.-I. (eds) (2009) Micro Process Engineering: Devices Reactions and Applications, vol. 2, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. Dummann, G., Quittmann, U., Groschel, L., Agar, D.W., Worz, O., and Morgenschweis, K. (2003) The capillarymicroreactor: a new reactor concept for the intensification of heat and mass transfer in liquid-liquid reactions. Catal. Today, 79 (1-4), 433–439. Abdallah, R., Fumey, B., Meille, V., and de Bellefon, C. (2007) Micro-structured reactors as a tool for chiral modifier screening in gas-liquid-solid asymmetric hydrogenations. Catal. Today, 125 (1-2), 34–39. Abdallah, R., Meille, V., Shaw, J., Wenn, D., and De Bellefon, C. (2004) Gas-liquid and gas-liquid-solid catalysis in a mesh microreactor. Chem. Commun., 10 (4), 372–373. Iwasaki, T., Kawano, N., and Yoshida, J. (2006) Radical polymerization using microflow system: numbering-up of microreactors and continuous operation. Org. Process Res. Dev., 10 (6), 1126–1131. Iwasaki, T. and Yoshida, J. (2005) Free radical polymerization in microreactors. Significant improvement in molecular weight distribution control. Macromolecules, 38 (4), 1159–1163. Janicke, M.T., Kestenbaum, H., Hagendorf, U., Schuth, F., Fichtner, M., and Schubert, K. (2000) The controlled oxidation of hydrogen from an explosive mixture of gases using a microstructured reactor/heat exchanger and Pt/Al2O3 catalyst. J. Catal., 191 (2), 282–293. Ueno, M., Hisamoto, H., Kitamori, T., and Kobayashi, S. (2003) Phase-transfer alkylation reactions using microreactors. Chem. Commun., 8, 936–937. Palde, P.B. and Jamison, T.F. (2011) Safe and efficient tetrazole synthesis in a continuous-flow microreactor. Angew. Chem. Int. Ed., 50 (15), 3525–3528. Maurya, R.A., Park, C.P., Lee, J.H., and Kim, D.P. (2011) Continuous in situ generation, separation, and reaction of

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microsystems. Chem. Commun., 36, 4509–4516. Song, H. and Ismagilov, R.F. (2003) Millisecond kinetics on a microfluidic chip using nanoliters of reagents. J. Am. Chem. Soc., 125 (47), 14613–14619. Cygan, Z.T., Cabral, J.T., Beers, K.L., and Amis, E.J. (2005) Microfluidic platform for the generation of organicphase microreactors. Langmuir, 21 (8), 3629–3634. Ahmed, B., Barrow, D., and Wirth, T. (2006) Enhancement of reaction rates by segmented fluid flow in capillary scale reactors. Adv. Synth. Catal., 348 (9), 1043–1048. de Bellefon, C., Abdallah, R., Lamouille, T., Pestre, N., Caravieilhes, S., and Grenouillet, P. (2002) High-throughput screening of molecular catalysts using automated liquid handling, injection, and microdevices. Chimia, 56 (11), 621–626. Kashid, M., Detraz, O., Moya, M.S., Yuranov, I., Prechtl, P., Membrez, J., Renken, A., and Kiwi-Minsker, L. (2013) Micro-batch reactor for catching intermediates and monitoring kinetics of rapid and exothermic homogeneous reactions. Chem. Eng. J., 214, 149–156. IDW (2005) http://idw-online.de/pages/ en/news135982 (accessed on July, 05, 2014). Jensen, K.F. (2001) Microreaction engineering – is small better? Chem. Eng. Sci., 56 (2), 293–303. Ramshaw, C. (1999) Process intensification and green chemistry. Green Chem., 1 (1), G15–G17. Fraunhofer ICT-IMM (2009) www.immmainz.de (accessed 31 March 2014). Renken, A., Hessel, V., Löb, P., Miszczuk, R., Uerdingen, M., and Kiwi-Minsker, L. (2007) Ionic liquid synthesis in a microstructured reactor for process intensification. Chem. Eng. Process. Process Intensif., 46 (special issue 9), 840–845. KIT (2009) http://www.fzk.de (accessed 31 March 2014). EHRFELD (2009) http://www.ehrfeld.com/ (accessed 31 March 2014). Kirschneck, D. and Tekautz, G. (2007) Integration of a microreactor in an

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2 Basis of Chemical Reactor Design and Engineering This chapter presents the fundamentals of chemical reaction engineering. It includes the basis of mass and energy balances, kinetics of homogenous reactions, including homogenous catalytic reactions in mono- and biphasic systems and kinetics of heterogenous catalytic reactions with special attention to the mass and heat transfer effects. Finally, the main design equations and comparative performance of three types of ideal reactors (Batch, Plug Flow, and Continuously Stirred Tank) are shortly summarized and discussed.

2.1 Mass and Energy Balance

The interactions between the chemical reaction and the simultaneously occurring transport processes for mass, energy, and impulse can be described by the fundamental conservation laws. At first the system boundaries must be specified. The volume enclosed by these boundaries is called system volume. The size of the system volume can be defined by natural boundaries, such as those of the phase boundary, the reactor, or by a small volume element of a phase through the defined limits of which mass, energy, and impulses can be exchanged (see Figure 2.1). For an unambiguous description, however, it is necessary to select the system volume in such a way that the conditions in it can be considered as uniform (e.g., constant temperature and concentrations). The design of any chemical reactor is based on material and energy balance. A material balance has to be set for all species participating in the reactions taking place within the selected system volume. The material balance for a component Ai can be formulated in the following manner: ⎧accumulation ⎫ ⎧rate of flow ⎫ ⎧rate of flow ⎫ ⎧rate of production or ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ of Ai in ⎬ = ⎨ of Ai into ⎬ − ⎨ of Ai out of ⎬ + ⎨ disappearance of Ai ⎬ ⎪ the system ⎪ ⎪the system⎪ ⎪the system⎪ ⎪ into the system ⎪ ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ dni dt

=

ṅ i,0

−

ṅ i

+

Pi

(2.1)

Microstructured Devices for Chemical Processing, First Edition. Madhvanand N. Kashid, Albert Renken and Lioubov Kiwi-Minsker. © 2015 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2015 by Wiley-VCH Verlag GmbH & Co. KGaA.

20

2 Basis of Chemical Reactor Design and Engineering

Figure 2.1

System volumes with differing selected sizes.

where ni represents the number of moles of species Ai in the system at time t, ṅ i is the molar flow rate, and Pi is the rate of Ai production or disappearance. The last term in the above equation is often referred to as source term. The rate of Ai production/disappearance corresponds to the product of the system volume, ΔV, and the transformation rate Ri of the component Ai (Equation 2.2). The size of the system volume is chosen in such a way that inside the volume, concentrations and temperatures are constant. Pi = Ri ⋅ ΔV

(2.2)

The rate of Ai transformation (Ri ) is the sum of the rates (rj ) of the reactions in which Ai participates: ∑ Ri = �i,j ⋅ rj (2.3) j

Much of this book deals with the finding of the expression that relates the Pi with different contacting patterns and various reaction parameters (intrinsic kinetic, reaction enthalpy, adiabatic temperature, etc.). Concerning the classification of chemical reactions, different principles can be applied. One of the most useful for the reactor design is the classification based on the amount and types of the phases involved, such as homogenous reactions (takes place only in one phase) and heterogenous reactions (involves two or more phases). Treating the kinetics of homogenous reactions in order to optimize the performance of an eventual reactor is easier than treating heterogenous reactions. If for homogenous reactions the temperature, pressure, and composition are the main variables affecting the rate of transformations, for the heterogenous reactions the situation becomes more complex. The reaction can take place within one or multiple phases, at the interface, and reactants and products may be distributed between different phases. This implies that material has to move from phase to phase influencing the overall rate of transformation. In addition, the heat transfer may also play an important role and for highly exothermic/endothermic reactions may limit the overall transformation rates. The mass and heat transfers become increasingly important with the increase in temperature where the intrinsic reaction rates are very high. The kinetics of homogenous and heterogenous reactions is discussed later in this chapter with a special attention on mass and heat transfer influence on the kinetics of heterogenous reactions. Applying the principle of conservation of energy leads to the energy balance that can be described as follows:

2.2

Formal Kinetics of Homogenous Reactions

⎧ ⎫ ⎧ rate of flow ⎫ ⎧ rate of work ⎫ ⎧ rate of energy ⎫ ⎧ rate of energy ⎫ ⎪rate of energy⎪ ⎪ of heat ⎪ ⎪ done by ⎪ ⎪ added to ⎪ ⎪ leaving the ⎪ ⎪accumulation ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ system ⎨ within ⎬ = ⎨to the system ⎬ − ⎨ the system ⎬ + ⎨ the system ⎬ − ⎨ ⎬ ⎪ ⎪ ⎪ from the ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ on the by mass flow by mass flow ⎪ ⎪ the system ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ surroundings⎭ ⎩ surroundings⎭ ⎩into the system⎭ ⎩out of the system⎭ dEsys dt

=

Q̇

−

Ẇ

Ė in

+

−

Ė out

(2.4)

The work term Ẇ is generally separated into flow work, Ẇ f , and shaft work, ̇ Ws . Shaft work is, for example, from the stirrer in a stirred tank or a turbine in a tubular reactor. When the shear stress can be neglected, the work term is ∑ ∑ ̂i |in + ̂i |out + Ẇ s (2.5) Ẇ = − ṅ i pV ṅ i pV i

i

̂i , the molar volume of the reactant Ai . with V Introducing these in Equation 2.4 results in dEsys dt

= Q̇ − Ẇ s −

∑

̂i )|in + (Ė i + ṅ i pV

∑

i

̂i )|out (Ė i + ṅ i pV

(2.6)

i

The energy Ei is the sum of the internal energy, the kinetic energy, the potential energy, and all other energies such as electric, magnetic, or light. For the description of the majority of chemical reactors, the kinetic, potential, and other energies can be neglected, resulting in ̂i Ė i ≅ ṅ i U

(2.7)

Introducing the enthalpy ̂ i + pV ̂i ̂i = U H

(2.8)

we obtain finally: dEsys dt

= Q̇ − Ẇ s −

∑

̂ i,0 + ṅ i,0 H

∑

i

̂i ṅ i H

(2.9)

i

̂ i are the molar ̂ i and U The subscript “0” indicates inlet conditions while H enthalpy and molar energy, respectively.

2.2 Formal Kinetics of Homogenous Reactions

Kinetics is a key discipline for chemical reaction engineering. It relates the rate at which a chemical transformation occurs to macroscopic process parameters, like pressure, concentrations, temperature. Moreover, it enables to find a link between the observed transformation rates to a reaction mechanism that describes intimate interactions between individual molecules. For solving chemical reaction engineering problems we are mostly interested in practical situations, where relatively large quantities of matter are transformed.

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2 Basis of Chemical Reactor Design and Engineering

In order to quantify the rate of a chemical transformation, we need to introduce some definitions. First, we distinguish between different types of reactions based on the form used to describe eventual chemical transformation, as single or multiple reactions. Usually this can be done from material balance after examining the stoichiometry between reacting materials and products. If a single stoichiometric equation can present the transformation, this is a single reaction. If more than one equation is necessary to present all observed components and their transformations, this it is a case of multiple reactions. The examples are as following: Single irreversible reaction: A1 + A2 → A3 + A4 Consecutive reactions (or reactions in series): A1 → A2 → A3 A1 → A2 Parallel reactions: A1 → A3 A1 + A2 → A3 Parallel-consecutive: A3 + A2 → A4 More complicated schemes are also possible and present a combination of the listed reactions. According to the International Union of Pure and Applied Chemistry (IUPAC) [1] the reaction rate in homogenous system is the change in the number of moles per unit of time and unit of volume because of the reaction divided by the stoichiometric coefficients. The reaction rate is always positive. 2.2.1 Formal Kinetics of Single Homogenous Reactions

Experimentally it was observed that the reaction rate depends on the concentrations of the reacting species and temperature. Very often a simple Power Rate Law (PRL) can be applied to describe this dependency. For the single irreversible reaction: A1 + A2 → products, the following equation results: n

n

r = k ⋅ c1 1 ⋅ c22

(2.10)

The exponents in Equation 2.10 are called the reaction orders. The reaction is n1 order with respect to reactant A1 and n2 with respect to A2 . The overall reaction order is given by: n = n 1 + n2 The decomposition of N2 O5 in the gas phase is the example of a formal first order reaction [2]: N2 O5 → NO2 + NO3 A1 → A2 + A3 r = k 1 ⋅ c1

(2.11)

An example for second order reactions is the formation of HI from hydrogen and iodine [2]

2.2

Formal Kinetics of Homogenous Reactions

H2 + I2 → 2 HI A 1 + A2 → 2 A 3 r = k2 c1 c2

(2.12)

In general, formal kinetic equations are empirical, valid only within a limited domain of concentrations and temperatures. The reaction rate constant (or coefficient) in Equation 2.10 is independent of the composition of the reaction mixture, but is strongly influenced by the temperature. In practically all cases, the rate constant can be described by the Arrhenius law: ) ( −Ea (2.13) k = k0 ⋅ exp RT where k 0 is the preexponential or frequency factor, and Ea is the apparent activation energy of the reaction. This expression fits well with the experiments over a wide range of temperatures. The frequency factor is much less temperaturesensitive than the exponential term, and, therefore, its variation with temperature is masked allowing in the Arrhenius law to consider k 0 = const. For most reactions, the activation energy lies in the range of 40–300 kJ mol−1 . Its value can be estimated (provided that Ea remains constant) from the reaction rates measured at constant concentrations but at two different temperatures using the Arrhenius law: ( ) k2 k (T2 ) Ea 1 1 ln = (2.14) = ln − k1 k (T1 ) R T1 T2 The knowledge of E a allows to predict the kinetic rate constant at different operation temperature (see example 2.1).

Example 2.1: Estimation of reaction rate constants at different temperatures. What will be the increase of the rate constant for a temperature rise of 10 K? Calculate for the reaction with the activation energy of 100 kJ mol−1 , supposing a base temperature of (a) 25 ∘ C and (b) 100 ∘ C. Solution: ]) ( [ −Ea 1 1 − k(T2 ) = k(T1 ) exp R T2 T1 [ ] [ ] k(T2 ) −Ea 1 1 1 ⋅ 105 1 1 ln = 1.31 = − =− − k(T1 ) R T2 T1 8.31 308 298 k(T2 ) = 3.71 for a temperature increase from 25 to 35 ∘ C k(T1 ) k(T2 ) = 2.32 for a temperature increase from 100 to 110 ∘ C ⇒ k(T1 )

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2 Basis of Chemical Reactor Design and Engineering

2.2.2 Formal Kinetics of Multiple Homogenous Reactions

As it has already been mentioned, often more than one chemical equation is necessary to present all observed components and their transformations. In this case one talks about complex or multiple reactions. One of the simple examples of multiple reactions is a reversible reaction occurring in the reactor, which can be presented in the following way: −−−−−−−−−−−−− → j = 1 ∶ �1,1 A1 + ν2,1 A2 ← − �3,1 A3 + �4,1 A4 − − − − − − − → j = 2 ∶ �3,2 A3 + �4,2 A4 ←−−−−−−− �1,2 A1 + �2,2 A2

(2.15)

The transformation rates (see Equation 2.3) for the species A1 , A3; are given by: R1 = �1,1 r1 + �1,2 r2 ; R3 = �3,1 r1 + �3,2 r2

(2.16)

The reaction rates depend on the concentrations of the reacting species and can be described by a PRL: ∏ n ci i (2.17) rj = kj ⋅ i

where ci is the concentration of reactant Ai and ni is the reaction order with respect to Ai . If |�i | = 1 and both reactions are of first order for all the reactants Ai , Equation 2.16 can be rewritten: R1 = −1 ⋅ k1 c1 c2 + 1 ⋅ k2 c3 c4 ; R3 = +1 ⋅ k1 c1 c2 − 1 ⋅ k2 c3 c4

(2.18)

A chemical reaction proceeds in the direction in which the free Gibbs energy, G, of the reaction mixture diminishes. When equilibrium is reached: R1 = R3 = 0 c∗ c∗ k k1 c∗1 c∗2 = k2 c∗3 c∗4 ; Kc = 1 = ∗3 ∗4 k2 c1 c2

(2.19) (2.20)

If the thermodynamic activities of the reactants correspond to their concentrations, the equilibrium constant Kc can be estimated from the second law of thermodynamics: ) ( −ΔG0 (2.21) Kc (T) ≅ K(T) = exp RT Taking the derivative of Equation 2.21, where ΔG0 = ΔH 0 − TΔS0

(2.22)

the van’t Hoff equation can be obtained: ( ) d ln K ΔH 0 d −ΔG0 = = dT dT RT RT 2 By integrating from the standard temperature (273 K) to the desired temperature T, we obtain the dependence of the equilibrium constant on the

2.2

Formal Kinetics of Homogenous Reactions

reaction enthalpy (the superscript “0” indicates standard conditions: T st = 273 K, pst = 105 Pa): T

ln K = ln K(273) +

ΔH 0 dT ∫273 RT 2

(2.23)

For gas phase reactions, the PRL can be expressed in partial pressures. If the ideal gas law can be applied, the reaction rate constants are related as follows: ∏ n ∏ n ci i = kj,p ⋅ pi i ; kj,p ⋅ (RT)ni = kj,c kj,c ⋅ i

i

n (2.24) with pi = i ⋅ RT = ci ⋅ RT V where V is the volume occupied by the reaction mixture, ni – is the total number of moles of Ai in the mixture, R is the gas constant, and T is the reactor temperature in Kelvin, ci is the concentration of Ai in the reactor. 2.2.3 Reaction Mechanism

The reaction mechanism describes intimate interactions between individual molecules and represents a network of elementary reactions involved in an overall transformation. It is often much more complex than the stoichiometry of the reaction as its formal kinetics in the form of PRL suggests. An example is the gas phase reaction between NO2 and CO described by the following stoichiometric equation: NO2 + CO → NO + CO2

(2.25)

Experimentally, a second order with respect to NO2 and a zero order with respect to CO is found: r = k c2NO c0CO = k c2NO 2

(2.26)

2

This PRL expression indicates that the reaction is not elementary meaning that it does not proceed via a collision between NO2 and CO molecules. The mechanism has been studied and proposed to consist of two consecutives elementary steps [3]: NO2 + NO2 → NO3 + NO; r1 = k1 c2NO

2

NO3 + CO → NO2 + CO2 ; r2 = k2 cNO3 cCO NO2 + CO → NO + CO2

(2.27)

The experimentally obtained formal kinetic equation (Equation 2.26) can be explained by a very fast second step compared to the first one (r 2 ≫ r1 ). In this case the overall transformation rate will be controlled by the rate of the first step as the slowest one being in agreement with the experimentally observed PRL equation. This method to derive a concentration term in the rate expression is called the “rate-determining step” approach.

25

26

2 Basis of Chemical Reactor Design and Engineering

Another useful and widely used approach is called the “quasi steady-state approximation” (QSSA). In this case we hypothesize the existence of at least one (or more) intermediates involved in the reaction mechanism whose concentration in the reacting mixture is very low and can be considered as quasi constant. In general, formal kinetic models are valid only within a limited domain of concentrations and temperatures. 2.2.4 Homogenous Catalytic Reactions

The following types of homogenous catalytic reactions can be distinguished:

• Acid/base catalysis: substrate activation by protonation or deprotonation • Nucleophilic/electrophilic catalysis: substrate activation by Lewis bases via electron pair donor complexes or by Lewis acids via electron pair acceptor complexes • Organometallic complex catalysis: substrate activation via coordinative interaction [4] • Enzyme catalysis: substrate activation by multifunctional interactions [5]. The kinetics of homogenous catalytic reactions are presented here for the acid/base and enzymatic catalysis as examples. Acid/base catalysis is probably the oldest type of homogenous catalytic reactions. Following the definitions by Brønstedt [6] and Lowry [7], the acids are proton donators and the bases are proton acceptors. Let us consider a bimolecular catalytic reaction with the equation presented as following: HA

A1 + A2 −−−→ A3

(2.28)

The mechanism of this acid catalyzed reaction is depicted in Scheme 2.1 If the product formation is slow compared to the protonation reactions (k 2 ≪ k 1 , k −1 ) the transformation depends only on the concentration of protons cH + and a fast preequilibrium can be considered. r2 = k2 ⋅ cX + ⋅ c2 the rate-determining step k c + with ∶ cX + = 1 H c1 k−1 Ka k 1 r = k2 1 c + ⋅ c1 ⋅ c2 = k ′ ⋅ cH + ⋅ c1 ⋅ c2 k−1 Ka H HA + A1 X+ + A2 H+ + A−

k1 k−1 k2

(2.29)

X+ + A− : formation of activated substrate X+ (protonation) (fast) A3 + H+ : irreversible rection of the protonated substrate with A2 (slow)

HA

: equilibrium (fast)

Scheme 2.1 Acid catalyzed bimolecular reaction A1 , A2 : reactants; A3 : product; HA: acid as catalyst.

2.2

Formal Kinetics of Homogenous Reactions

If the protonation of the substrate is rate determining (k 2 ≫ k 1 ), the reaction rate is directly proportional to the concentration of the Brønstedt acid. r = r1 = k1 ⋅ cHA ⋅ c1

(2.30)

Examples for reactions of fast protonations (Equation 2.29) are ester hydrolysis and alcoholysis, inversion of sucrose, and the hydrolysis of acetals. The mutarotation of glucose and the dehydration of acetaldehyde hydrate are examples of slow protonations described with Equation 2.30. In systems, homogenously catalyzed by organometallic complexes, the selectivity of the reaction can be controlled by the appropriate choice of ligands on the catalytic metallic center. Combining a catalytic active metal with ligands often allows the synthesis of organic compounds that are otherwise accessible only through complex multistep synthesis. Kinetics of homogenously catalyzed reactions is mostly described with the Michaelis–Menten model [8]. The model was first published in the field of enzyme kinetics in the beginning of the twentieth century. According to this model, the catalyst reacts with the substrate, A1 in a preequilibrium to form a catalyst substrate complex, X # , which reacts usually irreversibly to form the product, A2 . Besides enzymatic reactions, many homogenously catalyzed hydrogenations follow Scheme 2.2. An example is the asymmetric hydrogenation dehydroamino acid derivatives with rhodium or ruthenium catalysts. On the basis of Scheme 2.2 the following rate equation can be derived:

r=

k2 ⋅ ccat,0 ⋅ c1

KM + c1 c ⋅c k + k2 with KM = −1 = cat 1 ; k1 cX # k k−1 KM ≅ if k2 ≪ k−1 ; KM ≅ 2 if k2 ≫ k−1 k1 k1

(2.31)

For k 2 ≪ k −1, the Michaelis constant KM corresponds to the inverse stability constant of the catalyst substrate complex.

Cat + A1 X#

k2

k1 k−1

X # equilibrium (fast)

A2 + Cat the rate determining step (slow)

Ccat,0 + Ccat + CX # Scheme 2.2 Reaction sequence and catalyst mass balance of the simple Michaelis–Menten model.

27

2 Basis of Chemical Reactor Design and Engineering

For c1 ≫ KM , Equation 2.31 reverts to a zero order kinetics with regard to the reactant. For this situation the concentration of free catalyst is near zero and the reaction rate attains a maximum. (2.32)

rmax = k2 ⋅ ccat,0 ; cX # ≅ ccat,0

If, on the other hand, c1 ≪ K M , the rate of the product formation depends on the concentration of the catalyst as well as the reactant A1 , both of the first order. This is illustrated in example 2.2. (2.33)

r = k2 ⋅ ccat,0 ⋅ c1 = rmax ⋅ c1

Example 2.2: Enzyme catalysis (Michaelis-Menten model). The kinetics of glucose formation from lactose by means of ß-galactosidase was studied in a wide range of variables by Flaschel et al. [9]. The dependence of the reaction rate as the function of the lactose concentration is shown in Figure 2.2. 0.30 Reaction rate, r (mmol g–1 s–1)

28

0.25

rmax

0.20 rmax = 0.2442 mmol g–1s–1 KM = 0.0483 mol L–1

0.15 0.10 0.05 0.00 0.0 KM 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

–1

Lactose concentration, c1 (mol l ) Figure 2.2 Initial rate of enzymatic lactose hydrolysis [10] (T = 50 ∘ C, pH = 3.5; ccat, 0 = 0.125 g l−1 ).

The experimental results fit well to the rate equation derived from a Michaelis–Menten model as seen in Figure 2.2. The two-model parameters are easily obtained by fitting the model (Equation 2.31) to the experiments. For c1 = KM , the reaction rate corresponds to half of the maximum value. rc1 =KM =

k2 ⋅ ccat,0 ⋅ c1 KM + c1

=

k2 ⋅ ccat,0 ⋅ c1 c1 + c1

=

1 k ⋅c 2 2 cat,0

2.3

Ideal Reactors and Their Design Equations

2.3 Ideal Reactors and Their Design Equations

Most chemical reactors used in practice can be classified according to some common criteria and assigned to the so-called basic or ideal reactor types. On the basis of the characteristics of ideal reactors, the complex interactions of chemical reaction kinetics, mass, heat, and impulse transport can be discussed in a general way. The behaviors of many actually used reactors approach the ideal types so that their fundamental relationships can be applied at least for a first reactor design. In other cases, the reactor behavior of real systems must be described with the help of models often containing the ideal reactors as individual elements (see Chapter 3). For ideal reactors, highly simplified assumptions are used as the starting point, such as an ideal mixing down to the molecular level or a plug flow (piston type flow pattern). We distinguish between:

• the ideally mixed, batch-wise operated stirred tank reactor (BSTR), • the ideally mixed, continuously operated stirred tank reactor (CSTR), and • the ideal plug flow reactor (PFR). 2.3.1 Performance Parameters

Several terminologies are used in the chemical reaction engineering literature to represent the performance of both catalytic and noncatalytic chemical processes. The definitions that are commonly used are given as follows: Conversion: It indicates the progress of the reaction and is defined as the ratio of the amount of the limiting reactant transformed and the total amount fed to the reactor. For the following parallel reactions j = 1 ∶ v1,1 A1 + v2,1 A2 → v3,1 A3 + v4,1 A4 j = 2 ∶ v1,2 A1 → v4,2 A4

(2.34)

the conversion of A1 is given by X1 = X =

n1,0 − n1 n1,0

(2.35)

where, n1,0 and n1 are the initial and final numbers of moles of A1 , respectively. Yield: It is the amount of product formed in the reaction referred to the amount of the reactant fed to the reactor. Considering the reactions indicated in Equation 2.34 the yield of A3 with respect to reactant A1 is defined as: Y3,1 =

n3,0 − n3 �1,1 n1,0

�3,1

(2.36)

29

30

2 Basis of Chemical Reactor Design and Engineering

The yield of A3 referred to the reactant A2 is defined as Y3,2 =

n3,0 − n3 �2,1 n2,0

�3,1

(2.37)

Selectivity: The selectivity corresponds to the amount of the desired product formed with respect to the amount of the compound reacted: S3,1 =

n3,0 − n3 �1,1 n1,0 − n1 �3,1

(2.38)

Thus, from Equations 2.35, 2.36, and 2.38 one can find that Y3,1 = X1 ⋅ S3,1

(2.39)

Instantaneous selectivity: As the reaction rates depend on the reactant concentrations, the instantaneous yield and selectivity can change with time or the location in the reactor. The instantaneous or differential selectivity is defined as the ratio between the rate of product formation and the rate of reactant transformation: s3,1 =

� 1 R3 � dc = 1 3 � 3 R1 �3 dc1

(2.40)

2.3.2 Batch Wise-Operated Stirred Tank Reactor (BSTR)

In a batch reactor, there is no inflow or outflow of reactants. It is a commonly used apparatus in the fine and pharmaceutical industry as well as in laboratories because of its flexibility and multifunctionality. The ideal stirred tank reactor is characterized by complete mixing down to the molecular level. Therefore, no concentration or temperature gradients exist. The system volume (Figure 2.1) corresponds to the volume occupied by the reaction mixture as indicated in Figure 2.3. As reactants are neither added nor removed during the reaction time (batch time), the mass balance Equation 2.1 simplifies to { } { } rate of reactant rate of reactant = accumulation transformation ∑ dni vij rj (2.41) = V ⋅ Ri = V ⋅ dt j The volume V occupied by the reaction mass may change, if the density of the reaction mixture varies during the reaction time as a result of the changing product composition and of physical processes like heating or cooling. In contrast to mass, the BSTR can exchange heat through the reactor wall with the surroundings, resulting in a heat balance as: (C w + mcp )

∑ dT rj (−ΔHr,j ). = U ⋅ A ⋅ (Tc − T) + V dt j

(2.42)

2.3

Ideal Reactors and Their Design Equations

Tc, out

V

ci

Tc,0

Figure 2.3

Batch-wise operated stirred tank reactor (BSTR).

with U the global heat transfer coefficient, A the heat exchange surface area of the reactor, T c the mean temperature of the cooling/heating medium, and T the temperature of the reaction mixture. The total heat capacity of the reactor is designated as C w and is assumed to be independent of temperature. The same holds for the average specific heat cp of the reaction mixture, for which it is additionally assumed that it does not change with the product composition. Equations 2.41 and 2.42 are used to describe the behavior of the reactor during the reaction period and to determine the reactor performance. The reactor performance Lp is defined as the amount of product Ai produced per unit time. In batch-operated reactors, Lp depends on the entire reaction cycle t cycle . The cycle consists of the reaction time tR required to reach a desired degree of conversion and the shut-down time ta needed for charging, emptying, cleaning, heating, and cooling of the reactor. ni − ni,0 ni − ni,0 = (2.43) Lp = ta + t R tcycle The term ni,0 corresponds to the product present at the start of the cycle (usually ni, 0 = 0). The reaction time tR , which is needed to achieve the desired degree of conversion, is obtained by integrating Equation 2.45. For a single reaction (Ri = � i ⋅r) follows with Xf , the final degree of conversion of the key compound A1 : Xf =

n1,0 − n1,f n1,0

(2.44)

Xf

tR = n1,0

dX . ∫X0 (−R1 ) ⋅ V

(2.45)

31

32

2 Basis of Chemical Reactor Design and Engineering

The density of the reaction mixture may change during the transformation process. Examples are polymerizations, where often the density of the polymeric product is higher than the monomer. Density changes lead to a variation of the volume occupied by the reaction mixture, V, in the reactor. If the reaction volume is a linear function of the conversion, we obtain the following relationship: (2.46)

V = V0 ⋅ (1 + αX)

The expansion factor α corresponds to the fractional volume change on complete conversion as defined in Equation 2.47. �=

VX=1 − VX=0 VX=0

(2.47)

Introducing the expansion factor in Equation 2.45 we obtain: tR = n1,0

∫0

Xf

dX = c1,0 ∫0 (−R1 ) ⋅ V0 ⋅ (1 + �X)

Xf

dX (−R1 ) ⋅ (1 + �X)

(2.48)

For an irreversible nth order reaction with −R1 = k ⋅ cn1 = k ⋅ cn1,0 (1 − X)n , the reaction time is given by: tR =

1 n−1 ∫ k ⋅ c1,0 0

Xf

(1 + �X)n−1 dX, (1 − X)n

Xf t (1 + �X)n−1 n−1 or DaI = R = tR k c1,0 = dX ∫0 tr (1 − X)n

(2.49)

DaI is the first Damköhler number, which is defined as the ratio of the residence time in the reactor (tR ) to the characteristic reaction time tr as defined in Equation 2.50. c1,0 1 ; t = (nth-order reaction) (2.50) tr = n−1 (−R1,X=0 ) r k ⋅ c1,0 For a first order reaction the necessary reaction time, respectively, the Damköhler number for a required conversion, is independent of the expansion factor and we obtain with dX R1 = −kc1 = −kc1,0 (1 − X); = k(1 − X) (2.51) dt Xf n1,0 1 dX −1 (2.52) tR = = ln(1 − Xf ) ∫ V k ⋅ c1,0 0 (1 − X) k or Xf = 1 − exp(−k ⋅ tR ) = 1 − exp(−DaI)

(2.53)

The situation is different for reaction orders n ≠ 1. For a second order reaction, integration of Equation 2.49 results in DaI = tR k c1,0 =

∫0

Xf

(1 + �)Xf (1 + αX) + α ln(1 − Xf ) dX = 2 1 − Xf (1 − X)

(2.54)

2.3

Ideal Reactors and Their Design Equations

1.00 0.95

0.85 0.80

0.65

α=+

0.70

0.5

0.75 α=– 0.5 α=0

Conversion, X

n=2

n=1

0.90

1

10

100

Damköhler number, DaI Figure 2.4

Conversion as function of the Damköhler number and the expansion factor.

In Figure 2.4 the influence of the expansion on the conversion in BSTR is illustrated for a second order reaction. The expansion has no influence on first order reactions in BSTR. Selectivity and yield obtained in BSTR are discussed in Examples 2.3 and 2.4 for parallel and consecutive reactions.

Example 2.3: Selectivity in BSTR for parallel reactions. How the selectivity toward A2 changes with the conversion of A1 for parallel reactions and which value it will have if the rate constants are equal (k 1 = k 2 )? k1

�11 A1 −−→ �21 A2

�11 = �12 = −1;

�21 = �32 = +1

k2

�12 A1 −−→ �32 A3 n=1 Solution: S2,1 =

k 1 c1 k1 = k1 c1 + k2 c1 k1 + k2

The selectivity toward A2 , S2,1 does not depend on the conversion and if the constants are equal, S2,1 = 0.5.

Example 2.4: Yield of the intermediate product as function of conversion. Derive an expression and plot the concentration profiles for consecutive reactions (both of first order) with k 2 /k 1 = 0.5. How the concentration and yield of A2 changes with the conversion of A1 and which value it will have if the

33

2 Basis of Chemical Reactor Design and Engineering

constants are equal? k1

k2

A1 −−→ A2 −−→ A3 Also propose an expression to estimate the yield of A3 . Solution: k1

A1 −−→ A2

r1 = k1 c1

k2

A2 −−→ A3 −

r2 = k2 c2

dc1 = k1 c1 dt

c1 = c1,0 exp(−k1 t); ⇒ X = 1 − f1 = 1 −

c1 c1,0

dc2 = k1 c1 − k2 c2 dt c2 = c1,0 Y2,1 =

k1 [exp(−k1 t) − exp(−k2 t)] k2 − k1

c2 k 1 = [exp(−DaI1 ) − exp(−κ ⋅ DaI1 )] for κ = 2 ≠ 1 c1,0 κ−1 k1

c2 = k1 t ⋅ c1,0 exp(−k1 t) c k Y2,1 = 2 = DaI1 ⋅ exp(−DaI1 ) for � = 2 = 1 c1,0 k1 The yield of the intermediate product passes through a maximum value as function of the time and the first Damköhler number, respectively. This is shown for first order consecutive reactions in Figure 2.5. 1.0 f1 = c1/c1,0

0.8

ci /c1,0

34

Y3 = c3 /c1,0

0.6 0.4

Y2 = c2 /c1,0

0.2 0.0

0

2

4 6 DaI1 = t k1

8

10

Figure 2.5 Product yields and unreacted fraction of the key reactant f 1 as function of the first Damköhler number. First order irreversible reaction, � = 0.5.

2.3

Ideal Reactors and Their Design Equations

The maximum yield of the intermediate depends only on the ratio of the rate constants � = k2 ∕k1 as shown in Figure 2.6. Decreasing � results in an increasing maximum yield. ( ) k2 c2,max � k1 k2 −k1 Y2,1,max = = = � 1−� ; for � ≠ 1 c1,0 k2 c2,max 1 ≅ 0.368; for � = 1 Y2,1,max = = c1,0 exp(κ)

Yield of the intermediate, Y2,1 = c2/c1,0

1.0

κ = k2/k1 = 0.02 0.05

0.8

0.1 0.2

0.6 0.5

0.4

1 2

0.2 5

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Conversion of A1, X Figure 2.6 Yield of the intermediate product as function of the conversion and the ratio of rate constants. Irreversible first order reactions.

The yield of A3 may be found from the mass balance: c1 + c2 + c3 = c1,0 ⇒ Y3,1 = 1 − f1 − Y2,1

2.3.3 Continuous Stirred Tank Reactor (CSTR)

One of the most important parameters to characterize continuous flow reactors is the degree of backmixing. In the ideal mixed reactor the concentrations and the temperature within the reactor volume are uniform. In consequence, the whole volume occupied by the reaction mixture can be taken as the system volume for the mass balance (see Figure 2.1). In an ideal CSTR, the reactants fed to the reactor are instantaneously mixed up to the molecular level. The concentrations in the reactor correspond to the concentrations at the reactor outlet (Figure 2.7).

35

36

2 Basis of Chemical Reactor Design and Engineering

V0, c1,0 T0

Tc, out

V

ci

Tc,0 Vout , c1,out = c1 Tout = T Figure 2.7

Continuous stirred tank reactor (CSTR).

When the molar flow ṅ i is replaced by the volumetric flow V̇ and the concentration ci , we obtain: ∑ dni dc = ṅ i,0 − ṅ i + V ⋅ Ri = V i = V̇ 0 ci,0 − V̇ out ci,out + V vij rj ; (ci,out = ci ) dt dt j (2.55) The volume occupied by the reaction mixture is designated with V. In general, V corresponds to about three-fourth of the nominal reactor volume. The ratio of the reaction volume to the volumetric inlet flow V̇ 0 is known as the space time: τ=

V V̇ 0

(2.56)

The reciprocal value of � is often designated as the space velocity or, in biotechnology, the dilution rate. After a transient period that corresponds to about five times the space time, the reactor operates at steady state, that is, the composition of the reaction mixture is time invariant and the mass balance is reduced to a simple algebraic expression. ∑ vij rj = V̇ 0 ci0 − V̇ out ci,out + V ⋅ R1 = 0 (2.57) V̇ 0 ci0 − V̇ out ci,out + V j

Introduction of the degree of conversion for the key reactant A1 leads to ∑ v1,j rj −V ⋅ c1,0 ṅ 1,0 − ṅ 1,out j V ⋅ (−R1 ) V X (2.58) = = or = X= ̇ ̇ ̇ ⋅(−R1 ) ṅ 1,0 V0 ⋅ c1,0 V0 ⋅ c1,0 V0

2.3

Ideal Reactors and Their Design Equations

The space time necessary to achieve a required conversion is: �=

c1,0 X

(2.59)

−R1

For reactions with constant density (�0 = � = �out ) the volumetric inlet flow is identical to the volumetric flow at the outlet (V̇ 0 = V̇ out ) and the design equations for irreversible nth order reactions are readily found. First order reaction

Second order reaction

k1

k1

A1 −−−→ A2 ; r = k2 ⋅ c21 V̇ 0 ⋅ (c1,0 − c1 ) = V ⋅ k2 ⋅ c21 √

A1 −−−→ A2 ; r = k1 ⋅ c1 V̇ 0 ⋅ (c1,0 − c1 ) = V ⋅ k1 ⋅ c1 1−X =

c1 c1,0

1−X =

1 1+DaI

=

c1,out c1,0

=

1 1+k1 �

1−X = (2.60)

1−X =

−1+

1+4k2 c1,0 �

2k2 c1,0 � √ −1+ 1+4DaI 2DaI

(2.61)

The necessary volume of the reaction mixture for a required performance (LP , kmol s−1 ) depends on the conversion of the key reactant A1 . This is summarized in Equations 2.62 and 2.63 for single reactions with a product selectivity of S2,1 = 1 ⇒ LP = ṅ 2,out = ṅ 1,0 (1 − X) = V̇ 0 c1,0 (1 − X) V =

LP k1 c1,0 (1 − X)

(2.62)

V =

LP 2 k1 c1,0 (1 −

(2.63)

X)2

For reactions with changing density because of the transformation, the relations between conversion and space time depend also on the expansion factor � (Equation 2.47). For continuous flow reactors � is defined as follows: �=

V̇ X=1 − V̇ X=0 V̇ X=0

(2.64)

The volume occupied by the reaction mixture depends on the arrangement of the outlet tubes and is, therefore, independent of the density. But, the outlet flow V̇ out is a function of the expansion factor. V̇ out = V̇ 0 (1 + αX)

(2.65)

In consequence, because of the density change, the mean residence time of the reaction mixture t is not identical to the space time (t ≠ �). The design equations for CSTR with density change of the reaction mixture is based on the mass balance (Equation 2.57). For single nth order reaction follows: V̇ 0 ci0 − V̇ out ci,out + V ⋅ R1 = V̇ 0 ci0 − V̇ out ci,out + V ⋅ (−k cn1 ) = 0

(2.66)

37

2 Basis of Chemical Reactor Design and Engineering

The concentration of A1 at the reactor outlet, which is identical with the concentration within the reactor, corresponds to the ratio between the molar outlet flow and the volumetric outlet flow. c1 = c1,out =

ṅ 1,out ṅ 1,0 ⋅ (1 − X) (1 − X) = = c1,0 (1 + αX) V̇ out V̇ 0 ⋅ (1 + αX)

(2.67)

With the mass balance Equation 2.58, we can now determine the necessary space time, respectively, the necessary Damköhler number, for obtaining a required conversion. c1,0 c1,0 (1 + �X)n V = X= n X −R1 k c1,0 (1 − X)n V̇ 0 (1 + �X)n n−1 = X DaI = � k c1,0 (1 − X)n �=

(2.68)

It is important to emphasize the fact that even for first order reactions the density change influences the performance of continuously operated reactors in contrary to batch reactors. In Figure 2.8 the influence of the expansion factor on the conversion of first order reactions is demonstrated. The heat balance at the steady state of the CSTR is expressed as: ∑ rj (−ΔHr,j ) = 0. (2.69) V̇ 0 �0 cp0 T0 − V̇ �cp T + U ⋅ A(Tc − T) + V j

For the simple case of a single reaction we obtain with the mass balance for the key component A1 − R1 V = r ⋅ V = Xc1,0 V̇ 0 V̇ 0 �0 cp0 T0 − V̇ �cp T + U ⋅ A(Tc − T) + V̇ 0 c1,0 X(−ΔHR ) = 0

(2.70)

1.0 α = –0.5 α=0

0.9 Conversion, X

38

α = +0.5

0.8

0.7

0.6 0

5

10 DaI = τ/tr

15

20

Figure 2.8 Conversion in a CSTR as function of the Damköhler number and the expansion factor for first order reactions.

2.3

Ideal Reactors and Their Design Equations

or, at constant specific heat capacity (cp = cp,0 ) V̇ 0 �0 cp0 (T − T0 ) + U ⋅ A(T − Tc ) = V̇ 0 c1,0 X(−ΔHR ).

(2.71)

Together with the mass balance, Equation 2.71 serves for the design of the reactor, that is, to determine the operating parameters (V̇ 0 , T0 , ci,0 ) for a required reactor performance. It is important to note that CSTR, operating at steady state, can be operated isothermally. The cooling temperature Tc and the heat exchange area A can be adapted for known inlet temperatures T 0 , and the required temperature inside the reactor, T (Figure 2.7). 2.3.4 Plug Flow or Ideal Tubular Reactor (PFR)

In contrast to the ideal CSTR, backmixing is excluded in an ideal tubular reactor, characterized by a plug flow pattern of the fluid, with uniform radial composition and temperature. The material balance for a small volume system element (ΔV ) shown in Figure 2.9 at the reactor steady state is written as −

d(ci V̇ ) ∑ d(c u) + vij rj = − i + Ri = 0 dV dz j

(2.72)

Any disturbance in the inlet flow travels with the linear velocity u through the reactor. Therefore, a novel stationary axial concentration profile is reached after the space time � = V ∕V̇ 0 . For a single, stoichiometrically independent reaction we obtain d(ci u) dṅ i = = vi r = Ri . (2.73) dV dz After the introduction of the conversion for the key component A1 , the design equations for irreversible nth order reactions are readily found: −R1 −R1 dX = = . dV ṅ 1,0 V̇ 0 ⋅ c1,0

(2.74)

The reactor volume required to achieve a target conversion can be calculated by integration. XL

V = ṅ 1,0

ni,0

dX ∫X0 −R1

ni ΔV

ni +

(2.75)

∂ni ∂V

z z + Δz z Figure 2.9

Ideal plug flow reactor.

ΔV

ni,L

39

2 Basis of Chemical Reactor Design and Engineering

c1,0 /(–R1) (s)

40

τCSTR

τPR

Conversion, X Figure 2.10 Space time in CSTR and PFR.

The space time in the reactor is then given by X

�=

L V dX = c1,0 . ̇V0 ∫X0 −R1

(2.76)

The space time necessary for a required conversion corresponds to the hatched surface under the curve c1,0 ∕(−R1 ) = f (X) shown in Figure 2.10, which is for a reaction with positive order (n > 0). The space time in a CSTR is represented by the whole gray rectangle. Because of the low reactant concentration within a CSTR, the space time in the latter is considerably higher leading to a poor reactor performance. For reactions with zero order (n = 0), the performance of CSTR and PFR are equal for any conversion. For the reactions with negative order, the transformation rate increases with decreasing reactant concentration and the performance of a CSTR will be higher compared to a PFR. Reactions with constant density (�0 = � = �out ): If the density of the reaction mixture does not change throughout the reactor, the linear velocity of the reaction mixture remains constant and Equation 2.73 can then be transformed with d� = dz/u to u

∑ dc dci V̇ = i = vij rj = Ri .; with u = 0 = constant. dz d� Acs j

(2.77)

For single irreversible reactions we obtain: XL c1,L dc1 1 1 1 dc; ��� = � ⋅ k c(n−1) = dX = R1 = −k cn1 ; � = − n 1,0 ∫0 (1 − X)n d� k ∫c1,0 c1

(2.78)

2.3

Integration leads to: ( ) c � k = ��� = − ln c1,L = − ln(1 − XL ) 1,0 [( )1−n ] c1,L 1 n−1 � k c1 = ��� = n−1 −1 = c 1,0

Ideal Reactors and Their Design Equations

for n = 1; � = 0 (1−XL )1−n −1 n−1

for n ≠ 1; � = 0

(2.79)

The derived expressions for the ideal tubular reactor are the same as those for an ideal, batchwise operated stirred tank reactor. The reaction time tR is replaced by the space time �, that is, the conversion achieved in a batch reactor is identical to that in an ideal flow tube when the reaction time tR and the space time � are equal. However, this comparison is no longer valid when the reaction is accompanied by a density change, which leads to variations in the linear velocity. Because of the volume change of the reaction mixture, the reactant concentration changes not only by chemical transformation, but also by expansion. Supposing a linear dependency between reaction volume and conversion (Equation 2.64), the concentration of reactant A1 at any point of the reactor is given by: c1 =

ṅ 1,0 ⋅ (1 − X) ṅ 1 (1 − X) = c1,0 = (1 + � X) V̇ V̇ 0 ⋅ (1 + � X)

(2.80)

The concentration has to be introduced in the transformation rate expression and in the general design equation (Equation 2.76) for determining the space time for a required conversion. The space time can always be found by numerical or graphical integration. However, for simple kinetics analytical integration is possible. For the following nth order reactions analytical solutions are obtained: XL dX General expression (Equation 2.76): � = VV̇ = c1,0 . ∫X0 −R1 0 XL

Zero order reaction: −R1 = k; � = c1,0 ��� =

∫X0

dX . k

k� =X c1,0

(2.81)

k

First order reaction: A1 −−→ products; − R1 = k c1 = k c10

XL (1 + � X) (1 − X) ; � = k c1,0 dX ∫X0 (1 − X) (1 + �X)

��� = k� = −(1 + �) ln(1 − X) − �X

(2.82)

k

Second order reaction: 2A1 −−→ products; XL (1 − X)2 (1 + � X)2 ; � = k c1,0 dX 2 ∫X0 (1 − X)2 (1 + �X) X ��� = k c1,0 � = 2 � (1 + �) ln(1 − X) + � 2 X + (1 + �)2 (2.83) 1−X The designs of reactors with volume change are illustrated in Example 2.5 and 2.6.

− R1 = k c21 = k c21,0

41

42

2 Basis of Chemical Reactor Design and Engineering

Example 2.5: Design of a plug-flow reactor for reactions with increasing volumetric flow. According to K. M. Watson [11] the noncatalyzed dehydrogenation of butene to butadiene can be described by the following expressions: C4 H8 → C4 H6 + H2 −R1 = k ⋅ p1 (kmol ⋅ h−1 ⋅ m−3 ) ) ( −30 200 kmol ⋅ m−3 ⋅ h−1 ⋅ 10−5 Pa−1 k = 1.75 ⋅ 1015 exp T Determine the space time in an isothermal plug flow reactor to achieve 90% conversion (X = 0.9) of butene under the following conditions: T = 923 K ṅ 10 = 1 kmol ⋅ h−1 butene p = 105 Pa ṅ 1 = 1 kmol ⋅ h−1 water vapor Solution: The reaction is accompanied by a change in volume, as it is performed at constant pressure. The reaction mixture is assumed to behave as an ideal gas, so that the volume changes linearly with increasing butene conversion. As a mixture of butene and inert water vapor is employed, we obtain for the expansion factor: V − VX=0 3−2 = 0.5. = � = X=1 VX=0 2 The partial pressure of butene is then given by p1 =

ṅ (1 − X) ṅ 1 ⋅ RT 1−X ∶ = c10 ⋅ RT = RT 10 ̇V ̇V0 (1 + �X) 1 + �X

Substitution in the rate equation and the mass balance leads to XL

XL

(1 + �X) dX ∫0 (1 − X) 1 �= [−�XL − (1 + �) ln(1 − XL )]. k ⋅ RT Thus, after the insertion of the numerical values, we obtain: � = c10

�=

dX 1 = −R1 k ⋅ RT ∫0

105 ⋅ 3.00 = 3.62 ⋅ 10−3 h = 13 s. 1.75 ⋅ 1015 exp(−30 200∕923) ⋅ 8313 J ⋅ 923

2.3

Ideal Reactors and Their Design Equations

Example 2.6: Design equation for 1/2 order reaction with volume change. k

The kinetics of a homogenous cracking reaction A1 −−→ 3A2 can be described with the following rate equation: 5 5 −R1 = k c0.5 1 ; 450 K ≤ T ≤ 550 K; 4 ⋅ 10 Pa ≤ p ≤ 6 ⋅ 10 Pa

A PFR is operated at 500 K and 6 ⋅ 105 Pa with pure A1 at the entrance. The initial concentration is c1,0 = 0.528 kmol m−3 and the rate constant is found to be k = 0.02 kmol1/2 m−3/2 s−1 under the reaction conditions. Find the space time needed for a conversion of X = 0.8. Solution: For the given stoichiometry and with the pure reactant at the reactor entrance, one volume of the feed gas will give three volumes of product gas at full conversion. The expansion factor is � = (3 − 1)∕1 = 2. Therefore, the transformation rate is given by: ) ( 1 − X 1∕2 1∕2 −R1 = k c1,0 1+2⋅X For the design equation (Equation 2.76) we obtain XL ( ) 1 1 + 2 X 1∕2 dX � = c1,0 ∫0 k c1∕2 1 − X 1,0 or, by introducing the Damköhler-number: XL ( ) 1 + 2 X 1∕2 −1∕2 ��� = �k c1,0 = dX ∫0 1−X The necessary Damköhler number for a required conversion can be √ obtained by plotting (1 + 2X) ∕ (1 − X) as function of X and determining the area under the curve between the initial and final conversion as shown in Figure 2.11a. A simple way to estimate the area under the curve is to use the trapezoidal rule. In this case we break up the function into a number of trapezoids and calculate their areas. The area under the curve is then approximated by the sum of the trapezoids as shown in Figure 2.11b. The accuracy of the numerical integration increases with decreasing spaces between the points. In the present example with four trapezoids, the area under the curve is esti0.8 √ (1 + 2X)∕(1 − X) dX ≅ 1.54. mated to be ��� = ∫0 The required space time for obtaining a conversion of X = 0.8 is: √ √ c1,0 0.528 � = ��� ⋅ ≅ 1.54 ⋅ = 56 s k 0.02

43

2 Basis of Chemical Reactor Design and Engineering 4.0

4.0

3.5

3.5

(1 + 2 X )0.5 (1 – X )–0.5

(1 + 2 X )0.5 (1 – X )–0.5

44

3.0 2.5 2.0 1.5 1.0 0.5 0.0

(a)

0.8 (1 + 2 • X)1/2

Dal = ∫ 0

0.2

0.4

(1 – X)1/2 0.6

Conversion, X

dX 0.8

3.0 2.5 2.0 1.5 1.0 0.0

1.0 (b)

0.2

0.4

0.6

0.8

1.0

Conversion, X

Figure 2.11 (a, b) Determining the Damköhler number for a given conversion.

The heat balance for a volume element in an ideal PFR at steady state can be formulated as follows: dT ∑ dA ̇ p = . (2.84) rj (−ΔHr,j ) + U(Tc − T) mc dV dV j In accord with the previously made assumptions, the temperature over the cross section is constant and only a function of the axial position. The term (dA∕dV ) corresponds to the reactor surface per volume element (specific surface area, a). For circular tubes with a constant diameter dt follows: a=

dA 4 = dV dt

(2.85)

In general, the heat exchanged through the tube wall will be different from the heat generated or consumed by the reaction at the same axial position. As a consequence, an axial temperature profile develops. Exceptions are reactions with formally zero order, which doesn’t depend on the reactant concentration. To determine the axial temperature and concentration profiles, heat and mass balances must be solved simultaneously. For a single, stoichiometrically independent reaction and under stationary conditions, we obtain for the key component A1 −R1 ⋅ � dX = dZ c1,0 dT ̇ p � �(Tc − T) + r(−ΔHr ) − mc =0 dV ) ( U ⋅� dA r⋅� dT (Tc − T) + ΔTad = dZ dV c1,0 �0 ⋅ cp

(2.86)

(2.87)

Z = z/L (relative length of the reactor) The heat management of tubular reactors is discussed in detail in Chapter 5.

2.4

Homogenous Catalytic Reactions in Biphasic Systems

2.4 Homogenous Catalytic Reactions in Biphasic Systems

A drawback of homogenous catalytic processes is often the complex and costly separation and recycling of the catalyst. Therefore, considerable efforts are made to combine the easy separation of heterogenous catalysts with the high potential activity and selectivity of homogenous molecular catalysts. Different methods are proposed to facilitate the recovery of the catalyst. A very successful way is to use biphasic systems of two immiscible liquids. The catalyst should be soluble only in one phase, in which the transformation takes place, while the products and sometimes the reactants should be preferentially soluble in the second. The catalyst is thus “immobilized” within a “liquid support.” The immiscible liquids can be separated after the reaction and the catalyst is recycled. This can be done without any thermal or chemical treatment. As the reaction is carried out in the presence of dissolved catalyst, the advantages of homogenous catalysis are fully preserved. The liquid support may be water, supercritical fluids, ionic liquids, organic liquids or fluorous liquids [12]. The Shell higher olefin process (SHOP) and the Oxo synthesis (hydrofomylation) are examples of important industrial processes based on biphasic catalytic systems. As the reaction takes place in the catalyst containing phase, the reactants must, first of all, be transferred from the second and eventually gas phase to the reaction phase. Therefore, special attention has to be paid to the mixing and dispersion of one phase within the other and mass transfer efficiency between phases. The mass transfer rate between the different phases depends on the area of the interface and the mass transfer coefficient. Whether the reaction will take place in the bulk of the reaction phase or near the interface depends on the ratio between the characteristic reaction time (tr ) and the characteristic time for mass transfer (tm ). This ratio is known as the Hatta number (Ha). The discussion can be facilitated on the basis of the film model and by supposing a first order irreversible reaction in the reaction phase and neglecting the mass transfer resistance in the non-reactive phase I [13–15]. √ �a =

tm = �II tr

√ k′ = D1,II

√

k ′ D1,II kL,II

(2.88)

With � II : the thickness of the boundary layer; k ′ : the reaction rate constant, which is a function of the catalyst concentration (k ′ = k ⋅ ccat ); D1,II : the diffusion coefficient for the compound 1 in the second liquid phase; and kL : the mass transfer coefficient in this liquid phase (phase II). Depending on the value of Ha, different regimes can be distinguished (Figure 2.12): For Ha ≤ 0.3 the reaction rate is slow compared to the mass transfer and the reaction takes place in the bulk phase (Figure 2.12a). In this case, the mass transfer can be considered as an additional resistance in series to the

45

46

2 Basis of Chemical Reactor Design and Engineering

Gas-phase phase I Interface ∗ c1,II

Reaction phase phase II

Reaction phase phase II

∗ c1,II

Main body of liquid (bulk)

∗

∗ C1,G =

Gas-phase phase I Interface

P1 RT

∗

Liquid (bulk)

∗

C1,G =

P1 RT

c1,II

c1,II �II (a)

�II y

(Stagnant) liquid film

(Stagnant) liquid film

(b)

Gas phase phase I

y

Reaction phase phase II

Interface c∗1,II ∗ C1,G =

Liquid (bulk)

∗

P1 RT

�II

c1,II ; 0 y

(Stagnant) liquid film

(c)

Figure 2.12 Concentration profiles for mass transfer with pseudo first order chemical reaction (film model) (a) slow chemical reaction: Ha ≤ 0.3; (b) moderate chemical

reaction: 0.3 ≤ Ha ≤ 3.0; and (c) fast chemical reaction: Ha ≥ 3.0. (Adapted from Ref. [15], Figure 4.20 Copyright © 2012, Wiley-VCH GmbH & Co. KGaA.)

reaction. The effective (observed) reaction rate is given by: ( rov =

1 1 + kL a k ′

)−1 c1,II

(2.89)

For values of the Hatta number of 0.3 ≤ Ha ≤ 3, the reaction takes place partially in the boundary layer and in the bulk of phase II. This leads to the deformation of the concentration profile in the stagnant film from the straight line as presented in Figure 2.12b. The overall rate of reaction is given by the reaction in the bulk at reactant concentration c1,II and in the boundary layer. This leads to the following expression for the observable effective rate:

reff

[ ] c1,II 1 Ha 1− ∗ ⋅ ⋅ kL a ⋅ c∗1,II = tanh �a c1,II cosh Ha

(2.90)

2.4

Homogenous Catalytic Reactions in Biphasic Systems

The bulk concentration c1,II is a rather complex function of the intrinsic reaction rate, the mass transfer coefficient, and the area of the interface [15]. c1,II c∗1,II

=

1 cosh Ha[1 + Ha(1∕B − 1) ⋅ tanh Ha]

(2.91)

where B = A ⋅ �II ∕VII = Vfilm ∕VII corresponds to the ratio between the film volume (V film ) and the volume of the reacting phase (VII ). Practical values for B are found to be in the range between B = 0.1 (for highly efficient fluid/fluid contactors) and B = 2 ⋅ 10−4 . Therefore, for Ha > 1 the concentration in the bulk phase can be neglected (c1,II ∕c∗1,II ≅ 0) and the effective rate becomes: reff ≅

Ha ⋅ k a ⋅ c∗1,II , Ha > 1 tanh Ha L

(2.92)

A further increase of the intrinsic reaction rate at constant volumetric mass transfer coefficient (k L,II ⋅a) results in Hatta numbers greater than 3 (Ha > 3). The reaction rate can be considered as very fast compared to the mass transfer rate. As a consequence, the reactants do not reach the bulk phase (c1,II ≈ 0); the reaction takes place only in the boundary layer (Figure 2.12c). Under these conditions, the reaction rate increases proportionally with the specific interfacial area between the phases (a), the square root of the reaction rate constant, and the catalyst concentration as indicated in Equation 2.93. reff = kL,II ⋅ a ⋅ Ha ⋅ c∗1,II =

√ √ k ′ ⋅ D1,II ⋅ a ⋅ c∗1,II = k ⋅ ccat ⋅ D1,II ⋅ a ⋅ c∗1,II , Ha ≥ 3 (2.93)

Example 2.7: Influence of catalyst concentration on the effective reaction rate. Catalytic hydrogenation is carried out in a two-phase batch reactor, with the rate proportional to the catalyst concentration in the reaction phase. The intrinsic reaction rate was found to be k ′ = k ⋅ ccat = 1.12 ⋅ 102 s−1 . From the literature data it was found: kL = 5 ⋅ 10−4 m s−1 D1 = 5 ⋅ 10−9 m2 s−1 If one wants to double the effective hydrogenation rate, how should the concentration of the catalyst be changed? Solution: Ha =

√ D1 ⋅ kccat kL

= 1.5 reff ≅

Ha 1.5 ⋅ k a ⋅ c∗1,II , reff ≅ ⋅ k a ⋅ c∗1,II tanh Ha L 0.91 L

To double the hydrogenation rate, we obtain (From Equation 2.92 and 2.93): reff,2 = 2 ⋅ reff,1 ;

reff,2 reff,1

=

Ha2 ⋅ kL a ⋅ c∗1,II Ha1 tanh Ha1

⋅ kL a ⋅ c∗1,II

=2

47

2 Basis of Chemical Reactor Design and Engineering

Ha1 1.5 = 3.3 =2⋅ tanh Ha1 0.91 √ ccat,2 Ha2 ⇒ = 2.2 = ⇒ ccat,2 = 4.84 ⋅ ccat,1 Ha1 ccat,1

Ha2 = 2 ⋅

So, the catalyst concentration must be approximately fivefold higher in order to double the hydrogenation rate. In summary, high Ha values lead to low reactant concentration in the reacting bulk phase and, as a consequence, the available volume of the reacting phase is less and less utilized. This situation can be characterized by introducing an efficiency factor �. The efficiency factor is defined as the ratio between the observed effective rate and the maximum production rate (controlled by the intrinsic kinetics) referred to the reactor volume (VR ) and corresponding to the maximum reactant concentration in the reacting phase (c1,II = c∗1,II ): �=

reff V ; rmax = k ′ II c∗1,II rmax VR

(2.94)

The reactor efficiency depends on the Ha-number and the specific interfacial area. For a first order irreversible reaction the following relationship is obtained: ] [ a ⋅ VR ⋅ D1,II tanh (��) + (B−1 − 1)�� B (2.95) �= with B = �� 1 + (B−1 − 1)�� tanh(��) VII ⋅ kL The parameter B can be interpreted as the ratio between the film volume (V film ) and the volume of the reacting phase (VII ). In Figure 2.13 the efficiency factor as 1 Ha = 0.1

Reactor efficiency, η

48

0.1 Ha = 0.3

0.01 Ha = 1

0.001

Ha = 3

B= Ha = 5

0.0001 0.001

0.01

a • VR • D1,II VII • kL 0.1

B Figure 2.13 Effectiveness factor for fluid/fluid reactions as function of Ha and B. (Adapted from Ref. [16], Figure 4.21 Copyright © 2012, Wiley-VCH GmbH & Co. KGaA.)

2.5

Heterogenous Catalytic Reactions – +

Support –

+

L1 –

L2 Me

+ –

L2 –

Porous support

Liquid phase catalyst

+

L1

–

+

Packed bed

49

+

Ionic liquid/ homogenous catalyst

Figure 2.14 Schematic presentation of the concept of supported liquid phase catalysis. (Adapted from Ref. [16], Figure 4.22 Copyright © 2012, Wiley-VCH GmbH & Co. KGaA.)

function of Ha and B is shown. It clearly demonstrates that the reactor efficiency decreases with increasing Ha and with decreasing specific interfacial area a. These relations are strictly valid only for simple irreversible first order reactions, but appropriate models for more complex kinetics can accordingly be developed based on the film model. As shown above, fast chemical transformations characterized by Ha ≥ 3 occur mainly near the interface and thus are limited by the interfacial area, which must be continuously generated by vigorous stirring of the multiphase mixture. A possibility to overcome this drawback consists of immobilizing the liquid on a highly porous support. In this way a thin liquid layer is formed on the solid support leading to the desired high fluid/fluid interface. This approach is called supported liquid phase catalyst (SLPC) and combines the advantages of homogenous catalysis with a heterogenous fluid/solid system discussed in the following section. SLPC can be used like traditional heterogenous catalysts in packed bed reactors or even in fluidized beds. A schematic representation is shown in Figure 2.14. The main problem related to SLPC is the loss of solvent because of evaporation in a continuously operated catalytic reactors. This problem can be overcome by using ionic liquids as solvent [17–20]. Ionic liquids are molten salts and their partial pressure is low under conditions commonly used for hydroformylation and hydrogenation reactions. As generally observed for SLPC, the catalytic activity and product selectivity depends on the liquid loading and the nature of the porous support [21]. A detailed discussion can be found in [22]. In order to diminish internal diffusion resistances within the supported liquids by using microstructured supports with high porosity like foams or fibrous materials, are proposed for SLPC [23].

2.5 Heterogenous Catalytic Reactions

The advantage of the heterogenous catalysis over homogenous is the easy postreaction separation of the catalyst which can further be used after regeneration and

50

2 Basis of Chemical Reactor Design and Engineering

the possibility to apply open reactors (flow processes). The chemical transformation occurs through new reaction pathways, which usually have lower energies of activation compared to the noncatalyzed reaction, and through formation of adsorption complexes. In a few cases, the kinetics of heterogenous catalyzed reactions is based on a complete knowledge of the underlying reaction mechanisms. Generally, the kinetics of many commercially important reactions are derived from experimental investigations and are often based on simplified reaction models. 2.5.1 Rate Equations for Intrinsic Surface Reactions

No catalytic reaction can be elementary as at least three steps are always involved: adsorption of the reactant, surface reaction and desorption of the formed product. For a simple monomolecular reaction, for example, an isomerization, the steps involved are shown in Figure 2.15. To describe the catalytic reaction, the catalyst must be included in the catalytic cycle as a participating species. The simplest way to do so is to consider a solid catalyst as an ensemble of single active sites (*). The transformation from A1 to A2 can be presented as a sequence of elementary steps: k1

∗ −−−−−−−−−−−−− → A1 + ∗ ← − A1

reactant adsorption∕desorption

k−1

k2

A∗1

∗ −−−−−−−−−−−−− → ← − A2 k-2

A∗2

−−−−−−−−−−−−− → ← − A2 + ∗ product desorption∕adsorption

reversible surface reaction

k3

k−3

(2.96)

−−−−−−−−−−−−− → ← − A2

A1

The adsorption of the reactant is herein considered as a reaction with an empty site (*) to give an adsorbed intermediate A∗1 . All sites are considered as equivalent and each can be occupied by a single species only. Considering all steps as elementary reactions, expressions for the rate of each step can be obtained: Rad = r1 − r−1 = k1 p1 Ztot �v − k−1 Ztot �1 A1

A1

A2 ∗

A1

A2

∗

A2

Solid catalyst Figure 2.15 Schematic presentation of a catalytic reaction.

(2.97)

2.5

Heterogenous Catalytic Reactions

Rrx = r2 − r−2 = k2 Ztot �1 − k−2 Ztot �2

(2.98)

Rdes = r3 − r−3 = k3 Ztot �2 − k−1 p2 Ztot �v

(2.99)

The total concentration of active sites is represented by Z tot , and � 1 , � 2 , � v are the fractions occupied by A1 , A2 and the fraction of vacant sites, respectively. In order to derive the overall rate equation of this isomerization reaction, one should know the fraction of the sites occupied by each species, �1 and � 2 , which is called fractional surface coverage: �i =

Zi Ztot

(2.100)

The coverage of a catalyst surface by gaseous molecules at constant temperature depends on the partial pressure of this gas above the surface. The quantitative relationships are called isotherms. 2.5.1.1 The Langmuir Adsorption Isotherms

For describing the kinetics of heterogenous catalytic reactions, the Langmuir adsorption isotherms are used mainly. We now derive them for associative, dissociative, and competitive adsorption. The main assumptions are the following:

• The solid surface is uniform and contains a number of equivalent sites, each can be occupied by only one species of adsorbate;

• A dynamic equilibrium exists between the gas and the adsorbed molecules at constant temperature and pressure; adsorbate molecules from the gas phase are continually colliding with the surface. If they impact a vacant adsorption site, they may form a bond with the site and stick. If they strike a filled site, they are reflected back into the gas phase; • Once adsorbed, the molecules are localized • The enthalpy of adsorption per site remains constant irrespective of coverage (no lateral interaction between the adsorbed species). When molecules are hitting the surface, they can interact by bonding with an active site being attached for some time. This process can be considered as chemical reaction and are characterized by the rates of adsorption, rads , and desorptions, rdes . A1 +∗ → A∗1 rads = k1 p1 �v A∗1 → A1 +∗ rdes = k−1 �1

(2.101)

When the equilibrium is attained, rdes = rads , and after introducing the adsorption equilibrium constant K1 = k1 ∕k−1 , we can write the Langmuir adsorption isotherm for associative adsorption of gas A1 (without any dissociation on interaction with the surface) and only one adsorbing gas present (Example 2.8): �1 =

K1 ⋅ p1 1 + K1 ⋅ p1

(2.102)

51

52

2 Basis of Chemical Reactor Design and Engineering

Example 2.8: Adsorption isotherm. Determine the fraction of a catalyst surface occupied under equilibrium for a species A1 at the partial pressure of 1, 2, and 5 bar (K = 5 bar−1 ). Solution: The amount of surface occupied can be calculated using Langmuir adsorption isotherms given by Equation 2.102. �1 (p1 = 1 bar) =

K1 ⋅ p1 5⋅1 = 0.83 = 1 + K1 ⋅ p1 1+5⋅1

�1 (p1 = 2 bar) = 0.9 �1 (p1 = 5 bar) = 0.96

The common form to present this equation as a linear dependence of 1/� 1 against 1/p1 allows to find experimentally the constant of adsorption equilibrium, K 1 , and to verify a consistency of the Langmuir assumptions: 1 1 = +1 �1 K1 ⋅ p1

(2.103)

For dissociative adsorption (when molecules break their bonds on interaction involving two surface sites) the same considerations can be applied leading to the corresponding isotherm: ∗ −−−−−−−−−−−−− → A2 + 2 ∗ ← − 2 A1 √ K1 ⋅ p1 �1 = √ 1 + K1 ⋅ p1

(2.104)

This is a very important case because molecules like H2 or O2 (participating in catalytic hydrogenation and oxidation reactions) often dissociate on the catalytic surface adsorbing with fragmentation. This brings some consequences for the observed kinetics and optimization of the reaction conditions. Competitive adsorption takes place when two (or more) different molecules are in the gas phase and compete for the same sites. If each species adsorbs on one site only without dissociation, the corresponding Langmuir isotherm is as follows: �i =

K i pi ∑ 1+ Ki pi

(2.105)

i

where � i is the fractional coverage; Ki , the constant of adsorption equilibrium of molecule Ai . Data for the isotherm can be obtained experimentally from the equilibrium coverage of the surface at a particular temperature over a range of pressures and then presented in a linear form allowing finding Ki . From the Ki at different temperatures, the heat of adsorption (ΔH a,i ) can be estimated using the van’t Hoff

2.5

Heterogenous Catalytic Reactions

equation: ( Ki,T2 = Ki,T1 exp

( )) −ΔHa,i T1 − T2 RT1 T2

(2.106)

As the adsorption is always exothermic, Ki decreases with temperature. 2.5.1.2 Basic Kinetic Models of Catalytic Heterogenous Reactions

In general, a mechanism for any complex reaction (catalytic or non-catalytic) is defined as a sequence of elementary steps involved in the overall transformation. To determine these steps and especially to find their kinetic parameters is very rare if at all possible. It requires sophisticated spectroscopic methods and/or computational tools. Therefore, a common way to construct a microkinetic model describing the overall transformation rate is to assume a simplified reaction mechanism that is based on experimental findings. Once the model is chosen, a rate expression can be obtained and fitted to the kinetics observed. Some basic models often used for heterogenous catalytic reactions are described and the overall rate expressions are developed. Langmuir-Hinshelwood Model The main assumption of this model is that the catalytic reaction proceeds only via chemical adsorption of all reactants on the catalytic surface and the transformation takes place as a series of surface reactions ending up with a desorption of the products. Let’s first consider a monomolecular transformation, like the catalytic isomerization of hydrocarbons, as a simple example.

−−−−−−−−−−−−− → A1 ← − A2

(2.107)

The transformation can be described by considering three surface processes as shown in Equation 2.108: adsorption of the reactant, surface reaction, and desorption of the product. If the reaction is carried out in an open reactor under constant conditions, for example, in a catalytic packed bed ( ) reactor, the fractions occupied d� d� by A1 and A2 are time invariant dt1 = dt2 = 0 . With Equations 2.97–2.99 we obtain: d�1 = k1 Ztot p1 �v − k−1 Ztot �1 − k2 Ztot �1 + k−2 Ztot �2 = 0 dt d�2 = k2 Ztot �1 − k−2 Ztot �2 − k3 Ztot �2 + k−3 Ztot p2 �v = 0 dt

(2.108)

As the open reactor operates under stationary conditions, accumulation of products and reactants are excluded. In consequence, the transformation rate of

53

54

2 Basis of Chemical Reactor Design and Engineering

A1 corresponds to the production rate of A2 . − R1 = R2 or − k1 p1 �v + k−1 �1 = k3 �2 − k−3 p2 �v

(2.109)

� 1 + �2 + �v = 1

(2.110)

with We can eliminate the different occupied fractions of active sites and we get finally the dependence of the production rate as function of the partial pressures of A2 and A1 . R2 =

kZtot (p1 − p2 ∕K) 1 + kI p1 + kII p2

(2.111)

With: k=

k1 k2 k3 k−1 (k−2 + k3 ) + k2 k3

kI =

k1 (k2 + k−2 + k3 ) k−1 (k−2 + k3 ) + k2 k3

kII =

k−3 (k−1 + k2 + k−2 ) k−1 (k−2 + k3 ) + k2 k3

K=

k1 k2 k3 k−1 k−2 k−3

(2.112)

It is evident that the six individual rate constants cannot be obtained under steady-state reaction conditions. To estimate their values, independent measures of the adsorption and reaction behavior under transient (non-steady-state) conditions are necessary. The Quasi-Surface Equilibrium Approximation If we suppose that the adsorption and desorption processes are fast compared to the surface reaction, we can estimate the surface concentrations from the equilibrium constants. With the Langmuir adsorption isotherm, the following relations result for the simple monomolecular reaction presented in Equation 2.107. K 1 p1 �1 = 1 + K1 p1 + K2 p2

�2 =

K 2 p2 1 + K1 p1 + K2 p2

with ∶ K1 ≃

k1 k ; K2 ≃ 2 k−1 k−2

(2.113)

2.5

Heterogenous Catalytic Reactions

The transformation rate is then simply given by: ( ) k2 Ztot K1 p1 − p2 ∕Keq k K −R1 = k2 Ztot �1 − k−2 Ztot �2 = , Keq = 2 1 (2.114) 1 + K1 p1 + K2 p2 k−2 K2 For a quasi-irreversible reaction and negligible product adsorption (Keq → ∞, K2 ≪ K1 ), Equation 2.114 is further simplified to give: −R1 =

k2 Ztot K1 p1 1 + K 1 p1

(2.115)

If we divide denominator and nominator by K 1 we obtain the Michaelis– Menten equation, where KM corresponds to 1/K 1 . The Most Abundant Surface Intermediate (MASI) Approximation Catalytic transfor-

mations may include the formation of many intermediates on the catalyst surface, which are difficult to identify. In these cases, it is impossible to formulate a kinetic model based on all elementary steps. Often, one of the intermediates adsorbs much more strongly in comparison to the other surface species, thus occupying nearly all active sites. This intermediate is called the most abundant surface intermediate “masi” [24]. For a simple monomolecular reaction, A1 → A2 , the situation can be illustrated with the following scheme: k1

A1 + ∗ −−→ A∗1 k2

A∗1 −−→ I2∗ ⋅ ⋅ kn−1

∗ In−1 −−−−→ In∗ (masi) kn

In∗ −−→ A2 + ∗

(2.116)

Neglecting all intermediates having a very short lifetime on the catalyst results in: �v + �masi ≃ 1

(2.117)

The transformation rate of reactant A1 corresponds to the first step in Equation 2.116: −R1 = k1 Ztot p1 �v = k1 Ztot p1 (1 − �masi )

(2.118)

The final product is formed in the nth step and corresponds to the transformation to A2 and its desorption: R2 = kn Ztot �masi

(2.119)

As steady state holds, R2 = −R1 , and � v can be easily calculated and the final expression for describing the production rate is given by: R2 =

k1 Ztot p1 k ; with K = 1 1 + Kp1 kn

(2.120)

55

56

2 Basis of Chemical Reactor Design and Engineering

It is important to underline that the mathematical form of the obtained kinetic equations (Equations 2.111, 2.114, and 2.120) are quite similar, whereas the interpretation of the corresponding model parameters and the physical meaning of the constants are very different. Bimolecular Catalytic Reactions Supposing that the surface reaction is the ratedetermining step, we obtain for an irreversible bimolecular reaction the following relations:

A 1 + A2 → A3 − R1 = kZtot �1 �2

(2.121)

The surface fractions occupied by the reactants A1 and A2 are given by the Langmuir isotherm, supposing competitive adsorption and neglecting the coverage by the product. �1 =

K 1 p1 1 + K1 p1 + K2 p2

�2 =

K 2 p2 1 + K1 p1 + K2 p2

�3 ≃ 0 k ⋅Z K p ⋅K p −R1 = R3 = k3 ⋅ Ztot �1 ⋅ �2 = 3 tot 1 1 2 22 (1 + K1 p1 + K2 p2 )

(2.122) (2.123)

At constant pressure of the reaction partner A2 , the transformation rate as function of p1 passes through a maximum (demonstrated in Example 2.9). The maximum depends on p2 and the values of the adsorption constants K 1 and K 2 . The optimum pressure for A1 can easily be calculated with Equation 2.124. p1,op =

1 + K 2 p2 K1

(2.124)

Example 2.9: Maximum rate of bimolecular catalytic reaction. Investigate the surface coverage and normalized transformation rate (R1 /R1, max ) as a function of the mole fraction of A1 for a reaction given by Equation 2.121. The constants are K 1 = 2 bar−1 and K 2 = 3 bar−1 while p2 = 1 bar. Solution: The transformation rate is maximal for � 1 = � 2 . The surface coverage � 1 and � 2 can be calculated using Equation 2.122. To calculate the unknown p1 , the mole fraction, y1 = p1 /(p1 + p2 ), in the range of 0–1 can be assumed. The transformation rate is given by Equation 2.121. However, we have to also calculate

2.5

Heterogenous Catalytic Reactions

the maximum transformation rate R1, max . As the R1 is directly proportional to product � 1 � 2 , the maximum rate can be achieved when the product is maximum. Thus, R1 � � = 1 22 (2.125) R1,max �1 The results are illustrated in Figure 2.16.

Surface fraction, normalized rate (–)

1 R1/R1,max 0.8 θ2 0.6

0.4

0.2 θ1 0

0

0.2 0.4 0.6 0.8 Mole fraction, y1 = p1 /(p1 + p2) (–)

1

Figure 2.16 Surface coverage of A1 and A2 and the normalized transformation rate as function of the mole fraction of A1 .

2.5.2 Deactivation of Heterogenous Catalysts

In heterogenous catalytic reactions, a decrease in catalyst activity is often observed with increasing operation time. There are many reasons for this; the most important factors can be classified into three groups:

• Poisoning of the catalyst surface by irreversible adsorption and/or reaction of a chemical species, thus making the active centers required for the catalyzed reaction inactive. Example is CO adsorption on iron catalysts used for the ammonia synthesis. • Coverage of the surface with substances that leads to a mechanical blockage of the catalytically active surface. Example is deposition of coke in various hydrocarbon reactions such as isomerization, cyclization, and cracking.

57

58

2 Basis of Chemical Reactor Design and Engineering

• Decreasing of the active surface by sintering and recrystallization processes. Example is the decrease of the active nickel surfaces through recrystallization on alumina-supported nickel catalysts in hydrogenation reactions. For the course of a catalytic reaction whose kinetics can be described as: r=

kr′ Ztot K1 p1 k 1 p1 = 1 + K 1 p1 1 + K1 p1

(2.126)

the number Z tot of active surface sites can decrease with the operating time (also known as lifetime) t ′ of the catalyst, for example, by poisoning through components present in the system that do not take part in the reaction. The decreasing catalyst activity can be taken into account in the kinetic model by introducing an activity factor, a(t ′ ), which is a function of the operation time t ′ . a(t ′ ) =

kr (t′ ) kr (t ′ = 0)

(2.127)

Thus, the activity factor corresponds to the ratio between the rate constant after a certain time of operation referred and the initial value. It is important to underline that this way of including the deactivation is possible only if the deactivation kinetics is separable from the transformation kinetics, viz. the kinetic model for the transformation is not altered by the deactivation process. In the present case, the rate of deactivation rd corresponds to the change in the number of active surface sites with time of operation. The rate constants for catalysts undergoing deactivation can be formulated as follows: kr (t ′ ) = kr (t ′ = 0) ⋅ a(t ′ ).

(2.128)

The rate of deactivation can depend on the temperature, the activity factor a(t′ ) of the catalyst, the concentration of a component cdea (causing deactivation), and on the activation energy Ed of the deactivation process. −rd = kd0 e

( E ) − RTd

f (a, cdea )

(2.129)

If the deactivation process is slow compared to the rate of transformation, the activity is quasi uniform in the reactor and Equation 2.129 can be formulated simply as a power term, and we obtain da = kd an cm . (2.130) dea dt ′ For the case that m = 0, that is, there are no poisoning components in the reaction mixture but rather a sintering process is the cause of the deactivation, and n = 1, then Equation 2.129 simplifies to −rd = −

−rd = kd0 e

( E ) − RTd

a

(2.131)

If under the intrinsic conditions (without transport disguises) of a catalytic process, rd depends only on cdea , the poisoning may be supposed to be the main cause of deactivation. For more complex situations, like in the case of fast deactivation, “a” becomes not only a function of time but also a function of location within

2.6

Mass and Heat Transfer Effects on Heterogenous Catalytic Reactions

an eventual reactor. For detailed studies on deactivation kinetics, the reader is referred to [25, 26].

2.6 Mass and Heat Transfer Effects on Heterogenous Catalytic Reactions

Heterogenous catalytic reactions involve by their nature a combination of reaction and transport processes, as the reactants must be first transferred from the bulk of the fluid phase to the catalyst surface, where the reaction occurs. The combined reaction and transport processes are shown schematically in Figure 2.17. We suppose a porous catalyst particle with a large specific surface area surrounded by liquid or gaseous reaction mixture. For the transformation of the reactant A1 to the product A2 , the following steps are necessary:

1) 2) 3) 4) 5) 6) 7)

external diffusion of reactants (film diffusion) internal diffusion of reactants (pore diffusion) adsorption of the reactants on the surface catalytic reaction on the surface desorption of the products internal diffusion of products (pore diffusion) external diffusion of products (film diffusion). Reaction: reactant A1 →product A2

Boundery layer

Porous catalyst 2 3 4

1

A1

5 6

7

A2

Figure 2.17 Physical and chemical steps involved in heterogenous catalytic reactions. (Adapted from Ref. [16], Figure 4.1 Copyright © 2012, Wiley-VCH GmbH & Co. KGaA.)

59

60

2 Basis of Chemical Reactor Design and Engineering

If the rates of the chemical steps 3–5 are comparable or higher than the transport processes 1, 2 and 6, 7, significant concentration profiles of A1 and A2 inside the catalyst particle or in the surrounding layer will occur. If the intrinsic rates are very high as compared to the diffusion process in the pores, the reaction will take place only near the external surface, and the observed transformation rate will be controlled by the external mass transfer. The same situation is observed for nonporous pellets or so-called “egg-shell” catalysts, where the active phase is placed in a layer near the outer pellet surface. If the intrinsic reaction rate is comparable with the diffusion rate within the pores, a pronounced concentration profile of the reactant A1 within the pellet will develop. Simultaneously to the chemical transformation, heat is released or consumed in the case of exothermic or endothermic reactions. Consequently, temperature gradients inside and outside of the catalyst pellet will develop. The different situations are illustrated in Figure 2.18. As a consequence of the concentration profiles caused by the transfer phenomena, the observed (effective) reaction rates are modified compared to the rate, which would occur at constant bulk phase concentration. This effect is commonly characterized by an effectiveness factor as defined in Equation 2.132: �ov =

observed rate of reaction rate of reaction at bulk concentration and temperature

(2.132)

Besides the modification of the overall reaction rate, the product selectivity may be changed. This is discussed in detail in the following subsections. 2.6.1 External Mass and Heat Transfer

The first step in heterogenous catalytic processes is the transfer of the reactant from the bulk phase to the external surface of the catalyst pellet. If a nonporous catalyst is used, only external mass and heat transfer can influence the effective rate of transformation. The same situation will occur for very fast reactions, where the reactants are completely consumed at the external catalyst surface. As no internal mass and heat transfer resistances are considered, the overall catalyst effectiveness factor corresponds to the external effectiveness factor, � ex . For a simple irreversible reaction of nth order, the following relation results: ( ) k(Ts ) ⋅ cni,s k(Ts ) ci,s n = (2.133) �ov = �ex = k(Tb ) ⋅ cni,b k(Tb ) ci,b 2.6.1.1 Isothermal Pellet

The external mass transfer process can be described by the so-called film model as shown in Figure 2.19. According to the film model, a stagnant fluid layer of thickness � surrounds the external surface, where the total resistance to mass transfer is located. Accordingly, the concentration profile is confined to this layer. The molar flux of reactant Ai is proportional to the difference in concentration (the driving

2.6

Mass and Heat Transfer Effects on Heterogenous Catalytic Reactions

(a) Very slow reaction

Temperature profile within the pellet

Bulk concentration Boundary layer

Boundary layer

Bulk concentration

Surface temperature

Bulk concentration

Concentration profile within the pellet

Surface temperature Ts

(b) Fast reaction, internal and external concentration profils

Boundary layer

Boundary layer

Surface concentration

Bulk concentration (c) Very fast reaction, reaction occurs at the outer surface

Surface concentration cs = 0 Figure 2.18 Concentration profiles in porous catalysts for different reaction regimes. (Adapted from Ref. [16], Figure 4.5 Copyright © 2012, Wiley-VCH GmbH & Co. KGaA.)

61

2 Basis of Chemical Reactor Design and Engineering

Bulk concentration DaII = 1 Surface concentration DaII → ∞; cs → 0

Boundary layer

Bulk concentration

Boundary layer

62

Figure 2.19 External concentration profile according to the film model. (Adapted from Ref. [16], Figure 4.6 Copyright © 2012, Wiley-VCH GmbH & Co. KGaA.)

force) as given in Equation 2.134. (2.134)

Ji = km (ci,b − ci,s )

with km the mass transfer coefficient and ci, s , ci, b the concentration of Ai at the external surface and the bulk of the fluid, respectively. At steady-state condition, the molar flux of Ai is equal to the rate of transformation at the outer catalyst surface: −Ri,p Vp = ; Ji = km (ci,b − ci,s ) = −Ri,p ⋅ Ap ap with Ri,p = �i rp

(2.135)

where Ri is the transformation rate of Ai , per volume of catalyst pellet, rp , the intrinsic reaction rate, Vp, , Ap the pellet volume and outer surface, respectively, and ap the specific external surface area of the pellet. ap =

Ap Vp

; ap =

6 (sphere) dp

(2.136)

For an irreversible first order surface reaction with � 1 = −1, we obtain for the reactant A1 : J1 = km (c1,b − c1,s ) =

−R1,p ap

=

kr c1,s ap

(2.137)

The reactant concentration on the surface is given by: c1,s =

km ap km ap + kr

⋅ c1,b

(2.138)

2.6

Mass and Heat Transfer Effects on Heterogenous Catalytic Reactions

and for the effective (observed) transformation rate follows: −R1,eff = kr

km ap km ap + kr

(2.139)

⋅ c1,b

A similar development can be observed for irreversible nth order reactions. At steady state follows: km ap (c1,b − c1,s ) = −R1,p = kr cn1,s

(2.140)

Dividing by km ap c1,b leads to ) ) ( n−1 ( kr c1,b c1,s n c1,s n c1,s = = ���� 1− c1,b km ap c1,b c1,b

(2.141)

The second Damköhler number, DaII, is defined as the ratio between the characteristic mass transfer time tm = 1∕(km ap ) and the characteristic reaction time, n−1 tr = 1∕(kr c1,b ). n−1 kr c1,b tm = ���� = tr km ap

(2.142)

With Equation 2.133 we find for the external effectiveness under isothermal conditions: ) ( c1,s n (2.143) �ex = c1,b The external effectiveness factors as function of the second Damköhler number are obtained by solving Equation 2.141. This is done for reaction orders n = 1, 2, 1∕2, and −1 and displayed in Figure 2.20 [27]. n = 1 ∶ �ex =

1 1 + ����

)2 (√ 1 + 4 ���� − 1 n = 2 ∶ �ex = 2 ����

1 n = ∶ �ex = 2 n = −1 ∶ �ex =

[

2 + ���I 2 2

(

√ 1−

4 1− (2 + ���I 2 )2

2 ; for ���� < 0.25 √ 1 + 1 − 4 ����

)] 1 2

(2.144)

From Figure 2.20 we see that

• The effectiveness factor diminishes for the same DaII with increasing reaction order

• An effectiveness factor higher than one is obtained for reaction with reactant inhibition (negative reaction order)

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2 Basis of Chemical Reactor Design and Engineering

1.4

T = const.

n = –1

1.2 1.0 ηex

64

n = 0.5

0.8

n=1

0.6 n=2 0.4 0.2 0.0 0.01

0.1

1

10

DaII Figure 2.20 Isothermal external effectiveness factor as function of the Damköhler number. (Adapted from Ref. [16], Figure 4.7 Copyright © 2012, Wiley-VCH GmbH & Co. KGaA.)

• For large values of DaII the effectiveness is inversely proportional to DaII (�ex ≃ 1∕����) for all reactions with positive reaction order.

The observed reaction rate is given by: ′

rp,eff = keff cn1,b = �ex kr cn1,b

(2.145)

With increasing intrinsic reaction rate (increasing DaII) the observed rate constant approaches the volumetric mass transfer coefficient (keff → km ap ) and the reaction order changes from n to unity. Whereas Figure 2.20 is quite instructive, it is not of practical use for estimating the importance of the mass transfer influence from experimental data, as the intrinsic rate constant is normally unknown. Replotting the effectiveness factor as function of the ratio between observed reaction rate to the maximum mass transfer rate called as Carberry number (Ca) allows estimating the external effectiveness factor plotted in Figure 2.21. rp,eff km ap c1,b

= �ex

n−1 kr c1,b

km ap

= �ex ���� = ��

(2.146)

Isothermal Yield and Selectivity For a network of parallel and/or consecutive reac-

tions mass transfer may affect drastically the target product yield. For consecutive first order reactions and in the absence of mass transfer influence we obtain for the transformation rate of the reactant and the production rate of the intermediate (the target product): k1

k2

A1 −−→ A2 −−→ A3

(2.147)

−R1 = k1 c1,b R2 = k1 c1,b − k2 c2,b

(2.148)

2.6

Mass and Heat Transfer Effects on Heterogenous Catalytic Reactions

1.4

n = –1

T = const. 1.2

ηex

1.0

n = 0.5 n=1

0.8 n=2

0.6 0.4 0.2 0.0 0.01

0.1

eff-ex.jnb

1

ηex *DaII = Ca Figure 2.21 Effectiveness factor as function of the observable variable: the Carberry number (Adapted from Ref. [15], Figure 4.8 Copyright © 2012, Wiley-VCH GmbH & Co. KGaA)

The instantaneous or point selectivity for the intermediate product is obtained by dividing R2 by (−R1 ): s2,1 = −

k2 c2,b R2 =1− , |v1 | = |v2 | = 1 R1 k1 c1,b

(2.149)

Anticipating the mass transfer phenomena in the extreme case results in: km ap (c1,b − c1,s ) = k1 c1,s

(2.150)

km ap (c2,s − c2,b ) = k1 c1,s − k2 c2,s

(2.151)

Solving for the surface concentrations c1, s and c2, s we obtain for the instantaneous selectivity the following relations: (s2,1 )eff = − (s2,1 )eff

R2,s R1,s

=1−

k2 c2,s

k1 c1,s k (1 + ���I1 ) c2,b 1 = − 2 1 + ���I2 k1 (1 + ���I2 ) c1,b

(2.152)

with ���I1 = k1 ∕(km ap ) and ���I2 = k2 ∕(km ap ). Under the initial conditions at the reactor entrance the product concentrations are zero and the instantaneous selectivity becomes: (s2,1 )eff,0 =

1 1 = 1 + ���I2 1 + (k2 ∕km ap )

(2.153)

Obviously, the effective selectivity of the intermediate product depends on the ratio of the escape rate from the surface to its rate of the transformation to the consecutive product A3 on the catalytic surface. Low mass transfer rates as compared to the rate of the consecutive reaction is detrimental for the selectivity and

65

66

2 Basis of Chemical Reactor Design and Engineering

yield of the intermediate product. At contrast, high mass transfer rates (km ≫ k 2 ) increase the initial selectivity of the intermediate product approaching unity. For parallel reactions the influence of mass transfer depends on the individual reaction orders: k1

(2.154)

A1 −−→ A2 ; R2 = k1 cn1 −1 k2

(2.155)

A1 −−→ A3 ; R3 = k2 cn2 −1

The ratio between the products A2 and A3 depends on the rate constants and the reactant concentration. k (n −n ) R2 = 1 c1 1 2 R3 k2

(2.156)

As the concentration gradient around the catalyst leads to a lower surface concentration compared to the bulk, the observed alteration of rate ratio depends on the individual reaction order: ( ) (R2,s ∕R3,s ) c1,s (n1 −n2 ) = (2.157) (R2.b ∕R3,b ) c1,b As c1, s < c1, b we see that diffusion intrusion leads to

• a reduced selectivity for A2 , if n1 > n2 • an increased selectivity for A2 , if n1 < n2 • no change of the selectivity for n1 = n2 . Nonisothermal Pellet For highly endothermic or exothermic reactions, the tem-

perature of the catalyst surface can be considerably different from the temperature of the surrounding fluid. We evaluate the surface temperature by the heat balance at steady state conditions: (2.158)

(−ΔHr ) ⋅ rp,eff = h ⋅ ap (Ts − Tb ) with h, the heat transfer coefficient. We divide Equation 2.158 by km ap c1,b h ⋅ ap km ap c1,b or h ⋅ ap km ap c1,b

(Ts − Tb ) = (−ΔHr ) ⋅

rp,eff km ap c1,b

(Ts − Tb ) = (−ΔHr ) ⋅ �ex ���� = (−ΔHr ) ⋅ ��

(2.159)

Invoking Chilton-Colburn analogy between heat and mass transfer: h Pr 2∕3 = km Sc2∕3 � ⋅ cp

(2.160)

2.6

Mass and Heat Transfer Effects on Heterogenous Catalytic Reactions

we obtain the ratio between heat and mass transfer coefficient: ( )2∕3 h Sc = �cp km Pr

(2.161)

and finally with Equation 2.159: ( Ts = Tb + ΔTad

Pr ��

)2∕3

��

Ts ΔTad ( Pr )2∕3 =1+ �� = 1 + �ex ⋅ �� Tb Tb Sc With ΔTad =

(−ΔHr )c1,b �cp

the adiabatic temperature rise, Pr =

� Dm

(2.162)

� �

=

� �∕(�cp )

the Prandtl

the Schmidt number. number, and �� = For a given system the temperature difference between bulk and surface depends on the reactant concentration via ΔT ad , the ratio between Prandtl and Schmidt number, and the Carberry number. The temperature difference is maximum for reactions limited by mass transfer (Ca = >1). As for gases the Schmidt and Prandtl numbers are approximately unity (Pr ≃ �� ≃ 1), the temperature difference can reach the adiabatic temperature (Ts − Tb ≃ ΔTad ). The nonisothermal external effectiveness factor is

�ex =

rp,eff rp,b

=

( ) k(Ts ) c1,s n k(Tb ) c1,b

(2.163)

On the basis of Equations 2.143 and 2.144 we can estimate the surface concentration and obtain for a first order reaction: c1.s ∕c1,b = (1 + ����)−1 . The rate constant at the surface temperature, k(T s ), is given by the Arrhenius law: ( )) ( ( )) ( Tb Tb E k(Ts ) = k(Tb ) exp − −1 = k(Tb ) exp −� −1 (2.164) RTb Ts Ts with � = RTE , the Arrhenius number. The surface temperature is determined by b the adiabatic temperature rise and the ratio of Schmidt and Prandtl number as shown in Equation 2.162. In summary, the external effectiveness factor for a given ( )2∕3 ΔT . Ca depends on the Arrhenius number, � and the parameter �ex = T ad Pr Sc b The nonisothermal effectiveness as function of the Ca for different Arrhenius numbers and � ex are shown in Figures 2.22 and 2.23.

67

2 Basis of Chemical Reactor Design and Engineering

�=

10

E = 10 (Arrhenius number) RTb

Efficiency, ηex

�ex =

ΔTad Pr Tb Sc

βex = 1.0

�ex = 0.5

2/3

�ex = 0.2 1

�ex = 0

Ca = ηex DaII =

reff

�ex = –0.2

kmapc1,b �ex = –0.5

0.1 0.001

0.01

0.1

1

Ca = ηex DaII Figure 2.22 Nonisothermal external effectiveness factor as function of the parameter � ex and the Carberry number (� = 10). (Adapted from Ref. [16], Figure 4.9 Copyright © 2012, Wiley-VCH GmbH & Co. KGaA.)

100 �=

10 Efficiency, ηex

68

E (Arrhenius number) RTb

�ex =

ΔTad Pr Tb Sc

� = 20

2/3

= 0.5 � = 10 �=5 �=2

1

Ca = ηex DaII = 0.1 0.01

reff

�=0

kmapc1,b

0.1 Ca = ηex DaII

1

Figure 2.23 Nonisothermal external effectiveness factor as function of the Arrhenius number, � and the Carberry number (� ex = 0.5). (Adapted from Ref. [16], Figure 4.10 Copyright © 2012, Wiley-VCH GmbH & Co. KGaA.)

2.6

Mass and Heat Transfer Effects on Heterogenous Catalytic Reactions

We see that

• the effectiveness factor can be greater than unity for exothermic reactions • the Arrhenius number � is more important than the parameter � ex in determining � ex .

• at high values of Carberry number the effectiveness factor falls well below unity even for highly exothermic reactions. 2.6.2 Internal Mass and Heat Transfer

For most catalytic processes, porous catalysts with a high inner specific surface area are used. Therefore, the reactant has to be transported through the pores to the catalytically active sites as described in Figure 2.17. Because of the chemical reaction, a gradient of the reactant concentration in the fluid (gas or liquid) may develop from the outside to the center of the pellet. For the following discussion we assume isotropic particles and that the transport process can be represented by molecular diffusion. The molar flux of reactant A1 can be described by: dc1 (2.165) dz where De is the effective diffusion coefficient for reactant A1 , and z is the particle coordinate, defined as the distance from the center. Formally, Equation 2.165 corresponds to first Fick’s law. As the diffusion occurs within a porous media, an effective diffusion coefficient is introduced. The effective diffusion coefficient takes into account that pores occupy only fraction, �p , of the particle volume, and that the pores are not linear in z-direction. As a consequence, the diffusion path through the pores is longer than z. This is accounted for by introducing a tortuosity factor � p . With both corrections the effective diffusion coefficient can be estimated with the following expression: �p (2.166) De = D1 �p J1 = −De

with D1 , the molecular diffusion coefficient of reactant A1 . The particle porosity is in the order of 0.3 < �p unity. Therefore, the effectiveness factor for strong diffusional resistances is 1 �p ≃ (2.176) � Concentration profiles in slabs for different values of � are shown in Figure 2.24b. The results presented above are specific for a first order reaction and a catalyst in the form of a slab. For spherical particles the corresponding equation is [ ] 1 1 3 (2.177) − �p = �s tanh �s �s The corresponding solution for a cylinder is �p =

2 I1 (�c ) �c I0 (�c )

(2.178)

where I1 (�) and I0 (�) denote the modified Bessel functions of first and zero order, respectively. In Figure 2.25 the effectiveness factor as function of the Thiele modulus for different pellet shapes is shown. For small values of the Thiele modulus the effectiveness factor reaches unity in all cases. The reaction rate is controlled by the intrinsic kinetics, and the reactant concentration within the pellet is identical to the concentration at the outer pellet surface. This situation may be observed for low catalyst activity or very small particles as used in fluidized beds or suspension reactors. For large values of the Thiele modulus the dependency of � p approaches an asymptotic solution: �p = m∕� with m = 1, 2, 3 for a slab, a cylinder, and a sphere, respectively. This situation may occur for very fast reactions or large catalyst particles. The concentration in the center of the catalyst particles approaches zero for � p < 0.2. The observation that the slope of the asymptotic solution for �(�) becomes independent of the particle geometry suggests that the dependence of the effectiveness factor on the Thiele modulus can be described by a generalized relationship, valid

71

2 Basis of Chemical Reactor Design and Engineering

1 n=1 T = const. Efficiency factor, ηp

72

Sphere 0.1 Cylinder Flat plate

0.01 0.01

0.1

1 Thiele modulus, ϕ

10

100

Figure 2.25 Effectiveness factor as function of the Thiele modulus for different pellet shapes. (Adapted from Ref. [28], Figure 5 Copyright © 2008, Wiley-VCH GmbH & Co. KGaA.)

for arbitrary pellet shapes. This was in fact demonstrated by Aris [29] by defining a general Thiele modulus �gen based on the ratio of pellet volume to external surface as characteristic diffusion length. A further correction was proposed by Petersen to get a general effectiveness factor for a nth order reaction with a characteristic reaction time tr = (kr c(n−1) )−1 . The final definition is given in Equation 2.179. 1,s

�gen

√ √ (n−1) √ Vp √ √ kr c1,s n+1 ⋅ = Ap De 2

(2.179)

The effectiveness factor as function of the generalized Thiele modulus is shown in Figure 2.26 for a slab and a sphere. Both curves coincide exactly for �gen → ∞. The maximum deviations are in the order of 10–15%. In general, the intrinsic kinetic parameters of a catalytic reaction under study are unknown. Therefore, the relationships based on the Thiele modulus cannot be used to estimate the influence of inner mass transfer on the measured overall reaction rate. Observed is the experimentally accessible efficient reaction rate, rp, eff . In addition, the characteristic diffusion time in the porous pellet can be estimated. This allows to define a new modulus based on the characteristic effective reaction time t r, eff and the characteristic diffusion time in the particle, tD . The ratio of these two values is known as Weisz modulus. We obtain for spherical pellets:

�s2 =

tD tr,eff

=

R2sphere c s = �p �2s De rp,eff

(2.180)

2.6

Mass and Heat Transfer Effects on Heterogenous Catalytic Reactions

Effectiveness factor, �p

1 Sphere

Slab

0.1

0.01 0.1

1

10

100

Generalized Thiele modulus, �gen Figure 2.26 Effectiveness factor as function of the generalized Thiele modulus for different pellet geometries. (Adapted from Ref. [16], Figure 4.13 Copyright © 2012, Wiley-VCH GmbH & Co. KGaA.)

In analogy with the generalized Thiele modulus, we can define a Weisz modulus that applies to arbitrary pellet shapes and different reaction orders, n:

2 �gen

=

(

tD tr,eff

=

Vp

)2

Ap

n + 1 rp,eff = �p �2gen 2 De cs

(2.181)

In Figure 2.27 a plot of the effectiveness factor against the generalized Weisz module for different reaction orders is shown. Using this relation, the effectiveness factor can be estimated based on the experimental kinetic results and the estimated diffusion coefficient. 1

n=1

Effectiveness factor, �p

Flat plate T = const.

n=0

n=2

2

ψgen =

0.1 0.1

(Vp/Ap) reff n + 1 D1,e c1,s

2

2

= �p�gen

1 Generalized Weisz modulus, ψgen

10

Figure 2.27 Effectiveness factor as function of the generalized Weisz modulus for different reaction orders. (Adapted from Ref. [16], Figure 4.14 Copyright © 2012, Wiley-VCH GmbH & Co. KGaA.)

73

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2 Basis of Chemical Reactor Design and Engineering

Isothermal Yield and Selectivity The influence of transport phenomena on selectivity and yield is often more important than on the effective catalyst activity. The following analysis is restricted to two important schemes for complex reactions [28]: k2

k1

parallel reactions: A3 ←−− A1 −−→ A2 and k1

k2

consecutive reactions: A1 −−→ A2 −−→ A3 . We also neglect the effect of external mass transfer resistances and assume that the concentration at the pellet surface is identical to the bulk concentration (ci,s = ci,b ). In the case of parallel reactions the rate equations for the consumption of the reactant A1 and concomitant formation of the desired product A2 are given by: n

n

−R1 = k1 c11 + k2 c1 2 n

R2 = k 1 c 1 1

(2.182)

with k 1 and k 2 as the intrinsic rate constants. The instantaneous or point selectivity is defined as the ratio of the A2 production rate to the rate of reactant consumption: s2,1 =

R2 1 1 = = (n2 −n1 ) (n −n ) −R1 1 + k2 ∕k1 ⋅ c1 1 + �c1 2 1

(2.183)

There is no influence of the concentration profile on selectivity in the case of equal order kinetics for the two reaction paths. If n1 ≠ n2 , the effective selectivity will be influenced by internal diffusion. As the influence of the internal concentration profile becomes more pronounced with increasing reaction order, the product selectivity will diminish, if the desired reaction has a higher order than the undesired. Otherwise, if the desired reaction has a lower kinetic order, the selectivity will be improved with increasing internal mass transfer resistance. To discuss the influence of internal transport processes on consecutive reactions, we assume simple irreversible first order reactions. With k 1 and k 2 being the intrinsic rate constants, the production rate of A2 and the disappearance of A1 are given by: R2 = k1 c1 − k2 c2 −R1 = k1 c1

(2.184)

We obtain for the instantaneous selectivity in the kinetic regime: s2,1 =

c2,b k c2,b k R2 =1− 2 =1−� ; with � = 2 −R1 k1 c1,b c1,b k1

(2.185)

If transport resistances can be neglected, the concentration inside the catalyst pellet corresponds to the bulk concentration ci = ci, b . The instantaneous selectivity decreases with increasing conversion, X. In a catalytic fixed bed reactor with

2.6

Mass and Heat Transfer Effects on Heterogenous Catalytic Reactions

plug flow behavior (see Section 2.3.4), the product yield can be determined by integration: XL

Y2,1 =

∫0

Y2,1 =

1 [(1 − X)� − (1 − X)] 1−�

s2,1 dX (2.186)

The yield increases up to a maximum and finally reaches zero for X = 1 as shown in Figure 2.28. The maximum depends on the ratio of the two rate constants and is given by Y2,1,max = � �∕(1−�) at Xop = 1 − � 1∕(1−�) ; for � ≠ 1

(2.187)

To evaluate the influence of internal mass transfer on the product selectivity and yield, we have to solve the material balance for A1 and A2 in the porous catalyst. Assuming a flat plate and equal diffusion coefficient (D1, e = D2, e = De ), we obtain with ci,s = ci,b : √ d2 f1 k1 c c z = �21 f1 ; f1 = 1 = 1 ; Z = ; �1 = L dZ 2 c1,s c1,b L De √ ( ) d2 f2 k2 k2 1 c1,s 2 = �2 f2 − f1 ; � = �2 = L (2.188) 2 dZ � c2,s k1 De 0.7 0.6

φ2 = L

Yield, Y2,1

0.5

k2 De

κ = 0.25

0.4

ϕ2 = 0, ηp = 1 ϕ2 = 0.5, ηp = 0.762 ϕ2 = 1, ηp = 0.482

0.3

ϕ2 > 3, ηp < 0.2

0.2 0.1 0.0

0.0

0.2

0.4 0.6 Conversion, X

Figure 2.28 Effective product yield as function of conversion for consecutive reactions. Influence of internal mass transfer resistance. � = k2 ∕k1 = 0.25, initial product

0.8

1.0

concentration c2, 0 = 0. (Adapted from Ref. [16], Figure 4.15 Copyright © 2012, WileyVCH GmbH & Co. KGaA.)

75

76

2 Basis of Chemical Reactor Design and Engineering

With the concentration profile for the reactant A1 given by Equation 2.171, the solution of the differential equation leads to: √ k1 cosh (�1 Z) f1 = ; �1 = L cosh (�1 ) De ) ( √ c1,s 1 cosh (�2 Z�) c1,s 1 cosh (�1 Z) − ; �2 = � ⋅ �1 f2 = 1 + c2,s 1 − � cosh (�2 �) c2,s 1 − � cosh (�1 ) (2.189) The efficient instantaneous catalyst selectivity is given by the ratio of the efficient production rate of A2 and the rate of reactant A1 disappearance (see also Equation 2.172). √ ) ( √ tanh(�2 �) c2,s R2,eff (dc2 ∕dZ)Z=1 1 1 ⋅ � − =− = + s2,1,eff = −R1,eff (dc1 ∕dZ)Z=1 1−� c1,s 1 − � tanh(�2 ) (2.190) For strong diffusion resistance within the catalyst pellet, Equation 2.190 can be simplified to: √ c2,s √ 1 for �2 � > 3 (2.191) s2,1,eff = √ − � c1,s 1+ � The overall product yield is obtained by integration of Equation 2.190 over a range of conversion. For a product concentration at the reactor inlet c2, b, 0 = 0 the result is: 1 Y2,1,eff = [(1 − X)Δ� − (1 − X)]; c2,b,0 = 0 1−� √ tanh(�2 ) √ tanh(�2 ) = � (2.192) with Δ� = � √ tanh(�1 ) tanh(� ∕ �) 2

For very strong diffusional resistance Equation 2.192 can be simplified and the yield may be estimated from: √ 1 (2.193) [(1 − X) � − (1 − X)]; c2,b,0 = 0 1−� The integral product yield as function of conversion for different values of the Thiele modulus is shown in Figure 2.28 for � = k2 ∕k1 = 1∕4. It is obvious that internal diffusional resistance leads to a drastic decrease of the target product selectivity and yield. In the domain of practical interest with � < 1, the maximum obtainable yield for strong diffusion resistance (�2 ≥ 3, Equation 2.194) drops roughly to 50% of the value reached in the kinetic regime (Equation 2.187). At the same time the efficiency factor in the porous catalyst drops to ηp 0. (Tcenter − Ts )max = (−ΔHr )c1,s

De �e

(2.198)

Obviously, the maximum temperature difference will depend on the reaction enthalpy and the ratio between effective diffusion and effective thermal conductivity. If we refer the maximum temperature difference to the surface temperature, we get the dimensionless so-called Prater number, �. �=

(−ΔHr )c1,s De ΔTmax = Ts Ts �e

(2.199)

For exothermic reactions the temperature inside the pellet will be higher than the surface temperature. Because of the exponential increase of the reaction rate, the temperature effect can overcompensate the lower concentration in the pellet. An example is shown in Figure 2.29 where the effectiveness factor is plotted versus the Thiele modulus for an Arrhenius number of � = 20 and different Prater numbers. The curves shown were obtained by numerical integration by Weisz and Hicks [30]. Efficiency factors higher than one can be expected at relatively low Thiele modulus and high Prater and Arrhenius numbers. At large values of �, the

77

2 Basis of Chemical Reactor Design and Engineering 1000.0 500.0

�=

100.0 50.0

�=

E = 20 R • Ts (–ΔHr)c1,s De Ts

�e

10.0

Effectiveness factor, ηp

78

5.0

1.0

b= 0.8

0.5

0.6 0.4

0.1

0.3 0.2 0.1 0.0

0.05 –0.6

–0.2 –0.4

–0.8

0.01 0.005

0.001 0.1

0.5 1.0

5.0 10.0

50

100

500 1000

Thiele modulus, φgen Figure 2.29 Effectiveness factor as function of the Thiele modulus. Nonisothermal sphere, first order reaction [30]. (Adapted with permission from Elsevier.)

effectiveness factor becomes inversely proportional to the Thiele modulus, as observed at isothermal conditions. Besides the increase of the observed reaction rate because of the high internal temperature, multiple steady states are predicted for reactions with high Arrhenius numbers and high Prater numbers. In the region of multiple steady states, different temperature and concentration profiles for the same Thiele modulus may exist leading to different effectiveness factors. This behavior is shown in Figure 2.29 for 0.2 < � < 1 and � > 0.2. For majority of industrial catalysts the effective heat conductivity is in the order of 0.2 < �e < 0.5 and efficient diffusion coefficients for gas phase reactions are in the order of 10−5 to 10−6 m2 s−1 . Therefore, Prater numbers seldom exceed values of � = 0.1 and the maximum temperature in the pellet center is seldom higher than ΔT max = 10 K. In summary, temperature differences between gaseous bulk and catalyst surface are much more important as discussed in Section 2.6.1.2.

2.6

Mass and Heat Transfer Effects on Heterogenous Catalytic Reactions

2.6.2.3 Combination of External and Internal Transfer Resistances

In the previous chapters we discussed the influence of internal mass and heat transfer by neglecting external transport phenomena. Hence, we assumed that concentrations and temperature at the outer surface of the catalyst particle and the bulk of the fluid are the same. But this assumption is not justified under certain conditions and concentration and temperature profiles inside and outside the porous catalyst must be considered. 2.6.2.4 Internal and External Mass Transport in Isothermal Pellets

If the efficient reaction rate is high enough, the reactant concentration drops significantly across the external boundary layer as indicated in Figure 2.18. In this case the surface concentration is lower compared to the bulk of the fluid phase (c1, s < c1, b ). First we will neglect eventual heat effects and assume equal temperatures in the fluid and the catalyst particle (T = Ts = Tb ). To determine the concentration profile in the particle, we first have to calculate the concentration at the external surface. This will be done based on the mass balance for the reactant A1 . At steady state, the molar flux of A1 from the bulk to the external surface must be equal to the effective rate of transformation (see Equation 2.137). J1 = km (c1,b − c1,s ) =

−R1,p,eff

(2.200)

ap

For a simple irreversible first order reaction we obtain: km (c1,b − c1,s ) =

�p kr c1,s

with �p the internal effectiveness factor

ap

(2.201)

Solving Equation 2.201 for the unknown surface concentration: c1,s =

c1,b

(2.202)

1 + �p kr ∕(km ap )

If we introduce the ratio between the characteristic diffusion time in the pellet tD and the external mass transfer time tm we will get a clear physical interpretation of this relationship. The mentioned ratio is known as the mass Biot number, Bim . Bim =

L2 tD = c km ap with Lc the characteristic length of the pellet (2.203) tm De

Introducing the Biot number in Equation 2.202 yields: c1,s =

c1,b 1 + �p

kr L2c De

⋅

1 Bim

=

c1,b 2

1 + �p Bi�

(2.204)

m

The overall effectiveness factor is defined as the ratio between the effective transformation rate and the rate at constant bulk concentration. �ov =

−R1,eff −R1,b

=

−R1,eff kr c1,b

=

�p c1,s c1,b

(2.205)

79

2 Basis of Chemical Reactor Design and Engineering

In combination with Equation 2.204 we obtain: �ov =

�p 1+

2 �p Bi� m

=

1 1 �p

+

(2.206)

�2 Bim

For a catalyst in the form of a flat plate the effectiveness factor is given by �p = tanh �L ∕�L (Equation 2.174) and the overall effectiveness factor can be expressed as a function of the Thiele modulus and the Biot number. �ov =

(

tanh �

� 1+

�⋅tanh � Bim

)

(2.207)

The relationship shown in Equation 2.207 suffers from the fact that the Thiele modulus must be specified to estimate the catalyst efficiency. This is, in general, not possible as the intrinsic kinetics is not known. It is, therefore, more convenient to relate the overall effectiveness factor to the Weisz modulus, which is based only on observable parameters. The catalyst efficiency decreases strongly at small mass Biot numbers as seen in Figure 2.30. This is because of the reduced reactant concentration on the external pellet surface. In contrast, external mass transfer influences can be neglected at Bim > 100. In practice, catalytic particles are in the range of several millimeters and the mass Biot numbers are in the order of 100–200. Hence, the overall effectiveness factor is almost entirely determined by the intraparticle diffusion. 1 n=1 T = const. �ov

80

Flat plate

0.1

Bim = 1 0.1

Bim = 10 1 ψ=

Bim = 100 10

–R1,eff L2 c1,b

De

Bim = ∞ 100

= �ov �2

Figure 2.30 Overall effectiveness factor as a function of the Weisz modulus for different mass Biot numbers (isothermal, irreversible first order reaction in a porous slab). (Adapted from Ref. [16], Figure 4.17 Copyright © 2012, Wiley-VCH GmbH & Co. KGaA.)

2.6

Mass and Heat Transfer Effects on Heterogenous Catalytic Reactions

2.6.2.5 The Temperature Dependence of the Effective Reaction Rate

As pointed out, the influence of mass transfer on the observed reaction rate depends on the ratio between the characteristic reaction time and the characteristic time for mass transfer. By increasing the temperature, the intrinsic reaction rate increases more strongly (exponentially) than the rates of external and internal mass transfer. Consequently, the Thiele modulus and the second Damköhler number augment with increasing temperature, and transport phenomena become more and more important and will finally control the transformation process. In addition, the temperature dependence of the observed reaction rate will change as indicated. At low temperatures the process is controlled by the intrinsic chemical kinetics and the rate constant increases exponentially following Arrhenius law: ) ( −E k = k0 exp (2.208) RT with k 0 the frequency factor and E the intrinsic activation energy. The temperature dependence of the diffusion process is represented by proportionality to T 3/2 but can be also approximated by an Arrhenius equation: ) ( −ED ; with 5 < ED < 10 kJ mol−1 De = De,0 exp (2.209) RT This is not a theoretical dependence of De on temperature but is useful for the following discussions. At strong influence of internal diffusion on the reaction rate, the effectiveness factor was found to be inversely proportional to the Thiele modulus (e.g., Equation 2.176). Accordingly, the effective rate constant is given by: k 1√ = k ⋅ De (2.210) keff = √ L L k∕De For the temperature dependence follows: √ ) ( k0 De,0 E + ED keff = exp − L 2 ⋅ RT

(2.211)

Normally E ≫ ED , as diffusion is not very temperature sensitive, so the observed apparent activation energy is about one-half of the true value when pronounced internal concentration profiles are present (Figure 2.31). Further temperature increase will diminish the reactant concentration on the outer pellet surface as the influence of external mass transfer becomes important. Finally, interphase mass transfer will be the rate controlling step and the surface concentration drops to zero. Under those conditions, the apparent activation energy corresponds to ED . Besides the apparent activation energy, the effective reaction order changes during the transition from the kinetic to the diffusion controlled regime. A first order reaction will be observed under external mass transfer control. The effective reaction order observed approaches napp = (n + 1)∕2 for severe influence of intraparticle diffusion.

81

2 Basis of Chemical Reactor Design and Engineering

External mass Transition transfer region

slope =

In keff

82

ED 10

Internal diffusion

Transition region

Kinetic control

–ED R kJ mol

slope =

–Ea 2R slope =

–Ea R

1/T Figure 2.31 Arrhenius plot for heterogenous catalytic reactions. Transition from the kinetic regime to mass transfer controlled regime.

2.6.2.6 External and Internal Temperature Gradient

In the case of fast highly exothermic or endothermic reactions, temperature gradients inside the porous catalyst and temperature differences between the fluid phase and catalyst surface cannot be neglected. Depending on the physical properties of the fluid and the solid catalyst, important temperature gradients may occur. The relative importance of internal to external temperature profiles can be estimated based on the relationships presented in Sections 2.6.1.2 and 2.6.2.2. According to Equation 2.158 the temperature difference between bulk and outer pellet surface is: Ts − T b =

(−ΔHr ) ⋅ reff with reff = km ap (c1,b − c1,s ) h ⋅ ap

Ts − Tb = (−ΔHr )

km (c − c1,s ) h 1,b

(2.212)

With the Chilton-Colburn analogy we can replace the ratio km ∕h and obtain (see Equation 2.161) ( ) 1 Pr 2∕3 Ts − Tb = (−ΔHr ) (c1,b − c1,s ) (2.213) �cp Sc For large internal diffusional resistance, the concentration of the reactant in the pellet center drops to zero. In this situation the temperature difference between the outer surface and the center of a porous catalyst pellet is maximal and given by Equation 2.198. In Equation 2.214 the temperature difference between bulk and pellet surface is compared with the maximum internal temperature gradient. The ratio between

2.6

Mass and Heat Transfer Effects on Heterogenous Catalytic Reactions

Table 2.1 Physical properties of fluid/solid systems [13].

D or De (m2 ⋅s−1 ) � or �e (W⋅m−1 K−1 ) � cp (J⋅m−3 K−1 )

Gas

Liquid

Porous solid

10−5 –10−4 10−3 –10−1 102 –105

10−10 –10−9 10−2 –10 105 –107

10−7 –10−5 10−2 –1 106 –107

these two temperature differences depends on the ratio of the mass Biot to the thermal Biot numbers. k � c1,b − c1,s Bi c1,b − c1,s Ts − Tb = m e = m (Tcenter − Ts )max h De c1,s Bith c1,s k ⋅ L h⋅L with Bith = ; Bim = m (2.214) �e De In Table 2.1 the order of magnitude of some physical properties of fluid/solid systems are summarized. On the basis of these values we can conclude that for gas/solid systems the ratio of Bim /Bith is in the range of 10–104 . Hence, the temperature gradient in the external boundary layer is much more important than within the pellet under usual reaction conditions: (Ts − Tb ) >> (Tcenter − Ts ); gas∕solid system

(2.215)

In contrast, we expect a higher temperature difference within the pellet in liquid/solid systems. 2.6.3 Criteria for the Estimation of Transport Effects

For the catalyst development and optimization as well as for the correct reactor design, it is important to ascertain the influences of transport phenomena on the reaction kinetics. It is essential that criteria for estimating transport effects are based on what is measurable or observable [31, 32]. One way to estimate the influence of transport processes is to use directly experimental results observed under given experimental conditions. In general, the experimentalist has information concerning observed reaction rates, bulk reactant concentrations, and temperature, as well as the catalyst pellet form and dimensions. With these details at hand, the Weisz module can be estimated. For example, for spherical catalytic particles, see Equation 2.180. Each of the � s values is experimentally accessible and the effectiveness factor can be computed such as shown in Figure 2.32. Hence, a set of graphs can be prepared relating � s to � with the Arrhenius and Prater numbers as parameters and allowing estimation of the effectiveness factors directly from experimental results. An important number of criteria for estimating the influence of transport

83

2 Basis of Chemical Reactor Design and Engineering

�= 100

Ea = 20 RTs

�=

c1,s (–ΔHR)De �eTs

50

β = 0.8 Effectiveness factor, �

84

10 5

0.6 1.0

0.4

0.5

0.2

0.1 2

0.05

ψs =

rp reff De c1,s

0.0 –0.8

0.01 0.1

0.5 1.0

5

10

50 100

–0.2 500 1000

Weisz modulus (sphere), ψs Figure 2.32 Effectiveness factor in terms of the experimentally observable Weisz modulus. First order reaction, sphere [30]. (Adapted with permission from Elsevier.)

phenomena on catalytic reaction rates are published in the open literature. In general, these criteria are derived assuming that transport effects do not alter the true rate by more than ±5%. Because of the uncertainty involved in estimating the different parameters, the application of the criteria should be done in a conservative manner. The observed values should be at least several times or even an order of magnitude better than those proposed. The most general of the criteria (Equation 6 in Table 2.2) ensures the absence of any internal and external concentration and temperature gradient. But a problem may arise because of compensation between mass and heat transport. This situation will occur if � ⋅ � ≅ n. Therefore, it may be better to respect separately the criteria for isothermicity. It is disturbing that criteria for the absence of heat effects are based on the true activation energy, which is not observable, if mass transfer affects the rate of reaction. A critical discussion of the experimental results and a prudent application of the criteria are, therefore, indispensable.

2.7 Summary

In this chapter, the fundamentals of chemical reaction engineering are presented. The basic definitions along with the material balance of different types of ideal reactors and their design equations are discussed.

2.7 Summary

Table 2.2 Experimental criteria for the absence of inter and intra transport phenomena (0.95 < � < 1.05) for simple irreversible reactions. 1

Absence of interphase concentration gradients in isothermal systems

2

Absence of interphase temperature gradients. The criterion is independent whether intraparticle gradients exist or not

3

Absence of intraparticle/interphase gradients

rp,eff dp 2km ci,b

Absence of concentration profiles within an isothermal porous catalyst pellet

0.15 |n|

(−ΔHr )rp,eff dp 2h⋅Tb

rp,eff dp2 4ci,b De

�= 4


15; 10−3 < Rep < 103

(3.62)

3.5

Influence of RTD on the Reactor Performance 10

10

εbed·Peax = 0.2 + 0.011·Rep0.48

1

1 Peax

εbed·Peax

107

0.1

0.1

1 Peax,p

0.01 10–3

10–2

10–1

(a)

100 Rep

101

102

10–2

103

=

10–1

0.3 Rep Sc

100

(b)

+

0.5 1 + 3.8/(Rep Sc)

101 Rep Sc

102

103

Figure 3.14 Axial dispersion in fixed-bed reactors: (a) liquid flow and (b) gas flow [6]. Gray area represents experimental results. (Adapted from [6], Figure 27.24 Copyright © 2012, Wiley-VCH GmbH & Co. KGaA.) 10

10

Peax

Peax

1

1

0.1

Peax

=

1 Re • Sc

1 3 • 107 1.35 1 = + Peax Re2.1 Re0.125

+ Re • Sc 192

0.1 1

(a)

10

100

103

1000

Re Sc

(b)

104

105

Re

Figure 3.15 Axial dispersion in tubular reactors: (a) laminar flow and (b) turbulent flow. Gray area represents experimental results. (Adapted from [6], Figure 27.25 Copyright © 2012, Wiley-VCH GmbH & Co. KGaA.)

3.5 Influence of RTD on the Reactor Performance

For reactions with positive reaction order, the reactor performance will decrease with the broadening of the RTD at constant mean residence time. This can easily be demonstrated with Figure 3.16, where the conversion obtained in an ideal plug flow reactor is plotted as function of the residence time. For a residence time of 15 min the conversion in the ideal tubular reactor corresponds to X = 0.78 (point A in Figure 3.16). Supposing the fluid is distributed in two equal parts in plug flow reactors with residence times of 5 and 25 min, resulting in the same mean residence time of 15 min. The corresponding conversions obtained are 0.39 (point B1 ) and 0.92 (point B2 ). The mean conversion is indicated by point M on the line relating B1 and B2 . The mean conversion dropped to X = 0.66 as a result of the

106

3 Real Reactors and Residence Time Distribution (RTD)

1.0 B2

A

0.8 Conversion, X

108

M

0.6

0.4

B1

0.2

0.0

0

5

10

15

20

25

30

35

40

Mean residence time, t (min) Figure 3.16 Influence of RTD on the performance of tubular reactors. First order reaction, k = 0.1 min−1 .

RTD. The result can be generalized: RTD will diminish the conversion and the specific reactor performance for reactions with a positive reaction order (n > 0). 3.5.1 Performance Estimation Based on Measured RTD

In the case of known formal kinetics, the reactor performance can be determined directly from the RTD. We can imagine, for example, that the RTD in the reactor under consideration can be represented by a series of ideal plug flow reactors of different lengths arranged in parallel through which the reaction mass flows at equal rates (see Figure 3.17). The conversion at the end of an individual tube with a defined residence time can then be calculated easily. At the exit of the tubes, the various flows having different residence times are mixed; the result is an average conversion or, respectively, an average reactant concentration. When RTD and kinetics are known, it follows that: ⎫ ⎧ fraction of ⎫ ⎧ { } conversion in ∑⎪ ⎪ ⎪ ⎪ average conversion = ⎨ volume element ⎬ • ⎨ total flow with ⎬ at reactor outlet ⎪with residence time ti ⎪ ⎪residence time ti ⎪ ⎭ ⎭ ⎩ ⎩ (3.63)

Figure 3.17 Real reactor behavior modeled by ideal tubular reactors arranged in parallel.

3.5

X=

∫0

1

X(t)⋅dF ≅

1 ∑

Influence of RTD on the Reactor Performance

X(ti ) ⋅ ΔFi

0

or with dF = E(t)dt X=

∫0

∞

X(t) ⋅ E(t)dt ≅

∞ ∑

(3.64)

X(ti ) ⋅ E(ti )Δti

0

The presented method leads to exact values only for first order reaction (demonstrated in Examples 3.3–3.5) or for reactions in completely segregated systems (see Chapter 4). But, the proposed methods can be used also for a good estimation of reactor performances for reactions with n ≠ 1. Example 3.3: Estimation of conversion in real reactors. Estimate the conversion for the first order reaction in a nonideal tubular flow reactor. The residence time distribution is characterized by measured F function. The mean residence time can be calculated with Equation 3.14 applying the trapezoidal method. Compare the result with the conversion that could be obtained in ideal PFR and CSTR for the same mean residence of 10 min. −R1 = k ⋅ c1 ; k = 0.15 min−1 Solution:

1)

In Figure 3.18 the conversion is plotted as function of F(t) with values of Table 3.3. The mean conversion is obtained estimating the area under the X-F-curve. Numerical integration using the trapezoidal method results in 1 ∑ X= X(t) ⋅ ΔFi = 0.72 0

1.0 0.8

X(t)

0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6 F(t)

Figure 3.18

Conversion as a function of F(t).

0.8

1.0

109

110

3 Real Reactors and Residence Time Distribution (RTD)

2) Ideal plug flow reactor: X = 1 − exp(−k ⋅ �) = X = 1 − exp(−0.15 ⋅ 10) = 0.777 3) Continuous stirred tank reactor: X=

0.15 ⋅ 10 k⋅� = = 0.6 1−k⋅� 1 + 0.15 ⋅ 10

Table 3.3 Residence time of a tubular reactor. t (min)

0

5

7

8.75

10

15

20

25

30

F [−] X [−]

0 0

0.10 0.394

0.22 0.503

0.40 0.583

0.57 0.632

0.84 0.777

0.94 0.865

0.98 0.918

0.99 0.950

Example 3.4: Conversion in laminar flow tubular reactors. Estimate the conversion obtainable in a tubular reactor under laminar flow conditions neglecting radial diffusion for the reaction presented in Example 3.3. The mean residence time is t = 10 min. Solution:

The RTD in laminar flow reactors is given in Equations 3.36 and 3.38. 2

E(t)dt =

t dt; 2t 3

( )2 t E(t)dt = 1− ∫tmin 2t t′

F=

With Equation 3.64 we obtain: X=

∫0

∞

X(t) ⋅ E(t)dt =

∫0

2

∞

(1 − exp(−k ⋅ t) ⋅

t dt 2t3

Numerical integration between 0 < t < 100 min leads to a mean conversion of X = 0.69. 3.5.2 Performance Estimation Based on RTD Models

In the case of identical mean residence times for different tubular reactors, the conversion and selectivity of a complex reaction will depend on the RTD in the reactor. With increasing backmixing, the reactor approaches the behavior of an ideal CSTR. Accordingly, the performance of any tubular reactor will decrease with increasing RTD at a constant mean residence time for reactions with formally positive reaction orders.

3.5

Influence of RTD on the Reactor Performance

3.5.2.1 Dispersion Model

Backmixing in a tubular reactor has a direct influence on the axial concentration profile. With decreasing axial dispersion time compared to the space time (decreasing Bo) the concentration profile flattens and finally a uniform concentration results (Bo => 0). This is demonstrated in Figure 3.19 for an irreversible first order reaction at DaI = k ⋅ τ = 3. For reactions with positive reaction order, the flattening of concentration profile diminishes the mean reaction rate in the tubular reactor and the conversion will decrease at constant space time, constant DaI, respectively. On the basis of the dispersion model, the following mass balance for a small volume element results: dc1 d2 c − Dax 21 − R1 = 0 dz dz or, respectively, in dimensionless form:

(3.65)

u

−

D d2 X dX �(R1 ) − + ax =0 dZ c1,0 u ⋅ L dZ 2

(3.66)

Applying Danckwerts’ boundary conditions [2] Equation 3.66 can be solved for an irreversible first order reaction (Equation 3.67) [6]. 4a exp(Bo∕2) (1 + a)2 exp(aBo∕2) − (1 − a)2 exp(−aBo∕2) √ with a = 1 + 4DaI∕Bo

1−X =

(3.67)

The conversion is a function of DaI and the axial dispersion characterized by Bo as shown in Figure 3.20. With decreasing Bo the conversion diminishes at constant DaI (constant space time). At DaI = 5 a conversion of X = 0.99 is attained in a plug flow reactor (Bo = ∞), whereas the conversion drops to X = 0.83 for Bo = 0 (continuous stirred tank reactor). 1 Bo = 100 Bo = 40 Bo = 20 Bo = 10 Bo = 6 Bo = 4 Bo = 2

C/C0

0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Z = z/L Figure 3.19 Influence of Bo on the axial concentration profile. First order reaction, DaI = 3. (Adapted from Ref. [6], Figure 27.26 Copyright © 2012, Wiley-VCH GmbH & Co. KGaA.)

111

112

3 Real Reactors and Residence Time Distribution (RTD)

0.99 PF Bo = 20 Bo = 10 Bo = 4 Bo = 2 CSTR

X

0.95 0.90

0.50 0.10 0.1

1

100

10 DaI = k �

Figure 3.20 Conversion as function of DaI and Bo (first order reaction).

3.5.2.2 Tanks in Series Model

In Section 3.4.2, it was shown that the RTD in real reactors can be described with a series of ideally continuous stirred tank reactors. The scheme of such a cascade of continuous stirred tanks is shown in Figure 3.21. The total volume is divided in N equal sized stirred vessels. For reactions with positive order, the performance of such a cascade reactor has a specific function between an ideal plug flow reactor and a single CSTR. This can easily be understood comparing the reactant concentration as function of the reactor volume. In a PFR the concentration and, therefore, the transformation rate diminishes with increasing volume from the reactor entrance to the outlet. The low specific performance of a CSTR can be explained by the overall low concentration corresponding to the outlet concentration. In the cascade, the concentration diminishes stepwise from one vessel to the next. This is shown schematically for a series with N = 5 vessel in Figure 3.22. With increasing number of equal sized vessels the concentration profile approaches that of a PFR. The conversion in each vessel can be determined with the material balance for continuous ideally mixed vessels (see Section 2.2.3). V̇ 0 c1,0 − V̇ out c1,out = V (−R1 ) (steady state) •

•

V0 c1,0

•

V0 c1,1

Vi, �i

V0 c1,2

(3.68)

•

•

V0 c1,i

Vi, �i

Figure 3.21 Cascade of continuous stirred tank reactors.

•

V0 c1,N

V0 c1,(N–1)

Vi, �i

Vi, �i

3.5

Influence of RTD on the Reactor Performance

113

c1,0 PFR Concentration

c1,1 5 CSTR in series c1,2 c1,3 c1,4

CSTR

c1,out

c1,5

Volume of the reaction mixture

Figure 3.22 Concentration profile in a cascade with five stages in comparison with a PFR and a single CSTR.

Supposing an irreversible first order reaction and constant fluid density (� = 0), we get for the first vessel: V̇ 0 ( c1,0 − c1,1 ) = V1 ⋅ k c1,1 ; c1,0 − c1,1 = c1,1 c1,0

=

V1 ⋅ k c1,1 V̇ 0

1 1 + k�1

(3.69)

The outlet concentration of the first vessel corresponds to the inlet concentration of the second one. We finally find: c1,1 c1,0

=

c1,2 c1,1

c1,N c1,0

=

c1,3

=

c1,2

=1−X =

c1,i c1,i−1

···

c1,N c1,N−1

=

1 � ; � = �2 = · · · = � i = (3.70) 1 + k�i 1 N

1 1 = (1 + k�∕N)N (1 + DaI∕N)N

(3.71)

In Figure 3.23 the unreacted fraction of the key reactant (f1 = 1 − X) is plotted against the Damköhler number for different stirred tanks in series. For N → ∞ the final conversion corresponds to that of an ideal PFR. 1

0,1

N=1

0,01

N=3

N=2 N=5 0,001

0,0001

0,1

N=1 N=2 N=3 N=5 N = 10

N = 10 PFR

(a)

f1 = 1 – X

f1 = 1 – X

1

N = 20

PFR 0,01

0

2

4

6 Dal

8

10

12

(b)

0

20

40

60

80

Dal = kc1,0 �

Figure 3.23 Unreacted fraction as function of DaI and N: (a) first order reaction and (b) second order reaction.

100

120

114

3 Real Reactors and Residence Time Distribution (RTD)

Example 3.5: Estimation of conversion based on RTD-models. A first order reaction is carried out in the real reactor with an RTD characterized in Examples 3.1 and 3.2. Under reaction conditions the rate constant is found to be k = 2.67 ⋅ 10−3 s−1 . Estimate the conversion based on the kinetics and the experimental RTD and compare the results with predictions based on the cell-in-series and the dispersion model. Solution:

Following the model of parallel arranged plug flow reactors, the mean conversion in a real reactor can be estimated with Equation 3.64 ∞ ∑ X= X(t)⋅E(t)dt ≅ X(ti )⋅E(ti )Δti ∫0 In Example 3.1 we calculated C(�) and plotted it in Figure 3.3. To apply Equation 3.64 we have to multiply C(�) with the mean residence time and to calculate the conversion as function of time with the given kinetics. We suppose a closed/closed system. Thus C(t) = E(t). The result is shown in the following table: T (s)

0

120

240

360

480

600

720

840

960

E(t) ⋅ 103 X(t)

0 0

1.083 0.274

2.083 0.473

2.083 0.618

1.667 0.722

0.833 0.799

0.417 0.854

0.167 0.894

0 0.923

Applying Equation 3.64 results in X = 0.59. For a first order reaction the conversion can be calculated with the dispersion model (Equation 3.67). As the model supposes a closed/closed system, we obtain with Bo = 8 and t = 374.4 s ⇒ DaI = 2.67 ⋅ 10−3 ⋅ 374.4 = 1; X = 0.60. For the tanks in series model we estimate the number of cells with the Bo: N ≅ Bo∕2 = 4. Applying Equation 3.71 for N = 4 tanks in series we obtain with DaI = 1: 1 1 f1,N = 1 − X = ( )3.5 = 0.415; X = 0.59 )N = ( 1 k� 1 + 3.5 1+ N The results demonstrate that the predicted mean conversion can be calculated with one of the discussed methods giving roughly identical results. In complex reaction systems, axial dispersion will also affect the product yield and selectivity attainable in real tubular reactors. This will be demonstrated for first order consecutive reactions. k1 k2 k A1 −−→ A2 −−→ A3 ; κ = 2 k1

3.6

RTD in Microchannel Reactors

Yield of intermediate product, Y2

1.0 0.8 Y2,max

κ = k2 /k1 = 0.1

0.6 0.4

DaI1,op = k1τop =

0.2 Y2,max = 0.0 0.0

c2,max =κ c1,0

1 · ln (κ) κ–1 κ 1–κ

2.0 DaI1,op 4.0 6.0 DaI1 = k1 τ

8.0

10.0

Figure 3.24 First order consecutive reactions. Yield of the intermediate as function of space time. c

The yield of the intermediate product Y2 = c 2 depends on the space time and the 1,0 ratio of the rate constants, �. It follows with DaI 1 , the Damköhler number referred to the first reaction (Figure 3.24): Y2 =

1 [exp(−DaI1 ) − exp(−κDaI1 )] κ−1

(3.72)

The product yield first increases up to a maximum value (Y 2, max ) and falls down to zero for DaI 1 => ∞. The maximum is attained at the optimal time, respectively, Da-number DaI 1, op . At higher or lower values the yield diminishes. Therefore, it is evident that the highest yield can be reached only in an ideal plug flow reactor with a space time corresponding to DaI 1, op . Any RTD in real tubular reactors will never allow the maximum yield of the intermediate. This is demonstrated for three different values of � = k 2 /k 1 in Figure 3.25. The real tubular reactor is modeled with the cell model.

3.6 RTD in Microchannel Reactors

Flow in microchannels with diameters between 10 and 1000 μm is mostly laminar and has a parabolic velocity profile. Therefore, the molecular diffusion in axial and radial directions plays an important role in RTD. The diffusion in the radial direction tends to diminish the spreading effect of the parabolic velocity profile, while in the axial direction the molecular diffusion increases the dispersion [7, 8]. With the so-called Taylor-Aris correlation the axial dispersion coefficient can be predicted based on the molecular diffusion coefficient Dm , the mean velocity of the stratified flow, the hydraulic diameter of the microchannel, and the geometry

115

3 Real Reactors and Residence Time Distribution (RTD)

1.00 Maximum yield, Y2,max /Y2,max,PFR

116

0.95 0.90 κ = k2/k1 = 0.015 κ = k2/k1 = 0.04 κ = k2/k1 = 0.1

0.85 0.80 0.75 0.70 1

10

100

Number of cells in series, N Figure 3.25 Maximum yield of intermediate product referred to maximum yield in PFR as function of axial dispersion (cell model).

of the cross section: Dax = Dm + χ ⋅

u2 dh2 Dm

(3.73)

with � = 1/119 for square and � = 1/192 for circular cross section. The dispersion in tubular reactors can be estimated for stratified flow in microchannels by introducing Equation 3.73 in the Bo-number. 1

=

Bo

Dax d2 u D L = m + χ h 2 – L u u·L Dm L τ/ tD,ax tD,rad /τ

(3.74)

The first term in Equation 3.74 corresponds to the ratio between space time and characteristic axial molecular diffusion time (tD,ax = L2 ∕Dm ). The second term corresponds to the ratio between radial diffusion time and space time. Molecular diffusion coefficients are in the order of 10−5 m2 s−1 for gases and 10−9 m2 s−1 for liquids. For microchannels with the length of several centimeter and mean residence times of seconds, axial diffusion can be neglected. In consequence, the dispersion in the channel is determined mainly by the ratio between the mean residence time in the reactor and the characteristic radial diffusion time. It follows: 1 Dm L ⋅ ⋅ ; χ dh2 u D � L ⋅ = 48 , circular tube Bo ≅ 192 ⋅ m 2 u t dt D,rad D � L Bo ≅ 119 ⋅ m ⋅ = 30 , square channels 2 u t dh D,rad Bo ≅

(3.75)

3.6

RTD in Microchannel Reactors

3.6.1 RTD of Gas Flow in Microchannels

Axial dispersion can be neglected (Bo ≥ 100), if the space time is at least two times the radial diffusion time. Accordingly, axial dispersion of gases in microchannels can be neglected, if their diameters are less than 1000 μm and the space time is longer than 0.1 s. This could also be proved experimentally. The approach can also be used for multichannel reactors. Because of the small volume of a single channel, many channels have to be used in parallel to obtain sufficient reactor throughput. A uniform distribution of the reaction mixture over thousands of microchannels is necessary to obtain an adequate performance of the microstructured reactor. Flow maldistribution will enlarge the RTD in the multitubular reactor and lead to a reduced reactor performance along with reduced product yield and selectivity. Therefore, several authors have presented design studies of flow distribution manifolds [9–13]. Besides maldistribution, small deviations in the channel diameter introduced during the manufacturing process cause an enlargement of the RTD. The deviations may also be because of a nonuniform coating of the channel walls with catalytic layers. If the number of parallel channels is large (N > 30), a normal distribution of the channel diameters with a standard deviation � can be assumed. The relative standard deviation, � ̂d = �d ∕dt influences the pressure drop over the micro-reactor [11]: Δp =

128 ⋅ � ⋅ V̇ tot ⋅ L 4

(3.76)

� ⋅ N ⋅ dt ⋅ (1 + 6̂ �d2 ) The relation (3.76) shows that a variation of the channel diameter leads to a decrease of the pressure drop at a constant overall volumetric flow. As the pressure drop for each channel is identical, the variation of the diameter results in a variation of the individual flow rates, V̇ i , and the residence time, �i = Vi ∕V̇ i . Supposing plug flow in each channel (Boi → ∞), the overall dispersion is inversely proportional to the relative standard deviation and can be estimated by Equation 3.77 [11]: Boreactor ≅

dt2 2�d2

(3.77)

In consequence, the plug flow behavior in a multichannel micro-reactor (Boreactor ≥ 100) can be assumed only if the relative standard deviation is �d ≤ 0.07. dt In conclusion, narrow RTD in multichannel microreactors can only be expected, when the design of gas distributer in front of the microchannel array and the design of the collector behind the channels are optimized. A difficulty for the experimental characterization is the fact that the fluid distribution in the distributer and collector are comprised in the experimental distribution curve. As the flow in the inlet and outlet regions can be complex, correct

117

3 Real Reactors and Residence Time Distribution (RTD)

modeling of the complete microdevice with the presented models is hardly possible [14]. An example for measured RTD of gas flow in a stainless steel microstructured device (Figure 3.26) is shown in Figure 3.27. The experiments were carried out with an array of 340 rectangular channels of 300 × 240 μm, which were coated with an alumina catalyst. The results prove that the coating was very regular and did not deteriorate the flow behavior. The experimental results can be described satisfactorily with the dispersion model with a Bodenstein number of Bo = 70. Compared to predicted RTD for single channels with the Taylor-Aris correlation, the Bo is quite low. This indicates the important influence of the inlet and outlet regions on the overall dispersion. A detailed study of the influence of the gas distributer and collector design on the RTD confirms the discussed findings [14]. 3.6.2 RTD of Liquid Flow in Microchannels

Whereas radial diffusion times for gases (Dm ≅ 10−5 m2 s−1 ) in microchannels is in the order of 10−2 s, the radial diffusion time for liquids (with Dm ≅ 10−9 m2 ≅s−1 ) is in the order of seconds even in microchannels with diameters of 100 μm. To reach a narrow RTD (Bo ≥ 100) in stratified flow, long residence times of � ≥ 8 ⋅ 108 ⋅ dh2 (in seconds) are necessary. But, in contrast to the estimations based on the TaylorAris correlation (Equation 3.75), experimentally determined RTD are often much Collector E(t)coll

Microchannel array E(t)channel

Distributor E(t)dist

Outlet

20 mm

118

Inlet

20 mm

2.5 mm

Figure 3.26 Drawing of a microstructured multichannel reactor. Channel: 300 × 240 μm. 34 channels/plate; 10 plates. (Institut für Mikrotechnik Mainz, IMM) [15]. (Adapted with permission from Elsevier.)

3.6

RTD in Microchannel Reactors

Residence time distribution, E(t/τ)

2.5 Uncoated channels

2.0

Coated: γ-Al2O3 Bo = 70, Dispersion model

1.5 1.0

0.5 0.0 0.0

0.5

1.0 1.5 2.0 Dimensionless time, t/τ

2.5

Figure 3.27 Measured residence time distribution in a microstructured device (Figure 3.26): 340 microchannels, space time � = 2.5 s. [14] (Adapted with permission from Elsevier.)

Oil bath 2 1 4

3

4

2 1 1. Syringe pump;

T

T

2. preheating coil; 3. micromixer; 4. delaypipe

Figure 3.28 Set-up of a microchannel system for ionic liquid synthesis. 1. Syringe pump, 2. preheating coil, 3. micromixer, 4. delaypipe. Adapted from Ref. [16] with permission from Elsevier.)

more narrow. An example is the experimentally determined RTD obtained in a microstructured device for the synthesis of ionic liquids [16]. The experimental set-up used is shown in Figure 3.28. The installation is typical for small-scale continuous chemical synthesis. It consists of high-precision pumps for dosing the reactants, preheater, a micromixer, and a delay channel. An efficient micromixer is essential to ensure fast mixing down to the molecular scale at very short residence times to avoid preliminary reactions eventually accompanied with an uncontrolled temperature increase. The transformation of the reactants occurs in the following delay pipe, where residence times of several minutes can be attained. The microtubular reactor consists of a 1.13 m long capillary in the form of a coil with an inner diameter of 1.8 mm. As the studied reaction is of second order and high conversion is warranted, uniform residence time is indispensable for high product yield and reactor performance.

119

3 Real Reactors and Residence Time Distribution (RTD)

1.0 0.8

F = c/c0

120

0.6 t = 402 s

0.4

Bo ≅ 150 0.2 0.0 0

200

400 600 Time, τ (s)

800

1000

Figure 3.29 F-curve measured by a step-stimuli response and predicted with dispersion model (Equation 3.78. Experimental values taken from Ref. [16].)

The RTD in the tubular reactor was determined experimentally with water as fluid and Brilliant Blue dye as tracer. The tracer was introduced at the reactor inlet in the form of a step function. The concentration of the dye was measured with an UV-vis spectrometer and the response curve is given as F-curve. As the experimental F-curve shown in Figure 3.29 is very steep, a low axial dispersion can be expected. Therefore, RTD will be described with the dispersion model supposing small deviation from plug flow (Equation 3.50). The F-curve valid for small dispersion (Bo ≥ 100) can be obtained by integrating the RTD given by E(�) (Equation 3.50). { [ )] ( } � √ √ t 1 ′ ′ erf F= − 1 − erf( Bo∕4) Bo∕4 ⋅ (3.78) E(� )d� = ∫0 2 t The mean residence time, t, and the Bo can be obtained by fitting the F-curve (Equation 3.78) to the experimental results. For the example shown in Figure 3.29 a mean residence time of 402 s ± 0.5% and a Bo of 150 ± 13% is obtained. This confirms the small dispersion and allows considering the reactor as an ideal plug flow reactor. On the basis of the Taylor-Aris correlation (Equation 3.75) a Bodenstein number of Bo ≅ 24 is expected. The experimental results suggest that efficient radial mixing occurs, which may be explained by the used capillary shaped as a coil provoking enhanced radial mixing. Narrow RTD in an array of plastic capillaries coiled in a spiral form were reported by Hornung and Mackley [17]. The array consisted of up to 19 capillaries in parallel with an inner diameter of 223 μm and a length of 10 m. The space time was varied between 30 s and 1.5 h. Experiments using optical fibers for the detection of a tracer dye at the entrance and outlet of the capillaries confirmed near plug flow behavior with Bo up to 220 depending on the flow rate, which could be predicted from the Taylor-Aris correlation for single tubes. Remarkable

3.6

RTD in Microchannel Reactors

is that the inlet flow is evenly distributed over the 19 capillaries thus avoiding maldistribution and broadening of the RTD (Figure 3.30). As mentioned above, radial mixing is crucial to get narrow RTD. Therefore, the use of passive mixer helps equalizing the radial concentration in the laminar flow domain. It is known that static mixer allows to obtain narrow RTD in tubular reactors even with high viscous media [18]. The beneficial effect of radial mixing can also be expected in microstructured mixers. Boškovi´c et al. [19, 20] studied the RTD in three different mixing devices: serpentine channel, split and recombine, and staggered herringbone reactors (SHR) (Table 3.4). They developed and applied an impulse-response technique to characterize the mixers in a wide range of Re between 0.3 < Re < 110. Serpentine channel reactor (SCR) and split and recombine reactor (SAR) demonstrated a similar behavior. With increasing volumetric flow the variance diminishes. This is shown for the SCR in Figure 3.31 as an example. Radial mixing in SCR and SAR becomes important mainly at Re > 30. The Bo-number at Re < 30 is relative low with values in the range of 20 < Bo < 30 (Figure 3.32). Within this domain, radial mixing seems to be mainly governed by molecular diffusion. For Re > 30 Bo increases drastically and reaches values of Bo ≅ 100 at Re ≅ 100. In spite of the fact that the space times under these conditions are in the order of 1 s, plug flow behavior is obtained. This fact is quite important for fast chemical transformations leading to high conversions at short residence times. The SHR shows different RTD characteristics compared to the previous discussed micromixers. Even at very low Re, the main peak of the distribution is quite narrow and symmetric. But, a flat and long tailing is observed [20]. The behavior suggests the presence of a dynamic phase with near plug flow behavior and a 1.0 0.8 Bo = 180 ± 6.5

0.6 F(θ)

19 capillaries 0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

θ = t/τ Figure 3.30 Cumulative RTD in a 19-capillary spiral microdevice at 1 ml min−1 . Experimental results compared to the dispersion model. (Values taken from Ref. [17].)

121

3 Real Reactors and Residence Time Distribution (RTD)

Table 3.4 Characteristics and flow conditions for RTD studies [20].

Hydraulic diameter dh (μm) Total channel length Lc (mm) Flow rate V̇ o

(ml min−1 ) Re [−]

Split and recombine reactor

Serpentine channel reactor

Staggered herringbone reactor

400

600

400

164

233

100

0.025 ≤ V̇ o ≤ 3

0.01 ≤ V̇ o ≤ 3

0.01 ≤ V̇ o ≤ 3

0.3 ≤ Re ≤ 83

0.4 ≤ Re ≤ 111

0.4 ≤ Re ≤ 111

Adapted with permission from Wiley.

4 0.010 ml min–1 0.250 ml min–1 0.500 ml min–1 1.500 ml min–1 3.000 ml min–1

3

E (θ)

122

2

1

0 0.5

1.0

1.5 Dimensionless time θ

2.0

2.5

Figure 3.31 Experimentally obtained RTDs of the serpentine channel reactor (SCR). (Adapted from Ref. [20] with permission from Wiley).

very small stagnant phase located in the grooves of the structure. The exchange between these dead zones and the dynamic phase is governed by molecular diffusion, which explains the long tailing. 3.6.3 RTD of Multiphase Flow in Microchannels

Because of the laminar flow in microchannels and the small diffusion coefficient in the order of 10−9 m2 s−1 , narrow RTD in straight channels at low Re-numbers are not easy to obtain. As the radial diffusion time is long, high axial dispersion is particularly important at short residence times.

3.6

RTD in Microchannel Reactors

Serpentine channel reactor Split and recombine reactor

100

Bo

80 60 40 20 0 1

10 Re

100

Figure 3.32 Evolution of Bo as function of Re for SAR and SCR micro-mixer. (Data taken from Ref. [20].)

A way to overcome problems related to RTD is the use of multiphase flow (gas–liquid, liquid–liquid). Laminar flow in microchannels permits the easy formation of fluid–fluid slugs (see Chapter 7). Under these conditions, the reaction mixture is present in the form of segments, which are separated by gas bubbles or by a second liquid, immiscible with the reaction phase. In this way, the reacting segments behave as a series of small batch reactors traveling through the channel, thus eliminating the problem of axial dispersion as found in laminar single-phase flow reactors at short residence times. But, the indicated situation is strictly true only if the different segments are completely disconnected from each other. Therefore, the reacting phase must be dispersed in a continuous carrier phase, which is wetting the microchannel walls. This situation is illustrated in Figure 3.33. A cross-junction with three inlets and one outlet is a suitable device for generating regular slug flow. The reactants are introduced in two opposite inlets while the immiscible carrier fluid is introduced through the third inlet generating slug flow. The experimental proof of this concept is presented by Trachsel et al. [22]. They used a microstructured device composed of meandering channels with rectangular cross section, 0.4 mm wide, 0.115 mm high, and 1063 mm long as shown in Figure 3.34. The authors used a fluorescently labeled tracer, which was injected as impulse. The response of the inlet signal could be followed at different distances downstream from the injection point. Two-phase experiments were carried out by injecting gas as a separating fluid and the results were compared with RTD obtained with single-phase flow. Typical RTD curves for single and multiphase flow are reproduced in Figure 3.35. The experimental results are fitted to the dispersion model supposing open/open boundary conditions (see Figure 3.11, Equation 3.45).

123

124

3 Real Reactors and Residence Time Distribution (RTD)

Reactant, A1

(a)

Carrier 500 μm

Reactant, A2 μ-batch reactor

Internal circulations

(b)

Figure 3.33 Biphasic flow in microchannel for narrow RTD: (a) schematic illustration and (b) monochrome snapshot of slug flow. (Adapted from Ref. [21] with permission from Elsevier.)

Outlet

Gas inlet

Liquid inlet

10 mm

Figure 3.34 Scheme of the microchannel device for RTD studies. (Adapted from Ref. [22] with permission from Elsevier.)

The two-phase flow leads to significant narrower RTD with a Bodenstein number of Bo = 308 ± 3, whereas Bo = 83 ± 2 was estimated from the results obtained for single cell flow (Figure 3.35). In the presented study, liquid segments are separated by gas bubbles. But the liquid wets the channel wall and forms a small film, which allows communication between the segments and, as a consequence, enables axial dispersion. The influence of wall film on the RTD in segmented flow was studied in detail by Kuhn et al. [23]. The authors used microreactors with a square section of 0.4 × 0.4 mm and a length of 750 mm. The walls of the silicon-based microdevices were modified by growing a thin silicon oxide layer to get a hydrophilic surface

3.6

Segmented flow

5 4

C (θ)

RTD in Microchannel Reactors

Fit

3 2 Single phase flow 1 0 0.6

0.8

1.0 θ = t /τ

1.2

Figure 3.35 Comparison of measured RTD with dispersion model (Equation 3.45). Segmented flow: superficial velocity uliquid = 3.6 mm s−1 ,

1.4

ugas = 25.2 mm s−1 , � = 59 s. Single-phase flow: uliquid = 14.9 mm s−1 , � = 71 s. (Adapted from Ref. [22] with permission from Elsevier.)

Hydrophilic SiO2 surface Aqueous wall film Toluene

Water, wetting phase

Toluene

Hydrophobic PTFE surface

Toluene wall film Water dispersed phase

Toluene, wetting phase

Water dispersed phase

Figure 3.36 Sketch of the phase behavior depending on the wettability of the surface.

or by coating it with a thin PTFE (polytetrafluoroethylene) layer for obtaining a hydrophobic surface. Toluene and water were used as biphasic system. The RTD of the water phase was studied by injecting a pulse of a sodium benzoate solution at the reactor entrance. The response curve was determined by UV-vis spectroscopy. In the silicon-oxide-coated microchannels, water constitutes the continuous phase in which toluene is dispersed. The water segments, separated by toluene slugs, form an aqueous wall film, which allows communicating with each other through axial dispersion. In contrast, in the PTFE-coated microchannel, water is dispersed in the continuous toluene phase. In consequence, the water

125

126

3 Real Reactors and Residence Time Distribution (RTD) 7

6

6

Bo = 330 ± 18

5

E(θ)

4

E(θ)

Bo = 520 ± 33

5

3 2

4 3 2

1

1 0

0 0.6

0.8

1.0

1.2

θ

(a)

0.6

1.4 (b)

Figure 3.37 Residence time distributions for (a) silicon oxide and (b) PTFE coated microchannels. Flow rate: 12.5 μl min−1 water, 12.5 μl min−1 toluene. Solid lines represent

0.8

1.0

1.2

1.4

θ the fit of the dispersion model Equation 3.50. (Values taken from Ref. [23]. Adapted with permission Kuhn et al. [23]. Copyright (2011) American Chemical Society.)

segments are isolated and can no longer communicate with each other. The described situations are illustrated schematically in Figure 3.36. In Figure 3.37 the measured RTD for hydrophilic and hydrophobic microchannels are shown. The experimental results can be described with the dispersion model valid for small dispersion (Equation 3.50). The resulting Bo-numbers estimated by curve fitting was found to be Bo = 300 ± 18 for the hydrophilic channels and Bo = 520 ± 33 for the hydrophobic channels. The results confirm the beneficial use of segmented flow for realizing plug flow behavior in microstructured reactors. Narrow RTD can be obtained even for short residence times, allowing high performance and product yields for fast chemical reactions. 3.7 List of Symbols

Symbols

Significance

Unit

c(t) E(t) E F g(t) N n, ninj

Concentration at time t Probability function Dimensionless probability function F-curve in RTD Convolution function Number of tanks in cascade Number of moles, amount of non-reacting tracer injected

mol m−3 s−1 — — — — mol

References

Symbols

Significance

Unit

ṅ i , ṅ i,0 , ṅ i,L

Molar flow rate of species i, at reactor inlet, at length L (outlet) Skewness Mean residence time, mean residence time from measured RTD curve for open or semi-open systems Average velocity over cross section, velocity at radial position r, velocity in the center of the tube in laminar flow Mean conversion for multichannel reactor Ratio of radial distance to tube radius Time interval Moments of the distribution density function variance of the distribution, variance from measured RTD curve for open or semi-open systems variance of the distribution, variance from measured RTD curve for open or semi-open systems Dimensionless time from measured RTD curve for open or semi-open systems Space time of reactor i Geometrical constant

mol s−1

sk t, t c u, u(r) , umax

X y Δt �1 , �2 � 2 , �c2 2 ��2, ��c

�c �i �

— s m s−1

— — s — s2 — — s —

References 6. Renken, A. and Kashid, M.N. (2012) in Catalysis: From Molecular to Reacempirical equations for the residence tor Design (eds M. Beller, A. Renken, time distributions in disperse sysand R.A.v. Santen), Wiley-VCH Verlag tems – part 1: continuous phase. Chem. GmbH, Weinheim, pp. 563–628. Eng. Technol., 27 (11), 1172–1178. 7. Aris, R. (1955) On the dispersion of a Baerns, M., Behr, A., Brehm, A., solute in a fluid flowing through a tube. Gmehling, J., Hofmann, H., Onken, Proc. R. Soc. London, Ser. A, A 235, U., and Renken, A. (2006) Technische 67–77. Chemie, 4th edn, Wiley-VCH Verlag 8. Taylor, G. (1953) Dispersion of soluble GmbH, Weinheim, 733 p. matter in solvent flowing slowly through Danckwerts, P.V. (1953) Continuous flow a tube. Proc. R. Soc. London, Ser. A, A systems: distribution of residence times. 219, 186–203. Chem. Eng. Sci., 2 (1), 1–13. Baerns, M., Hinrichsen, K.-O., Hofmann, 9. Aubin, J., Prat, L., Xuereb, C., and Gourdon, C. (2009) Effect of microchanH., and Renken, A. (2013) in Chemische nel aspect ratio on residence time Reaktionstechnik. Technische Chemie (eds distributions and the axial dispersion M. Baerns, A. Behr, A. Brehm, et al.), coefficient. Chem. Eng. Process., 48 (1), Wiley-VCH Verlag GmbH, Weinheim, 554–559. 736 pp. 10. Commenge, J.-M., Falk, L., Corriou, J.-P., Pallaske, U. (1984) Kritik an zwei and Matlosz, M. (2002) Optimal design bekannten Verweilzeitformeln für das for flow uniformity in microchannel Diffusionsströmungsrohr. Chem. -Ing. reactors. AIChE J., 48 (2), 345–358. Technol., 56, 46–47.

1. Ham, J.H. and Platzer, B. (2004) Semi-

2.

3.

4.

5.

127

128

3 Real Reactors and Residence Time Distribution (RTD) 11. Delsman, E.R., de Croon, M.H.J.M.,

12.

13.

14.

15.

16.

Elzinga, G.D., Cobden, P.D., Kramer, G.J., and Schouten, J.C. (2005) The influence of differences between microchannels on microreactor performance. Chem. Eng. Technol., 28 (3), 367–375. Mies, M.J.M., Rebrov, E.V., De Croon, M.H.J., Schouten, J.C., and Ismagilov, I.Z. (2006) Inlet section for providing a uniform flow distribution in a downstream reactor comprises upstream and downstream passage with specifically positioned elongated parallel upstream and downstream channels. WO Patent 2006107206-A2 Rebrov, E.V., Schouten, J.C., and de Croon, M.H.J.M. (2011) Single-phase fluid flow distribution and heat transfer in microstructured reactors. Chem. Eng. Sci., 66 (7), 1374–1393. Wibel, W., Wenka, A., Brandner, J.J., and Dittmeyer, R. (2013) Measuring and modeling the residence time distribution of gas flows in multichannel microreactors. Chem. Eng. J., 215-216, 449–460. Rouge, A., Spoetzl, B., Gebauer, K., Schenk, R., and Renken, A. (2001) Microchannel reactors for fast periodic operation: the catalytic dehydration of isopropanol. Chem. Eng. Sci., 56 (4), 1419–1427. Hu, S., Wang, A., Löwe, H., Li, X., Wang, Y., Li, C., and Yang, D. (2010) Kinetic study of ionic liquid synthesis in a microchannel reactor. Chem. Eng. J., 162 (1), 350–354.

17. Hornung, C.H. and Mackley, M.R.

18.

19.

20.

21.

22.

23.

(2009) The measurement and characterisation of residence time distributions for laminar liquid flow in plastic microcapillary arrays. Chem. Eng. Sci., 64 (17), 3889–3902. Flaschel, E., Nguyen, K.T., and Renken, A. (1985) Proceeding of the 5th European Conference on Mixing, Würzburg, BHRA, Bedford, p. 549. Boškovi´c, D. and S.L. (2008) Modelling of the residence time distribution in micromixers. Chem. Eng. J., 135 (Suppl. 1), S138–S146. Boškovi´c, D., Loebbecke, S., Gross, G.A., and Koehler, J.M. (2011) Residence time distribution studies in microfluidic mixing structures. Chem. Eng. Technol., 34 (3), 361–370. Kashid, M., Detraz, O., Moya, M.S., Yuranov, I., Prechtl, P., Membrez, J., Renken, A., and Kiwi-Minsker, L. (2013) Micro-batch reactor for catching intermediates and monitoring kinetics of rapid and exothermic homogeneous reactions. Chem. Eng. J., 214, 149–156. Trachsel, F., Günther, A., Khan, S., and Jensen, K.F. (2005) Measurement of residence time distribution in microfluidic systems. Chem. Eng. Sci., 60 (21), 5729–5737. Kuhn, S., Hartman, R.L., Sultana, M., Nagy, K.D., Marre, S., and Jensen, K.F. (2011) Teflon-coated silicon microreactors: impact on segmented liquid-liquid multiphase flows. Langmuir, 27 (10), 6519–6527.

129

4 Micromixing Devices 4.1 Role of Mixing for the Performance of Chemical Reactors

Mixing is one of the basic unit operations involved in chemical transformations. The problems associated with bad mixing may have a decisive impact on the product distribution and the performance of a chemical reactor. Mixing is a general definition applied for different physical processes. It can be subdivided into two main classes: 1) Microscopic mixing: mixing of individual molecules, like homogenization of two miscible fluids, or dissolution of a solid in a liquid without the formation of any concentration (temperature) gradients. 2) Macroscopic mixing: mixing of groups or aggregates of molecules, such as a. Suspension of a solid in a liquid or slurry formation b. Dispersion (or emulsification) of two immiscible liquids c. Dispersion of a gas in a liquid (or foam formation). In general, good mixing at the macroscopic scale can be easily attained, but it is much more difficult to obtain an intimate mixing on the molecular scale. Nonuniform reactant concentrations on the molecular scale may have a significant influence on the effective transformation rate and the product distribution, especially when complex and rapid chemical reactions are involved. In this chapter we mainly consider the microscopic mixing between miscible fluids. This means that two completely miscible reacting fluids are brought together. If the characteristic mixing time defined as the time required for two fluids to become homogenous on the molecular scale is in the same order of magnitude as the characteristic reaction time or even longer, the product distribution is strongly affected, especially in the case of complex networks with parallel and/or consecutive reactions [1, 2]. To avoid the negative effect on reactor productivity and yield of the desired product (often an intermediate in a complex reaction network), the characteristic mixing time should be at least 10-fold shorter than the characteristic reaction time. To describe the mixing process and its influence on the performance of chemical reactors, different models were developed [3, 4]. Herein we discuss the influence of the micromixing process using the concept of segregation as proposed by Baldyga Microstructured Devices for Chemical Processing, First Edition. Madhvanand N. Kashid, Albert Renken and Lioubov Kiwi-Minsker. © 2015 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2015 by Wiley-VCH Verlag GmbH & Co. KGaA.

4 Micromixing Devices

[5]. In addition, we restrict the discussion to ideal tubular reactors supposing plug flow behavior. The concept of segregation in chemical reactors was first introduced by Danckwerts in 1953 [6]. He defined an intensity of segregation, I s, varying between one and zero. For Is = 0, complete mixing on the molecular scale is attained. The situation of complete segregation is illustrated in Figure 4.1. It shows a system with two reactants A1 and A2 , which are initially present in two separated volumes (expressed as volumetric fractions). Segregation also may be observed in tubular reactors with separate feeds of the reacting fluids (Figure 4.2). Danckwerts defined the intensity of segregation in turbulent flow, Is , in terms of the mean square of the concentration fluctuation as shown in Figure 4.3 and given in Equation 4.1. Is = (

Δc21 ) Δc21

(4.1) 0

c1,0

c2,0

α1

1 – α1

Figure 4.1

·V1,0 = α1·V0

Complete segregation (Is = 1) of reactants A1 and A2 in two volume fractions.

c1,0

c2,0 Figure 4.2

c1

·V0 ci,L

VR

·V2,0 = (1 – α1)··V0

Concentration, c1

130

z=0

z=L

Feeding of unmixed reactants to a tubular reactor. Constant fluid density.

Δc1

Time Figure 4.3 Time-variant concentration fluctuations around the mean value c1 . (Reproduced from Ref. [5]. Copyright © 2013, Wiley-VCH GmbH & Co. KGaA.)

4.1

Role of Mixing for the Performance of Chemical Reactors

where (Δc21 )0 corresponds to the initial mean square of concentration fluctuations. The mean square of the concentration variations is given in Equation 4.2 Δc21 = (c1 − c1 )2

(4.2)

For the situation shown in Figure 4.1 the initial mean square fluctuations are given by (Δc21 )0 = c21,0 ⋅ �1 (1 − �1 )

(4.3)

For separate feed to a continuously operated reactor, the volume fraction � 1 corresponds to the volumetric flow containing A1 referred to the total inlet flow. α1 =

V̇ 1,0 V̇ 1,0 + V̇ 2,0

=

V̇ 1 V̇ 0

(4.4)

Depending on the mixing intensity in the reactor, the segregation will diminish with increasing residence time. The rate of decay of the concentration variance can be supposed to be a first order process as indicated in Equation 4.5 rd = −bs ⋅ Δc21 = −

1 ⋅ Δc21 tmx

(4.5)

The parameter bs is equivalent to the inverse mixing time, t mx , and is a function of the power dissipation per volume and the geometry of the mixing device. In plug flow reactors (PFRs) under steady state, the segregation intensity will decay with the distance from the reactor inlet, respectively, with the residence time, �. (

Δc21 ) = Is = exp(−bs �) = exp(−bs �PFR Z) Δc21 0

with �PFR = VR ∕V̇ 0 ; Z = V ∕VR = z∕L (constant diameter)

(4.6)

The transformation rate of an irreversible second order reaction in a partially segregated fluid can be expressed in terms of the mean concentrations and the concentration fluctuations of the reactants [7]. R1 = −k ⋅ c1 c2 = −k ⋅ (c1 ⋅ c2 + Δc1 Δc2 ) dc1 = R1 = −k ⋅ (c1 ⋅ c2 − Is c1,0 ⋅ c2,0 ) d� with ∶ Δc1 Δc2 = −Is (c1,0 ⋅ c2,0 )

(4.7)

With the following dimensionless variables: fi = ci ∕c1,0 , c2,0 = c2,0 ⋅ (1 − �1 ); c1,0 = c1,0 ⋅ �1 , DaI = M = c2,0 ∕c1,0 , kc1,0 �PFR The material balance for the reactant A1 can be expressed in dimensionless form as follows: df1 = −DaI ⋅ [f1 ⋅ (f1 + M − 1) − Is M] (4.8) dZ

131

4 Micromixing Devices

For given reaction conditions and residence time in a PFR, the axial concentration profile and the outlet conversion depends strongly on the ratio between mixing time and characteristic reaction time. This ratio can be interpreted as a second Damköhler number for mixing DaII mx . DaIImx =

tmx = k ⋅ c1,0 ⋅ tmx (second order reaction) tr

(4.9)

This is shown in Figure 4.4, where the residual concentration of A1 is plotted as function of the reactor length for different values of DaII mx and constant DaI and equimolar reactant feed (M = 1). Whereas for immediate mixing of the reactants at the reactor entrance (DaII mx = >0) the conversion of A1 corresponds to X = 0.833 (f 1 = 0.167) at DaI = 5, the conversion drops to X = 0.345 (f 1 = 0.655) for DaII mx = 5. A roughly four times higher residence time in the reactor is needed to get the same conversion as for DaII mx = 0. In general, long mixing times as compared to the characteristic reaction time diminish the reactor performance for reactions with positive reaction orders (n > 0). This leads to the increased reactor size in order to attain the same degree of conversion. Even more important is the influence of slow mixing on the performance of complex reactions. If the reaction between two fluids takes place to an appreciable extent before the homogeneity is attained, segregation can affect product distribution. This is discussed for consecutive competing reactions represented by Equation 4.10. k1

A1 + A2 −−→ A3 k2

(4.10)

A3 + A2 −−→ A4 Residual concentration of A1, f1 = 1 – X

132

1.0

0.8

Dallmx = 5.0

0.6 Dallmx = 0

0.2

0.5

1.0

2.0

0.4

0.2 Final concentration for Dallmx = 0 0.0

0.0

0.2

0.4

0.6

0.8

1.0

Reactor length, Z = z/L Figure 4.4 Estimated residual concentration of A1 as function of reactor length for different DaIImx . DaI = 5; M = 1.

4.1

Role of Mixing for the Performance of Chemical Reactors

Examples of such consecutive competing reactions are chlorination, nitration of hydrocarbons, or the addition of alkene oxides (e.g., ethylene oxide) to amines or alcohols. If the mixing is fast enough, so that the reacting fluid is homogenous before the reaction takes place, the maximum yield of the desired intermediate A3 will be controlled by the ratio of k 2 /k 1 . Let us suppose irreversible second order reactions, so the following relations describe the transformation rates of the involved reactants. R1 = −k1 ⋅ c1 c2 R2 = −k1 ⋅ c1 c2 − k2 ⋅ c2 c3 R3 = k1 ⋅ c1 c2 − k2 ⋅ c2 c3 (4.11)

R4 = k2 ⋅ c2 c3

In batch or ideal PFR the concentration of the intermediate product A3 will first increase with increasing residence time, pass through a maximum, and finally disappear at long residence times, if the initial reaction partners are fed in stoichiometric ratios. For the simple reaction scheme shown by Equation 4.10, M = c2,0 ∕c1,0 must be M = 2 for complete transformation of A1 and A2 . The concentration profiles can be calculated by solving the corresponding mass balances. For a PFR the following mass balances are obtained: dc1 d� dc2 − d� dc3 d� dc4 d� −

= k 1 ⋅ c1 c2 = k1 ⋅ c1 c2 + k2 c2 c3 = k1 ⋅ c1 c2 − k2 c2 c3 (4.12)

= k 2 c2 c3

Initial conditions: � = 0 ∶ c1 = c1,0 ; c2 = c2,0 ; c3 = c4 = 0 By dividing the third equation by the first one, the residence time can be eliminated and we get a relationship between the reactant concentrations. −

dc3 k c =1− 2 ⋅ 3 dc1 k1 c1

(4.13)

After integration, a relation between the yield of the intermediate A3 and the conversion of the key component A1 is obtained. Y3,1 =

(1 − X)� − (1 − X) ; 1−�

Y3,1 = (X − 1) ⋅ ln(1 − X);

k2 ≠1 k1 k �= 2 =1 k1 �=

(4.14)

133

4 Micromixing Devices

The intermediate yield as function of conversion is dependent only on the ratio of rate constants. This is also true for the maximum attainable yield Y 3, 1 . �

Y3,1,max = � 1−� , � = Y3,1,max =

k2 ≠1 k1

k 1 = 0.368, � = 2 = 1 exp(1) k1

(4.15)

Decreasing the ratio of the rate constants leads to higher maximum yield of the intermediate, and the maximum value is shifted to higher reactant conversion as illustrated in Figure 4.5. For understoichiometric ratio of the reactants (M < 2), A1 cannot be completely converted and the final yield and conversion is indicated by the intersection of the dashed lines with the corresponding curve (Figure 4.5). The transformation rates in terms of the intensity of segregation and mean concentrations are given by neglecting the influence of the covariance Δc2 ⋅ Δc2 [8]: R1 = −k1 ⋅ (c1 ⋅ c2 + Δc1 Δc2 ) R2 = −k1 ⋅ (c1 ⋅ c2 + Δc1 Δc2 ) − k2 ⋅ c2 ⋅ c3 R3 = k1 ⋅ (c1 ⋅ c2 + Δc1 Δc2 ) − k2 ⋅ c2 ⋅ c3 (4.16)

R4 = k 2 ⋅ c 2 ⋅ c 3

If the reactant A1 and A2 are fed separately to the PFR and mixing is slow as compared to the chemical transformation, significant yield losses for the intermediate product A3 can be expected, whereas the formation of the final product A4 will increase. The effect will become more pronounced by increasing the mixing time referred to the characteristic reaction time, which corresponds to DaIImx = k1 ⋅ c1,0 ⋅ tmx . In Figure 4.6 the influence of DaII mx on the intermediate 1.0 0.8

κ = 0.02 A1 + A2 A3 + A2

Yield, Y3,1

134

k1 k2

0.05

A3

κ = k2 /k1

0.1

A4

0.2

0.6 0.5

0.4 1 2

0.2 10

0.0 0.0

M = 0.5

0.2

M = 1.0

0.4 0.6 Conversion, X

5 M = 1.5

0.8

1.0

Figure 4.5 Yield of the intermediate product A3 as a function of conversion and the ratio of the rate constants, �.

4.1

Role of Mixing for the Performance of Chemical Reactors

0.8

DaIImx = 0

Intermediate yield, Y3,1

0.6

c2,0

M=

c1,0

0.2

0.5

κ = k2 /k1 = 0.1

DaIImx = 1

=2

DaIImx = 2

0.4 DaIImx = 5 0.2

0.0 0.0

0.2

0.4 0.6 Conversion, X

0.8

1.0

Figure 4.6 Yield of the intermediate product A3 as a function of conversion for different dimensionless mixing times (DaIImx = tmx /tr ); M = 2.

A3 yield as a function of the residence time, respectively, the first Damköhler number, is illustrated for � = k 2 /k 1 = 0.1 as an example. In summary, with increasing mixing time as compared to the characteristic reaction time, the maximum yield of the desired intermediate is significantly diminished and shifted to longer residence times, respectively, to higher DaI values. At the same time the maximum yield is attained at lower reactant conversion (Figure 4.7). The consequence is a drastic decrease of the reactor performance concomitant with an increased formation of the unwanted by-product, which is often considered as waste. 0.8 0.2

DaIImx = 0

Intermediate yield, Y3,1

0.5 0.6

DaIImx = 1 DaIImx = 2

0.4 DaIImx = 5 0.2 M= 0.0

0

5

c2,0 c1,0

= 2 κ = k2 / k1 = 0.1

10 DaI = k1 c1,0 τPFR

15

20

Figure 4.7 Yield of the intermediate product A3 as function of the dimensionless time (DaI) for different dimensionless mixing times (DaIImx = tmx /tr ).

135

136

4 Micromixing Devices 800 μm 100 μm

Solution 2 0 10

μm

Solution 1

300 μm

15

00

μm

100 μm 200 μm

Figure 4.8

T-shaped micromixer [14]. (Adapted with permission from Wiley.)

4.2 Flow Pattern and Mixing in Microchannel Reactors

Microchannel reactors are characterized by small hydraulic diameters of dh < 1 mm. Therefore, laminar flow with Reynolds numbers Re < 2000 prevails in microchannels. But, in spite of the laminar flow, vortices and engulfment processes may develop and promote greatly the mixing process. Extensive theoretical and experimental studies of the flow pattern and mixing efficiency in T-shaped micromixer [9–14] confirm that only in smooth channels and at low Re less than roughly 100, a stratified flow is formed. The studies were carried out with a T-mixer as shown in Figure 4.8. Within the stratified flow regime, separately fed miscible solutions flow side by side and the mass transfer perpendicular to the main flow direction is controlled by molecular diffusion only. In consequence, the mass transfer is very slow because of the rather thick liquid layers corresponding to half of the channel height. The spatial structure of the parallel flow of pure water and water colored with rhodamin is illustrated in Figure 4.9. The concentration profile is measured by confocal laser scanning microscopy [14].

Figure 4.9 Parallel flow in a T-mixer (200 × 100 × 100 mm) system water/water rhodamin, T = 20 ∘ C, Re = 120 [14]. (Adapted with permission from Wiley.)

4.3

Theory of Mixing in Microchannels with Laminar Flow

About 3 μm

Figure 4.10 Parallel flow in a T-mixer (200 × 100 × 100 mm) system water/water rhodamin, T = 20 ∘ C, Re = 240 [14]. (Adapted with permission from Wiley.)

Z 100 mm

500 mm

1500 mm

Numerical simulation Experimental result Figure 4.11 Specific interphasial area at different distances from the mixing point. Re = 186 [14]. (Adapted with permission from Wiley.)

For higher Re the “engulfment regime” sets in. Two different vortices develop within the different layers as seen in Figure 4.10. The stretching and thinning liquid lamellae and the wrapping of lamellae downstream from the mixing point yield in an enlarged interfacial area between the liquids. In this flow regime, fluid elements penetrate in the form of small lamellae of several μm thicknesses in the opposite parts of the channel. This, in turn, greatly improves the mass transfer rate. The development of the specific contact area along the mixing channel is shown in Figure 4.11. Tracer profiles obtained by numerical simulation for different distances from the mixing point are compared with experimental results [14].

4.3 Theory of Mixing in Microchannels with Laminar Flow

The mixing in a microchannel takes place by diffusion and convection depending on the flow pattern and the operating conditions used [15, 16]. The final mixing on the molecular scale, where the reaction takes place, occurs only by molecular diffusion. The time for diffusion in an elementary structure is defined as

137

138

4 Micromixing Devices

follows [3]: ( )2 ′

tD = A

l 2

(4.17)

Dm

with l the thickness of the aggregate, Dm – the molecular diffusion coefficient, and A′ – a shape factor. A′ =

1 ; p = 0(slab), p = 1 (cylinder), p = 2 (sphere) (p + 1)(p + 3)

(4.18)

In liquids, the value of molecular diffusion coefficient is ∼Dm = 10−9 m2 s−1 . In consequence, mixing by diffusion in liquids is a very slow process. In the case of very low Re ( ca. 100 the dimension of the characteristic fluid structure decreases in the direction orthogonal to the elongation as discussed in Section 4.2. Therefore, diffusion and convection contribute simultaneously to the mixing process. As diffusion is slow, convection is the dominant process for large structures, and diffusion becomes the controlling step of mixing at small scale. To estimate the overall mixing time, t mx , of intertwined lamellae the following relation is proposed [19]: ) ( 0.76 ⋅ �̇ ⋅ 2δ20 1 (4.19) tmx = tD+shear = arcsinh 2�̇ Dm where � 0 is the original thickness of the lamella and �̇ is the shear rate. The mean shear rate in laminar flow depends on the kinematic viscosity, � of the fluid, and the specific power dissipation, �, expressed in W kg−1 : ( )1 � 2 �̇ = (4.20) 2ν The specific power dissipation is proportional to the flow rate and the pressure drop. The pressure drop through open channels with laminar flow is given by the Hagen-Poiseuille equation [20]: Δp = 32�

�u Lt dt2

(4.21)

where � is a geometric factor, which is 1 for circular tubes and it depends on the height (H) to width (W ) ratio for rectangular channels. The correction factor becomes 0.89 for quadratic channels and assumes the asymptotic value 1.5

4.3

Theory of Mixing in Microchannels with Laminar Flow

when the ratio goes to zero, which corresponds to parallel plates. An empirical correlation is given by the following expression [21]: ( ) H � = 0.8735 + 0.6265 exp −3.636 (4.22) W It follows for the specific power dissipation: �=

V̇ ⋅ Δp 32 ⋅ ν ⋅ u2 = �⋅V dt2

(circular channel)

(4.23)

If the thickness of the lamella corresponds initially to the half diameter of the microchannel, we obtain the following relationship for the characteristic mixing time [17]: d2 ∕Dm dt arcsinh(0.76 ⋅ Pe) = t arcsinh(0.76 ⋅ Pe) 8⋅u 8 ⋅ Pe u ⋅ dt with the Péclet number ∶ Pe = Re ⋅ Sc = Dm

tmx =

(4.24)

For Pe > 20 the arcsinh (0.76⋅Pe) can be replaced by ln (1.52 ⋅ Pe) and Equation 4.24 becomes: ( )1 d2 ∕Dm 1 ν 2 tmx = t ln(1.52 ⋅ Pe) = √ ln(1.52 ⋅ Pe) (4.25) 8 ⋅ Pe 2 � On the basis of Equation 4.25, the mixing time in microchannels with different diameters between 50 and 1000 μm was estimated to be a function of the specific power dissipation. Physical properties correspond to water at room temperature and atmospheric pressure. Because of the damping effect of the logarithmic function, the influence of the Pe-number diminishes and the mixing time can be estimated with the simplified relation in Equation 4.26 (see Figure 4.13). tmx ≅ 0.0075 ⋅ �−0.5 (water)

(4.26)

The constant in the above equation possess an unit of m ⋅ s−0.5 . The mixing characteristics of many different microstructured mixers were studied by using the well-known “Villermaux-Dushman” reaction that takes place in aqueous solutions as described in [22]. The numerous published results collected by Falk and Commenge are added to Figure 4.13. In spite of the large scatter of the experimental results, it can be concluded that for all micromixers the obtainable mixing time seems to depend only on power dissipation. The construction geometry of the mixing device is of minor importance, if at all. It is also evident from the comparison with theoretical predictions that the experimentally observed mixing times are more than 1 order of magnitude longer. The experimental results can be roughly represented by the following relation: tmx ≅ 0.21 ⋅ �−0.5 (water)

(4.27)

The constant has unit of m ⋅ s−0.5 . This corresponds to an energy efficiency of 3–4% (explained in section 4.6). In Example 4.3 the characteristic mixing time is estimated supposing engulfment flow.

141

4 Micromixing Devices Triangular Tangential Mikroglas IMTEK

T-mixer

Caterpillar IMM

Starlam IMM

100

10–1

Mixing time, tmx (s)

142

10–2

Experimental

10–3

Predicted

10–4

10–5 100

101

102

103

104

Specific power dissipation, ε (W kg–1)

Figure 4.13 Predicted (Equation 4.25) and experimentally determined mixing time as function of the specific power dissipation. (Experimental values taken from Ref. [15]. Adapted with permission from Elsevier.)

The mixing in microchannels with laminar flow in the engulfment regime (validity of Equations 4.19 and 4.24) is summarized as

• Energy dissipation seems to be the only relevant parameter to be taken into consideration for a design of the efficient mixer.

• Multilamination improves mixing by reducing the striation thickness, requires additional mechanical power to create fine multilamellae before contacting.

• It is common that flow fields and concentration fields do not match. Mechanical energy is used to create a flow in the devices, but in zones of pure component with no interface with another component, mechanical energy does not contribute to mixing.

Example 4.3: Mixing time in the engulfment regime. Estimate the mixing time and the specific power dissipation in a cylindrical channel with a diameter of dt = 1 mm, a linear velocity of u = 0.2 m ⋅ s−1 . Dm = 10−9 m2 s−1 , � = 10−6 m2 s−1 . Calculate the mixing time and specific power dissipation for a channel with dt = 0.5 mm supposing the same volumetric flow. Solution:

dt = 0.5 mm:

V̇ ⋅Δp �⋅V

32⋅ν⋅u2 (circular channel) dt2 � = 82 W kg−1 ; t mx = 0.023 s.

dt = 1 mm: � =

=

⇒ � = 1.3 W kg−1 ; t mx = 0.18 s

4.4

Types of Micromixers and Mixing Principles

4.4 Types of Micromixers and Mixing Principles

As presented before, short radial diffusion lengths decrease the mixing time in microchannels even at low flow rates in the laminar flow regime. The mixing time can further be reduced using convective mixing by creating eddies. The convective diffusion enhancement is commonly employed in mixing devices using various methods. A lot of efforts have been made to develop efficient micromixers, and different mixer concepts have been proposed [23, 24]. In general, two types of mixers can be distinguished: passive and active mixers. The former works essentially with energy provided by pumping while the latter uses an external energy source such as acoustic fields, electric fields (electrokinetic instability), or microimpellers. Compared with active micromixers, passive mixers have the advantages of low cost, and easy integration in the microfluidic systems, no complex control units, and no additional power input. For chemical reactions, micromixers are mounted in front of a residence time unit that provides the required residence time for the reaction to complete. The choice of the micromixer to be used depends on the characteristic reaction time. The mixing time should be at least 10 times shorter compared to the reaction time to avoid losses in reactor performance and product selectivity as discussed in Section 4.1. Besides mixing efficiency, heat transfer potential is an important issue for fast exothermic reaction as is discussed in Chapter 5. A series of micromixers with different types of mixing elements have been developed using different mixing principles as listed in Table 4.1 and depicted in Figure 4.14. In the case of passive mixing, the flow that is caused by pumping or hydrostatic potential is restructured in order to get faster mixing. Thin lamellae are created in special feed arrangements, termed interdigital. A commonly used method to enhance passive mixing in microchannels is to distribute the flow into compartments and reduce diffusion paths beyond the geometric dimensions of the mixing microchannel. For instance, splitting and recombining the feed streams or the injection of substreams via a special microstructure can break the laminar profile and better mixing can be achieved [25]. Chaotic mixing Table 4.1 Passive and active mixing techniques used in micromixers. Passive mixing

Active mixing

Interdigital multilamellae arrangements Split-and-recombine concepts (SAR) Chaotic mixing by eddy formation and folding Droplet binding in two-phase environment Nozzle injection in flow Specialties, e.g., Coanda effect

Ultrasound Acoustically induced vibrations Electrokinetic instabilities Periodical variation of pressure field Electrowetting-induced joint of droplets Magnetohydrodynamic action Small impellers Piecoelectrically vibrating membrane Integrated micro valves/pumps

143

144

4 Micromixing Devices Decrease of diffusion path

Injection of substreams Splitting and recombination

Forced mass transport

Injection of substreams to a main stream

High energy collision

Contacting

Periodic injection

Figure 4.14 Schematic representation of selective passive and active mixing principles used in micromixers [26]. (Adapted with permission from the authors.)

creates eddy-based flow patterns that provide high specific interfaces, though they pose a danger of being spatially inhomogenous. Besides, the injection of many substreams, for example, via nozzles, into one main stream and collision of jets provides a means for turbulent mixing. Finally, a number of specialty flow guidance has been described as, for example, the Coanda effect, relying on a microstructure for redirecting the flow. In the case of active mixers, external energy sources such as ultrasound, acoustic, bubble-induced vibrations, electrokinetic instabilities, periodic variation of flow rate, electrowetting-induced merging of droplets, piezoelectric vibrating membranes, magnetohydrodynamic action, small impellers, integrated microvalves/pumps are used. 4.4.1 Passive Micromixer 4.4.1.1 Single-Channel Micromixers

The simplest passive mixers are often used for characterizing mixing under welldefined conditions (typical geometries are shown in Figure 4.15). If a T-mixer is considered, three flow regimes are observed (Section 4.2): Stratified flow: a clear interface separates the two flows (lamination) and mixing occurs entirely by diffusion Vortex flow: the vortex creates secondary surface area Engulfment flow: the axial symmetry of the flow breaks up (see Section 4.2).

4.4

(a)

(b)

Types of Micromixers and Mixing Principles

(c)

Figure 4.15 Different types of MSR with various contacting geometries and cross sections [27]: (a) T-square, (b) Y-rectangular, (c) concentric. (Adapted with permission from Elsevier.)

The transition of regimes depends on Re and reactor geometry. In T-mixer, stratified flow is observed at low Re. Engulfment flow is observed at Re > about 100. Engler et al. [11] proposed a parameter (K) to estimate the flow regime in T-mixers. K = dh ∕λK

(4.28)

where �K is the Kolmogorov length denoting the smallest eddy in a fully developed turbulent flow and is defined as ( 3 )0.25 ν (4.29) λK = � The first transition from stratified to vortex flow was observed at about K = 15, whereas the transition to engulfment flow happened at K = 40. The corresponding Re for two transitions in a 600 × 300 × 300 μm T-mixer are 45 and 150, respectively, which corresponds roughly to the observations presented in Section 4.2. Another way of increasing the mixing efficiency for Re in the range of a few hundred is the design of curved channels [28]. Here the mixing quality is improved by creating secondary flow pattern for Dean number (De) > 140, where ( )0.5 d De = Re ′′h (4.30) R where R′′ is the mean curve radius of the microchannel (see Figure 4.16). The same effect is observed for zig-zag channels, where secondary flow pattern appears at Re in the range of a few hundred [29]. In Figure 4.16 the interface stretching factor, λ, is defined as the interface length at a certain position divided by the initial interface length [28]. As can be seen in Figure 4.16, no stretching occurs for De = 100, whereas for De = 200 the stretching increases exponentially with the number of curved elements in series. Another type of mixer where both fluids are mixed tangentially or radially (Figure 4.17) are cyclone type mixers that offer high mixing performances [30]. The fluids to be mixed enter the cyclone mixing chamber tangentially via two entry channels creating a swirl motion in the mixing chamber enhancing mixing. The combined fluids exit the cyclone mixing chamber through the exit channel. Mixers with feed channels arranged radially were intended to avoid the swirl motion of the fluids. A similar device was proposed by Arsani et al. by using a nonaligned T-mixer [31].

145

4 Micromixing Devices

16000 14000 Interface stretching factor, λ

146

Meander mixer

De = 200

12000 10000 8000 6000 4000 2000

De = 100

0 1

2

3 4 Number of loops

5

6

Figure 4.16 Interface stretching factor in meander mixer as function of the number of loops and Dean number [28]. (Adapted with permission from Wiley.)

(a)

(b)

Figure 4.17 Sketch of tangential (a) and radial (b) feed positions for mixers with two and four entry channels [30]. (Adapted with permission from Elsevier.))

4.4.1.2 Multilamination Mixers

As we have seen before, stratified flow (two laminations) can be achieved in singlechannel micromixers. The benefit of the geometrical focusing can be easily seen by looking at the characteristic mixing time, which in case of a straight rectangular channel with a height, H, comprising two fluid lamellae equals tmx = tD =

(H∕2)2 Dm

(4.31)

Thus a reduction of the channel height corresponding to a reduction of the hydraulic channel diameter leads to a decrease of the mixing time. By compressing the fluid lamellae to a few micrometers, the diffusion distances are reduced, corresponding to liquid mixing in the milliseconds range. Similarly, geometric compression, that is, reduction of the flow cross section, can lead to lamellae thinning. To replicate such a flow in a single device, that is, multilamination, structures with alternate feeds can be created by interdigital (see Figure 4.18a) or bifurcation structures. These lamellae of alternate concentrations decrease the striation

4.4

Types of Micromixers and Mixing Principles

500 μm

Flow distribution zone

Mixing channel for generation of multilamellae

(a) Slit-type interdigital micromixer

(b) Superfocus micromixer

(c) Flow pattern in a cyclone micromixer

(d) Star lamination micromixer

Figure 4.18 Different types of multilamination micromixers. (a) liquid flows 10–1000 ml h−1 , (b) 138 microchannels flow: 350 l h−1 at 3.5 bar, (c) 20 ml h−1 , and (d)

snapshot of mixer for throughput 1). If heat transfer is referred to the exchange area of the reactor channel, the specific heat exchange surface a is given by a = Aex ∕VR . For channels surrounded by the cooling medium, that is, in a shell and tube heat exchanger reactor, the specific surface area corresponds to a = Aex ∕VR = 4∕dh . Assuming constant cross section, Equation 5.31 can be written as (−R1 ) ⋅ �PFR ⋅U ⋅ �PFR ⋅ 4 dT = ΔTad − (T − Tc ) dZ c1,0 �0 ⋅ cp d h with Z = V ∕VR ; �PFR = VR ∕V̇ 0 ;

(5.34)

ΔTad the adiabatic temperature rise: ΔTad =

c1,0 ⋅ (−ΔHr ) �0 ⋅ cp

(5.35)

with c1,0 as the inlet concentration of the limiting reactant. From Equation 5.32 it follows: −R1 ⋅ �R dX = dZ c1,0

(5.36)

To determine the axial temperature profile in the tubular reactor, Equations 5.34 and 5.36 must be solved simultaneously by numerical integration. The design of microchannel reactors is discussed in detail in Example 5.3 and 5.4

Example 5.3: Overall heat evacuation for a type B reaction. The microchannel reactor described in Example 5.1 is previewed for a type B reaction with a characteristic reaction time of tr = 5 s at 323 K. The reaction is irreversible and of first order. The reaction enthalpy

199

200

5 Heat Management by Microdevices

is ΔHr = −150 000 kJ kmol−1 and the activation energy is found to be Ea = 80 kJ kmol−1 . The reaction will be carried out at 323 K in toluene as solvent and a residence time of �PFR = 15 s. The inlet concentration is fixed to c1,0 = 0.7 kmol m−3 . The temperature of the cooling fluid is 10 K below the reaction temperature at the reactor inlet to evacuate the reaction heat. Estimate the reactor performance and the axial temperature profile. Solution:

The maximum total heat produced (X = 1) is given by q̇ r =

c1,0 ⋅ (−ΔHr ) �PFR

=

0.7 ⋅ 150 000 = 7 ⋅ 103 kW m−3 15

With a volumetric global heat transfer coefficient of Uv = 1.06 ⋅ 103 kW m−3 K−1 and a temperature difference of ΔT = 10 K, the maximum total heat generated can be evacuated. q̇ ex = Uv ⋅ ΔT = 10.6 ⋅ 103 kW m−3 The adiabatic temperature rise can be calculated as: ΔTad =

c1,0 ⋅ (−ΔHr ) �0 ⋅ cp

=

0.7 ⋅ 150 000 = 70.4 K 867 ⋅ 1.72

In Figure 5.17 the reactor temperature and the conversion as function of the distance from the reactor entrance is shown. In spite of the low cooling temperature, the temperature increases within the first 20% of the reactor length. The maximum temperature is roughly 20 K higher than the inlet temperature. The steep temperature increase is accompanied with a drastic increase of the conversion from X = 0 to X = 0.8. The diminished reactant concentration leads to low transformation rates and low heat production, and the reactor temperature decreases and attains nearly the temperature of the cooling fluid at the reactor outlet. It is important to note that measuring the outlet temperature gives no indication of the reaction control within the reactor. The conversion finally reaches 0.973 at the reactor outlet. The specific reactor performance reaches Lp, v = 45.41 mol m−3 s−1 . The influence of the inlet temperature on the axial temperature profile in the microchannel reactor is illustrated in Figure 5.18. All other reaction conditions are kept constant. The initial temperature difference between the cooling fluid and the reaction mixture at the inlet is in all cases identical (T 0 –Tc = 10 K). At the reactor outlet the temperature reaches nearly the cooling temperature for all inlet conditions. But, the maximum temperature increases drastically with an increase of T 0 . The important increase of the maximum temperature because of a small change of the inlet conditions is called parametric sensitivity and can be observed for fast and exothermic reactions in tubular reactor. In the domain of high parametric sensitivity the reactor is difficult to control and important temperature excursions cannot be avoided. High local temperatures may lead to important

5.3 Temperature Control in Chemical Microstructured Reactors

1.0

345

0.8

335 0.6

330 325

0.4

Conversion, X

Reactor temperature, T (K)

340

320 0.2 315 310 0.0

Cooling temperature 0.2

0.4

0.6

0.8

1.0

0.0

Reactor length, Z = z/Lt

Temperature difference, (T – Tc)

Figure 5.17 Axial temperature and conversion profile (Example 5.3).

40

T0 = 325 K Initial temperature difference: T0 – Tc = 10K

30 T0 = 323 K 20

10

0 0.0

T0 = 321 K

0.2

0.4

0.6

0.8

1.0

Reactor length, Z = z/Lt Figure 5.18 Influence of the inlet temperature on the axial temperature profile. (Data taken from Example 5.3.)

losses of product selectivity and quality, or may even damage the used catalyst or reactor. Therefore, high parametric sensitivity of chemical reactors must be avoided. 5.3.2 Parametric Sensitivity

For given reaction kinetics the thermal behavior of tubular reactors depends on three different parameters:

201

202

5 Heat Management by Microdevices

The Arrhenius number: �=

E R ⋅ Tc

(5.37)

The heat production potential: S′ =

ΔTad (−ΔHr ) ⋅ c0 E ⋅� = ⋅ � = ΔTad � ⋅ cp ⋅ Tc Tc R ⋅ Tc2

(5.38)

and the ratio of the characteristic reaction time to the cooling time: N′ = tr =

tr tc

� ⋅ cp 1 ;t = n−1 c Uv k(Tc ) ⋅ c1,0

(5.39)

The characteristic reaction time is calculated with the temperature of the cooling medium (Tc ) respectively the inlet temperature (T0 ). In the following discussion the inlet and the cooling temperature are supposed to be identical (T 0 = Tc ). To facilitate the discussion on the influence of the above-defined parameters (Equations 5.37–5.39) on the reactor behavior and the parametric sensitivity, Equations 5.34 and 5.36 are given in a dimensionless form. According to the studies of Barkelew [25] the mean residence time is referred to the characteristic reaction time and the temperature is given in the form of a relative temperature difference normalized with the Arrhenius number (Equation 5.41). ) ( � −E n−1 �R′ = = � ⋅ k0 exp ⋅ c1,0 (5.40) tr R ⋅ Tc The dimensionless residence time, �R′ , can be interpreted as a first Damköhler number, defined with the characteristic reaction time calculated with the temperature of the cooling medium. ′

ΔT =

(T − Tc ) (T − Tc ) E ⋅� = Tc Tc2 R

(5.41)

With these definitions the conversion and the temperature as function of the dimensionless channel length can be calculated. ) ( ΔT ′ dX ⋅ (1 − X)n ≅ exp(ΔT ′ ) ⋅ (1 − X)n = �R′ ⋅ exp (5.42) dZ 1 + ΔT ′ ∕� dΔT ′ = �R′ ⋅ S′ ⋅ exp dZ

(

ΔT ′ 1 + ΔT ′ ∕�

) ⋅ (1 − X)n − �R′ ⋅ N ′ ⋅ ΔT ′

≅ �R′ ⋅ S′ ⋅ exp(ΔT ′ ) ⋅ (1 − X)n − �R′ ⋅ N ′ ⋅ ΔT ′

(5.43)

where N ′ is the ratio of characteristic reaction time (tr ) to cooling time (tc ) and S′ is the heat production potential as defined in Equations 5.39 and 5.38.

5.3 Temperature Control in Chemical Microstructured Reactors

203

The Arrhenius numbers are in general high, � > 20. Therefore, the term ΔT ′ /� is small and can mostly be neglected and Equations 5.42 and 5.43 are often given in a simplified form: dX ≅ exp(ΔT ′ ) ⋅ (1 − X)n dZ

(5.44)

dΔT ′ ≅ �R′ ⋅ S′ ⋅ exp(ΔT ′ ) ⋅ (1 − X)n − �R′ ⋅ N ′ ⋅ ΔT ′ dZ

(5.45)

The temperature change along the dimensionless reactor length, Z, is a result of the difference between the heat produced and removed (Figure 5.19a). In the first part of the channel, the heat produced by reaction exceeds the heat evacuated through the channel wall. As a consequence, the reaction mass is heated up to a maximum. At temperature maximum, commonly called “hot spot,” the produced heat by reaction is entirely removed by the cooling system �R′ ⋅ S′ ⋅ exp(ΔT ′ ) ⋅ (1 − X)n = �R′ ⋅ N ′ ⋅ ΔT ′ . As the main amount of the reactant is transformed within the region of the temperature peak (Figure 5.19b) the concentration drops down and the reaction rate diminishes. In the second part of the channel, heat production is low compared to the heat transfer performance and the reaction mass is cooled down close to the cooling temperature, Tc . It is important to note that decreasing the cooling intensity from N ′ = 45 to 30 and further to N ′ = 25 has a moderate influence on the hot spot formation. The situation changes drastically for values of N ′ < 25. In this range, even small variations of the cooling intensity leads to tremendous variations of the maximum temperature. The reactor is operated in a region of high parametric sensitivity. The reactor behavior of a second order reaction shown in Figure 5.20 corresponds to the behavior of a first order reaction (Figure 5.19). But, the region of 1.0

8

0.6

N′ = 22 N′ = 22.5 N′ = 23 N′ = 25 N′= 30 N′ = 45

4

X

ΔT′

0.8

S′ = 15, n = 1 γ = Ea/(R Tc) = 40

6

0.4

2

0 0.0 (a)

0.2

0.2

0.4

0.6

0.8

Reactor length, Z = z/Lt

0.0 0.0

1.0 (b)

N′ = 22 N′ = 22.5 N′ = 23 N′ = 25 N′ = 30 N′ = 45

S′ = 15, n = 1 γ = Ea/(R Tc) = 40 0.2

0.4

0.6

0.8

Reactor length, Z = z/Lt

Figure 5.19 Axial temperature (a) and conversion profile (b) in a cooled microchannel for different cooling intensity, N′ (n = 1, T 0 = Tc ).

1.0

204

5 Heat Management by Microdevices 1.0

8

0.6

N′ = 15 N′ = 16 N′ = 17 N′ = 18 N′ = 20 N′ = 25 N′ = 30

4

2

X

ΔT ′

0.8

S′ = 15, n = 2 γ = Ea/(R Tc) = 40

6

0.4

0.2 S′ = 15, n = 2 γ = Ea/(R Tc) = 40 0.0 0.0

0 0.0 (a)

0.2

0.4

0.6

0.8

1.0

Reactor length, Z = z/Lt

0.4

0.2

0.6

N′ = 15 N′ = 16 N′ = 17 N′ = 18 N′ = 20 N′ = 25 N′ = 30 0.8

1.0

Reactor length, Z = z/Lt

(b)

Figure 5.20 Axial temperature (a) and conversion profile (b) in a cooled microchannel for different cooling intensity, N′ (n = 2, T 0 = Tc ).

high parametric sensitivity is shifted to lower values of N ′ for the same heat production potential (S′ = 15). The dimensionless reactor length, Z, can be eliminated by dividing Equation 5.45 by Equation 5.44 to get the reaction temperature as function of the conversion. dΔT ′ = S′ − N ′ ⋅ ΔT ′ ⋅ exp dX

(

−ΔT ′ 1 + ΔT ′ ∕�

)

′ 1 ′ ′ ′ exp(−ΔT ) ≅ S − N ⋅ ΔT (1 − X)n (1 − X)n (5.46)

′

For the temperature maximum follows: dΔT = 0. This allows estimating the condX version at the hot spot neglecting ΔT ′ ∕� compared to 1 in the denominator of the exponential function. ′

exp(−ΔTmax ) dΔT ′ ′ =0 = S′ − N ′ ⋅ ΔTmax dX (1 − Xmax )n N′ ′ ′ (1 − Xmax )n = ′ ⋅ ΔTmax exp(−ΔTmax ) S ]1 [ ′ ) n ( N ′ ′ Xmax = 1 − ⋅ ΔTmax exp −ΔTmax S′

(5.47)

The ratio between N ′ and S′ is an important parameter for the general discussion of the stability and parametric sensitivity of chemical reactors. N ′ /S′ corresponds to the reciprocal Semenov number [26] Uv Uv RTc2 RTc2 1 1 N′ = = = n−1 ΔT E S′ Se � ⋅ cp k(Tc )c1,o (−ΔHr )k(Tc )cn1,o Ea ad a

(5.48)

5.3 Temperature Control in Chemical Microstructured Reactors

205

5 1.4 1.2

0.6

3 ΔT′

0.8

N/S′ = 2.0

4

S′ = 100 S′ = 50 S′ = 25 S′ = 10 S′ = 5 ΔT′max = f(Xmax)

1.0 ΔT′

S′ = 27.34

N′/S′ = exp(1)

Maximum temperature

2 S′ = 27

0.4

1

0.2 0.0 (a)

S′ = 27.33

S′ = 21 S′ = 10

0 0.0

0.2

0.4

0.6 X

0.8

1.0

0.0 (b)

0.2

0.4

0.6

0.8

X

Figure 5.21 Temperature profile and maximum temperature in tubular reactors. First order reaction (Adapted from Ref. [28], Figure 7.33. Copyright © 2013, Wiley-VCH GmbH & Co. KGaA.)

N ′ /S′ can be interpreted as the ratio between the time to maximum rate, t mr, under adiabatic conditions supposing zero order and the characteristic cooling time, tc [27]. In Figure 5.21a the temperature as function of the conversion and the predicted maximum temperature (Equation 5.47) are shown for the parameter N ′ /S′ = exp(1). It follows from this figure that the maximum dimensionless temperature difference cannot exceed the value of ΔT ′ = 1, as long as the temperature difference at the reactor inlet is smaller than 1 (ΔT ′ < 1). With the definition of the dimensionless temperature difference (Equation 5.41), we can estimate the maximum difference between the cooling temperature and the temperature of the reaction mass: T − Tc = Tc2 ⋅ R∕E. It is evident from Figure 5.21a that no parametric sensitive region will be reached. The reactor is stable for all values of the heat production potential, S′ , and reaction orders n ≥ 0 as long as N ′ /S′ ≥ exp(1). The situation changes for N ′ /S′ < exp(1). In Figure 5.21b the temperature profile for N ′ /S′ = 2 and different values for the heat production potential, S′ , is shown. Starting with low values, S′ is increased by increasing the inlet reactant concentration and in consequence the adiabatic temperature rise or by increasing the cooling temperature. For the supposed kinetics of an irreversible first order reaction, the maximum temperature in the reactor first increases slowly with S′ , but reaches a region of very high parametric sensitivity for S′ > 27. A similar behavior can be observed for all reactions with positive reaction order (n > 0). Different criteria are proposed for estimating the region of high parametric sensitivity based on accessible criteria [27]. Most of the corresponding publications are restricted to first order reactions. The published results are slightly different

1.0

5 Heat Management by Microdevices

2.5 Stable reactor behavior

2.0

N′/S′

206

1.5

n = 0.5 High parametric sensitivity “reactor runaway”

1.0

0.5

n=1

T 0 = Tc

n=2 10

100 S′

Figure 5.22 Parametric sensitivity of plug flow reactor for reactions of n = 0.5, 1.0, and 2.0. Area under the corresponding curves indicate high parametric sensitivity. (Adapted from Ref. [28], Figure 7.34. Copyright © 2013, Wiley-VCH GmbH & Co. KGaA.)

and are reproduced as a hatched band in Figure 5.22 separating the regions of high and low parametric sensitivity. A very simple criterion for estimating the region of stable reactor behavior is based on a maximum temperature peak. If the maximum temperature difference is fixed to ΔT ′ = 1.2, the following correlation is obtained for separating the region of stable operation from that of high parametric sensitivity (see Figure 5.22): N′ B B ≥ exp(1) − √ = 2.72 − √ S′ S′ S′ The coefficient B depends on the reaction order:

(5.49)

n=0∶B=0 n = 0.5 ∶ B = 2.60 n = 1 ∶ B = 3.37 m = 2 ∶ B = 4.57 The parameter N ′ contains the overall heat transfer coefficient, while the heat production potential is dependent on the adiabatic temperature rise, the (apparent) activation energy, and the cooling temperature. With Equation 5.49 a minimum value for N ′ can be estimated to assure stable reactor operation. The application of the relations presented above for the design of MSR is illustrated in Example 5.4 and 5.5.

5.3 Temperature Control in Chemical Microstructured Reactors

Example 5.4: Reaction conditions for the synthesis of an ionic liquid in a slit-like MSR. The ionic liquid, 1-ethyle-3-methyle imidazolium ethyl sulfate, is synthesized by alkylation of methyllimidazole as follows [29] O N +

N

A1

O

S

O O

N+ N

– O

S

O

O

A2

A3

O

The reaction is highly exothermic and is carried out without any solvents. The reaction kinetics is fast with characteristic reaction times between several minutes to several seconds depending on the reaction temperature. The heat management is of major concern to attain high quality product and to avoid thermal runaway [30]. Preliminary studies show that reaction temperatures higher than 100 ∘ C lead to coloration of the reaction mixture because of degradation reactions. Therefore, the hot spot temperature must be kept lower. The microstructured slit reactor as presented in Example 5.1 is used for the solvent free synthesis of the ionic liquid. Estimate (1) the adiabatic temperature rise, (2) the maximum cooling/inlet temperature with T c = T 0 , (3) the maximum temperature in the reactor, and (4) the temperature and conversion profiles along the reactor length. The relevant reaction and reactor data are summarized in Table 5.4. Suppose established temperature and laminar velocity profile. Table 5.4 Physical properties and reaction conditions. Reaction rate constant at T = 313 K Activation energy Reaction enthalpy Density Viscosity Specific heat capacity Heat conductivity (293 K ≤ T ≤ 353 K) Inlet concentrations Overall volumetric heat transfer coefficient Required conversion

k 313 = 1.5⋅10−3 m3 kmol−1 s−1 [29] Ea = 86 000 J mol−1 [29] ΔHr = −102 000 kJ kmol−1 [29] � = 1254–0.598⋅(T/K-273) (kg m−3 ) [31] � = 10.81 ⋅ 10−4 ⋅exp(3.38 ⋅ 103 /T) (mPa s) [31] cp = (5.827 ⋅ 102 –6.161 ⋅ 104 /T)/236.3 (kJ kg−1 K−1 ) [32] �f = 0.18 10−3 kW m−1 K−1 [33] c1,0 = c2,0 = 4.7 kmol m−3 Uv = 1.066 ⋅ 103 kW m−2 K−1 (Example 5.1) X ≥ 0.75

Solution:

1) The adiabatic temperature rise is calculated using Equation 5.35 as ΔTad =

c1,0 ⋅ (−ΔHr ) �0 ⋅ cp

=

4.7 ⋅ 102, 000 = 239 K (T = 313 K) 1230 ⋅ 1.63

207

5 Heat Management by Microdevices

2) Stable reactor operation can be supposed, if the following conditions are fulfilled (see Equation 5.49): √ (5.50) N ′ = 2.72 ⋅ S′ − B S′ For second order reactions the coefficient B is found to be B = 4.57. Both N ′ and S′ depend on temperature. √ Uv Ea E 1 ⋅ = 2.72 ΔTad 2 − 4.57 ΔTad a2 ( )) ( E �0 cp k exp − a 1 − 1 RTc RTc c1,0 313 R Tc 313 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ RHS N′

(5.51) The maximum cooling temperature for stable operation can easily be found by plotting N ′ and the right-hand side (RHS) of Equation 5.51 as function of Tc . The result is shown in Figure 5.23. Obviously, a maximum temperature of Tc = T 0 = 318 K is allowed for stable operation. 3) The characteristic reaction time can easily be calculated with the Arrhenius law: 1 tr,319 = = 84 s ( ( )) Ea 1 1 k313 exp − R 318 − 313 ⋅ c1,0 1000

N′

100 N′, RHS

208

RHS 10

1 300

310

320 330 340 Temperature, Tc (K)

350

360

Figure 5.23 Graphical solution of Equation 5.51.

A minimum conversion of X = 0.75 is required. For a second order reaction the conversion in an ideal plug flow reactor depends on the first Damköhler number, DaI, as shown in Equation 5.52 (see Chapter 2):

5.3 Temperature Control in Chemical Microstructured Reactors

X=

� DaI ; with DaI = R = k ⋅ c1,0 ⋅ �R 1 + DaI tt

(5.52)

With a characteristic reaction time of tr = 84 s we need for the required conversion a residence time of � R = 252 s. The maximum temperature in the reactor can be estimated with the defini′ = tion of the dimensionless temperature (Equation 5.41), knowing that ΔTmax 1.2. With the kinetic data summarized in Table 5.4 we obtain: −T )E (T R ′ ΔTmax = max 2 c a ⇒ Tmax ≅ Tc + Tc2 ≅ 328 K R Ea Tc 4) The simulated axial temperature and conversion is calculated by numerical integration of Equations 5.34 and 5.36 with the data of Table 5.4. The results are reproduced in Figure 5.24. 330

1.0 0.8

326 0.6

324

0.4

322 320

0.2

318 0.0

Conversion, X

Temperature, T (K)

328

Cooling temperature

0.2

0.4

0.6

0.8

0.0 1.0

Reactor length, Z = z/Lt Figure 5.24 Axial temperature and conversion profile (Example 5.4).

The simulation results confirm the estimated maximum temperature allowing safe reactor operation. The conversion is slightly higher (X = 0.78) than predicted on the basis of an isothermal reactor operation at 318 K. This is because of the temperature peak near the reactor entrance. To complete the reaction a second reactor operating at higher temperature is needed. As the inlet concentration of the second reactor is relatively low (c1, 0 ≅ 1 kmol m−3 ) temperature control is facilitated.

Example 5.5: Design of a multichannel MSR for the high performance production of an ionic liquid. It is planned to use a multichannel microreactor for the synthesis of the ionic liquid, 1-ethyle-3-methyle imidazolium ethyl sulfate [C2 mim] [C2 SO4 ] (see

209

210

5 Heat Management by Microdevices

Example 5.4) at high temperature to intensify the production process. For this purpose a micro heat exchanger with square channels as illustrated in Figure 5.25 should be used. The reactor channels are alternatively assembled with cooling channels of the same diameter. 50 stainless steel foils with 50 channels each are piled in between 51 foils with 50 cooling channels. In total 1250 reactor channels and 1300 cooling channels are assembled. Water is used as cooling medium. The walls of the channels have a thickness of e = 100 μm.

Tc

Cooling channels

T

Reactor channels

e

Tc

W

H

Figure 5.25 Micro heat exchanger for the synthesis of [C2 mim] [C2 SO4 ]. Total number of channels: 2550, reactor length Lt = 0.2 m.

To avoid any risk of reactant or product degradation, the maximum hot spot temperature is fixed to T max = 95 ∘ C. The following operational parameters must be estimated for the final design of the reactor: 1) The cooling temperature Tc = T 0 . 2) The mean residence time to get the required conversion of X ≥ 0.75, supposing plug flow behavior in the channels. 3) The overall volumetric heat transfer coefficient, Uv . 4) The hydraulic channel diameter, which corresponds to the height of the channel (dh = H = W ). 5) The total throughput (kg h−1 ) and the pressure drop in the channels. Solution:

1) The cooling temperature can be estimated from the definition of the T −T E ′ = maxT 2 c Ra = 1.2. With dimensionless maximum temperature: ΔTmax c T max = 368 K a cooling temperature of 353.5 K is calculated. For the safe reactor design we will take T c = T 0 = 353 K. The hot spot temperature is estimated to reach T max = 367.5 K. 2) To estimate the mean residence time in a plug flow reactor we suppose a � � X ; tR = 0.75 = 3. constant reactor temperature. This leads to DaI = tR = 1−X 0.25 r

r

5.3 Temperature Control in Chemical Microstructured Reactors

The characteristic reaction time is defined with the inlet condition. 1 1 ⇒ tr,353 = = 3.35 s tr = ( )) ( Ea 1 k(Tc ) ⋅ c1,0 ⋅ 4.7 k exp − − 1 313

R

353

313

�R = DaI ⋅ tr = 3 ⋅ 3.35 s = 10 s 3) The estimation of the overall volumetric heat transfer coefficient is based on the relation in Equation 5.49: √ � ⋅ cp E t ′ Nmin = r = 2.72 ⋅ S′ − 4.57 ⋅ S′ ; S′ = ΔTad a 2 ; tc = tc Uv R ⋅ Tc c1,0 (−ΔHr )

4.7 ⋅ 102, 000 = = 229 K �353 cp,353 1210 ⋅ 1.73 E 86, 000 = 19 S′ = ΔTad a 2 = 229 ⋅ 8.314 ⋅ (353)2 R ⋅ Tc

ΔTad =

′ Nmin =

Uv =

√ tr t = 2.72 ⋅ 19 − 4.57 ⋅ 19 = 31.8 ⇒ tc = ′r = 0.105 s tc Nmin

� ⋅ cp tc

=

1210 ⋅ 1.73 = 1.99104 ≅ 2 ⋅ 104 kW m−3 K−1 0.105

4) To determine the hydraulic diameter of the channel, we will use the relationship given in Equation 5.33 Nu ⋅ �f dh 1 1 = + Rth = + Rth ⇒ U = CRth ⋅ hr = CRth U hr Nu ⋅ �f dh The resistance to heat transfer in the channel walls is very low because of the small wall thickness of 100 μm and can be neglected in the further calculations: Rth ≅ 1∕hex = dh ∕(Nu ⋅ �f ,ex ). We suppose established temperature and laminar velocity profile in the reactor and cooling channels. Therefore, the same asymptotic Nusselt number can be used. From Figure 5.4 we find Nu∞ = 3.7 for square channels. ( )−1 dh dh 1 Nu 1 1 1 = + Rth = + ; U= + U hr Nu ⋅ �f Nu ⋅ �f ,c dh �f �f ,c U = CRth ⋅ hr ⇒ CRth =

1∕�f 1∕0.18 U = 0.79 = = hr 1∕�f + 1∕�f ,c 1∕0.18 + 1∕0.66

The heat conductivity of the cooling water at Tc = 353 K is �f , c = 0.66 W m−1 K−1 [8].

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5 Heat Management by Microdevices

The overall volumetric heat transfer coefficient is given by Uv = U ⋅ CRth

Nu⋅�f dh

⋅

Aex VR

with

lows Uv = CRth

Aex VR

Nu⋅�f dh

⋅

2 dh

=

for channels cooled on two sides, it fol√ Nu⋅� ⋅2 ⇒ dh = CRth U f

=

2 dh

2⋅H H2

Aex VR

=

v

√ dh =

0.79

3.7 ⋅ 0.18 ⋅ 10−3 ⋅ 2 = 229 ≅ 230 μm 2 ⋅ 10−4

5) The pressure drop in the reaction channels is given by the Hagen–Poisseuille law as L u⋅� Δp = 32 ⋅ � ⋅ 2 Lt ; � = 0.89 for square channels; u = t = 0.02 m s−1 �R dh The viscosity of ([C2 mim] [C2 SO4 ]) at 353 K is � = 15.57 10−3 Pa s (Table 5.4) 0.02 ⋅ 15.57 ⋅ 10−3 0.2 = 3.35 ⋅ 104 Pa = 0.335 bar (2.3 ⋅ 10−4 )2 The mass flow through the channel is ṁ channel = V̇ ⋅ � = u ⋅ Acs ⋅ � = 0.02 ⋅ (2.3 ⋅ 10−4 )2 ⋅ 1.21 ⋅ 103 = 1.28 ⋅ 10−6 kg s−1 Total mass flow: ṁ = 1250 ⋅ ṁ channel = 1.6 ⋅ 10−3 kg s−1 = 5.76 kg h−1 Δp = 32 ⋅ 0.89 ⋅

5.3.3 Multi-injection Microstructured Reactors

As discussed in chapter 5.3.2, theoretically the temperature of every fast and exothermic reaction can be controlled in microchannel reactors. The simplest solution is to adapt the reactor dimensions according to the reaction properties like adiabatic temperature rise, characteristic reaction time, and activation energy, thus working under stable conditions (see Equation 5.49). By reducing channel diameter, the channel volume for constant reactor length decreases proportionally to diameter square. For exothermic quasi-instantaneous reactions, channel sizes below 100 μm may be needed, which cannot be operated on an industrial scale because of high friction losses and high risk to channel clogging. For the described limits, one possible solution to combine high-throughput with good thermal management is a multi-injection microchannel reactor, where one reactant is injected along the reactor. This concept is used as one of the approaches for scale-up [34, 35]. Distributed feeding of one reactant in multiple locations reduces the local heat power released depending on the number of injection points. The concept corresponds to the semibatch operation of conventional reactor vessels.

5.3 Temperature Control in Chemical Microstructured Reactors

Reactant, A2 V2,0; c2,0

V1,0; c1,0

V2,0 V V V c2,0 2,0 c2,0 2,0 c2,0 2,0 c2,0 N N N N

j=1 Reactant, A1

j=2

j=3

Product

j=N

Figure 5.26 Scheme of a multi-injection reactor with N injection points.

5.3.3.1 Mass and Energy Balance in Multi-injection Microstructured Reactors

For a multi-injection MSR with a total of N injection points (Figure 5.26), the mass balance can be derived in a similar approach as for a plug flow reactor with only one inlet (Equation 5.27) [35]. The only difference is a sudden change in temperature and reaction mass at each injection point j, which can be described by using a Dirac pulse �(z) and the Heaviside function �(z). Reactant A2 contained in flow 2 is injected in deficit to reactant A1 into flow 1 before reaching the last point, where the stoichiometric balance is attained. It is assumed that the volume of the injected reactant is equal at each injection point and that it mixes instantaneously with the main stream. ( ) ṅ 2,0 ) dṅ i L ( = Acs ⋅ Ri + � z − t ⋅ j − 1 ⋅ dz N N

(5.53)

(−R1 ) ⋅ (−ΔHr ) dT = Acs ⋅ ( ) ̇ ) ( N dz ∑ V2,0 ) Lt ( ̇V1,0 + ⋅ � ⋅ cp ⋅ j−1 ⋅ � z− N N j=1 U ⋅ a ⋅ (T − Tc ) ) ̇ ) ∑ V2,0 ) Lt ( ̇V1,0 + ⋅ � ⋅ cP ⋅ j−1 ⋅ � z− N N j=1 ) ( V̇ 2,0 ) Lt ( (Tinj − T) ⋅ j−1 ⋅ +� z− N V̇ 1,0 ⋅ N − Acs ⋅ (

N

(

(5.54)

The third term of heat balance (Equation 5.54) is the heat added to the system because of the eventual temperature difference between the injected flow V̇ 2 and the main flow denoted as (T inj − T). The above equations can be solved using a simple ordinary differential equation solver for each interval between two injection points. In this case, the boundary conditions of the jth interval have to be adapted considering the reaction mass injected at point j and its temperature as well as the temperature and concentrations at the end of the interval j − 1. If instantaneous mixing and reactions are considered, the heat is produced at the injection point in the reactor. The heat evacuation time can be calculated by

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5 Heat Management by Microdevices

following heat balance equation: U dT U ⋅a = (Tc − T) = v (Tc − T) d� � cp � cp

(5.55)

Integration of Equation 5.55 allows to determine the residence time between two injection points (� j ), respectively, the distance (Lj ) to cool the reaction mass from T in, j to T out, j (Equation 5.56). ) ( Tin,j − Tc � cp �j = ln Uv,j Tout,j − Tc ( ) � cp uj Tin,j − Tc ln Lj = (5.56) Uv,j Tout,j − Tc where uj is the velocity at jth injection and U v, j the overall volumetric heat transfer coefficient in the jth section. The inlet temperature for each injection point differs and so does the outlet temperature. For the first injection, the inlet temperature is the sum of the cooling temperature and the adiabatic temperature rise at first point as given in the following: Tin,1 = To + ΔTad,1 = Tc + ΔTad,1 ;

for To = Tc

j=1

(5.57)

If we consider equal flow splitting for all injection points, the total volumetric flow rate and corresponding adiabatic temperature rise would be given as V̇ 1 = V̇ 1,0 + ΔTad,1 =

V̇ 2,0 N

(ṅ 2,0 ∕N)(−ΔHr ) (� V̇ 1 )⋅cp

j=1

(5.58)

where ṅ 2,0 is the inlet molar flow rate of A2 (ṅ 2,0 = V̇ 2,0 ⋅ c2,0 ). According to Equation 5.56, the residence time required to remove the heat completely (T out = Tc ) is infinity and, therefore, an adjustment needs to be made. If we assume that 90% of the heat is removed within one section and 10% will be carried to the next injection point, the corresponding outlet temperature can be calculated with Equation 5.59. Tout,1 = Tc + 0.1(Tin,1 − Tc )

(5.59)

Further, temperature at the second injection for constant heat capacity is V̇ 2,0 ∕N V̇ 1,0 + V̇ 2,0 ∕N Tout,1 + Tc + ΔTad,2 V̇ tot,2 V̇ tot,2 V̇ 2,0 ∕N V̇ 1 = Tout,1 + Tc + ΔTad,2 V̇ tot,2 V̇ tot,2

Tin,2 =

and

(5.60)

5.3 Temperature Control in Chemical Microstructured Reactors

V̇ tot,2 = V̇ 1,0 + 2

V̇ 2,0 N

; ΔTad,2 =

(ṅ 2,0 ∕N)(−ΔHr ) (� V̇ tot,2 ) ⋅ cp

(5.61)

For a multi-injection MSR with equal injection at all points, a generalized equation can be established to investigate the temperature at jth injection:

Tin,j =

V̇ j−1 V̇ j

Tout,j−1 +

(ṅ 2,0 ∕N)(−ΔHr ) V̇ 2,0 ∕N Tc + ΔTad,j ; ΔTad,j = (5.62) ̇Vj (� V̇ j ) ⋅ cp

V̇

with V̇ j = V̇ 1,0 + j ⋅ N2,0 . The axial temperature profile in multi-injection microchannels is presented in Example 5.6.

Example 5.6: Characteristics of a multi-injection microchannel reactor A second order quasi-instantaneous reaction (heat of reaction = −120 000 kJ kmol−1 ) is carried out in a microstructured plug flow reactor. To reduce the hot spot temperature, the reaction partner, A2, is evenly distributed over several injection points along the reactor length. The reactor conditions and properties are summarized in Table 5.5: Table 5.5 Physical properties and reaction condition. Reaction enthalpy Density Specific heat capacity Inlet concentrations Inlet flow rate

ΔH r = −120 000 kJ kmol−1 � = 1142 kg m−3 cp = 1.7 kJ kg−1 K−1 c1,0 = c2,0 = 1.56 kmol m−3 V̇ 1,0 = V̇ 2,0 = 0.9 ⋅ 10−7 m3 s−1

Considering three configurations of 4, 6, and 10 injections and assuming instantaneous mixing at each contacting junction, investigate the adiabatic temperature rise for each injection point. Solution:

The total adiabatic temperature rise (ΔT ad ) is given as ΔTad =

ṅ 2,0 (−ΔHr ) 1.4 × 10−7 × 120000 = = 48.08 K (5.63) ̇ ̇ (V1,0 + V2,0 ) ⋅ � cP 1142 × 0.18 × 10−6 × 1.7

The adiabatic temperature rise corresponding to total concentrations is 48.08 K. Let us consider an MSR with four injection points (N = 4); the flow rate and adiabatic temperature rise at each injection is given by Equation 5.62 (Table 5.6).

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5 Heat Management by Microdevices

Table 5.6 Temperature rise at different injection point in multi-injection reactor. Injection point

1 2 3 4 5 6 7 8 9 10

N=6 V̇ j (m3 s−1 )

1.05 × 10−7 1.20 × 10−7 1.35 × 10−7 1.50 × 10−7 1.65 × 10−7 1.80 × 10−7 — — — —

N = 10 �T ad,j (K)

V̇ j (m3 s−1 )

�T ad, j (K)

14.3 12.5 11.1 10 9.1 8.35 — — — —

9.90 × 10−8 1.08 × 10−7 1.17 × 10−7 1.26 × 10−7 1.35 × 10−7 1.44 × 10−7 1.53 × 10−7 1.62 × 10−7 1.71 × 10−7 1.80 × 10−7

9.1 8.35 7.70 7.15 6.68 6.26 5.89 5.56 5.27 5.01

For injection j = 1: −7

0.9 ⋅ 10 = 1.125 × 10−7 m3 s−1 V̇ 1 = 0.9 × 10−7 + 1 4 (1.4 × 10−7 ∕4) × 120000 = 19.23 K ΔTad,1 = 1142 ⋅ 1.125 ⋅ 10−7 ⋅ 1.7 Similarly, for injection point j = 2: V̇ 1 = 1.35 ⋅ 10−7 m3 s−1 ; ΔTad,2 = 16.02 K For injection j = 3: V̇ 1 = 1.57 ⋅ 10−7 m3 s−1 ; ΔTad,2 = 13.77 K For injection j = 4: V̇ 1 = 1.80 ⋅ 10−7 m3 s−1 ; ΔTad,2 = 12.01 K The temperature rise at each injection point decreases along the length of the reactor because of the increase in the flow rate.

In spite of the fact that in multi-injection microchannels the temperature profiles are not well established because of the injection along the length that disturbs each time the velocity and temperature profile increasing the heat transfer performance, we use a constant overall volumetric heat transfer coefficients for simplicity. Keeping the residence time in each section constant, the outlet temperatures in each section are identical (see Equation 5.56). But, because of the increasing volumetric flow, the lengths of the sections vary as shown in Figure 5.27. Practically, the mixing at the injection point is not instantaneous. The rates of fast reactions are influenced by mixing. The mixing time varies from milliseconds

5.3 Temperature Control in Chemical Microstructured Reactors

375

Temperature, T (K)

370

365

360

355

Cooling temperature

j=1 350

0.0

j=2 0.2

j=4

j=3 0.4

0.6

0.8

1.0

Dimensionless length, Z = z/Lt Figure 5.27 Typical temperature profile in a multi-injection microchannel (N = 4). Ninety percentage of the heat produced at each injection is removed within each section j.

Referred temperature difference, ΔTj /ΔTad

to a few seconds (Chapter 4); therefore, in practice a significant amount of time should be considered for mixing and then for heat evacuation. The influence of the transformation rate, respectively the characteristic reaction time on the temperature profile in multi-injection reactors, is shown in Figure 5.28 for a second order reaction. For nearly all instantaneous reactions, the

Adiabatic temperature rise Quasi instantaneous reaction, tr = 0.008 s

0.4

Fast reaction, tr = 0.67 s 0.3

0.2

0.1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Dimensionless length, Z = z /Lt

Figure 5.28 Temperature profile in a multi-injection reactor with N = 4 injection points. Influence of the effective characteristic reaction time. T 0 = T inj = Tc .

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5 Heat Management by Microdevices

maximum temperature at the first injection points reaches the values predicted with Equations 5.58 and 5.62. Because of the decreasing concentrations of reactant A1 and the dilution following the injected flow 2, the reaction rate diminishes and the predicted temperature rise at the fourth injection point is not attained. For a fast reaction with a characteristic reaction time of 0.67 s the maximum temperature is considerably lower than the adiabatic temperature. Because of the high overall volumetric heat transfer coefficient, efficient heat evacuation in the mixing zone reduces the hot spot temperature. 5.3.3.2 Reduction of Hot Spot in Multi-injection Reactors Number of Injection Points The hot spot within a multi-injection reactor is con-

trolled mainly by the amount of injection points (N). For a first approximation of the temperature rise at each injection point, a simplified system can be considered [34]. For the case of instantaneous mixing and reaction with an equally distributed flow among the injection points (V̇ 2,1 = V̇ 2,2 = · · · = V̇ 2,N = V̇ 2,0 ∕N), the temperature rises quasi-adiabatically at each injection point. In order to avoid a high temperature rise, the heat produced at each injection point j is removed before reaching the next injection point j + 1. For such a system, Equation 5.62 can be rewritten as ΔTad,j =

(V̇ 2,0 ∕N) c2,0 ⋅ |−ΔHr | c2,0 ⋅ |ΔH| 1 = ( (5.64) )⋅ ̇ 1,0 V ̇ ̇ � ⋅ cp � ⋅ cp (V1,0 + j(V2,0 ∕N)) N ⋅ V̇ + j 2,0

The adiabatic temperature rise for a single-injection reactor (ΔT ad ) is defined as ΔTad =

V̇ 2,0 ⋅ c2,0 ⋅ (−ΔHr ) (V̇ 1,0 + V̇ 2,0 )� ⋅ cp

(5.65)

The temperature rise at each injection point referred to the adiabatic temperature rise obtained in reactors with only one inlet (N = 1) is expressed in Equation 5.66. ) (̇ V1,0 +1 ΔTad,j V̇ 2,0 (5.66) = ( ) V̇ ΔTad N ⋅ V̇ 1,0 + j 2,0

It is obvious that the highest temperature rise occurs at the first injection point as shown in Figure 5.29. A further reduction in hot spot can be obtained by maximizing the flow ratio V̇ 2,0 ∕V̇ 1,0 . The influence of the number of injection point and the ratio V̇ 2,0 ∕V̇ 1,0 is shown in Figure 5.29. To apply the simple relation (Equation 5.66), one has to choose carefully the distances between consecutive injection points to avoid the accumulation of heat in the reactor (see Equation 5.55). Rewriting Equation 5.66, the maximum number of injection points required to avoid a temperature rise higher than �T ad, N, j = 1 can be estimated as a function of F = V̇ 2,0 ∕V̇ 1,0 and the adiabatic temperature rise of the reaction �T ad, N = 1 in the

Temperature reduction, ΔTad, N, j =1 /ΔTad

5.3 Temperature Control in Chemical Microstructured Reactors

1.0

0.8

0.6 V1,0 0.4

V2,0

= 0.5

V1,0

0.2 V1,0 V2,0

0.0 1

2

3

V2,0

= 10

4 5 6 7 8 Number of injection points, N

9

=1

10

Figure 5.29 Reduction of temperature rise at the first injection point (j = 1) as a function of the total amount of injection points N and inlet flow ratios F = V̇ 2,0 ∕V̇ 1,0 .

case of a single injection point reactor (N = j = 1): N = (1 + F)

ΔTad,N=1 ΔTad,N,j=1

− F; with F =

V̇ 2,0 V̇ 1,0

(5.67)

Again, one has to choose carefully the characteristic cooling time to avoid the accumulation of heat in the reactor. Unequal Flow Partition In the previous sections, the flow through the injection channels was assumed to be equal. It was shown that this type of design results in a temperature profile as given in Figure 5.27, which leads to a maximum temperature at the first injection point. In order to further reduce this maximum temperature in a multi-injection reactor, one can design a microchannel reactor to obtain N equally high hot spots by increasing the injected volume along the length. The optimal flow distribution can be calculated using the model above. For a multi-injection reactor with N injection points, the adiabatic temperature rise at each injection point has to be equal as (1)

(2)

(3)

(N−1)

ΔTad,N,1 = ΔTad,N,2 = ΔTad,N,3 = … = ΔTad,N,N

(5.68)

It leads to a set of N − 1 equations for N unknown normalized injection flow rates Fj = V̇ 2,j ∕V̇ 1,0 . Solving Equation 5.68 leads to the following expression for the normalized injection flow rates as a function of F 1 : Fj = F1 ⋅ (F1 + 1)j−1

(5.69)

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5 Heat Management by Microdevices

Thus the growth factor for the volumetric flow rate between two injection points j and j + 1 is (F1 + 1). Assuming a constant density along the length: F=

N ∑

(5.70)

Fj

j=1

From above equation, Fj can be determined for a known value of F. Using above example (F = 1 and N = 4), the injection flow rate at j = 1 is F 1 = 0.182 resulting in a growth factor of 1.182. The adiabatic temperature rise at any injection point can be calculated according to Equation 5.64: ΔTad,N,j = ΔTad,N,1 =

V̇ 2,1

c2,0 ⋅ (−ΔHr )

(V̇ 1,0 + V̇ 2,1 )

ρ ⋅cp

=

c2,0 ⋅ (−ΔHr ) F1 ⋅ (5.71) 1 + F1 ρ ⋅c p

The adiabatic temperature rise for a single-injection reactor (ΔT ad ) is calculated with Equation 5.65. Thus, the temperature rise at each injection point referred to the adiabatic temperature rise obtained in reactors with only one inlet (N = 1) is expressed in Equation 5.72 ΔTad,N,j ΔTad

=

ΔTad,N,1 ΔTad

=

V̇ 2,1 V̇ 1,0 + V̇ 2,1

V̇ 1,0 + V̇ 2,1 F1 1 + F = 1 + F1 F V̇ 2,0

(5.72)

Compared to an equally distributed multi-injection reactor with N = 4 a 20% reduced temperature rise at the first injection point is obtained. Especially in cases with high F, this kind of channel design can be beneficial (see Example 5.7).

Example 5.7: Multi-injection microtubular reactor with identical temperature rises. The reaction described in Example 5.6 can be carried out in a microtubular reactor with N = 2 injections. Identical adiabatic temperature rises are required (ΔTad,2,1 = ΔTad,2,2 ). Determine the injecting flows V̇ 2,1 , V̇ 2,2 and the resulting adiabatic temperature rise. Estimate the reactor length to evacuate 90% of the produced heat (Table 5.7).

Table 5.7 Physical properties and reaction condition. Reaction enthalpy Density Specific heat capacity Inlet concentrations Inlet flow rate Tubular reactor Volumetric heat transfer coefficient

ΔHr = −120 000 kJ kmol−1 � = 1142 kg m−3 cp = 1.7 kJ kg−1 K−1 c2,0 = 2.33 kmol m−3 ; c1,0 = 5 kmol m−3 V̇ 1,0 = 1.2 ⋅ 10−7 m3 s−1 ; V̇ 2,0 = 0.6 ⋅ 10−7 m3 s−1 dt = 1 mm Uv = 1464 kW m−3 K−1

5.4

Case Studies

Solution:

The ratio between inlet flow 2 to 1 is F = V̇ 2,0 ∕V̇ 1,0 = 0.6∕1.2 = 0.5. From Equations 5.70 and 5.69 follows: F = F1 + F2 = F1 + F1 (F1 + 1) = 0.5; ⇒ F1 = 0.225 F2 = F1 (F1 + 1) = 0.276 ΔTad,2,1 = ΔTad,2,2 =

F1 1 + F 0.225 1.5 ΔTad ; ⋅ 48 = 26.5 K 1 + F1 F 1.225 0.5

5.4 Case Studies

Three examples of new designs of single-injection MSR and one example of multi-injection reactor applied for fast and exothermic liquid phase reactions are presented. Finally, it is demonstrated that the process intensification has been achieved using these microreactors. 5.4.1 Synthesis of 1,3-Dimethylimidazolium-Triflate

This is a quasi-instantaneous second order reaction (Figure 5.30) [36]. The important characteristic of this reaction is that at temperatures above 100 ∘ C, product coloration is observed indicating its decomposition and hence has to be avoided. A cooling system was integrated in the MSR and toluene was used as a coolant. Because of its low boiling point, toluene evaporates creating an additional effect on channel wall (two-phase cooling). The evaporated coolant is condensed in a 120 W-heat pipe system, which is normally used for cooling the CPU unit of computer. Because the reaction is of the second order, the highest heat production occurs in the mixer located at the entrance of the residence time loop. Temperature is monitored by two thermocouples placed on the outer surface of the static mixer and the reactor outlet, respectively. The maximum tolerable temperature O H3C

N

N

+

F3C

S

–

O

O

CH3

+

H3C

N

CF3SO3 N

CH3

(1)

(2)

(3)

82.11 g mol–1

164.1 g mol–1

246.21 g mol–1

Figure 5.30 Reaction scheme for the synthesis of the ionic liquid 1,3-Dimethylimidazoliumtriflate [36].

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5 Heat Management by Microdevices

is given by reactant 2 (see Figure 5.30) whose boiling point lies at 100 ∘ C. When MSR is working just under ambient cooling (natural convection/heat conduction), the maximal attainable flow rate lies at about 0.6 ml min−1 . The total flow rate was increased up to 9.7 ml min−1 by applying the two-phase cooling and thus confirming significant process intensification. 5.4.2 Nitration of Dialkyl-Substituted Thioureas

The target product is symmetrically substituted N,N ′ -dialkyl-N,N ′ -dinitro-urea, which is conventionally obtained at yields of about 10% (Figure 5.31) [37]. The reaction was carried out in silicon MSR with channels of 250 μm width. In order to increase the throughput, after the mixing of both reactants, the initial flow was distributed to nine identical channels cooled from both sides. The temperature gradient inside the channels was monitored by the means of infrared thermometry (see Figure 5.32). Because of the small channel size and O

S R

N H

C N H

R

Nitrating agent HNO3 HNO3 / H2SO4

R

N NO2

R

C NH2

R = CH4, C2H6

NO2

Figure 5.31 Nitration of dialkyl substituted thio-urea.

34.0 °C

30

27.0 °C

34.0 °C

34.0 °C

30

30

27.0 °C

27.0 °C

Figure 5.32 Infrared thermometry of the microreactor used to carry out nitration of dialkyl-substituted thioureas [37]. (Adapted with permission from Elsevier.)

5.4

Case Studies

the relatively high thermal conductivity of silicon (� = 148 W mK−1 ), temperatures inside the channel were almost the same (close to the isothermal operation). The temperature rise at the hot spot was about 5 ∘ C and it was mostly located in the section for distributing the liquids to the channels and in the collecting part. This example demonstrates that by changing the mixers from macro to micro scale, the mixing properties as well as the temperature control of this exothermic reaction was improved allowing a close to isothermal operation and resulting in the yield increase from 10 up to 85% at a residence time of 3 min. 5.4.3 Reduction of Methyl Butyrate

In this reaction, the initial reactant methyl butyrate is reduced by diisobutylaluminum hydride (Dibal-H) toluene to an unstable organometallic intermediate [38]. In a second step, the intermediate is further reduced to butyraldehyde, which is the desired product. A further reduction leads to the unwanted formation of butanol (Figure 5.33). The reaction scheme can be simplified to: k2

k1

k3

A −−−−−−−−−→ B∗ −−−−→ C −−−−→ D Dibal-H Dibal-H,toluene

(5.73)

In this reaction, the rate constant k 1 > k 2 . As the carbonyl group of aldehydes (C) is more reactive than that of esters (A), the rate constant k 3 > k 1 > k 2 . Hence, the concentration of the target product C is very low during the reaction, which is usually carried out at low temperatures (around 238 K). In addition, B* is unstable at higher temperatures, which considerably reduces the yield of C at higher temperatures in batch mode. Because of the high exothermicity of the reaction and its instantaneous character, its implementation in an industrial scale using conventional semibatch reactor would result in extremely high dosing times and thus poor performance. Therefore, an MSR with four different mixing properties and cooling capacities were used [38]. This consecutive reaction shows high sensitivity to mixing as well as to temperature variations. Whereas in a lab-scale batch reactor 63% of the target product C is obtained only at temperatures lower than 218 K, similar selectivity

CO2Me A

k1

OMe

Dibal-H, toluene

H O—Al(iBu)2 B k2

OH

k3 CHO

Dibal-H D

C

Figure 5.33 Reaction scheme of the reduction of methyl butyrate [38].

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5 Heat Management by Microdevices

could be obtained in MSR for higher temperatures (up to 253 K) following an improved thermal control. It was shown that better yields are obtained by injecting the reactant at several points along reactor length. In this case, the single hot spot is divided in several hot spots. At temperatures above 253 K, the reaction rates are increased to a point where characteristic reaction time is lower than characteristic mixing time. In this regime, the yield becomes a function of the mixing properties of the reactor. The comparison of four MSR showed the best selectivity (90% surface area of product C in the gas chromatograph), which were obtained with ER-25 MSR at 218 K, which is a multilamination mixer with sheets of 25 μm opening and is equipped with integrated cooling. Lower yields were obtained with micromixers without integrated cooling or with longer characteristic mixing times. 5.4.4 Reactions with Grignard Reagent in Multi-injection Reactor

The reactions of this class are instantaneous (mixing controlled) and highly exothermic and so the selectivity is sensitive to mixing and temperature control [23, 34]. In the reaction involving organometallic reagents, phenylethyl magnesium bromide, and 2-chloropropionylchloride, a productivity of about 100 g min−1 was achieved in a 35 mm3 continuous multi-injection reactor. It requires a batch reactor with 1 m3 volume. The glass reactor (Figure 5.34) was HE in T1

Feed 1

HE out

HE in T3

DT

HE out

MF Glass micro-structure MANIFOLD function Split feed 1 into four Provide temperature management

MF HE in T2

DT Glass micro-structure Heat/cool the feed at set-up temperature

MJ Glass micro-structure Reacts feed 1 with feed 2 Provides mixing, residence time and temperature management

MJ

DT

HE out

Feed 2

4 HE in T4

HE out DT Glass micro-structure Heat/cool the feed at set-up temperature

Product outlet Figure 5.34 Scheme of a modular multi-injection reactor [34]. (Adapted with permission from Wiley.)

5.4

Case Studies

225

designed in a modular flexible manner for use in multiple processes. For each specific function required for an optimal performance in a multi-injection reactor, a separate module was built:

• DT: precooling of the feed. • MF: flow distribution, splitting one flow into four using channel length to create equal pressure drop. Equal flow distribution is attained by ±8% relative variation. • MJ: multi-injection unit, containing four injection points. Each injection point is followed by a mixing zone containing mixing elements. Each mixing zone leads into a residence time channel of 5 mm width, 0.6 m height, and 0.8 m length designed to evacuate the heat produced before reaching the following injection point. It has to be highlighted that the different zones in the multi-injection unit specifically avoid an unwanted temperature rise and accumulation of reactants. In further experiments with the model Grignard reaction it was demonstrated that the increased number of injection points, N, increases the yield (Figure 5.35) [23]. The biggest gain in the yield (an increase from 22 to 30%) is obtained by switching from one injection point to two injection points. By adding more injection points, the relative yield increase diminishes. An increase of the number of injection points to more than six results in a maximum yield (of 37%). By repeating the same experiments with insufficient cooling between the injection points, a clear loss of selectivity can be seen. In order to increase the heat produced at each injection point, the flow rate is doubled from 18 to 36 g min−1 resulting in a further drop in yield down to 32% with six injection points. An identical behavior to that of low flow rate for insufficient cooling was observed at high flow rates. 40% 38%

T = 20 °C

36%

Product (area)

34% 32% 30% 28%

18 g min–1, sufficient cooling

26%

18 g min–1, insufficient cooling

24%

36 g min–1, sufficient cooling

22%

36 g min–1, insufficient cooling

20% 0

1

2

3

4

5

6

Number of grignard injection points Figure 5.35 Effect of the number of injection points and cooling on product formation [23]. (Adapted with permission from Wiley.)

7

226

5 Heat Management by Microdevices

5.5 Summary

In this chapter, the general design criteria of MSR for fast and highly exothermic reaction are presented. For most reactions with characteristic reaction time over a few minutes, the temperature profile in MSR can be considered as nearly isothermal. For fast reactions with characteristic reaction time 50 ⋅ dp , plug flow behavior is obtained (Bo ≥ 100). To avoid excessive pressure drop, a cross-flow microreactor for catalyst testing and kinetic studies is proposed (Figure 6.1) [5]. As the reactor is short, a nearly uniform axial concentration can be assumed corresponding to complete backmixing. The short packed bed reactor was microfabricated in silicon. The complex cross-flow design achieves uniform flow distribution over a 25.5 mm wide but shallow catalyst bed (400 μm long, 500 μm deep) to realize sufficiently high conversions and to allow monitoring of the reactants with conventional analysis techniques. The use of catalyst particles (dp = 53–71 μm) implies that conventional synthesis procedures can be used and experimental results can be translated to catalysts in macroscopic reactors. A set of complex, shallow microfabricated channels maintains a spatially uniform pressure drop irrespective of the variations in catalyst packing and allows uniform flow distribution. Quantitative analysis of transport effects indicated that temperature and concentration gradients in the catalyst bed can be neglected making this reactor a useful experimental tool for

233

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6 Microstructured Reactors for Fluid–Solid Systems

Thermocouple wells

Flow Inlet

64 bifurcated inllet channels Backed bed: 0.4 × 25.5 × 0.5 mm

Outlet 2.5 cm

256 pressure drop channels and catalyst retainers

Figure 6.1 Picture of the silicon cross-flow microreactor [5]. (Reproduced from Ref. [5] with kind permission of Springer Science+Business Media.)

studying kinetics and to optimize the reaction conditions, which was proved by the experiments of CO oxidation that compared well with parameters previously determined in macroscale systems. An integrated packed bed heat exchanger was developed for different heterogeneous catalytic reactions like methanol [6] and dimethyl ether synthesis [7]. The reactor is made of stainless steel and can be operated at pressures up to 100 bar and temperatures up to 350 ∘ C. The device consists of rectangular slits 8.8 mm width, 0.8 or 1.5 mm height, and 60 mm length. The slit channels are filled with catalyst particles forming a micro fixed bed. The reactor channels are sandwiched between cross-flow cooling channels for effective heat transfer. Different fractions

Cooling channels Reaction channel (a)

(b) 1mm (c) Figure 6.2 (a) Reaction slit and cooling channels arrangement, (b) pillar structure at the ends of the slit, (c) pillar structure in the whole reactor slit [7], and (d) Scanning

(d) electron microscope (SEM) picture of the pillar structure [6, 7]. (Adopted with permission from Elsevier.)

6.3

Microstructured Reactors for Catalytic Gas-Phase Reactions

of particles between 50 μm and up to 200 μm were used to study the external mass transfer and pressure drop in the reactor. To facilitate a uniform gas distribution over the channel cross section, cylindrical pillars were introduced at the entrance and at the end, or even in the whole reactor (Figure 6.2). Near isothermal operation could be obtained with maximum temperature differences of less than 3 ∘ C close to the entrance. 6.3.2 Structured Catalytic Micro-Beds

The drawback of randomly packed microreactors is the high pressure drop. In multitubular micro fixed beds, each channel must be packed identically or supplementary flow resistances must be introduced to avoid flow maldistribution between the channels, which leads to a broad residence time distribution in the reactor system. Initial developments led to structured catalytic micro-beds based on fibrous materials [8–10]. This concept is based on a structured catalytic bed arranged with parallel filaments giving identical flow characteristics to multichannel microreactors. The channels formed by filaments have an equivalent hydraulic diameter in the range of a few microns ensuring laminar flow and short diffusion times in the radial direction [10]. A microstructured string-reactor was designed for the autothermal production of hydrogen by oxidative steam reforming of methanol [10] (Figure 6.3). The main difficulty in carrying out the oxidative steam reforming is the much faster exothermic methanol oxidation compared to the endothermic reforming reaction. As a consequence, heat is generated mostly at the reactor entrance, whereas the heat consumption occurs in the middle and rear of the reactor. Metal-based catalysts with high thermal conductivity can help to integrate the exothermic combustion of methanol and the endothermic steam reforming avoiding important axial temperature profiles. The design of the microstructured string-reactor is based on catalytically active wires placed in parallel into a tube forming small channels of diameter about 100 μm. The heat generated during methanol oxidation at the reactor entrance is efficiently transferred to the reactor zone of the endothermic steam reforming. Brass metal wires (Cu/Zn = 4/1) were used as precursors for the

Catalytic filament Microchannel

dw

1

2

ø 0.1 − 0.3 dw Figure 6.3 Schematic presentation and photograph of the microstructured string reactor [9]. (Adapted with permission from Elsevier.)

235

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6 Microstructured Reactors for Fluid–Solid Systems

preparation of string-catalysts. The brass wires have high thermal conductivity (120 W m−1 K−1 ) and the chemical composition is similar to the active phase of the Cu/ZnO/Al2 O3 in traditional catalysts for the steam reforming of methanol. The catalyst is developed by metal/aluminum alloy formation on the outer surface of wires followed by an acid treatment leaching out aluminum. This treatment leads to an increase of the specific surface area (SSA) because of the formation of porous outer layer on the wire surface [11]. In a second example, a filamentous microstructured catalyst was used in a membrane reactor specially developed for the continuous production of propene from propane via nonoxidative dehydrogenation. The catalytic filaments with a diameter of about 7 μm consisted of a silica core covered by a γ-alumina porous layer on which an active phase of Pt/Sn is supported [8]. The catalytic filaments were introduced into the tubular reactor in the form of threads. A bundle of ∼100 filaments with a diameter of ∼7 μm each formed threads of diameter of about 0.5 mm. The catalytic threads were placed in parallel into the tube to form a cylindrical catalytic bed of several centimeters’ length. This arrangement gives about 300 threads per cm2 within the tube cross section with a porosity of �struc = 0.8. The specific surface per volume is in the order of 108 m2 m−3 and, thus, about 50 times higher compared to washcoated tubes of the same inner diameter [8]. The performance comparison under identical experimental conditions with randomly packed beds with particles of silica and γ-alumina of different shapes and sizes showed significantly broader residence time distribution compared to the structured filamentous packing with about five times lower pressure drop for the same hydraulic diameter and comparable gas flow rates. Reactor channels filled with metallic or ceramic foams as catalyst supports demonstrate several advantages compared to randomly packed beds [12]. Open cell foams consist of a network of interconnected solid stunts-building cavities

1 mm

31372 (a)

1000 μm

20 μm

1 mm (b)

Figure 6.4 (a) Photograph of metallic foam and (b) metallic foam washcoated with catalyst [13]. (Adapted with permission from Elsevier.)

6.3

Microstructured Reactors for Catalytic Gas-Phase Reactions

(cells). Metallic foams have porosities of up to 95% and the porosity of ceramic foams lay between 75 and 85%. Examples are shown in Figure 6.4. Foams were proved to be highly suitable as catalytic carrier when low pressure drop is mandatory. In comparison to monoliths, they allow radial mixing of the fluid combined with enhanced heat transfer properties because of the solid continuous phase of the foam structure. Catalytic foams are successfully used for partial oxidation of hydrocarbons, catalytic combustion, and removal of soot from diesel engines [14]. The integration of foam catalysts in combination with microstructured devices was reported by Yu et al. [15]. The authors used metal foams as catalyst support for a microstructured methanol reformer and studied the influence of the foam material on the catalytic selectivity and activity. Moritz et al. [16] constructed a ceramic MSR with an inserted SiC-foam. The electric conductive material can be used as internal heating elements and allows a very rapid heating up to temperatures of 800–1000 ∘ C. In addition, heat conductivity of metal or SiC foams avoids axial and radial temperature profiles facilitating isothermal reactor operation. Slit-like channels can be filled with highly porous foams integrated in MSR allowing near isothermal operation for fast exo- or endothermic reactions. Compared to wall coated catalytic multichannel reactors, the foam catalyst can be easily changed in case of catalyst deactivation. An example for MSR with integrated foam is shown in Figure 6.5. The foam plates are 60 mm wide, 200 mm long, and about 1.5 mm high. They are sandwiched between plates provided with rectangular parallel cooling channels in the submillimeter range.

Uncoated

Catalyst coated

Figure 6.5 Microstructured reactor with integrated exchangeable catalytic foam plates. (Courtesy Fraunhofer ICT-IMM, Germany.)

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6 Microstructured Reactors for Fluid–Solid Systems

300 μm

Sintered metal fiber 00019953

100 μm

IMVT

00019958

30 μm

IMVT

Figure 6.6 (a) Schematic presentation of a “sandwich” reactor and (b) SEM images of sintered metal fiber (SMF) catalyst. (Reproduced from Ref. [22]. Copyright © 2008, John Wiley & Sons.)

An alternative to foams are fibrous materials in the form of tissues or sheets [17–19]. Of particular interest for MSR are sintered metal fibers (SMF) sheets [18, 19]. They have an open and homogeneous structure with porosities of 70–90% and a high thermal conductivity, which ensures homogeneous temperatures in the catalytic bed. SMF consist of thin metallic fibers of 10–20 μm diameter sintered together in the form of thin (300–600 μm) sheets, thus, ensuring high external fluid–solid mass transfer performance [20]. The fibers can be covered by a homogeneous layer of zeolites [20], or oxide washcoat, which can be impregnated with an active material [21]. A SMF catalyst integrated in an MSR in the form of a sheet “sandwiched” between two metallic plates is shown in Figure 6.6 [21]. The resulting packed microchannel is 400 μm deep, 20 mm wide, and 20 mm long [21], which was used for CO oxidation under forced oscillating temperatures. 6.3.3 Catalytic Wall Microstructured Reactors

Small dimensions of the channels, a high surface-to-volume ratio together with the integrated heat exchange are the key features of the multichannel microreactors as compared to traditional reactors. The major characteristics of these reactors are discussed to quantify the potential gain in the reactor performance by structuring the fluid flow into parallel channels and by accelerating the heat supply and the heat removal. The typical configuration of a microstructured multichannel reactor with a wall catalyst is shown in Figure 6.7. The multichannel reactor is characterized by the wall thickness (e, esw ) and the height (H) and width (W ) of the channels. The catalytic wall reactor with channel diameter in the range of 50–1000 μm and a length dependent on the reaction time required circumvents the shortcomings of micro packed beds. This is discussed in more detail in Section 6.5.4. However, in most of the cases, the catalytic surface area provided by the walls alone is insufficient for the chemical transformation and, therefore, the SSA has to be increased by the chemical treatment of the channel walls, or by coating them with highly porous support layers. The thickness of the layer � cat depends on catalytic activity. In general, the layer thickness is sufficiently small to avoid internal heat and mass transfer influences. Catalytic layers can be obtained by using a

6.4

W

Hydrodynamics in Fluid–Solid Microstructured Reactors

esw

H δcat

e 200 μm

Figure 6.7 (a) Scheme of a wall coated microstructured multichannel reactor. (b) Scanning electron microscopy image of a cross section of a diffusion welded and

subsequently anodically oxidized microstructured aluminum reactor. (Reproduced with permission from Ref. [23]. Copyright CHIMIA.)

Table 6.2 Different techniques used to increase surface area in catalytic wall mictrostructured reactors. Coating layer

Alumina [25–27] Alumina [23] Alumina [28], silica [29], and titanium oxide [30] Alumina [31] γ-Alumina [32] Al2 O3 , ZnO, and CeO2 [33] Zeolite [34, 35] Zeolite [36–38] Alumina [39] Au/TiO2 porous catalyst [40] Carbon coating [41] Carbon nanofibers [42]

Technique

Metal oxide coatings Anodic oxidation of aluminum Anodic oxidation of AlMg microstructured reactor wall Sol–gel technique High-temperature treatment of Al containing steel (DIN 1.4767 FrCrAlloy) Wash-coating Electrophoretic deposition Zeolite-coated microchannel reactors Direct formation of zeolite crystals on metallic structure Chemical vapor deposition (CVD) Flame spray Carbon based coatings Carbonization of polymers Thermal chemical vapor deposition

variety of techniques such as sol–gel, electrophoretic, and chemical vapor deposition (CVD) or physical vapor deposition (PVD) [24]. Different methods of the deposition of film on the solid surface are presented in Table 6.2 as examples. 6.4 Hydrodynamics in Fluid–Solid Microstructured Reactors

The pressure drop during the passage of a fluid through a reactor is an important parameter related to the optimization of the energy consumption. Pressure

239

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6 Microstructured Reactors for Fluid–Solid Systems

drop is considered with the assumption of noncompressible fluids and standard continuum mechanics. For example, gas properties at temperatures up to ∼600 K, at a minimum pressure of 0.1 MPa with fluid velocities 5%, and thus the resulting criterion for effective isothermal catalytic wall behavior is given by: √ 2 R �eff Ts (6.19) �cat,max ≤ 0.3 E |ΔHr |reff where Ts corresponds to the temperature of the catalyst surface and R is the gas constant. In general, the thickness of the catalytic layer is kept sufficiently small to avoid the influence of internal mass transfer on the kinetics. In this way, only the transfer of the reactants from the bulk of the fluid to the catalytic wall and the reaction rate per unit of the outer surface of the catalytic layer must be considered. Because of the small diameters of microchannels, laminar flow can be assumed. The radial velocity profile in a single channel develops from the entrance to the position where a complete Poiseuille profile is established. The length of the hydrodynamic entrance zone (Le ) in a circular tube depends on the Re (= �udt ∕�) and can be estimated from the following empirical relation [51, 52]: Le ≤ 0.06 ⋅ Re ⋅ dt

(6.20)

Within the entrance zone, the concentration profile can be developed and the mass transfer coefficient diminishes reaching a constant value. The

6.5

Mass Transfer in Catalytic Microstructured Reactors

Table 6.3 Mass transfer characteristics for different channel geometries [53]. Geometry

Sh∞

Circular Ellipse (W /H = 2) Parallel plates Square Equilateral triangle Sinusoidal Hexagonal

3.66 3.74 7.54 2.98 2.47 2.47 3.66

dependency can be described with (Equation 6.21) in terms of Sherwood numbers, Sh = km dh ∕Dm [53, 54]: )0.45 ( d Sh = Sh∞ 1 + 0.095 h Re ⋅ Sc L

(6.21)

where, Sc is Schmidt number (Sc = �∕�Dm ). Sh∞ is the asymptotic Sh for constant concentration at the wall, which is identical to the asymptotic Nusselt number Nu, characterizing the heat transfer in laminar flow at constant wall temperature. The asymptotic Sh depends on the geometry of the channel as summarized in Table 6.3. For rectangular channels, Sh∞ depends on the ratio between channel height, H, and width, W (Figure 6.7), and can be estimated with the following correlation: ) ( H Sh∞ = 2.8932 + 4.6482 exp −4.4754 (6.22) W It follows for a circular-shaped reactor: Sh∞ = 3.66; for L ≥ 0.05Re ⋅ Sc ⋅ dh (constant wall concentration) (6.23) The entrance length estimation is demonstrated in Example 6.3. Thus, the mass transfer coefficient can be calculated as km =

Sh∞ Dm dh

(6.24)

Example 6.3: Length of entrance zone to achieve asymptotic Sh-number. Estimate the length of the entrance zone to achieve asymptotic Sh for constant concentration at the wall for rectangular (H/W = 0.5) and square channels, assuming identical operating conditions and Sh is 5% higher than Sh∞ The gas flows with a velocity of 0.1 m s−1 and the diffusion coefficient is 10−5 m2 s−1 . Data: � = 1 m2 s−1 , dh = 0.4 mm.

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6 Microstructured Reactors for Fluid–Solid Systems

Solution:

As the Sh is assumed to be 5% higher than Sh∞ , Sh = 1.05Sh∞ . The length is investigated from Equation 6.21. Let us first solve for product Re⋅Sc as required udh udh � = � Dm Dm 0.1 × 0.4 ⋅ 10−3 = =4 1 ⋅ 10−5 and the length, say for rectangular channel with height/width ratio = 0.5, is Re ⋅ Sc =

L=

0.095 ⋅ dh ⋅ Re ⋅ Sc 0.095 ⋅ 0.4 ⋅ 10−3 ⋅ 4 = ( ) 1 ( ) 1 0.45 Sh 1.05×3.39 0.45 − 1 −1 Sh 3.39 ∞

= 0.32 ⋅ 10−3 m as the ratio of Sh/Sh∞ is constant for all channels, that is, 1.05. The length of the entrance zone to achieve asymptotic Sh is identical for all channels for identical operating conditions. The specific performance of the MSR under mass transfer limitations depends on the mass transfer coefficient and the SSA of the channel (a): a=

4 dh

(6.25)

4⋅A

with dh = l cs hydraulic diameter circ The product (km ⋅a) is called volumetric mass transfer coefficient (demonstrated in Example 6.4), which determines the maximal reactor performance for very fast catalytic reactions. Its value increases with 1∕dh2 .

Example 6.4: Volumetric mass transfer coefficient for different microchannel geometries Estimate and plot the volumetric mass transfer coefficient (kG a) for different microchannel geometries such as slit, rectangle (H/W = 0.25), circular, and square for different hydraulic diameter ranging from 50 to 1000 μm neglecting the influence of entrance zone. The diffusion coefficient of gas (Dm ) is 10−5 m2 s−1 . Solution:

The volumetric mass transfer coefficient for different hydraulic diameters can be investigated assuming asymptotic values of Sherwood number (Sh∞ ) using Equation 6.24, the specific interfacial area using Equation 6.25. If the entrance zone in the tube is neglected, the mass transfer constant is given by

6.5

Mass Transfer in Catalytic Microstructured Reactors

Sh∞ . Sh∞ can be estimated with Equation 6.22 and dt is replaced by dh Sh∞ Dm dh 4 for all channels. a= dh

km =

The calculated values are plotted in Figure 6.12.

Vol. mass transfer coeff., kma (s−1)

105

Slit 104

103

H/W=0.25

Square

Circular 102 200

400 600 800 Hydraulic diameter, dh (um)

1000

Figure 6.12 Volumetric mass transfer coefficient as function of the hydraulic diameter in microchannels (Dm = 10−5 m2 s−1 ).

If the mass transfer is accompanied by a chemical reaction at the catalyst surface on the reactor wall, the mass transfer depends on the reaction kinetics [55]. For a zero-order reaction, the rate is independent of the concentration and the mass flow from the bulk to the wall is constant, whereas the reactant concentration at the catalytic wall varies along the reactor length. For this situation the asymptotic Sh in circular tube reactors becomes Sh′∞ = 4.36 [55]. The same value is obtained when reaction rates are low compared to the rate of mass transfer. If the reaction rate is high (very fast reactions), the concentration at the reactor wall can be approximated to zero within the whole reactor and the asymptotic value for Sh is Sh∞ = 3.66. As a consequence, the Sh in the reacting system depends on the ratio of the reaction rate to the rate of mass transfer characterized by the second Damköhler number defined in Equation 6.11. Villermaux [55] proposed a simple relation to estimate the asymptotic Sh for mass transfer with chemical reaction (Sh′′∞ ) as function of DaII

249

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6 Microstructured Reactors for Fluid–Solid Systems

) ( 1 1 1 DaII 1 (6.26) − = ′ + Sh′′∞ Sh∞ DaII + 1.979 Sh∞ Sh′∞ Thus, the mass transfer coefficient in multitubular MSR depends besides the molecular diffusion coefficient, on the channel diameter dt and the second Damköhler number, DaII (refer Example 6.5): Sh′′ D kG = ∞ m (6.27) dt

Example 6.5: asymptotic Sh and mass transfer coefficient (with chemical reaction) Estimate the asymptotic Sh for mass transfer with chemical reaction for a circular cross section microchannel with the diameter of 500 μm for DaII equal to 0.1, 1, and 100. Also, estimate the mass transfer coefficient for a gas-phase system with molecular diffusivity (Dm ) of 10−5 m2 s−1 . For circular tube Sh∞ = 3.66 and Sh′∞ = 4.36. Solution:

The asymptotic Sh for mass transfer with chemical reaction is estimated using Equation 6.26. For DaII = 0.1: asymptotic Sh is ( ) 1 1 DaII 1 1 = + − Sh′′∞ Sh′∞ DaII + 1.979 Sh∞ Sh′∞ ( ) 1 0.1 1 1 = 0.23 + − = 4.36 0.1 + 1.979 3.66 4.36 ⇒ Sh′′∞ = 4.32 and the mass transfer coefficient can be calculated using Equation 6.27 as Sh′′∞ Dm dt 4.32 × 1 ⋅ 10−5 = = 8.64 ⋅ 10−2 m ⋅ s−1 500 ⋅ 10−6 Similarly, For DaII = 1: Sh′′∞ = 4.09, kG = 8.19⋅10−2 m ⋅ s−1 For DaII = 100: Sh′′∞ = 3.67, kG = 7.34⋅10−2 m ⋅ s−1 kG =

The above example indicates that with increase in DaII, Sh′′∞ decreases and attends a value close to the case of pure mass transfer. An identical behavior can be observed for mass transfer coefficient. On the other hand, the example of mass transfer with very high transformation rate in packed bed and microchannel reactor is demonstrated in Example 6.6. In addition to appropriate mass transfer rates, sufficiently rapid heat transfer is essential to control the behavior of chemical reactors. For example, if the local

6.5

Mass Transfer in Catalytic Microstructured Reactors

rate of heat removal does not match the rate of heat produced by the chemical reaction, hot spots will be formed. As the reaction rate depends exponentially on temperature, reactor performance, that is, product yield and selectivity, is strongly influenced by nonisothermicity. In the case of exothermic reactions, a steep local temperature rise can lead to reactor runaway. The stability criteria mentioned in Chapter 5 can be applied to investigate the reactor stability.

Example 6.6: Performance comparison of packed bed microreactor and microchannel reactor A toxic compound in air has to be eliminated by catalytic oxidation. The concentration of the toxic must be reduced from initially 1% to � m > 0.1, the values for micro packed beds is found to be between 0.006 > � m > 0.002. The highest effectiveness can be obtained with microchannel 1 Mass transfer efficiency, ηm = DaIm/Eu

254

Slit channel, av = astruc = 4000 m2 m−3 Square channel, av = astruc = 4000 m2 m−3 0.1 Foam: εfoam = 0.87, av = 2160 m2 m−3 0.01

0.001 0.0

Packed bed: εbed = 0.4, av = 4000 m2 m−3

0.2

0.4

0.6

0.8

1.0

Interstitial velocity, uv (m s−1) Figure 6.13 Mass transfer effectiveness for different microstructured reactors. Gas-phase, physical properties of air at 20 ∘ C, 0.1 MPa.

6.6

Case Studies

reactors. The effectiveness increases with decreasing aspect ratio, H/W , of the rectangular channels. Highest values are obtained for slit-like geometries.

6.6 Case Studies

Fluid–solid MSR have been extensively used particularly for gas–solid reactions such as catalytic partial oxidations, selective hydrogenations, dehydrogenations, dehydrations, and reforming processes [57, 58]. Similarly to the other reactions carried out in MSR, the main objective was to achieve better temperature control in order to prevent selectivity loss, catalyst deactivation, hot spot formation and, thus, allowing safe processing with high throughput. Some of the examples with a short description of reactor design, channel dimensions, methods of microreactor fabrication, and key results of their testing are described in the following. 6.6.1 Catalytic Partial Oxidations

Many partial oxidations are carried out using pure oxygen at elevated pressures. By this, space time yields can be increased. One of the initial studies was on ammonia oxidation in a chip-like microreactor [59] with the aim of demonstrating the feasibility of decentralized and safe production of hazardous chemicals. The reaction is highly exothermic and has several series and parallel reaction pathways allowing for selectivity studies. The desired product of the reaction is NO, while production of N2 represents a product loss. A single T-shaped microchannel with integrated temperature and flow sensors, etched by KOH into a silicon wafer, was covered by a SiN membrane carrying a thin-film Pt layer for heating on the outer side and a Pt catalyst layer on the inner side. A temperature as high as 800 ∘ C was achieved in the reaction zone because of selective heating of the reactor parts; however, the experiments were limited to 650 ∘ C, because high temperatures caused deformations and rupture of the membrane. Increasing the heating power leads to a higher NO/N2 ratio. Higher residence time led to complete conversion and it also decreased the NO/N2 ratio as the NO had time to re-adsorb and decompose. In ammonia-rich feed, conversion vanished because of an inhibition of NO desorption by ammonia. The small reactor had a production capacity of 10 g NO per day. Another silicon membrane microreactor, composed of an aluminum bottom plate, a microstructured silicon layer carrying the channel system, and a 3 μm thick SiN membrane as a cover of the reactor, was developed [60]. Pt as an active component was put on the membrane either by wet chemistry or by PVD on a Ti adhesion layer. The reactor was manufactured by photolithography and plasma etching. The channels were introduced either by wet-etching or deep reactive ion etching. By increasing the thickness of the membrane from 1 to 1.5 and 2.6 μm,

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6 Microstructured Reactors for Fluid–Solid Systems

and switching to pure silicon, the ability of the reactor membrane to dissipate heat was increased by an order of magnitude [61]. At this high temperature, a high selectivity to NO was found. The thicker membranes allowed for lower operation temperatures and thus lower NO selectivity. Introducing silicon membranes into the reactor increased the heat conductivity by 25 times, which improved the temperature uniformity. By using SiO2 and Si3 N4 as membrane material, stress-compensated membranes of 0.25 and 0.3 μm thicknesses could be manufactured. A silicon-based MSR for the study of the intrinsic kinetics of the catalytic partial oxidation of methane was used [62]. The single microchannel was 30 mm long and a cross section of 500 × 500 μm, manufactured by photolithography, was put into a housing made of aluminum. The channel was covered by a 1.9 μm thick silicon sheet, which allowed good thermal contact between the Rh catalyst beneath it and the 5 Pt heating wires and the 12 Pt temperature sensors on top of it. Isothermal conditions of gas inlet temperature and wall side and bottom temperature generated highest Nusselt numbers and, therefore, heat transfer coefficients, which was favorable for the minimization of heat transfer of the reaction under investigation. Microchips with catalytic wire have been applied for H 2 /O2 oxidation reaction in a single-channel reactor [63]. The reactor was designed as a modular and flexible tool for various high-temperature reactions. Stainless steel housing took up the silicon chips that were carrying the microchannels. The wafers were coated with silicon oxide (400 nm thickness) and silicon nitride by low-pressure chemical vapor deposition (LPCVD) alternatively. The chips were manufactured by photolithography and etching. The catalyst (for this application Pt) was introduced as a wire (150 μm thickness), which was heated resistively for igniting the reaction. The ignition of the reaction occurred at 100 ∘ C and complete conversion was achieved at a stoichiometric ratio of the reacting species generating a thermal power of 72 W. The same reaction was performed in a quartz glass microreactor with a diameter of 600 μm and 20 mm length [64]. The ceramic housing of the reactor and the reactor itself were stable for temperatures exceeding 1100 ∘ C. Again, a Pt wire of 150 μm diameter was used as a catalyst and electrically heated for start-up. Residence times down to 50 μs were achieved. The fact that no homogeneous reactions, which are explosive, could be detected demonstrating the possibility of separating homogeneous and heterogeneous reactions in microreactors because of the higher surface area to the volume ratio of this reactor type. A microstructured device consisting of a preheating unit, a mixer, a reactor, and a quenching zone was used for the exothermic oxidative dehydrogenation of methanol to formaldehyde [65]. The reactor was manufactured by photolithography and etching followed by the catalyst deposition and anodic bonding of the Pyrex glass cover. Silver was introduced as catalyst by sputtering. The reaction was carried out at residence times between 4 and 25 ms, temperatures between 430 and 530 ∘ C, inlet methanol concentration of 8.5–8.6 vol%, and pressure slightly above atmospheric pressure. CO2

6.6

Case Studies

and formic acid were found as products, the carbon monoxide formation being successfully suppressed. Increasing the residence time from 4 to 24 ms increased the conversion, but hardly affected the selectivities. Generally, in the deeper channels, higher conversion was achieved. Further, a chip-like reactor for the same reaction at temperatures exceeding 600 ∘ C was used [66] with the motivation of achieving extremely fast heating and cooling and ultrashort residence times. In an integrated heater/reactor/cooler system, the heater was composed of silicon microstructured by KOH etching and capped with an SiN membrane. The channel was 7.4 mm long and had a trapezoidal cross section 1.3 mm wide and 380 μm deep. Pt–Ta filaments were used to heat the bottom made of Pyrex. At a flow rate of 30 std cm3 min−1 , which corresponds to a residence time of 4 ms, a heating power as low as 1.7 W was able to increase the exit temperature of the test gas nitrogen up to 400 ∘ C. The heat exchanger for cooling had a counterflow design. All channels were approximately 300 μm deep and 20 mm long. The cooling fluid, nitrogen, with a rate of 500 std cm3 min−1 was cooled from 500 to 54 ∘ C in less than 1 ms by the cooler. 6.6.2 Selective (De)Hydrogenations

Hydrogenations and dehydrogenation reactions are usually highly exo/endothermic, involve hydrogen, a very reactive compound, and, therefore, precise control of reaction conditions and safety are important issues. Membrane MSR for the dehydrogenation of cyclohexane to benzene were designed [67]. This is an endothermic reaction whose equilibrium conversion is 18.9% at 200 ∘ C. The conversion can increase beyond equilibrium up to 99% if the hydrogen is removed from the system. Therefore, a Pd-membrane with microchannels has been used to continuously remove hydrogen out of the reaction zone in order to enhance the conversion. The reactors were made of silicon using photo-etching technique, and Pt was used as a catalyst, which was sputtered onto the reaction chamber [67]. Out of two reactors, one example is shown in Figure 6.14. A chip-like silicon-based process-engineering device consisting of a preheater, a reactor, and an integrated thermocouple mainly applying photolithographic techniques (see Figure 6.15) was fabricated and used for benzene hydrogenation [68]. The top and bottom part of the reactor was made of glass, which is attached to the silicon core by anodic bonding. The reactor was heated by an integrated Pt wire at the bottom as shown in the bottom view of Figure 6.15. Both sputtering of the silicon surface and wet impregnation of alumina precipitated by the sol–gel method were used to introduce Pt as the active component. After activation in H2 at 500 ∘ C, the catalyst was tested for benzene hydrogenation at temperatures between 100 and 150 ∘ C, a flow rate of 1 cm3 min−1 , and residence times from 100 to 600 ms. First order kinetics were found for the reaction.

257

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6 Microstructured Reactors for Fluid–Solid Systems

Cross section

Roof

Products outlet

Hydrogen outlet

Reactants inlet Membrane Silicon membrane

Reactor

Silicon reactor Palladium

Base

AI

Ti

Pt

Figure 6.14 Membrane microreactor designed. (Adapted from Ref. [67].)

Top view Preheater

Reactor Thermocouple well Glass Si

Side view Pt heater

Glass

Bottom view

Figure 6.15 A silicon-based microchip MSR with Pt heater. (Reproduced from Ref. [68] With kind permission of Springer Science+Business Media.)

Two silicon reactors with different channel systems manufactured by photolithography and KOH etching were used for cyclohexene (de)hydrogenation over Pt catalysts [69]. In the first reactor, 39 channels of 100 μm width revealing a total surface area of 2 cm2 were incorporated, the second one had 780 channels of 5 μm width manufactured by ion coupled plasma (ICP) etching with a total surface area of 28 cm2 . The Pt catalyst was introduced by sputtering. Higher conversions were obtained in smaller reactors because of the higher surface area and thus catalyst mass of the smaller channel system, which results in a higher modified residence time (catalyst mass/flow rate). The above channel systems were modified using dip-coating, spin-coating, and drop-coating to introduce silica as a porous layer [70]. Pt was then introduced

6.6

Case Studies

by both sputtering and wet impregnation. The resulting surface areas increased from 1000 to 15 000 times. Selectivity toward benzene was favored at temperatures exceeding 150 ∘ C. The lifetime of the supported catalyst was 3.5 times higher compared to their unsupported counterparts. 6.6.3 Catalytic Dehydration

A glass MSR was used to perform the dehydration of ethanol. The microchannel of size 200 × 80 μm deep × 30 mm (in a “Z” shaped configuration) was produced by photolithographic etching [71]. A sulfated zirconia catalyst immobilized over the surface of the top cover block. In addition, a NiCr wire was immobilized in the reactor cover as a heating device. At a reaction temperature of 155 ∘ C and a flow rate of 3 μl min−1 the main products were 68% ethylene, 16% ethane, and 15% methane. A further increase of the residence time resulted in a reaction progress beyond dehydration to almost complete cracking of the ethanol to methane. 6.6.4 Ethylene Oxide Synthesis

The reactor used for ethylene oxide synthesis consisted of a two-piece housing that was sealed by graphite gaskets [72]. In the recesses, stacks of platelets of microchannels were inserted. The first recess contained a stack of mixer platelets made by a combination of Laser-LIGA (briefed in chapter 1) and electroforming. These platelets had curved microchannels that made the fluid turn by 90∘ . Two types of mirror-imaged platelet designs allowed in an alternating stack arrangement for creating gas multilamellae. The second recess with a stack of silver reaction platelets made by LIGA and electroforming and chemically etched silver and milled “aluchrom” (aluminum containing stainless steel) platelets were used. Heating the aluchrom material to 1100 ∘ C with oxygen creates an α-Al2 O3 surface, which is the only alumina species active for the ethylene reaction. The surface of the silver reaction channels was enhanced by a factor of 2–3 by means of oxidation. The silver catalyst was introduced by sputtering. The reactor demonstrated safe operation of a highly explosive reaction mixture (explosion of ethylene oxide in air 2.6–100 vol% at ambient pressure) [73]. Contrary to the industrial process that applies alumina, supported pure silver was used as catalyst for the reaction with external heating and operated at 300 ∘ C and 25 bar. The space time yields achieved were 0.18–0.67 t m3 h−1 , which is superior compared to the industrial process (Figure 6.16). Further [73] silver wafers were also used for ethylene oxide synthesis [74]. Easy handling and fast exchange of catalyst platelets were fabricated by means of thin-wire μEDM (micro electrical discharge machining) in aluminum or aluminum alloys such as Dural, AlMg3 , and AlMg 4.5Mn. The rough surface of the microchannels regarded as beneficial for the coating techniques

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6 Microstructured Reactors for Fluid–Solid Systems

Inner housing for microstructured wafers Outer shell

Copper gasket

Copper gasket Flange

25

30

123

60

Outer shell

Flange

10

+

30

10

7

Diffusor (2x) 30

260

10 34

Inner housing (front view) Figure 6.16 Modular microchannel reactor for gas–solid reactions. (Adapted from Ref. [74].)

applied subsequently. Both anodic oxidation and the sol–gel method were used to get a porous alumina carrier. The active compound, silver, was introduced by sputtering in both cases. A stepwise increase of the silver layer thickness from 100 to 400 nm increased conversion at constant catalyst selectivity. 6.6.5 Steam Reforming

A silicon-chip in a stainless steel housing, electrically heated and sealed with graphite, for hydrogen production from methanol steam reforming was used [75]. The microchannel was a long serpentine of 1000 μm width and 230 μm depth fabricated by photolithography and KOH etching. Cu catalyst was sputtered to a thickness of 33 nm onto the chip. Preliminary simulations revealed nonuniform temperature distribution in the reactor housing pointing at the importance of proper insulation especially in low power systems. At a feed composition of 76 mol% methanol in steam less than 7% conversion was achieved at 250 ∘ C. In another reactor carrying microstructured plates, a copper-based low temperature water–gas-shift catalyst was applied [76]. The reactor took up 20 plates made of FeCrAl alloy with channel size 200 × 100 μm. The kinetic measurements were carried out and expressions were determined for both a tubular fixed bed reactor containing 30 mg catalyst particles and the microreactor coated with the

6.7 Summary

catalyst particles. The reaction rate was on an average 34% higher for the coated catalysts. 6.6.6 Fischer–Tropsch Synthesis

A silicon-chip-based reactor was applied for the Fischer–Tropsch synthesis using an iron catalyst [77]. The chips had outer dimensions of 1 × 3 cm with channel dimensions of 5 or 100 μm width at 50–100 μm depth. The reaction was carried out at a H2 /CO ratio of 3, and flow rates of 0.4 std. cm3 min−1 between 200 and 250 ∘ C. Conversions between 50 and 70% were found after 12 h activation of the catalyst under reaction conditions.

6.7 Summary

In this chapter, the various characteristics of MSR for fluid–solid reactions are presented. It is clear that microreactors are mostly suitable for reactions that have fast intrinsic kinetics and require rapid transport, are carried out at high temperatures and pressures, and, therefore, ensure inherent safety. Effective exploitation of the full chemical potential of catalysts through high rates of heat and mass transfer provides an excellent means for identifying novel synthesis routes that are both economically attractive and environmentally benign. The time available for chemical transformation in the MSR is very short because of their small size, which results in low hold-ups, on one hand, but necessitates highly efficient mass/heat transfer, on the other. The amount of power dissipation for multiphase reactions per unit of interfacial area is very low, leading to significant reductions in the energy consumption. Several examples reported show precise control of the operating conditions resulting in increased selectivity toward the target compound. Higher conversion rates were achieved by processing at high pressure and high temperature, often in the explosive regime, increasing space time yield as compared to conventional reactors. Many examples of partial oxidations were described with the processes of utmost industrial importance. Within consecutive processes, high selectivity was achieved for species that are thermodynamically not the most stable molecule of all species serially generated [57]. In many cases, the better reactor performance as compared to fixed bed technology was achieved. Nevertheless, there are several constraints hampering the use of microstructured devices for fluid–solid reactions. In the catalytic reactions, the performance is very adversely affected by catalyst deactivation. Effective in situ catalyst regeneration thus becomes necessary, as the simple catalyst change practiced in conventional reactors is usually no longer an option. The thickness of the catalytic wall is often greater than the internal diameter of the channel and, therefore, may impede heat transfer for highly exothermic reactions leading to nonisothermal behavior.

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Reactions involving highly viscous materials or suspended particles are difficult to carry out in the microreactor.

6.8 List of Symbols Ap , Afoam a, abed , astruc , afoam , ap , aR , av av,foam

ACS, bed , ACS, struc DaIm dc , d p , d w ds Lbed , Lstruc , Le Lcirc Rep , Ref Shp , Shf Sh′∞ , Sh′′ ∞

V v, foam uv � cat �bed , �foam , �struc �p �m

Outer surface area of pallet, outer surface area of foam Catalytic surface area per reactor volume, of randomly packed bed, of microchannels, specific surface area per foam volume, specific surface area of the pellet, specific surface area referred to reaction volume, specific surface area referred to void volume, specific surface area of foam referred to void volume Cross-section area of packed bed, of structured bed First Damköhler number based on mass transfer rate Cell diameter, diameter of pallet or pore, diameter of wire (catalytic filament) Mean strut thickness Length of randomly packed bed, length of structured bed, length of the entrance zone Perimeter Particle Reynolds number, foam Reynolds number, particle Sherwood number, foam Sherwood number, asymptotic Sherwood number at constant mass flow (with reaction) from the bulk to the wall, at mass transfer with chemical reaction Void volume of foam Velocity in the void volume or interstitial velocity Thickness of catalytic layer Porosity of randomly packed bed, porosity of foam, porosity of structured bed or multichannel reactors Internal effectiveness factor in isothermal catalyst layer Mass transfer effectiveness

m2 m2 m−3

m2 — m m m m — — —

m3 m s−1 m — — —

References 1. Henkel, K.-D. (2012) Reactor types

and their industrial applications, in Ullmann’s Encyclopedia of Industrial Chemistry, Wiley-VCH Verlag GmbH, Weinheim.

2. Cybulski, A. and Moulijn, J.A. (eds)

(2006) Structured Catalysts and Reactors, 2nd edn, Taylor & Francis Group, Boca Raton, FL, London, New York.

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267

7 Microstructured Reactors for Fluid–Fluid Reactions In this chapter, fluid–fluid flow patterns and mass transfer in microstructured devices are discussed. The first part is a brief discussion of conventional fluid–fluid reactors with their advantages and disadvantages. Further, the classification of fluid–fluid microstructured reactors is presented. In order to obtain generic understanding of hydrodynamics, mass transfer, and chemical reaction, dimensionless parameters and design criteria are proposed. The conventional mass transfer models such as penetration and film theory as well as empirical correlations are then discussed. Finally, literature data on mass transfer efficiency at different flow regimes and proposed empirical correlations as well as important hydrodynamic design parameters are presented.

7.1 Conventional Equipment for Fluid–Fluid Systems

Fluid–fluid systems are widely used in chemical, petroleum, pharmaceutical, hydrometallurgical, and food industries. Commercially important examples of gas–liquid mass transfer with or without reaction include gas purification, oxidation, halogenations, hydrogenation, and hydroformylation to name but a few. Important liquid–liquid reactions include nitration, phase transfer catalysis (PTC), cyclization, emulsion polymerization, homogenous catalyst screening, enzymatic reactions, extraction, precipitation, crystallization, and cell separation. Conventionally, a wide variety of equipment is used for fluid–fluid applications involving gas–liquid and liquid–liquid systems: stirred tanks, bubble columns, centrifugal-, packed-, plate columns, Buss loop reactors, straight or coiled tubular reactors, static mixer reactors, and film reactors. The schematics of these equipments are depicted in Figure 7.1. The contacting principles are bubbling, filming, spraying of one fluid into the other or disturbing the two-phase flow stream to create turbulence. The conventional equipments work well for most slow and moderately fast reactions. However, for fast intrinsic kinetics the overall transformation rate is controlled by mass transfer. Microstructured Devices for Chemical Processing, First Edition. Madhvanand N. Kashid, Albert Renken and Lioubov Kiwi-Minsker. © 2015 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2015 by Wiley-VCH Verlag GmbH & Co. KGaA.

268

7 Microstructured Reactors for Fluid–Fluid Reactions L

G

L

(e) (f) (a)

Mixer-settler

Centrifugal (L–L)§

(b)

(c)

(d)

Columns

Figure 7.1 Conventional equipment used for fluid–fluid reactions (columns: (a) multistage agitated column, (b) packed column, (c) sieve tray column, (d) buss loop reactor, (e) tubular reactor, and (f ) static mixer)

Tubular reactor, static mixer

Film reactor (G–L)§

§ G – Gas, L – liquid. (Adapted with permission from Ref. [1]. Copyright (2009) American Chemical Society.)

The advantages and limitations of fluid–fluid reactors are listed in Table 7.1. Stirred tanks are the most commonly used for reactions involving fluid–fluid systems. Bubble columns are used for gas–liquid reactions, while centrifugal reactors are used for liquid–liquid systems with low density differences. The third variety of equipment, the columns, is commonly used in chemical industries in their countercurrent mode of operations. Tubular contactors offer a number of advantages because of their flexibility, simplicity, and wide range of operating windows. To intensify the mixing in the tubular reactor, internals like static mixers are useful. Such equipment is used for mixing immiscible liquids in a compact configuration and is found to be effective [2]. In falling film contactors, a thin film is created by a liquid falling under gravity pull. The liquid flows over a solid support, which is normally a thin wall or stack of pipes. In conventional falling film devices, a film with a thickness of 0.5–3 mm is generated [3]. The film flow becomes unstable at high throughput and the film may break up into rivulets, fingers, or a series of droplets at high flow rates. A common drawback of all the above mentioned equipments is the inability to condition the drop or film size precisely and to avoid the nonuniformities that arise because of the complex hydrodynamics. This leads to uncertainties in the design and often imposes severe limitations on the optimal performance. Multiphase microstructured devices can potentially be used to diminish the limitations of conventional reactors. They generally take advantage of their large interfacial area reducing the mass transfer resistances. 7.2 Microstructured Devices for Fluid–Fluid Systems

Microstructured devices for fluid–fluid systems exist in a number of configurations. They can be roughly classified into three types based on the contacting principles [1]: micromixer, microchannels, and falling film microreactors. The first two types of devices are used for all fluid–fluid applications while the microstructured falling film reactor is used only for gas–liquid systems. Depending on the application, these microstructured devices can also be used in series, for example,

7.2 Microstructured Devices for Fluid–Fluid Systems

Table 7.1 Different types of fluid–fluid reactors, their advantages, and limitations. Reactors

Advantages

Limitations

Stirred tank

Low maintenance cost Intense mixing gives higher performance Simple construction Mixing because of sparged gas – requires low energy Works at low density differences between two fluids Less solvent volume required for extraction Rapid mixing and separation can enhance product recovery and cost Easy to operate

Difficult to condition the drop/bubble size Difficult to separate small density difference fluids Very complex hydrodynamics Applicable only for slow reactions

Bubble column

Centrifugal contactor

Static mixers

Loop reactor

Tubular reactor

Falling film

Satisfactory performance at lower cost Better temperature control Higher productivity Behavior close to plug flow High heat and mass transfer rates – suitable for highly exothermic/endothermic reactions Low pressure drop High interfacial area

Difficult to scale up Mechanical complexity and high maintenance cost

Performance completely dependent on packing/internals Good performance only in a limited range of flow rates Difficult to handle rapid fouling materials High probability of residence time distribution in the loop Not suitable for slow reactions Difficult to use high viscous fluids

Unstable at high throughput Thick liquid film results in higher mass transfer resistance for three-phase reactions

the first mixer for creating dispersion and second mixer for allowing the transport and/or chemical transformation. 7.2.1 Micromixers

These are the devices in which fine dispersion occur because of high flow rates or static internals. Three types of mixers are available: mixer-settler [4, 5], cyclone [6], and interdigital mixer [3] (see Figure 7.2). The principle of fluid contacting in micromixer-settler is almost similar to the conventional mixer-settler assembly. However, because of the reduced size of the equipment, the moving part of the mixer is replaced by the static. Two-fluid streams are introduced from the top of mixer and biphasic mixture is taken out from the top central line, which is further introduced to the mini-settler where the phases disengage based on their density

269

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7 Microstructured Reactors for Fluid–Fluid Reactions

α β

(d) (g) G

(a) (e)

(b)

L L

G

L

L

L L

(h)

(f) (i)

(k)

(c) Micromixer

(j) Microchannels

Figure 7.2 Schematic representation of fluid–fluid microstructured reactors: (a) micromixer settler, (b) cyclone mixer, (c) interdigital mixer, microchannels with (d) inlet Y, (e) T-shaped inlet, (f ) concentric inlet, (g) microchannel with partial overlap, (h) microchannel with membrane or metal

Microstructured film reactor (gas-liquid)

contactor, (i) microchannel with static internals bed, (j) parallel microchannels with internal redispersion units, (k) microstructured falling film reactor. (Adapted with permission from Ref. [1]. Copyright (2009) American Chemical Society.)

difference [4]. The advantage of mixer-settler over channel reactors is that drop size and specific interfacial area can be changed over a wide range in a given reactor. While there is an increase in the inlet flow velocity, the drop size decreases and thus specific interfacial area increases. The maximum values of specific interfacial area are reached within less than 1 s, being up to fivefold higher than other microstructured reactors [5]. It is also possible to use arrays of multiple static elements in the mixer to extend the throughput. However, similar to conventional contactor it is difficult to control the bubble/drop sizes precisely and thus the interfacial area. In the cyclone mixer (Figure 7.2b), two phases are dosed through two different nozzles. The bubble size can be influenced by the arrangement of gas and liquid injection nozzles (either parallel or vertical) [6]. The spiral patterns of the gas bubbles similar to cyclone vortex are formed in the liquid. Another type of mixer – interdigital mixer (Figure 7.2c) – induces the immiscible fluid streams to merge with or without prior splitting of the different phases into finer substreams. The reaction channel downstream of the mixing section is of sufficiently large diameter so that the small bubbles generated in the mixing section flow in the form of foam or emulsion [3].

7.2 Microstructured Devices for Fluid–Fluid Systems

7.2.2 Microchannels

Microchannels use mainly Y-shaped, T-shaped or concentric contacting geometry (Figure 7.2d–f ), for a two-phase contact, which further flows through a channel with or without structured internals. These microchannels can be divided into various types such as microchannels with partial overlapping (Figure 7.2g), microchannels with mesh contactors (Figure 7.2h) (porous membrane, sieve-like structure, etc.), microchannels with static mixer (Figure 7.2i), and multichannel contactors with intermediate redispersion units (Figure 7.2j). 7.2.2.1 Microchannels with Inlet T, Y, and Concentric Contactor

In this case the contacting of two fluids is restricted to uniting only the fluid streams using Y, T- types of junctions, or concentric inlets. These contactors are used as a laboratory tool for the application requiring precise definition of flow regime and specific interfacial area for mass transfer. Depending on the flow mixer geometry, physical properties of fluids, and operating conditions, different flow regimes are observed. The flow regime and mass transfer performance of such single channel reactors are discussed in detail in the next section. 7.2.2.2 Microchannels with Partial Two-Fluid Contact

In this type of channel, anodically bonded silicon/glass plates, each carrying a single channel with rectangular and semicircular cross section, are fitted in order to form partially overlapping channels (see Figure 7.2g). The advantage of this microstructured reactor (MSR) is that the contact between two fluids can be adjusted depending on the application. Partially overlapping channels MSR were developed for liquid–liquid extraction by Central Research Laboratory (CRL), UK [7]. The concept was tested for large throughput and 120 identical contactors were operated in parallel in one device [8]. Out of this work a platform for the use of microstructured contactor for liquid–liquid extraction was created. 7.2.2.3 Microchannels with Mesh or Sieve-Like Interfacial Support Contactors

Similarly to partially overlapping channels, microchannels with mesh contactors (Figure 7.2h) are used to create the partial contact of fluids. The advantage of these contactors is that both modes of operation, cocurrent and countercurrent, can be applied. Besides, the flow is stabilized because of the solid support between two fluids. The solid contactors are porous membrane [9, 10] and metal sheets with sieve-like structure [11]. Similarly to parallel flow, the mass transfer in both cases is only by diffusion and the flow is under laminar flow regime dominated by capillary forces. The membrane contactor has the advantage of being flexible with respect to the ratio of two fluids. In addition to flow velocities, the mass transfer is a function of membrane porosity and thickness. In another type of microextractor, two microchannels are separated by a sieve-like wall architecture to achieve the separation of two continuous phases. However, the hydrodynamics in both types of contactors is more complex because of interfacial support and bursting of fluid

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from one channel to the other at higher pressure drop in case countercurrent flow limits their applications. 7.2.2.4 Microchannels with Static Mixers

In this case, contacting of two fluids is achieved either by uniting, dividing and recombining fluid streams or using static internals in the microchannels (Figure 7.2i). A large number of small reaction volumes can be handled serially in a flow channel known as digitalization [12]. Different T-shaped and Y-shaped contactors can be used to split the flow stream into two equal flow streams of biphasic mixture and change the size of the drops. In addition, by changing the diameter of the channel and creating resistance to the flow, multiple drops can be united and the interfacial area can be decreased. An example for microchannels with structured internals are caterpillar micromixer, where the surfaces consist of ramp-like structures, moving the fluid constantly up and down and the fluid contact is achieved by a sequence of repeated splitting-reshaping-recombination processes [5]. Sometimes simple foam stacks are used as static mixer for dispersing two immiscible phases. 7.2.2.5 Parallel Microchannels with Internal Redispersion Units

In this case, the redispersion units are placed along the length of the parallel microchannels and are made of a metal sheet having multiple channels Figure 7.2j or with metal foam. The aim is to provide continuously large surface area by repetitive formation and breaking of the drops [13]. 7.2.3 Microstructured Falling Film Reactor for Gas–Liquid Reactions

The falling film MSR is one of the most commonly used devices for gas–liquid reactions (examples are given in gas–liquid reactions section). The liquid flows downward because of gravity in the form of film and gas flows through the open space that lies in the top cover of the housing. The falling film contactor consists in general of a stainless steel plate with open channels, typically 300 μm deep, separated by about 100 μm thick walls. The role of open microchannels is to prevent the breakup of the liquid film. Because of capillary force and small channels’ width, surface wetting liquids are pulled along the walls, thus forming a flowing meniscus. With increasing flow rate, the thickness of the film increases and its surface becomes flatter. The specific gas–liquid interfacial area can attain up to 20 000 m2 m−3 , which is 2–3 orders of magnitude larger than conventional bubble columns and agitated vessels (200 m2 m−3 ). The main drawback of the microstructured falling film reactor is the short residence time of the liquid in the channels, which typically varies between 5 and 20 s, depending on physical properties of the liquid and the operating conditions. The residence time can be increased by lengthening the microchannel or by decreasing the angle of descent, which can be achieved by helicoidal microchannel falling

7.3

Flow Patterns in Fluid–Fluid Systems

film reactor. The residence time could be increased by a factor of about 50 in a microchannel with an angle of descent decreased from 90∘ to 7.5∘ [14]. 7.3 Flow Patterns in Fluid–Fluid Systems 7.3.1 Gas–Liquid Flow Patterns

Depending on reactor geometry and operating conditions, different flow regimes such as bubbly flow, Taylor flow, slug bubbly flow, slug annular, churn flow, and annular flow are observed in microstructured reactors [15–17]. On the basis of the forces acting on the gas–liquid flows, these flow regimes can be classified as surface tension dominated and transition and inertia dominated as shown in Figure 7.3. The main problem for controlling the flow pattern is its dependence on many experimental parameters such as flow velocity, flow ratio of phases, fluid properties, channel geometry, microchannel material, wall roughness, pressure, and temperature. All these parameters influence the relative importance of the different forces. Different flow regimes in gas–liquid flows in microstructured reactors are discussed in the following section. 7.3.1.1 Bubbly Flow

Bubbly flow in microstructured reactors occurs in different types of geometries as shown in Figure 7.4. It is characterized by the bubbles with diameters less than or equal to the capillary diameter. In a microchannel, this flow pattern typically occurs at relatively high liquid velocities and low gas velocities [16]. For a given set of operating conditions, the gas inlet channel of the two fluids mixing element decides the bubble size. This limitation is circumvented by flow focusing geometries that consist of a gas-feeding nozzle positioned upstream of an orifice (50–200 μm) through which a liquid stream is forced [18, 19]. For a gas dispersion smaller than 20%, the bubble size is as small as 10 μm and is always smaller than the orifice (ratio of bubble size to orifice ≈ 0.1–0.6). The gas dispersion is further increased using multilamination mixer in which both gas and liquid inlet channels are increased and fed into one outlet channel [22, 23]. If the maldistribution is avoided in inlet channel and geometry is optimized, a bubble size of 30–50 μm with a variation of 10% can be obtained. The flow behavior varies from plug flow to partially backmixed flow depending on the width of the channel and flow velocity. Backmixing increases with increased total flow rate and gas–liquid ratio and is affected by the reactor tube orientation with vertical tubes showing up to five times higher backmixing compared to the horizontal tube [24]. The bubbly flow in a single channel at low gas hold-up (Figure 7.4a) shows well-defined specific interfacial area, which is, however, a way below to other flow regimes. At high gas hold-up, the small size of the bubble provides very high

273

7 Microstructured Reactors for Fluid–Fluid Reactions uL Bubbly Slug-bubbly Churn

Parallel/Annular

Transition

Inertia dominated

Taylor

Surface tension dominated

(a)

uG

10

BUBBLY

1

uL (m s–1)

274

CHURN TAYLOR Bubbly

Taylor

0.1

0.01 0.01

Churn

ANNULAR Annular

Horizontal universal transition lines Vertical universal transition lines

0.1

(b)

1

10

100

uG (m s–1)

Figure 7.3 Classification of flow patterns. (a) General classification and (b) comparison of flow pattern transition lines in horizontal and vertical channels – a universal map for gas–liquid systems (dh = 0.1–1 mm) [17].

Regimes in capital are for horizontal flow map, while regimes in lower case are for vertical flow map. (Adapted with permission from Elsevier.)

specific interfacial area (Figure 7.4b–d). However, nonuniform drop size, flow instability, backmixing, and difficulties in the characterization of performance parameters limit its use. 7.3.1.2 Taylor Flow

This flow regime is the most commonly observed in microchannels and is also referred to as slug and train flow. As mentioned before, the bubble size is restricted by channel dimensions. With smaller channels and relatively low liquid velocity, elongated cylindrical bubbles longer than the channel diameter are formed because of pressure squeezing mechanism in the surface tension dominated region. In some cases this flow is further introduced in a wider channel, referred to as delay tube, to form bubbly flow with higher gas hold-up of up to 90% as shown in Figure 7.4b–d [20].

7.3

Flow Patterns in Fluid–Fluid Systems

εL = 0.91

εL = 0.57

εL = 0.09

(a)

Air-water

Air-ethanol

Air-oil

(b)

Liquid Gas Liquid 60 ml h–1 nitrogen, 30 ml h–1 glycerol

Schematic diagram Liquid

120 ml h–1 nitrogen, 30 ml h–1 glycerol

Gas Liquid

Optical micrograph (c)

500 μm

(d)

Figure 7.4 Bubbly flow in different types of devices. (a) Vertical glass capillary. (Adapted with permission from Ref. [16]. Copyright (2005) American Chemical Society.) (b) Cross flow mixing element for foam flow

180 ml h–1 nitrogen, 30 ml h–1 glycerol

generation. (Adapted from Ref. [20].) (c) Flow focusing device [21]. (Adapted with permission from Nature Publishing Group.) (d) Foam formation in slit-shaped glass mixer [22]. (Adapted with permission from Elsevier.)

Five steps of Taylor flow generation, adapted from liquid–liquid systems [25], are briefed (Figure 7.5): (i) Initially the dispersed phase penetrates the main stream and the bubble begins to grow. (ii) As the tip of the dispersed phase grows, it blocks almost the whole cross section of the main channel, which in turn builds pressure upstream. The radius of the tip curvature is limited by the dimensions of the channel. The continuous phase flows in the gap between the wall and the dispersed phase drop with higher flow velocity. (iii) Because of the shear exerted by the continuous fluid and the pressure drop along the channel length, the bubble elongates and grows downstream. The interface approaches the downstream inlet of the dispersed phase and a slug is formed in the main channel.

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7 Microstructured Reactors for Fluid–Fluid Reactions

Stage I

Stage II

Stage III

Stage IV

Stage V Figure 7.5 Five steps of Taylor (slug) flow generation. Dispersed gas phase is introduced through side channel while wetting continuous phase is introduced through main channel.

(iv) The neck of the immiscible liquid gets squeezed further, that is, the radial curvature dominates the axial and the slug separates from the dispersed phase stream. (v) The processes of bubble penetration and separation from the dispersed phase continue and produce a well-defined slug flow. Depending on flow rates and fluid properties, the bubbles often have hemispherical shaped tops and flattened tails [16]. The wide applications of Taylor flow for reactions and separations come from its stability and ability to provide well-defined high specific interfacial area. The recirculation within the liquid slugs improves heat and mass transfer from liquid to wall and interfacial mass transfer from gas to liquid [26]. It reduces axial dispersion and enhances radial mixing. The radial mixing can further be enhanced using meandering channels as shown in Figure 7.6c [27]. In Taylor flow, liquid phase forms a thin wall film providing lubricating action to the enclosed bubbles and, as a consequence, bubbles flow at relatively higher velocity than the liquid. The interfacial gas–liquid area thus comprises two parts: the lateral part (that of the wall film) and the perpendicular part (between the bubble and the adjacent liquid slug). Often the length of the bubble is many times greater than the channel diameter resulting in higher contribution of lateral part than the perpendicular part. The thickness of the wall film in Taylor flow in capillaries is mainly dependent on the ratio of viscous to interfacial forces, which is given by the capillary number, Ca. � ⋅u (7.1) Cai = i b ; with ub , the bubble velocity � Ca is mostly referred to the continuous phase (i = C); this corresponds to the liquid phase (i = L) in gas–liquid systems. Different correlations can be found in the literature for estimating the thickness of the wall film. The majority of the studies suggest that the film thickness (� film ) is a function of capillary diameter (dt ) and capillary number (Ca). Two correlations, Bretherton [28] and Aussillous and Quere [29], are widely used.

7.3

Gas bubble

Lslug /2

Flow Patterns in Fluid–Fluid Systems

Static film

Dynamic liquid

Lslug /2

Lb LUC

(b)

(a)

Gas

(c)

Gas

Gas

Gas

Straight channel

Meandering channel

Figure 7.6 Experimental snapshots and schematic presentation of Taylor flow in different configurations. (a) Taylor flow in vertical capillary. (b) Schematic presentation of Taylor flow in horizontal capillary (Lb – length of bubble and LUC – unit slug length). (Adapted with permission from

Ref. [16]. Copyright (2005) American Chemical Society.) (c) Comparison between flow behavior in the liquids slugs of Taylor flow in straight and meandering channel. (Adapted from Ref. [27] with permission of The Royal Society of Chemistry.)

Bretherton [28]: �film = 0.67 dt CaL3 for CaL ≤ 3 ⋅ 10−3 2

(7.2)

Aussillous and Quere [29]: �film = dt

2∕ 0.67CaL 3 2∕ 1 + 3.35CaL 3

for CaL < 1

(7.3)

It is important to note that the definition of Ca is based on bubble velocity (ub ). The bubble velocity in vertical capillaries was found to depend on the two-phase superficial velocity u and the capillary number (CaL,u ), which is calculated with the two-phase superficial velocity. On the basis of the experimental results obtained with different capillary diameters and liquids, Liu et al. [16] proposed the following relationship: ub =

� u u ; with CaL,u = L ; u = uG + uL 0.33 � 1 − 0.61 ⋅ CaL,u

(7.4)

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7 Microstructured Reactors for Fluid–Fluid Reactions

uG , uL are the superficial velocities of the gas and the liquid, respectively. The bubble velocity is related to the superficial gas velocity by: ub =

uG ; �G is the volumetric fraction of the gas phase �G

(7.5)

The bubble velocity and film thickness is estimated in Example 7.1.

Example 7.1: Bubble velocity and wall film thickness Estimate the capillary number for air–water system and wall film thickness for two-phase superficial velocity between 0.05 ≤ u ≤ 1.2 m s−1 and uG ∕uL = 1. The capillary diameter is dt = 1.0 mm. The physical properties are summarized in Table 7.2. Table 7.2 Physical properties of the air/water system. Density, � (kg m−3 )

Air Water

Viscosity, � (Pa s)

Interfacial tension, 𝝈 (N m−1 )

1.78 × 10−5 10−3

— 0.072

1.2 998.2

Capillary number, Ca (–) 10–4

50 Wall film thickness, δfilm (μm)

1.4 1.2 uG, ub (m s–1)

278

1.0

Bubble velocity, ub

0.8 0.6 0.4 0.2 0.0 0.0

(a)

Superficial gas velocity, uG

0.2

0.4

0.6

0.8

1.0

Two phase superficial velocity, u (m s–1)

(b)

10–2

40 30 Aussillous and Quere (2000)

20 10 Breherton (1961)

0 0.001

1.2

10–3

0.01 0.1 Bubble velocity, ub (m s–1)

1

Figure 7.7 (a) Estimated bubble velocity versus two-phase superficial velocity; (b) liquid wall film thickness versus bubble velocity and capillary number.

Solution:

Using Equation 7.4, the bubble velocity can be estimated. In Figure 7.7 the bubble velocity and the superficial gas velocity is plotted as the function of the two-phase superficial velocity u. It is evident that the bubble velocity is much higher than the calculated gas velocity and exceeds even the two-phase superficial velocity. Figure 7.7b shows the estimated thickness of the liquid wall film � film based on Equations 7.2 and 7.3. For low capillary number

7.3

Flow Patterns in Fluid–Fluid Systems

(Ca ≤ 3 ⋅ 10−3 ) both relations result in the same values. Aussillous and Quere corrected Bretherton’s relation to fit experimental results obtained for higher capillary numbers. For noncircular channels, the general practice is to use the hydraulic diameter dh in the above equation though the situation is more complex. A long bubble in a circular tube acts as a tight-fit piston, while in a polygonal tube, particularly at higher flow velocities, it behaves like a leaky piston. The fluid prefers to bypass the bubble through the corners because of the large drag of the bubble. The corner flow could be an order faster than the bubble that loses symmetry with the microchannel axis. This transition in square channel occurs at CaL ≅ 0.04 [30].

7.3.1.3 Slug Bubbly Flow

The slug bubbly flow is a transition regime that occurs between bubbly and Taylor flows. In most of the literature, this regime is considered as a part of Taylor flow. However, it differs from Taylor flow because of the presence of small bubbles in the continuous liquid phase as shown in Figure 7.8a. Small and elongated bubbles are separated from each other by liquid slugs. This regime is relatively unstable and the transition from Taylor flow to slug bubbly flow occurs by increasing the liquid flow rate at constant gas flow rate. 7.3.1.4 Churn Flow

Churn flow occurs at very high gas velocities. It consists of very long gas bubbles and relatively small liquid slugs as shown in Figure 7.8b. In churn flow, because of high gas velocity, a wave or ripple motion is often observed at the bubble tail with tiny gas bubbles entrained in the liquid slug [16]. In a microchannel, two types of churn flow are observed: either showing streaks and swirls that trail the gas slug or interfacial structures resembling a serpentine-like gas core moving through the

(a)

(b)

(c)

Figure 7.8 Flow regime at elevated gas and liquid velocities in vertical capillary. (a) Slug-bubbly flow, (b) churn flow, (c) annular flow. (Adapted with permission from Ref. [16]. Copyright (2005) American Chemical Society.)

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Table 7.3 Difference between slug and parallel flow in fluid-fluid MSR. Slug or Taylor flow

Parallel flow

Slug flow is a flow of a series of slugs (plugs) of one phase separated by the other Each slug serves as an individual processing subvolume. In liquid–liquid systems, one phase acts as continuous while other discrete Mass transport is because of convection within each slug and diffusion between two adjacent slugs Relatively higher interfacial area, which can be changed in a given reactor by varying the flow rates Intensity of internal circulations increases with flow and thus diffusive penetration between two phases Downstream separation is difficult

Parallel flow is a flow of two parallel streams Both phases are continuous

Mass transfer is because of diffusion only

Relatively low interfacial area and is constant in a given microchannel No effect of flow velocity on diffusive penetration Downstream separation is relatively easy

liquid [31, 32]. With increased flow velocity, it leads to the merging of the bubbles that is often referred to as slug annular flow [32]. 7.3.1.5 Annular and Parallel Flow

In a microchannel, at excessively high gas velocity and very low liquid velocity, two flow regimes are observed: annular and parallel flow. The former is produced in a flow symmetric contacting (concentric) geometry while the latter is formed in flow asymmetric geometries (T or Y type). In an annular flow, a continuous gas phase is present in the central core with the liquid phase being displaced to form an annulus between the wall and the gas phase [16] (Figure 7.8c), while in parallel flow, both phases flow parallel. The comparison of annular (or parallel) flow with Taylor flow, another stable regime, is presented in Table 7.3. The additional advantages of slug flow over parallel flow allow it to use for wide range of applications. There are some case-specific applications where annular flow is used. Falling film reactor is one of the examples. In this case, the liquid flows downward because of gravity in the form of film and gas flows through the open space, which lies in the top cover of the housing. 7.3.2 Liquid–Liquid Flow Patterns

The flow regimes observed in liquid–liquid flow in microchannels such as drop, slug, slug-drop, deformed interface, annular, parallel, and dispersed flow are depicted in Figure 7.9. The different flow regimes are presented and discussed in the following sections.

7.3

Flow Patterns in Fluid–Fluid Systems

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 7.9 Flow regimes observed for liquid–liquid systems in microstructured reactors. (a) Drop flow in a Y-junction capillary microchannel, (b) Slug flow in a concentric microchannel, (c) Slugdrop flow in a concentric microchannel, (d) Deformed interface flow in a concentric

microchannel, (e) Annular flow in a concentric microchannel, (f ) Parallel flow in a wedge-shaped microchannel, (g) Slugdispersed flow in a caterpillar mixer attached capillary, (h) Dispersed flow in a caterpillar mixer (u = 3.2 m s−1 ) attached capillary [33]. (Adapted with permission from Elsevier.)

7.3.2.1 Drop Flow

This flow is identical to the bubbly flow of gas–liquid systems. It is characterized by the drops with diameters less than or equal to the microchannel diameter. In a microchannel, this flow pattern typically occurs at relatively high continuous flow velocities and low dispersed phase velocities. The drop size is restricted by channel dimensions. By varying the microchannel dimensions, the flow regimes can be changed from drop flow to slug flow and vice versa [12]. 7.3.2.2 Slug Flow

In this flow regime, one liquid flows as a continuous phase while the other flows in the form of slugs (droplets) longer than the diameter of the microchannel. In this case, the surface tension between one of the liquids and the wall material is higher than the interfacial tension between two liquids. The high surface tension phase flows in the form of enclosed slugs while the other phase flows as a continuous phase forming a thin wall film of a few micrometers in size. If the flow does not satisfy this condition, then both liquids flow alternatively without forming a wall

281

282

7 Microstructured Reactors for Fluid–Fluid Reactions Internal circulations

Stagnant zone

(a)

Internal circulations

(b)

Figure 7.10 Schematics of internal circulations within the slug. (a) Without surfactant and (b) with surfactant added to one of the liquids.

film. In both the cases, the surface tension at the liquid–liquid interface must be sufficiently high in order to avoid the destruction of slugs by the shear [34]. This flow pattern occurs at relatively low and approximately equal flow rates of both liquids where interfacial forces dominate. This is a commonly observed stable flow regimes in microchannels [35–37]. As both phases move alternatively, each slug of the dispersed phase serves as an individual processing subvolume, which is highly regular (see Figure 7.9b) and guarantees well-defined interfacial area for mass transfer. A key feature of this type of flow is the ability to manipulate the two principal transport mechanisms: convection within the individual slug of each liquid phase (Figure 7.10a) and interfacial diffusion between adjacent slugs of different phases. The shear between capillary wall and dispersed slug axis generates intense internal circulations within the slug, which in turn reduces the thickness of interfacial boundary layer and thereby augments the diffusive penetration. The mass transfer behavior depends on the slug geometry and circulation patterns, which vary with the physical properties of liquids as well as with operating parameters such as flow rates, mixing element geometry, and the capillary dimensions. These internal circulations can be hindered by adding surface active agents, for example, cationic surfactant, that accelerate the movement of the interface resulting in local convection pattern. Thus, the mass transfer performance can be increased. 7.3.2.3 Slug-Drop Flow

In this regime, the dispersed phase flows in the forms of irregular slugs and drops. This flow occurs during the transition to or from slug flow: either at low volumetric flow rate of the dispersed phase compared to the continuous phase, or at the flow rate higher than slug flow. 7.3.2.4 Deformed Interface Flow

This flow regime is also referred to as intermittent or irregular flow. At relatively high flow velocity, the dispersed phase flows to a certain distance in the form of either parallel or annular flow and then produces irregular droplets. The length of the parallel flow region increases with flow velocity. If the volumetric flow rate of the dispersed phase is higher as compared to the continuous phase, then the slugs of the former flow close to each other. The deformation of hemispherical caps of the slug becomes more pronounced and it tends to form bridges between

7.3

Flow Patterns in Fluid–Fluid Systems

adjacent slugs, which may lead to the formation of larger slugs by coalescence. This regime is less stable and acts as a transition between the slug-drop flow regime and the parallel flow. 7.3.2.5 Annular and Parallel Flow

As in gas–liquid systems, this flow is formed based on the type of microchannel geometry used: flow symmetric geometry forms annular flow, while flow asymmetric forms parallel flow. This flow regime is observed at elevated flow rates in the microchannel without static internals – the higher the flow velocity, the better the stability. The shear force of the continuous phase is dominant over the surface tension force and, therefore, the dispersed phase flows straight forming annular or parallel flow. The parallel flow can also be formed at low flow velocity, especially in rectangular cross section microchannel, if the interfacial tension between two fluids is very low, for example, wedge-shaped parallel flow contactor [38], or modifying the wettability of the channel walls (see Figure 7.28). The flow could be stabilized by placing membranes or sieve plates inside the channels [9, 39]. 7.3.2.6 Slug-Dispersed Flow

To create fine dispersion in microcapillaries, a micromixer (e.g., caterpillar mixer) needs to be attached upstream. At elevated flow velocity the static internals create dispersion, and as a result, part of the continuous phase flows in the form of small droplets in the dispersed phase. 7.3.2.7 Dispersed Flow

This flow regime is observed when the flow velocity is further increased in the microchannel with structured internals. Very fine droplets of one phase into the other are created. The flow regime transition in liquid–liquid flow could be explained by applying the dimensionless numbers. The flow patterns in liquid–liquid systems depend on the volume fraction of dispersed phase (�D ) and hydraulic diameter of the microchannel in addition to the dispersed phase Reynolds number uD dh (7.6) ReD = �D Here uD is the superficial velocity of the dispersed phase. The investigation of volume fraction of dispersed phase in the microchannel is not trivial. Equating volume fraction (εD ) to volumetric flow fraction �D = V̇ D ∕(V̇ D + V̇ C ), a new group, ReD dh /�D , is introduced to characterize the flow pattern of liquid–liquid systems in capillaries [40]. The following criteria were obtained for the toluene/water system: Surface tension dominated (slug flow) ∶ ReD ⋅ dh ∕�D < 0.1 (m) Transition (slug-drop, deformed interface flow) ∶ 0.1(m) < �eD ⋅ dh ∕�D < 0.35(m) Inertia dominated region (annular, parallel flow) ∶ ReD ⋅ dh ∕�D > 0.35 (m)

(7.7)

283

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7 Microstructured Reactors for Fluid–Fluid Reactions

As the criterion possesses dimension, it is advised to use SI units. It is important to note that the above criterion is valid for microchannels without structured internals.

Example 7.2: Flow pattern of a liquid-liquid system Investigate the proportion of two fluids in a microchannel with circular cross section to operate under slug flow regime for extraction of acetone from continuous water to dispersed toluene with V̇ D = 2 ml min−1 . Data: �toluene = �D = 867 kg m−3 , �toluene = �D = 0.6 ⋅ 10−3 kg ms−1 , �water = �C = 998.2 kg m−3 , �water = �C = 1 ⋅ 10−3 kg ms−1 , � = 0.036 N m−1 . Solution: According to the criteria proposed in Equation 7.7, the stable slug flow regimes is observed at ReD ⋅ dh /εD < 0.1 m. Thus, equating the term to 0.1, one can estimate the diameter of a microchannel. Thus, �D uD dh d = 0.1 m �D �D h As the velocity and dimensions of microchannel are not given, they can be written in terms of flow rate as V̇D = �dh2 uD ∕4 or uD dh2 = 4V̇ D ∕� resulting in �D 4V̇ D = 0.1 m �D �D � −6

⋅ 867 4 ⋅ 2⋅10 4 ⋅ V̇ D ⋅ �D 60 ⇒ �D = = = 0.61 0.1 ⋅ � ⋅ �D 0.1 ⋅ � ⋅ 0.6 ⋅ 10−3 Thus, volumetric flow fraction of toluene is 0.61 resulting in water flow fraction of 0.39, which is necessary to obtain slug flow behavior.

7.4 Mass Transfer

For the applications involving multiphase reactions and separations, the mass transfer of a solute from one phase to the other or of a pure phase into another is necessary. The mass transfer rates are different in nonreactive and reactive chemical systems. In nonreactive (separation/extraction) case, the mass is transferred from the phase with higher chemical potential (partial pressure or concentration) to the lower until the equilibrium is reached. In reactive systems, the mass transfer is enhanced because of the consumption of transferring species from one phase to the other.

7.4

Mass Transfer

7.4.1 Mass Transfer Models

Different approaches have been used to model mass transfer performance of reactors. They comprise two main parts: the micromodel, describing the mass transfer between two phases, and the macromodel, describing the mixing pattern within the individual phase. The micromodels assume two types of interfacial behavior: stagnant films or dynamic absorption in small elements at the contact surface. Let us consider the gas–liquid mass transfer. In the stagnant film model, it is postulated that mass transfer proceeds via steady-state molecular diffusion in a hypothetical stagnant film at the interface with thickness � int while the bulk of the liquid is well mixed [41]. Though this model incorporates the important features of the real system and is simple to use, the prediction of hydrodynamic parameter � int is difficult as it depends on the contactor geometry, liquid agitation, and physical properties. The penetration model, proposed by Higbie [42], assumes that every element of surface is exposed to gas for the same time (� c ) before being replaced by a liquid of bulk composition. The exposure time (� c ) is investigated using hydrodynamic properties such as interface velocity and its length. Film and penetration models are most commonly used defining the liquid-side mass transfer coefficients (kL ) as follows: √ Penetration model ∶ kL = 2 Film model ∶ kL =

Dm �int,II

Dm ��c (7.8)

where Dm is molecular diffusivity of solute in liquid phase and � int, II is hypothetical interfacial film thickness in the liquid phase. The application of suitable models to various systems must be determined on a case-by-case basis. This could be judged from the behavior of experimental mass transfer coefficient with respect to the contact time of two phases. For dynamic systems, the penetration model is physically more realistic than the stagnant film model. However, the mixing in different phases is important to describe the overall mass transfer performance, and, therefore, the above models are usually combined with fluid flow models, which includes detailed flow description. Further, a film-penetration model is also used to include resistance on both sides, yielding a two-parameter model combining the stagnant film and penetration models, as shown in Figure 7.11b [43, 44]. Frequently applied micromodels assume the presence of a liquid bulk. However, some systems are without liquid bulk as in falling film reactors where a very thin layer of liquid flows over a solid surface.

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7 Microstructured Reactors for Fluid–Fluid Reactions

δint c1,I, p1,I

c1,I, p1,I

Pure

Pure Diluted

c∗int,II δint,I

(a)

Phase I ∗ cint,I , p∗int,I (film)

Phase II (film)

Phase I ∗ ∗ (film) cint,I, pint,I

Phase II (penetration) c1,II

Diluted

c1,II

t

δint,II x=0

c∗int,II

δint,I x=∞

x=0

(b)

Figure 7.11 Schematic representation of concentration profiles in gas–liquid and liquid–liquid systems assuming (a) two-film model and (b) combined film-penetration model. Here, phase I is gas for gas–liquid

x=∞

and liquid for liquid–liquid systems while phase II is always a liquid forming the continuous phase. The superscript * indicates the equilibrium values while subscript int indicates interface values.

7.4.2 Characterization of Mass Transfer in Fluid–Fluid Systems

When a solute transfers from phase I (gas phase) to phase II (liquid) with the phase equilibrium at the interface, the individual transfer rates at steady state can be written as ) ( ∗ pint,1,II ∗ JI = kG (p1,I − pint,1,I ) = JII = kL − c1,II (7.9) H where p1, I is partial pressure of a gaseous solute while kG and kL are the gas and liquid phase mass transfer coefficients, respectively. The ratio of partial pressure of a solute in phase I to its corresponding bulk concentration in phase II at equilibrium is called as Henry’s constant (H). Equating both fluxes (JI = JII = J) and deriving the overall mass transfer coefficient (k ov ), Equation 7.9 can be written as ( ( ) ) p1,I p1,I HkG kL J= − c1,II = kov − c1,II HkG + kL H H HkG kL 1 = 1 (7.10) where, kov = HkG + kL + 1 HkG

kL

The mass transfer performance of any reactor for a given system (reactor/contactor and solute) is characterized by overall volumetric mass transfer coefficient (k ov a). A microstructured reactor is an open system and its performance can be compared to ideal plug flow or backmixed flow reactor. Generally, continuous flow microstructured reactors are considered as ideal plug flow reactors with narrow residence time distribution (see Chapter 3). In the case of slug or Taylor flow, both fluids flowing through microchannels exchange mass between the same compartments (e.g., between same gas and liquid slugs), and, therefore, one driving force

7.4

Mass Transfer

is required to define the mass transfer rate. For a solute transferring from phase I (gas) to II (liquid) and assuming that the resistance lies in the liquid phase (phaseII), the change of its concentration in the liquid with respect to residence time (�) can be written as: dc1,II d�

= JII a = kL a(c∗1,II − c1,II ); with kL ≅ kov

(7.11)

where c∗1,II is the equilibrium concentration in phase II corresponding to bulk concentration in phase I. On integration of the above equation from initial time (� = 0) ) to outlet (cout ), the overall to residence time (�) and concentration from inlet (cin 1,II 1,II volumetric mass transfer coefficient (k ov a) becomes ) ( ∗ c1,II − cin 1,II 1 (7.12) kov a = ln � c∗1,II − cout 1,II where a is the specific interfacial area, defined as the interfacial area per unit volume of the dispersed phase. 7.4.3 Mass Transfer in Gas–Liquid Microstructured Devices

Mass transfer takes place from the gas phase to the liquid phase as well as in the reverse direction and chemical reactions may occur in the gas and/or in the liquid phase, respectively. The mass transfer performance of any gas–liquid reactor depends on two-phase flow patterns that define the interfacial area. The mass transfer performance of gas–liquid microstructured reactors has been investigated under different flow regimes, which are discussed in the following section. 7.4.3.1 Mass Transfer in Taylor Flow

The experimental works on investigation of gas–liquid mass transfer in microstructured reactors are listed in Table 7.4. Most of the experimental results were obtained for Taylor flow in capillaries with diameters between 1 and 3 mm. The influence of experimental conditions on mass transfer is very complex. This explains why most of the published relations describing gas–liquid mass transfer are empirical. A relatively simple model to describe the gas–liquid mass transfer in circular channels with slug flow pattern was proposed by van Baten and Krishna [47]. For their fundamental model the authors considered an idealized geometry of the Taylor bubbles as shown in Figure 7.12. The bubbles consist of two hemispherical caps and a cylindrical body. The Higbie penetration model was applied to describe the mass transfer process of a compound from the gas phase to the liquid (Equation 7.8). For a rising bubble, the liquid will flow along the bubble surface of the cap. The average distance

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Table 7.4 Literature on gas–liquid mass transfer in Taylor flow. Regime and system

Conditions and definition

Global volumetric mass transfer coefficient

Irandoust et al. [45] Slug(Taylor) flow – vertical reactor Absorption of oxygen from air into water, ethanol, and ethylene glycol (EG)

k ov a = 0.01–0.29 s−1 dt = 1.5, 2 mm �(d −� )uy +D Sh d L = 0.6 m kov a = 4 h film 2 m m L b ym = dt LUC u = 0.092–0.56m s−1

cm −cin 2 c∗2 −cin 2

(7.13)

Bercic and Pintar [26] Slug(Taylor) flow – vertical reactor Nonreacting system Methane–water

dt = 1.5, 2.5, 3.1 mm L = 1.12 m u = 0.02–0.43 m s−1

k ov a = 0.005–0.115 s−1 kov a =

0.111 (uL +uG )1.19 ((1−�G )LUC )0.57

(7.14)

Vandu et al. [46] Slug flow – vertical reactor Nonreacting system Air–water

dt = 1, 2, 3 mm L = 0.2–1.4 m u = 0.22–0.43 m s−1

k ov a = 0.08–0.47 s−1 √ Dm uG 1 kov a = 4.1 L d

dh = 667 μm uG = 0–2 m s−1 uL = 0.09–1 m s−1

k ov a = 0.3–21 s−1 Slug flow: ShL adh = 0.084Re0.213 Re0.937 Sc0.5 L L G

UC

(7.15)

t

Yue et al. [32] Slug flow, slug annular, churn flow Reacting system CO2 /buffer solution of 0.3 M NaHCO3 , 0.3 M Na2 CO3 , NaOH

ShL = G or L

kL d h ; Rei Dm

=

�i ui dh ; ScL �i

=

(7.16)

�L ;i �L Dm

=

traveled by the liquid packets will be one-half of the bubble circumference: lc = � ⋅ db /2. As the thickness of the wall film is small compared to the capillary diameter (see Figure 7.7), it can be neglected and we obtain lc = � ⋅ dt /2. The average contact time of the liquid with the bubble cap will be: �c,cap =

� ⋅ dt ; with ub the bubble velocity 2 ⋅ ub

(7.17)

The penetration model for mass transfer yields: √ kL,cap = 2

√ √ Dm ⋅ ub ⋅ 2 2 2 Dm ⋅ ub = � dt � 2 ⋅ dt

(7.18)

7.4

Mass Transfer

δfilm

db Lslug

Luc

Lfilm

db/2 dt Figure 7.12 Taylor bubble (schematic).

The volumetric mass transfer of the bubble caps is obtained by multiplying the mass transfer coefficient with the specific surface of the two caps referred to the volume of a cell unit V UC . acap =

2 Acap VUC

kL,cap acap

=

2 �db2 ∕2 �∕4 ⋅ db2 ⋅ LUC

=

4 LUC

√ √ 2 2 Dm ⋅ ub 4 ⋅ = � dt LUC

(7.19)

(7.20)

Accordingly, the penetration model can be used to predict the film mass transfer coefficient. √ Dm 2 (7.21) kL,film = √ � � c,film The specific interfacial area of the film afilm can be approximated with [46]: afilm =

4 ⋅ �G ; with �G the gas hold-up (volume fraction) dt

(7.22)

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7 Microstructured Reactors for Fluid–Fluid Reactions

In summary, the gas–liquid mass transfer in Taylor flow has two contributions (see Figure 7.6b): (i) the caps (assumed to be hemispherical) at both ends of the bubble and (ii) the liquid film surrounding the lateral sides of the bubble. Considering these two contributions and assuming resistance in the liquid phase, the relationship for the overall mass transfer coefficient (k ov a ≅ kL a) is given in the following: kL a = kL,cap acap + kL,film afilm √ √ 2Dm ub 4 Dm 4�G +2 kL a = 2 ��c,film dt � 2 dt LUC

(7.23)

The model provided excellent agreement with computational fluid dynamic simulations for capillaries with 1.5, 2, and 3 mm diameters and idealized bubble geometry as shown in Figure 7.12 [47]. The major contribution to mass transfer is in the film (k L, film afilm ) as long as the concentration of the solute is low in the liquid film. Under these conditions, the model (Equation 7.23) can be simplified taking only the film contribution into account as follows [46]: √ √ L Dm �G Dm ub �G 8 kL a ≅ √ = 4.5 ; �c,film = b (7.24) � d L d u � c,film t b t b On the basis of the experimental studies of Taylor flow in capillaries of Heiszwolf et al. [48], an empirical correlation for estimating liquid slug lengths (Lslug ) was proposed by Kreutzer [30] : �L Lslug = dt (7.25) ; with �L = (1 − �G ) −0.00141 − 1.55�2L ln(�L ) The bubble length and the length of the liquid slug can be approximated with [46]: (7.26)

Lb ≅ LUC ⋅ �G ; Lslug ≅ LUC ⋅ (1 − �G )

In Equation 7.24 we replace the bubble length with Lb ≅ LUC ⋅ �G and the bubble rise velocity with the superficial gas velocity (Equation 7.5) ub = uG ∕�G and obtain: √ √ Dm uG �G Dm uG 1 = 4.5 (7.27) kL a ≅ 4.5 LUC dt �2G LUC dt √

D u ∕L

m G UC A plot of experimental kL a values versus showed a straight line dt with a slop of 4.5, thus confirming the theoretical value from the model for the film contribution. The agreement between the model and the experiment is reasonably good for both circular and square capillaries showing dependency of k√ L a on capillary diameter. The validity of the model was found to be well

for ((uG + uL )∕Lslug ) > 3, which corresponds to a short film contact time and a dominant film contribution [46]. Below this range, the film contribution to mass transfer diminishes as the liquid in the film begins to approach saturation.

7.4

Mass Transfer

A simple criterion for the effectiveness of the interfacial film area was proposed by Pohorecki [49]. The criterion is based on the characteristic diffusion time of the species in the film t D, film and the film contact time � c, film . For physical absorption, the film can be considered as far from saturation, if �c, film ≪ t D, film . The criterium is summarized in Equation 7.28. �c,film tD,film

=

Lb Dm ⋅ 2 ≪1 ub �film

(7.28)

Estimation of mass transfer using equation 7.15 is demonstrated in Example 7.3.

Example 7.3: Mass transfer in gas-liquid MSR Estimate the volumetric mass transfer coefficient in a Taylor flow capillary with an internal diameter of 1 mm and a volumetric gas flow of V̇ G = 2.71 cm3 min−1 and a volumetric liquid flow of V̇ L = 2.0 cm3 min−1 . Use the simplified model presented in Equation 7.27 and compare the values with those predicted with the empirical model Equation 7.14. Use the physical properties for air and water presented in Example 7.1. The molecular diffusion coefficient in the liquid phase is approximated with Dm = 10−9 m2 s−1 . Solution: The superficial velocities of gas and liquid are given by ui = V̇ i ∕Acs with Acs = 7.85 ⋅ 10−7 m2 , the cross section area. Superficial velocities: u=

V̇ G +V̇ L Acs

=

4.52⋅10−8 +3.33⋅10−8 7.85⋅10−7

= 0.1 m s−1 uG = 0.0575 m s−1 ; uL = 0.0425 m s−1

Bubble rise velocity (Equation 7.4): � u 10−3 ⋅ 0.1 u = 1.39 ⋅ 10−3 ; with CaL,u = L = 0.33 � 0.072 1 − 0.61 ⋅ CaL,u 0.1 ub = = 0.107 m s−1 1 − 0.61 ⋅ (1.39 ⋅ 10−3 )0.33 ub =

Volumetric fraction of the gas phase (gas hold-up) Equations 7.5 and 7.26: u 0.0575 = 0.537 � G = G ; �G = ub 0.107 Slug length and length of a unit cell (Equation 7.25): �L Lslug = dt ; with �L = 1 − �G = 0.463 −0.00141 − 1.55�2L ln(�L ) Lslug =

10−3 ⋅ 0.463 = 1.82 ⋅ 10−3 m −0.00141 − 1.55 ⋅ 0.4632 ⋅ ln(0.463)

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7 Microstructured Reactors for Fluid–Fluid Reactions

LUC ≅

Lslug (1 − �G )

=

1.82 ⋅ 10−3 = 3.9 ⋅ 10−3 m 0.463

Volumetric mass transfer coefficient (Equation 7.27) [46]: √ √ Equation 7.27 is applicable if: ((uG + uL )∕Lslug ) > 3; (0.1∕(1.82 ⋅ 10−3 )) = 7.4 √ √ Dm uG 1 10−9 ⋅ 0.0575 1 = 4.5 = 0.546 s−1 kL a ≅ 4.5 LUC dt 3.9 ⋅ 10−3 10−3 Volumetric mass transfer coefficient estimated with the empirical equation (Equation 7.14) [26]: kov a =

0.111 (uL + uG )1.19 0.111 (0.1)1.19 = 0.262 = ((1 − �G )LUC )0.57 (0.463 ⋅ 3.9 ⋅ 10−3 )0.57

Remark: Comparing the estimation based on the widely used empirical Equations 7.14 and the semi-empirical Equation 7.27, the latter predicts a roughly two times higher volumetric mass transfer coefficient. This may be because of the fact that Equation 7.14 does not include the capillary diameter. In general, predictions must be taken with caution because the two-phase systems are complex and none of the models include all practical experimental conditions.

7.4.3.2 Mass Transfer in Slug Annular and Churn Flow Regime

The mass transfer performance under slug annular and churn flow regime is depicted in Table 7.5. From the reported values of mass transfer coefficients, it is observed that kL a is higher in churn flow compared to Taylor flow. However, the flow irregularity and low stability of these regimes limit their use for mass transfer.

Table 7.5 Operating conditions used for mass transfer under churn flow regime by Yue et al. [32]. Regime and system

Conditions and definition

Slug annular/churn flow Reacting system CO2 /buffer solution of 0.3 M NaHCO3 , 0.3 M Na2 CO3 ,NaOH

dh = 667 μm uG > 2 m s−1 uL > 0.5 m s−1

Global volumetric mass transfer coefficient

Physical absorption Slug annular and churn flow: 0.344 ShL adh = 0.058 ��G ��0.912 Sc0.5 L L k d

ShL = DL h ; ��i = m i = G or L

�i ui dh ; ScL �i

=

�L ; �L Dm

7.4

Mass Transfer

7.4.3.3 Mass Transfer in Microstructured Falling Film Reactors

In falling film contactors, a thin film is created by a liquid falling under the influence of gravity. The liquid flows over a solid support, which is normally a thin wall or a stack of pipes. In conventional falling film devices, a film with a thickness of 0.5–3 mm is generated. This rather thick liquid film results in a significant mass transfer resistance for gaseous reactants. Furthermore, the film flow becomes unstable at high throughputs, and it may break up into rivulets, fingers, or droplets. These problems can be overcome by microstructuring the solid wall. The microstructured falling film reactor consists of open microchannels, which are typically less than 1 mm wide, about 100 μm deep, and about 80 mm long. The channels are separated by 100 μm wide walls (Figure 7.13). Inflow and outflow of the liquid occur through boreholes that are connected via one large slit to numerous small orifices at the top of the channels. A structured heat-exchanger plate is inserted beneath the falling film plate for heat removal, and nearly isothermal operation can be achieved even for highly exothermic reactions such as the direct fluorination of organics with gaseous fluorine. The main drawback of the microstructured falling film reactor is the short residence time of the liquid in the channels, which typically varies from 5 to 20 s, depending on the physical properties of the liquid and the operating conditions. The residence time can be increased by lengthening the channels or by decreasing the angle of descent, which can be achieved with a helicoidal microchannel falling film reactor. The residence time was found to be increased by a factor of about 50 in a microchannel when the angle of descent was decreased from 90∘ to 7.5∘ [14]. The mass transfer data on falling film microstructured reactor are presented in Table 7.6. Bottom housing section with microstructured channels

Contact-zone mask Top housing section with open space

Reaction plate

Figure 7.13 A falling film microreactor with a viewing window [6]. (Adapted with permission from Wiley.)

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Table 7.6 Literature on mass transfer in falling film microchannel. Regime and system

Zhang et al. [50] Complete falling flow regime Nonreacting system CO2 /deionized water, 5.2 and 12 wt% EG solutions Sobieszuk et al. [51] Reacting systems Absorption of CO2 to NaOH and monoethanolamine (MEA) solution

Conditions and definition

Global volumetric mass transfer coefficient

Gas chamber depth – 0.8, 1.45, and 3.0 mm 3 PMMA 20 channels MSR (1000 μm × 300 μm × 60 mm) V̇ G = 0.76, 3.05 ml ⋅ s−1 V̇ L = 0.033 − 0.66 ml ⋅ s−1 T = 23–25∘ C

kL = 5.83 ⋅ 10−5 –13.4 ⋅ 10∗−5 m s−1 for ReL < 150 Empirical correlation: 0.57 (7.29) ShL = 0.0145 ��0.69 L ScL

A stainless steel plate with 29 straight, open, vertical, parallel channels (78.3 mm × 0.3 mm × 0.6 mm) V̇ G = 3.3 ml ⋅ s−1 V̇ L = 0.255, 0.379, 0.627 ml ⋅ s−1 Inlet concentration of CO2 in N2 : 12–97%

ShL = ScL =

kL d h ; ��L Dm �L Dm

=

4�film ufilm ; �L

kL = 7.22 ⋅ 10−5 –12.5 ⋅ 10−5 m ⋅ s−1

In falling film microstructured reactors, Zhang et al. [50] proposed an empirical relation to estimate the mass transfer coefficient as shown in Equation 7 (Table 7.6). The Reynolds number is defined as: ��L =

4ufilm �film vL

(7.30)

with the film velocity ufilm =

2 g�film

the film thickness √ �film =

(7.31)

3vL

3

3V̇ L vL �L ≅ n ⋅ W ⋅ g ⋅ sin � (�L − �G )

√ 3

3V̇ L vL n ⋅ W ⋅ g ⋅ sin �

(7.32)

where W, n, g, and � are width of microchannel, the number of microchannels, gravitational acceleration, and inclination angle from horizontal, respectively. The mass transfer coefficient could be described with the penetration model supposing that the liquid residence time � L is very short compared to the diffusion time in the liquid film tD . Fo =

�L L Dm ≪1 = 2 tD ufilm �film

(7.33)

7.4

Mass Transfer

Example 7.4 investigates the mass transfer in microstructured falling film reactor.

Example 7.4: Mass transfer in microstructured falling film reactor In a vertically placed 20 parallel channel microstructured falling film reactor of 60 mm length, 1 mm width, and 0.3 mm depth, gas and liquid flows with 46 ml ⋅ min−1 and 3.6 ml ⋅ min−1 , respectively, estimate: (1) thickness of the wall film, (2) mean velocity of liquid film, (3) Fourier number, (4) Reynolds number, and (5) mass transfer coefficient, kL . Given: liquid kinematic viscosity � L = 8.97 ⋅ 10−7 m2 ⋅ s−1 and Schmidt number ScL = 452. Solution: The first parameters are calculated using Equation 7.32 1) Wall film thickness for n = 20, W = 1 mm, g = 9.81 m ⋅ s−2 , V̇ L = 3.6 ml ⋅ min−1 √ √ ̇ � 3 V v 3V̇ L vL 3 3 L L L �film = ≅ n ⋅ W ⋅ g ⋅ sin � (�L − �G ) n ⋅ W ⋅ g ⋅ sin � √ √ −6 √ 3 ⋅ 3.6⋅10 ⋅ 8.97 × 10−7 3 √ 60 = 9.37 ⋅ 10−5 m = 93.7 μm = 20 ⋅ 1 ⋅ 10−3 ⋅ 9.81 ⋅ sin 90 2) Mean velocity of liquid film (Equation 7.31) ufilm =

2 g �film

3 vL

=

9.81 ⋅ (93.7 ⋅ 10−6 )2 = 0.03 m ⋅ s−1 3 ⋅ 8.97 ⋅ 10−7

3) Fourier number: The diffusion coefficient required for Fo can be calculated from ScL as ScL = � L /Dm �L 8.97 ⋅ 10−7 = 1.98 ⋅ 10−9 = ScL 452 Dm L 1.98 ⋅ 10−9 ⋅ 60 ⋅ 10−3 = Fo = = 0.45 2 ufilm �film 0.03 ⋅ (93.7 ⋅ 10−6 )2

Dm =

4) Reynolds number ��L =

4ufilm �film 4 ⋅ 0.03 ⋅ 93.7 ⋅ 10−6 = = 12.53 vL 8.97 ⋅ 10−7

5) Mass transfer coefficient: The volumetric mass transfer coefficient is investigated using the correlation presented in terms of Sherwood number as a function of Reynolds number and Schmidt number in Table 7.6. 0.57 ShL = 0.0145 ��0.69 L ScL

= 0.0145 × 12.530.69 × 4520.57 = 2.7

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7 Microstructured Reactors for Fluid–Fluid Reactions

To investigate kL from ShL , the hydraulic diameter dh (=4Acs /Lcir ), where Acs is cross sectional area and Lcir is circumference of the channel, is needed. Thus, 4 ⋅ (1 ⋅ 10−3 × 0.3 ⋅ 10−3 ) = 4.6 ⋅ 10−4 m 2 ⋅ (1 ⋅ 10−3 + 0.3 ⋅ 10−3 ) Sh D 2.7 × 1.98 ⋅ 10−9 = 1.16 ⋅ 10−5 m ⋅ s−1 kL = L m = dh 4.6 ⋅ 10−4

dh =

Mass transfer in falling film has been considered mainly with the dependence of liquid-side mass transfer. Recent research activities are concentrated on the potential increase of mass transfer in the liquid film to enhance the reactor performance. For this purpose, different modifications of the surface structure of the reaction plates are proposed. One way to structure the surface is to mimic a regular porous network. Rhombic structures are used to modify and disturb the laminar liquid flow [52]. So-called “streamlined fins” are arranged horizontally in different rows (Figure 7.14a). The rows are shifted in such a way that one rhombus is positioned in the center of the space between two neighbored rhomb of the up- and downstream rows. This arrangement forces the redirection of the flow, thus ameliorating the mixing of the fluid in a split and recombine manner. Intensification of liquid mass transfer can also be achieved by structuring the channels of falling film plates in the form of staggered grooves in herring bone arrangements (Figure 7.14b) [52, 53] as presented in Section 4.4.1. This kind of chaotic mixers are very efficient at low Reynolds numbers and allow, in addition, to narrow the residence time distribution of laminar flow (Section 3.6.2). 7.4.4 Mass Transfer in Liquid–Liquid Microstructured Devices 7.4.4.1 Slug Flow (Taylor Flow)

The literature on mass transfer in liquid–liquid slug flow is presented in Table 7.7. The mass transfer is investigated for both reacting and nonreacting systems. Most of the results show that the mass transfer coefficient increases with increasing flow velocity. Identical to gas–liquid Taylor flow, the mass transfer in liquid–liquid slug flow has two contributions: film and slug caps. The specific interfacial area, ratio of surface area of the slug per unit its volume, can be written by neglecting the film thickness as a = acap + afilm ≅

2 ⋅ dt2 dt2

⋅ Lb

+

4 ⋅ dt ⋅ Lb dt2

⋅ Lb

=

2 4 + Lb dt

(7.34)

7.4

2400 μm

(a)

Mass Transfer

2400 μm

(b)

Figure 7.14 Photograph of the microstructured falling film plates [52]. Insets show the structure in detail. (a) Reaction plate with rhomb structure. (b) Reaction plate with herring bone structure. Courtesy Fraunhofer ICT-IMM, Germany

where Lb is the length of the dispersed liquid slug. In the case of mass transfer without chemical reaction, the wall film could be saturated and only acap (= 2/Lb ) is utilized for mass transfer. In this case, the volumetric mass transfer coefficient is inversely proportional to the slug length in the capillary. 7.4.4.2 Slug-Drop and Deformed Interface Flow

The mass transfer performance of slug-drop and deformed interface flow is given in Table 7.8. The velocity range in the table shows that this flow regime acts as a transition between slug flow and deformed interface flow. When these k ov a values are compared with those obtained in slug flow regime in an identical microchannel (Table 7.7), the slug-drop shows relatively high values because of larger specific interfacial area formed by small droplets besides regular slugs. 7.4.4.3 Annular and Parallel Flow

The mass transfer coefficients investigated for annular and parallel flow are depicted in Table 7.9. As explained in the flow regime section, this flow exists over a wide range of flow velocities giving k ov a from 0.07 to 17.35 s−1 .

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7 Microstructured Reactors for Fluid–Fluid Reactions

Table 7.7 Literature review on mass transfer in liquid–liquid slug flow MSR. Regime and system

Conditions

Volumetric mass transfer coefficient

Burns and Ramshaw [35] Glass chip reactor Reacting system Kerosene/acetic acid/water + NaOH

dh = 380 μm Cacetic acid,or = 0.65 mol ⋅ l−1 CNaOH,aq = 0.1–0.4 mol ⋅ l−1 u ≤ 35 mm ⋅ s−1

kov a = 0.5 s−1 Order of magnitude

Kashid et al. [54] Teflon® Y-junction and capillary tubing Nonreacting system Kerosene/acetic acid/water

dt = 0.5–1 mm L = 100 mm cacetic acid,or = 0.03 mol ⋅ l−1 u = 10–70 mm ⋅ s−1

dt = 0.5 mm ∶ kov a = 0.4–1.4 s−1 dt = 0.75 mm ∶ kov a = 0.4–1.1 s−1 dt = 1 mm ∶ kov a = 0.4–1 s−1

Dessimoz et al. [36] T and Y-junction glass chip reactor Reacting system Hexane/trichloroacetic acid/water + NaOH

dh = 400 μm L = 56 mm Cacid,or = 0.6 mol ⋅ l−1 CNaOH,aq = 0.15–0.3 mol ⋅ l−1 u ≤ 20 mm ⋅ s−1

kov a = 0.2–0.5 s−1

Kashid et al. [55] T-junction/square channel Nonreacting system Water-acetone-toluene

TS ∶ dh = 400 μm L = 56 mm, u = 0.1–0.42 m ⋅ s−1 cacetone,aq = 3.5 wt%

kov a = 0.11–0.74 s−1

Table 7.8 Mass transfer literature on slug-drop and deformed interface flow. Regime and systema)

Slug-drop flow Deformed interface flow

Conditionsa)

u = 0.42 m ⋅ s−1 u = 0.63–1.04 m ⋅ s−1

Volumetric mass transfer coefficient

kov a = 0.47 s−1 kov a = 0.66–1.05 s−1

a) T-junction/square channel, Nonreacting system: water-acetone-toluene. Source: Adapted from Ref. [55].

7.4.4.4 Slug-Dispersed and Dispersed Flow

The mass transfer performance of a caterpillar micromixer under slug-dispersed flow regime is given in Table 7.10. The fine dispersion results in very high specific interfacial area leading to k ov a as high as 2.25 s−1 . Unlike for gas–liquid systems, no efforts have been made to develop mass transfer models based on either film or penetration theory for liquid–liquid MSR. The

7.4

Mass Transfer

Table 7.9 Mass transfer literature data on annular and parallel flow. Regime and system

Conditions

Volumetric mass transfer coefficient

Zhao et al. [56] Annular/parallel flow Nonreacting system Water-succinic acid-n-butanol

L = 45, 60 mm dh = 0.4 mm, u = 0.01–2.5 mm ⋅ s−1 dh = 0.6 mm, u = 0.005–2 mm ⋅ s−1 cacid,or = 1 wt%

kov a = 0.067–17.35 s−1

dh = 269 μm L = 40 mm cacid,or = 0.6 mol ⋅ l−1 cNaOH,aq = 0.1–0.2 mol ⋅ l−1 u = 0–50 mm ⋅ s−1

kov a = 0.2–0.5 s−1

u = 1.25–1.88 m ⋅ s−1

kov a = 1.12–1.27 s−1

Dessimoz et al. [36] T and Y-junction glass chip reactor Parallel flow Reacting system Toluene/trichloroacetic acid/water + NaOH Kashid et al. [55] Parallel flow T-junction/square channel Nonreacting system Water-acetone-toluene

Table 7.10

Mass transfer literature on slug-dispersed and dispersed flow.

Regime and systema)

Conditionsa)

Volumetric mass transfer coefficient

Slug-dispersed flow Dispersed flow

u = 5.92–8.88 m ⋅ s−1 u = 10.37–13.33 m ⋅ s−1

kov a = 1–1.5 s−1 kov a = 1.61–2.25 s−1

a) Concentric-junction/circular channel, Nonreacting system: water-acetone-toluene. Source: Adapted from Ref. [55].

complexity in the liquid–liquid systems is because of the resistance on both liquid phases, whereas in gas–liquid systems the main resistance to mass transfer is mostly in the liquid phase. 7.4.5 Comparison with Conventional Contactors

The mass transfer coefficients obtained in microstructured devices and in conventional gas–liquid contactors are listed in Table 7.11. The liquid-side kL a and interfacial area in microstructured devices are at least 1 order of magnitude higher than those in conventional contactors such as bubble columns and packed columns, being up to 21 s−1 .

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7 Microstructured Reactors for Fluid–Fluid Reactions

Table 7.11 Comparison of gas–liquid microstructured devices with conventional contactors. Type of contactor

Bubble columns Couette–Taylor flow reactor Impinging jet absorbers Packed columns, concurrent Packed columns, countercurrent Spray column Static mixers Stirred tank Tube reactors, horizontal, and coiled Tube reactors, vertical Gas–liquid microchannel

kL × 105 (m s−1 )

a (m2 m−3 )

kL a (s−1 )

10–40 9–20 29–66 4–60 4–20 12–19 100–450 0.3–80 10–100 20–50 40–160

50–600 200–1200 90–2050 10–1700 10–350 75–170 100–1000 100–2000 50–700 100–2000 3400–9000

0.005–0.24 0.03–0.21 0.025–1.22 0.0004–1.02 0.0004–0.07 0.015–0.022 0.1–2.5 0.03–0.4 0.005–0.7 0.02–1 0.3–21

Source: Adapted from Yue et al. [32].

Table 7.12 Comparison of liquid–liquid microstructured devices with conventional equipment. Contactor

Agitated contactor [57] Packed bed column (Pall/Raschig ring, Intalox saddles) [58] RTL extractor (Graesser raining bucket) [59] Air operated two impinging jet reactors [60] Two impinging jets reactor [61] Capillary microchannel (ID = 0.5–1 mm)

a (m2 m−3 )

k ov a (s−1 )

32–311 80–450 90–140 350–900 1000–3400 830–3200

0.048–0.083 0.0034–0.005 0.0006–0.0013 0.075 0.28 0.88–1.67

The volumetric mass transfer coefficients found in the liquid–liquid microstructured devices at various flow rates were compared with those for conventional equipment in Table 7.12. Identical to gas–liquid devices, the mass transfer coefficients found in liquid–liquid microstructured devices are well above those of conventional contactors.

7.5 Pressure Drop in Fluid–Fluid Microstructured Channels

Besides the mass transfer coefficient, pressure drop plays an important role in the design of microstructured devices. The discussion here is focused on Taylor flow and annular flow.

7.5

Pressure Drop in Fluid–Fluid Microstructured Channels

7.5.1 Pressure Drop in Gas–Liquid Flow

Two models can be proposed [62]: 1) the homogenous model with mean flow velocity similar to single-phase flow, and 2) the separated flow model with an artificially separated gas and liquid flow. Model (1): One of the most commonly used models to characterize the pressure drop in microchannels is that proposed by Lockhart and Martinelli [63] for gas–liquid horizontal flow in pipes, which is used for all regimes. It employs two friction multipliers for gas and liquid, Φ2G and Φ2L , as given by the following equation: ( ) ) ( Δpf Δpf = Φ2G (7.35) L 2p L G or

(

Δpf L

)

( 2p

= Φ2L

Δpf L

) (7.36) L

where the index 2p indicates two-phase flow. The two equations given above are correlated in terms of a dimensionless number called the Lockhart–Martinelli parameter (Ψ). It is the ratio of the singlephase pressure drop of liquid to that of the gas and given by ( ) Ψ= (

Δpf L

Δpf L

)

L

(7.37)

G

Here (Δpf ∕L)L and (Δpf ∕L)G are the frictional pressure drop gradients when liquid and gas are assumed to flow in the microchannel alone, respectively. They are calculated as ) ( Δpf 2fL ṁ 2total (1 − xG )2 = L L dh �L ) ( Δpf 2fG ṁ 2total x2G = L G ACS dh �G ṁ G ṁ G = (7.38) xG = ṁ G + ṁ L ṁ total The liquid friction factor fL and liquid Reynolds number (and vapor friction factor fG and gas Reynolds number with the gas viscosity) are obtained from fi =

d ṁ 0.079 ; ��i = total h 0.25 ACS �i ��i

(7.39)

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7 Microstructured Reactors for Fluid–Fluid Reactions

The relationship between Φ2L and Ψ can be obtained from the widely used Chisholm’s equation [64]: C 1 (7.40) + Ψ Ψ2 where C is a constant, ranging from 5 to 20, depending on the flow pattern of gas and liquid in the channel. In the case of microchannels, the Reynolds number for both the liquid and gas phases are less than 1000 and the constant C is considered to be about 5 [64]. However, from experimental measurements [32], it is reported for the CO2 -water system in rectangular microchannels that the friction multiplier cannot be predicted reliably with a single value of C. It was found to become greater with increasing mass flux and, therefore, a new correlation was proposed with a standard deviation of 9.2%: Φ2L = 1 +

C = 0.185Ψ−0.0942 Re0.711

(7.41)

Thus, the steps in estimating the pressure drop using Equation 7.36 are: (i) estimation of the frictional pressure drop of each individual phase, (ii) calculation of the Lockhart–Martinelli parameter (Ψ), and (iii) estimation of the friction multiplier for liquid, and Φ2L . Model (2): A flow-regime-dependent relationship for estimating the total pressure drop in a two-phase vertical capillary flows was reported [16]. Initially a single-phase vertical tube with liquid flowing in the laminar regime was considered. The total pressure drop (Δptot ) is composed of two contributions: (i) the pressure drop because of frictional effects of the liquid flow (Δpf ) and (ii) the hydrostatic pressure of the liquid: Δptot = Δpf + �L g ⋅ L

(7.42)

For laminar flow, the frictional pressure drop is given by the Hagen–Poiseuille equation 32�L uL L (7.43) Δpf = dt2 Combining both equations, the total pressure drop is given by: [ ) ( ] dt2 32�L L uL + �L g Δptot = 32�L dt2

(7.44)

By comparing Equations 7.43 and 7.44 a gravity equivalent liquid velocity (ug ) in the capillary can be introduced that would result in a pressure loss equal to the hydrostatic pressure exerted by the liquid phase. Assuming laminar flow, the gravity equivalent liquid velocity becomes: ) ( dt2 (1 − �G )�L g (7.45) ug = 32�L The gas hold-up �G in the capillary can be estimated with Equations 7.4 and 7.5:

7.5

Pressure Drop in Fluid–Fluid Microstructured Channels

The total mixture velocity (utot ) is defined as the sum of the superficial velocities of the two phases and the gravity equivalent velocity (ug ): utot = uG + uL + ug

(7.46)

A dimensionless two-phase pressure factor f tot can be defined, analogous to the Fanning friction factor [65]: ftot =

Δptot ∕L 1∕2�L u2tot (4∕dt )

(7.47)

In a situation where both the gas- and liquid phase flows are laminar, the pressure factor can be expected similarly to the Fanning friction factor as: ftot =

� u d C with Retot = L tot t Retot �L

(7.48)

The constant C depends on the channel geometry and has values of 16 and 14.2 for circular and square channels, respectively. For uG /uL < 0.5 the pressure factor can be estimated with Equation 7.48. At uG /uL > 0.5 and uG /(uG + uL ) < 0.5 the slip ratio Rslip between bubble and liquid velocity influences the friction factor: Rslip =

uG ∕�G uL ∕(1 − �G )

(7.49)

To predict the pressure drop for this flow regime, an empirical correlation for estimating the friction factor was obtained from experimental data in the range of 0.008 < (uG + uL ) < 1 ms−1 : ftot =

] ( ) 1 [ C exp −0.02Retot + 0.07Re0.34 √ tot Retot R slip

(7.50)

The pressure drop results using above equation are demonstrated in Example 7.5. Example 7.5: Pressure drop in upward gas-liquid flow in a circular capillary Estimate the pressure drop in upward gas–liquid flow in a circular capillary for air–water system as a function of gas flow velocity (range 1–30 mm s−1 ) for different liquid flow velocities of 30, 50, and 75 mm s−1 . The hydraulic of the channel is 2 mm. Solution: The pressure drop in upward gas–liquid flow in a square capillary is investigated using Equation 7.47. ( ) Δptot 4 1 = ftot �L u2tot (7.51) L 2 dt

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7 Microstructured Reactors for Fluid–Fluid Reactions

The friction factor is approximated with Equation 7.50. The pressure drop as a function of the superficial gas velocity is plotted in Figure 7.15. 110 Total pressure drop, Δp/Lc (hPa m–1)

304

100 90 80

uL = 0.075 m s–1

70

uL = 0.05 m s–1

60 uL = 0.03 m s–1

50 40 0.005

0.010

0.015

0.020

0.025

0.030

Gas superficial velocity, uG (m s–1) Figure 7.15 Pressure drop in upward gas–liquid flow in a circular capillary of dh = 2 mm.

7.5.2 Pressure Drop in Liquid–Liquid Flow

There are two fundamental differences between gas–liquid and liquid–liquid slug flow in microchannels [37].

• In the liquid–liquid slug flow system, because of close physical properties of both fluids it might be possible that there is no wall film and both fluids flow alternatively through the capillary; this is not observed in gas–liquid systems. • In the case of a wall film in the horizontal microchannel, because of considerable shear of the discrete liquid phase on the continuous phase, the latter moves with finite velocity while film in the gas–liquid system is considered stagnant. 7.5.2.1 Pressure Drop – Without Film

In the case of without film, the pressure drop in liquid–liquid slug flow comes from two main contributions: the hydrodynamic pressure drop of the individual phases and the pressure drop because of capillary phenomena, pc . If we consider the single flow unit shown in Figure 7.16a, the overall pressure drop along its length can be written as: Δptot = Δp1 + Δp2 + pC

(7.52)

7.5

Δp1

Pressure Drop in Fluid–Fluid Microstructured Channels

Δp2

Δpfilm

ΔpCP

θw (a)

(1-ε1)LUC

ε1LUC

(b)

εdLUC

(1-εd)LUC

Figure 7.16 Pressure drop along a single slug unit. (a) Without film and (b) with film.

The single-phase hydrodynamic pressure drop can be calculated using the Hagen–Poiseuille equation, while the capillary pressure is obtained from the Young–Laplace equation for a cylindrical tube as given by the following equations: Δp1 =

32�1 u �1 LUC

where, u =

dt2

; Δp2 =

32 �2 u (1 − �1 )LUC

V̇ 1 + V̇ 2 V̇ 1 , �1 = Acs V̇ 1 + V̇ 2

dt2

and pc =

4� cos �w dt (7.53)

Here Acs is the cross sectional area of the circular channel. Assuming a constant dynamic contact angle and slug lengths with an equal number of slugs of both phases under similar operating conditions and neglecting end effects, the overall pressure across for a given length of the capillary is the summation of pressure drops across all slugs and the capillary pressure at all interfaces. 7.5.2.2 Pressure Drop – With Film

For theoretical predictions, it is assumed that the pressure drop along the length of the capillary is because of the film region only. A model for pressure drop in the pipeline flow of slugs (referred to as “capsules”) is given by Charles [66], which relates the pressure drop in the slug region, (Δp∕L)film , to that of single-phase flow of the continuous phase, (Δp∕L)CP . According to this model, the pressure drop along the length of the film can be determined by the following equation: ( ) ) ( Δp ) ( d R − �film Δp 1 ;R = t (7.54) = ;� = L film L CP R 2 1 − �4 where L and R is length and radius of the tube, respectively. In the above model, it was assumed that slugs follow each other sufficiently closely so that the fluid between them can be considered as part of the slug stream. However, in the liquid–liquid slug flow for chemical engineering applications, this assumption is usually not valid and will apply only when the enclosed slug has a length of several times more than the other slug. The slug that forms the film may, however, be longer depending on the inlet flow ratio for both phases. It is, therefore, necessary to consider the phase fraction of both liquids to calculate the pressure drop for a given length of the liquid–liquid slug flow microchannel. In addition, the film thickness is very small compared to the radius of the slug, which justifies

305

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7 Microstructured Reactors for Fluid–Fluid Reactions

the assumption that the length of the film region for a given length of capillary is nothing more than the corresponding phase fraction times the total length. Assuming phase fraction inside the capillary equal to volumetric flow fraction (for dispersed phase, �D ), the pressure drop along the film region for a given pipe length can thus be written as [37]: ) ( ( � ) ( Δp ) Δp Δp D = = L L film 1 − �4 L CP with �D , the volume fraction of the dispersed phase (7.55) To calculate the pressure drop using the above equation, the film thickness is crucial. It can be estimated using Bretherton’s or Aussillon and Quere’s correlations depending on the capillary number (Equations 7.2 or 7.3) as demonstrated in Example 7.6.

Example 7.6: Pressure drop in liquid-liquid slug flow A liquid–liquid system flows in a circular cross section capillary with dt = 0.8 mm diameter forming a continuous water phase and a dispersed toluene phase. The flow rate of the continuous phase is 5 ml min−1 while the flow rate of the dispersed phase is 4 ml min−1 . Estimate the pressure drop in the microchannel assuming dispersed slug velocity equals the two-phase velocity. Data: �toluene = �D = 867 kg ⋅ m−3 , �toluene = �D = 0.6 ⋅ 10−3 Pa s, �water = �C = 998.2 kg ⋅ m−3 , �water = �C = 1 ⋅ 10−3 Pa s, � = 0.036 N m−1 . Solution: As the two-phase systems form dispersed and continuous phase, Equation 7.53 can be used. It requires single-phase pressure drop and �. The singlephase pressure drop can be calculated using Hagen–Poisseuille equation (Equation 7.42). ( ) � V̇ � u Δp = 32 C2 = 128 C 4 for circular channels L CP dt �dt The single phase pressure drop of the continuous phase is calculated with the velocity of the two-phase system (u = uD + uC ) or flow rate two phases. ) ( � V̇ Δp = 128 C 4 L CP � dt ( ) 1 ⋅ 10−3 ⋅ 5+4 ⋅ 10−6 60 = 128 ⋅ = 14920 Pa ⋅ m−1 � ⋅ (0.8 ⋅ 10−3 )4 To determine the film thickness the capillary number must be known: � ⋅u (7.1) Cai = i b �

7.6 Flow Separation in Liquid–Liquid Microstructured Reactors

Here the bubble velocity corresponds to the velocity of the slug of the dispersed flow, which was assumed to be identical with the twophase flow velocity (ub = uD,slug ≅ u). ( ) 5+4 −6 × 10 60 V̇ = 0.30 m ⋅ s−1 u= � 2 = π −3 2 d × (0.8 ⋅ 10 ) 4 h 4 CaC =

�C ⋅ u 10−3 ⋅ 0.3 = = 8.3 ⋅ 10−3 � 0.036

For capillary number greater than 3 ⋅ 10−3 the relation of Aussillons and Quere is used to estimate the film thickness (Equation 7.3): 2∕3

�film = dt

0.67CaC 1+

2∕3 3.35CaC

= 0.8 ⋅ 10−3

0.67 ⋅ (8.3 ⋅ 10−3 )2∕3 = 1.93 ⋅ 10−5 m 1 + 3.35 ⋅ (8.3 ⋅ 10−3 )2∕3 −3

−5

−1.94⋅10 = 0.9515 The value of � is found to be � = Rfilm = 0.4⋅100.4⋅10 −3 Thus, the two-phase pressure drop can be determined with Equation 7.55: ) ( 4 ) ( Δp ( �D ) Δp 5+4 = × 14920 = L 1 − �4 L CP 1 − (0.9515)4 R−�

= 3.67 ⋅ 104 Pa ⋅ m−1 = 0.367 bar ⋅ m−1

7.5.2.3 Power Dissipation in Liquid/Liquid Reactors

Power input, a decisive parameter for benchmarking technical reactors, has been investigated using the experimental pressure drop and compared with conventional contactor as shown in Table 7.13. The power input for continuous reactors is investigated in terms of kJ m−3 of liquid, product of pressure drop and volumetric flow rate. The comparison reveals that the liquid–liquid slug flow microreactor requires much less power than the alternatives to provide large interfacial area – as high as a = 5000 m2 m−3 in 0.5 mm capillary microreactor, which is above the values in a mechanically agitated reactor (a ≈ 500 m2 m−3 ). The specific interfacial area obtained for conventional liquid contactor is summarized in Table 7.12.

7.6 Flow Separation in Liquid–Liquid Microstructured Reactors

The separation of dissolved components by liquid–liquid extraction has been discussed in the mass transfer section. In this section, flow separators required for splitting biphasic mixture are presented.

307

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7 Microstructured Reactors for Fluid–Fluid Reactions

Table 7.13 Power input requirement for various liquid–liquid contactors [67]. Contactor type

Agitated extraction column Mixer-settler Rotating disk impinging streams contactor Impinging streams Impinging stream extractor Centrifugal extractor Liquid–liquid slug flow

Power input, (kJ m−3 ) of liquid

0.5–190 150–250 175–250 280 35–1500 850–2600 0.2–20

7.6.1 Conventional Separators

Separation of two liquid phases after the mass transfer is an important step in the liquid–liquid extraction. Therefore, when designing an extraction unit, a question always arises as how to achieve the separation of two phases immediately following phase contact in the extraction zone. In conventional extraction equipment, the operations associated with mixing and separation of two liquids is usually at least partially distinct. Gravity based separation, dependent on the density difference between two phases, is the most commonly used method of separation. Difficulties that often occur in the separation of immiscible liquids include poor or slow phase separation, emulsion or rag layer formation, and poor process control, especially in batch systems. Some liquid–liquid dispersions take hours to separate in conventional systems resulting in poor performances of the extraction units. In addition to the single-stage operation, as discussed above, multistage operations are common in liquid–liquid extraction for large-scale production and effective use of chemicals. Depending on the selectivity of the solvent and the amount of mass transfer required to achieve the desired solute recovery, several stages of extraction may be required. In this case, countercurrent contact is the most efficient extraction method as it conserves the mass transfer driving force and, therefore, gives optimal performance. Most of the countercurrent operations in the laboratory practice use batch processes and they are carried out using milliliter amount of feed and solvent in the flasks. 7.6.2 Types of Microstructured Separators

Flow separation after the mass transfer zone in microstructured devices has been a topic of research for a long time. The schematics of the phase separators used for liquid–liquid separation are depicted in Figure 7.17. Three principles are used: geometrical modifications, wettability based separation, and gravity based separators. These are discussed in the following section.

7.6 Flow Separation in Liquid–Liquid Microstructured Reactors

(a)

(c)

Figure 7.17 Schematics of different types of flow splitters for separation of liquid–liquid two-phase flow. (a) Geometrical modifications (e.g., Y-separator), (b) wettability based (e.g., membrane separator), and (c) gravity based separator (e.g., settler).

7.6.2.1 Geometrical Modifications

In this case, either geometry or liquid properties are modified or selected to generate the parallel flow of two liquids. The surface tension forces pinning the liquid to the channel walls are generally strong enough to resist buoyancy and viscous shear forces. In such cases, it is also possible that lighter liquid may flow under the denser fluid. The pressure drop in each outlet can be calculated using Hagen–Poiseuille equation in terms of flow ratio: Δpout = 32�

�u � V̇ L L = 128� 2 dh �dh4

(7.56)

where � is a geometric factor, which is 1 for a circular tube and it depends on the height (H) to width (W ) ratio for a rectangular channel. Here L refers to the length of separator outlets. The correction factor becomes 0.89 for quadratic channels and assumes the asymptotic value 1.5 when the ratio goes to zero. An empirical correlation is given by the following expression [68]: ( ) H � = 0.8735 + 0.6265 exp −3.636 (7.57) W Ideally, the pressure drop in each outlet should be equal along each outlet (Δpout ) with only one phase per outlet [69]: ( ) ( ) 128 � L 128 � L ̇ ̇ = �2 V2 (7.58) Δpout = �1 V1 �dh4 �dh4 1

2

and the flow ratio S for ideal splitting can be calculated: S=

4 V̇ 2 � (128 � L∕dh )1 = 1 �2 (128 � L∕d4 )2 V̇ 1 h

(7.59)

The characterization of the splitter is done in terms of contamination (C ′ ) in each outlet of the splitter: C′ =

Volume of undesired sample Total volume collected

(7.60)

′ And the average contamination (Cav ) in two outlets is investigated by following equation: ] [ V1,2 V2,1 1 ′ (7.61) = + Cav 2 V1,2 + V2,2 V1,1 + V2,1

309

310

7 Microstructured Reactors for Fluid–Fluid Reactions

Output 1

Input 1 Liquid 1

(a)

Output 2

Flow Output 2

(b)

V2 / V1 < S Contamination

Liquid 2 Output 1

V2 /V1 = S Ideal split

Input 2

Liquid 2

Liquid 1

V2 / V1 > S Contamination

Figure 7.18 A wedge-shaped flow splitter to split parallel flow. (a) Inlets and outlets of wedge shapes separator and (b) schematic presentation of contamination and ideal split [69]. (Adapted with permission from Elsevier.)

where V is the volume collected at the outlet at a certain time. The subscript i, j indicates fluid and outlet numbers, respectively. A wedge-shaped flow splitter to split parallel flow has already been applied for the separation of the parallel flow of kerosene (+dye) and aqueous solution of propylene glycol as shown in Figure 7.18. In this example, the lighter kerosene phase flowed under the dense aqueous solution. The viscosity of fluids varied by adding propylene glycol to water and it was observed that at low viscosity ratio of 0.56 and 3.06, the flow was stabilized while at higher viscosity ratio of 22.1 disrupted flow patterns were observed. A contamination level below 2% was achieved. By changing the flow ratio, higher or lower, than its ideal value S (Equation 7.59), one outlet could be made contamination free. This may be the preferred mode of operation for many systems as it provides one stable clean output at the expense of slightly higher contamination in the other (Figure 7.18). 7.6.2.2 Wettability Based Flow Splitters

Every liquid has an ability to maintain contact with some solid materials, which is often referred to as wetting. The degree of wetting is represented in terms of contact angles (� w ), angle at which the liquid–vapor interface meets the solid–liquid interface, as shown in Figure 7.19a. It indicates that wetting is poor at � w > 90∘ , good at � w ≪ 90∘ , and complete at � w ∼ 0∘ . If two materials with preferential wettability of two liquids are considered, the liquids have a tendency to flow along the surface of the material to which they have the greatest affinity. The behavior of drop on the zone-selectively modified with hydrophilic and hydrophobic material is shown in Figure 7.19b. The liquid drop was placed on the boundary of hydrophilic and hydrophobic zones on a plate immersed in another immiscible liquid. In the case of octane drop for the plate immersed in water, it was moved toward the hydrophobic zone and settled and when the two-zone plate was dipped in octane and a water drop was placed on the boundary, and the drop settled in the hydrophilic zone

7.6 Flow Separation in Liquid–Liquid Microstructured Reactors

0.5 mm

Octane

Water

95°

311

Poor wetting

Octane

15°

Water

Good wetting

0° (a)

Complete wetting

Hydrophilic

Hydrophobic

Hydrophilic

(b)

Figure 7.19 Contact angle [70]. (a) Contact angle and wetting of solids. (b) Behavior of three-phase contact angle when the liquid drop placed on boundary of hydrophilic and hydrophobic material for octane drop on the

plate immersed in water and water drop on the plate immersed in octane. (Adapted from Ref. [70]. Copyright © 2013, Wiley-VCH GmbH & Co. KGaA.)

[71]. The photographs suggest that the boundary between water and octane could be pinned along the boundary between the hydrophilic and hydrophobic regions. These preferential displacements can be used to separate two liquid phases. Different types of wettability based separators have been used for the separation of two immiscible liquids. Some of the examples are shown in Figure 7.20. The Yshaped separator was used for aqueous-organic systems comprising one inlet and two outlets. The splitter consists of PTFE with a steel needle having an internal diameter equal to the Y-junction, being fitted into one of the outlets. The aqueous phase has a strong affinity toward steel, whereas the organic phase has an affinity toward PTFE. This difference in the affinity can be harnessed for the separation of the two phases. The results show that the average contamination was about 5%. The results show that there is no significant effect of flow rate and capillary size on flow splitting for a given splitter. The minor problems observed with phase cross contamination could be resolved by modifying the splitting geometry. One of the possible modifications could be the transition of slug flow regime to parallel flow in the beginning of flow splitter. A flow splitter made up of PTFE and stainless steel with rectangular cross section is depicted in Figure 7.21. The slug flow becomes parallel because of rectangular cross section and preferential wettability of the separator enhancing the separation. Further modifications are used in a flow splitter that consists of a set of grooves on two surfaces having different surface properties (e.g., one could be glass and the other an organic material) and fixing those surfaces on to each other to form channels in between them [73]. The experiments performed with a chip consisting of two layers, one coated with polar (glass) and nonpolar (a silicon monomer mix) materials for separating isooctane and water showed excellent results. The presented technique is suitable for use in microscale equipment, in which surface tension forces, rather than gravitational force, dominate. The wettability based separation of two immiscible liquids is explained in more detail for the membrane separator [72].

Hydrophobic

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7 Microstructured Reactors for Fluid–Fluid Reactions

Outlet 2 PTFE

Outlet 2

Inlet Inlet Aqueous phase Organic phase

Outlet 1 steel

Outlet 1

(a) Outlet 1

A - B mixture inlet

Inlet

Outlet 1

Liquid A

Membrane

Liquid B Outlet 2 (b)

Outlet 2 Figure 7.20 Schematics of wettability based separators. (a) Y-shaped separator. (Adapted with permission from Ref. [54]. Copyright (2007) American Chemical Society.) (b) Membrane separator. (Adapted from Ref. [72] with permission of The Royal Society of Chemistry.)

The pressure drop in the separator outlets should be identical because the twofluid streams exit at the same ambient pressure (Equation 7.58). However, it is proposed that the design of the microseparator should be based on a worst case criteria where the pressure drop in dispersed phase outlet should be calculated on the total flow rate [72]. In the case of slug flow, the movement of slugs to the nondesired outlet can be restricted with the capillary pressure [72] (Figure 7.20b). The correct design of the membrane separator allows to get noncontaminated liquid flows at outlet 1 and 2. Let us assume a slug flow of an organic liquid (B), for example, hexane, toluene from an aqueous phase (A). The mixture enters the separator equipped with a hydrophobic membrane, for example, PTFE. The aqueous phase leaves the separator at outlet 1 whereas the organic phase, which is wetting the membrane, flows through the pores leaving the device at outlet 2. The maximum pressure drop through outlet 1 is obtained when the whole flow leaves the device at position 1. This is the worst case applied for the design. Using Hagen–Poiseuille relation for laminar flow we obtain: Δp1 =

128 ⋅ �1 ⋅ V̇ tot � ⋅ d14

L1 ; V̇ tot = V̇ 1 + V̇ 2

(7.62)

7.6 Flow Separation in Liquid–Liquid Microstructured Reactors

Outlet-1

Steel

Inlet

PTFE

Outlet-2 Figure 7.21 A new flow splitter developed for the separation of biphasic mixture. (Adapted from Ref. [70]. Copyright © 2013, Wiley-VCH GmbH & Co. KGaA.)

with � 1 the dynamic viscosity of the aqueous phase. The pressure drop in the channel (connected to membrane) on the aqueous side of the separator devices is given by: Δpsep =

128 ⋅ �1 ⋅ V̇ tot 4 � ⋅ dsep

Lsep

(7.63)

In most practical situations the pressure drop in the separator will be small compared to Δp1 and can be neglected: Δpsep ≪ Δp1

(7.64)

The pressure drops through outlet 1 and the sum of the pressure drop through the membrane and outlet 2 must be equal. Δp1 = Δpmem + Δp2 128 ⋅ �1 ⋅ V̇ tot 128 ⋅ �2 ⋅ V̇ 2 128 ⋅ �2 ⋅ V̇ 2 ⇒ L = L + L2 1 pore � ⋅ d14 n ⋅ � ⋅ dp4 � ⋅ d24

(7.65)

The diameter of the membrane pores and the membrane thickness are given by dp and Lmem , respectively; n indicates the number of parallel pores in the membrane. Its estimation is demonstrated in Example 7.7. To avoid the nonwetting aqueous phase penetrating the pores, the pressure drop through the membrane must be smaller than the capillary pressure: Δpc =

4� cos �w dp

(7.66)

where � is interfacial tension and � w the wetting angle. Δpc > Δpmem = Δp1 − Δp2 32 ⋅ �2 ⋅ V̇ 2 � cos �w > Lpore n ⋅ � ⋅ dp3

(7.67)

Equation 7.67 must be respected to avoid contamination of liquid in outlet 2.

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7 Microstructured Reactors for Fluid–Fluid Reactions

To assure that liquid will not leave through outlet 1, the flow resistance for liquid B through outlet 1 must be higher than the sum of the resistance through the membrane and outlet 2.

128 ⋅ �2 ⋅ L1 �⋅ ⇒

L1 d14

d14 ≫

≫

Lpore dp4

128 ⋅ �2 ⋅ Lpore n⋅�⋅ +

dp4

+

128 ⋅ �2 ⋅ L2 � ⋅ d24

L2

n⋅ d24 Δp 128 ⋅ � ⋅ L = with ∶ the fluidic resistance � ⋅ d4 V̇

(7.68)

Example 7.7: Design of membrane liquid-liquid separator A two-phase liquid–liquid mixture is to be separated using a wettability based membrane separator with pore size (dp = 10 μm). Estimate the capillary pressure considering that the dispersed phase A forms a contact angle of 101∘ (nonwetting liquid) with the membrane material. Also, estimate the number of pores (n) required for the membrane with a length of 20 mm and a width of 1 mm. The thickness of the membrane, corresponding to the length of the pores is � mem = Lpore = 150 μm. The lengths of the outlets are 50 mm on each side. The dispersed phase outlet 1 has a diameter 0.2 mm while the continuous side outlet 2 has diameter 0.4 mm. The total liquid flow is V̇ tot = 1 ⋅ 10−8 m3 s−1 with V̇ 1 = V̇ 2 . Data: – liquid B (toluene) �toluene = �2 = 867 kg m−3 , �toluene = �2 = 0.6 ⋅ 10−3 Pa s, – liquid A (water) �water = �1 = 998.2 kg m,−3 �water = �1 = 1 ⋅ 10−3 Pa s, � = 0.036 N m−1 . Solution: The dispersed aqueous phase forms a contact angle of 101∘ with the membrane material. The capillary pressure is given by Equation 7.66 as Δpc =

4� 2 ⋅ 0.036 cos �w = |cos(101)| = 2747.5 Pa dp 5 ⋅ 10−6

The outlet of the aqueous phase is 50 mm long. The maximum pressure drop in outlet 1 with V̇ 1 = V̇ tot using Hagen–Poiseuille equation: Δp1 = 128

�1 ⋅ V̇ ⋅ L1 �⋅

d14

= 128 ⋅

1 ⋅ 10−3 (1 ⋅ 10−8 ) ⋅ 50 ⋅ 10−3 = 12.73 kPa � ⋅ (0.2 ⋅ 10−3 )4

The outlet 2 has 0.4 mm ID and 50 mm length. The pressure drop is: Δp2 = 128 ⋅

0.6 ⋅ 10−3 ⋅ (0.5 ⋅ 10−8 ) ⋅ 50 ⋅ 10−3 = 0.24 kPa � ⋅ (0.4 ⋅ 10−3 )4

7.7 Fluid–Fluid Reactions in Microstructured Devices

Thus, from Equation 7.65, the maximum pressure drop through the membrane is Δpmem = Δp1 − Δp2 = 12.73 − 0.24 = 12.5 kPa = 128

�2 V̇ 2 Lpore n � dp4

To assure that the organic liquid will not leave through outlet 1 and contaminate the aqueous flow, the flow resistance for liquid B through outlet 1 must be higher than the sum of the resistance through the membrane and outlet 2 (Equation 7.68). To satisfy Equation 7.68 we require a factor 10 times higher flow friction in outlet 1, compared to the flow friction in the membrane and outlet 2. ) ( Lpore L1 L2 = 10 + d14 n ⋅ dp4 d24 ⇒n≥

10Lpore ∕dp4 L1 ∕d14 − 10 ⋅ L2 ∕d14

≅ 12800

7.6.3 Conventional Separator Adapted for Microstructured Devices

Often microstructured reactors are used for high flow velocities where the inertial forces dominate the surface forces. In this case, a separation principle identical to conventional equipment is used. The gravity based separation, based on the density difference between two phases, is the most commonly used method of separation. The most commonly used separator is IMM settler (see Figure 7.22) [74]. It consists of a glass tube, attached with special fittings on both ends. The biphasic mixer settles in the horizontal tube with the lighter liquid on the top and the denser liquid at the bottom. The level in a settler is adjusted by a flexible tube siphon. These settlers can be used in series for further purification of one of the liquids. This settler tested successfully for flow rates of up to 150 ml min−1 . 7.7 Fluid–Fluid Reactions in Microstructured Devices

General aspects of fluid–fluid reactions are discussed in detail in Section 2.4 within the context of homogeneous catalytic reactions in biphasic systems. Mostly, the reaction takes place only in one phase and the reactant must be transferred from the nonreactant phase, for example, the gas phase to the reaction phase. In consequence, the mass transfer between the different phases plays an important role on the overall kinetics and may strongly influence the

315

316

7 Microstructured Reactors for Fluid–Fluid Reactions

Figure 7.22 IMM settler for separation of liquid–liquid systems [74] (Courtesy Fraunhofer ICT-IMM, Germany).

transformation rate. The influence of mass transfer on the overall reaction rate is characterized by the ratio between the characteristic mass transfer time tm and the characteristic reaction time tr . This ratio is known as the Hatta number (Ha). On the basis of the film model and by supposing a pseudo first reaction in the reaction phase we obtain: √ √ k ′ D1,II tD = (7.69) Ha = tr kL with k ′ , the rate constant of the pseudo first reaction, D1, II , the molecular diffusion coefficient of reactant A1 in the reaction phase (phase II), and kL the mass transfer coefficient in the reaction phase. Depending on the value of Ha, different situations can be distinguished (see Figure 2.12): For Ha ≤ 0.3 the reaction rate is slow compared to the mass transfer and the reaction takes place in the bulk phase. For values of the Hatta number Ha > 3, the reaction rate is very fast compared to the mass transfer rate and the reaction takes place only in the fluid film of the reaction phase near the interfacial area. Under these conditions, the transformation increases proportionally with the specific interfacial area between the phases (a) and the square root of the reaction rate constant (Equation 2.93): √ reff = kL a Ha c∗1,II = a ⋅ c∗1,II k ′ ⋅ D1,II ; with c∗1,II ∶ equilibrium concentration at the interphase.

(7.70)

Therefore, high transformation rates can be obtained in fluid–fluid devices with high interfacial area. Microstructured multiphase reactors are characterized by interfacial areas, which are at least 1 order of magnitude higher compared to conventional contactors, and, therefore, suited particularly for very fast reactions.

7.7 Fluid–Fluid Reactions in Microstructured Devices

A further advantage for exothermic reactions is the generally high heat transfer performance of microstrutured reactors. 7.7.1 Examples of Gas–Liquid Reactions

Especially fast reactions benefit from the excellent mass transfer characteristics of microstructured devices. In addition, heat management for highly exothermic reactions is greatly facilitated because of efficient removal of heat produced during the reaction. Selective examples of different gas–liquid reactions that have been studied in the microstructured reactors are listed in Table 7.14. 7.7.1.1 Halogenation

Fluorination has been carried out using conventional reactors as a multistep process, for example, the Schiemann reaction [75]. Microstructured devices have been employed for this type of reactions as direct fluorination can reduce significantly the number of steps in the synthesis of fluorinated compounds. The first application of microchannels for selective fluorination was demonstrated using the annular flow regime in order to form a film of liquid over a solid material [83]. Further, direct fluorination of toluene, pure, or dissolved in either acetonitrile or methanol was carried out using elemental fluorine [75]. Two types of reactors were employed: microstructured falling film reactor and a micro bubble column (Figure 7.23). The microstructured falling film reactor was oriented vertically and a thin liquid film was fed through orifices into the reaction channels (with 100 × 300 μm cross section) generating a relatively large surface for contact with the gas. In the micro bubble column the liquid and Table 7.14

Examples of gas–liquid reactions carried out in the microstructured reactors.

Reaction

Direct fluorination of toluene and nitrotoluene [75]

Reactor

Microstructured falling film and micro bubble column reactor Selective fluorination of 4-nitrotoluene, 1,3-dicarbonyl, Single micro channel operating in and heterocyclic compounds [76] annular flow regime Chlorination of acetic acid [77] Microstructured falling film reactor Photochlorination of toluene-2,4-diisocyanate (TDI) Microstructured falling film reactor [78] Interdigital mixers and micromixer Nitration of naphthalene using N2 O5 [79] with the split-recombine technique Oxidation of alcohols and Baeyer–Villinger oxidation Single microchannel operating in of ketones using elemental fluorine [80] annular flow regime Sulfonation of toluene with gaseous sulfur trioxide [81] Microstructured falling film reactor Mesh microreactor Asymmetric hydrogenation of Z-methylacetamidocinnamate (mac) with rhodium chiral diphosphine complexes [82]

317

318

7 Microstructured Reactors for Fluid–Fluid Reactions

id Liqu

Orifices

(a)

tant

reac

Mini heat exchanger

Gas supply Gaseous reactant

Withdrawal zone

Static micromixer

Liquid supply

Reaction channel array

(b)

Figure 7.23 Schematic presentation of (a) microstructured falling film reactor and (b) micro bubble column (b) [75]. (Adapted with permission from Elsevier.)

gaseous reactants are contacted through a static micromixer, and, subsequently, fed into the reaction channels. For the reaction, two types of channels, narrow (50 × 50 μm) and wide (300 × 100 μm), were used. The specific interfacial areas achieved in these channels were about 27 000 m2 m−3 for the falling film (10 μm film thickness) and about 9800 and 14 800 m2 m−3 for the micro bubble column reactors. The microstructured falling film reactor has also been used for chlorination reactions [77]. Chlorination of acetic acid at a temperature over 140 ∘ C was carried out and the by-product (dichloroacetic acid) was reduced significantly, meaning that there is no need for additional costly and time-consuming separation processes. Further, a photochemical gas–liquid reaction by the selective photochlorination of toluene-2,4-diisocyanate (TDI) was demonstrated [78]. 7.7.1.2 Nitration, Oxidations, Sulfonation, and Hydrogenation

Considering the corrosivity of nitrating agents and the explosive potential of nitroproducts, microstructured devices are more suitable for such reactions compared to conventional reactors. Antes et al. [79] carried out the nitration of naphthalene using N2 O5 as a nitrating agent. For intense contacting of the two fluids, interdigital mixers as well as microstructures with the split-recombine technique were applied. Nitration in conventional batch operation requires low temperatures to avoid thermal explosion; in microreactors the nitration was carried out at a temperature up to 50 ∘ C and eightfold excess of N2 O5 with high selectivity for mononitro naphthalene without any risk of reaction runaway. A two-phase capillary reactor was used for the oxidation of aromatic alcohols and Baeyer–Villiger oxidation of ketones using elemental fluorine [80]. The substrate in an appropriate solvent (acetonitrile or formic acid) was injected at a controlled rate by a syringe pump into the reaction channel. Compared to batchwise operation the yield and conversion was comparable or better using microstructured devices. The sulfonation of toluene is one of the complex reactions compared to other elemental reactions as numerous side and consecutive reactions are possible [3].

7.7 Fluid–Fluid Reactions in Microstructured Devices

Sulfonation of toluene with gaseous SO3 was carried out and very encouraging results observed [81]. With increasing SO3 /toluene mole ratio, the selectivity of the undesired by-products decreases while the selectivity of sulfonic acid stays nearly constant. A well-known gas–liquid asymmetric hydrogenation of Z-methylacetamidocinnamate with rhodium chiral diphosphine complexes was carried out using a mesh microreactor [82]. The reactor has two 100 μm deep cavities (100 μl) separated by a micromesh in which the upper cavity is fed with the reacting gas while the other, the reacting chamber, contains the reacting liquid. Porosity of the mesh is 20–25%, which leads to a gas–liquid interfacial area of ∼2000 m2 m−3 of liquid. Mesh MSR tests can be applied to very active catalysts such as the Rh/diop complex by operation in the continuous flow mode enabling short residence times of about 1 min. If longer residence times are required, the mesh microreactor can be operated batchwise by interrupting the liquid flow using appropriate valves. 7.7.2 Examples of Liquid–Liquid Reactions 7.7.2.1 Nitration Reaction

Nitration of benzene is one of the most widely used reactions for benchmarking chemical reactors for liquid–liquid application. This reaction is fast and highly exothermic (−145 kJ mol−1 ). In consequence, the reactor performance is strongly influenced by mass and heat transfer. The reaction is catalyzed by sulfuric acid that creates nitrating ions (NO+2 ) from HNO3 [69]. All reactions are assumed to take place within the acid phase. Poor mixing in this process reduces the transfer rates and leads to a buildup of dissolved nitrated products near the interface, which can be further nitrated to form often unwanted dinitro and trinitro compounds [84]. The reaction was carried out in a simple experimental microreactor setup [69]. Narrow bore capillary tubes with various lengths (50–180 cm), diameters (127–254 μm), and materials (316 stainless steel and PTFE) were used. Both phases containing reactants, acid and organic, were introduced through a T-connector to which the capillary tube was attached downstream. The tube was coiled within a controlled temperature bath. The experiments were carried out at 63–85% for H2 SO4 , 2.6–4.9% for HNO3 , 2–20 cm s−1 flow velocities and 60–90 ∘ C temperature. The small reactor showed higher organic transformation rates than the conventional reactor. The reaction rate was improved for higher flow velocities with high H2 SO4 concentration where the production rate should depend on the mass transfer performance because of improved mixing. Nitration of toluene is another exothermic nitration performed in 150 μm bore PTFE tube for varying acid strengths, reactor temperatures, and flow ratios [84]. Here too a strong influence of flow velocity on transformation rates of toluene nitration was observed: Results from both benzene and toluene nitration have indicated that reaction rate constants in the range of 0.5–20 min−1 can be achieved in a capillary reactor

319

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7 Microstructured Reactors for Fluid–Fluid Reactions

with significant reduction in by-products formation compared to conventional processes. This may be because of isothermal environment and fundamental differences in contacting the two phases. 7.7.2.2 Transesterification: Biodiesel Production

Biodiesel is a mixture of esters of high molecular fatty acids and aliphatic alcohols produced by alcoholysis of biological feedstocks with a high content of triacylglycerols. The triacylglycerol reacts stepwise with the alcohol, mostly methanol, or ethanol, to form finally the corresponding fatty acid esters and glycerol. The reaction is catalyzed by strong bases like KOH or NaOH. Initially, a biphasic mixture of triacylglycerol and alcohol is obtained. As a consequence, the first phase of the transformation is strongly influenced by mass transfer, and intensive mixing greatly accelerates the transesterification rate. Schwarz et al. [85] studied the efficiency of different microstructured mixers followed by microchannels and their influence on the space time for obtaining high product yields. With increasing mass transfer performance of the micromixer and decreasing channel diameter of the microchannel reactors, shorter reaction times of several minutes at lower reaction temperatures compared to conventional batch reactor were obtained. Similar observations are reported for the synthesis of biodiesel in capillary microreactors [86] and in zigzag microchannels [87]. 7.7.2.3 Vitamin Precursor Synthesis

Cyclization of pseudoionone to β-ionone is an important reaction used in the synthesis of vitamin A. Conventionally, pseudoionone is slowly dosed to a stirred tank reactor containing a biphasic mixture of concentrated sulfuric acid and an organic solvent to control the temperature of the highly exothermic reaction [88]. The reaction takes place in the acid phase, where by-products are formed very quickly. The by-product formation is observed to increase with increasing temperature. The product yield obtained in conventional semibatch reactors is in the range of 70%. Important issues in this reaction are short residence time, isothermal reaction, and defined reaction time. This can be achieved by fast mixing of the two viscous phases, the instantaneous quenching of the reaction by dilution with water after the reaction, and rapid separation of biphasic mixture. A suitable microreactor system corresponding to the above mentioned requirements was developed by Wörz et al. [89]. Their installation consisted of 32 stainless steel channels of 900 × 60 μm size separated by cooling channels (Figure 7.24). Reactant and the acid were mixed extremely fast in these microchannels and cooled simultaneously. As the product is sensitive for consecutive reactions, it is obvious that the absence of backmixing increases the product yield. At a temperature of 20 ∘ C, a maximum yield of 90–95% could be achieved with a residence time of 30 s. The reaction is quenched by diluting the concentrated sulfuric acid-reactant mixture with water. The dilution of concentrated sulfuric acid has an even higher exothermicity and must carried

7.7 Fluid–Fluid Reactions in Microstructured Devices Coolant Reactant 2 Reactant 1

Figure 7.24 Reactor sketch for fast, highly exothermic reactions [89]. (Adapted with permission from Elsevier.)

out in a micromixer that is embedded in an additional cooling layer. Careful construction of the microreactors is required for such application because of the danger of blockage following high viscosity and the eventual formation of polymeric products. 7.7.2.4 Phase Transfer Catalysis (PTC)

PTC is a common approach used to accelerate a biphasic reaction by ensuring a ready supply of necessary reagent to the phase in which the reaction occurs [90, 91]. Each reactant is dissolved in the appropriate solvent, which may be immiscible and then a phase transfer catalyst is added to promote the transport of one reactant into the other phase. Hydrolysis of p-nitrophenyl acetate to p-nitrophenyl acetate 2 was carried out in PTFE tubing (300 μm diameter, 400 mm length) using segmented flow conditions (organic phase: toluene) [92]. A phase transfer catalyst, 10 mol% tetrabutylammonium hydrogen sulfate (Bu4 NHSO4 ), was used at 20 ∘ C under segmented conditions. With PTC, increased reaction rate was observed compared to the case of slug flow without phase transfer catalyst. Further, sono-chemical technique was applied in which the microchannel tubing was immersed in the ultrasound bath during the reaction time. During sonication, some irregular-sized segments (1–10 μm length) were formed together with some emulsion, which increased the interfacial area. The highest reaction rates were observed when combining segmented flow, phase transfer catalyst, and sonication. Another reaction, alkylation reactions of �-keto esters, is an important carbon–carbon bond-forming reactions in organic synthesis. An example is the benzylation of ethyl 2-oxocyclopentanecarboxylate (1) with benzyl bromide (2) in the presence of 5 mol% of tetrabutylammoniun bromide (TBAB) as a phase transfer catalyst as shown in Figure 7.25 [93]. The reaction was carried out in a setup depicted in Figure 7.26. The performance of the microchannel is compared to those obtained with round bottom flask for

321

322

7 Microstructured Reactors for Fluid–Fluid Reactions

O

O

O OEt

(1)

+ PhCH2Br

O

5 mol% TBAB

OEt Ph

CH2Cl2 : 0.5 N NaOH aq. = 1 : 1

(2)

(3)

Figure 7.25 The phase transfer benzylation reaction of ethyl 2-oxocyclopentanecarboxylate (1) with benzyl bromide (2).

0.3 M 1 0.45 M 2

5 mol% TBAB 0.5 N NaOH aq.

Microreactor

Teflon tube (10 cm)

NH4Cl aq.

Figure 7.26 Benzylation reaction. MSR used (channel: 200 μm width, 100 μm depth, and 45 cm length and Teflon tube: diameter 200 μm, length 10 cm). (Adapted from Ref. [93]. With permission of The Royal Society of Chemistry.)

different stirrer speed as presented in Figure 7.27. In the microchannel, the reaction proceeded smoothly, and the desired alkylation product (3) was obtained in 57% yield at 60 s, which was increased to over 90% after 300 s. A much lower yield was obtained using standard batch systems even with vigorous stirring. The result confirms the high mass transfer performance in microchannels, which are not achieved in classical batch equipment. 7.7.2.5 Enzymatic Reactions

The enzymatic dehalogenation of p-chlorophenol was studied by Maruyama et al. [94] in a microchannel device with two-phase liquid–liquid flow. The microchannel (100 μm width, 25 μm depth) were fabricated on a glass plate (70 × 38 mm). The enzyme (laccase) was solubilized in a succinic aqueous buffer and the substrate (p-chlorophenol) was dissolved in isooctane. The surface of the microchannel was partially modified with octadecylsilane groups to make it hydrophobic, which allows a stable parallel flow of the aqueous and organic phases. The degradation of p-chlorophenol occurs only in the enzyme containing aqueous phase or at the aqueous–organic interface. At the reactor outlet a perfect phase separation could be obtained allowing the recycling of the enzyme catalyst (Figure 7.28).

7.8 Summary

100 90

Microreactor

80

Round-bottomed flask (1350 rpm)

Yield (%)

70 60 50

Round-bottomed flask (400 rpm)

40 30 20

Round-bottomed flask (0 rpm)

10 0 0

100

200 300 400 Mean residence time, t (s)

500

600

Figure 7.27 Profile of product yield in a microreactor and standard batch systems with different mixing intensity. (Adapted from Ref. [93]. With permission of The Royal Society of Chemistry.)

Aqueous phase Organic phase

Inlet

Aqueous phase

Organic phase

Outlet

Figure 7.28 Photographs of the parallel two-phase flow in the microchannel reactor. (Adapted from Ref. [94]. With permission of The Royal Society of Chemistry.)

7.8 Summary

In this chapter, different aspects of fluid–fluid systems in microstructured devices have been described. The disadvantages of conventional reactors have been clearly

323

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7 Microstructured Reactors for Fluid–Fluid Reactions

mentioned and the corresponding microreactor types and their advantages have been highlighted. Different flow regimes and corresponding mass transfer characteristics are discussed with a focus mainly on slug and parallel flow. The main parameters controlling the flow pattern are the microreactor geometry, fluid viscosity, and interfacial tension. Mass transfer performance can be interrelated using penetration model to a certain extent. The mass transfer coefficient data and correlations provided could help in the designing of microchannel reactors. These reactors are suitable for reactions with fast intrinsic kinetics, requiring high mass and heat transport leading to ameliorated reactor safety.

7.9 List of Symbols cint cm c* C′ cin , cout CaL, u db fL , f tot H Lb lc Lcir Lpore Lslug LUC n Rslip ReD S tD , film ug utot V UC V̇ , V̇ D

C

xi � film � int � mem �C , �D , �G , �L

Interfacial concentration Mixed-cup solute concentration Equilibrium concentration Contamination in flow separation Concentration at inlet and outlet Capillary number based on two phase velocity (u) Diameter of dispersed slug Liquid friction factor, two phase friction factor Henry’s constant Length of dispersed slug Average distance traveled by the liquid packets along the bubble surface of the cap Circumference of the channel Length of the pores Length of continuous (liquid) slug Length of unit slug (pair of gas–liquid or liquid–liquid slugs) Number of microchannels Slip ratio between bubble and liquid velocity Dispersed phase Reynolds number Flow ratio Diffusion time of solute in the film Gravity equivalent liquid velocity Total mixture velocity Volume of unit cell Volumetric flow rate of dispersed, of continuous phase Mass fraction of phase i Thickness of wall film Thickness of hypothetical stagnant film at the interface Thickness of the membrane Volume fraction of continuous phase, dispersed phase, of gas, of liquid phase

mol m−3 mol m−3 mol m−3 — mol m−3 — m — Pa m3 mol−1 m m m m m

— — — — s m s−1 m s−1 m3 m3 s−1 — m m

—

References

� �c �c,cap �c,film �w �D , �C �G, �L Φ2G ,

Φ2L

𝛹 ( Δp ) f

L )L ( Δp

,

f

L

Inclination angle from horizontal in falling film reactor Exposure time of solute at the interface Average contact time of the liquid with the bubble cap Average contact time of the liquid with the film Wall wetting angle Volumetric flow fraction of dispersed phase, of continuous phase Residence time of gas, of liquid phase

∘

Friction multipliers for gas and liquid in Lockhart and Martinelli equation Lockhart–Martinelli parameter

—

Frictional pressure gradients when liquid and gas are assumed to flow alone

Pa m−1

s s s ∘ — s

—

G

Pressure drop due to frictional effects

Pa

Δp , ( L )film Δp L CP

Pressure gradient in the slug region, single phase pressure gradient of the continuous phase

Pa m−1

Δpc Δpout Δpsep Δpmem

Pressure drop due to capillary phenomena Pressure drop in the outlets of the separator Pressure drop in the straight channel of separator Pressure drop across the membrane

Pa Pa Pa Pa

Δp ( f)

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8 Three-Phase Systems 8.1 Introduction

Three-phase reactions comprise gas–liquid–solid and gas–liquid–liquid reactions. Gas–liquid reactions using solid catalysts represent a very important class of reactions. Conventionally, they are carried out in slurry reactors, (bubble columns, stirred tanks), fluidized beds, fixed bed reactors (trickle beds with cocurrent downflow or cocurrent upflow, segmented bed, and countercurrent gas–liquid arrangements) and structured (catalytic wall) reactors. For gas–liquid–liquid reactions equipment similar to that used for liquid–liquid reactions are employed. The hydrodynamics in these reactors is extremely complex because of the three phases and their convoluted interactions. An example is the “grazing” behavior of small solid particles enhancing mass transfer at gas–liquid interfaces. The scale-up from laboratory to the production site thus poses numerous problems with respect to the reactant’s mixing, temperature control (heat removal), catalyst selectivity, and its deactivation [1]. The performance of such processes can be predicted analytically only to a limited extent for reactors with well-defined flow patterns. In this chapter, different types of microstructured devices for three-phase reactions are described. The characterization of mass transfer for gas–liquid–solid systems is presented. Finally, literature examples of both gas–liquid–solid and gas–liquid–liquid reactions are briefed. 8.2 Gas–Liquid–Solid Systems 8.2.1 Conventional Gas–Liquid–Solid Reactors

The gas-liquid–solid reactions are carried out in various types of reactors, such as packed beds, fluidized/slurry, and catalytic wall reactors (Figure 8.1). The advantages and limitations of these reactors are described in Table 8.1. Compared to fluid–solid systems, an additional phase makes it difficult to predict flow patterns Microstructured Devices for Chemical Processing, First Edition. Madhvanand N. Kashid, Albert Renken and Lioubov Kiwi-Minsker. © 2015 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2015 by Wiley-VCH Verlag GmbH & Co. KGaA.

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8 Three-Phase Systems

Table 8.1 Different types of gas–liquid–solid reactors, their advantages, and limitations. Reactors

Advantages

Limitations

Packed bed reactor (trickle bed reactor)

Easy to operate

Flow maldistribution

Can accommodate more than 60 (volumetric) catalyst Suitable for slow reactions

High pressure drop

Fluidized bed reactor

Good heat and mass transfer Pressure drop independent of fluid throughput

Slurry reactor

Effective utilization of catalyst Good liquid–solid mass transfer Good heat transfer

Catalytic wall reactors

Effective utilization of catalyst following structuring Low pressure drop

Risk of hot spot formation Thermal instabilities Good performance in the limited range of flow rates Complex hydrodynamics Broad residence time distribution Catalyst abrasion Moderate gas–liquid mass transfer Catalyst separation is difficult and a filtration step is required Low conversion and selectivity in continuous mode following backmixing Heat/mass transfer limits the performance for fast reaction for large diameter reactors

L

(a)

(b)

(c)

G

L

(d)

Figure 8.1 Schematics of Conventional reactors used for gas–liquid–solid reactions: (a) packed bed, (b) fluidized bed, (c) slurry reactor (bubble and agitated), (d) film reactor (catalytic wall).

because of their convoluted interactions. In conventional fixed bed reactors, catalyst particles of various sizes are randomly distributed, which may lead to inhomogenous flow patterns. Near the reactor walls, the packing density is lower than the mean value, and faster flow of the fluid near the wall is unavoidable. As a result, reactants may by-pass the catalyst particles, and the residence time distribution will be broadened. Moreover, the nonuniform access of reactants to the catalytic surface diminishes the overall reactor performance and can lead to unexpected hot spots and even to reactor runaway in the case of exothermic reactions. Pressure drop in three-phase packed beds, and thus energy consumption, is a crucial matter in process economics, in particular when large quantities of raw

8.2

Gas–Liquid–Solid Systems

materials have to be converted. On the other hand, when reactions are conducted in suspensions such as in fluidized beds or slurry reactors, pressure drop is minimized, but catalyst abrasion and catalyst recovery are critical issues. Fluidized beds give relatively higher performance, but within a narrow operating window. Another type of reactors, the slurry reactor, effectively utilizes the catalyst because of their small particle size in the micrometer range. However, catalyst separation is difficult and a filtration step is required to separate fine particles from the product. Moreover, when applied in the continuous mode, backmixing lowers the conversion and usually the selectivity [2]. Conventional continuous tubular reactors are used as falling film or wall reactor with catalyst coated on the wall; however, supply/removal of heat and often broad residence time distribution because of large reactor diameters are two main drawbacks commonly encountered with such reactors. Structured catalysts may be used to overcome the drawbacks of conventional catalytic reactors [3]. These are reactors with monolithic converters, with catalystcoated static mixers and arranged packings as applied in distillation and absorption columns. 8.2.2 Microstructured Gas–Liquid–Solid Reactors

Different types of gas–liquid–solid microstructured reactors (MSR) have been developed, using different gas–liquid contacting principles [4]. These principles can be classified as

• Continuous phase contacting, where the fluid phases are separated. Examples are microstructured falling film and mesh reactors.

• Dispersed phase contacting that is obtained when one of the fluid phases is dispersed into the other phase. 8.2.2.1 Continuous Phase Microstructured Reactors

In falling film contactors a thin film is created by a liquid falling under gravity pull. The liquid flows over a solid support (see Figure 8.2), which is normally a thin wall or stack of pipes. In conventional falling film devices, a film with a thickness of 0.5–3 mm is generated [4]. This rather thick liquid film results in an important mass transfer resistance for the gaseous reactant diffusing to the solid catalyst on the reactor wall. In addition, the film flow becomes unstable at high throughput and the film may break up into rivulets, fingers or a series of droplets. The mentioned problems can be overcome by microstructuring the solid wall [5, 6]. The microstructured falling film reactor consists of microchannels, which are typically 300 μm wide, 100 μm deep, and about 80 mm long. The channels are separated by 100 μm wide walls. The microstructured falling film reactor has been described in detail in the previous chapter. In a mesh microcontactor, the gas and liquid flow through separate channels. To provide stable operation the fluids are separated by a thin mesh of typically 5 μm thickness. The fluids are in contact through holes with diameters of about

333

334

8 Three-Phase Systems

Gas flow Reactor housing (bottom) Heat exchanger plate

Falling film plate

Liquid feed in

Teflon and stainless steel gaskets Reactor housing (top)

Figure 8.2 Components and schematic of the microstructured falling film reactor [7]. (Adapted with permission from Elsevier.)

5 μm [8]. In contrast to microstructured falling film reactors, the velocity of the fluids can be varied without changing the interfacial area, which is given by the porosity of the membrane. Interfacial forces help to stabilize the fluid interface within the openings, while fluid layers are thin enough to enhance mass transfer. The meniscus shape at the interface between the two phases defines the available area for mass transfer and is a function of contact angle, pore geometry, and pressure difference between phases. The open area of the micromesh contactor can be as high as 40% while the mesh to wall distances can be set, generally, to 80–140 μm providing chamber volumes of ∼100 μl. A quadrant-reactor configuration with a deep outlet channel at the circumference provides radial flow and minimizes flow resistances ensuring even the flow of the fluidic streams (and consequently residence time) in the reactor (see Figure 8.3). The mesh is fabricated in nickel using a two-stage electroplating method. It is placed between two glass layers that form the chambers for the two fluids. The struts that help preserve the physical structure of the mesh fabricated on the mesh align with pillars on the glass to provide the necessary channel width of the reactant channels. For gas–liquid–solid reactions, the bottom glass insert in the reaction chamber can be coated with a catalytic layer. 8.2.2.2 Dispersed Phase Microstructured Reactors

In gas–liquid systems (Chapter 7) different flow regimes for two phase systems are observed. One of the most stable and commonly applied flow regime in which one fluid flow in the form of dispersed phase is segmented (Taylor) flow. Taylor flow is characterized by gas bubbles that are too large to retain a spherical shape and are deformed to fit inside the channel. Surface tension provides the driving force for the bubbles to attain a spherical shape, and the bubbles try to expand toward the channel wall such that only a thin film of liquid remains between the

8.2 Metal block

Gas-in

Gas-out

Nickel mesh

Glass spacer post Liquid-in

Gas–Liquid–Solid Systems

Mesh

Gasket Catalyst layer

(a)

Inlet

Liquid-out (b)

Figure 8.3 Mesh microcontactor. (a) Schematics of mesh contactors. (Adapted from Ref. [8]. With permission of The Royal Society of Chemistry.) (b) Fully assembled

Gas supply

micromesh reactor and microstructured mesh. (Adapted with permission from Ref. [4]. Copyright (2005) American Chemical Society.)

Mini heat exchanger: medium supply and withdrawal

Static micromixer

Reaction channel array Liquid supply

Figure 8.4 Microbubble column with integrated cooling channels. (Reproduced from Ref. [9]. With kind permission of Springer Science + Business Media.)

gas and the wall. The liquid slugs are entrapped between the bubbles. This flow regime is also used for three phase reactions. In segmented flow gas–liquid–solid reactors, the liquid usually flows over the solid surface while the gas flows through the liquid in the form of bubbles or annular flow (see Figure 8.4), depending on the MSR geometry and the catalyst arrangement. The hydrodynamic characteristics of these reactors, such as pressure drop and residence time distribution, can be determined from those for fluid–solid and fluid–fluid reactors. The difference between the gas–liquid and gas–liquid–solid systems is that because of the reaction at the surface of the catalyst, there is always a concentration gradient in the liquid phase in the latter case. Microstructured packed beds have been used for gas–liquid–solid reactions. An advantage of microstructured packed beds for heterogenous catalytic processes stems from the fact that active and selective catalysts are commercially

335

8 Three-Phase Systems

available. In addition, the particle size of these catalysts used in suspension reactors is in the micrometer range and fit well for use in microchannels. However, a proper design of the reactor is required to maintain an acceptable pressure drop. To avoid an excessive pressure drop, an MSR consisting of a microfluidic distribution manifold and a microchannel array were constructed [10]. Multiple reagent streams (specifically, gas and liquid streams) were mixed on-chip, and the fluid streams were brought into contact by a series of interleaved, high-aspectratio inlet channels. These inlet channels deliver the reactants continuously and cocurrently to 10 reactor chambers containing standard catalytic particles in the diameter range of 50–75 μm. 8.2.2.3 Mass Transfer and Chemical Reaction

The global transformation rate of a gas–liquid reaction catalyzed by a solid catalyst is influenced by the mass transfer between the gas–liquid and the liquid–solid. The two mass transfer processes and the surface reaction are in series and for fast chemical reactions, mass transfer will influence the reactant concentration on the catalytic surface and, as a consequence, influence the reactor performance and the product selectivity. Compared to gas–solid catalytic reactions as discussed in Section 2.5, an additional resistance in the liquid must be considered (Figure 8.5). The general discussion of the reaction in gas–liquid–solid systems is based on the simple film model. In addition, we consider an irreversible reaction between a gaseous reactant (A1 ) and a reactant in the liquid phase (A2 ), which is in large excess: �1 A1 (gas) + �2 A2 (liquid) → �3 A3 (liquid) with c1 ≪ c2

(8.1)

If the thickness of the catalytic layer on the reactor wall is sufficiently small, internal mass transfer resistances can be neglected and only external resistances in the fluid phases are considered. The reaction rate per unit of the outer surface of the catalytic layer is described by a pseudo first order reaction (mol m2 s−1 ): rs = ks c1,s with ∶ c2,s = c2,L ≫ c1,s Gas

p1

Porous catalyst

Liquid

c2,L

p1∗ c∗1

Interface gas-liquid

Figure 8.5

(8.2)

Wall

Concentration

336

c1,L

c1,s Interface liquid-solid

Concentration profiles of reactants in the gas, liquid, catalyst phase.

8.2

Gas–Liquid–Solid Systems

The reaction rate per unit volume of the liquid (continuous) phase is given as (8.3)

r = ks aLS c1,s = kc1,s

where aLS corresponds to the interfacial liquid/solid area: aLS = ALS ∕VL . The mass transfer rate (rm ) from the liquid phase to the surface of the catalytic layer is proportional to the concentration gradient between the two phases: (8.4)

rm,LS = kLS aLS (c1,L − c1,s )

The rate of the volumetric mass transfer (rm ) from gas phase to liquid is proportional to the overall concentration gradient between the two phases: ) (p (8.5) rm,GL = kGL aGL 1 − c1,L H where H is the Henry constant, p1 is the partial pressure of A1 in the bulk gas phase, k GL is the overall gas/liquid mass transfer coefficient, and aGL corresponds to the specific gas/liquid interfacial area referred to the liquid phase aGL = AGL ∕VL . Under stationary conditions the reaction rate and mass transfer rates must be identical: rm,LS = rm,GL = r. From Equations 8.5 and 8.3 follows: k c1,s = kLS aLS (c1,L − c1,s ) kLS aLS k 1 ⇒ c1,s = c = c ; DaIILS kLS aLS + k 1,L 1 + DaIILS 1,L kLS aLS

(8.6)

Here DaII LS is the ratio of reaction rate constant to the liquid–solid mass transfer coefficient. Similarly, the relation between gas and liquid phase concentrations can be established by equating Equations 8.4 and 8.5 as ) (p kGL aGL 1 − c1,L = kLS aLS (c1,L − c1,s ) H (p ) kGL aGL 1 (8.7) ⇒ c1,L = 2 (kLS aLS ) H k a +k a − GL GL

LS LS

kv +kLS aLS

Finally, the concentration at the catalyst surface can be calculated as function of the partial pressure of the gaseous reactant A1 with Equations 8.6 and 8.7: (p ) (p ) 1 1 1 1 c1,s = = (8.8) H 1 + DaIIGL + DaIILS H 1+ k + k kGL aGL

kLS aLS

The observed (effective) reaction rate follows: (p ) k 1 rs,eff = 1 + DaIIGL + DaIILS H

(8.9)

If we consider a microstructured falling film reactor, the liquid phase is reduced to a small liquid layer between the gas phase and the solid catalyst as shown schematically in Figure 8.6a. The thickness of this film (� film ) depends on the liquid flow as discussed in Chapter 7. Because of the surface forces the wetting liquids form a meniscus within the channel. With increasing flow rate the meniscus becomes more and more flat. This is demonstrated on Figure 8.6b for a channel with a width of 300 μm and a height of the side walls of 100 μm [11]. The channel

337

8 Three-Phase Systems

Porous catalyst

Gas c2,L p1∗

δfilm

c∗1

wall

Concentration

p1

c1,s Liquid film

(a)

Channel profile/film thickness, δfilm (μm)

338

100 80 60 40 20 0

0.5 ml/min 0.7 ml/min 1 ml/min Channel profile

0

50

100

150

200

250

300

Channel width (μm)

(b)

Figure 8.6 (a) Concentration profile in a catalytic microstructured film reactor (schematic) and (b) average liquid film thickness profile across a channel at various flowrates [11] (Adapted with permission from Elsevier.)

profile itself is also curved because of the procedure of catalyst coating. In spite of the observed radial film profile, which depends on the fluid and solid properties the average film thickness will be estimated based on the simple Nusselt falling film theory given in Equation 8.10 (Chapter 7). √ �film =

3

√ 3V̇ L vL 3V̇ L vL �L 3 ≅ ; � ≤H n ⋅ W ⋅ g ⋅ sin � (�L − �G ) n ⋅ W ⋅ g ⋅ sin � film

(8.10)

where � is an inclination angle from horizontal and g is the gravitational acceleration. As flow rate increases, the liquid film thickness increases up to the maximum value given by the height of the channel walls. An increase in film thickness will increase the diffusion time for solutes across the liquid film to the catalyst surface, thus increasing the mass transfer resistance and decreasing the effective reaction rate and diminish the conversion. Higher flow rates will decrease the mean residence time of the fluid, thus reducing the conversion as well. An eventual influence of the reactant diffusion in the porous catalytic layer can be respected by adding an efficiency factor � to the rate constant (see Chapter 2), and the observed reaction rate is given by Equation 8.11. rob =

(p )

1 1 kGL aGL

+

1 kLS aLS

1

+

1 �k

H

=

(p )

�k 1+

�k kGL aGL

1

+

�k kLS aLS

H

(8.11)

In many practical situations, the main mass transfer resistance is concentrated in the liquid phase and the mass transfer resistance in the gas phase can be neglected. The mass transfer coefficient in the liquid film can be estimated based on the film model to be: kLS =

Dm �film

(8.12)

8.2

Gas–Liquid–Solid Systems

The specific interfacial area (aLS ) can be estimated as aLS =

W ⋅L 1 Interfacial surface area ≅ = liquid volume �film ⋅ W ⋅ L �film

(8.13)

The estimations of film thickess and mass transfer coefficient are demonstrated in Example 8.1.

Example 8.1: Film thickness and mass transfer coefficient of falling film MSR. Estimate the film thickness, mass transfer coefficient, volumetric mass transfer coefficient, and specific interfacial area for a vertically placed falling film MSR with the dimensions: W = 600 μm 1 mm, H = 200 μm, L = 100 mm. The liquid flow rate is fixed to V̇ L = 0.1 cm3 min−1 0.1 ml/min. Data: �L = 1000 kg m−3 , � L = 10−6 m2 s−1 , Dm = 10−9 m2 s−1 , n = 5 Solution: The film thickness is estimated using Nusselt falling film theory given by Equation 8.10. √ ( ) √ √ √ 0.1⋅10−6 3 ⋅ ⋅ 1 ⋅ 10−6 √ 3 ̇ 3V L ⋅ v L 60 √ 3 = 46.7 μm = �film ≅ n ⋅ W ⋅ g ⋅ sin � 5 ⋅ (1 ⋅ 10−3 ) ⋅ 9.81 ⋅ sin(90) Mass transfer coefficient (kLs ) and specific interfacial area (aLs ) given by Equation 8.12: Dm 1 ⋅ 10−9 = = 2.14 ⋅ 10−5 m ⋅ s−1 �film 46.7 ⋅ 10−6 1 1 = ≅ = 2.1 ⋅ 104 m2 ⋅ m−3 �film 46.7 ⋅ 10−6

kLS = aLS

Thus, the volumetric mass transfer coefficient is kLS aLS = 2.14 ⋅ 10−5 × 2.1 ⋅ 104 = 0.45 s−1

From the above example, it is clear that the specific interfacial area and thus the volumetric mass transfer coefficients are high in falling film reactors and are suitable for fast catalytic reactions. In a mesh microcontactor, the meniscus shape at the interface between the two phases defines the available area for mass transfer and is a function of contact angle, pore geometry, and pressure difference between phases. The open area of the micromesh contactor can be as high as 40%, which leads to gas–liquid interfacial area of 2000 m2 m−3 . This high gas–liquid interfacial area combined with the small fluid layer thickness in the order of 100 μm results in high mass transfer coefficients. The volumetric mass transfer coefficient reported for the very fast

339

340

8 Three-Phase Systems ∗ c1,L

Gas layer

Mesh 5 μm 5 μm δfilm

Liquid film

Porous catalyst

δcat

Wall

Figure 8.7

Mesh reactor (schematic) [12]. (Adapted with permission from Wiley.)

hydrogenation of α-methylstyrene over a Pd/γ-alumina catalyst was found to be in the range of 0.8–1.6 s−1 [12, 13] (Figure 8.7). In segmented flow gas–liquid–solid reactors, the liquid usually flows over the solid surface while the gas flows through the liquid in the form of bubbles or annular flow, depending on the channel geometry and the catalyst arrangement. The hydrodynamic characteristics of the three-phase reactors, such as pressure drop and residence time distribution, can be determined from those for fluid–solid and fluid–fluid reactors. For the gaseous reactant three mass transfer steps can be identified [14]: (i) the transfer from the bubble through the liquid film to the catalyst (kGS aGS ), (ii) the transfer from the caps of the gas bubbles to the liquid slug (kGL aGL ), and (iii) the transfer of dissolved gas to the catalytic surface (kLS aLS ). Steps (ii) and (iii) are in series and in parallel with respect to step (i) (Figure 8.8). The overall resistance using resistances in series and parallel can be written as 1 1 1 = + Rov RGS RGL + RLS

(8.14)

Replacing the resistance with volumetric mass transfer coefficients, the following expression can be obtained for overall volumetric mass transfer (k ov a): kov a = kGS aGS +

1 kGL aGL

1 +

(8.15)

1 kLS aLS

S

Solid

kLSaLS

Gas kGSaGS

(a)

kGLaGL Liquid

RGS G (b)

RLS RGL

L

Figure 8.8 Mass transfer in dispersed phase microstructured reactors where gas solute diffuses through liquid toward solid surface. (a) Schematic representation. (b) Resistance model.

8.2

Gas–Liquid–Solid Systems

Various attempts were made to determine the individual mass transfer coefficients in Equation 8.15 in nonreactive systems. However, because the concentration profiles in the liquid surface film and in the slugs are strongly affected by fast chemical reactions, caution must be exercised in extending the results to reactive systems. The mass transfer resistance through the caps of the bubble and the liquid phase are particularly difficult to estimate as assured models are missing. The transfer from the gas bubble through the wall film can be estimated. Ignoring the thickness of the film, the specific interfacial area is given by aGS =

4 � dt G

(8.16)

with � G the volume fraction of the gas phase. The mass transfer resistance is given mainly by the diffusion through the thin wall film. With the film model we obtain: D kGS = m (8.17) �film The film thickness can be estimated with the relations presented in Chapter 7 [15, 16]. 2∕ �film = 0.67 dt CaL 3 for CaL ≤ 3 ⋅ 10−3 2∕ 0.67CaL 3 for CaL < 1 �film = dt 2∕ 1 + 3.35CaL 3

(8.18)

The mass transfer model presented in Equation 8.15 was applied to interpret experimental results characterizing the hydrogenation of 4-nitrobenzoic acid to 4-aminobenzoic acid [17]. The reaction was conducted in a capillary with a circular cross section and a washcoat incorporating an alumina-supported palladium catalyst. Flow regimes in the microreactor were characterized visually for different flow rates and gas to liquid flow ratios. For low liquid and gas velocities, bubbles were formed at the entrance and were carried by the liquid through the packed bed. Under these conditions the hydrogenation of cyclohexene was studied and used as a model reaction to measure the mass transfer resistances. Overall, mass transfer coefficients (kov a) were measured in the range of 5 to 15 s−1 , which is nearly 2 orders of magnitude larger than values reported in the literature for standard laboratory-scale reactors [18]. 8.2.2.4 Reaction Examples

Most of the reported microstructured gas–liquid–solid reactors concern catalytic hydrogenations (Table 8.2). This is because hydrogenation reactions represent about 20% of all the reaction steps in a typical fine chemical synthesis. Catalytic hydrogenations are fast and highly exothermic reactions. Consequently, reactor performance and product selectivity are strongly influenced by mass transfer, and heat evacuation is an important issue. Both problems may be overcome using microstructured devices.

341

342

Reaction/example

Type of reactor

Performance/yield/selectivity

Hydrogenation of cyclohexene [19, 20]

Packed bed MSR (catalyst powder different sizes: 53–75, 36–38, or